v  / / **■ 
       National
Library of Medicine
FOUNDED 1836
Bethesda, Md.
II. S. Department of Health,
  Education, and Welfare
   PUBLIC HEALTH SERVICE 
A
TREATISE   ON  OPTICS.
BY
     SIR DAVID BREWSTER, LL. D, F. R. S. L. & E.

CORRESPONDING  MEMBER OF THE INSTITUTE OF FRANCE, HONORARY
    MEMBER OF THE IMPERIAL ACADEMY OF PETERSBURG, AND OF
        THE ROYAL ACADEMY OF SCIENCES OF BERLIN, STOCK-
            HOLM, COPENHAGEN, GOTTINGEN, &C. &C.
FIRST AMERICAN EDITION
WITH AN APPENDIX, CONTAINING AN ELEMENTARY VIEW OF THE APPLICA-
        TION OF ANALYSIS TO REFLEXION AND REFRACTION,
              BY A.  D. BACHE, A.M.
PROFESSOR OF NAT. PHILOS. AND CHEMISTRY  IN THE UNIVERSITY OF
   PENNSYLVANIA, ONE OF THE SECRETARIES OF AM. PHILOS. SOC,
        MEM. ACAD. NAT. SC, HON. MEM. ASSOCIATE SOC.
                    OF NEWPORT, R. I.
                  itytla&elpftia:
CAREY, LEA, & BLAJNCHARD—CHESNUT STREET.
1833. 
     /S33

     e.l



MOfflM. UK*** Of «»«»«
  . BEIHESDM4, MD. 
(



    NOTE BY THE AMERICAN EDITOR.
   1y object in undertaking the revision of the Treatise
 a Optics by Dr. Brewster was, principally, to introduce
an Appendix, containing such a discussion of the subjects
of Reflexion and Refraction, as might adapt the work to
use in those of our colleges in which considerable exten.
sion  is given to the course of Natural Philosophy.   In
this revision, I  have thought it best, without specially
calling the attention  of the reader to them, to  correct
such errors as my comparatively limited knowledge of
the subject assured me,  would not have  been passed
over  by the author in a second Edition.
                                  A.  D. BACHE.
   Philadelphia, Jan., 1833. 
 
CONTENTS.
Introduction.......................................Page   11

                          PART I.

         ON Tire REFLEXION AND REFRACTION OF LIGHT.

                       CATOPTRICS.

                           CHAP. I.
Reflexion by Specula and Mirrors  ...........................   13
    Reflexion of Rays from Plane Mirrors ...................   15
    Reflexion of Rays from Concave Mirrors.................   16
    Reflexion of Rays from Convex Mirrors..................   20

                          CHAP. II.
Images formed by Mirrors..................................   22

                        DIOPTRICS.

                          CHAP. III.
Refraction................................................   26

                          CHAP. IV.
Refraction through Prisms and Lenses........................   31
    On the total Reflexion of Light  .........................   34
    Refraction of Light through Plane Glasses ...............   36
    Refraction of Light through Curved  Surfaces.............   37
    Refraction of Light through Spheres  .....................   38
    Refraction of Light through Convex  and Concave Surfaces .   40
    Refraction of Light through Convex  Lenses ..............   41
    Refraction of Light through Concave Lenses  .............   41
    Refraction of Light through Meniscuses and Concavo-convex
        Lenses...........................................   45

                           CHAP. V.
On the Formation of Images by Lenses, and on their magnifying
        Power...........................................   46

                          CHAP. VI.
Spherical Aberration of Lenses and Mirrors  ..................   51
    Spherical Aberration of Mirrors.........................   57
    On Caustic Curves formed by Reflexion and Refraction ---   58
                              A2 
6                        CONTENTS.


                          PART II.

                   PHYSICAL OPTICS.

                         CHAP. VII.
On the Colors of Light, and its Decomposition  ................   63
    Decomposition of Light by Absorption ...................   67

                         CHAP. VIII.
On the Dispersion of Light .................................   "70
                          CHAP. IX.
On the Principle of Achromatic Telescopes...................   74
                          CHAP. X.
On the Physical Properties of the Spectrum...................   78
    On the Existence of Fixed Lines in the Spectrum  .........   ib.
    On the Illuminating Power of the Spectrum..............   80
    On the Heating Power of the Spectrum..................   81
    On the Chemical Influence of the Spectrum..............   82
    On the Magnetizing Power of the Solar Rays.............   83
                          CHAP. XI.
On the Inflexion or Diffraction of Light  ......................   86
                         chap. xn.
On the Colors of Thin Plates ...............................   90
    Table of the Colors of Thin Plates of Air, Water, and Glass   93
                         CHAP. XIII.
On the Colors of Thick Plates...............................   97
                         CHAP. XIV.
On the Colors of Fibres and grooved Surfaces.................   101
                          CHAP. XV.
On Fits of Reflexion and Transmission, and on the Interference of
    Light................................................   Ill
                         CHAP. XVI.
On the Absorption of Light.................................   120
                         CHAP. XVII.
On the Double Refraction of Light  ..........................   125
    On Crystals with one Axis of Double Refraction  ..........   128
    On the Law of Double Refraction in Crystals with one Nega-
        tive Axis.........................................   130 
                          CONTENTS.                         7

    On the Law of Double Refraction  in Crystals with one  Posi-
        tive Axis.........................................  132
    On Crystals with two Axes of Double Refraction..........  133
    On Crystals with innumerable Axes of Double Refraction ...  134
    On Bodies to which Double Refraction may be communicated
        by Heat, rapid Cooling, Pressure, and Induration......  135
    On Substances with Circular Double Refraction...........  136

                         chap. xvra.
On the Polarization of Light................................  137
    On the Polarization of Light by Double Refraction........  138

                          CHAP. XIX.
On the Polarization of Light by Reflexion ....................  142
    On the Law of the Polarization of Light by Reflexion  .....  146
    On the partial Polarization of Light by Reflexion..........  149

                           CHAP. XX.
On the Polarization of Light by ordinary Refraction  ...........  152

                          CHAP. XXI.
On the Colors of Crystallized Plates in Polorized Light.........  157

                          CHAP. XXII.
On  the System of Colored Rings in Crystals with one Axis  .....  165
     Polarizing Intensities of Crystals with one Axis ...........   172

                          CHAP. XXIII.
On the Systems of Colored Rings in Crystals with two Axes---.   ib.
     Polarizing Intensities of Crystals with two Axes...........   179

                          CHAP. XXIV.
Interference  of Polarized Light—On the Cause of the Colors of
     Crystallized Bodies.....................................   lb-
                           CHAP. XXV
On the Polarizing Structure of Analcime....................■■   183

                          CHAP. XXVI.
On Circular Polarization  ...................................   ^
     Circular Polarization in Fluids..........................  188
     Crystals which turn the Planes from Right to Left.........    lb.
     Crystals which turn the Planes from Left to Right.........    ib.

                          CHAP. XXVII.
 On Elliptical Polarization, and on the Action of Metals upon Light  190
     On Elliptical Polarization  ..............-...............  191
     Order in which the Metals polarize most Light in the Plane
          of Reflexion......................................    lb- 
 viii                      COMMENTS.

                         CHAP. XXVIII.
 On the Polarizing Structure produced by Heat, Cold, Compression,
         Dilatation, and Induration  ..........................  197
     1. Transient Influence of Heat and Cold.................   ib.
         (1.) Cylinders of Glass with one  positive Axis of Double
             Refraction....................................   ib.
         (2.) Cylinders of Glass  with  a negative Axis of Double
             Refraction....................................  198
         (a) Oval  Plates of Glass with two  Axes of Double Re-
             fraction ......................................   ib
         (4.) Cubes of Glass with Double Refraction...........  199
         (5.) Rectangular Plates of Glass with Planes of no Double
             Refraction....................................   ib.
         (6.) Spheres of Glass, &c. with an infinite  Number of
             Axes of Double Refraction  .....................  201
         (7.) Spheroids of Glass with one Axis of Double Refrac-
             tion along the Axis of  Revolution and two  Axes
             along the Equatorial Diameters  .................   ib.
         (8.1 Influence of Heat on regular Crystals  ............  202
    2. On the permanent Influence of sudden Cooling.........  ib.
    3. On the Influence of Compression and Dilatation.........  203
    4. On the Influence of Induration .......................  205
                         CHAP.  XXIX.
Phenomena of Composite or Tesselated Crystals  ..............  20G
                          CHAP. XXX.
On the Dichroism.or Double Color, of Bodies; and the Absorption
        of Polarized Light................................." 210
    Colors of the two Images in  Crystals with one Axis........  211
    Colors of the two Images in  Crystals with two Axes  .......  212
    General Observations on Double Refraction...............  214
                          PART  III.
 ON  THE APPLICATION OF OPTICAL  PRINCIPLES TO THE EXPLANATION
                    OF NATURAL PHENOMENA.

                         CHAP. XXXI.
On unusual Refraction .....................................   215
                        CHAP. XXXII.
On the Rainbow .........................................   223
                        CHAP. XXXIII.
On Halos, Coronae, Parhelia, and Paraselense  ..................   227
                        CHAP. XXXIV.
On the Colors of Natural Bodies  ............................   235 
                        CONTENTS.                       9

                       CHAP. XXXV.
On the Eye and Vision...................................... 240
    On the Phenomena and Laws of Vision................... 243

                       CHAP. XXXVI.
On Accidental Colors and Colored Shadows.................... 254
                         PART IV.

                   ON OPTICAL INSTRUMENTS.
                       chap, xxxvn.
On Plane and Curved Mirrors................................ 261
   Kaleidoscope  .......................................... 262
   Plane Burning Mirrors .................................. 264
   Convex and Concave Mirrors  ............................ 265
   Cylindrical Mirrors..................................... 266

                      CHAP, xxxvni.
On Single and Compound Lenses............................. 267
   Burning and niuminating Lenses  ......................... 268

                       CHAP. XXXIX.
On Simple and Compound Prisms............................. 270
   Prismatic Lenses  .......................................  ib.
   Compound and Variable Prisms .......................... 271
   Multiplying Glass  ...................................... 273

                         CHAP. XL.
On the Camera Obscura and Camera Lucida................... 274
   Magic Lantern ......................................... 276
   Camera Lucida  ........................................ 277

                        CHAP. XLI.
On Microscopes............................................ 279
   Single Microscopes  .....................................   ib.
   Compound Microscopes  ................................. 283
   On Reflecting Microscopes  ............................... 286
   OnTestObjecte  ........................................ 287
   Rules for Microscopic Observations .......................   ib.
   Solar Microscope  ....................................... 288

                        CHAP. XLII.
On Refracting and Reflecting Telescopes......................  289
   Astronomical Telescope .................................   ib. 
10
CONTENTS.
   Terrestrial Telescope....................................  290
   Galilean Telescope  .....................................  2'.)l
   Gregorian Reflecting Telescope...........................  ib.
   Cassegrainian Telescope.................................  293
   Newtonian Telescope  ...................................  294
   Sir William Herschel's Telescope.........................  296
   Mr. Ramage's Telescope.................................  297

                         CHAP. XLin.
On Achromatic Telescopes..................................  297
   On Achromatic Eye-pieces ...............................  300
   Prism Telescope..........................................  302
   Achromatic Opera Glasses with Single Lenses...............  304
   Mr. Barlow's Achromatic Telescope  .......................  ib.
   Achromatic Solar Telescopes with Single Lenses............  305
   On the Improvement of imperfectly Achromatic Telescopes... 306
                           NOTES.
On the absolute Refractive powers of Bodies................... 315
   Absorptive Power of Water..............................  ib.
   Sir David Brewster's analysis of the Spectrum .............  ib.
   Melloni's  experiments on the heating Powers of the Spectrum 316
   Hypothesis of Undulations applied to explain Young's principle
      of Interference....................................... 318
   On the translucency of gold leaf.......................... 320
   Classification of Colored Bodies, by Sir David Brewster......  ib.
   Duration of Impressions on the Retina..................... 321
   lasensibility of the eyes of certain persons to particular Colors  ib. 
A TREATISE  ON OPTICS.
INTRODUCTION.
  (1). Optics, from a Greek word which signifies to see, is
that branch of knowledge which  treats  of the properties of
light and of vision, as performed by the human eye.
  (2). Light is an emanation, or something which proceeds
from bodies,  and  by means of which we are  enabled to see
them by the eye.   All visible bodies may be divided into two
classes—self-luminous and non-luminous.
  Self-luminous bodies, such as the stars, flames of all kinds,
and bodies which  shine  by being  heated or rubbed, are  those
which possess in themselves the property of discharging light.
Non-luminous bodies are those which have not the power of
discharging light of themselves,  but which throw back the
light which falls upon them  from  self-luminous bodies.   One
non-luminous body may receive light from another non-lumi-
nous body, and discharge it  upon  a third; but in every case
the light  must originally come  from  a self-luminous body.
When a lighted candle is brought into a dark room, the form
of the  flame is seen by the light which  proceeds from the
flame itself;  but the objects in the room  are seen by the light
which they receive from the candle, and again throw  back;
while other objects, on which the  light of the  candle does not
fall, receive light from the white  ceiling and walls, and thus
become visible to  the eye.                        .
  (3)  All  bodies, whether self-luminous or non-luminous, dis-
charge light of the same color with themselves.  A red flame
or a red-hot body discharges  red light; and a piece ot red
cloth discharges  red  light,  though it  is illuminated by the
white light of the sun.
  (4).  Light is emitted from every visible point of a luminous
or of an illuminated body, and in every direction in which the
point is visible.   If we look  at the flame of a candle, or at  a
sheet of white paper,  and magnify them ever so much, we
shall not observe  any points destitute of light. 
12                  INTRODUCTION.

   (5.) Light moves in straight lines, and consists of separate
and  independent parts, called rays of light.  If we admit the
light of the sun into a dark room through a small hole, it will
illuminate a spot on the wall exactly opposite to the sun,—the
middle of the spot, the middle of the hole, and the middle of
the sun, being all in the  same straight line.  If there is dust
or smoke in the room, the progress of the light in straight
lines will be distinctly seen.   If we stop a very small portion
of the admitted light, and allow the rest to pass, or if we stop
nearly the whole light, and allow only the smallest portion to
pass, the part which passes is not  in  the  slightest degree af-
fected by its separation from  the  rest.   The smallest portion
of light which we can either stop or allow to pass is called a
ray of light.
   (6).  Light moves with  a velocity  of 192,500 miles in a
second of time.   It travels from the sun to the earth in seven
minutes and a half.   It moves through a space equal  to  the
circumference of our globe in  the 8th part of a second, a
flight which the swiftest bird could not perform  in  less than
three weeks.
   (7).  When light falls upon any body whatever, part of it is
reflected or driven back,  and part of it enters the body, and is
either  lost within it  or  transmitted through  it.  When  the
body is bright and well polished like silver, a great part of the
light is reflected, and the remainder  lost within the  silver,
which can  transmit light only when  hammered out into  the
thinnest  film.   When the body is transparent, like  glass or
water, almost all the light is transmitted, and only a small
part of it reflected.  The light which is  driven  back from
bodies is reflected according to particular laws, the considera-
tion  of which forms that branch  of optics  called catoptrics;
and the light which is transmitted through transparent bodies
is  transmitted  according to particular  laws, the  consideration
of which constitutes the  subject of dioptrics. 
CHAP. I.      INCIDENCE AND REFLEXION.            13
                   PART I.

ON THE REFLEXION AND REFRACTION OF LIGHT.
                    CATOPTRICS.

  (8.) Catoptrics is that branch of optics which treats of
the progress of rays of light after they are reflected from sur-
faces either plane  or curved, and of the formation of images
from objects placed before such surfaces.


                        CHAP. I.

           REFLEXION BY SPECULA AND MIRRORS.

   (9.) Ant substance of a regular form employed for the pur-
pose of reflecting light, or of forming images of objects, is
called a speculum or mirror.   It is  generally made of metal
or glass, having a highly polished surface.   The name of mir-
ror is  commonly given  to reflectors that  are made of glass;
and the glass is always quicksilvered on the back, to make it
reflect more light.   The word speculum is used to describe a
reflector which is metallic, such as those made of silver, steel,
or of grain tin mixed with copper.
  (10.) Specula  or mirrors are  either plane, concave, or
convex.
  A plane speculum is one which is perfectly flat,  like a look-
ing-glass ; a concave speculum is one which is hollow like the
inside of a watch-glass;  and a convex speculum is one which
is round like the outside of a watch-glass.
  As the light which falls upon glass mirrors  is  intercepted
by the glass before it is reflected from the quick-silvered sur-
face, we shall suppose all our mirrors to be formed  of polished
metal, as they are in almost all optical instruments.
  (11.) When a ray of light, A D,fig.  1.,  falls upon a  plane
       jy  x         speculum, M N, at the point D,  it will
          '  '         be reflected or driven back in a direction
       .^-—j—*v.      D B, which is  as much inclined to E D,
   y<Q        yA  a line perpendicular to M N, as the ray
   /   \.    /  \  AD was; that is, the angle B D E is
  /     \ /      \  equal to A D E, or the circular arc B E
M'-  ■  ''   t> '    " ■  fo is  equal to E A.
                            B 
 14             A  TREATISE  ON OPTICS.        PART I.

   The ray A D is called the incident  ray,  and D B the  re-
 flected ray, A D E the angle of incidence, and  B D E the
 angle of reflexion; and a plane passing through A D and
 D B, or the plane in which these two lines lie, is  called the
 plane of incidence, or the plane of reflexion.
   (12.) When the speculum is concave, as M N, fig. 2., then
         Fig. 2.         if C  be the centre  of the circle  of
          c           which M N is a part, the incident ray
                      AD and the  reflected ray D B will
      ^-— ^"\     form equal angles with the line  C D,
    *J\        /\    which  is  perpendicular  to  the small
 MbL   >v    /   \** Porti°n °f the speculum  on which the
   ^fciBii^ ' «itt^r    ray fe-Hs at D.  Hence in this case also
     ^^^^00^^    the angle of incidence A D E is equal
 to the angle of reflexion B D E.
   (13.) When  the speculum is convex, as M N,fig. 3., let C
                    be the centre of the circle of which M N
                    forms  a  part, and C  E a  line  drawn
                    through D; then the angle of incidence
                    A D E will be equal  to the angle of re-
                    flexion B D E.
                       These results are found to be true by
                    experiment;  and  they may  be easily
                    proved by admitting a ray of the sun's
                    light through a hole in the window-shut-
                    ter,  and  making it fall on the  mirrors
                    M N in the direction A D, when it will
                    be seen  reflected  in the direction D B.
 If the incident ray A D is made to approach the perpendicular
 D E, the reflected ray D B will also approach the perpendicu-
 lar D E; and when the ray A D falls in the direction E  D, it
 will be reflected in  the  direction D  E.   In like  manner
 when the  ray A D approaches to D N, the ray D B will ap-
 proach to D M.
   (14.) As these results are true under all circumstances, we
 may consider it as a general law, that when light falls upon
 any surface, whether plane or curved, the angle of its reflex-
 ion is equal to the angle of its incidence.
   Hence we have a method of universal application for  find-
 ing the direction of a reflected ray when  we know the direc-
 tion of the  incident ray.  If A D, for example, figs. 1, 2, 3.,
 is  the direction in which the incident ray falls upon the mirror
 at D, draw the perpendicular D E in Jig.  1., and in Jig. 2  or
Jig. 3. draw a line from D to C, the centre of the curved sur-
face M N;  and, having described a circle M B E A N round
 D  as a centre, take  the distance A E in  the  compasses and
 
CHAP. I.
PLANE  MIRRORS.
15
carry it from E to B, and having drawn a line from D to B, D B
will be the direction of the reflected ray.


          Reflexion of Rays from Plane Mirrors.

  (15.) Reflexion of parallel rays. When parallel or equidis-
tant rays, A D, A' I)', Jig. 4., are incident upon  a plane mir-
                          Fig. 4.
ror, M N, they will continue to be parallel after reflexion. By
the method already explained, describe arches of circles round
D, D' as centres, and make  the arch from E towards B equal
to that between A D and D E, and also the arch from E' to-
wards B' equal to that between A' D' and D' E'; then drawing
the lines D B, D' B', it will be found that these  lines are par-
allel.   If the space between  A D and A' D' is filled with other
rays parallel to A D, so as  to constitute  a  parallel beam  or
mass of light, A A' D' D, the reflected rays will be all parallel
to B D, and will constitute a parallel reflected beam.  The
reflected beam, however, will be inverted;  for the side A D,
which  was uppermost  before reflexion, will be  undermost,  as
at D B, after reflexion.
  (16.) Reflexion of  diverging rays. Diverging  rays are
those which proceed from a  point, A, and  separate as they ad-
vance, like A D, A D', A D". When  such rays fall upon a
 
16              A  TREATISE ON OPTICS.         FART I.

plane mirror M N,fig. 5., they will be reflected in directions
D B,  D'  B', D" B", making the angles B  D E,  B' D'  E',
B" D" E" respectively, equal to A D E, AD' E', A D" E";
the lines D E, D' E,' D" E"  being drawn  from the points
D, D', D", where the rays are incident, perpendicular to M N;
and by continuing the reflected rays backwards, they will  be
found to meet at a point A' as far behind the mirror M N as
A is before it;  that is, if A N A' be drawn  perpendicular to
M N, A' N will be  equal to A N.  Hence the rays will have
the same divergency after reflexion as  they had before it.   If
we consider A D" D as a divergent beam of light included
between A D   and A D", then the reflected beam included
between D B and D" B" will diverge from A', and will be  in-
verted after reflexion.
  (17.) Reflexion of converging rays.   Converging  rays
are those which proceed from several points A A' A", Jig. 6.,
towards  one point  B.  When such rays  fall upon a  plane

                          Fig. 6.
mirror, M N, they will be reflected in directions D B', D' B',
D" B', forming the same angles with the perpendiculars D E,
D' E', D" E", as the incident rays did, and converging to a point
B' as far before the mirror as the point B is behind it.   If we
consider A D D" A" as a converging beam of light, D" B' D
will be its form after reflexion.
  In all  these cases the  reflexion  does nothing  more than
invert the incident beam of light, and shift its point of diver-
gence or convergence to the opposite side of the mirror.


        Reflexion of Rays from Concave Mirrors.
  (18.) Reflexion of parallel rays.  Let M N, fig. 7., be  a
concave mirror whose centre of concavity is C ;  and let A D,
AM, A N be parallel rays, or a parallel  beam of light falling 
CHAP. I.
CONCAVE MIRRORS.
17
upon it, at and near to the vertex D.   Then, since C M, C N
are perpendicular to the surface of the mirror at the points M
and N, C M A, C N A will be the angles of incidence of the
rays A M, A N. Make the angles of reflexion C M F, C N F
equal to C M A, C N  A, and it  will be found that the  lines
M F, N F meet at F in the line A D, and these lines M F,
N F will be  the reflected rays.  The ray A C D being per-
pendicular to the mirror at D, because it passes through the
                          Fig. 7.
If ^-  .^
        Kn§££-——-----------------------.—3%-

centre C, will be reflected in an opposite direction D  F; so
that all the three rays, A M, A D, and A N, will meet at one
point, F.   In like manner it will be found that all other rays
between A M and A N, falling upon other points of the mirror
between M and N, will be reflected to the same point F. The
point F, in which a concave mirror collects the rays which fall
upon it, is called the ybcws, or fire-place, because the rays thus
collected have the  power of burning any inflammable body
placed there.  When the rays which the mirror collects are
parallel, as in the present case, the point F is called its prin-
cipal focus, or its focus for parallel rays.  When we consider
that the rays which form the beam A M N A occupy a large
space before they fall upon the mirror M N, and by reflexion
are condensed upon a small space at F, it is easy to understand
how they have the power of burning bodies placed at F.
  Rule.—The distance of the focus F from the nearest point
or vertex  D of the  mirror M N  is in spherical mirrors, what-
ever be their substance, equal to one half of C D, the  radius
of the mirror's concavity.  The distance F  D is called the
principal focal distance of the mirror.  The truth of this rule
may be found by projecting Jig.  7. upon a large scale, and by
taking the points M N near to D.
  (19.) Reflexion  of diverging  rays.  Let M N, Jig. 8.,
be  a  concave mirror,  whose  centre  of  concavity  is  C;
and let rays A  M,  A D, A  N, diverging or radiating from
the point A,  fall upon the mirror at the points, M,  D,  N,
and be reflected from these points; M and N being near
to D.  The  lines C M, C D, and  C N being  perpendicu-
lar to the mirror at the points  M, D,  and N, we shall find
                           B2 
18
A  TREATISE  ON  OPTICS.

          Fig. 8.
PART I-
IN
the reflected rays M F, N F, by making the angle F M C
equal to A M C, and F N C equal to A N C; and the point F
where these rays meet will be the focus where the diverging
rays A M, A N are collected.   By comparing^. 7. with^.
8  it is obvious that, as the incident ray A M mfig. 8. is nearer
the perpendicular C M than the same ray is mj«^. 7, the  re-
flected ray M F will also  be  nearer the perpendicular C M
than the same  ray mfig. 7.;  and as the same is true  of the
reflected ray N F, it follows that the point F must be nearer
C in Jig. 8. than in Jig. 7.; that is, in the reflexion of diverg-
ing rays the focal distance  D F of the  mirror is greater than
its focal distance for parallel rays.
   If we suppose the point of divergence A, Jig.  8.,  or  the
radiant point, as it  is called, to approach to C, the incident
rays A M, A N will approach to the perpendiculars C M, C N,
and consequently the reflected rays M F, N F will also  ap-
proach to C M, C N; that is, as the radiant point A approaches
to the centre of concavity C, the focus F also  approaches to
it,  so that when A reaches C, F will  also reach C; that is,
when rays diverge  from  the  centre, C, of a concave mirror,
they will all be reflected to the same point.
   If the radiant point A passes C towards D, then the focus F
will pass C towards A; so that if the light now diverges from
F it will be collected in A, the  points  that were formerly the
radiant points being now the foci.   From this relation, or in-
terchange, between the radiant points  and the foci, the points
A and F have been called conjugate  foci,  because if either
of them be the radiant point  the other will be the focal point.
   If in Jig. 7. we suppose F to be the radiant  point, then the
focal point A will be at an infinite distance; that is, the  rays
 will never meet in a focus, but will be  parallel, like M A,  N A
 in Jig. 7.
   In like manner it is obvious, that if the point F is at/, as in
 Jig. 9., the reflected rays will be M a, N a; that is, they will
 diverge from some point, A',  behind the mirror M N ; and  as 
CHAP. I.           CONCAVE MIRRORS.                 19

                          Fig. 9.












/approaches to D, they will diverge more and more, as if the
point A', from  which they seem  to  diverge,  approached to
D.  The point A' behind the mirror, from which the rays M a,
N a seem to proceed, or at  which they would meet if they
moved backwards in the directions a M, a N is called  their
virtual focus, because they only tend  to meet in that focus.
  In all these cases the distance of the  focus F may be deter-
mined either by projection or by the following rule, the radius
of the concavity of the  mirror, C D, and  the  distance, A D,
of the radiant point, being given.
  Rule. Multiply the distance, A D,  of the radiant point from
the mirror by the radius, C D, of the mirror, and divide this
product by the difference between  twice the distance of the
radiant point and the radius of the mirror, and the quotient
will be F D, the  conjugate focal distance required.
  In applying this rule we must observe, what  will be readily
seen from the figures, that if twice A D is less than C D (as
at / Jig. 9.), the rays will not meet before the mirror, but will
have  a virtual focus behind it,  the distance of  which from D
will be given by  the rule.
  (20.) Reflexion of converging rays. Let M N,fig.  10., be
a concave mirror whose  centre of concavity is C, and let rays
A M, A D, A N, converging to a point  A' behind the mirror, fall
upon  the mirror at the points M, D, and  N,  and suffer reflexion
at these points; M and N being  near to D. The lines C M, C D,
and C N being perpendicular to the mirror at the points M, D,
and N, we shall find the reflected rays M F and N F by making
the angle F M C equal to A M C, and F N C equal to A N C;
and the point F, where these rays meet, will be the focus where
the converging rays A M, A N are collected.   By comparing
fig. 10. with Jig. 7. it will be manifest, that, as the incident ray
A M  in fig. 10.  is  farther from the  perpendicular C M than
the same ray A M mfig. 7., the reflected ray M F in fig. 10. 
20              A TREATISE ON OPTICS.         PART I.

                         Fig. 10.
J£<^----©a—7F-----J".E>C ■ A-
will also be farther from the perpendicular C M than the same
ray in Jig. 7.; and as the same is true of the reflected ray N F,
it follows that the point F must be farther  from C in Jig. 10.
than infig. 7.;  that is,  in  the reflexion  of  converging rays,
the conjugate focal distance D F of the mirror is less than its
distance for parallel rays.
  If we suppose the point of convergence A.', fig. 10., to ap-
proach to D, or  the  rays A M, A N to become  more  conver-
gent, then the incident rays A M, A N will recede from the
perpendiculars CM, C N; and  as the reflected rays M F,
N F will also recede from C M, C N, the focus F will like-
wise approach to D; and  when A' reaches D,  F will  also
reach D.
  If the rays A M, A N become less convergent, that is,  if
their point of convergence A' recedes farther from D to the
left, the focus F will  recede  from D to the  right; and when
A' is infinitely distant, or when A M, A N are  parallel, as  in
fig. 7., F  will be half-way between D and C.
   In these cases the place of the focus F will be found by the
following  rule.
   Rule.  Multiply the distance of the point of convergence
from the mirror by the radius of the mirror, and divide this
product by the sum of twice the distance of the radiant point
and the radius C D, and the quotient will be the distance  of
the focus, or F D, the focus F being always in  front of the
mirror.

          Reflexion of Rays from Convex Mirrors.

   (21.) Reflexion of parallel rays.  Let M N, fig.  11., be a
convex mirror whose  centre is C, and let A M, A D, A N  be
parallel rays falling upon it. Continue the lines C M and C N
to E, and  M E, N E will be perpendicular to the surface  of 
CHAP. I.
CONVEX MIRRORS.
21
the mirror at the points M and N.  The rays A M, A N will
therefore  be  reflected  in directions M B, N B, the angles of
reflexion E M B, E N B being equal to the angles of incidence
E M A, E N A.  By continuing the  reflected rays B M, B N
backwards, they will be found to meet at F, their virtual focus
behind the mirror; and the focal distance D F for parallel rays

                          Fig. 11.
will be almost exactly one  half of the radius of convexity
C D, provided the points M and N are taken near D.
  (22.)  Reflexion of diverging rays. Let M N,fig. 12., be a
convex mirror, C its centre of convexity, and A M, AN rays
diverging from A, which  fall upon the mirror at the points
M, N.   The lines C M E and C N E will be, as before,  per-
pendicular to the mirror at M and N;  and consequently,  if
we make the angles of reflexion E M B, E N B equal to the
angles of incidence E M A,  E N A, M B, N B will be the re-
flected rays which, when  continued backwards, will meet  at 
22
A  TREATISE  ON OPTICS.         PART I.
F, their virtual focus behind the mirror.  By comparing fig.
12. with fig. 11., it is  obvious  that the ray A M, fig. 12., is
farther from M E than in fig. 11., and consequently the  re-
flected ray M B must also  be farther from it.   Hence, as the
same is true of the ray N B, the point F, where these rays
meet, must be nearer to D in fig. 12. than mfig. 11.; that is,
in the reflexion of diverging rays, the virtual focal distance
D F is less than for parallel rays.
   For the same reason, if we suppose the point of divergence
A to approach the mirror, the virtual focus F will also approach
it; and when A arrives at D, F will also arrive at D. In like
manner, if A recedes from the mirror, F will recede from it;
and when  A is infinitely distant, or  when  the rays become
parallel, as in fig. 11., F will be  half-way between D and C.  In
all these cases, the focus is a virtual one behind the mirror.*
                        CHAP.  II.
               IMAGES FORMED BY MIRRORS.
  (23.)  The image  of any object is a picture of it formed
either in the air, or in the bottom of the eye, or upon a white
ground, such as a sheet of paper.  Images are generally form-
ed by mirrors or  lenses; though they may be formed also by
placing a screen, with a small aperture, between the  object
and the  sheet of paper  which is to receive the image.   In
order to understand this, let C D be a screen or window-shut-
ter with a small aperture, A, and E F a sheet of white paper
placed in a dark room.  Then, if an illuminated object, RGB,
is placed on the outside of the shutter, we shall observe an in-
verted image of this object painted on the paper at r g b.   In
order to understand how this takes  place, let us  suppose the
object R B to have three distinct colors, red at R, green at G,
and blue at B;  then it  is plain that the red light  from R  will

  * For a discussion of the subjects in this chapter, see (in the College Edi-
tion) the Appendix of American Editor, Chapter I. 
CHAP. II.    IMAGES FORMED BY MIRRORS.           23

pass in  straight lines through the  aperture A, and fall upon
the paper E F at r.  In like  manner the green light from G
will fall upon the  paper at g, and  the blue light from B will
fall upon the paper at b; thus painting upon the paper an in-
verted image, rb, of the object, R B.  As every colored point
in the object R B has a colored point corresponding to it, and
opposite to  it on the  paper E F, the image o r will be an ac-
curate picture  of the object R B, provided the aperture A is
very small.   But if we increase the aperture,  the  image will
become less distinct; and \t will be nearly obliterated  when
the aperture is large.  The reason of this is,  that, with a
large aperture, two adjacent points of the object will throw
their light  on the  same point of the paper, and thus create
confusion in the image.
  It is obvious  bom fig. 13., that the size of the image b r will
increase with the distance of the paper E F behind  the hole
A.   If A g is  equal to A G, the image will  be  equal to the
object;  if Ag is less than A G, the image will  be less than
the object; and if A g  is greater than A. G,  the  image will
be greater than the object.
  As each  point of an object  throws out rays in all directions,
it is ^manifest that  those only which fall upon  the small aper-
ture at A concur in forming the image b r; and  as the num-
ber of these rays is very small, the  image b r  must have very
little light,  and therefore cannot be used for any optical pur-
poses.   This evil is completely remedied  in the formation of
images  by mirrors and lenses.
  (24.)  Formation of images by concave mirrors. Let A B,
fig. 14., be a concave mirror whose centre is  C, and let M N
be an object placed at some distance before  it.   Of all  the
                          Fig. 14.
rays emitted in every direction by the point M, the mirror re-
ceives only those which lie between M A and M B, or a cone
of rays MA B whose base is the spherical mirror, the section
of which  is A B.  If we draw the reflected rays A m, B m,
for  all the incident  rays M A, M  B,  by the methods already
described, we shall find that they will all meet at the point m, 
24             A TREATISE  ON  OPTICS.        PART I.

and will there paint the extremity M of the object.   In like
manner, the cone of rays NAB flowing from  the other ex-
tremity N of the object will be  reflected to a focus at n, and
will there paint that point of the  object.  For the same reason,
cones of rays flowing from intermediate  points between M
and N will be reflected to intermediate points  in the image
between m and n, and m n will be an exact inverted  picture
of the object M N.  It will also be  very  bright,  because a
great number of rays concur in forming  each point of the
image.   The distance of the image from the mirror is found
by the same rule which we have given for finding the focus of
diverging rays, the points  M, m in fig. 14. corresponding with
A and F mfig. 8.
  If we measure the relative sizes of the object M N and its
image m n, we shall find  that in every case the size of the
image is to the size of the object as the distance of the image
from the mirror is to the distance  of the object from it.
  If the concave mirror A B is large, and if the object  M N
is very bright, such as a plaster of Paris statue strongly illu-
minated, the image m n will appear suspended in the air; and
a series  of instructive experiments may be made by varying
the distance of the object,  and observing the variation in the
size and place of the  image.  When the object is placed at
m n, a magnified representation of it will be formed at M N.
  (25.)  Formation of images by convex mirrors. In concave
mirrors  there is, in all cases, a real image of the object formed
in front  of the mirror, excepting when the object is placed be-
tween the principal focus  and the  mirror, in  which  case  it
gives  a  virtual image formed behind  it; whereas  in  convex
mirrors  the image  is always a virtual  one formed behind the
mirror.
  Let A B,fig. 15., be a convex mirror whose centre is C, and
        Fig. 15.        M N an object placed before it;  and let
                     the eye of the observer be situated any-
M vY    A   **$ •  where m front of tne mirror, as at E.
    \V   :'\ /D<V'  ®nt or"tne great number of rays which
    \  \J\^/       are emitted in every direction from the
    \^-^^L.       points M, N of the object, and  are sub-
aT^C^S-  ^b  sequently reflected from  the mirror, a
      \   ;           few only can enter the eye at E. Those
       \ i           which do enter the eye, such as D E,
        \ i           F E  and G E,  H E, will be reflected
        (u            from the portions D F, G H of the mir-
                    ror so situated with respect to the eye
and the  points M, N that the angles of incidence and reflexion
will be equal.  The ray M D will  be reflected in a direction 
CHAP. II.      IMAGES IN  PLANE MIRRORS.          25
D E, forming the same  angle that M D does with the perpen-
dicular C N, and the ray N G in the direction G E.  In like
manner, F E, H E will be the reflected rays corresponding to
the incident ones M F, N H. Now, if we continue backwards
the rays D E, F E, they will meet at m; and they will there-
fore appear to the eye to have come from the point m as their
focus.   For the same reason the rays G E, H E will appear to
come from  the point n as their focus, and m n will be the vir-
tual image of the object M N.  It is called virtual because it
is not formed by the actual union of rays in a focus, and cannot
be received upon paper.  If the  eye E is placed in any other
position before the mirror, and if rays are drawn from M and
N, which after reflexion enter the eye, it will be found that
these rays continued backwards will have their virtual foci  at
m and n.  Hence, in every position of the eye before the mir-
ror, the image will be seen in the same spot m n. If we draw
the lines C M, C N from the centre  of the mirror, we shall
find that the points m, n are always in these lines.  Hence it
is obvious that the image m n is  always erect,  and less than
the object.  It will  approach to the mirror as the object M N
approaches  to it, and it  will recede from  it as M N recedes ;
and  when M N is  infinitely distant, and the  rays which it
emits become parallel,  the  image m n will be half-way be-
tween C and the mirror.  In other  positions of the object the
distance of the image will be found by the rule already given
for diverging rays falling upon convex mirrors.   The  size of
the image is to the  size of the object, asCm,  the distance of
the image from  the centre  of the mirror, is to C M, the dis-
tance of the object.   In approaching  the mirror,  the image
and object approach to equality; and when they touch it, they
are both of the same size.   Hence it follows that objects are
always seen diminished in convex mirrors, unless when they
actually touch the mirror.
   (26.)  Formation  of images by plane mirrors.   Let  A B,
      Fig. 16.      fig. 16., be a plane mirror or looking-glass,
                  M N an object  situated before it, and  E
                  the place of the eye; then, upon the very
                  same prhxiples which we have explained
                  for a convex mirror, it will be found that
                  an image of M N will be formed  at m n,
                  the virtual foci m, n being determined by
                  continuing back  the  reflected  rays D E,
                  F E till they meet at m, and G E, H E till
                  they meet at n.  If we join the points
                  M, m and N, n,  the lines M m, N n will
                            C
 
26
A  TREATISE ON OPTICS.         PART I.
be perpendicular to the mirror A B, and consequently parallel;
and the image will be at the same distance, and have the .same
position behind the mirror that the object has before it. Hence
we see the reason why  the images of all objects  seen  in a
looking-glass have the same form  and distance as the objects
themselves.*
DIOPTRICS.
  (27.)  Dioptrics is that branch of optics which treats of the
progress of those rays of light which enter transparent bodies
and are  transmitted through their substance.

                       CHAP. III.
                       REFRACTION.
  (28.)  When light passes through a drop of water or a piece
of glass, it obviously suffers some change in its direction, be-
cause it does not illuminate  a piece of paper placed  behind
these bodies in the same manner as it did before they were
placed in  its way.  These bodies have  therefore  exercised
some action, or produced some change upon the  light, during
its progress through them.
  In order to discover the nature of this change, let A B C D
         Fig. 17.        ^ be an empty vessel, having  a hole
                     \^ H in one of its sides B D, and let a
                        lighted candle S  be placed within a
                        few feet of it,  so that a ray of its
                        light S H may fall upon the  bottom
                        C D of the vessel, and form a round
                   <2=§[j spot of light at  a.  The  beam of
  Ca,j,c N       d^ light SHR a will be  a straight
line.  Having marked  the point  a which the ray from  S
strikes, pour water into the vessel till it rises to the level E F.
As soon as the surface of the water has become smooth, it will
be seen that the round spot  which was formerly at  a is now
at b, and that the ray S H R b is bent at R;  H R and R b
being two straight  lines meeting at R, a point in the surface
of the water.  Hence it follows, that all objects seen under
water are not seen in their true direction by a person whose
eye is not immersed in  the water.  If a fish, for example,  is
lying at b,fig. 17, it will be seen by an eye at S in the direc-
tion S a, the direction of the refracted ray R S;  so that, in
 * For the formation of images by mirrors, see (in the College edition) the
Appendix of Am- ed. chap. ii. 
CHAP. III.
REFRACTION.
27
 order to shoot it with  a ball, we must direct the gun to a
 point nearer us than the point a.  For the same reason, every
 point of an object  under water appears  in  a place  different
 from its true place; and the difference between the real and
 apparent place of  any  point of an object increases with its
 depth beneath the  surface,  and with the obliquity of the ray
 R S by which it is  seen.  A straight stick, one half of which
 is immersed in water, will  therefore appear crooked or bent
 into an angle at the point where it  enters  the  water.   A
 straight rod S R a, for example, will appear bent like S R b;
 and a rod bent will, for a like reason, appear  straight.  This
 effect must have been  often observed  in the  case of an oar
 dipping into transparent water.
   If hi place of  water  we  use  alcohol, oil, or glass, the sur-
 faces of all these  bodies coinciding with the line E F, we shall
 find that they all have the power of bending the ray of light
 S R at the point R; the alcohol bending it  more  than the
 water, the oil more than the alcohol, and  the glass more than
 the oil.  In the case of glass, the ray would be bent into the
 direction R c. The  power which thus bends or changes the
 direction of a ray of light is called refraction,—a name de-
 rived from a Latin  word, signifying breaking  back,—because
 the ray S R a is broken at R, and the water is said to refract,
 or break the ray, at R.  Hence we may conclude  that if a
 ray of light, passing through air, falls in an oblique or slanting
 direction on the surface of solid or fluid bodies that are trans-
 parent,  it will be refracted towards a line, M N, perpendicular
 to the surface E F at the point R, where the ray enters it;
 and  that the  quantity of this refraction, or  the  angle a R  b,
 varies with the nature of the  body.   The  power by which
 bodies produce this effect is  called their refractive power, and
 bodies that produce it  in different  degrees  are said to have
 different refractive  powers.
   Let the vessel A B C D be now emptied, and let  a bright
 object, such as a sixpence, be cemented on the  bottom of it at
 a.  If the observer places himself a few feet from the vessel,
 he will  find a position where he will see the sixpence at  a
 through the hole H.  If water be now poured  into the vessel
 up to E F, the observer will no  longer see the sixpence; but
 if another  sixpence is placed at a, and is moved towards b,  it
 will become visible when it  reaches b.  Now, as the ray from
the sixpence at b reaches the eye, it must come out of the
water at a point, R, in the surface, found by drawing a straight
line, S H R, through the eye and the hole H;  and conse-
quently b R must be the direction of the ray, which makes
the sixpence visible,  before its  refraction at R.   But if this 
28
A  TREATISE ON OPTICS.         PART I.
ray had moved onwards in a straight line, without being re-
fracted at R, its path would have been b h; whereas, in  con-
sequence of the refraction, its path is R H. Hence it follows,
that when a ray of light, passing through any dense medium,
such as water, &c, in a direction oblique or slanting to its
surface, quits  the medium at any point,  and enters a rarer
medium, such  as air, it is refracted from the line perpendicu-
lar to the  surface at the point where it quits it.
   When the ray S H R from the  candle falls,  or is incident
upon the surface E F of the water, and is refracted in the di-
rection R b, towards the perpendicular M N, the angle M  RII
which it makes with the perpendicular, is called the angle of
incidence ; and the angle N R b, which  the  ray R b bent or
refracted  at R makes with the same perpendicular, is called
the angle of refraction.  The ray H R is called the incident
ray, and R b the refracted ray.  But  when the light comes
out of the water from the  sixpence at  b, and is refracted at R
in the direction  R H, b R is the incident ray  and R H the
refracted  ray.   The angle N R 6 is the angle  of incidence,
and M R H the angle of refraction.
   Hence it follows, that when light passes out of a rarer  into
a denser medium, as from air to water, the angle of incidence
is greater than the angle of refraction; and when light
passes out of a denser into a rarer medium, as out of water  into
air, the  angle  of incidence is less  than  the angle of refrac-
tion:  and  these  angles are so related to one  another,  that
when the ray  which was  refracted in the one  case becomes
the incident ray,  what was formerly the incident ray becomes
the refracted ray.
   (29.)  In order to discover the law, or rule, according to
        Fig. 18.        which the  rays of light enter or  quit
                      water, or other refracting media, so
                      that we may be able to determine the
                      refracted ray when we know the di-
                      rection of the  incident ray, describe a
                      circle  M  N  upon  a  square  board
                      A B C D, fig. 18. standing  upon a
                      heavy  pedestal P, and  draw the  two
                      diameters M N, E F perpendicular to
                      one another, and also to the sides, A B,
                      A C of the piece of wood.  Let a
                      small tube, H R, be  so made that it
                      may be attached to  the board along
                      any radius H R, H' R, or, what would
                      be still  better,  that it  may  move
freely round R as a centre. Let the board with its pedestal be
 
CHAP. III.        LAW  OF REFRACTION.
29
placed in a pool or tub of water, or in a glass vessel of water, so
that the surface of the wTater may coincide with the line E F
without touching the end R of the tube H R.  When the tube
is in the position M R, perpendicular to the surface E F of the
water, admit a ray of light down the tube, and it will be seen
that it enters the water at R, and passes  straight on to N,
without suffering  any change  in  its direction.  Hence it fol-
lows, that a ray  of light incident perpendicularly on a re-
fracting surface experiences no refraction or change in its
direction.   If we now place a sixpence at N, we shall see
it through the tube M R;  so that  the  rays from the sixpence
quit the water at R, and  proceed in the  same  straight line
N R M.   Hence a ray of  light quitting a refracting surface
perpendicularly undergoes no refraction or change of direc-
tion.  If we now bring  the tube  into the  position H R, and
make a ray of light pass along it,  the  ray will be refracted at
R in some direction  R b, the angle of refraction N R  b being
less than the angle of incidence M R H.   If  we now  with a
pair of compasses, take the shortest distance  b n of the  point
b from the perpendicular  M N, and  make a scale of equal
parts,  of which b n  is one part, the  scale being divided into
tenths and hundredths, and if we set the distance H m  upon
this scale, we shall find it to be 1-336 of these parts, or 1|
nearly.  If this experiment is repeated at any other position,
H R, of the tube where R b'  is  the refracted ray, we  shall
find that on a new scale, in which b' n' is one part, H' m' will
also be 1-336 parts.   But the lines H m, H'm' are called the
sines of the angles  of incidence H R M, H' R  M, and b n,
b' n' the  sines  of the angles of refraction b R N, b' R N.
Hence it follows, that in water the sine of the angle of inci-
dence is to the  sine of the angle of refraction as 1-336 to 1,
whatever be the position of the ray with respect to the surface
E F of the water.  This truth is called by optical writers the
constant ratio  of the sines.  By  placing a sixpence at b, we
shall find that it will be seen  through the  tube when it has
the position H R;  and placing  it at b', it will be seen through
it in the position H R.  Hence, when light quits the surface
of water, the sine of its angle of incidence b  R N will be to
the sine of its angle of refraction H R M as 1  to 1-366, as
these are the measures of the sines b n, Hm;  and since these
are  also the measures of b' n' H' m! upon another  scale, in
which b' n' is  unity, we  may conclude  that,  when light
emerges from water into air, the  sines of the angles of  inci-
dence and refraction are in the constant ratio of 1 to 1-336.
   If we make the same experiment  with other bodies, we
shall obtain different degrees of refraction at the same angles;
                           C2 
30
A  TREATISE ON OPTICS.         PART I.
but in every case the sines of the  angles  of incidence and
refraction will be found to have a constant ratio to each other.
  The number 1-336, which expresses this  ratio for water, is
called the index of refraction  for water, and sometimes  its
refractive power.
  (30.) As philosophers have determined the index of refrac-
tion for a great variety of bodies, we are able, from  those de-
terminations, to ascertain the direction of any ray  when  re-
fracted at any angle of incidence from the surface of a given
body, either in entering or quitting it.  Thus, in the case  of
water, let it be required to find the direction  of a ray, HR,^.
18., after it is refracted at the surface E F of water: draw R M
perpendicular to E F at the point R, where the ray H R enters
the water, and from H  draw  H m perpendicular to M R.
Take H m in the compasses, and make a scale in which this
distance occupies 1-336 parts, or \\ nearly.  Then, taking 1
on the  same scale, place one  foot  of the  compasses in the
quadrant N F, and move  that foot towards or from N till the
other foot falls upon some one  point n in the perpendicular
RN, and in no other point of it.  Let b be the point on which
the first foot of the compasses is placed when the second falls
upon n, then the  line R 6 passing through  this point will  be
the refracted ray corresponding to the incident ray H R.
  (31.) Table I. (Appendix) contains the index of refraction
for some of the substances most interesting in optics.
  (32.) As the bodies  contained in these tables  have all dif-
ferent densities,  the  indices of refraction  annexed  to their
names cannot be considered as  showing  the relation of their
absolute  refractive powers,  or the refractive powers of their
ultimate particles.  The  small refractive index of hydrogen,
for example, arises from its particles being at  so great a distance
from one another; and,  if we  take its  specific  gravity into
account, we shall find that, instead of having a less refractive
power than all other bodies, its ultimate particles exceed  all
other bodies in their absolute action upon light.
  Sir Isaac Newton has shown, upon the supposition that the
ultimate particles of bodies are equally heavy,  and  that the
law of the forces which different media  exert is  of  the same
form in all, that the absolute refractive power is equal to the
excess of the square of the index of refraction above unity,
divided by the specific gravity of the body.
  In this way Table  II. (Appendix) has been calculated.
  Mr. Herschel has justly remarked, that if, according to the
doctrines of modern chemistry, material bodies  consist of a
finite number of atoms,  differing in their actual weight  for
every differently compounded substance, the intrinsic refrac-
tive  power of the atoms  of any given  medium  will be the 
   CHAP. IV.                LENSES.                       31

   product arising from multiplying the number for the medium,
   in Table II. by the weight of its atom.
     (33.) In examining Table II., it appears that the substances
   which contain fluoric acid have the least absolute refractive
   power, while  all inflammable bodies have the greatest   The
   high absolute  refractive power of oil of cassia, which is placed
   above  all other fluids, and even above diamond, indicates the
   great inflammability of its ingredients, f


                         CHAP. IV.*
^       REFRACTION THROUGH PRISMS  AND LENSES.
 .'   (34). By  means of the  law  of refraction explained in the
  preceding pages, we are enabled to  trace a ray of light in its
  passage through any medium or body of any figure, or through
  any number of bodies, provided we can always find the incli-
  nation of the incident ray to that small  portion of the surface
  where  the ray either enters or quits the body.
    The  bodies  generally used in optical  experiments, and in
  the  construction of optical instruments, where the effect is
  produced by refraction,  are  prisms, plane glasses, spheres,
  and lenses, a  section of each of which is shown in the  an-
  nexed figure.
                            Fig.  19.
   1. The most common optical prism, shown at A, is a solid
 having two plane surfaces A R, A S, which are  called its re-
 fracting surfaces.   The face R S, equally inclined to A R
 and A S, is called the base of the prism.
   2. A plane glass, shown  at B, is a plate  of glass with two
 plane surfaces, ab, cd, parallel to each other.
   3. A spherical lens, shown at C, is a sphere, all the points
 in its surface being equally distant from the centre O.
   4.  A double convex lens,  shown at D, is a solid formed by
 two  convex spherical  surfaces, having their centres on oppo-
 site sides of the lens.  When the radii of its two surfaces are
 equal, it is said to be equally convex;  and when  the radii are
 unequal, it is said to be an unequally convex lens.
  * For the subjects treated in this and in the preceding chapter, see (in the
College edition) the Appendix of Am. ed. chap. iii.
  f See Note No. I. at the close of author's Appendix. 
32
A TREATISE ON OPTICS.         PART I.
  5. A plano-convex lens,  shown at E, is a lens having one
of its surfaces convex and the other plane.
  6. A double concave lens, shown at F, is a solid bounded by
two concave spherical surfaces, and may be either equally or
unequally concave.
  7. A plano-concave lens, represented at G, is a lens one of
whose surfaces is concave and the other plane.
  8. A meniscus, shown at H, is a lens one of whose surfaces
is convex and the other concave, and in which the two surfaces
meet if continued.  As the convexity exceeds the concavity,
it may be regarded as a convex lens.
  9. A concavo-convex lens, shown at I, is a lens one of whose
surfaces  is concave and the other convex, and in which the
two  surfaces will  not meet though continued.  As the con-
cavity exceeds the convexity, it may be regarded as a concave
lens.
  In all  these lenses a line, M N, passing through the centres
of their curved surfaces, and  perpendicular to their  plane sur-
faces, is  called the axis.  The figures represent only the sec-
tions of  the lenses, as if they were  cut by a plane  passing
through  their axis; but the reader will understand that the
convex surface of a lens is like  the outside of a watch-glass,
and the concave surface like the inside of a watch-glass.
  In showing the  progress of light through such lenses, and
in explaining their properties, we shall still use  the sections
shown in the above figure; for since every section of the same
lens passing through its axis has exactly the same form, what
is true of the rays passing through one  section must be true
of the rays passing through every section, and consequently
through the whole surface.
  (35.) Refraction of light through prisms.  As prisms are
introduced into several optical instruments, and are essential
parts of the apparatus used for decomposing light and exam-
ining the properties of its component parts, it is  necessary
that the reader should  be able to trace the progress of light
throughtheir two refracting surfaces.  Let A B C be a prisnfof
        Fig. 20.        plate glass whose index of refraction
          A          is 1-500, and let H R be a ray of light
                      falling obliquely upon its first surface
                      A B at  the  point R.   Round  R as  a
                      centre,  and with any radius H R, de-
                      scribe the circle H M b.  Through R
                    -^ draw M R N perpendicular to A B,
                      and H m perpendicular to M R.  The
                      angle IIR M will be the ano-le of in-
cidence of the ray H R, and H m its sine, which in the present
 
 CHAP. IV.       REFRACTION BY PRISMS.             33

 case is 1500.  Then having made  a scale in which the dis-
 tance H m is 1-500, or 1-^ parts, take 1 part or unity from the
 same scale, and having set  one foot of the compasses on the
 circle  somewhere about b, move it to different points of the
 circle  till the other foot strikes only one point n of the line
 R N;  the  point b thus found will be that through which the
 refracted ray passes, R 6 will be the refracted ray, and n R 6
 the angle of refraction, because the sine b n of this angle has
 been made  such that its ratio to H m, the sine of the angle of
 incidence, is as 1 to 1-500.  The ray R b thus refracted will
 go da in a straight line till it meets the  second surface of the
 prism at R', where it will again suffer refraction in the direc-
 tion R' b'.  In order to determine this direction, make R' H'
 equal to R  H, and, with  this distance as radius, describe the
 circle H' b'.  Draw R' N perpendicular to A C, and H' m'
 perpendicular to R' N, and form a scale on which H' m!  shall
 be 1  part, or 1-000, and divide it into tenths and hundredths.
 From this scale take in the compasses the  index of refraction
 1*500,  or 1^ of these parts; and  having set one foot some-
 where in the line R' n', move it to different parts of it till the
 other foot falls upon some part of the circle  about 6', taking
 care  that the point b' is such, that when one foot of the com-
 passes is placed there, the other foot will touch the line R' n',
 continued, only in one place.  Join  R' b'.  Then, since H' R' m!
 is the angle of incidence  on the second surface AC, and H'm'
 its sine, and since n' b', the sine of the angle b' R' n', has been
 made to have to H m' the ratio of 1-500 to 1, b' R' n' will be
 the angle of refraction, and R' b' the refracted ray.
   If we suppose the original ray H R to proceed from a  can-
 dle, and if we place our  eye at b'  behind the prism so as to
 receive the  refracted ray b' R', it will  appear as if it came
 in the direction D R' b', and the candle will be seen in that di-
 rection ; the angle H E D representing its angular change of
 direction, or the angle of deviation, as it is called.
   In the construction of Jig. 20., the ray H R has been made
 to fall upon the prism at such an angle that the refracted ray
 R R' is equally inclined to the faces A B, A C, or is parallel to
 the base B C of the  prism; and  it will be found that the angle
 of incidence on the face of the  prism, H R B is  equal  to the
 angle  of emergence b' R' C.  Under these circumstances we
 shall find, by making  the angle H R B  either greater or less
 than it is in the figure, that the angle of deviation H E D is
 less  than at any other angle of incidence.   If we, therefore,
 place  the eye behind  the prism at  b',  and turn the prism
round in the plane B A C, sometimes bringing A towards the
eye and sometimes pushing it from it, we shall easily discover 
34
A  TREATISE  ON  OPTICS.
PART I.
the position where the image of the candle seen in the direc-
tion 6' D has the least deviation.  When this position is found,
the angles H R B and b' R' C are equal,  and R R' is parallel
to B C, and perpendicular to A  F, a line bisecting the refract-
ing angle B A C  of the prism.   Hence  it may be shown by
the similarity of  triangles, or proved by  projection, that the
angle of refraction b R n at the first surface is equal to B A F,
half the refracting angle of the prism.  But  since B A F is
known,  the angle of refraction b R n is also known;  and the
angle of incidence H R M being found by the preceding methods,
we may determine the index of refraction for any prism by the
following analogy.  As the sine of the angle of refraction is to
the sine of the angle of incidence, so is unity to the index of
refraction; or the  index of refraction is equal  to the  sine of
the angle of incidence divided  by the sine of the angle of re-
fraction.
   (36.)  By this method, which is very simple in practice, we
may readily measure the refractive powers of  all bodies.  If
the body be solid, it must be shaped into a prism ; and if it  is
soft or fluid, it must be placed in the angle B A C of a hollow
      Fig.21.        prism ABC, fig.21.,  made by cement-
            ^nc '  mg together three pieces of plate glass,
        ^y0\    A B, A C, B C.   A  very simple hollow
    ^r   jl  \p  prism for  this purpose may be made by
&i^__\__%J    fastening  together  at  any angle  two
   ...........................................i3L„.-B pieces  of  plate glass, A B, A C, with a
bit of wax, F.  A drop of the fluid may then be placed in the
angle at A, where it will be retained by the force of capillary
attraction.
   When light is incident upon  the  second surface of a prism,
it may fall so obliquely that the surface is incapable of refract-
ing it, and therefore  the incident  light is totally reflected from
the second surface.  As this is  a curious property of light, we
must explain it at some length.

              On the total Reflexion of Light.

   (37.)  We have already stated, that when light falls upon
the first  or second surfaces  of transparent bodies, a certain
portion of it is reflected, and another and much greater portion
transmitted.  The light is in this case said to be partially re-
flected.   When the light, however, falls very obliquely upon
the second surface of a transparent body, it is wholly reflected,
and not a single ray  suffers refraction, or is transmitted by the
surface.  Let A B C be a prism  of glass, whose index of re-
fraction is 1-500:  let a ray of light G K,fig. 22., be refracM 
CHAP. IV.         TOTAL  REFLEXION.                35
             Fia. 22            at K by the first surface A B, so
    «                         as to fall on the point R of the
    fc^_                     second surface very obliquely,
    jllllil^^               and in the direction K R. Upon
                ^EL     —G  R as a centre, and with any ra-
                      Ea^,_    dius, R H, describe the circle H
     il    Ir    Jf         a M E N F; then, in order to find
       \v     ./              the refracted ray corresponding
          n                  to H R,  make a scale on which
                              H m is equal to 1, and  take in
 the compasses 1-500 or 1£ from that scale,  and  setting one
 foot in the quadrant E N, try to find some point in it, so that
 the other foot may fall only in one point of the radius R N.  It
 will soon be seen that there is no such point, and that 1-500 is
 greater even than E R, the sine of an angle E R N of 90°.   If
 the distance 1-500 in the compasses  had been less than E R,
 the ray would have been refracted at R; but as  there is no
 angle  of refraction whose  sine is  1-500,  the  ray  does not
 emerge from the  prism, but suffers total reflexion  at R in the
 direction R S, so  that the angle of reflexion M R S is equal to
 the angle of incidence M R H.  If we construct fig. 22. so as
 to make the incident ray H R take different positions between
 M R and F R, we shall find  that the  refracted ray will take
 different positions between R N  and  R E.  There  will be
 some position  of  the incident ray about H R, where the  re*
 fracted ray will just coincide with RE; and that will happen
 when the quantity 1-500, taken from the scaje on which H m
 is equal to 1, is exactly equal to R E, or radius.  At all posi-.
 tions of the incident ray between this line and F R, refraction
 will be  impossible,  and the ray incident at R will be totally
 reflected.   It will also be found that the sine of the angle  of
 incidence at R, at which the light begins to be totally reflect-
 ed, is equal to T.^, or -666, or f, which is the sine of 41° 48',
 the angle of total reflexion for plate glass.
   The passage from partial to total reflexion may be finely
 seen, by exposing one side, A C,  of a prism A B C,fig. 20,, to
 the light of the sky, or at night to the  light reflected from  a
 large sheet of  white paper.  When the eye is placed behind
 the other side,  A B,  of the prism, and looks  at the image of
 the sky, or the paper,  as reflected from  the  base, B C, of the
 prism, it will see when the angle of incidence upon  B C is
 less than 41° 48',  the faint light produced by partial reflexion;
 but by turning the prism round, so as to  render the incidence
 •gradually more oblique, it will see  the  faint light pass sud-
denly into a bright light, and separated from  the faint light by 
36
A  TREATISE  ON  OPTICS.
PART I.
a colored fringe, which marks the boundary of the two reflex
ions at an angle of 41° 48'.   But, at all angles of incidence
above this, the light will suffer total reflexion.

        Refraction of Light through Plane  Glasses.
  (38.) Let M N,fig. 23., be the section of a plane glass with
         „  23         parallel faces;  and let a ray of light,
          s' *          A B, fall upon the first surface  at B,
                       and be refracted  into the direction
                    «' B C: it will again be  refracted  at its
                       emergence  from the second surface
                       at C, in a direction, C D, parallel to
                       A B; and to an eye at D it will ap-
                       pear to have proceeded in a direction
                       a C, which  will be found  by continu-
ing D C backwards.  It will thus appear to come from a point
a below A, the point from which it was really emitted.   This
may be proved by projecting the figure by the method already
described; though it will be obvious also from  the considera-
tion, that if we suppose the refracted ray to become the inci-
dent ray, and to move backwards, the incident ray will become
the refracted ray.  Thus the refracted ray B C,  falling at equal
angles upon the two surfaces of the plane  glass, will suffer
equal refractions at B and C, if  we suppose it  to move in op-
posite directions; and consequently  the angles which the  re-
fracted rays B A, C D form  with the two refracting  surfaces
will be equal, and the rays parallel.
  If  we suppose another ray,  A' B', parallel  to A B, to fall
upon the point B',  it will suffer  the same refraction at B' and
C, and will emerge in the direction C  D', parallel to C  D, as
if it came from a point a'.   Hence parallel rays falling upon
a plane glass will retain  their parallelism after passing
through it.
   (39). If rays  diverging from  any point, A, fig. 24., such as
           Fig. 24.            a  B, A B', are incident  upon a
                            plane glass, M N,  they will be
                            refracted into the directions B C,
                            B'  C by the first  surface, and
                            C D, C D' by  the  second.  By
                            continuing C B, C B' backwards,
                            they will be found to meet at a,
                            a   point farther from  the glass
                            than A.  Hence, if we suppose
the surface B B' to be that of standing water, placed horizon-
 
CHAP. IV.         CURVED SURFACES.                 37

tally, an eye within it would see the point A removed to a,
the divergency of the rays B C, B' C having been diminished
by refraction at the surface B B'.  But when the rays B C,
B' C suffer a second refraction, as in the case of a plane glass,
we shall find, by continuing D C, D' C backwards, that they
will meet at b,  and  the  object  at A will seem to  be  brought
nearer to the glass; the rays C D, C D', by which it is seen,
having been rendered more divergent by the two  refractions.
A plane glass, therefore, diminishes the distance of the diver-
gent point of diverging  rays.
   If we  suppose  D C, D' C to be rays converging to b, they
will be made to converge to A by the refraction  of the two
surfaces;  and  consequently  a  plane glass causes to  recede
from it the convergent point of converging rays.
   If the  two surfaces B B', C C are equally curved, the one
being convex and the other concave, like a watch-glass, they
will act upon light  nearly like  a plane glass;  and accurately
like a plane glass, if the convex and  concave sides are so re-
lated that the rays B A,  C D are  incident at equal angles  on
each surface: but this is not the case when the surfaces have
the same centre,  unless  when the radiant point  A is in their
common  centre.  For these reasons,  glasses with parallel sur-
faces are used in  windows and for watch-glasses, as they pro-
duce very little change upon the form  and  position of objects
seen through them.
       Refraction of Light through Curved Surfaces.

   (40). When we consider the inconceivable  minuteness of
the particles of light, and  that a single ray consists of a suc-
cession of those particles, it is obvious that the small part of
any curved  surface on which it falls,  and which is  concerned
in refracting it, may be regarded as a plane.  The  surface of
a lake,  perfectly still, is known to be a curved  surface of the
same radius as that of the earth, or about 4000 miles;  but a
square  yard of it, in which it is  impossible to discover any
curvature, is larger  in  proportion to  the radius of the earth
than the small space on the surface of a lens occupied by a ray
of light is  in relation to the radius  of that surface.   Now,
mathematicians have demonstratedthat a line touching a curve
at any point may be safely regarded as coinciding with an in-
finitely small part of the curve ;  so that when a ray of light,
AB,fig.25., falls upon  a  curved refracting surface at B,  its 
38              A  TREATISE  ON  OPTICS.         PART I
        Fig. 25.         angle of incidence must be considered
                      as  A B D,  the angle which  the  ray
                      A B forms with a line D C, perpendic-
                      ular to a line M N, which touches, or
                      is a tangent to, the curved surface at
                      B.   In all spherical surfaces,  such as
                      those  of  lenses, the tangent  M N is
                      perpendicular to the radius C B of the
                      surface.  Hence, in spherical  surfaces
the consideration of the tangent M N is unnecessary;  because
the radius C D,  drawn  through the point of incidence B, is
the perpendicular from which the angle of incidence  is to be
reckoned.

           Refraction of Light through Spheres.
  (41.)  Let M N be the  section  of a sphere of glass whose
centre is C, and whose index of refraction is 1-500;  and let
parallel rays, fig. 26.,  H R, H' R',  fall  upon it at equal  dis-
                          Fig. 20.
tances on each side of the axis G C F.   Tf the ray H R is in-
cident at R, describe the circle IID b round R; through C and
R draw the line CRD, which will be  perpendicular to the
surface at R, and draw II m perpendicular to R D.  Draw the
ray R b r through a point b found by the  method already ex-
plained, and so that  the  sine b n of the angle of refraction
6RC may be 1 on the  same scale on which II m is 1-500, or
1^; then R b will be the  ray  as refracted by the first surface
of the sphere.   In like manner draw R' r' for  the refracted
ray corresponding to H' R'.
   If we continue the  rays R  r, R' r', they will meet the axis
at E, which will be the focus of parallel rays for a single con-
vex surface RPR'; and the focal distance P E may be found
by the following rule.
   Rule for finding the principal focus of a single convex
surface.   Divide the  index of refraction  by its  excess  above
unity, and the quotient will be the principal focal distance,
P E;  the  radius of the surface, or C K,  being 1.   If C R id 
CHAP. IV.  CURVED  REFRACTING  SURFACES.          39

given in inches, then we have to multiply the result by that
number of inches.  When the surface is that of glass, of which
the index of refraction is 1-5, then the focal distance, P E, will
always be equal to thrice the radius, C R.
  Round r as a centre, with a radius equal to RII, describe
the circle D' b' h, and, by the method formerly explained, find
a point b' in the  circle, such that b' n', the sine  of the angle
of refraction b'rn1, is 1-500 or 1^ on the same scale on which
hm', the sine of the angle of incidence, is 1 part, and r b'F
will be the ray refracted at the second  surface.   In the same
manner we shall find r' F to be the refracted ray correspond-
ing to the incident ray R' r', F being the point where r b' cuts
the axis  G E.   Hence the point F will  be  the focus of paral-
lel rays for the sphere of glass M N.
  If diverging rays  fall upon  the points R, R', it  is quite
clear, from  the inspection of the figure, that their focus will
be on some point of the axis G F  more remote from the sphere
than F, the distance of their focus  increasing as the radiant
point  from which they  diverge approaches to  the sphere.
When the radiant point is as far  before the sphere as F is be-
hind it, then the rays will be refracted into parallel directions,
and the focus will be infinitely distant   Thus, if we  suppose
the rays F r, F r' to diverge from F, then they will emerge
after refraction in the parallel directions R H, R' H'.
  If converging rays fall upon  the  points R R', it is equally
manifest  that their  focus will  be at some point of the axis,
G F, nearer the sphere than its  principal  focus F; and their
convergency may be so great that their focus will fall within
the sphere.  All these truths may be rendered more obvious,
and would be more deeply impressed upon the mind, by tracing
rays  of different degrees of divergency and  convergency
through the sphere,  by the methods already so fully explained.
  (42.) In  order  to  form an idea of the effect of a sphere
made of substances of different refractive  powers, in bringing
parallel  rays to a focus, let us suppose the  sphere to have a
radius of one inch, and let the focus F be determined as mfig.
26., when the substances are,
           ~—;■    F      Index of       Distance, F d, of the
                  *j     Refraction.      Focus from the Sphere.
           Tabasheer  -   1-11145    -    4  inches.
           Water      -  1-3358    -    1   —
           Glass       -  1-500       -    h   —
           Zircon      -  2-000     -    0   —
Hence we find that in tabasheer  the distance F Q, is 4 inches;
in water, 1 inch; in glass, half an inch;  and in zircon, nothing;
that is, r and F coincide with Q, after a single refraction atR. 
 40              A TREATISE ON OPTICS.         PART I.

   When the index of refraction  is greater than 2-000, as in
 diamond and  several other substances, the ray  of light  R r
 will cross the axis at a  point somewhere  between C and Q.
 Under certain circumstances the  ray R r will  suffer total re-
 flexion at r, towards another part of the sphere, where it will
 again suffer total reflexion, being carried round  the circum-
 ference of the sphere, without the power of making its escape,
 till the ray is lost by absorption.  Now, as this  is true of every
 possible section of the sphere, every such ray, R r, incident
 upon it in a circle equidistant from the axis, G F, will suffer
 similar reflexions.
   Rule for finding the focus F  of a sphere.  The distance
 of the focus, F, from the centre, C, of any sphere  may be thus
 found.   Divide the index  of refraction by twice its  excess
 above 1, and the quotient is the distance, C F,  in  radii  of  the
 sphere.   If the radius of the sphere is 1 inch, and its refrac-
 tive power 1-500, we  shall have C F  equal to 1J inches, and
 Q F equal to half an inch.


 Refraction of Light through Convex and Concave Surfaces.
  (43.) The method of tracing the progress of a ray which
 enters a convex surface, is shown in Jig. 26. for the  ray H R,
and of tracing one entering a concave surface of a rare me-
dium, or quitting a convex surface of a dense  one, is shown
for the ray R r, in the same figure.
  When the ray enters the concave  surface of a dense me-
dium, or quits a similar surface, and enters the  convex surface
of a rare medium, the method  of tracing its progress is shown
in fig. 27., where M N is a dense medium (suppose  glass)
                          Fig.<n.
with two concave surfaces, or a thick concave lens.  Let C, C
be the centres of the two surfaces lying in the axis  C C, and
H R, H' R' parallel rays incident on the first surface. As C R
is perpendicular to the surface at R, IIR C will be the angle
of incidence; and  if a  circle  is described with  a  radtus
355 
CHAP. IV.   REFRACTION THROUGH LENSES.           41

R h, h m will be the  sine of that angle.  From a scale on
which hm is 1-500, take ir/the compasses 1, and find  some
point, b, in the circle wher6, when one foot of the compasses
is placed, the other will fall only on one  point, n, of the per-
pendicular C R:  the line  R b drawn  through this point will
be the refracted ray.  By continuing this ray 6 R backwards,
it will be found that it meets the  axis at  F.   In  like manner
it will be seen that the ray H R'  will be refracted in the di-
rection R' r', as if it also diverged from F.  Hence F will be
the virtual focus of parallel rays refracted by a single concave
surface,  and  may be found by the  following rule.
   Rule for finding the principal focus  of a single concave
surface.  Divide the  index of refraction  by its excess above
unity,  and the quotient will be the principal focal distance
F E, the radius of the  surface, or C E, being  1.  If the radius
C E is given in  inches, we  have only to multiply C F,  thus
obtained by  that number of inches, to have the value of F E
in inches.
   If, by  a similar method, we find the refracted ray r d at the
emergence of the ray r b from the second surface r r' of the
lens, and continue it backwards, it will be found to meet the
axis at /; so that the divergent rays  R r, R' r' are rendered
still more divergent by the second surface, and/will be the
focus of the  lens M N.

        Refraction of Light through  Convex Lenses.

   (44). Parallel rays. Rays of light falling upon  a  convex
lens parallel to its  axis are refracted  in precisely  the  same
manner as those which fall upon a sphere; and  the refracted
ray may be found by the very same methods.  But as a sphere
has an axis  in every  possible  direction, every  incident ray
must  be parallel to an axis of it; whereas, in a lens which
has only one axis, many o£_the  incident rays  must be oblique
to that axis.  In every case, whether of spheres or lenses, all
the rays that pass along the axis suffer no refraction, because
the axis  is always perpendicular to the refracting surface.
   When parallel rays,  R L, R C, R L, fig. 28., fall upon a
double convex lens, L L, parallel to its  axis R F, the ray R C
which coincides with the axis will pass through  without suf-
fering  any refraction, but the other rays, R L, R L, will be
refracted at each of the surfaces of the lens; and the refract-
ed rays  corresponding  to them, viz. L F, L F, will  be found,
by the method already given, to meet at some point,  F, in the
axis.
  When the rays are  oblique to  the  axis, as S L, S L,  T L,
                           D2 
42
A  TREATISE  ON  OPTICS.
PART I.
Fig. 28.
 T L, the rays S C, T C, which pass through the centre, C, of
 the lens, will suffer refraction at each surface; but as the two
 refractions are equal, and in opposite directions, the finally re-
 fracted rays C/, C/' will be parallel to S C, T C.  Hence, in
 considering oblique rays, such as S I,, T L, we may regard
 lines S/ T/' passing through the centre, C, of the  lens as the
 directions of the refracted rays corresponding to S C, T C. By
 the methods already explained, it will be found that S L, S L will
 be refracted to a common point,/, in the direction of the cen-
 tral ray S/ and T L, T L, to the point/'.  The focal distance
 F C, or fC, may be found numerically by the following rule,
 when the  thickness of the lens is so small that  it may be
 neglected.
  Rule for finding the principal focus, or the focus of paral-
 lel rays, for a glass lens unequally  convex.  Multiply the
 radius of the one surface by the radius of the other, and divide
 twice this product by the sum of the same radii.
  When the lens is of glass and equally convex, the focal dis-
 tance will be equal to the radius.
  Rule for  the principal focus of a plano-convex lens of
glass.  When the convex side is exposed to parallel rays, the
distance  of the focus from the  plane  side, will be equal to
twice  the  radius of the convex surface, diminished by  two
thirds of the thickness of the lens.
  When the plane side is exposed to parallel rays, the dis-
tance of  the focus from  the convex side will be equal to twice
the radius.
  (45.) Diverging rays.  When diverging rays,  R L, R L,
fig. 29., radiating from the point R, fall upon the double con-
vex lens  L L, whose principal focus is at O and O', their focus
will be at some point F  more remote than O. If R approaches
to L L, the focus F will recede^from L L.  When R comes
to P, so that P C is equal to twice the principal focal distance 
CHAP. IV.   REFRACTION  THROUGH  LENSES.          43
                          Fig. 29.



        R^---1   J>>F--P<

                               i.
C O, the focus F will be at P' as far behind the lens as the
radiant point P is before it.   When R  comes to O', the focus
F will be infinitely distant, or the  rays  L F, L F will be par-
allel ; and when R is between O' and  C,  the refracted rays
will diverge  and have a virtual focus before  the lens.  The
focus F of a glass lens, when the  thickness is small, will be
found by the following rule.
  Rule for finding the focus of a convex  lens for diverging
rays.   Multiply twice the product of the radii of the two sur-
faces  of the  lens by the distance, R C, of the radiant point,
for  a dividend.  Multiply the sum  of  the two radii  by the
same distance R C, and from this product  subtract  twice the
product of the radii, for a divisor.  Divide  the above dividend
by the divisor, and the quotient will be the  focal distance, C F,
required.
  If the lens is  equally convex, the rule will be this.  Multi-
ply the distance of' the radiant point, or  R C, by the radius of
the surfaces, and divide that product by the difference between
the same distance and the radius, and the quotient will be the
focal length, C F, required.
  When the lens is plano-convex, divide twice the product of
the distance of the radiant point multiplied by the  radius  by
the difference between that distance and twice the radius, and
the quotient will be the distance of the  focus from the centre
of the lens.
  (46.)  For converging rays.  When  rays,  R L, R L, con-
verging to a point f,fig. 30., fall upon a convex lens L L, they
                          Fig. 30.
r-----—He ys1  o-----^£*>*
will be so refracted as to converge to a point or focus F nearer
the lens than its  principal focus O.  As the point of con-
vergence/recedes from the lens, the point F will also recede 
44
A  TREATISE ON OPTICS.
PART I.
 from it towards O, which it just reaches when the point/ be-
 comes infinitely distant.  \Vhen / approaches the  lens,  F
 also  approaches  it.   The focus  F  of  a glass lens  may be
 found when the thickness is small, by the following rule :—
   Rule for finding the focus of converging rays.  Multiply
 twice the product of the radii of the two surfaces  of the lens
 by the distance/C of the point of convergence,  for a divi-
 dend.  Multiply the sum of the  two radii  by the same  dis-
 tance/C, and to this  product add twice the product of the
 radii, for a divisor.  Divide the above dividend by  the divisor,
 and the quotient will be the focal distance C F required.
   If the  lens  is equally convex, multiply the distance/C by
 the radius of the surface, and divide  that product by the sum
 of the same distance and  the radius, and the quotient will be
 the focal  length F C required.
   When the lens is plano-convex, divide twice the  product
of the distance/C multiplied by the  radius by the  sum of
that distance and twice the radius, and  the quotient  will be
the focal  distance F C required.

       Refraction of Light through Concave Lenses.
  (47.) Parallel rays.  Let L L be a double concave lens, and
                           £•31.


              B
R-
I.' ,--'
R L, R L parallel rays incident upon  it; these rays will di-
verge after  refraction  in  the  directions L r,  L r, as if they
radiated from a point F, which is the virtual focus of the lena
The rule for finding F C is the same as for a convex lens
  (48.) Diverging  rays.  When the  lens L L receives the
Fig. 32.
L
rays R L, R L diverging from R, they will be  refracted into 
CHAP. IV.   REFRACTION  THROUGH LENSES.          45
lines, L r, L r, diverging from  a focus F, more remote from
the lens than the principal focus O, and the focal distance,
F C, will be found by the following rule:—
  Rule for finding the focus of a concave lens of glass, for
diverging rays.  Multiply twice the product of the radii by
the distance, R C, of the radiant point  for a dividend.  Mul-
tiply the sum  of the radii by the distance R C, and add to this
twice the product of the radii, for a divisor.  Divide the divi-
dend by the divisor, and the quotient will be the focal distance.
  If the lens  is equally concave, the rule will be this.  Mul-
tiply the distance of the radiant point by the radius, and divide
the product by the sum of  the  same distance and  the radius,
and the quotient will be the focal distance.
  When the lens is plano-concave,  multiply twice the radius
by the distance of the  radiant point, and  divide this  product
by the sum of the same distance and twice the radius; the
quotient will be the focal distance.
  (49).  Converging rays.  When rays,  R L, R  L, fig. 33.,

                          Fig. 33.
    F -<^ o---------lie------6---^~S=5~;f


           K'.-----------    ^        ~       r
converging to a point /, fall upon a concave lens, L L, they
will be refracted so as to have their virtual focus at F, and the
distance F C will be found by the rule given for convex lenses.
The rule for finding the focus of converging rays is exactly
the same as that for diverging rays in a double convex lens.
  When the lens  is  plano-concave, the  rule  for finding the
focus  of converging rays is the same as for diverging rays on
a plano-convex lens.

Refraction of Light  through Meniscuses and  Concavo-con-
                   vex Lenses of Glass.
  (50.) The general effect of a meniscus in refracting paral-
lel, diverging, and converging rays, is the same as that of a
convex lens of the same focal length;  and the  general  effect
of a concavo-convex lens is the same as that of a concave lens
of the same focal length.
  Rule for a meniscus with parallel rays.   Divide twice the
product of the two radii by their difference, and the quotient
will be the focal distance required. 
46              A TREATISE ON OPTICS.         PART I.

   Rule for a meniscus with diverging rays.  Multiply twice
the distance of the  radiant point by the product of the two
radii for a dividend.  Multiply the difference between the two
radii by the same distance of the radiant point,  and  to this
product add twice the product of the radii for a divisor.  Divide
the above dividend by this divisor, and the quotient will be the
focal distance required.
   The same rule is applicable for converging rays.
   Both the  above  rules are applicable to concavo-convex
lenses; but the focus is a virtual one in front of the lens.
   The truth of the preceding rules and observations is ca-
pable of being demonstrated mathematically;  but the reader
who has not studied mathematics may obtain an ocular  demon-
stration of them, by projecting the rays and lenses in  large
diagrams, and determining the course of the rays after refrac-
tion by the methods already described. We would  recommend
to him also to submit the rules and observations to the  test of
direct experiment with the lenses themselves.
                        CHAP. V.

ON  THE  FORMATION OF  IMAGES BY LENSES,  AND  ON THEIR
                   MAGNIFYING POWER.*

  (51.) We have already described, in Chapter II., the  prin-
ciple of the formation  of images by small apertures,  and by
the convergency of rays to foci by reflexion from mirrors.
Images are formed, by refraction, by lenses in the very  same
manner as they are formed, by reflexion, in mirrors; and it is
a universal rule,  that when an  image is formed by a convex
lens, it is inverted in position relatively  to the  position of the
object, and its magnitude is to that of the object as its distance
from the lens is to the distance of the object from the lenp.
  If M N is an object placed before a convex lens, L 'h,fig.
34., every point  of it will send  forth rays in every direction.
Those rays which fall  upon  the lens L L will be refracted to
foci behind the lens,  and at such distances from it as  may be
determined by the Rules in the last chapter.   Since the  focus
where any point of the object is represented in its image is in
the straight line drawn from that point of the object through
the middle point C of the lens, the upper end M of the object
will be represented somewhere in the line M C m, and the
lower  end N  somewhere in  the line N C n, that is, at the

       See, in the College edition, Appendix of Am. ed. chap. iv. 
CHAP. V.     IMAGES  FORMED BY  LENSES.            47

points m, n, where the rays L m, L m, L n, L n cross the lines
M C m, N C n.  Hence m will represent the upper, and n the
lower end of the  object M N.   It is also evident, that in the
Fig. 34.

                    M
 two triangles  M C N, m C n, m n, the length of the image
 must be to M N the length of the object as C m, the distance
 of the  image,  is to C M, the  distance of the object from the
 lens.
   We  are enabled, therefore, by a lens, to form an image of
 an object at any distance behind the lens we please, greater
 than its principal focus, and  to  make this image as large as
 we please, and in any proportion  to  the object.   In  order to
 have the image large, we must bring the object near the lens,
 and in  order to have it small, we must remove the object from
 the lens; and  these effects we can vary still farther,  by using
 lenses of different focal lengths or distances.
   When the lenses have the same, focus, we may increase the
 brightness of the image by increasing the  size of the lens or
 the area of its surface.  If a lens has an  area of 12 square
 inches, it will obviously intercept twice as many rays proceed-
 ing from every point of the object as if its  area were only 6
 square  inches; so  that, when it  is out of our power to in-
 crease the brightness of the object by illuminating it, we may
 always increase the brightness of the image by using a larger
 lens.
   (52.) Hitherto we have supposed the image m n to be re-
 ceived upon white paper, or stucco, or some  smooth and white
 surface on which a  picture  of it is distinctly formed ; but if
 we receive it upon ground glass, or transparent paper, or upon
 a plate  of glass one of whose sides is covered with a dried
 film of  skimmed milk, and if we place our eye 6 or 8 inches
 or more behind this semi-transparent ground  interposed at m n,
 we shall  see the inverted  image m n as distinctly as before.
If wc keep the eye in- this position, and remove the seini- 
48             A TREATISE ON OPTICS.           PART I.

transparent ground, we shall see an image in the air distinctly
and more bright than before. The cause of this will be readily
understood, if we  consider that all the  rays which form by
their convergence the points m, n of the image m n, cross one
another at in, n, and diverge from these points exactly in the
same manner as they would do from a real object of the same
size and brightness placed in m n.  The  image m n therefore
of any object  may be  regarded as a  new  object;  and by
placing another lens behind it, another  image of the image
m n would be  formed, exactly of the  same size and  in the
same place as it would have been had m n been a real object.
But since the new image of m n must be inverted, this  new
image  will now be an erect image of the object M N, obtained
by the  aid of two lenses; so that, by using one or more lenses,
we can obtain direct or inverted images of any object at plea-
sure.   If the object M N is a movable one,  and within our
reach,  it is unnecessary to use two lenses to obtain an  erect
image  of it:  we have only to turn it  upside down, and we
shall obtain, by means of one lens, an image erect in reality,
though still inverted in relation to the object.
  (53.) In order to explain the power of lenses in magnifying
objects and bringing them  near us, or rather in giving mag-
nified  images  of objects,  and bringing the images  near us,
we must examine the different circumstances under which
the same object appears when placed at different distances
from the eye.   If an eye placed at E looks at a man a b, Jig.
35., placed at a distance, his general outline only will be seen,
                         Fig. 35.
and neither his age, nor his features, nor his dress will  be re-
cognized.  When he is brought gradually nearer to us, we dis-
cover the separate parts of his dress, till at  the distance of a
few feet we perceive his features; and when brought still
nearer,  we can count his  very eye-lashes,  and observe the
minutest lines  upon  his skin.  At the distance E b the man
is seen under the  angle b E a, and at the distance  E B he is
seen under the greater angle B E A or b E A', and  his appa-
rent magnitudes, a b, A' b, are measured  in  those different
positions by  the angles b E a, B E A, or b E A'.  The appa-
rent magnitude of the smallest object may, therefore, be equal 
CHAP. V.   MAGNIFYING POWER OF LENSES.        49

to the  apparent magnitude of the greatest.  The head of a
pin, for example, may be brought so near the eye that it will
appear to cover a whole mountain, or even the whole visible
surface of the earth, and in this case the apparent magnitude
of the pin's head is said to be equal to the apparent magnitude
of the mountain, &c.
  Let us now suppose the  man a b to be placed at  the dis-
tance of 100 feet from the eye at E, and that we place a con-
vex glass of 25 feet  focal distance, half-way between the ob-
ject a b and the eye, that is 50 feet from each;  then, as we
have previously shown, an inverted image of the man will be
formed 50 feet behind the lens, and of the very same size as
the object, that is,  six feet  high.  If this object is looked at
by the eye, placed 6 or 8 inches behind it, it will be seen ex-
ceedingly distinct, and nearly as well as if the man had  been
brought nearer from the distance of 100 feet to the distance
of 6 inches, at which we can  examine minutely the details
of his personal appearance,  Now, in  this  case,  the  man,
though not  actually magnified,  has been apparently  magni-
fied, because his apparent magnitude is greatly increased, in
the proportion nearly of 6 inches to 100 feet, or of 200 to 1.
  But if, instead of a lens of 25 feet focal length, we make
use of a lens of a shorter focus, and place it in such a position
between the eye and the man, that its conjugate foci  may be
at the distance of 20 and 80 feet from  the lens, that is, that
the man is 20  feet before the  lens, and  his image 80  feet be-
hind it, then the size of the image is four times that of the
object, and  the eye placed 6 inches behind this magnified
image will  see it with  the greatest distinctness.  Now in
this case the image is magnified 4 times directly by the lens,
and 200 times by being brought 200 times nearer the eye;
so that its apparent magnitude will be  800 times as large as
before.
  If, on the other  hand, we use a lens of a still smaller focal
length, and  place it in such  a position  between  the eye and
the man, that  its conjugate foci may be at the distance of 75
and 25 feet from the lens, that is, that the man is 75  feet be-
fore the lens, and his image 25 feet behind it, then the size of
the image will be  only one third of the size of the  object;
but though the  image is thus diminished  three times  in size,
yet its apparent magnitude is  increased  200 times by being
brought within 6 inches of the eye, so  that it is still  magni-
fied, or its apparent magnitude  is increased 2§°, or 67 times,
nearly.
  At   distances less than the  preceding, where  the  focal
length of the  lens forms a considerable part of the whole dis- 
 50              A  TREATISE  ON  OPTICS.         PART I.

 tance, the rule for  finding the magnifying power of a lens,
 when the eye views, at the distance of 6  inches, the image
 formed by the lens, is as follows.  From the distance between
 the image and the object in  feet, subtract the focal distance
 of the lens in feet, and divide the  remainder by the same focal
 distance.  By this quotient divide twice the  distance of the
 object in feet, and the new quotient will be  the magnifying
 power, or the number of times that the apparent magnitude
 of the object is increased.
   When the focal length of the  lens is quite inconsiderable,
 compared with the  distance  of the object, as it  is in most
 cases, the rule becomes this.  Divide the focal length of the
 lens by the distance at which the  eye looks at the image ; or,
 as the eye will generally look at it at the distance of 6 inches,
 in order to see it most distinctly, divide the focal length by 6
 inches;  or,  what is the same thing, double the focal length in
 feet, and the result will be the magnifying power.
   (54.)  Here,  then, we have the principle  of  the simplest
 telescope; which consists of a  lens, whose focal length ex-
 ceeds six inches, placed  at one end of a tube whose length
 must always be six inches greater than the  focal length of the
 lens.   When the eye is placed at the other end of the tube,
 it will see an inverted image of distant objects, magnified in
 proportion to the focal length of the lens.  If the lens has a
 focal  length  of 10 or 12 feet, the magnifying power will be
 from 20  to 24 times, and the satellites  of Jupiter will be dis-
 tinctly seen  through this  single  lens telescope.  To a very
 short-sighted person, who sees objects distinctly at a distance
 of three inches, the magnifying power would be from 40 to 48.
   A single concave  mirror is, upon the same  principle, a re-
flecting  telescope, for it is of no consequence  whether the
 image of the object is formed by  refraction or reflexion.   In
 this case, however, the image m n, fig. 14., cannot be looked
 at without standing  in the way of the object; but  if the re-
 flection is made a little obliquely, or if the mirror is sufficiently
large, so as not to intercept all the  light from the object, it may
be employed  as a telescope. By using his great mirror, 4  feet
in diameter and 40 feet in focal  length, in the way now men ■
tioned, Dr. Herschel  discovered one of the satellites of Saturn.
   But there  is still  another way  of increasing the apparent
magnitude of objects, particularly of those  which are within
our reach, which  is of great importance in optics.  It will be
proved, when we come to treat of  vision, that a good eye seed
the visible outline  of an  object  very  distinctly  when it is
placed at a great  distance, and that,  by a particular  power in
the eye, we  can accommodate it to perceive objects at differ- 
CHAP. V.   MAGNIFYING POWER  OF LENSES.         51

ent distances.  Hence, in order to  obtain distinct vision of
any  object,  we have only to cause  the rays  which proceed
from  it to enter  the eye  in parallel  lines, as  if the object
itself was very distant.   Now, if we  bring an  object, or the
image of an object, very near to the eye, so as to give it great
apparent magnitude, it becomes indistinct; but  if we can, by
any contrivance, make the rays which proceed  from it enter
the eye nearly parallel, we shall necessarily see it distinctly.
But  we have already shown that when rays diverge from the
focus of any lens, they will  emerge from it parallel.  If we,
therefore, piace an object, or an image of one, in the focus of
a lens held close to the eye, and having a small focal distance,
the rays will  enter the eye parallel, and we shall see the ob-
ject very distinctly,  as it will be magnified in the proportion
of its present short  distance  from the eye to the distance of
six inches, at which we see small objects most distinctly.  But
this  short distance is equal to the focal length of the lens, so
that the magnifying power produced by the lens will be equal
to six inches divided  by the focal length of the  lens.  A lens
thus used to  look at or magnify any object is a single micro-
scope ; and  when such a lens is used to magnify the magnifi-
ed image produced  by another lens, the two lenses together
constitute a compound microscope.
  When such a  lens is used to magnify the image produced
in the  single lens  telescope from a  distant object, the  two
lenses together constitute what is called the astronomical re-
fracting telescope;  and  when it  is  used  to magnify the
image produced by a concave mirror from a distant object, the
two constitute a reflecting telescope, such as that used by Le
Maire and  Herschel: and when it is  used to magnify an en-
larged  image, M N, Jig.  14.,  produced from an  object m n,
placed  before a concave  mirror, the two constitute a reflect-
ing  microscope.   All these  instruments will be more  fully
described in a future chaDter.
                        CHAP.  VI.

     SPHERICAL ABERRATION OF  LENSES AND MIRRORS.*

  (55.) In the preceding chapters we have supposed that the
rays refracted at spherical surfaces meet exactly in a focus;
but this is by no means strictly true: and if the reader has in
any one case projected the rays by the methods described, he

  * For a discussion of these  subjects, see (in the college edition) the. Ap-
pendix of Am. ed. chap. v. 
52             A  TREATISE  ON  OPTICS.         PART I.
must have seen that the rays nearest the axis of a spherical
surface, or cf a lens, are refracted to a  focus  more remote
from the lens than those which are incident at a distance from
the axis of the lens.  The rules which we have given for the
foci of lenses and surfaces are true for rays very near the axis.
  In order to understand the  cause of spherical aberration, let
L L be a plano-convex lens one of whose surfaces is spherical,
and let its plane surface LmLbe turned towards  parallel
rays R L, R L.   Let R' L', R' L' be rays very near the axis
A F of the lens, and let F be  their focus after refraction.  Let
R L, R L be parallel rays incident at the very margin of the
lens, and it will be found by the method of projection that the
                         Fig. 36.
corresponding refracted rays L/,  L/ will meet at a point/
much nearer the lens than F.  In like manner intermediate
rays between R L and R' L' will have their foci intermediate
between /and F.   Continue the rays L/, L/, till they meet
at G and H a plane passing through F, and perpendicular to FA.
The distance /F is called the longitudinal spherical aberra-
tion, and G H the lateral spherical aberration of the  lens. In
a plano-convex lens placed like that in the figure, the longitu-
dinal spherical aberration/F is no less than 4i times m n the
thickness of the lens.  It is obvious that such a lens cannot form
a distinct picture of any object in its focus F.  If it is exposed
to the sun, the central part of the lens L' m L' whose focus is
at F, will form a pretty bright image of the sun at F ;  but as
the rays of the sun which pass through the outer part L L of
the lens have their foci at points between/and F, the rays will,
after arriving at those points, pass on to the plane G H, and
              occupy a circle whose diameter is G H;  hence
              the  image of the sun  in the focus F will be a
              bright disc surrounded and rendered indistinct
              by a broad halo  of light growing  fainter and
              fainter from F to G and H.   In like  manner,
               every  object  seen through such a lens, and
               every  image  formed by it,  will be rendered
               confused and indistinct by spherical  aberration.
   These results may be illustrated experimentally by taking
 
CHAP. VI.      SPHERICAL  ABERRATION.
53
a ring of black paper, and  covering up the  outer parts of the
face L L of the lens.  This will diminish the halo G H, and
the indistinctness of the image, and  if we cover up all the
lens excepting a small part in the centre, the image will be-
come perfectly distinct, though less bright than before, and the
focus will be at F.   It; on  the contrary, we cover up all the
central part, and leave  only  a narrow ring at  the circumfe-
rence of the lens, we  shall have a very distinct image of the
sun formed about/
   (56.) If the reader will draw a very large diagram of a
plano-convex and of a double  convex lens,  and  determine the
refracted  rays at different distances from the axis where par-
allel rays fall on each  of the  surfaces of the  lens, he will  be
able to verify the following results for glass lenses.
   1.  In a plano-convex lens, with its plane side turned to par-
allel rays as in Jig. 36., that is, turned to distant objects if it is
to form an image behind it, or turned to the eye if it  is to  be
used  in magnifying a near object, the spherical aberration will
be 4^ times the thickness, or  4|  times m n.
   2.  In a plano-convex lens,  with its convex side turned  to-
wards parallel rays, the  aberration is only l^ths of its thick-
ness. In using a plano-convex lens, therefore, it should always
be so placed that parallel rays either enter the convex surface
or emerge from it.
   3.  In a double convex lens with equal convexities, the aber-
ration is lT^ths of its thickness.
   4.  In a double convex lens having its radii as 2 to 5, the
aberration will be the same as in a plano-convex  lens in Rule
1, if the side whose radius is 5 is turned  towards parallel rays;
and the same as the plano-convex  lens  in  Rule 2, if the side
whose radius is 2 is turned to parallel rays.
   5.  The lens which has the least spherical aberration is a
double convex one, whose radii are as 1 to 6.  When the face
whose radius is 1 is turned towards parallel  rays, the aberra-
tion is only l^ths of its thickness; but when the side with
the radius 6 is turned towards parallel rays, the aberration is
■3/irVths of its thickness.
   These  results are equally true of plano-concave and double
concave lenses.
   If we suppose the lens of least spherical aberration to have
its aberration equal to 1, the aberrations of the  other lenses
will be as follows :—
   Best form, as in Rule 5..........J"J0J
   Double convex or concave, with  equal curvatures   .    Iv>ol
   Plano-convex or concave in best position, as in Rule 2.  1-UJJ
   Plano-convex or concave in worst position, as in Rule 1.  4-~0l>
                        E2 
54              A  TREATISE  ON  OPTICS.         PART I.
  (57.) As the central parts of the  lens L L,fig. 36., refract
the rays too little, and the marginal parts too much, it is  evi-
dent that if we could increase the convexity at n and diminish
it gradually towards L, we should remove the spherical aber-
ration.  But the ellipse and the hyperbola are curves of this
kind, in which the curvature  diminishes  from n to L;  and
mathematicians have shown how spherical aberration may be
entirely removed, by lenses  whose sections are ellipses or hy-
perbolas.  This curious discovery we owe to Descartes.

                          Fig. 38.
   If A L D L, for example, is an ellipse whose greater axis
A D is to the distance between its foci F,/as the index of re-
fraction is to unity, then parallel rays R L, R L incident upon
the elliptical  surface L A L will be refracted  by the single
action of that surface into lines, which would meet exactly in
the focus F, if there were no second surface intervening be-
tween L A L and F.  But as every useful lens must have two
surfaces, we have only to describe a circle L a L round F as a
centre, for the second surface of the lens L L. As all the rays
refracted at the  surface L A L converge accurately to F, and
as the circular surface L a L is perpendicular to every one of
the refracted rays, all  these rays will  go on to F without suf-
fering any refraction at the circular surface.  Hence it follows
that a meniscus  whose convex surface is part of an ellipsoid,
and whose concave  surface is part of any spherical  surface
whose centre is in  the farther focus, will  have no spherical
aberration, and will refract parallel rays incident on its convex
surface to the farther focus.
  In like manner a concavo-convex lens, L L, whose concave
r-	Fig. 39.	
	.*Sl "^-^ V	
	^ft,--"""' /	
V	-""""""fiX._____^y	
 
CHAP. VI.   LENSES WITHOUT ABERRATION.
55
surface L A L is part of the  ellipsoid A L D L, and whose
convex surface lalis a,  circle described  round the farther
focus of the  ellipse, will  cause parallel rays R L, R L to di-
verge in directions L r, L r, which when continued backwards
will meet exactly in the focus F, which will be its virtual
fOCUS.                                          TAT     *■
   If a plano-convex lens has its convex  surface, L A L, part

                           Fig. 40.
of a hvperboloid formed by the revolution of a hyperbola whose
greater axis is to the distance between the  foci as unity is to
the index of refraction; then parallel rays, RL,RL, falling
perpendicularly on the plane surface will be refracted without
aberration to the farther focus of the hyperboloid.  The same
property belongs to a  plano-concave  lens,  having a similar
hyperbolic surface,  and receiving parallel  rays on  its plane

8U Am-eniscus with spherical surfaces has the property of re-
fracting all converging rays to its focus, if its first surface  is

                           Fig. 41.
>~E
convex, provided the distance of the point of convergence or
arXence from the centre of the first surface is to the radius
of tn? first surface as the index of refraction  is to unity. 
 56              A  TREATISE  ON  OPTICS.         PART I.

 Thus, if M L L N is a meniscus, and R L, R L rays  converg-
 ing to the point E, whose distance E C from the centre of the
 first surface L A L of the  meniscus is  to the radius C A or
 C Las the index of refraction is to unity, that is as 1-500 to 1,
 in glass; then if F is the focus of the first surface, describe
 with any radius less than F A a circle M a N for the second
 surface of the lens.  Now it will be found by projection that
 the rays R L, R L, whether near the axis A E or remote from
 it, will be refracted accurately to the focus F, and us all these
 rays fall perpendicularly on the second surface M, they will
 still pass on without refraction to the focus F.   In like manner
 it is obvious that rays F L,  F L diverging from F will be re-
 fracted into R L, R L, which diverge accurately from the vir-
 tual focus.
   When these properties of the ellipse and hyperbola, and of
 the solids generated by their revolution, were  first discovered,
 philosophers exerted all their ingenuity in grinding and polish-
 ing lenses with elliptical and hyperbolical surfaces, and various
 ingenious mechanical contrivances were proposed for this pur-
 pose.   These, however,  have not succeeded, and the  practical
 difficulties wThich yet require to  be overcome are so great, that
 lenses with spherical surfaces  are  the only ones  now in use
 for optical instruments.
   But though  we cannot remove  or diminish the spherical
 aberration of single lenses beyond lT(|lTths of their thickness,
 yet by combining two or more lenses, and making opposite
 aberrations correct each  other, we can remedy this defect to a
 very considerable extent in  some cases, and in other cases re-
 move it altogether.
   (58.) Mr. Herschel  has  shown,  that  if two plano-convex
 lenses A B, C D, whose focal lengths are as 2-3 to 1, are placed
 with their convex sides together, A B the least convex being
 next the eye when the combination is to  be used as a micro-
 scope,  the aberration will be only 0-248, or one fourth of that
                         of a  single lens in its best form.
                         When this lens is used to form an
                         image, A B must be turned to the
                         object.  If the focal  lengths of the
                         two lenses are equal, the spherical
                         aberration will  be 0-603, or a  little
                         more than one-half of a single lens
                         in its best form.
                           Mr.  Herschel has also shown that
                         the spherical  aberration  may  be
                         wholly removed  by  combining  a
meniscus C D with a double convex lens A B, as in figs. 43.
and 44., the lens A B being  turned to the  eye  when it is used
B 
CHAP. VI.   LENSES WITHOUT  ABERRATION.          57

for a microscope, and to the object when it is to  be used for
forming images, or as a burning-glass.
   The  following are the radii of curvature for these lenses,
as computed by Mr. Herschel; the first supposes, as a condi-
tion, that the focal length of the compound lens shall be as
near 10-000 as is consistent  with correcting the aberration;
and the second, that the  same focal length shall be the least
possible:—
                                        Fig. 43.     Fig. 44.
   Focal length of the double convex )  ,  nnnnn i  mnnn
    lens A B.......\ +  1UU0U + 10U00
   Radius of its first or outer surface   -f-   5-833 -f-  5-833
   Radius of its second surface  .   .   —  35-000 — 35-000
   Focal length of the meniscus C D  +  17-829 -f-  5-497
   Radius of its first surface  .  .  .  -f   3-688 -f  2-054
   Radius of its second surface  .  .  +   6-291 -f-  8-128
   Focal length of the compound lens -j-   6-407 +  3-474

             Spherical Aberration of Mirrors.

  (59.)  We have already stated, that when parallel rays, A M,
A N,  are incident  on a spherical mirror, M N, they are re-
fracted to the same focus, F, only when they are incident very
near the axis, AD.   If F is the focus of those  very near the
                          Fig. 45.
axis, such as Am, then the focus of those more remote, such
as A M, will be at / between F  and  D, and/F will be the 
58
A  TREATISE  ON  OPTICS.         PART I.
longitudinal spherical aberration, which will obviously increase
with the diameter of the mirror when its curvature remains
the same, and with  the curvature when its diameter is con-
stant.   The images, therefore, formed by mirrors will be in-
distinct, like those formed by spherical lenses, and  the  indis-
tinctness will arise from the same cause.
   It is manifest that if M N were a curve of such a nature
that a line, A M, parallel to its axis, A D, and  another line,
/M drawn from M to a fixed point, / should always  form
equal angles with a line,  C M,  perpendicular  to  the curve
M N, we should  in  this case have a surface which would re-
flect parallel rays exactly to a focus /, and form  perfectly dis-
tinct images of objects.   Such a curve is the parabola; and,
therefore, if we  could  construct mirrors of such a form that
their section M N is a parabola, they would have the invalua-
ble property of  reflecting  parallel  rays  to  a  single  focus.
When the curvature  of the mirror is very small, opticians
have  devised methods of  communicating to it a parabolic
figure ;  but when the  curvature is great, it has  not yet been
found practicable to  give them this figure.
   In the same manner it may be shown, that when diverging
rays fall upon a concave mirror of a spherical form, they will
be reflected to different points of the axis; and  that if a sur-
face could be formed so that the incident and reflected  rays
should form equal angles  with a line  perpendicular to the sur-
face at  the point of incidence,  the  reflected rays would all
meet in a single  point as their focus.  A  surface whose sec-
tion is an  ellipse has  this  property; and  it may be proved
that rays  diverging  from one focus  of  an  ellipse will be re-
flected  accurately to the other  focus.  Hence  in  reflecting
microscopes the  mirror should be a portion  of an  ellipsoid;
the axis of the mirror being the  axis of the ellipsoid, and the
object being placed  in the focus nearest the mirror.

 On Caustic Curves formed by Reflexion and Refraction.*
   (60.)  Caustics formed by reflexion.—As the rays incident
on different points of a reflecting surface at different distances
from its axis are reflected  to different  foci in that axis, it is
evident  that the rays thus reflected must cross one another at
particular points, and wherever the  rays cross they will illu-
minate  the white ground on which they are received with
twice as much light as falls on other parts  of the ground.
These luminous intersections form curve lines, called caustic
lines or caustic  curves;  and their nature  and form will, of
     * See (in the College edition) the Appendix of Am. ed. chap. v. 
CHAP. VI.          CAUSTIC CURVES.                  59

course, vary with the aperture of the mirror, and the distance
of the radiant point.
  In  order to explain their  mode of formation and general
properties, let M B N be a concave spherical  mirror, fig. 46.,
whose centre is C, and whose  focus for parallel and central

                          Fig. 46.
rays is F.  Let R M B be a diverging beam  of light falling
on the upper  part, M B, of the mirror at the points 1, 2, 3, 4,
&c.  If we draw lines perpendicular to all these points from
the centre C, and  make  the angles of reflexion equal to the
angles of incidence, we shall obtain the directions and foci of
all  the reflected rays.   The ray R 1, near the axis R B, will
have  its conjugate focus at / between F and the centre C.
The next ray, R 2, will cut the axis nearer F, and so on with
all the rest, the foci advancing from /to B.  By drawing all
the reflected rays to these foci, they will be found to intersect
one another as in the figure, and to form by their intersections
the caustic curve  M/  If the light had  also been incident
on the lower part  of the mirror, a similar caustic shown by a
dotted line would also have been formed between N and /
If we suppose, therefore, the point of incidence to move from
M to  B, the conjugate focus of  any two  contiguous rays, or
an infinitely slender  pencil diverging from R, will move along
the caustic from M to /
  Let us now suppose  the convex surface  M B N of the mir-
ror to be polished,  and  the radiant point R to be placed as far
to the right hand of B as it is now to the left, it will be found,
by drawing the incident and reflected rays, that they will di-
verge after reflexion; and that  when continued backwards
they will intersect unv another, and form an imaginary caustic 
60             A TREATISE  ON  OPTICS.         PART I

situated behind  the  convex surface, and  similar to the real
caustic.
   If we suppose the convex mirror M B N to be  completed
round  the  same centre, C, as at M A N, and the pencil  of
rays still  to radiate from R, they  will form the  imaginary
caustic M/N  smaller than M/N, and uniting  with it at
the points M, N.
   Let the radiant point It be  now supposed to  recede from
the mirror M B N, the line B/, which is called the tangent
of the real caustic M/N,  will obviously diminish, because
the conjugate focus / will approach to F; and, for the same
reason, the tangent A /' of the imaginary caustic will in-
crease.  When R becomes infinitely distant, and the incident
rays parallel, the points /,/', called the cusps of the caustic,
will both coincide with F and F', the  principal foci, and will
have the very same size and form.
   But if the radiant point R  approaches to the mirror, the
cusp / of the real caustic will  approach to the centre C, and
the tangent B / will increase,  the  cusp /'  of the  imaginary
caustic will approach A, and its tangent A/', will diminish;
and when the radiant point arrives at the circumference at A,
the cusp /' will also arrive at A, and the imaginary caustic
will disappear.  At the same  time, the  cusp / of the real
caustic will be a little to the right  of C, and its two opposite
summits will meet in the radiant point at A.
   If we suppose the radiant  point R now to enter  within the
circle A M B N, as shown in fig. 47., so that R C is less than
Fig. 47.
R A,  a remarkable  double caustic will be formed.  This
caustic will consist of two short ones of the common kind, 
CHAP. VI.          CAUSTIC  CURVES.                 61
ar, br, having their common cusp at r, and of two long
branches, af, bf, which meet in a focus at /  When R C is
greater than R A, the curved branches that meet at / behind
the mirror will  diverge, and have a virtual  focus within the
mirror.  When R coincides with F, a point half-way between
A and C, and the virtual principal focus of the convex mirror
MAN, these curved branches become parallel  lines;  and
when  R coincides with the centre C, the  caustics disappear,
and all the light is condensed into a single  mathematical point
at C, from which it  again diverges, and is again  reflected to
the same point.
   In virtue  of the principle on which  these phenomena de-
pend, a spherical mirror has, under certain circumstances, the
paradoxical  property of rendering rays diverging from a fixed
point either parallel, diverging, or converging; that is, if the
radiant point  is a little way within the principal focus of a
mirror, so that*rays very near the axis are  reflected into par-
allel lines, the  rays which are incident still nearer the axis
will be rendered  diverging, and those incident farther from
the axis will be  rendered  converging.   This property may be
distinctly exhibited by the projection of the reflected rays.
  Caustic curves  are frequently seen in a very distinct and
beautiful manner at the bottom of cylindrical vessels of china
or earthenware that  happen to be exposed  to the  light of the
sun or of a candle.  In these cases the rays generally fall too
obliquely on their cylindrical surface, owing to their depth;
but this depth may be removed, and the caustic curves beau-
tifully  displayed,  by inserting a  circular piece  of  card or
white paper about an inch or so beneath their upper edge, or
by filling them to that height with milk or any white and
opaque fluid.
                    The following method,  however,  of ex-
                  hibiting  caustic curves I have found ex-
                  ceedingly convenient and instructive. Take
                  a piece of steel spring highly polished, such
                  as a watch-spring, M N, fig. 48., and hav-
                  ing bent it into a concave form as  in the
                  figure, place it vertically  on its edge upon
                  a piece of  card or white paper A B.  Let
                  it  then be exposed either to  the rays  of
                  the sun,  or those of any other luminous
                  body,  taking care that the plane of the
                  card or the paper passes nearly through
                  the sun; and the two caustic curves shown
                  in the  figure will be finely displayed.  By
                  varying the size of the spring, and bending
                            F
 
62
A  TREATISE  ON  OPTICS.
 it into curves of different shapes, all the variety of caustics,
 with their cusps and points of contrary flexure, will  be finely
 exhibited.  The steel may be bent accurately into  different
 curves by applying a portion of its breadth to the  required
 curves drawn upon a piece of wood, and  either cut or burned
 sufficiently deep in the  wood to allow the edge of the thin
 strip of metal to be inserted in it.   Gold or silver foil answers
 very well; and when the light is strong,  a thin strip of mica
 will  also answer the purpose.   The best substance  of all,
 however, is a thin strip of polished silver.
   (61.)  Caustics formed by refraction.  If we expose  a
 globe of glass filled  with water, or a solid spherical lens, or
 even the belly of a round decanter, filled with water, to the
 rays  of the sun, or to the  light  of a lamp or candle, and re-
 ceive the refracted light on white paper  held almost parallel
 to the axis of the sphere,  or  so that its  plane passes  nearly
 through the luminous body, we  shall perceive on the paper a
 luminous figure bounded by two bright caustics, like af and
 b  f, Jig. 47, but placed behind the sphere, and forming a
 sharp  cusp or angle at the point /, which is the  focus of re-
 fracted rays.  The production of these curves depends upon
 the intersection of rays, which, being incident on the sphere
 at different distances from the axis, are refracted to foci at dif-
 ferent points of the axis, and  therefore cross one  another.
 This  result is so easily understood, and may be exhibited so
 clearly, by projecting the  refracted rays, that it  is unneces-
 sary to say any more on the subject.
   Some of the phenomena of caustics produced by refraction
 may be illustrated experimentally in the following manner:—
 Take a shallow  cylindrical vessel  of lead, M N, two or three
 inches in diameter, and cut its upper margin, as shown in the
 figure, leaving two opposite projections, a c, b d, forming each
 about 10° or 15° of the whole circumference.   Complete the
 circumference by cementing on  the vessel two strips  of mica,
        F;> 49         so as to substitute for the lead that has
                       been removed two transparent cylin-
                       drical surfaces.  If this vessel is filled
                       with water, or any other  transparent
                       fluid, and  a  piece of card or white
                       paper, A B C D,  is held almost paral-
                       lel to the  surface of the water, and
                       having  its  plane  nearly  passing
 through the sun or the candle, the caustics A F, 1) F will be
 finely  displayed.  By altering  the curvature  of  the vessel,
and that of the strips of mica, many interesting variations of
 the experiment may be made.
 
CHAP. VII. ANALYSIS OF LIGHT BY THE PRISM.        63
PART II.
                 PHYSICAL OPTICS.

  (62.)  Physical Optics is that branch of the science which
treats of the physical properties of light.  These properties
are exhibited in the decomposition and recomposition of white
light; in its decomposition  by absorption; in the inflexion or
diffraction of light; in the colors of thick  and  thin plates; and
in the double refraction and polarization of light.
                       CHAP. VII.

    ON THE COLORS OF LIGHT, AND ITS DECOMPOSITION.
  (63.) In the preceding chapters we have regarded  light as
a simple substance, all the parts of which had the same index
of refraction, and therefore suffered the same  changes when
acted  upon by transparent media.  This, however, is not its
constitution.  White light, as emitted from the sun or from
any luminous body, is  composed of seven different kinds of
light, viz., red, orange, yellow, green, blue, indigo, and violet;
and this compound substance may be decomposed, or analyzed,
or separated  into its elementary parts, by two different  pro-
cesses, viz., by refraction and absorption.
  The first of these processes was that which was employed
by Sir Isaac Newton, who discovered the composition of white
light.    Having  admitted  a beam of  the sun's  light, S II,
through a small hole,  H, in the window-shutter, E F, of a
darkened room,  it will go on in a straight  line  and form a
round white spot at P.  If we now interpose a prism, B A C,
whose refracting angle is B A C, so that  this beam of light
may fall on its  first surface C A, and emerge  at the same
angle from its second surface B A in the direction g G, and if
we receive the refracted beam on the opposite wall, or rather
on a white screen, M N, we should expect, from the principles
already laid down, that the white beam which previously fell
upon P would suffer only a change in  its direction,  and fall
somewhere upon M N, forming there a round  white  spot ex-
actly similar to that at P.   But this is not the case.   Instead
of a white spot,  there will be formed upon the  screen M N an 
64
                A TREATISE ON  OPTICS.        PART II.

oblong image K L of the sun, containing seven colors, viz.
red, orange, yellow, green, blue, indigo, and violet, the whole
Fig. 50.
beam of light diverging from its emergence out of the prism
at g, and being bounded by the lines g K, g L.  This length-
ened image of the sun is called the solar spectrum, or the
prismatic spectrum.  If the aperture H is small, and the dis'
tance g G considerable, the colors of the spectrum will be very
bright.  The lowest portion of  it at L is a brilliant red.  This
red shades off by imperceptible gradations into orange, the
orange into yellow, the yellow into green, the green into  blue,
the blue into a pure indigo, and the indigo into a violet.   No
lines are seen  across the spectrum thus produced;  and it ia
extremely difficult for the sharpest eye to point out the bound-
ary of the  different colors.  Sir Isaac Newton,  however, by
many trials, found the lengths of the  colors to be as follows,
in the  kind of glass of which his prism was made.  We  have
added  the results obtained by Fraunhofer with flint glass.
                           CTC'*^    Newton.  Frounhofor.
              Red......45     56
              Orango.....27     27
              Yellow.....40     27
              Green.....60     46
               Blue......60     48
               Indigo.....48     47
               Violet.....80     109
Total length   360    360 
CHAP. VII.      RECOMPOSITION  OF LIGHT.            65

These colors are not equally brilliant.   At the lower end, L,
of the spectrum the red is comparatively faint, but grows
brighter  as  it approaches  the  orange.   The light increases
gradually to the middle of the yellow space,  where  it is
brightest; and from this it gradually declines to the upper or
violet end, K, of the spectrum, where it is extremely faint.
  (64.) From the phenomena which we have now described,
Sir Isaac Newton concluded that the beam of white light, S,
is compounded of light of seven different colors,  and that for
each of these different  kinds of light, the glass,  of which his
prism was made, had different indices of refraction ; the index
of refraction for the red light being the least, and that of the
violet the greatest.
  If the prism is made of crown glass, for  example, the  in-
dices of refraction for the different colored rays will be as fol-
lows :—
                       Index of                         Index of
                       Refraction.                        Infraction.
Red   ......1-5258 I Blue.......1-5360
Orange.....1-5268 | Indigo......1.5417
Yellow.....1-5296 I Violet......1-5466
Green......1-5330 |

If we now draw the prism, B A C, on a great scale, and de-
termine the progress of the refracted rays,  supposed to be in-
cident upon the same point of the first  surface C A, by using
for each ray the index of refraction in the preceding table, we
shall find them to diverge as in the preceding figure, and to
form the different colors in the order of those in the spectrum.
  In order to examine  each color separately, Sir Isaac made
a hole in the screen M N, opposite the  centre of each colored
space; and he allowed that particular  color to fall upon a
second prism, placed behind the hole.   This light, when  re-
fracted by the second prism, was not drawn out into an oblong
image as before, and was not refracted  into any other colors.
Hence he concluded that the light of each  different color had
the  same index  of refraction; and he called such light homo-
geneous, or simple, white  light being regarded  as heteroge-
neous or compound.   This  important doctrine is  called the
different refrangibility of the rays of light.  The  different
colors as existing in the spectrum are called primary colors;
and any mixtures or combinations of any of them  are called
secondary colors, because we can easily separate  them into
their primary coloYs by refraction through the prism.
  (65.) Having thu3  clearly established the composition of
white light, Sir Isaac also proved, experimentally, that all the
seven colors, when again combined and made to fall  upon the
                         F2 
 66              A TREATISE ON OPTICS.         PART II.

 same spot, formed or recomposed white light.  This important
 truth he established by various experiments; but the following
 method of proving it is so satisfactory, that no farther evidence
 seems to be wanted.   Let the screen M N,fig- 50., which re-
 ceives the spectrum, be  gradually brought nearer the prism
 B A C, the spectrum K L will gradually diminish; but though
 the colors begin to mix, and  encroach upon one another, yet,
 even when it is brought close to the face B A of the prism, we
 shall recognize the separation of the light  into its  component
 colors.  If we now take a prism, B a A, shown by dotted lines,
 made of the same  kind of glass  as B A C, and having its re-
 fracting angle A B a  exactly equal to  the refracting angle
 B A C of the other prism; and if we place it in  the opposite
 direction, we shall find that  all  the seven differently colored
 rays which fall upon the second prism, A B a, are again com-
 bined into a single beam of white light g P, forming a white
 circular spot at P, as if neither of the prisms had  been inter-
 posed.   The very same  effect will be produced, even if the
 surfaces, A B, of the two prisms are joined by a transparent
 cement of the same refractive power as the glass, so as to re-
 move entirely the refractions at the common surface A B. In
 this state the two prisms combined are nothing more than a
 thick piece of glass, B C A a, whose two sides, A C, a B, are
 exactly parallel; and the decomposition of the light by there-
 fraction of the first surface, A C, is counteracted by the oppo-
 site and equal refraction  of the second surface, a B; that is,
 the light decomposed by the first surface is recomposed by the
 second surface.   The  refraction and  re-union  of the rays in
 this experiment may be well exhibited by placing a thick plate
 of oil of cassia  between  two parallel  plates of  glass,  and
 making a narrow beam of the sun's light fall upon  it very ob-
 liquely.  The spectrum formed by the action of the first sur-
 face will be distinctly  visible, and the re-union of the  colors
 by the second will be equally distinct.  We  may, therefore,
 consider the  action of a plate of parallel  glass on  the sun's
 rays,.that is, its property of transmitting  them  colorless, as a
 sufficient proof of the recomposition of light.
  The same doctrine  may be illustrated  experimentally by
 mixing  together seven different powders having  the  same
 colors as those of the  spectrum, taking as much of each as
seems to be proportional to the  rays in  each colored space.
The union of these colors will be a sort of grayish-white, be-
cause it  is impossible to obtain powders of the  proper  colors.
The same result will be obtained, if we take a circle of paper
and divide it  into sectors of the  same  size as the colored 
CHAP. VII.     NEW  ANALYSIS  OF  LIGHT.
67
 spaces;  and when this circle is made to revolve rapidly, the
 effect of all the colors when combined will be a grayish-white.


           Decomposition of Light by Absorption.

   (66.)  If we measure  the quantity of light which is reflected
 from the surfaces and transmitted through the substance of
 transparent bodies, we  shall  find that the sum of  these quan-
 tities is always  less than the  quantity of light  which falls
 upon the body.  Hence we may conclude that a certain por-
 tion  of light is lost in  passing through  the most  transparent
 bodies.   This loss arises from two causes. A part of the light
 is scattered  in all directions by irregular reflexion from the
 imperfectly polished  surface of particular media, or from the
 imperfect union of its parts; while another,  and generally a
 greater portion, is absorbed, or stopped by the particles of the
 body.  Colored  fluids, such as black and  red ink, though
 equally homogeneous, stop or absorb different kinds of rays,
 and when exposed to the sun they become heated  in different
 degrees; while  pure water seems  to transmit all the rays
 equally,* and scarcely receives any heat from the passing light
 of the sun.
   When we examine  more minutely the action  of colored
 glasses and colored fluids in absorbing light, many remarkable
 phenomena present themselves, which throw much light upon
 this curious subject.
   If we take a piece of blue glass, like that generally used
 for finger  glasses, and transmit through it a beam of  white
 light, the light will be a fine deep blue.  This blue is not a
 simple homogeneous color, like the blue Or indigo of the spec-
 trum, but is a mixture of all the colors of white light which
 the  glass has  not  absorbed; and the colors which the glass
 has  absorbed are  those which the blue wants of white light,
 or which, when mixed with this blue, would form white  light.
 In order to determine  what  these colors are, let us transmit
 through  the blue glass  the prismatic  spectrum K L, fig. 50.;
 or, what is the  same thing, let the observer place his eye be-
 hind  the prism B A C, and  look  through it  at the  sun, or
 rather at a circular aperture  made in the window-shutter of
 a dark room.  He will  then  see through the prism the  spec-
 trum K L as far below the aperture as it was above the spot P
 when shown in the screen.  Let the blue glass be  now inter-
posed between the eye and the prism, and a remarkable  spec-
trum will be seen, deficient in a certain number of its differ-
* See Note II., of Am. ed., which follows the author's Appendix. 
68
A  TREATISE ON OPTICS.        PART II.
 ently colored rays.  A particular thickness absorbs the middle
 of the red space, the whole of the orange, a great part of the
 green, a considerable  part of the blue, a little of the indigo,
 and  very little of the violet.  The yellow  space, which has
 not been much absorbed, has increased in breadth.   It occu-
 pies  part of the space formerly covered by the orange on one
 side, and part of the space formerly covered by the green on
 the other.  Hence it follows, that the blue glass has absorbed
 the red  light, which, when mixed with the yellow light, con-
 stituted orange, and has absorbed also the blue light, which,
 when mixed  with  the yellow,  constituted the  part of the
 green space next to the yellow.  We have therefore, by ab-
 sorption, decomposed  green light into yellow and blue, and
 orange  light into yellow and red; and it consequently follows,
 that  the orange and green rays of the spectrum, though they
 cannot be decomposed by prismatic  refraction, can be decom-
 posed by absorption,  and actually  consist  of  two  different
 colors possessing the same  degree of refrangibility.   Differ-
 ence of color is therefore not a test of difference of refrangi-
 bility, and  the conclusion deduced  by Newton is no longer
t admissible as a general truth: " That to the same degree of
 refrangibility ever  belongs  the same color, and to the same
 color ever belongs the same degree of refrangibility."
   With the view of obtaining a complete analysis of the spec-
 trum, I have examined the spectra produced by various bodies,
 and  the changes which they  undergo by  absorption when
 viewed  through various  colored  media, and I find  that the
 color of every part of the. spectrum may be changed  not only
 in intensity, but in  color, by the  action of particular media;
 and from these observations, which it would be out of place
 here to detail,  I conclude that the solar spectrum consists of
 three spectra  of equal lengths, viz. a red spectrum, a yellow
 spectrum, and  a blue spectrum.   The primary red spectrum
 has its  maximum  of intensity about the  middle of the red
 space in the solar spectrum, the primary yellow spectrum has
 its maximum  in the  middle of  the  yellow space,  and the
 primary blue  spectrum has its maximum between the blue
 and the  indigo space.   The two minima of each of the three
 primary spectra coincide at the two  extremities of the solar
 spectrum.
   From this view  of  the constitution of the solar spectrum
 we may draw the following conclusions:—
   1.  Red, yellow, and blue light exist at every point of the
 solar spectrum.
   2.  As a certain portion of red, yellow, and blue constitute
 white light, the color of every point  of the spectrum  may be 
CHAP. VII.      NEW ANALYSIS OF  LIGHT.            69

considered as consisting of the predominating color at any
point mixed with white  light.  In the  red space there is more
red than is necessary to make white light with the small por-
tions of yellow  and blue which exist there;  in  the yellow
space tho§e is more yellow than is necessary to make white
light with the red and blue ; and in the part of the blue space
which appears  violet there is more  red  than yellow,  and
hence the excess of red forms a violet with the blue.   ->
   3. By absorbing the excess of any color at any point of the
spectrum above what is necessary to form white  light, we
may actually cause white light to appear at that point, and
this white light will possess the remarkable property of re-
maining white after any number of refractions, and of being
decomposable only by absorption.  Such a white light I have
succeeded in developing  in different parts of the  spectrum.
These views harmonize in a remarkable manner with the
hypothesis of three colors, which has been  adopted by many
philosophers, and which others had rejected from  its incom-
patibility with the phenomena of the spectrum.
   The existence of three primary colors in the spectrum, and
the mode in  which they produce by their combination the
seven secondary or compound  colors which  are developed by
the prism, will be understood from fig. 51. where M N is the
prismatic spectrum, consisting of three primary spectra of the
same lengths, M N, viz. a red, a yellow, and a blue spectrum.
The red  spectrum has its maximum  intensity at R; and this
intensity may be represented  by the  distance of the point R
from M N.   The intensity declines rapidly to M and slowly
to N, at both of which  points it vanishes.   The yellow  spec-
                          Fig-. 51.
trum has its maximum intensity at Y, the intensity declining to
zero at M and N;  and the blue has its maximum intensity at
B,  declining to nothing at M and N.   The general curve
which represents the total illumination at any point will be
outside of these three curves, and its ordinate  at any point
will be equal to the sum  of the three ordinates  at the same
point.  Thus the ordinate of the general curve at the point Y
will be equal to the  ordinate of the yellow curve, which we 
70             A TREATISE  ON  OPTICS.        PART II.

may suppose to be 10, added to that of the red curve, which
may be 2, and that of the blue, which may be 1.   Hence the
general ordinate will be 13.   Now,  if we suppose that 3 parts
of yellow, 2 of red, and 1 of blue make white, we shall have
the color  at Y equal to 3 -f- 2 -f 1,  equal to 6 parts of white
mixed with 7  parts of yellow;  that is, the  compound tint at
Y will be a bright yellow  without  any trace of red or blue.
As these colors all occupy the same place in the spectrum,
they cannot be separated by the prism; and if we could find
a colored glass which would absorb  7 parts of the yellow, we
should obtain  at the point Y a white light which the prism
could not decompose.*
                      CHAP. VIII.

              ON  THE DISPERSION  OF LIGHT.

  In the preceding observations, we have considered the pris-
matic spectrum, K L, fig. 50., as produced by a prism of glass
having a given refracting angle, B A C.   The green ray, or
g G, which, being midway between g K  and g L, is  called
the mean ray of the spectrum,  has been refracted  from P to
G, or through an angle of deviation,  P g G, which is  called
the mean refraction or deviation, produced by the prism.   If
we now increase the angle B A C of the  prism, we shall in-
crease the  refraction.  The mean  ray g G will be refracted
to a greater distance from P, and the  extreme rays g L, g K,
to a greater distance in the same proportion;  that is,  if gG
is refracted twice as much, g L  and g K will also  be refracted
twice as much, and  consequently the length of the spectrum
K L will be twice as great.  For the same  reason,  if we
diminish the angle B A C of the prism, the mean  refraction
and the  spectrum will diminish in the same proportion; but,
whatever be the angle of the prism, the length K L will  al-
ways bear the same proportion to G P, the mean refraction.
  Sir Isaac Newton supposed  that prisms made of all sub-
stances whatever produced spectra bearing the same propor-
tion to the mean refraction as prisms of glass; and it is a re-
markable circumstance, that  a  philosopher of  such sagacity
should have overlooked a fact  so palpable, as that different
bodies produced spectra whose  lengths were different, when
the mean refraction was the same.
  The prism  B A C being  supposed  to be  made of crown

      * See Note III., by Am. ed., following the author's Appendix. 
CHAP. VIII.      DISPERSION OF  LIGHT.
71
glass, let us take another of flint glass or white crystal, with
such a  refracting angle that,  when placed in  the position
B A C, the light enters  and  quits it at equal angles, and re-
fracts the mean ray to the same point G.  The  two prisms
ought, therefore, to have  the same mean refraction. But when
we examine the spectrum produced by the flint glass prism,
we shall find that it extends beyond K and L, and  is evidently
longer than the  spectrum produced  by the crown  glass prism.
Hence flint glass is said to have a greater dispersive power
than crown glass, because at  the same  angle of mean refrac-
tion it separates the extreme  rays of the spectrum, g L, g K,
farther from the mean ray g G.
   In order to explain more clearly what is the real measure
of the dispersive power of a body, let us  suppose  that in the
crown glass prism, B A C, the index of refraction for the ex-
treme violet ray, gK, is  1-5466, and that  for the extreme red
ray, g L, 1-5258;  then the difference of  these  indices, or
•0208, would be a measure of the  dispersive power of crown
glass, if it and all other bodies had the same mean refraction:
but as this is far from being the  case, the  dispersive power
must  be measured by the relation between -0208  and the
mean refraction, or 1-5330,  or to the excess of this above
unity, viz., -5330, to which the mean refraction is always pro-
portional.  For  the purpose  of making this  clearer, let it be
required to  compare  the dispersive powers of diamond and
crown glass.  The index of  refraction of  diamond for the ex-
treme violet ray is 2-467, and for the extreme red, 2-411, and
the difference of these is -0560, nearly three times as great as
•0208, the same difference for crown glass;  but then the dif-
ference between the sines of incidence and refracl^on, or the
excess of the index of refraction above unity, or 1-439, is also
about three times as  great  as the same  difference in crown
glass, viz., -5330; and, consequently, the  dispersive power of
diamond is very little greater than that of crown  glass.  The
two dispersive powers are as  follows:—
                                        Dispersive Powers.
         Crown Glass   .  .   .   :§f°| = 0-0386
         Diamond   ....   T°|f-° = 0-0388
   This similarity of dispersive power might be proved experi-
mentally, by taking a prism  of diamond, which, when placed
at BA C in Jig. 50., produced the same mean refraction as the
green ray g G.  It would then be seen that the spectrum
which it produced was  of the  same length as that produced
by the  prism  of crown glass.  Hence  the  splendid colors
which distinguish diamond from every other precious  stone 
 72              A TREATISE ON OPTICS.        PART II.

 are not owing  to its  high dispersive power,  but to its great
 mean refraction.
   As the indices of refraction given in our table of refractive
 powers are nearly suited to the mean ray of the spectrum,
 we may, by the second column of the Table of the Dispersive
 Powers of Bodies, given  in the Appendix, No.  I., obtain the
 approximate indices of refraction for the extreme red and the
 extreme violet rays, by adding half of  the number in the col-
 umn to the mean  index of refraction for the index of refrac-
 tion of the violet, and subtracting half of the same number
 for the index of the red ray.   The measures in the table are
 given for the ordinary light of day.  When the  sun's light is
 used, and when the eye is screened from the middle  rays of
 the spectrum,  the red and  violet may be traced to  a much
 greater distance from the mean ray of  the spectrum.
   When the  index of refraction for the  extreme ray is thus
 known, we may  determine  the  position and length of the
 spectra produced by prisms of different substances, whatever
 be their refracting angle, whatever be  the positions of the
 prism,  and whatever  be the distance of the screen on which
 the spectrum is received.
   If we take a prism of crown glass, and  another of flint
 glass, with such refracting angles that they produce a spec-
 trum of precisely the same length, it will be found, that when
 the two prisms are placed together with their refracting angles
 in opposite directions, they will not restore the refracted pencil
 to the state of  white light, as happens in the combination of
 two equal prisms of crown or two  equal prisms of flint  glass.
 The white light P, fig. 50., will be tinged on one  side  with
purple, and on  the other with green light.  This is called the
 secondary spectrum, and the colors secondary colors; and it
 is manifest that they must arise from the colored spaces in the
 spectrum of crown glass not being equal to those in the spec-
 trum of flint glass.
   In order  to  render this curious property of the spectrum
 very obvious to the  eye, let two spectra of equal  length be
 formed by  two hollow prisms, one containing oil of cassia,
 and the other sulphuric acid.  The oil  of cassia spectrum will
 resemble A B,fig. 52., and the sulphuric  acid spectrum C D.
 In the former, the red, orange, and yellow spaces are less than
 in the latter, while  the  blue, indigo, and violet spaces are
 greater; the least refrangible rays being, as it were, contract-
 ed in the former and expanded in  the  latter, while the  most
 refrangible rays are expanded in the one and contracted in the
 other.  In consequence of this difference in the colored spaces,
 the middle or mean ray m n does not pass through the same 
CHAP. VIII.       DISPERSION  OF LIGHT.
73
color in both spectra.   In the oil of cassia spectrum it is in
the blue space, and in the sulphuric acid spectrum it is in the

                          Fig. 52.
                      A         C
                       B          D
green space.   As the colored spaces have not the same ratio
to one another as the lengths of the spectra which they  com-
pose, this property has been called the irrationality of disper-
sion, or of the colored spaces in the spectrum.
   In order  to  ascertain whether any prism contracts or ex-
pands the least refrangible rays more  than  another, or which
of them acts most on green  light,  take a prism of each with
such angles that  they correct each other's dispersion as much
as possible,  or that they produce  spectra of the same  length.
If, through the prisms placed with their refracting angles in
opposite directions, we look at the bar of the window parallel
to the base of the prism, we  shall  see  its edges  perfectly free
from color, provided the two prisms act equally upon green light.
But if they act differently  on green light, the bar will  have a
fringe of purple  on one side, and a fringe of green  on the
other; and the green fringe will always be on the same side
of the bar as the  vertex of the prism which contracts the yel-
low space and expands  the blue and violet ones.  That is, if
the prisms are flint and crown glass,  the uncorrected green
fringe will be on  the lower side of the bar when the vertex of
the flint glass prism points  downwards. Flint glass, therefore,
has a less action upon green  light  than crown glass, and con-
tracts in  a greater degree the red  and  yellow spaces.  See
Appendix, No. II.
                            G 
74
A  TREATISE ON  OPTICS.        PART II.
                       CHAP. IX.
      ON THE PRINCIPLE OF  ACHROMATIC TELESCOPES.

  In treating of the progress of rays through lenses, it was
taken for granted that the light was homogeneous, and that
every ray that had the same  angle of incidence had  also the
same angle of refraction;  or, what is  the same thing, that
every ray which fell upon the lens had the same index of re-
fraction.  The  observations  in  the  two  preceding  chapters
have, however, proved that this is not true, and that, in the
case of light falling upon  crown  glass, there are rays with
every possible index of refraction from 1-5258, the index  of
refraction for the red, to 1-5466, the index of refraction for the
violet rays.  As the light of  the sun, by which all the bodies
of nature are rendered visible, is white, this property of light,
viz. the different refrangibility of its parts, affects greatly the
formation of images by lenses of all kinds.
  In order  to explain this, let L L be a convex lens of crown
glass, and R L, R L rays of white light incident upon it par-

                          Fig. 53.

         B-----^tfL^^---1        -^

         R---eM---J^^^^^r*^
allel to its axis R r.   As each ray R L of white light consists
of seven differently colored rays having different degrees of
refrangibility or different indices of refraction, it is evident
that all  the rays which compose R L cannot possibly be re-
fracted in the same direction, so as to fall upon one point. The
extreme  red rays, for example, in R L, R L, whose index of
refraction is 1-5258, if traced  through the lens by the method
formerly given, will be found to have their focus in r, and C r
will be  the focal  length of the lens  for red  rays.  In like
manner the extreme  violet rays, which have a greater index
of refraction, or 1-5466, will be  refracted  to a focus v much
nearer the lens,  and C v will  be the focal length of the lens
for violet rays.   The distance v r is called the chromatic aber-
ration, and the circle whose diameter is a 6 passing through
the focus of the mean refrangible rays  at o, is called the circle
of least aberration.
   These effects may be shown experimentally by exposing the 
CHAP. IX.      CHROMATIC  ABERRATION.             75

lens L L to the parallel rays of the  sun.  If we receive the
image of the sun on a piece of paper placed between o and C,
the  luminous circle on the  paper will have a red border, be-
cause it is a section of the cone L a b L, the exterior rays of
which La, Lb are red; but  if the paper is placed at any
greater distance than o, the luminous circle on the paper will
have a violet border, because it is a section of the cone I  a bl',
the  exterior rays of which a I, b V are violet, being a contin-
uation of the violet rays L v, L v.  As the spherical aberration
of the lens is here  combined  with  its chromatic aberration,
the  undisguised  effect of the latter will be better seen by
taking a large convex lens L L, and covering up all the cen-
tral part, leaving only a  small rim round its circumference at
L L, through which the  rays of light may pass.   The refrac-
tion of the differently colored rays will be  then finely dis-
played by viewing the image of the sun on the different sides
of a b.
   It is clear from these observations that the lens will form a
violet image of the sun at v, a red image at r, and images of
the other colors in the  spectrum at intermediate points be-
 tween r and v; so  that if we place  the  eye behind  these
 images, we shall see a  confused image,  possessing none of
 that sharpness and distinctness which it would  have had if
 formed only by one kind of rays.
   The same observations are true of the  refraction of white
 lio-ht  by a concave lens; only in this  case the parallel rays
 which such a lens refracts diverge, as if they proceeded from
 separate foci, v and r, in front of the lens.
   If we now place behind L L a concave  lens G G  of the
 same glass, and  having its  surfaces ground to the same cur-
 vature,  it is obvious that since v is its virtual focus for  violet,
 and r its virtual focus for red rays, if the paper is held  at a b,
 the focus of the mean refrangible rays, where  the violet and
 red rays cross at a and b, the image will be more distinct than
 in any other position; and when rays converge to the focus of
 any concave lens,  they will be refracted  into parallel direc-
 tions ; that is, the concave lens will refract these converging
 rays into the parallel lines Gl, G I, and they will again form
 white light.  That the  red and  violet  rays  will be  thus re-
 united in one, viz. G I, may be proved by projecting them; but
 it is obvious also from the consideration that the two  lenses
 L L, G G actually form a piece of parallel glass, the outer
 concave surface of G G being parallel to the outer convex sur-
 face of L L.
   (67.) But though we  have thus corrected the color produced
 by  L L, by means of the  lens G G, we have done  this by a 
76              A  TREATISE ON OPTICS.        PART II.

useless combination; since the two together act only like  a
piece of plane glass, and are incapable of forming an image.
If we make the concave lens G G, however, of a longer focus
than L L, the two together will act as a convex lens,  and will
form images behind it, as the rays G I, G I will now converge
to a focus behind L L.  But as the chromatic aberration of the
lens  G G will now be less  than that of L L,  the  one  will not
correct or compensate  the other; so  that  the difference be-
tween the two aberrations  will still remain.  Hence  it is im-
possible,  by means of two  lenses of the same glass,  to  form
an image which shall be free from color.
  As Sir Isaac Newton believed that all substances whatever
produced the same  quantity of color, or had the same chro-
matic aberration when formed  into lenses, he concluded that
it was impossible, by the combination of a concave with a con-
vex glass, to produce refraction without color.   But we  have
already seen that the premises from which this conclusion was
drawn are incorrect, and that bodies  have different dispersive
powers, or produce different degrees of color at the same mean
refraction.  Hence it follows that different lenses  may produce
the  same degree  of  color when they  have  different  focal
lengths;  so that if the lens L L is made of crown  glass, whose
index of refraction is 1-519, and dispersive  power 0-036, and
the lens G G of flint glass, whose index of refraction is 1-589,
and  dispersive power 0-0393, and if the focal length of the
convex crown-glass lens is made 4J inches, and that of the
concave flint-glass lens 7f  inches, they will form  a lens with a
focal length of 10 inches, and will  refract white light to a
single  focus free of color.  Such a  lens is  called an achro-
matic lens; and when used as a telescope, with  another glass
to magnify the colorless image which it forms of distant ob-
jects,  it constitutes the  achromatic telescope, one of the
greatest  inventions of the last century.   Although  Newton,
reasoning from  his imperfect knowledge of the dispersive
power of bodies,  pronounced such an invention to be hopeless;
yet,  in a  short time  after the death of that great philosopher,
it was accomplished by a Mr. Hall, and afterwards by Mr.
Dollond,  who brought it to a high degree of perfection.
  The image formed by an achromatic lens  thus constructed
would have been perfect if the equal spectra formed by the
crown and flint glass  were in every respect  similar: but  as
we have seen  that the colored spaces in the one  are not equal
to the colored spaces  in  the other,  a secondary spectrum  is
left ; and therefore the images of all  luminous objects, when
seen through such a lens,  will be bordered on  one side with a
purple fringe, and  on the other with a green fringe.  If two 
CHAP. IX.      ACHROMATIC TELESCOPE.            "il

substances could be found of different refractive and dispersive
powers, and capable of producing equal spectra, in which the
colored spaces were equal, a perfect achromatic lens would be
produced: but, as  no such substances have yet been  found,
philosophers  have endeavored to remove the imperfection by
other means; and Doctor Blair had the merit of surmounting
the difficulty.  He  found that muriatic acid had the property
of producing a primary spectrum,  in which the green rays
were among the most refrangible, something like C B,fig. 52.,
as in crown glass.  But as muriatic acid has too low a  refrac-
tive and dispersive  power to fit it for being used as  a concave
lens  along with  a  convex one of crown glass, he  therefore
conceived the idea  of increasing the refractive and  dispersive
powers of the muriatic  acid,  by mixing it  with metallic solu-
tions, such as muriate of antimony ; and he found he could do
this to the requisite extent without altering its law of  disper-
sion, or the proportion  of the colored spaces in its  spectrum.
By inclosing, therefore, muriate  of antimony, L L, between
two convex  lenses of crown glass, as A B, C D in fig. 54,
Doctor Blair succeeded in refracting parallel  rays RA, RB
                     BLD
to a single focus F, without the least trace  of secondary
color.  Before  he discovered this property of the muriatic
acid, he  had contrived another, though a  more complicated
combination, for producing the same effect; but as he prefer-
red the combination which we have described, and employed
it in the  best aplanatic object-glasses which he constructed, it
is unnecessary to dwell any longer upon the subject.
  In these  observations, we  have supposed that the  lenses
which  are combined have no spherical aberration; but though
this is  not the case, the combination  of concave with  convex
surfaces, when  properly adjusted, enables us completely to
correct the  spherical along with the chromatic aberration of
IpfiSGS
  In the course of an  examination of the secondary spectra
produced by different  combinations, I was  led to  the  conclu-
sion that there  may be  refraction without color, by means, at
two prisms,  and that two lenses may converge  white  light to
                           G 2 
78             A TREATISE ON OPTICS.          PART II.

one focus, even though the prisms and the lenses are made of
the same kind of glass.  When one prism of a different angle
is thus made to correct the dispersion of another prism, a ter-
tiary  spectrum  is produced, which depends  wholly on the
angles at which the light  is refracted at the two surfaces of
the prisms.  See  Treatise on  New Philosophical  Instru-
ments, p. 400.
                        CHAP.  X.

      ON THE PHYSICAL PROPERTIES OF THE SPECTRUM.

   (68.)  In the preceding chapter we have considered  only
those general properties of the solar spectrum on which the
construction of achromatic  lenses depends.  We shall  now
proceed to take a general view of all its physical properties.

     On the Existence of Fixed Lines in the Spectrum.

   In the year 1802, Dr. Wollaston  announced that  in the
spectrum formed by a fine prism of flint glass, free from veins,
when the luminous object was a slit, the twentieth of an inch
wide, and viewed at the distance  of 10 or 12 feet, there were
two fixed dark lines, one in  the green and the other in the
blue space.   This discovery did not excite any attention, and
was not followed out by its ingenious author.
   Without a  knowledge of Dr. Wollaston's observation, the
late celebrated M. Fraunhofer, of Munich,  by viewing through
a telescope the spectrum formed from a narrow line of solar
light by the finest  prisms of flint glass,  discovered  that the
surface  of  the spectrum was  crossed throughout its whole
length by dark lines of different breadths.  None of these lines
coincide with  the boundaries of the colored spaces.  They are
nearly 600 in number: the largest of them subtends an angle
of from 5" to 10".  From their distinctness, and the facility
with which they may be found, seven of these lines, viz.
B, C, D, E, F, G, H, have been particularly distinguished by
M. Fraunhofer.  Of these B lies in the  red space, near its
outer end;  C, which is broad and black, is beyond the middle
of the red; D is in the orange, and is a  strong  double  line,
easily seen, the two lines being nearly of the same size, and
separated by a bright one; E is in the  green, and consists of
several, the middle one being the strongest; F is in the blue,
and is a very strong line; G is in  the  indigo, and H  in  the 
CHAP. X.     ON LINES IN  THE  SPECTRUM.          79

violet.   Besides these lines there are others which deserve to
be noticed.  At A is a well  defined dark line within the red

                         Fig. 55.
space, and half-way between A and B is a group  of seven or
eight, forming together a dark band.  Between B  and C there
are 9 lines; between  C and D there are 30; between D and
E there are 84 of different sizes.  Between E and b there are
24, at b there are three very strong  lines, with a fine clear
space  between the two widest;  between b and  F there are
52;  between F  and  G 185;  and between G and H 190,
many being accumulated at G.
  These lines are seen with equal distinctness in  spectra pro-
duced by all solid  and  fluid bodies, and, whatever  be the
lengths of the  spectra and the  proportion of their colored
spaces,  the lines preserve the  same  relative position  to the
boundaries of the colored  spaces; and therefore their  propor-
tional distances  vary  with the  nature of the prism by which
they are produced.  Their number, however, their order, and
their intensity are absolutely invariable, provided light coming
either directly*  or indirectly  from  the sun be employed.
Similar bands are perceived in the light of the planets and
fixed stars, of colored flames, and of  the electric  spark. The
spectra from the light of Mars and from that of  Venus con-
tain the lines D, E, b, and F in the  same positions as in sun-
light.  In  the spectrum  from  the  light of Sirius, no fixed
lines could be perceived in the orange and yellow spaces; but
in the green there was a very strong streak, and two  other
very  strong ones in the  blue.   They had  no  resemblance,
however, to  any of the lines in planetary  light. The star
Castor  gives a  spectrum exactly  like  that of  Sirius, the
streak in the green  being in the very same place.  The
streaks were also seen in the blue,  but Fraunhofer could not
ascertain their place.   In the spectrum of Pollux there were
many weak but fixed lines, which looked like those in Venus.
* Fraunhofer found the very same lines in moonlight. 
80
A  TREATISE ON OPTICS.        PART II.
It had the line D, for example, in the very same place as in
the light of the planets.  In the  spectrum of Capella the
lines D and  b are seen as in the sun's light.  The  spectrum
of Betalgeus contains numerous fixed lines sharply defined,
and those at D and b are precisely in the same places as in
sun-light.  It resembles the spectrum of Venus.  In the spec-
trum of  Procyon Fraunhofer  saw the line D in the orange;
but though  he observed other lines, yet he could not deter-
mine their place with any degree of accuracy.  In the spec-
trum of electric light there is a great number of bright lines.
The spectrum from the light of a lamp contains none  of the
dark fixed lines seen in the spectrum from sun-light; but
there is in the orange a bright line which is more distinct
than the rest  of the spectrum.   It is a double line, and oc-
curs at  the  same  place where D is found  in the solar spec-
trum.  The spectrum from the light of a flame maintained by
the blowpipe contains several distinct bright lines*
  (69.)  One of  the  most important practical results  of the
discovery of these fixed lines in the solar spectrum is, that
they enable us  to take the  most accurate  measures of the
refractive and  dispersive powers of bodies,  by measuring
the distances  of the lines B, C, D, &c.   Fraunhofer com-
puted the table  of the indices of refraction of different sub-
stances, given in the Appendix, No. III.  From the numbers
in the  table here  referred to we may  compute the ratios of
the dispersive powers of any two of the substances,  by the
method already explained in a preceding chapter.

       On the Illuminating Power of the  Spectrum.
  (70.)  Before the time of M. Fraunhofer, the illuminating
power of the different parts of the spectrum had been given
only from a rude estimate.   By means of a photometer he ob-
tained the following  results:—
  The place of maximum illumination  he  found to be at M,
fig. 55.,  so situated  that D M was about one  third or one
fourth of D E; and therefore this place is at the boundary of
the orange and yellow.  Calling the illuminating power at M,
where  it is  a maximum, 100, then the light of other  points
will be as follows:—
Light at the red extremity -  0*0
------B......3-2
------C......9-4
------D......64-0
Maximum light at M    -   100-0
Light at E.....48-0
Light at F  -  -  -  - 17-00
------ G  .  -  .  .  3-10
------ H  -  -  -  -  0-56
------■ the violet ex- \  « m
  tremity  ...    {
* See The Edinburgh. Journal qf Science, No. XV. p. 7. 
CHAP. X.  HEATING  RAYS  IN THE  SPECTRUM.       81

  Calling the intensity of the light in the brightest space D E
100, Fraunhofer found the light to have the following intensity
in the other spaces :—
Intensity of light in BC  -  2-1
-----------------C D   29-9
-----------------DE  100-0
Intensity of light in E F  32-8
               — FG  18-5
                  GH   3-5
  From  these results it follows that,  in the spectrum exam-
ined by Fraunhofer, the most luminous ray is nearer the red
than the violet extremity in the proportion of 1 to 3-5/ and
that the  mean ray is almost in the middle of the blue space.
As a great part, however, of the violet extremity of the spec-
trum  is not seen under ordinary circumstances, these results
cannot be  applied  to spectra  produced  under such circum-
stances.


         On the Heating Power of the Spectrum.

  (71.) It had always been supposed by philosophers that the
heating power in the spectrum would be  proportional to the
quantity  of light;  and  Landriani,  Rochon,  and Sennebier,
found the yellow to  be the warmest of the colored spaces. Dr.
Herschel, however, proved by a series of experiments that the
heating power gradually increased from the violet to the red
extremity of the spectrum. He found also that the thermome-
ter continued to rise when placed beyond the red end of the
spectrum, where  not a single ray of  light could be perceived.
  Hence he drew the important conclusion,  that there were
invisible  rays in the light of the sun, which had  the power of
producing heat, and which had a less degree of refrangibil-
ity than red light.  Dr. Herschel was desirous of ascertaining
the refrangibility of the extreme invisible ray which possessed
the power of heating, but he found this to  be impracticable;
and he satisfied himself with determining that,  at a point 1^
inches distant from  the extreme red ray, the invisible rays ex-
erted  a considerable heating power, even though  the  ther-
mometer was placed at the distance of  52 inches  from the
prism.
  These  results were confirmed by Sir Henry Englefield, who
obtained the following measures:—
                     Temperature.           i        Temperature.
    Blue.....56°    I  Red   ....   72°
    Green.....58       Beyond red  .  .   79
    Yellow   ....   62     |
  When  the thermometer was  returned from beyond the red
into the red, it fell again to 72°. 
82
A  TREATISE  ON  OPTICS.        PART  II.
   M. Berard obtained analogous measures; but  he found that
 the maximum of heat was  at  the very extremity of the red
 rays when  the bulb of the thermometer was  completely cov-
.ered by them, and  that beyond the red  space  the heat was
 only one fifth above that of the ambient air.
   Sir Humphry Davy ascribed Berard's results to his using
 thermometers with circular bulbs, and of too large a size; and
 he therefore repeated the experiments in Italy and at Geneva,
 with very slender thermometers, not more than one twelfth of
 an inch in diameter, with very long bulbs filled with air  con-
 fined by a colored fluid.  The result of these experiments was
 a confirmation of those  of Dr.  Herschel*
   M. Seebeck, who has more recently studied this subject, has
 shown that the place of maximum  heat in the spectrum varies
 with the substance  of which the prism is made.   The follow-
 ing are  his results:—
      Substance of the Prism.             Colored space in which the heat la
     Water........ .  .   Yellow.
     Alcohol.........  Yellow.
     Oil of turpentine......Yellow.
     Sulphuric acid concentrated   .  .   Orange.
     Solution of sal-ammoniac   . .  .   Orange.
     Solution of corrosive sublimate   .  Orange.
     Crown glass........Middle of the red.
     Plate glass........   Middle of the red.
     Flint glass   ........Beyond the red.
   The observations on alcohol and oil  of  turpentine were
 made by M. Wunsch.f

         On the Chemical Influence of the Spectrum.
   (72.)  It was long ago noticed  by the celebrated Scheele,
 that muriate of silver is rendered much blacker by the violet
 than by any of the  other rays  of the spectrum.   In 1801, M.
 Ritter  of Jena, while repeating the  experiments  of Dr. Her-
 schel, found that the muriate of silver became very soon black
 beyond  the violet extremity of the spectrum.   It became a
 little less blackened in the violet itself, still  less  in the blue,
 the blackening  growing  less  and less  towards  the red ex-
 tremity.  When muriate of  silver  a little blackened was used,
 its color was partly  restored when  placed in the  red space, and
 still more  in the space of the invisible rays beyond the red.
 Hence he concluded that there are two sets  of invisible  raya

   * See Edinburgh Encyclopedia, vol. x. p. 69., where  they were first pub-
 lished, as communicated to me by Sir Humphry.
   t For the recent observations of Signor Melloni, see Note IV. of Am. ed.
 which follows author's Appendix. 
CHAP. X.
MAGNETIC RAYS.
83
in the solar spectrum, one on the red side which favors oxy-
genation, and the other  on the violet side which favors dis-
oxygenation.  M. Ritter also  found that  phosphorus  emitted
white fumes in the invisible red; while in the invisible violet,
phosphorus in a state  of oxygenation was instantly extin-
guished.
   In repeating the experiments with muriate of silver, M.
Seebeck found that its color varied with the colored space in
which it was held.  In and beyond the violet, it was reddish
brown ; in the blue, it was blue or bluish grey; in the yellow,
it was white, either unchanged or faintly tinged with yellow;
and in and beyond the red it was red.  In prisms of flint glass,
the  muriate  was decidedly colored beyond  the  limits of the
spectrum.
   Without knowing what had been done by Ritter, Dr. Wol-
laston obtained the very same  results  respecting the action of
violet light on muriate  of silver.  In  continuing his experi-
ments, he discovered some new chemical effects of light upon
gum guaiacum.   Having dissolved some of this gum in alco-
hol, and washed a card with the tincture, he exposed it in the
different colored  spaces  of the spectrum without observing
any change of color. He then took a lens 7 inches in diame-
ter,  and having covered the central part of it so as to leave
only a ring of one tenth of an inch at its circumference, he
could collect the rays of any color in a focus, the focal  dis-
tance being  about 24^  inches  for yellow  light.  The card
washed with guaiacum was then cut in small pieces, which
were placed in the different rays concentrated by the lens.  In
the  violet  and blue rays  it acquired a green color.  In  the
yellow no effect was produced.   In the red rays, pieces of the
card already made green lost their green color, and were re-
stored to their original hue. The guaiacum card, when placed
in carbonic acid  gas, could not be rendered green at any dis-
tance from the lens, but was speedily restored from green to
yellow by the red  rays.   Dr.  Wollaston also  found that  the
back of a heated  silver spoon removed the green  color as ef-
fectually as the red rays.

      On the Magnetizing Power of the Solar Rays.

   (73.)  Dr. Morichini, more than twenty years  ago, announced
that the violet rays of the solar spectrum had the power of
magnetizing  small steel needles that were entirely free from
magnetism. This effect was produced by collecting the violet
rays in  the focus of a convex lens, and carrying the focus of
these rays from the middle of one half of the needle to  the 
84             A TREATISE  ON  OPTICS.        PART  II.

extremities of that half, without  touching the  other  half.
When  this operation  had been performed for an hour, the
needle had acquired perfect polarity.  MM. Carpa and Ridolfi
repeated this experiment with perfect success; and Dr. Mon-
chini magnetized several needles  in the presence of Sir H.
Davy, Professor Playfair, and other English philosophers.  M.
Berard at Montpelier,  M. Dhombre Firmas at Alais,  and pro-
fessor Configliachi at  Pavia, having  failed in producing the
same effects, a doubt was thus cast over the accuracy of pre-
pPQirjO* TGSGELrctlGS
   A few years ago, Dr. Morichini's experiment was restored
to credit by some  ingenious  experiments by Mrs. Somerville.
Having covered with paper half of a sewing needle, about an
inch long, and devoid of magnetism, and exposed  the  other
half uncovered to the violet rays,  the  needle  acquired  mag-
netism in about  two hours,  the exposed end exhibiting north
polarity.   The indigo rays produced nearly the same  effect,
and the blue and  green produced it in a less degree.  When
the needle was exposed to the yellow, orange, red, or calorific
 rays beyond the red, it did not receive the slightest magetism,
 although the exposures lasted for three days.   Pieces of clock
 and watch springs gave similar results; and when  the  violet
 ray was  concentrated with a  lens,  the needles, &c.,  were
 magnetized  in a  shorter time.   The same effects were pro-
 duced by exposing the needles  half covered with paper to the
 sun's rays transmitted through glass colored blue with cobalt.
. Green glass produced the same effect.   The light of the sun
 transmitted through blue and green riband produced the same
 effect as through  colored glass.   When the needles thus cov-
 ered had hung a day in the sun's rays behind a pane of glass,
 their exposed ends were north poles, as formerly.
   In repeating Mrs. Somerville's experiments, M. Baumgart-
 ner of Vienna discovered  that a steel wire, some parts of
 which were polished,  while  the rest were without lustre, be-
 came magnetic by exposure to  the white light of the sun; a
 north pole appearing at each polished part, and a south pole at
 each unpolished  part.  The effect was hastened by concen-
 trating the solar rays  upon the  steel wire.  In this way he ob-
 tained 8 poles on  a wire eight inches long.  He was not able
 to magnetize needles  perfectly  oxidated, or perfectly polished,
 or having polished lines in the direction of their lengths.
   About the same time, Mr. Christie of Woolwich found that
 when a magnetized needle, or a needle of copper or glass, vi-
 brated by the force of torsion  in  the white light of the sun,
 the arch of vibration was more  rapidly  diminished in the sun's
 light than in the shade.  The effect was greatest on the mag- 
CHAP. X.
MAGNETIC  RAYS.
85
 netized needle.   Hence he concludes that the compound solar
 rays possess a very sensible magnetic influence.
    These results have received a very remarkable confirmation
 from the  experiments of M. Barlocci  and M.  Zantedeschi.
 Professor Barlocci found that an armed natural loadstone,
 which could carry 1^  Roman pounds, had its power nearly
 doubled by twenty-four hours' exposure  to the strong light of
 the sun.  M. Zantedeschi found  that an artificial horse-shoe
 loadstone, which carried  13J oz., carried  3£ more by three
 days' exposure, and at  last  sifpported  31 oz., by  continuing it
 in the  sun's light.   He  found, that while the  strength  in-
 creased in  oxidated  magnets, it  diminished  in  those which
 were not oxidated, the diminution becoming insensible when
 • the loadstone was highly polished.  He now concentrated the
 solar rays upon the loadstone by means  of a lens; and  he
 found that, both in oxidated and polished  magnets, they ac-
 quire strength when their north pole is  exposed  to the sun's
 rays, and  lose strength when the south pole is exposed.  He
- found likewise  that  the  augmentation  in the first case ex-
 ceeded  the diminution in the second.  M. Zantedeschi re-
 peated the experiments of Mr. Christie on needles vibrating
 in the sun's light; and he found  that, by exposing the north
 pole of a needle  a foot long, the semi-amplitude  of the last
 oscillation was 6° less than the first; while, by  exposing the
 south pole, the last oscillation became greater than the first.
 M. Zantedeschi admits that he often encountered inexplicable
 anomalies in these experiments.*
    Decisive as these results  seem  to be in  favor  of the mag-
 netizing power  both  of violet and white light, yet a series of
 apparently very well conducted experiments have been lately
 published by MM. Riess and Moser,f which cast a doubt over
 the  researches  of preceding philosophers.  In these experi-
 ments, they examined the number of oscillations  performed in
 a given time before and after the needle was submitted to the
 influence of the violet  rays.  A focus of violet light concen-
 trated by a lens 1*2 inches in diameter, and 2-3 inches in focal
 length, was made to traverse one half of the needle 200
 times;  and though this experiment was repeated with differ-
 ent  needles, at different seasons of the year, and different
 hours of the day, yet the duration  of a given number of oscil-
 lations was almost exactly the same after as before the experi-
 ment.   Their attempts to verify  the results of Baumgartner
 were equally fruitless; and they therefore consider themselves
* Edinburgh Journal of Science, New Series, No. V., p. 76.
tld. No. IV., p. 225.
                      H 
86             A  TREATISE ON OPTICS.        PART  II.

as entitled  to reject totally a discovery, which for seventeen
years has at different times disturbed science.   " The small
variations," they observe, " which are found  in  some of our
experiments, cannot constitute a real action of the nature of
that which was observed by MM. Morichini, Baumgartner,
&c, in so clear and decided a manner."
                       CHAP. XI.
       ON THE INFLEXION OR  DIFFRACTION OF  LIGHT.

  (74.) Having thus described the changes which light expe-
riences when refracted by the surfaces of transparent bodies,
and the properties which it exhibits when thus  decomposed
into its elements, we shall now proceed to consider the phe-
nomena which it presents when passing near the edges of
bodies.  This  branch of optics is called the  inflexion  or the
diffraction of light.
  This curious property of light was first described by Gri-
maldi in 1665, and afterwards by Newton; but it is to the late
M. Fresnel that we are indebted for a most successful and able
investigation of the phenomena.
  In  order  to  observe  the action of bodies upon the light
which passes  near them, let a lens L L, of very short focus,
fig. 56., be fixed in the window-shutter, M N, of a dark room;

                          Fig. 56.
and let R L L be a beam of the sun's light, transmitted through
the lens.  This light will be  collected into a focus at F, from
which it will diverge in  lines FC, FD, forming a circular
image of light on the opposite wall.  If a  small hole, about
the fortieth of an inch in diameter, had been fixed in the win-
dow-shutter in place  of the  lens, nearly the same  divergent 
CHAP. XI.         INFLEXION OF LIGHT.               87
beam of light would have been obtained.  The shadows of all
bodies whatever held in this light  will be found to be sur-
rounded with three fringes of the following colors, reckoning
from the shadow:—
  First fringe.—Violet, indigo, pale blue, green, yellow, red.
  Second fringe.—Blue, yellow, red.
  Third fringe.—Pale blue, pale yellow, pale red.
  In  order to examine these fringes, we may either receive
them on a smooth white surface as  Newton did, or adopt the
method of Fresnel, who looked at  them with a  magnifying
glass,  in  the  same  manner as if  they  had been an image
formed by a lens.  This last method is decidedly the best, as
it enables the observer to measure the fringes, and ascertain
the  changes which they undergo  under  different circum-
stances.
  Let a body B be now placed at the distance B F from the
focus, and let its shadow be received on the screen C D, at a
fixed distance from  the body B, and the following phenomena
will be observed :—
  1.  Whatever be the nature of the body B with regard to its
density or refractive power, whether it is platma or the pith
of a  rush, whether  it is tabasheer or chromate of lead, the
fringes surrounding its shadow will be the very same in mag-
nitude and in color, and the  colors will be those given above.
  2.  If the light R L is homogeneous light of the different
colors in the  spectrum, the fringes will be  of the same color
as  the light RL; and they will be  broadest in red light,
smallest in violet, and of intermediate sizes in the interme-
diate  colors.
   3.  The body B continuing  fixed, let us either  bring the
screen C D nearer to B, or bring  the lens with which we
view the fringes nearer to B, so as to see them  at different
 distances behind B.   It will be found that they grow less and
 less  as they approach the edge of B, from which  they take
 their rise.  But if we measure the distances of any one fringe
 from the shadow at different distances behind B, we shall find
 that the  line joining  the same point of the  fringe is not a
 straight line, but a hyperbola whose vertex is at the edge of
 the body; so  that the same fringe  is not formed by the same
 light at all distances from the body, but resembles a caustic
 curve formed by the intersection of different  rays.   This cu-
 rious fact we have endeavored  to represent in the  figure by
 the hyperbolic curves joining the edge of the body B and the
 fringes which are shown by dotted  lines.
   4. Hitherto we have supposed that B has  been held at the
 same distance from  F; but  let  it now be brought to b, much 
88              A  TREATISE  ON  OPTICS.        PART II.

nearer F, and let the screen C D be brought  to c d, so that
b g is equal B G.   In this new position, where nothing has
been changed but  the distance from F, the fringes will be
found greatly increased  in  breadth, their relative distances
from each other and from the margin of the shadow remain-
ing the same.  The  influence  of distance from the radiant
point  F on  the  size  of  the fringes,  or on the quantity of
inflexion,  is shown in the following results obtained  by  M.
Fresnel:—
	Distance of the inflecting body B behind the ra-	Distance B G or bg behind the body B or 6, where the in-	Angular inflexion of the rod ray* of the first fringe.
F b FB	4 inches. 20 feet	39 inches. 39----	12' 6" 3 55
    When we consider that the fringes are largest in red, and
 smallest in violet light, it is easy to understand  the  cause
 of their colors in white light; for the colors  seen in this case
 arise from the superposition of fringes of all the seven colors;
 that is, if the eye could receive all the seven differently color-
 ed  fringes  at once, these colors would form by  their mixture
 the actual  colors in the fringes seen by white  light. Hence
 we see why the color of the first fringe is violet near the
 shadow, and red at a greater  distance; and why the blending
, of the colors beyond the third fringe forms white light, in-
 stead of exhibiting themselves in separate tints.
    Upon measuring the proportional breadths  of the fringes
 with great care, Newton found  that they were as the num-
 bers 1,  ./A,  y/\, ,/i, and their intervals in the same pro-
 portion.
    Besides  the  external fringes  which surround all bodies,
 Grimaldi discovered within the shadows of long and narrow
 bodies a number of parallel streaks or fringes alternately light
 and dark.   Their number grew smaller as the body tapered;
 and Dr. Young remarked  that the  central  line was always
 white, so that there must always be an odd number of  white
 stripes,  and an even  number of dark ones.  At the ano-ular
 termination of bodies  these fringes widen and become convex
 to the central  white line; and when  the  termination is rect-
 angular, what are called the  crested fringes  of Grimaldi are
 produced.
   The  phenomena exhibited  by substituting  apertures of
 various  forms in place of the body  B are very interesting.
 When the aperture is circular, such as that formed in a piece
 of lead  with a  small pin, and when a lens is placed behind it
 so as to view the shadow at different distances,  the aperture
 will be  seen surrounded with distinct rings, which contract 
CHAP. XI.
INFLEXION OF LIGHT.
89
and dilate, and change their  tints in the most beautiful man-
ner.  When the aperture is one thirtieth of an  inch, its dis-
tance F B from the luminous point 6 feet 6 inches, and its
distance from  the focus of the eye-lens, or B G,  24 inches,
the following series of rings was observed:—
   1st order.  White, pale.yellow, yellow, orange, dull red.
   2d order.  Violet, blue, whitish,  greenish  yellow, yellow,
bright orange.
   3d order.  Purple, indigo blue, greenish blue, bright green,
yellow green, red.
   4th  order.   Bluish green, bluish white, red.
   5th  order.   Dull green, faint bluish white, faint red.
   6th  order.   Very faint green, very faint red.
   7th  order.   A trace of green and red.
   When the aperture B is  brought nearer to  the eye-lens
whose focus is supposed to be at G, the  central  white  spot
grows less and less till it vanishes, the rings gradually closing
in upon it, and the  centre assuming  in succession the most
brilliant tints.  The  following were the  tints  observed by
Mr. Herschel; the distance between the radiant point F and
the focus  G of the  eye-lens remaining constant,  and  the
aperture, supposed to be at B, being gradually brought nearer
toG:—
Distance of aper-ture B from the eyeJlens.	Color of the Central spot.	Character of the ringa which earrotwd the central .pot.
24 in. 18 13 5 10 9-25 910 8-75 8-36 800 7-75 700 6-63 6-00 5-85 5-50 500 4-75 4-50 4-00 3-85 3-50	White. White. \ Yellow. \ Intense orange. Deep orange red. Brilliant blood red. Deep crimson red. Deep purple. Very sombre violet. Intense indigo blue. Pure deep blue. Sky blue. Bluish white. Very pale blue. Greenish white. Yellow. Orange yellow. Scarlet-Red. Blue. Dark blue.	Rings as described above. First two rings confused. Red of 3d, and green of 4th order, splendid. Inner rings diluted. Red and green of the outer rings good. All the rings much diluted. Rings all very dilute. Rings all very dilute. Rings all very dilute. Rings all very dilute. A broad yellow ring. A pale yellow ring. A rich yellow. A ring of orange, with a sombre space. Orange red, with a pale yellow space. A crimson red ring. Purple, with orange yellow. Blue, orange. Bright blue, orange red, pale yellow, white. Pale yellow, violet, pale yellow, white. White, indigo, dull orange, white. White, yellow, blue, dull red. Orange, light blue, violet, dull orange.
H2 
90              A TREATISE ON OPTICS.        PART II.

   When tivo small apertures are used instead of one, and the
rings examined by  the  eye-lens as  before,  two systems of
rings will  be seen, one round  each centre; but, besides the
rings, there is another set of fringes which, when the aper-
tures are equal,  are parallel  rectilineal fringes  equidistant
from the two centres, and  perpendicular to the line joining
these centres.  Two other sets of parallel rectilineal fringes
diverge in the form of a St. Andrew's cross from the middle
point between the two centres, and forming equal angles be-
tween the  first set of parallel fringes.  If the apertures are
unequal, the two systems of rings are unequal, and the first
set of parallel fringes become hyperbolas, concave  towards
the smaller system of rings, and having the aperture in then-
common focus.*
   The finest experiments on this subject are those of Fraun-
hofer; but  a proper view of them would require more  space
than we can spare. |
                       CHAP. XII.

             ON THE COLORS OF THIN PLATES.

  (75.) When light is either reflected from  the surfaces of
transparent  bodies, or transmitted  through portions of them
with parallel  surfaces, it is invariably white, for all the dif-
ferent  thicknesses of such bodies as we are in the habit of
seeing.  The thinnest films of blown glass, and the thinnest
films of mica generally met with, will both reflect and trans-
mit white light.   If we diminish, however, the  thickness of
these two bodies to a certain degree, we shall  find that, in-
stead of giving white light by reflexion and transmission, the
light is in both cases colored.
  Mr.  Boyle seems first to have observed that thin bubbles of
the essential  oils,  spirit of  wine,  turpentine, and soap and
water,  exhibited beautiful colors; and he succeeded in blow-
ing glass so thin as to show  the same tints.   Lord Brereton
had observed the colors of the thin  oxidated films which the
action  of the weather produces upon glass;  and Dr. Hooke
obtained films so equally thin that  they exhibited over  their
whole  surface  the same brilliant color.  Such pieces of mica
may be  produced  at the edges of plates quickly detached
from a mass;  but  they  may be more  readily obtained by
* Herschel's Treatise on Light, § 735.
J See Edinburgh Encyclopedia, art. Optics, Vol. XV., p. 55f>. 
CHAP. XII.      COLORS OF THIN PLATES.
91
sticking one side of a plate of mica to a piece of sealing-wax,
and tearing it away with  a sudden jerk.   Some extremely
thin films will then  be  left on the wax, which will exhibit
the liveliest colors by reflected  light.   If we could produce a
film of mica with only one tenth part of the thickness of that
which produces a bright blue color, this film would reflect no
light at all, and would appear black if viewed by reflexion against
a black body.  But though no such film has ever been obtained,
or is likely to  be obtained by any means with which we are
acquainted,  yet accident on one occasion produced solid  fibres
as thin, and actually incapable of reflecting light.  This very
remarkable  fact occurred in a crystal  of quartz of a smoky
color, which was broken in two. The two surfaces of fracture
were  absolutely black; and the blackness  appeared, at first
sight, to be owing  to a thin  film of opaque matter which had
insinuated  itself into the  crevice.  This  opinion, however,
was untenable, as every part of the surface was black, and the
two halves of the  crystals could not have stuck together had
the crevice extended across the whole section. Upon examin-
ing this specimen with care, I found that the surface was per-
fectly transparent by transmitted light, and that the blackness
of the surfaces arose from their being entirely composed of a
fine down of quartz, or of short and slender filaments, whose
diameter was  so exceedingly small that they were incapable
of reflecting a single ray of the strongest light. The diameter
of these  fibres was so small, that, from principles which we
shall  presently explain, they could not exceed the one third
of the millionth part of an inch.  This curious specimen is in
the cabinet of her  grace  the duchess of Gordon*  I have
another small specimen in my  own possession; and I have no
doubt that fractures of quartz and other minerals will yet be
found which shall exhibit a fine down of different colors de-
pending on their size.
   The colors  thus produced by thinness, and hence called the
colors of thin plates, are best observed in fluid bodies of a
viscous  nature. If we blow a soap-bubble,  and  cover it with
a clear glass to protect it from currents of air, we shall ob-
serve, after it has grown  thin by standing a little, a  great
many concentric colored rings round the top of it.  The color
in the centre of the  rings will vary with  the thickness; but
as the bubble  grows thinner the rings will dilate, the central
spot will  become  white, then  bluish, and  then  black,  after
which the bubble will burst, from its extreme thinness at the
place of the black spot.  The same change of color with the
* See Edinburgh Journal of Science, No. I., p. 108. 
93              A TREATISE ON OPTICS.        PART II.

thickness may be seen by placing a thick film of an evapora-
ble fluid upon a clean plate of glass, and watching the effects
of the diminution of thickness which take place in the course
of evaporation.
  The method used by Sir Isaac Newton for producing a thin
plate of air, the colors of which he intended to investigate, is
shown in fig. 57.,  where L L  is a plano-convex  lens, the
Fig. 57.
radius  of whose convex surface is  14 feet, and 11 a double
convex lens, whose convex surfaces have a radius of 50 feet
each. The plane side of the lens L L was placed downwards,
so as to rest upon one of the  surfaces of the lens 11.  These
lenses  obviously touch at  their middle points;  and if the
upper one is slowly pressed against the under one, there will
be seen round the  point of contact a system of circular color-
ed rings, extending wider and wider  as the  pressure  is in-
creased.  In  order to examine these rings under  different
degrees of pressure,  and when the lenses L L,  II axe at
different distances, three clamp-screws, p,p,p, should be em-
ployed, as shown  in fig. 58.,  by turning which we may pro-
duce a regular and equal pressure at the point of contact.
   When we  look at these rings through the upper lens, so as
to see those formed by the light reflected from the plate  of air
        Fig. 53.        between the lenses, we  may observe
                      seven  rings,  or rather seven circular
                      spectra or orders of colors, as described
                      by Newton in the first  two columns of
                      the following Table ; the colors  being
                      very distinct in the first three spectra,
                      but growing more and  more diluted in
                      the others, till they almost entirely dis-
                      appear in the seventh spectrum.
   When we view the plate of air by looking through the un-
 der lens 11 from  below, we observe another  set  of rings or
 spectra formed in the transmitted light.  Only five of these
 transmitted rings  are distinctly seen,  and their Colors,  as ob-
 served by Newton, are given in the third column of the fol-
 lowing table; but they are much more faint than those seen
 by reflexion.   By comparing the colors seen by reflexion with
 those seen by transmission, it will  be observed that the color
 transmitted is always  complementary to the one reflected, or
 which, when mixed with it, would make white light.
 
CHAP. XII.      COLORS  OP  THIN PLATES.             93
   Table of the Colors of Thin Plates of Air, Water, and Glass.
Spectra, or Orders o Colors, reckoned from the centre.	Colors produced at the thicknesses in the last three columns.		Thicknesses in millionths of an inch.		
f First Spectrum J or order of Colors. Seconu Spectrum or order of Colors. Thirh Spectrum or order * of Colors. Fourth Spectrum J or order j of Colors. Fifth ( Spectrum J or order 1 of Colors. [ Sixth ( Spectrum J or order | of Colors. [ Seventh ( Spectrum I or order | of Colors. [	Reflected.	Transmitted.	Air.	Water.	Glass.
	Very black Black Beginning ) of black \ Blue White Yellow Orange Red	White Yellowish red Black Violet Blue	1 I' 2 2J 5* I* 9	3 3 4 u 1! 3^ 5£ 6 6|	10 1& 31 If 1^ Ai 5'
	Violet Indigo Blue Green Yellow Orange Bright red Scarlet	White Yellow Red Violet Blue	HI 12f 14 16 f 17f 181 19|	8J 9| 10^ Hi 121 13 13J U3r	71 8rV 9 9? 10| Ill llf 12|
	Purple Indigo Blue Green Yellow Red Bluish red	Green Yellow Red Bluish green	21 22to 23| 25| 271 29 32	15J 16i 1^ 18tJo 201 21f 24	13ii 141 15rV 16* 171 18* 20|
	Bluish green Green Yellowish \ green \ Red	Red Bluish green	34 35| 36 404	251 27 301	22 • 22£ 23| 26
	Greenish ) blue \ Red	Red	46 52i	341 39|	29| 34
	Greenish ) blue \ Red	"	58f 65	44 48|	38 42
	Greenish > blue S Ruddy ) white )		71 7,7	53£ 57|	45| 49f
 
94
A  TREATISE  ON OPTICS.         PART II.
   The preceding colors are those which are  seen when light
is reflected and  transmitted nearly perpendicularly; but Sir
Isaac  Newton found that when the light was  reflected and
transmitted obliquely, the rings increased in size,  the same
color requiring a greater thickness to produce it.   The  color
of any film, therefore, will descend to a  color lower in, or
nearer the beginning of, the scale, when it is seen obliquely.
   Such are the general phenomena of the colored rings when
seen by white light.  When we place  the lenses in homoge-
neous light, or make the different colors of the solar spectrum
pass in succession over the lenses, the rings, which are always
of the same color as the  light, will be  found to be largest in
red light, and to contract gradually as they are seen in all the
succeeding colors,  till  they reach their smallest  size in the
violet rays.  Upon  measuring  their diameters, Newton found
them to have the following ratio in the different colors at their
boundaries:—
  > Extreme Red.   Orange.  Yellow.   Green.  Blue.  Indigo.  Violet.   Extreme.
       1       0-934    0-880   0-825   0-763  0-711   0-681    0-630
Since white light is composed of all  the preceding colors, the
rings seen by it will consist of all the seven  differently color-
ed systems of rings superposed as it were,  and forming, by
their  union,  the  different  colors  in  the Table.   In order to
explain this, we  have constructed the annexed diagram, Jig.
59., on the supposition that each ring  or spectrum has the
Fig. 59.
same breadth in homogeneous light which it actually  has
when  it is formed  between surfaces nearly  flat, or  when
the thickness of the  plate varies with the distance from the
point of contact.*  Let us then suppose  that we form such a
* This supposition is made in order to simplify the diagram. 
CHAP. XII.     COLORS  OP THIN  PLATES.             95

system of rings with the seven colors of the spectrum,  and
that a sector is cut out of each system, and placed, as in the
figure, round  the same centre C.   Let the angle of the  red
sector be 50°, of the  orange 30°, the yellow 40°, the green
60°, the blue 60°, the indigo 40°,  and the violet 80°, being
360°  in all, so as to complete the circle.  From the centre C
set off the first, second, and third rings in all the sectors, with
radii  corresponding to the  values  in the preceding small
Table.   Thus, since the proportional  diameters of the ex-
treme red and the extreme orange  are 1 and 0-924, the mid-
dle of the red will be  in the middle  between  these  numbers,
or 0-962; and consequently the proportional diameter, or the
radius of the first red  ring for the middle of the red space R,
will be 0-962.  In like manner, the radius for the orange  will
be 0-904, for the  yellow  0-855, for the  green 0-794, for the
blue 0-737, for the indigo 0-696, and for the violet 0-655.  Let
the red rings be colored red as they appear in  the experiment,
the orange rings orange, and so on, each color  resembling that
of the spectrum as nearly as possible.   If we  now suppose all
these  colored sectors  to  revolve rapidly  round C as a centre,
the effect of them all,  thus mixed, should  be the production of
the colored rings as seen by white light.  As  the diameter of
each ring varies from the beginning of the  red space to the
end of it, and so on  with all the  colors, the portion of the
ring  in  each  sector should be part of a  spiral, and all these
separate parts should  unite in forming a single spiral, the red
forming the commencement, and the violet the termination of
the spiral for each ring.
   This diagram enables us to ascertain the composition of any
of the rings seen in white  light.  Let it be required, for ex-
ample, to determine the color of the ring at the distance C m
from the centre, m being in the middle of the  second red ring.
Round C as a centre,  and with the radius C m, describe a cir-
cle, mnop, and it will be seen from  the  different colors
through  which it passes what  is its  composition.   It passes
nearly through the very brightest* part of the  second red ring,
at  m, and through a pretty bright part of the orange.  It
passes nearly through the bright part of the yellow, at n;
through the brightest  part of the green; through a less bright
part  of the blue;  through a dark part of the  indigo, at p;
and through the darkest part of the third violet ring.  If we
knew the exact law according to which the brightness of any
fringe varied from  its darkest to its brightest point, it would
thus  be easy to ascertain with accuracy the  number of rays
* In the figure, the brightest part is the most shaded. 
96              A TREATISE ON OPTICS.         PART II.

of each color which entered into the  composition of any of
the rings seen by white light.                          .
  In order to determine the thickness of the plate of air by
which each color was produced, Newton found the squares of
the diameters of the  brightest parts  of each to be  in  the
arithmetical progression of the odd numbers, 1, 3,  5, 7, 9, &c,
and the squares of the  diameters  of the obscurest parts in the
arithmetical progression of the even numbers, 2, 4, 6, 8, 10;
and as one of the glasses was  plane, and the other spherical,
their  intervals at these rings must be in the  same progression.
He then measured the diameter of the fifth dark ring, and
found that the thickness of the air at  the darkest part of the
first dark ring, made by perpendicular rays, was the ^,1^
part of an inch.   He then multiplied this number by the pro-
gression 1, 3, 5,  7, 9, &c, and 2, 4, 6, 8, 10, and obtained the
following results:—
                   Thickness of the air at the          Thickness of the air at lb*
                     most luminous part.             moot obscure part.
First Ring   -  -  -  TTsvrnrcr  " "    "   t^^ows or -pg^
Second Ring  -  -  -  j^-i^^  -  -    -    y^-gvo-Tjo
Third Ring  -  -  -  Yi-iwos  "  "    "    t^wd
Fourth Ring -  -  -  -p^otTcr  -  -    -    Trsvcrtnr
  When  Newton admitted  water between the lenses, he
found  the  colors to become fainter, and the rings smaller;
and upon measuring the .thicknesses  of water at which the
same  rings were produced, he found them to be nearly as the
index  of refraction for air is  to  the index  of refraction for
water, that is, nearly as 1-000 to  1-336.  From these  data he
was enabled to compute the three last columns of the Table
given  in page 93, which show the thicknesses in millionth
parts  of an inch at which  the colors are produced in plates of
air, water, and glass.  These columns are  of extensive use,
and may be regarded as presenting us with a micrometer for
measuring minute thicknesses of transparent bodies by their
colors, when all  other methods would be inapplicable.
  We have already seen that when the thickness of the film
of air is about y^f^dth of an inch, which corresponds to the
seventh ring, the colors cease to become visible, owing to the
union of all the separate colors forming white light; but when
the rings are seen in homogeneous light, they appear in much
greater numbers, a dark and a colored ring  succeeding each
other  to a  considerable  distance from the point of contact.
In this case, however, when the rings are  formed between
object glasses, the thickness of the plate of air  increases so
rapidly that the outer rings  crowd upon one another, and
cease to become  visible from this cause.  This effect would 
CHAP. XIII.     COLORS  OF THICK PLATES.           97

obviously not be produced if they were  formed by a solid film
whose thickness varied by slow gradations.  Upon this prin-
ciple, Mr. Talbot has pointed out a very beautiful method of
exhibiting  these rings with plates  of  glass and other sub-
stances even of a tangible thickness.  If we blow a glass ball
so thin that  it bursts* and hold any of the fragments in the
light of a  spirit lamp with a salted wick, or in the  light of
any of the monochromatic  lamps which I have elsewhere de-
scribed, all of which discharge  a  pure  homogeneous yellow
light, the surface  of these films will be seen  covered  with
fringes alternately yellow and black, each fringe marking out
by its windings the lines of equal thickness in the glass  film.
Where the thickness varies slowly, the fringes will be broad
and easily seen; but where the variation takes place  rapidly,
the fringes are crowded together, so as to require  a micro-
scope to render them visible?  If we suppose any of the films
of glass  to be only the thousandth part of an inch thick,  the
rings which it exhibits will belong to the 89th order; and if
a large rough plate of this glass could  be got with its thick-
ness descending to the millionth part of an  inch by slow gra-
dations, the whole  of those  89 rings, and probably many more,
would be distinctly visible to the eye.   In order to  produce
such effects, the light would  require to  be  perfectly homoge-
neous.
  The rings seen between the two lenses are equally visible
whether air or any other gas is used, and even when  there is
no gas at all; for  the  rings are visible  in the exhausted re-
ceiver of an air-pump.
                       CHAP.  xm.

             ON THE COLORS OF THICK PLATES.
   (76.) The colors of thick plates were first observed  and
described by Sir Isaac Newton, as produced by concave glass
mirrors.   Admitting  a beam of solar light, R,  into a dark
room, through an aperture a quarter of an inch  in  diameter
formed in the window-shutter M N, he allowed it to  fall upon
a glass mirror, A B, a quarter of an inch thick, quicksilvered
behind, having its axis in the direction R r, and the  radius of
the curvature of both its surfaces being equal to its distance
behind the aperture.  When a sheet of paper was placed  on
the window-shutter M N, with a hole in it to allow  the sun-
* Films of mica answer the purpose still better. 
98              A TREATISE ON OPTICS.        TART II.

beam to pass,  he observed the  hole  to be surrounded with
four or five colored  rings, with sometimes traces of a sixth

                          Fig. 60.



           R
and seventh.  When the paper was held at a greater or a less
distance than the centre of its concavity, the rings became
more dilute, and gradually vanished.   The colors of the rings
succeeded one another like those in the transmitted system in
thin plates,  as given in column 3d of the Table in  page 93.
When the light R was red the rings were red, and so on with
the other colors,  the  rings being largest in red and  smallest
in violet light.   Their diameters preserved the same propor-
tion as those seen between the object  glasses; the squares of
the diameters of the most luminous parts (in homogeneous
light) being as the numbers 0, 2, 4, 6,  &c, and the squares of
the diameters of the darkest parts  as the intermediate num-
bers 1, 3, 5, 7, &c.  With mirrors of greater thickness  the
rings  grew less, and their  diameters  varied inversely as  the
square roots of the thickness of the mirror. When the quick-
silver was removed, the rings became fainter; and when the
back surface of the mirror was covered with a mass of oil of
turpentine, they disappeared altogether.   These facts clearly
prove that the posterior surface of the  mirror concurs with the
anterior surface in the production of the rings.
   When the mirror A B is inclined to the incident beam R r,
the rings grow larger and larger as the inclination increases,
and so also does the white round spot; and new rings of color
emerge  successively out  of  their common centre,  and  the
white spot becomes a white ring accompanying them, and the
incident and reflected beams  always fall upon the opposite
parts  of this white ring, illuminating its perimeter  like two
mock suns in the  opposite parts of an iris.   The colors of
these  new rings were in a  contrary order to those of  the
former.
   The Duke de Chaulnes observed similar rings upon the sur-
face of the mirror when it was covered with gauze or muslin,
or with a skin of dried skimmed milk; and Sir W.  Herschel
noticed analogous phenomena when he scattered hair-powder
'M
yd
 
CHAP. XIII.    COLORS  OF THICK PLATES.
99
in the air before  a concave mirror on which a beam of light
was incident, and received the reflected light on a screen.
  (77.)  The method which  I  have found to be the most sim-
ple for exhibiting these colors,  is to place the eye immediately
behind a small flame from a minute wick fed with oil or wax,
so that we can examine them even at a perpendicular inci-
dence.  The colors of thick plates may be seen even with a
common candle  held  before the eye at the distance of 10 or
12 feet from a common pane of crown  glass in a window that
has accumulated a little fine dust upon its surface, or that has
on its surface  a fine deposition of moisture..  Under these
circumstances they are very bright, though they may be seen
even when the pane of glass is clean.
  The colors of thick plates may, however, be best displayed,
and their theory best studied, by using two plates of glass of
equal thickness.  The phenomena thus produced, and which
presented themselves to me in 1817, are highly beautiful, and,
as Mr.  Herschel  has shown, are admirably fitted  for illus-
trating the  laws of this  class of phenomena.   In order to ob-
tain plates of exactly the same thickness, I formed  out of the
same piece of parallel glass two plates, A B, C D, and having
placed  between them two pieces of soft wax, I pressed them
                         to the distance of about one tenth
                         of an inch from each other; and by
                         pressing above  one piece of wax
                         more than  another, I was  able to
                         give the two plates any small incli-
                         nation I chose.  Let A B, C D then
                         be a section of the two plates, thus
                          inclined, at right angles to the com-
                          mon section of their surfaces, and
                          let R S be a ray of light incident
                          nearly  in a vertical direction and
                          proceeding from a candle, or, what
                          is  better, from  a circular  disc of
                          condensed light subtending an an-
  gle of 2° or 3°.  If we place the eye behind the plates, when
  they are parallel we shall see only an image of the circular
  disc; but when they are  inclined, as in  the figure, we shall
  observe  in  the  direction V R several reflected images m a
  row besides the  direct image.  The first or the brightest of
  these will be seen crossed with fifteen or sixteen beautiful
  fringes or bands of color.   The three central  ones consist of
  blackish or whitish stripes; and the exterior ones of brilliant
  bands of red and  green light.  The direction  of these bands
  is always  parallel to the common  section  of the  inclined
Fig. 61.
 
100            A TREATISE ON OPTICS.        PART II.

plates.  These colored bands increase in breadth by diminish-
ing the inclination of the plates, and diminish by increasing
their  inclination.   When the light of the luminous circular
object falls obliquely on the first plate, so that the plane of in-
cidence is at right angles to the section  of the plates,  the
fringes  are not distinctly visible across any of the images;
but "their  distinctness is a maximum  when the plane of inci-
dence is parallel to  that  section.  The reflected images of
course become more bright, and  the  tints  more  vivid, as  the
angle of incidence becomes greater; when the angle of inci-
dence increases from 0° to 90°, the images that have suffered
the greatest number of reflexions are crossed by other fringes
inclined to them at a small angle.  If we conceal the  bright
light  of the first image so as to perceive the image formed by
a second reflexion within the first plate, and if we view the
image through a  small  aperture, we  shall observe colored
bands across the first image far  surpassing in precision of
outline  and richness of coloring any analogous  phenomenon.
When these fringes are  again  concealed, others are seen on
the image immediately behind them, and formed by a third
reflexion from the interior of the first plate.
  If we bring the plate C D a little  farther to the right hand,
and make the ray R S fall  first upon  the plate C D, and be
afterwards reflected back upon  the  first plate A B, from  both
the surfaces of  C D, the same colored bands will be seen.
The progress of the rays through  the two plates is shown in
the figure.
  When the two plates have the form of concave and convex
lenses, and are combined, as  in the double and triple achro-
matic object glass, a series of the most splendid systems of
rings are developed;  and these  are sometimes crossed by
others of a different kind. I have not yet had leisure to  pub-
lish an account of the numerous observations I have made on
this curious class of phenomena.
  In  viewing films  of blown glass  in homogeneous yellow
light, and even in common day-light, Mr. Talbot has observed
that when two films are placed together, bright  and obscure
fringes, or colored fringes of an irregular form, are produced
between them, though exhibited by neither of them separately. 
CHAP. XIV.        COLORS OP FIBRES.                101


                      CHAP.  XIV.

    ON THE  COLORS OF FIBRES  AND GROOVED SURFACES.

  (78.) When we look at a candle or any other luminous body
through a plate of glass covered with vapor or with dust in a
finely divided state, it is surrounded with a corona or ring of
colors, like a halo round the sun or moon. These rings increase
as the size of the particles which produce them is diminished;
and their brilliancy and number  depend on the uniform size of
these particles.  Minute fibres, such as those of silk and wool,
produce the same series of rings, which increase as the diameter
of the fibres is less; and hence Dr. Young  proposed an in-
strument  called an eriometer, for measuring the diameters of
minute particles and  fibres, by ascertaining the diameter of
any one of the series of rings which they  produce.  For this
purpose, he  selected the limit of the first red and green ring
as the one  to be  measured.  The  eriometer is formed of a
piece of card or a plate of brass, having an aperture about the
fiftieth of an inch  in diameter in the centre of a circle about
half an inch  in diameter, and  perforated with about eight
small holes.   The fibres  or particles to be measured are fixed
in a slider, and the eriometer being placed before  a strong
light, and the eye  assisted by a lens applied behind the small
hole, the  rings of colors will be seen.  The slider must then
be drawn out or pushed in till the limit of the red and green
ring coincides with the circle  of perforations, and the index
will then show on the scale the  size of the particles or fibres.
The seed of the  lycoperdon bovista was found by  Dr. Wol-
laston to be the 8500dth part of an inch in diameter;  and as
this substance gave rings which indicated 3^ on the scale, it
follows that 1 on the same scale was the  29750th part of an
inch, or the 30,000dth part.   The  following Table contains
some of Dr. Young's measurements, in thirty-thousandths of
an  inch:—
Milk diluted indistinct...  3
Dust of lycoperdon bovista   3J
Bullock's blood   .   .  .   .  4£
Smut of barley   .   .  .   .  6^
Blood of a mare  .   .  .   .  6£
Human blood diluted with
  water.......6
Pus........H
Silk........12
Beaver's wool   ....  13
Mole's  fur.....16
                           I
 Shawl wool......19
 Saxon wool......22
 Lioneza wool.....25
 Alpacca wool.....26
 Farina of laurestinus   .  .   26
 Ryeland Merino wool   .  .   27
 Merino South Down    .  .   28
 Seed of lycopodium ...   32
 South Down ewe ....   39
 Coarse wool.....46
 Ditto from some worsted  .   60
2 
102
A  TREATISE ON OPTICS.        PART II.
  (79.)  By observing the colors produced by reflexion from
the fibres which compose the crystalline lenses of the eyes of
fishes and other animals, I have been able to trace these fibres
to their origin, and to determine the number of poles or septa
to which they are related.  The  same mode of observation,
and  the measurement of the  distance of  the  first colored
image from the white image, has enabled me to determine the
diameters of the fibres, and to prove that they all taper like
needles, diminishing gradually from  the equator to the poles
of the lens, so as to allow  them to pack into a spherical su-
perficies as  they converge to their poles or points of origin.
These colored images, produced by the fibres of the lens, lie
in a line perpendicular to the direction  of the fibres, and by
taking an impression on wax from an indurated lens the colors
are communicated to the wax.  In several lenses I observed
colored  images at a great distance from the common image,
but lying in a direction coincident with that of the fibres; and
from this I  inferred, that the fibres were crossed by joints or
lines, whose distance was so small as the ll,000dth part of an
inch ; and I  have lately found, by the use of very powerful
microscopes, that each fibre  has  in this case teeth like those
of a rack, of extreme minuteness, the  colors being produced
by the lines which form the  sides of each tooth.
  (80.)  In  the same  class of phenomena we must rank  the
principal colors of mother-of-pearl.   This substance, obtained
from the shell of the pearl oyster, has been long employed in
the arts, and the fine play of its colors is therefore well known.
In order to observe its colors, take a plate of regularly formed
mother-of-pearl, with  its surfaces nearly parallel, and grind
these surfaces upon a hone  or upon a plate of glass with the"
powder of schistus, till the  image of a candle reflected from
the surfaces is of a dull reddish-white color.  If we now place
the eye near the  plate, and look at this reflected image, C, we

                          Fig. 62.
shall see on one side of it a prismatic image, A, glowing with
all the colors of the rainbow, and forming indeed a spectrum
of the candle  as distinct as  if it had been formed by an equi-
lateral prism of flint glass.   The blue side of this image is 
CHAP. XIV.   COLORS  OF MOTHER-OF-PEARL.        103

next the image C, and the distance  of the red part of the
image is in one specimen 7° 22'; but this angle varies even
in the same specimen.   Upon first looking into the mother-of-
pearl, the  image A may be above or below C, or on any side
of it; but, by turning the specimen  round, it may be brought
either to  the right or left hand  of C.  The distance A C is
smallest when  the light of the candle falls nearly perpen-
dicular on the surface, and increases as the  inclination of the
incident ray is  increased.   In one  specimen it was 2° 7' at
nearly a perpendicular incidence, and  9° 14' at a very great
obliquity.
  On the outside  of the image A there  is invariably seen a
mass, M, of colored light, whose distance M C is nearly double
A C.   These three images are always nearly  in a straight
line, but the  angular distance of M varies  with the angle of
incidence  according to a  law different from  that of A.  At
great angles  of incidence the nebulous mass is of a beautiful
crimson color ;  at  an angle of about 37° it becomes green;
and nearer the perpendicular it becomes yellowish-white,  and
very luminous.
  If we now polish the surface  of the mother-of-pearl,  the
ordinary image  C will become brighter and  quite white, but a
second prismatic image, B, will start up on the other side of
C, and at  the same distance from it.
  This second  image  has in all other  respects the same pro-
perties  as the first.  Its brightness increases with the polish
of the surface,  till it is nearly equal to that of A, the lustre
of which  is  slightly impaired by  polishing.  This  second
image is  never accompanied, like  the first, with a nebulous
mass M.  If we remove the polish, the image B vanishes,  and
A resumes its brilliancy. The lustre of the nebulous mass M
is improved by polishing.
  If we repeat these experiments on the opposite  side of the
specimen, the very same phenomena will  be observed, with
this difference only, that the images A and M are on the op-
posite side of C.
  In looking through  the  mother-of-pearl,  when  ground  ex-
tremely thin, nearly the same  phenomena will be  observed.
The colors and the distances of the  images  are the same ; but
the nebulous mass M is never seen by transmission.  When
the second image,  B, is invisible by reflexion, it is exceedingly
bright when seen by transmission, and vice versa.
  In  making these experiments, I had  occasion to fix  the
mother-of-pearl to  a goniometer with a cement of  resin  and
bees'-wax; and  upon removing it, I was  surprised to see the
whole surface of the wax shining with the prismatic colors of 
104            A  TREATISE  ON  OPTICS.        PART II.

the mother-of-pearl.  I at first thought that a small film of the
substance had  been left  upon the wax; but this was soon
found to be a mistake, and it became manifest that the mother-
of-pearl really impressed upon  the  cement its own power of
producing the Colored spectra.   When the unpolished mother-
of-pearl was impressed  on the wax, the wax gave only one
image, A; and when the polished surface was used,  it gave
both A and B: but the nebulous image M was never exhibited
by the wax.  The images seen in the wax are always on the
opposite  side of C, from what they are in the surface that is
impressed upon it.
  The colors of mother-of-pearl, as communicated to a soft
surface, may be best seen by using black wax; but I have
transferred  them also  to balsam  of Tolu,  realgar, fusible
metal, and to clean surfaces of lead and tin by hard pressure,
or the blow of a hammer.  A solution of gum  arable or of
isinglass, when allowed  to indurate  upon a surface of mother-
of-pearl, takes a most perfect impression from it, and exhibits
all the communicable colors in the finest manner, when seen
either by reflexion or transmission.  By placing the isinglass
between two finely polished surfaces  of good  specimens of
mother-of-pearl, we shall obtain a film  of artificial mother-of-
pearl, which when seen by single lights, such as that of a
candle, or by an aperture in the window, will shine with the
brightest hues.
  If, in this experiment, we could  make the grooves of the
one surface of  mother-of-pearl exactly parallel to the grooves
in the other, as in the shell itself, the images, A and B, formed
by each surface, would  coincide,  and only two would be ob-
served by transmission and reflexion: but, as this cannot be
done, four images  are  seen through the  isinglass film, and
also four by reflexion; the two new ones being formed by re-
flexion from the second  surface of the film.
  From these experiments  it is obvious that the colors under
our consideration are produced by a particular  configuration
of surface, which, like  a seal, can  convey a reverse  impres-
sion of itself to any substance capable of receiving  it.  By
examining this  surface with  microscopes, I discovered  in
almost every specimen  a grooved structure, like the delicate
texture of the skin  at the top of an infant's finger, or like the
section of the annual growths of wood,  as seen upon a dressed
plank of fir. These may sometimes be  seen by the naked eye,
but they are often so minute that 3000 of them  are contained
in an inch.  The direction of the  grooves is always at right
angles to the line M A C B, fig. 62.; and hence  in irregularly
formed mother-of-pearl,  where the grooves are often circular, 
CHAP. XIV.   COLORS  OF MOTHER-OF-PEARL.        105

and having every possible direction, the colored images A, B
are irregularly scattered round the common image C.   If the
grooves were, accordingly, circular,  the series of prismatic
images, A B, would form a prismatic ring round C, provided
the grooves retained the same distance.  The general distance
of the grooves is from the 200th to the 5000th of an inch, and
the distance  of the prismatic images from C increases as the
grooves become  closer.   In a specimen with 2500 in an inch,
the distance A C  was  3° 41'; and in  a specimen of about
5000  it was about 7° 22'.
   These  grooves  are  obviously the sections  of all the  con-
centric strata of the shell.  When we use the actual surface
of any stratum, none of the colors A, B are seen, and  we ob-
serve only the mass of  nebulous light M occupying the place
of the principal image C.  Hence we see the  reason why the
pearl gives none of the images A, B, why it communicates
none of its colors to wax, and why it shines with that delicate
white light which  gives it all its value.  The pearl is formed
of concentric spherical  strata, round a central nucleus, which
Sir Everard Home conceives to be one of the  ova of the fish.
None of the  edges of its strata are visible, and as the strata
have parallel surfaces, the mass of light M is reflected exactly
like the  image  C, and occupies its place; whereas  in the
mother-of-pearl it is reflected from surfaces of the strata, in-
clined to  the general surface  of the specimen which reflects
the image C.  The mixture of all these diffuse masses of
nebulous light, of a pink and green hue, constitutes the beau-
tiful white of the pearls.  In bad pearls, where the colors are
too blue  or  too pink,  one or other of these  colors has  pre-
dominated.   If we make an oblique  section of a pearl, so  as
to exhibit a sufficient number  of concentric strata, with their
edges tolerably close, we should observe all the communicable
colors of mother-of-pearl.*
   These  phenomena may be observed  in many  other shells
besides that of the  pearl-oyster; and  in every case we may
distinguish communicable from incommunicable colors,  by
placing a film of fluid or cement between the surface and a
plate  of  glass.   The communicable colors will all disappear
from the  filling  up of the grooves, and the mcommunicable
colors will be rendered more brilliant.
   (81.) Mr. Herschel has discovered in very thin plates of
mother-of-pearl  another pair  of nebulous prismatic images,
more  distant from C than A and B, and also a pair of  fainter
nebulous  images, the line joining which is  always at right
* See Edinburgh Journal of Science, No. XII., p. 277. 
106
A  TREATISE ON OPTICS.
PART II.
angles to the line joining the first pair.*  These images are
seen by looking through a thin  piece  of mother-of-pearl, cut
parallel to the natural  surface of the shell,  and between the
70th and the 300dth of an inch thick.  They are much larger
than A and B;  and Mr. Herschel found that the line  joining
them was always perpendicular to a veined structure which
goes through its substance.  The distance of the red  part of
the image from C was found to  be 16° 29', and  the veins
which produced these colors were so small that  3700 of them
were contained in an  inch.  We have represented them in
fig. 63. as crossing the ordinary grooves which give the com-
municable colors.   Mr. Herschel  describes  them as crossing

                          Fig. 63.
these grooves at all angles, " giving the whole surface much
the appearance of a piece of twilled silk, or the larger waves
of the sea intersected with minute ripplings."  The second
pair of nebulous images seen by transmission must arise from
a veined structure exactly perpendicular to the  first, though
the structure has not yet been  recognized by the microscope.
The structure which  produces the lightest  pair Mr. Herschel
has found to be in all cases coincident with the plane passing
through the centres of the two systems of polarized rings.
  The principle  of the production of color by grooved sur-
faces, and of the  communicability of these  colors by pressure
to various substances, has been happily applied to the arts by
John Barton, Esq.  By means of a delicate engine, operating
by  a screw of the most accurate workmanship, he has suc-
ceeded in cutting grooves upon steel at the distance  of from
  * In a specimen now before us, the line joining the two faintest nebulous
images is at right angles to the line joining A and B. 
CHAP. XIV.  COLORS OF GROOVED SURFACES.        107
the 2000th to the 10,000th of an inch.  These  lines are cut
with the point of a diamond ;  and such is their  perfect paral-
lelism and the uniformity of their distance, that while in
mother-of-pearl we see  only one prismatic  image, A, on each
side of the common image, C, of the candle,  in the grooved
steel  surfaces 6, 7, or 8 prismatic images are  seen, consisting
of spectra, as perfect as those produced by the finest prisms.
Nothing in nature or in art can surpass this  brilliant display
of colors; and Mr. Barton conceived the idea  of forming but-
tons for  gentlemen's dress, and articles of female  ornament
covered  with grooves,  beautifully arranged in  patterns,  and
shining in the light of candles or lamps with all the hues of
the spectrum. To these he gave the appropriate name of Iris
ornaments.   In forming the buttons, the  patterns were drawn
on steel dies, and these, when duly hardened,  were used to
stamp their  impressions upon polished buttons  of brass.   In
day-light the colors on these buttons are not easily distinguish-
ed, unless when the surface reflects the margin of a dark ob-
ject seen against a light one; but in the light of the sun, and
that of gas-flame or candles, these colors are scarcely if at all
surpassed by the brilliant flashes of the diamond.
  The grooves thus made upon steel are, of course, all trans-
ferable to wax, isinglass, tin, lead, and other substances;  and
by indurating thin transparent films of isinglass  between two
of" these  grooved  surfaces, covered with lines lying in all di-
rections, we obtain a plate which produces by transmission
the most extraordinary display of prismatic spectra that has
ever been exhibited.
  (82.) In examining the phenomena produced by some of
the finest specimens of Mr. Barton's skill, which he had the
kindness to execute for this  purpose, I have been led to the
observation of several curious properties of light.  In mother-
of-pearl, well polished, the central image, C, of  the  candle or
luminous object is always white, as we should expect it to be,
in consequence of being reflected from the flat and polished
surfaces between the  grooves.   In like  manner,  in many
specimens  of grooved  steel  the image C is also  perfectly
white, and the spectra on each side of it, to the  amount of six
or eight, are perfect  prismatic  images of the candle; the
image A, which is nearest C, being the least dispersed, and
all the rest in succession more and more dispersed, as  if they
were  formed  by prisms of greater  and  greater  dispersive
powers, or greater and greater refracting angles. These spec-
tra contain the fixed lines and all the prismatic colors; but the
red or least refrangible  spaces are greatly expanded, and the 
108
A  TREATISE ON OPTICS.        PART II.
violet or most refrangible spaces greatly contracted, even more
than in the spectra produced by sulphuric acid.
  In examining some of these prismatic images which seemed
to be defective in particular rays, I was surprised to find that,
in the specimens which produced them, the image C reflected
from the polished original surface  of the steel  was  itself
slightly colored; that its  tint varied with  the  angle of inci-
dence, and had some relation to the defalcation  of color in the
prismatic images.  In  order to observe  these phenomena
through a great range of incidence, I substituted for the can-
dle  a long narrow rectangular  aperture,  formed by nearly
closing  the window-shutters, and I then saw at one view  the
state of the ordinary image and all the prismatic images.  In
order  to understand  this,  let A B, Jig. 64., be  the  ordinary

                          Fig. 64.
	aE. < —ltrr	1,1 a o a id ,.,a„,t,q. n+i nor
	;' ™	,11 ! i Tt; r y "i I ■ : ""! 1*1
m		! 1 : i LI i \ \ 1 \ «
\	V	-vnAx^vliiiiiArix. i"M_J
_Jir	-ii-rL	
                 b" b'  S  b B b  6  V ft
image of the aperture reflected from the flat surface of the
steel which lies between the grooves, and a b, a' b', a" b", &c,
the prismatic images on each side of it, every one of these
images  fcrming a complete  spectrum  with all  its different
colors.   The image A B was  crossed in a direction perpen-
dicular  to its length with broad colored fringes, varying in 
CHAP. XIV.  COLORS  OP GROOVED  SURFACES.       109

their tints from 0° to 90° of incidenca  In a specimen with
1000 grooves in an inch, the  following were the colors dis-
tinctly seen at different angles of incidence:—
Blue - - - -	Angle of incidence. - 56° 0'	
Bluish green	-	54 30
Yellowish green	.	53 15
Whitish green -	.	51 0
Whitish yellow -	.	49 0
Yellow - - .	.	47 15
Pinkish yellow -	.	41 0
Pink red - - -	.	36 0
Whitish pink	.	31 0
Green - - -	-	24 0
Yellow - - -	.	10 0
; Reddish - - -	-	0 0
  These colors are those of the reflected rings in thin plates.
If we turn the steel plate  round in azimuth, the very same
colors appear at the same angle of incidence, and they suffer
no change either by varying the distance of the  steel plate
from the luminous aperture, or the distance of the eye of the
observer from the grooves.
  In the preceding  table there are  four orders of colors; but
in some specimens  there are only  three, in others two, in
others one, and in some only one or two tints of the first order
are developed.   A specimen of 500 grooves in an inch gave
only the yellow of the first order through the whole quadrant
of incidence.  A specimen of 1000 grooves gave only one
complete order, with a portion of the next.   A specimen of
3333 grooves gave only the yellow of the first order.   A spe-
cimen of 5000 gave a little more than one order; and a spe-
cimen of 10,000 grooves  in an  inch gave also a little more
than one order.
  In fig. 64. we have represented the portion of the quadrant
of incidence from about 22° to  76°.  In the first spectrum,
a b a b,  v v is the violet side of it, and r r the red side of it,
and between these are arranged all the other colors.   At m,
at an incidence of 74°, the violet light is obliterated from the
spectrum a b; and at n, at an incidence of  66°, the red  rays
are obliterated; the intermediate colors, blue, green, &c, being
obliterated at intermediate  points between m and n.   In the
second  spectrum, a' b' a' b', the  violet rays are obliterated at
m' at an incidence of 66° 20', and the red at n'  at an  inci-
dence of 56°.  In the  third spectrum, a" b" a" b", the violet
rays are obliterated at m"  at 57°, and the  red at n" at 41°
                            K
White    -  ---   90° 0'
Yellow   ---  -   80 30
Reddish orange  -  -   77 30
Pink.....76 20
Junction of pink  and ) -,,- .^
  blue    -  -  -   \
Brilliant  blue    -  -   74 30
Whitish   ....   71  0
Yellow   -  -  -  -   64 45
Pink.....59 45
Junction of pink  and > ^ ■.«
  blue    -  -  -   <0S iU 
110
A  TREATISE  ON  OPTICS.
PART II.
35'; and  in the fourth  spectrum, the violet  rays  are  oblit-
erated at m'" at 48°, and the red at »'" at 23° 30".  A simi-
lar succession of obliterated  tints takes place  on all the pris-
matic images at a lesser incidence, as shown  at v-v, ^V; the
violet being obliterated at p. and /*', and the red at v and v', and
the  intermediate colors  at intermediate  points.   In  this
second  succession the line pv begins and ends  at  the  same
angle of incidence as  the line m" n" in the third  prismatic
image a" b", and the  line p' v' in the second prismatic image
corresponds with m'" n'" on  the  fourth prismatic image.  In
all these cases, the tints obliterated in the direction mn jiv,
&c, would, if restored,  form a complete prismatic spectrum
whose length is mn pv,  &c.
   Considering the ordinary image as white, a similar oblitera-
tion of tints takes place upon it.   The violet is obliterated at
o about  76°, leaving pink, or what the violet wants of white
light; and the red is obliterated at p at 74°, leaving a bright
blue.  The violet is obliterated at q and s,  and the red at r
and t, as may be inferred from the preceding  Table of colors.
   The analysis of these  curious and apparently complicated
phenomena becomes very simple when they are examined by
homogeneous light.  The effect produced  on red light is re-
presented in fig. 65., where  A B is the image  of the narrow
                     aperture  reflected  from the original
                     surface of the steel, and the four images
                     on each side of it correspond  with the
                     prismatic  images.  All  these  nine
                     images, however, consist of homogene-
                     ous red light, which is  obliterated, or
                     nearly so, at the fifteen  shaded rectan-
                     gles, which are the minima of the new
                     series  of periodical colors which cross
                     both the ordinary and the lateral images.
                     The centres p, r, t,  n, v, &c., of these
                     rectangles correspond  with the points
                     marked with the same  letters  in fig.
                     64.; and if we  had drawn  the same
                     figure  for violet  light, the centres of
                     the rectangles  would have  been all
                     higher up  in the  figure,  and would
                     have corresponded with  o,  q, s, m, n,
                     •fee. in  fig. 64.   The rectangles should
                     have been shaded off to represent the
                     phenomena  accurately,  but  the  only
                     object  of  the  figure  is  to  show to
                     the eye  the position and  relations of
                     the minima.
Fig. 65.
  A.
\P 
CHAP. XV.          THEORY OF FITS.                Ill

  If we cover the surface of the grooved steel with a fluid,
so as to diminish the refractive power of the surface, we de-
velope  more orders  of colors on the ordinary image,  and a
greater number of minima on the lateral images, higher  tints
being produced at a given incidence.   But, what  is very re-
markable,  in  grooved surfaces when  the ordinary image is
perfectly white, and when the spectra are complete without
any obliteration of  tints, the application of fluids  to the
grooved surface developes  colors on the ordinary image, and a
'corresponding obliteration of tints on the lateral images.  The
following Table contains a few of the  results relative to the
ordinary image:—
Number of grooves in	Maximum tint without a fluid.	Maximum tint with fluids.
312. 3333 L	Perfectly white. ( Gamboge yellow I of the first order.	( 1. Water, tinge of yellow. < 2. Alcohol, tinge of yellow. ( 3. Oil of cassia, faint reddish yellow. f 1. Water, pinkish red (first order). J 2. Alcohol, reddish pink. 1 3. Oil of cassia, bright blue (second l_ order).
  Phenomena analogous to those above described  take place
upon the  grooved surfaces of gold, silver, and  calcareous
spar; and upon the surfaces of tin, isinglass, realgar, &c,
to which the grooves have  been transferred from steel.   For
an account of the phenomena exhibited by several of these
substances, I must refer the reader to the original memoir in
the Philosophical Transactions for 1829.
                       CHAP. XV.

  ON  FITS OF REFLEXION AND  TRANSMISSION, AND ON THE
                 INTERFERENCE OF LIGHT.

  (83.) In the preceding chapters we have described a very
extensive class of phenomena, all of which  seem to have the
same  origin.   From his experiments on the colors of thin and
of thick plates, Newton inferred that they were produced by
a sino-ular property of the particles  of light, in  virtue of
which they possess, at different points of their path, fits or
dispositions to be reflected from or transmitted by transparent
bodies.  Sir Isaac does  not pretend* to explain  the  origin of
these  fits, or the cause which produces  them; but we may
form a tolerable idea of them by supposing that each particle 
112             A TREATISE ON OPTICS.        PART II.
of light, after its  discharge  from a luminous body, revolves
round an axis perpendicular to the direction of its motion, and
presenting alternately to the  line of its motion an attractive
and a repulsive pole, in virtue of which it will be refracted if
the attractive pole is nearest any refracting surface on which
it falls, and reflected if the repulsive pole is nearest that sur-
face.  The disposition  to be  refracted and  reflected will  of
course increase and diminish as the  distance of either pole
from the  surface of the body is increased or diminished.  A
less scientific idea may be formed  of this hypothesis, by sup-«
posing a body with a sharp and a blunt end passing through
space, and successively presenting its sharp and blunt ends to
the line of its motion.   When the sharp end encounters any
soft body  put in its way, it will  penetrate it; but when the
blunt end  encounters  the  same body,  it will be  reflected  or
driven back.
  To explain this more clearly,  let R,j%. 66.,  be a ray  of
light falling  upon  a refracting surface M N, and transmitted
                  by that  surface.  It is clear  that it must
                  have met  the  surface M N when  it was
                  nearer its fit of transmission than its fit  of
                  reflexion;  but whether it was exactly at its
                  fit of transmission, or a little from it, it ia
                  put, by the action of the  surface, into the
                  same state as if it had begun its fit of trans-
                  mission at t.  Let us suppose that, after it
                  has moved  through  a space  equal to t r, its
                  fit of reflexion takes place, the fit of trans-
      t"Ly       mission always recommencing at 11', &c.
and that of reflexion at r r', &c.; then it is obvious, that  if
the ray  meets a second transparent surface at t V, &c, it will
be transmitted, and if it meets it at r r', Sic, it will  be reflected.
The spaces t V, t' t" axe called  the intervals of the  fits of
transmission, and  r r',  r' r" the  intervals  of the fits of re-
flexion.   Now, as  the spaces 11', rr\ Sic. are supposed equal
for light of the same colors, it is manifest that, if M N be the
first surface  of a  body, the  ray  will  be transmitted  if the
thickness  of the body is t V, 11'^ Sec.;  that is, tt',2t V, 3 t V,
4111, or any  multiple whatever of the interval of a fit of easy
transmission.   In like manner the ray will be reflected if the
thickness  of  the body is t r, t r'; or, since 11' is equal to rr',
if the thickness of the body is 4 t V, 14 t V, 2)- t V,  Z\ t V.
If the body M N,  therefore, had  parallel "surfaces, and  if the
eye were  placed above it so as to receive the rays reflected
perpendicularly, it would, in every case, see the surface M N
by the portion of light  uniformly reflected from that surface;
 
CHAP. XV.           THEORY OP PITS.               113
but when the thickness of the body was tt',2 t V, 3 11', 4 t V,
or 1000 t V, the eye would receive no rays from the second
surface, because they are all transmitted; and in like manner,
if the thickness was \ 11, \\ I V, 2\ 11', or  1000| t V, the eye
would receive all the light reflected  from the second surface,
because  it is all reflected. • When this reflected light meets
the  first  surface M N, on its  way to the eye, it is all trans-
mitted, because it is then in its fit of transmission.   Hence,
in the first case, the eye receives no light from the second
surface,  and in the second case, it receives all  the light from
the second surface.  If the body had intermediate thicknesses
between t V and  2 t V, Sic, as £ t V,  then a portion of the
light would be reflected from  the second surface, increasing
as the thickness increased from t V  to \\ 11', and diminishing
again as the thickness increased from 1^ t V to 2 ft'.
   But let us now suppose that the plate whose surface is M N
is unequally thick, like the plate  of air  between the  two
lenses, or a film of blown glass.  Let it have  its thickness
varying like a wedge M N F,fig. 67.  Let 11', rr' be the in-
tervals of the fits, and  let the eye be placed above the wedge
as before.  It  is quite clear that near the point N the light
that falls upon the second surface N P will be all transmitted, as
it is in a fit of transmission; but at the thickness t r the light
R will  be reflected by the  second surface, because it is  then
in its fit of«reflexion.   In like  manner the light will be trans-
mitted at t', again reflected at  r', and again transmitted at t";
so that  the  eye above M N will see a series of dark  and
luminous bands, the middle of the dark ones  being at N, t', t"
in the line N P, and of-the  luminous ones at r, r', &c. in the
same line.  Let us suppose that the figure is suited  to red
homogeneous light, 11' being the interval  of a fit for that
species of rays;  then in violet light, V, the interval of the fits
will  be less, as rp.  If we therefore  use violet light, the in-
                           K2 
 114            A  TREATISE  ON OPTICS.        PART II.
 terval of whose  fits is rp, a smaller series of violet and 06-
 scure bands or fringes will be seen, whose obscurest points
 are at N, ¥, t", &c, and whose brightest  points are at p, 'p,
 &e.  In  like manner, with  the  intermediate  colors of  the
 spectrum, bands  of intermediate magnitudes will be formed,
 having their obscurest points between t' and V, t" and t", and
 their brightest points  between p  and r, p  and r',  Sic.; and
 when white light is used, all these  differently colored bands
 will be seen forming fringes of'the different orders of colors
 given in  the Table in  page 93.  If M N P, in place of being
 the section of a prism, were the section  of one half of a
 plano-concave lens, whose centre is N, and whose concave sur-
 face has an oblique direction somewhat like N P, the direction
 of  the colored bands  will  always be perpendicular  to the
 radius N M, or will be regular circles.  For the same  reason,
 the colored bands are  circular in the concave lens of air be-
 tween the object glasses; the same colors always appearing
 at the same thickness of the medium, or at the same distance
 from the centre.
  By the  same means  Sir Isaac Newton explained the colora
 of thick plates, with this difference, that the fringes are not
 in that case produced by the light regularly refracted and re-
 flected at  the two surfaces of the concave mirror, but by the
 light irregularly  scattered by the first surface of the mirror
 in consequence of its  imperfect polish; for, as he  observes,
 " there is  no glass or speculum, how well soever polished, but,
 besides  the light which it  refracts and  reflects  regularly,
 scatters every  way irregularly a faint  light,  by means  of
 whicii the polished surface, when  illuminated in a dark room
 by a beam of the  sun's light,  may be easily seen in all posi-
 tions of the eye."
  The same theory of fits  affords  a ready explanation of the
 phenomena of double and equally thick plates, which we have
 described  in another chapter.  There are  other phenomena
 of colors,  however, to which it is not  equally applicable ; and
 it has accordingly been, in  a  great measure, superseded by
 the doctrine of interference, which  we  shall now proceed
 to explain.
  (83.) In examining the black and white stripes within  the
 shadows of bodies as formed  by inflexion, Dr. Young found
 that when he placed an opaque screen either a few inches be-
 fore or a few inches behind one side of the inflecting body, B,
fig. 56., so as to intercept all the light on that side by receiv-
ing the edge of the shadow on the screen, then all the fringes
in the shadow constantly disappeared,  although the light still
passed by the other  edge of the body as before.  Hence he 
CHAP. XV.    ON  THE INTERFERENCE  OP LIGHT.    115

concluded that the light which passed on both sides was ne-
cessary to the production of the fringes;  a conclusion which
he might have deduced also from the known fact, that when
the body was above a certain size, fringes never appeared in
its  shadow.  In reasoning upon  this conclusion,  Dr. Young
was led to  the opinion, that the fringes within the shadow
were produced by the interference of^the rays  bent into the
shadow by one side of the body B with the rays bent into the
shadow by the other side.
  In order to explain the law of interference indicated in this
experiment, let us suppose two pencils of light to radiate from
two points very close to each other, and that this light falls
upon  the same spot of a piece of paper held parallel to the
line joining the points, so that the spot is directly opposite the
point  which bisects the distance between the two radiant
points.   In this case they may be said to interfere with one
another; because the pencils would  cross one another at that
spot if the paper were removed, and would diverge from one
another.  The spot will, therefore,  be  illuminated  with the
sum of their lights; and in this case the  length of the paths
of the two pencils of light is exactly the same, the spot on
the paper being equally distant from both the radiant points.
Now, it has been found that when there is a certain minute
difference between the lengths of the paths of the two pencils
of light, the spot upon the paper where the two lights inter-
fere is still a bright spot illuminated by the sum  of the two
lights.  If we call this difference in the lengths of their paths
d, bright spots will be formed  by the interference  of the two
pencils when the  differences in the lengths of the paths are
d, 2 d, 3 d, 4 d, Sic  All this  is nothing  more than what is
consistent with daily observation; but, what is truly remark-
able and altogether unexpected, it has  been clearly demon-
strated that if the two pencils interfere at intermediate points,
or when the difference in the lengths of the paths of the two
pencils is | d, 1^ d, 2A d, 3^- d, Sic instead  of adding  to one
another's intensity, and producing an illumination equal to the
sum of their  lights, they destroy each  other, and produce a
dark spot.   This curious property is analogous to the beating
of two musical sounds nearly in unison with each other; the
beats taking place when the  effect of the two sounds is equal
to the sum ef their separate intensities, corresponding to the
luminous spots or fringes where the effect of the two lights is
equal to the sum of their separate intensities, and the cessa-
tion of sound between the beats when the two sounds destroy
each other,  corresponding to the dark spots or fringes where
the two lights produce darkness. 
116             A  TREATISE OX  OPTICS.        PART II.

  By the aid of this doctrine the phenomena of the inflexion
of light, and those of thin  and thick plates, may be well ex-
plained.   With regard to the interior fringes, or  those in the
shadow, it is clear that as the middle of the shadow is equally
distant from the edges of the inflecting body B, fig. 56., there
will be no difference in the length of the paths of the pencils
coming from each side  of the body, and consequently along
the middle of the whole  length of every narrow shadow there
should be a white stripe  illuminated with the sum of the two
inflected pencils;  but at a  point at such a distance from the
centre of the shadow that the difference  of the two paths of
the pencil from each side of the body is equal to \ d, the two
pencils will destroy each other, and give a dark stripe. Hence
there will be a dark stripe on each side of the central bright
one.  In like manner it may be shown, that at a point at such
a distance from the centre of the shadow that  the difference
in the lengths of the paths is 2 d, 3 d, there will  be bright
stripes; and  at intermediate points, where the difference in
the  lengths  of the  paths is 1^ d, 2\ d,  there  will be  dark
stripes.*
  In order to explain the origin of the  external fringes, both
Dr. Young and M. Fresnel ascribed them to the interference
of the direct rays with other rays reflected from the margin
of the inflecting body;  but M. Fresnel  has found that the
fringes  exist when no such reflexion can take place; and he
has, besides, shown the  insufficiency of the explanation, even
if such  reflected rays did exist.  He therefore ascribes the ex-
ternal fringes to the interference of the direct rays with other
rays which pass at a sensible distance from the inflecting body,
and which are made to deviate from their primitive direction.
That such rays do exist,  he proves  upon the undulatory theory,
which we shall afterwards explain.
  The phenomena of thin plates  are admirably explained by
the  doctrine of interference.   The light  reflected from the
second surface  of the plate interferes with the light reflected
from the first, and as these two pencils of light come from dif-
ferent points of space, they must reach the eye with different
lengths of paths. Hence they will, by their interference, form
luminous fringes when  the difference of the paths is d, 2 d,
3 d, Sic, and obscure fringes when that difference is 4 d, U d,
2±d,3±d,&DC
  "In accounting for the colors  of thick  plates observed by
Newton, the light scattered irregularly from  every point of
the  first surface of the  concave mirror falls diverging on the

     *See Note No. V., by Am. ed., following the author's Appendix. 
CHAP. XV.   ON  THE INTERFERENCE  OF LIGHT.     117

second surface, and being reflected from this surface  in  lines
diverging from a point behind, they will suffer refraction in
coming out of the first surface of the mirror, being made to
diverge as if from a point still nearer the mirror, but behind
its surface.   From this last point, therefore, the screen M N,
in fig. 60., is illuminated by the  rays originally scattered on
entering the first surface.  But when the regularly reflected
light, after reflexion  from the second surface, emerges from
the first, it will  be scattered  irregularly from each  point on
that surface, and radiating from  these points will illuminate
the paper screen  M N.  Every point, therefore, in the paper
screen is illuminated by two kinds of scattered light, the one
radiating from each  point of the first surface, and the other
from points behind the second surface; and hence bright and
obscure  bands will  be formed when the differences  of the
lengths of their  paths are such as have been already  de-
scribed.
   The colors of two  equally thick and inclined plates are also
explicable by the law of interference.  Although the light re-
flected by the different surfaces of the plate  emerges parallel
as shown in fig. 61., yet in consequence of the inclination of
the  plates it reaches  the eye by paths of different lengths.
   The colors of  fine fibres, of minute particles, of mottled and
 striated surfaces, and of equidistant parallel lines, may be all
 referred  to  the   interference  of different portions of light
 reaching the eye by paths of different lengths; and though
 some difficulties  still exist in the application of the doctrine to
 particular phenomena that have not been sufficiently studied,
 yet there can be no doubt that these difficulties will be re-
 moved by closer investigation.
   As all the phenomena of interference are dependent upon
 the quantity d, it becomes interesting  to ascertain its exact
 magnitude for the differently colored rays, and, if possible, to
 trace its  origin to  some primary  cause.  It is  obvious, as
 Fraunhofer  has remarked, that this quantity  d is a real abso-
 lute magnitude, and whatever meaning we may attach to it,
 it is demonstrable that one half of it, in reference to the phe-
 nomena produced by it, is opposed in its properties to the other
 half; so that if the anterior half combines accurately with
 the posterior half, or interferes with it in this manner under a
 small angle, the effect which would have been produced by
 each separately is destroyed,  whereas  the same effect  is
 doubled  if two  anterior or two posterior halves of this mag-
 nitude combine  or interfere in a similar manner.
    (84.) In  the  Newtonian theory of light,  or the  theory of
 emission, as it is called,  in which light  is supposed to consist 
118
A  TREATISE  ON  OPTICS.        PART II.
of material particles emitted by luminous bodies, and moving
through space with a velocity of 192,000 miles  in a second,
the quantity d is  double the interval of the fits of easy re-
flexion and transmission ; while in the undulatory theory it is
equal to the breadth of an undulation or wave of light.
  In  the  undulatory theory, an exceedingly thin and elastic
medium, called ether,  is supposed to fill all space, and to oc-
cupy the intervals between the particles of all material bodies.
The ether must be so extremely rare as to present no appre-
ciable resistance to the planetary bodies which move freely
through it.
  The particles of this ether are, like those of air, capable of
being put into vibrations by the agitation of the particles of
matter, so that waves or vibrations can be propagated through
it in all directions.  Within refracting media it is less elastic
than in vacuo, and its  elasticity is less in proportion to the re-
fractive power of the body.
  When any vibrations or undulations are propagated through
this ether, and reach the nerves of the retina, they excite the
sensation  of light, in  the same  manner  as the sensation of
sound is excited in the nerves of the ear by the vibrations of
the air.
  Differences of color are  supposed to arise from differences
in  the frequency of the etherial undulations; red being pro-
duced by a much  smaller number of undulations in  a given
time than blue,  and intermediate colors by  intermediate num-
bers of undulations.
  Each of these two theories of light is beset with difficulties
peculiar to  itself; but the theory of  undulations has  made
great progress  in modern times, and derives such  powerful
support from an extensive class of phenomena, that it has been
received by many of our most distinguished philosophers.
   In a work like this it would be in vain  to attempt to give
a particular account of the principles of  this theory.  It may
be  sufficient at present to state,  that the doctrine of inter-
ference is in complete accordance with the theory of undula-
tion.   When similar waves are combined,  so that the eleva-
tions and depressions of the one  coincide wTith those of the
other, a wave of double magnitude will be produced; whereas,
when the elevations of the one coincide with the depressions
of  the other, both, systems of waves will be totally destroyed.
" The spring and neap tides," says Dr. Young, " derived  from
the combination of the  simple soli-lunar  tides,  afford a mag-
nificent  example  of the interference of  two immense waves
with each other;  the spring tide being the joint result of the
combination when they  coincide in time and place, and the 
CHAP. XV.       UNDULATORY THEORY.
119
neap  tide where they succeed each other at the distance of
half an interval, so as to leave the effect of their difference
only sensible.   The tides of the port of Batsha, described and
explained by Halley and Newton, exhibit a different modifica-
tion of the  same opposition of undulations;  the ordinary pe-
riods  of high  and low water being altogether superseded  on
account of the different lengths of the two channels by which
the tides arrive, affording  exactly the half interval which
causes the  disappearance of the alternation.   It may also be
very easily observed, by merely throwing two equal stones
into a piece of stagnant water, that the circles of waves which
they occasion  obliterate each  other, and leave the  surface of
the water smooth in  certain lines of a hyperbolic form, while
in other  neighboring parts the surface  exhibits the agitation
belonging to both series united."
  The following Table given by Mr. Herschel contains  the
principal data  of the undulatory theory:—
Colors of the Spectrum.	Lengths of an Un-dulation in parts of an Inch in air.	Number ol Undulations	Number of Undulations in s Second.*
Extreme red . Red . . . Intermediate . Orange . . Intermediate . Yellow . . Intermediate . Green Intermediate . Blue . . . Intermediate . Indigo . . Intermediate . Violet . . Extreme violet	0-0000266 0-0000256 0-0000246 0-0000240 0-0000235 0-0000227 • 0-0000219 0-0000211 0-0000203 0-0000196 0-0000189 0-0000185 0-0000181 0-0000174 0-0000167	37640 39180 40720 41610 42510 44000 45600 47460 49320 51110 52910 54070 55240 57490 59750	458,000000,000000 477,000000,000000 495,000000,000000 506,000000,000000 517,000000,000000 535,000000,000000 555,000000,000000 577,000000,000000 600,000000,000000 622,000000,000000 644,000000,000000 658,000000,000000 672,000000,000000 699,000000,000000 727,000000,000000
   " From this Table,"  says Mr. Herschel, " we see that the
sensibility of the eye is confined within much narrower limits
than that of the ear; the ratio of the extreme vibrations being
nearly 1-58:1, and therefore less than an  octave, and about
equal to a minor sixth.  That man should be able to measure
with certainty such minute portions  of space and time, is not
a little wonderful; for  it may be observed, whatever theory
of light we adopt, these periods and these spaces have a real
existence, being in  fact deduced by Newton from direct  mea-
surements, and involving nothing hypothetical but the names
here given them."

      ♦Taking the velocity of light at 192,000 miles per second. 
120
A  TREATISE  ON  OPTICS.        PART II.
                      CHAP.  XVI.

              ON THE ABSORPTION OF LIGHT.

  (85.) One of the most curious properties of bodies in their
action  upon light, and one  which  we  are persuaded will yet
perform a most important part  in the explanation of optical
phenomena, and  become a ready instrument in optical re-
searches, is their  power  of absorbing light.   Even the most
transparent bodies innature, air and water, when in sufficient
thickness, are capable of absorbing  a great quantity of light.
On the summit of the highest mountains, where their light
has to  pass through a much less extent of air, a much greater
number of stars is visible to the eye than in the plains below;
and through great depths of water objects become almost in-
visible.  The  absorptive power of air is finely  displayed in
the color  of the morning and  evening clouds;  and that of
water in the red  color of the meridian sun, when seen from
a diving-bell at a great depth in the sea.  In both these cases,
one class of  rays is absorbed more readily than another in
passing through the absorbing  medium, while the rest make
their way in  the one case to the clouds, and in the other to
the eye.
  Nature presents us with bodies of all degrees of absorptive
power, as shown in the following brief enumeration:—
Charcoal.
Coal of all kinds.
Metals in general.
Silver.
Gold.
Black hornblende.
Black pleonaste.
Obsidian.
Rock crystal.
Selenite.
Glass.
Mica.
Water and transparent fluids.
Air and gases.
  Although charcoal is the most  absorptive of all bodies, yet,
when it exists in a minutely divided  state, as in some  of the
gases and flames, or in a particular state of aggregation, as in
the diamond, it is highly transparent.  In like manner, all
metals are transparent in a state of solution; and even silver
and gold, when  beaten into thin films, are translucent, the
former transmitting a beautiful blue, and the latter a beautiful
green light.*

  * See  Note No. VI. of Am. ed., in  the notes following the author'!
Appendix. 
CnAP. XVI.      ABSORPTION OF LIGHT.
121
  Philosophers have  not  yet ascertained  the nature of the
power by  which  bodies  absorb  light.  Some have  thought
that the particles of light are reflected in all directions by the
particles of the absorbing body, or turned aside by the  forces
resident in  the particles;  while others are  of opinion that
they are detained by the body,  and assimilated to  its sub-
stance.   If the  particles of light  were reflected  or merely
turned out of their direction by the action  of  the particles, it
seems to  be quite demonstrable  that  a portion  of the most
opaque matter, such  as charcoal, would, when exposed to a
strong beam of light, become actually phosphorescent during
its illumination, or would at least appear white ; but as all the
light which enters it is never  again visible, we  must believe,
till we have evidence of the  contrary, that the light is actu-
ally stopped by the particles of the body, and  remains within
it in the form of imponderable matter.
  Some idea may be formed of the law according to  which a
body absorbs light, by supposing  it to consist of a given num-
ber of equally thin plates, at the  refracting surfaces of  which
there is no light lost by reflexion.  If the  first  plate has the
power of absorbing TlTth of the  light which enters it, or 100
rays out of 1000;  then j^ths of the original light, or 900 rays,
will fall upon the second plate;  and y^th  of these, or 90, be-
ing absorbed, 810 will fall upon the third plate, and so on.
Hence it is obvious that the quantity of light transmitted by
any number of films is equal to the light transmitted through
one film multiplied as often  into  itself as there are films
Thus, since 900 out of 1000 rays are transmitted by one film
nr x ttt x To equal to  T^9o, or 729  rays, will be  the  quantity
transmitted by three films;  and  therefore the  quantity absorb-
ed  will be  271 rays.  Of the  various bodies  which  absorb
light copiously, there are few  that absorb all the colored rays
of the spectrum in equal proportions.   While certain  clouds
absorb the  blue rays  and transmit the red, there are  others
that absorb all the rays in equal  proportions, and exhibit the
sun and the moon when seen through them perfectly  white.
Ink diluted is a fine example of a fluid which  absorbs all the
colored rays in  equal proportions; and it  has on this account
been applied by  Sir  William Herschel as a  darkening sub-
stance for obtaining a  white image  of the  sun.,  Black  pleon-
aste and obsidian afford  examples of solid substances  which
absorb all the colors of the spectrum proportionally.
  (86.)  All  colored  transparent  bodies,  however,  whether
solid or fluid, do not necessarily absorb the colors proportion-
ally ; for it is only in consequence of an  unequal absorption
that they could appear colored by transmitted light.   In order 
122
A  TREATISE  ON  OPTICS.        PART II.
to exhibit this absorptive power, take a thick piece of the blue
glass that is used for finger  glasses, and which is sometimes
met  with in cylindrical rods of about  f^ths  of  an inch in
diameter, and shape it into  the form of a wedge.   Form a
prismatic image of the candle, or, what is better, of a narrow
rectangular aperture in the window  by a prism, and examine
this  prismatic image  through  the  wedge of colored  glass.
Through the thinnest edge the spectrum will be seen nearly
as complete as before the interposition of the wedge; but as we
look at it through greater and greater thicknesses, we  shall
see particular parts or colors of the  spectrum become fainter
and fainter, and gradually disappear, while others suffer  but a
slight diminution of their brightness.  When the thickness is
about the twentieth part of  an inch, the spectrum will  have
the appearance shown in fig. 68., where the middle R of the
red space is  entirely absorbed, the  inner red that is left is
weakened  in  intensity;  the  orange is entirely absorbed; the
yellow Y is left almost insulated; the green G on the side of
         Fi„ 68          the  yellow is very much absorbed;
  ^^_^  "'   ________  and a  slight absorption takes  place
  nil j| lljilj             along the green and blue space. At
  1 111 111 llllllli           I  a  greater  thickness still, the  inner
   R Y~  G         V  .  re(j diminishes rapidly, and also the
yellow, green, and  blue; till,  at a  certain thickness, all the
middle colors of the spectrum  are absorbed, and nothing left
but the  two  extreme  colors, the red R and the violet  V, as
shown in fig. 69.  As the red  light R has much greater in-
          mg.  69          tensity than  the violet,  the  glass
                         has at this thickness the appearance
                         of being  a  red glass;  whereas at
                       „ small thicknesses it had the appear-
  "-                v     ance of being a blue glass.
  Other colored media, instead of absorbing the spectrum in
the middle, attack it, some at one extremity, some at another,
and others at both.   Red glasses, for example, absorb the blue
and violet with  great  force.   A thin plate of native yellow
orpiment  absorbs the  violet  and refrangible blue  rays very
powerfully, and leaves the  red, yellow, and green but little
affected.   Sulphate of copper attacks  both ends of the  spec-
trum at  once, absorbing the red and violet rays with  great
avidity.  In consequence of  these different powers of absorp-
tion, a very  remarkable phenomenon may be exhibited.  If
we  look  through the blue glass so  as to see the spectrum in
fig.  69.,  and  then look at  this  spectrum again  with a thin
plate of  sulphate of copper,  which  absorbs the extreme rays
at R and V, the two substances thus  combined will be  abso; 
CHAP. XVI.       ABSORPTION OP LIGHT.            123

lutely opaque, and not a ray of light will reach the eye. The
effect is perhaps more striking  if we look at a bright white
object through the two media together.
  (87.) In attempting  to ascertain the influence of heat on
the absorbing  power of colored  media, I was surprised to ob-
serve that it produced opposite effects upon different glasses,
diminishing the absorbing  power in some and increasing it
in others.  Having brought to a red heat a piece of  purple
glass, that absorbed the greater part of the green, the yellow,
and  the interior or most refrangible red,  I held it before a
strong light;  and when its red heat had  disappeared, I  ob-
served that the transparency of the glass was increased, and
that it transmitted  freely  the  green,  the yellow, and  the
interior red, all of which it had  formerly, in a great measure,
absorbed.  This effect, however, gradually disappeared, and
it recovered  its former  absorbent  power,  when completely
cold.
  When yellowish-green glass was heated in a similar man-
ner, it lost its  transparency almost entirely.  In recovering its
green color, it passed through various shades of olive green;
but its tint, when cold,  continued less green than it was  he-
fore the experiment.  A part of the glass had received in
cooling a polarizing  structure, and  this part could be easily
distinguished from the other part by a difference of  tint.
  A plate of deep red glass, which gave a homogeneous  red
image of the  candle, became very opaque when heated,  and
scarcely transmitted the light of the candle after its red heat
had  subsided.   It recovered,  however,  its transparency to a
certain degree;  but when cold, it was more opaque than  the
piece from which it was broken.  I have observed  analogous
phenomena in mineral bodies.  Certain specimens of topaz
have their absorbing power permanently changed by heat.
In subjecting the Balas ruby to high degrees of heat, I  ob-
served that its red color changed into green, which gradually
faded  into brown as the cooling advanced, and resumed by
degrees its original red color.   In  like manner, M. Berzelius
observed  the  spinelle to become brown  by heat, then to grow
opaque as the  heat increased,  and to pass through a fine olive
green before it recovered its red color.  A remarkable change
of absorbent power is exhibited  by heating very considerably,
but so as not to inflame it, a plate of yellow native orpiment,
which absorbs the violet and blue rays..  The heat  renders it
almost  blood  red, in consequence  of" its now absorbing  the
greater part of the  green and  yellow  rays.  It resumes its
former  color,  however,  by cooling.  A still more striking
effect may be produced with pure  phosphorus, which is of a 
124
A  TREATISE ON OfTICS.        PART II.
slightly yellow color, transmitting freely almost all  the color-
ed rays.  When melted, and suddenly cooled, it acquired the
power of absorbing all the colors of the spectrum at thick-
nesses at which it formerly transmitted them all.  The black-
ness produced upon pure  phosphorus  was first observed by
Thenard.   Mr. Faraday observed, that  glass tinged purple
with manganese had its absorptive power altered by the mere
transmission through it of the solar rays.
 . By the method above described  of  absorbing  particular
colors in the spectrum, I was led to propose a new method of
analyzing white light.   The experiments with the blue glass
incontestably prove that the orange  and green colors in solar
light are compound colors, which, though they cannot be de-
composed by the prism, may be decomposed by absorption, by
which we may exhibit alone the red part of the orange and
the blue part of the green, or the yellow part of the orange
and the yellow  part  of the green; and, by submitting the
other colors of  the spectrum  to the scrutiny  of  absorbent
media, I was led to the conclusions  respecting the spectrum
which are explained in Chapter VII.
  We have already seen  that in the solar spectrum, as de-
scribed by Fraunhofer, there are dark  lines, as if rays of par-
ticular refrangibilities had been absorbed in their course from
the sun to the earth.   The absorption is  not likely to have
taken  place  in  our atmosphere, otherwise  the same lines
would have been wanting in the spectra from the fixed stars,
and the rays of solar light reflected from the moon and planets
would  probably  have  been  modified  by their  atmospheres.
But as this is not the case, it is probable that the rays which
are wanting in the spectrum have been absorbed by the sun's
atmosphere, as Mr. Herschel has supposed.
  (88.) Connected with the preceding phenomena  is the sub-
ject of colored  flames,  which,  when examined by a prism,
exhibit spectra  deficient in particular rays, and resembling
the solar spectrum examined by colored glasses.  Pure hy-
drogen gas burns with  a blue flame, in which many of the
rays of light are wanting.   The flame of an oil lamp contains
most  of the rays which are wanting in sun-light.  Alcohol
mixed with water, when heated and  burned, affords a flame
with  no other rays but yellow.  Almost all  salts  communi-
cate to flames a peculiar color, as may be seen by introducing
the powder of these salts into the exterior flame of a candle,
or into the wick of a spirit lamp.  The following results, ob-
tained  by  different authors, have been given  by Mr.  Her-
schel :— 
CHAP. XVII.      DOUBLE REFRACTION.
125
    Salts of soda,   ......Homogeneous yellow.
    ------ potash, *.....Pale violet.
    ------ lime,......Brick red.
    ------ strontia,.....Bright crimson.
    ------ lithia,......Red.
      ----- baryta,......Pale apple green,
     ------ copper,......Bluish green.

  According to Mr. Herschel the muriates succeed best on
account of their volatility.
                      CHAP. XVII.

          ON THE  DOUBLE  REFRACTION OP LIGHT.

  (89.) In the preceding  chapters of this work it has always
been supposed, when treating of the  refraction of light, either
through surfaces, lenses, or  prisms, that the transparent or re-
fracting body had the same  structure, the same  temperature,
and the same density in every part of it, and  in every direc-
tion  in which the ray could enter it.   Transparent bodies of
this kind are gases, fluids, solid bodies,  such as different kinds
of glass, formed by fusion, and slowly and equally cooled, and
a numerous  class  of crystallized bodies, the form of whose
primitive crystal is the cube, the regular octohedron,  and the
rhomboidal dodecahedron.   When  any of these bodies have
the same temperature and  density, and are not subject to any
pressure, a single pencil of  light incident upon any single sur-
face of them, perfectly plane, will be refracted into a single
pencil according to the  law of the sines explained in Chap-
ter III.
  In almost all other bodies, including salts and crystallized
minerals not having  the primitive forms  above mentioned;
animal bodies, such as hair, horn, shells, bones, lenses of ani-
mals and elastic integuments; vegetable bodies, such as cer-
tain leaves,  stalks, and  seeds;  and artificial  bodies, such as
resins, gums, jellies, glasses quickly and unequally cooled, and
solid bodies having unequal density either from unequal tem-
perature  or  unequal pressure;—in all such bodies a single
pencil of light incident upon their surfaces will be refracted
into two different pencils, more or less inclined to one another,
according to the nature and state of the body, and according
to the  direction in which the pencil  is incident.  The separa-
tion  of the  two pencils is sometimes very great, and in most
eases easily  observed and measured; but in other cases it is 
126
A TREATISE ON OPTICS.
PART II.
not visible,  and its  existence  is inferred only from  certain
effects which  could  not arise except from two refracted pen-
cils.  The refraction of the two pencils  is called double re-
fraction, and the bodies which produce it are  called doubly
refracting bodies or crystals.
   As the phenomena of double refraction were first discovered
in a transparent mineral substance called Iceland spar, calca-
reous spar, or carbonate of lime, and as this substance is ad-
mirably fitted for exhibiting them, we shall begin by explain-
ing  the  law of double  refraction as it exists in this mineral
Iceland spar is composed of 56 parts of lime, and 44 of car-
bonic acid.   It is found in almost all countries, in crystals of
various shapes, and often in huge masses;  but, whether found
in crystals or  in masses, we can always  cleave it or split it
into shapes like that represented in fig. 70., which is called a
                  rhomb of Iceland spar,  a solid bounded by
                  six equal  and similar rhomboidal surfaces,
                  whose sides are parallel, and  whose angles
                  B A C, A C D are 101° 55' and 78° 5'. The
                  inclination of any face A B C D  to any of
                  the adjacent faces that meet at A is 105° 5',
                  and to any of the .adjacent faces that meet
                  at X 74° 55'.  The line A X, called the
axis of the rhomb or of the  crystal, is equally inclined to each
of the six faces at an angle of 45° 23'.   The angle between
any of the three edges, BA, CA, EA, that meet at A, or of
the three that meet at X, and the axis A X is 66° 44' 46", and
the angle between any  of the six edges and the faces is 113°
15' 14" and 66° 44'  46".
   (90.) Iceland spar is very transparent, and  generally color-
less.  Its natural faces,  when it is split, are  commonly even
and perfectly polished ;  but when they are not so, we may, by
a new cleavage, replace the imperfect face by a  better one, or
we may grind and polish any imperfect face.
   Having procured  a rhomb of Iceland spar like that in the
figure, with smooth and well polished faces, and so large that
one of the edges A B is at least an inch long,  place one of
its faces upon a sheet  of  paper, having  a black line MN
drawn upon it, as shown in fig. 71.  If we then look through
the upper surface of the rhomb with  the eye about R, we
shall  probably  see  the  line  M N double; but  if  it  is not,
it will become double by turning the crystal a little round.
Two  lines,  MN, mn, will then be distinctly  visible; and
upon  turning  the crystal round,  preserving the same side
always upon the paper,  the  two lines will coincide with one
another, and appear to form one  at two opposite  points during 
CHAP. XVII.       DOUBLE  REFRACTION.
127
a whole  revolution of the crystal; and at two other opposite
points, nearly at right angles to the former, the lines will be
at their greatest distance.  If we place a  black spot at O, or
a luminous aperture, such as a pin-hole in a wafer, with light
passing through the  hole, the spot or aperture will appear

                          Fig. 71.
double, as at O and E;  and by turning the crystal round as
before, the two  images will be seen separate  in all positions;
the one, E, revolving, as it were, round the other, O.
   Let a ray or pencil of light, R r, fall upon the  surface of
the rhomb at r, it will be refracted by the action  of the  sur-
face into  two pencils, r 0,rF,  each of which, being again
refracted  at  the second surface at the points O, E, will move
in the directions O o, E e, parallel to one another and to the
incident ray R r.  The  ray R r has therefore been doubly re-
fracted by the rhomb.
   If we now examine and measure the angle  of refraction of
the ray r  O corresponding to different angles of incidence, we
shall find that, at 0° of incidence,  or a perpendicular inci-
dence, it  suffers no refraction, but moves straight through the
crystal in one unbroken  line ; that at all other angles of inci-
dence the sine  of the angle  of refraction is  to that of inci-
dence as 1 to 1-654;  and that the refracted ray is always in
the same  plane  as that  of the incident ray.  Hence it is ob-
vious that the ray r O is refracted according to the ordinary
law of refraction, which we  have already explained.  If we
now examine in the same way the ray r E, we shall find that,
at a perpendicular incidence, or one of 0°, the angle  of re-
fraction, in place of being 0°, is actually 6° 12'; that at other
incidences the angle of refraction is not such as to follow the
constant ratio of the sines;  and, what is still more extraordi-
nary, that the refracted ray r E  is bent  to one side, and lies
■entirely out of the plane of incidence.   Hence it follows that 
128
A  TREATISE  ON  OPTICS.       PART  II.
 the pencil r E is refracted according to some new and extra-
 ordinary law of refraction.   The  ray r O is therefore called
 the ordinary ray, and r E the extraordinary ray.
   If we cause the  ray R r to be  incident in various different
 directions, either on  the natural faces of the  rhomb or on
 faces cut and polished artificially, we shall find that in Iceland
 spar there is one direction, namely,  A X, along which, if the
 refracted pencil passes, it is  not refracted into two pencils, or
 does not suffer double  refraction.  In other crystals  there are
 two such directions, forming  an  angle with each  other. In the
 former case the crystal is said to have one axis of double re-
 fraction, and in the latter case two axes of double refraction.
 These lines are called axes of double refraction, because the
 phenomena are related to these lines.  In some bodies there
 are  certain planes, along which, if the refracted  ray passes, it
 experiences no double refraction.
   An  axis of double refraction, however, is not, like the  axis
 of the earth, a fixed line within the rhomb or crystal.  It is
 only a fixed  direction: for if we divide,  as we can do, the
 rhomb ABC, fig. 70., into two or more rhombs,  each of these
 separate rhombs will have their  axes of double refraction; but
 when these rhombs are again put  together, their axes will be
 all parallel to A X.  Every line, therefore, within the rhomb
 parallel to A X, is an  axis of double refraction; but as these
 lines have all one and the same direction in space, the crystal
 is still said to have only one axis of double refraction.
   In making experiments with  different crystals, it is found
 that in some the  extraordinary ray  is refracted towards the
 axis A X, while in others it is  refracted from the axis A X.
 In the first case the axis is called  a positive axis of double
 refraction, and in the second case  a negative axis of double
 refraction.

      On Crystals with one Axis of Double Refraction.

  (91.) In  examining the phenomena of double  refraction in
 a great number of crystallized  bodies, I found that all those
 crystals whose  primitive or simplest form had only one axis
of figure, or one pre-eminent line round which the figure  was
symmetrical, had also one axis of  double refraction ; and that
their axis of figure was also the axis  of double refraction. The
primitive forms which possess this property are as follows :—
           The rhomb with an obtuse summit.
           The rhomb with an acute summit.
           The regular hexahedral prism.
           The octohedron with a square base.
           The right prism with a square  base. 
chap. xvn.
DOUBLE  REFRACTION.
129
  (92.) The following Table contains the crystals which have
one axis of double refraction, arranged under their respective
primitive forms, the sign + being prefixed to those that have
positive double refraction, and — to those that have  negative
double refraction.

                                                Fig. 74.
r<XT>i
1.  Rhomb with obtuse summit, fig. 72.
-Carbonate of lime (Iceland
 spar.
-Carbonate of lime and iron.
-Carbonate of lime and mag-
 nesia.
-Phosphato-arseniate of lead.
-Carbonate of zinc.
-Nitrate of soda.
—Phosphate of lead.
—Ruby silver.
—Levyne.
—Tourmaline.
—Rubellite.
—Alum stone.
—Dioptase.
—Quartz.
Rhomb with acute summit, fig. 73.
               —Cinnabar.
               —Arseniate of copper.
—Corundum
—Sapphire.
—Ruby.

          3.
—Emerald.
—Beryl.
—Phosphate of lime (apatite)

         4.  Octohedron with a square base, fig. 75.
4- Zircon.                    —Molybdate of lead.
-f Oxide of tin.               — Octohedrite.
-fTuno-state of lime.          —Prussiate of potash.
__Mellite.                   —Cyanide of mercury.
Regular Hexahedral Prism, fig. 74.
                —Nepheline.
                —Arseniate of lead.
                -f-Hydrate of magnesia. 
130
  A  TREATISE ON OPTICS.

Fig. 75.                   Fig. 76.

                             AT
PART II.
5. Right Prism with a square base, fig. 76.
—Sulphate of nickel and cop-
  per.
—Hydrate of strontia.
+ Apophyllite of utoe.
+ Oxahverite.
+ Superacetate  of copper and
  lime.
-f-Titanite.
-flee (certain crystals).
—--Idocrase.
—Wernerite.
—Paranthine.
—Meionite.
—Somervillite.
—Edingtonite.
—Arseniate of potash.
—Sub-phosphate of potash.
—Phosphate of ammonia and
  magnesia.
  In all the preceding  crystals, and in the primitive forms to
which  they belong, the line A X is the axis of figure and  of
double refraction, or the only direction along which there is
no double refraction.

  On the Law of Double Refraction in Crystals with'pne
                      Negative Axis.              f"
  (93.) In order to give a familiar explanation of the law of
          Fig. 77.     _      double refraction, let us suppose
                           that a rhomb of Iceland spar is
                           turned in a lathe to the form of
                           a  sphere, as shown in fig. 77.,
                           A X being  the  axis of both the
                           rhomb and the sphere.
                             If we now make a ray pass along
                           the axis A X, after grinding or
                           polishing a small flat  surface at
A and  X, perpendicular to A X, we shall find that there is no
double refraction;  the ordinary and extraordinary ray forming
a single ray.   Hence,
  The index of refraction along
    the axis A X will be
1-654 for ordinary ray.
1-654 for extraordinary ray.
0000 difference. 
I
CHAP. XVII.   NEGATIVE  DOUBLE REFRACTION.     131

  If we do the same at any point, a, about 45° from the axis,
we shall have
The index of refraction along the line } 1-654 for ordinary ray.
  R a b O,  which is  nearly perpen- > 1-572 for extraordinary
  dicular  to the face of the  rhomb,  )          ray.
                                      0.082 difference.
   If we do the same at any point of the equator C D, in-
 clined 90° to the axis, we shall have
   The index of refraction per- > 1-654 for ordinary ray.
      pendicular to the axis,   \ 1.483 for extraordinary ray.

                                0-171 difference.
   Hence it follows that the index of extraordinary refraction
 decreases from the axis A X to the equator C D, or to a line
 perpendicular  to the  axis,  where  it  is the  least.     The
 index of extraordinary refraction  is the same at all  equal
 angles  with the axis A X; and hence, in every part of a cir-
 cle described on the surface of the sphere round the pole A or
 X, the index  of extraordinary refraction has the same value,
 and  consequently the double  refraction or  separation of the
 rays will be the same.  In crystals, therefore, with one axis
 of double refraction, the lines of equal double refraction are
 circles  parallel to  the equator or  circle  of greatest double
 refraction.
   The  celebrated Huygens, to whom we owe the discovery
 of the law of double  refraction in crystals with one  axis, has
 given the following method of determining the index of ex-
 traordinary refraction at any  point of the sphere, when the
 ray of light is incident in a plane passing  through the axis of
 the crystal A X:—
   Let it be required, for example,  to  determine the index of
 refraction for the extraordinary ray Rab, fig.  77., A X being
 the axis, and C D  the equator of the crystal; the  ordinary
 index of refraction being known, and  also the  least  extra-
 ordinary index of refraction, or that which takes place in the
 equator.  In calcareous spar these numbers  are  1-654 and
 1.483.  From O set off in the  lines O C, O D  continued, O c,
 0 d,  so that O C or O D is to O c or O d  as j.^^  is to t.tVj>
or as -604 is to -674;  and through the points A, c, X, d, draw
an ellipse, whose greater axis is c d, and whose lesser axis is
A X.   The radius  O a of the ellipse  will be  what is called
the reciprocal of the index of refraction at a;  and as we can
find O a, either by projecting the ellipse on a large  scale, or
by calculation, we have only to divide 1 by O a to have  that 
132
A  TREATISE  ON  OPTICS.        PART II.
 index.  In the present case O a is -636, and .^ is  equal  to
 1-572, the index required.
   As the index of extraordinary refraction thus found always
 diminishes from the pole A to the equator C 1), and is always
 equal  to  the index of ordinary refraction minus another
 quantity depending on the difference between the radii of the
 circle  and those of the ellipse, the crystals in which this
 takes place may be properly  said to have  negative double
 refraction.
   In  order to determine the direction of the  extraordinary
 refracted ray, when the plane of incidence is oblique  to a
 plane passing through the axis, the process, either by projec-
 tion or calculation,  is too  troublesome to be given in an ele-
 mentary work.
   In every case the force  which produces the double refrac-
 tion exerts  itself as if it proceeded from the axis.
   Every plane passing through the axis is called a principal
 section of the crystal.

   On the Law of Double Refraction in Crystals with one
                      Positive Axis.
   (94.)  Among  the crystals  best  fitted for exhibiting the
phenomena of  positive double refraction  is rock crystal or
quartz, a  mineral  which  is  generally found in  six-sided
  Fig. 78.  prisms, like fig. 78., terminated with six-sided pyra-
         mids, E, F.
           If we  now grind  down  the summits A and X,
         and replace  them by faces well polished, and per-
         pendicular to the axis A X;  and if we transmit a
         ray through  these faces,  so  that it may pass along
         the axis A X, we shall find that there is no double
         refraction, and that the index of  refraction  is as
         follows:—
   Index of refraction along ) 1-5484 for ordinary ray.
    the axis  AX   -   -   } 1-5484 for extraordinary ray

                            0-0000 difference.
   If we now transmit the ray perpendicularly through the
parallel faces E F, which are inclined 38° 20' to the axis A X,
the plane of its  incidence passing through A X,  we  shall
obtain the following results:—
 Index of refraction per pen- ) , K/lo. c     ,.
    dicular  to the  faces of { J'^JJ £or ordinary ray.
    the pyramid    -  -  -   ) * 'oo4A for extraordinary ray.
00060 difference. 
CHAr. XVII.       DOUBLE REFRACTION.              133
  In  like manner, it will be found  that when the ray passes
perpendicularly through the faces C D, perpendicular to the
axis  A X,  the  index  of extraordinary  refraction is  the
greatest, viz.
 Index of refraction per pen- I.-.q,  .     ,.
    dicular  to  the faces of U*'^ J>r ordinary ray.
    the prism CD--   ) 1-5582 for extraordinary ray.

                             0-0098 difference.
  Hence it  appears that in quartz the index of extraordinary
refraction increases  from the  pole A  to  the equator C D,
whereas it diminished in calcareous spar, and the  extraordi-
nary ray appears to be drawn to the axis.
  In  this case the variation  of the  index of extraordinary
refraction will be represented by an  ellipse, Ac Xd, whose
                       greater  axis coincides with the axis
                       A X of double  refraction, as in fig.
                       79., and O C will be to Oc as y.^^
                       is to T.^2, or  as  -6458 is  to -6418.
                       By determining, therefore, the radius
                       O a of the ellipse for any ray R b a,
                       and dividing 1 by  it,  we shall have
                       the  index of extraordinary  refraction
                       for that  ray.
  As the index of extraordinary refraction  is always equal to
the index of ordinary refraction, plus  another quantity de-
pending on the difference between the radii of the circle and
the ellipse, the crystals in which this takes place may properly
be said to have positive double refraction.

      On Crystals with two Axes  of Double Refraction.
  (95.) The great  variety of crystals,  whether  they  are
mineral bodies or chemical  substances, have two axes of
double refraction, or two  directions  inclined to each  other
along which the  double  refraction is nothing.   This property
of possessing two axes  of double refraction I discovered in
1815, and I  found that it belonged  to all the crystals which
are included  in the prismatic system of  Mohs,  or  whose
primitive forms are,
         A  right prism, base a rectangle.
         ----------- base a rhomb.
         -----------base an  oblique parallelogram.
         Oblique prism, base a rectangle.
         -----------base a rhomb.
         ----------- base an  oblique parallelogram.
         Octohedron, base a rectangle.
         ----------banc a rhomb.
                            M
 
134
A  TREATISE  ON  OPTICS.         PART II.
  In all these  primitive forms there is not a single pre-emi-
nent line or axis about which the figure is symmetrical.
  The following is a list of some of the most important crys-
tals, with their primitive forms according  to Haiiy, and the
inclination of the  two lines or axes along  which there is no
double refraction:—
Glauberite  -  -  -   2° or 3° Oblique prism, base a rhomb.
Nitrate of potash  -   5° 20'   Octohedron, base a rectangle,
Arragonite  -  -  -  IS  18   Octohedron, base a rectangle.
Sulphate of baryta -  37  42   Right prism, base a rectangle.
Mica.....45   0   Right prism, base a rectangle.
c  ,  ,  ^  „ ,.        cr.   n  S Right prism, base an oblique
Sulphate of lime  ■  60   Oj   parallelogram.
Topaz    -  -  -  -  65   0   Octohedron, base a rectangle.
Carbonate of potash  80  30   Prismatic system of Mohs.
Sulphate of iron   -  90   0   Oblique prism, base a rhomb.
   In crystals with one axis of double refraction, the axis has
the same position  whatever be the color of  the pencil of light
which is used; but in crystals with two axes, the axes change
their position  according to the color of the  light employed, so
that the  inclination of the two axes varies with differently
colored rays.   This discovery we owe to Mr. Herschel, who
found  that  in  tartrate  of potash and soda (Rochelle  salts)
the  inclination of the axis for violet light was about 56°,
while  in red light it was about 76°.   In other crystals, such
as  nitre, the inclination of the axes  for  the  violet rays is
greater than  for  the red rays; but in  every case the line
joining the extremity of the axes for all the different rays is
a straight line.
   In examining the properties of Glauberite, I found that it
had two  axes for red  light inclined about 5°, and only one
axis for violet light.
   It was at first supposed that in crystals  with two axes, one
of the  rays was refracted according to the ordinary law of the
sines, and the other by an extraordinary law; but Mr. Fresnel
has shown that both the rays are refracted according to laws
of extraordinary refraction.

  On Crystals with innumerable Axes of Double Refraction.

   (96.) In the various  doubly refracting bodies  hitherto men-
 tioned, the double refraction is related to one or more axes;
 but I have found  that in analci/ne there are several planes,
 along  which  if the refracted ray passes,   it will  not suffer
 double refraction, however various be the directions in  which 
CHAP. XVII.      DOUBLE REFRACTION.
135
it is incident   Hence we may consider each of these planes
as containing an infinite number of axes of double  refraction,
or rather lines in which there is no double refraction.  When
the ray is incident in any other direction, so that the refracted
ray is not in one  of  these planes, it is divided  into two rays
by double refraction.   No other substance has yet been found
possessing the same  property.

On Bodies to which  Double Refraction may be communicated
     by Heat, rapid  Cooling, Pressure, and, Induration.

  (97.) If we  take  a cylinder of glass,  C D, fig.  80., and
  Fig. 80.   having brought it to a red heat, roll it along a plate
    l^   of metal upon its cylindrical surface till it is cold, it
  ^-r-v   will  acquire a permanent doubly refracting struc-
  s___y   ture, and it will become a cylinder with one positive
          axis of double refraction, A X, coinciding with the
          axis  of the  cylinder, and along which there  is no
     J   double refraction.   This axis differs from that in
    [X   quartz, as it is a fixed line in the cylinder, while it
          is only a fixed direction in the quartz ; that is, any
other line parallel to A X, fig. 80., is not an axis of double
refraction, but  the double  refraction along that line increases
as  it approaches  the  circumference  of the cylinder.   The
double refraction is a maximum in the direction  C D, being
equal  in  every line perpendicular to the axis, and passing
through it
  If,  instead of heating the glass cylinder, we had placed it
in a vessel and surrounded it with boiling oil or boiling water,
it would have acquired the very same doubly refracting struc-
ture when the heat  had reached the axis A X;  but this struc-
ture is only transient, as it disappears when the  cylinder is
uniformly heated.
   If we had heated  the cylinder uniformly in boiling oil, or at
a fire, so as not to soften the glass,  and had placed it in a cold
fluid,  it  would  have  acquired a transient doubly refracting
structure as before, when the cooling had  reached the axis
A X;  but its  axis  of double refraction  A X will now be a
negative  one, like that of calcareous spar.
   Analogous  structures may be produced by pressure and by
the induration of soft solids, such as animal jellies, isinglass, &c.
  If the cylinder in the preceding explanation is not a circular
one, but has its section perpendicular to the axis everywhere
an ellipse in place of a circle, it will have two axes of double
refraction.
  In  like manner,  if we use rectangular plates of glass in- 
136
A  TREATISE  ON OPTICS.         PART II.
stead of cylinders in the preceding experiments, we shall have
plates with two planes of double refraction ; a positive struc-
ture being on  one  side of each plane, and a negative one on
the other.
  If we use perfect spheres, there will be axes of double re-
fraction along  every diameter, and  consequently an infinite
number of them.
  The  crystalline lenses  of almost all animals, whether they
are lenses, spheres, or  spheroids, have one or more axes of
double refraction.
  All these phenomena will  be more fully explained when we
treat of the colors produced by double refraction.

      On Substances with Circular Double Refraction.

  (98.) When we  transmit  a  pencil  of light  along  the axis
A X, Jig. 78., of a crystal of quartz, it suffers no double re-
fraction ; but  certain  phenomena, which will be afterwards
described, are seen along this axis, which induced M. Fresnel
to examine the light which passed along the axis.   He found
that it possessed  a new kind of double refraction, and he dis-
tinctly observed the refraction  of the two pencils.  This kind
of double refraction has, from its properties, been called cir-
cular;  and it  is divided  into two kinis,—positive or right-
handed, and negative or left-handed.
  The  following substances  possess  this  remarkable prop
erty:—
        Positive Substances.
    Rock crystal, certain speci-
       mens.
    Camphor.
    Oil of turpentine.
    Solution of camphor in al-
       cohol.
    Essential oil of laurel.
    Vapor of turpentine.
  In  examining this  class  of phenomena, I  found  that the
amethyst possessed in the same crystal both the positive and
the negative circular double refraction.  This  subject will be
more fully treated when  we come to  that  of  circular polar-
ization*

  * For the formula referring to certain of the articles of this and of the
subsequent chapter, see (in the  College edition,) Appendix of Am. ed.,
Chap. VI.
Negative Substances.
Rock crystal,  certain speci-
  mens.
Concentrated syrup of sugar.
Essential oil of lemon. 
CHAP. XVIII.   POLARIZATION OP LIGHT.         137
                    CHAP. XVIII.

             ON THE POLARIZATION OF LIGHT.

   If we transmit a beam of the sun's light through a circular
 aperture into a dark room, and if we reflect it from any crys-
 tallized or uncrystallized body, or transmit it through a thin
 plate of either of them, it will be reflected and transmitted in
 the very same manner and with the same intensity, whether
 the surface of the body is held above or  below the beam, or on
 the right side or left, or on any other side of it, provided that
 in all these cases it falls upon the surface in the same manner;
 or, what amounts to the same thing, the beam of solar light
 has the same  properties on  all  its sides; and this is true,
 whether it is  white light as directly emitted from the sun, or
 whether it is red light, or light of any other color.
   The same property belongs  to light emitted from a candle,
 or any burning or self-luminous body, and all such light is
 called common light.  A section of such a beam of light will
 be a circle, like A C B D, fig. 81., and we shall distinguish

                       Fig. 81.

             a         <g          Efi




        €KDO
              B          B'

 the section of a beam of common light by a circle with two
 diameters, A B, C D, at right angles to  each other.
   If we now  allow the same beam of light to fall upon a
 rhomb  of Iceland spar, as in fig. 71., and examine the two
 circular beams O o, E e, formed by double refraction, we shall
 find,
  1. That the beams O o, E e, have different properties on
 different sides; so that each  of them differs, in this respect,
 from the beam of common light.
  2. That the beam O o differs from E e in nothing, excepting
 that the former has the same properties  at the sides A' and B'
 that the latter has at the sides C and D', as shown in fig. 81.;
or, in general,  that the diameters of the  beam, at the extremi-
ties of  which  the beam has similar properties, are at right
angles to each  other, as A' B' and C D', for example.
                       M2 
138
A  TREATISE ON OPTICS.        PART II.
  These two teams, O o, E e, fig. 81., are therefore  said to
be polarized, or to be beams of polarized liglit, because they
have sides or poles of different properties ;  and planes passing
through the lines A B, C D, or A' B', C D', are said to be the
planes of polarization of each  beam, because they have the
same property, and one which no other plane passing through
the beam possesses.
  Now, it is a curious fact, that  if we cause the two polarized
beams O o, E e to be united into one, or if we produce them
by a thin plate of Iceland spar, which is not capable of sepa-
rating them, we obtain a beam which has exactly the same
properties as the beam A B C D  of common light
  Hence we  infer, that a beam of common  light, A B C D,
consists of tico beams of polarized light, whose planes  of po-
larization, or whose diameters  of similar properties,  are at
right angles to one  another.   If O o is  laid upon E e, it
will produce a figure like A B C D,  and we, therefore, re-
present  common light by  such  a figure.  If we  place 0 o
above E e, so  that the planes of polarization A' B' and C D'
coincide, then we shall have a beam of polarized light  twice
as luminous as either O o  or  E e,  and  possessing exactly
the same properties;  for the lines of similar  property in  the
one beam coincide with the lines  of similar  property in  the
other.
  Hence it follows that there are three ways  of converting a
beam of common light, A B C D, into a beam or beams  of po-
larized light.
  1. We may separate the beam  of common  light, A B C D,
into its two component parts, O o and E e.
  2. We may turn round the planes of polarization, A B, C D,
till  they coincide or are parallel  to each other.   Or,
  3. We may absorb or stop one of the beams, and leave the
other, which will consequently be in a state of polarization.
  The first of these methods of producing polarized light is
that in which we employ a doubly refracting crystal, which
we  shall now consider.

    On the Polarization of Light by  Double Refraction.

  (99.) When  a beam  of light suffers double refraction by
a negative crystal, as Iceland spar, fig. 71., where the  ray
R r is incident in the plane of the principal section, or, what
is the same thing, in a plane passing through the axis, the two
pencils r O, r E are each polarized ; the plane  of polarization
of the ordinary ray, r O, coinciding with the principal section,
and the plane of polarization of the extraordinary ray, r E, being 
CHAP. XVIII.    POLARIZATION OF  LIGHT.
139
at right  angles to the principal section.   In fig. 82., if O be
made to denote a section of the ordinary beam r O, fig. 71., E,
the diameter of which is drawn at right angles to that of O,
will represent a section of the  extraordinary beam r E.
'Fig. 82.
Fig. 83.
  If the beam of light R r is incident upon a positive crystal,
like quartz, O of fig. 83., will be the symbol of the ordinary
ray, and E that of the extraordinary ray.          ...
  The phenomena which arise from this opposite polarization
of the two pencils may be well seen in Iceland spar. For this
purpose let A r X be the principal section of a rhomb  of Ice-
land spar, fig. 84.,  through the axis A X, and perpendicular
to one of the faces, and let A' F X'  be a similar section of an-
other rhomb, all  the lines of the one being parallel to all the
lines of the other.  A ray of light, Rr,  incident perpendicu-
larly at r, will be divided into two pencils; an ordinary one,
Fig. 84.
Fig. 85.
A		B-1*
N		0 I
c a?		D X Gr
o	-	e
Eo		K. Oe
r D, and an extraordinary one, r C.  The ordinary ray falling
on  the  second crystal at G, again suffers ordinary refraction,
and emerges  at K an  ordinary ray, Oo, represented by the
svmbol  O,  Us. 82.  In like manner the extraordinary ray, r C,
fallincr on thf second  crystal at F, again  suffers extraordinary 
140
A TREATISE ON OPTICS.        PART  II.
refraction, and emerges at H an extraordinary ray, E e, repre-
sented by E, fig. 82. These results are exactly the same as if the
two crystals had formed a single crystal by being united at their
surfaces C X, A' G, either by natural cohesion or by a cement.
  Let the upper crystal A X now remain fixed,  with the same
ray R r falling upon it, and let the second crystal  A' X' be
turned round 90°, so that its principal  section is perpendicular
to that of the  upper one, as shown in fig. 85.; then the ray
r D ordinarily refracted by the first rhomb will be extraordi-
narily refracted by the second, and the ray r C extraordinarily
refracted by the first rhomb will  be ordinarily refracted by the
second.
  The pencils  or images formed from the ray R r, in the two
positions shown in figs. 84. and 85., may be thus described as
marked in the figures:—
 O is the pencil refracted ordinarily by the first rhomb.
 E is the pencil refracted extraordinarily by the first rhomb.
 o is the pencil refracted ordinarily by the second rhomb.
 e is the  pencil  refracted  extraordinarily  by the second
rhomb.
  O o is the pencil refracted ordinarily by both rhombs in
fig. 84.
  E e is the pencil refracted extraordinarily by both rhombs
in fig. 84.
  0 e is the pencil refracted ordinarily by the first, and ex-
traordinarily by the second rhomb in fig. 85.
  E o is the pencil  refracted extraordinarily by the first, and
ordinarily by the second rhomb in  fig. 85.
  In  both  the  cases shown  in  figs.  84.  and 85., when the
planes of the principal sections  of the two rhombs are either
parallel, as in fig. 84.,  or perpendicular to each other, as in
fig. 85., the lower rhomb is not  capable of dividing  into two
any of the  pencils which fall upon  it; but  in every other po-
sition between the parallelism  and  the perpendicularity of the
principal sections,  each of the pencils formed  by  the first
rhomb will be divided into two by the  second.
  In order to explain the appearances in all intermediate po-
sitions, let  us suppose that the ray R r proceeds from a round
aperture, like one of the  circles at A, fig. 86., and  that the
eye is placed behind the two rhombs at H K,fig. 84., so as to see
the images of this aperture.  Let the  two images shown at A, fig.
86., be the appearance of the aperture at R, seen through one of
the rhombs by an eye placed behind C T>,fig. 84., then B, fig. 86.,
will represent the images seen through the two rhombs in the
position in fig. 84., their distance being doubled, from  suffering
the same quantity of double refraction twice. If we now turn 
CHAP. XVIII.    POLARIZATION  OF LIGHT.           141

the second rhomb, or that nearest  the  eye, from left to right,
two faint images will appear, as at C, between the two bright
ones,  which will now be a little  fainter.  By continuing to
A	B	C	D	Fig. 86. E F G H	I	K
©	• ©	o m 9		JO * •- if	as	o
turn, the four images will be  all equally luminous, as at D;
they will next appear as at E ; and when  the  second rhomb
has moved round  90°, as in fig. 85., there will be  only two
images  of equal brightness, as at F.   Continuing to turn the
second  rhomb, two faint  images will appear,  as  atG;  by a
farther  rotation,  they will be all equally bright, as at H;
farther  on  they will become unequal, as at I;  and at 1WU  ot
revolution, when the planes of the  principal section are again
parallel, and the axes A X, A' X'  at  right angles nearly to
each other, all the images will coalesce into one bright image,
as at K, ha vino- double the brightness of either of those  at A,
B, or F, and four times the brightness of any one of the four
at D and H.                       .                 ,
   If we now follow any one of the images A,  B from the po-
sition in fig. 84., where the principal sections  are inclined U°
to one another, to  the position in fig. 85., where it disappears
at F we shall find that its brightness diminishes as the square
of the  cosine of  the angle formed by the principal sections,
while the  brightness of any  image, from  its  appearance be-
tween B and C, fig. 86.,  to its greatest brightness at F, in-
creases as the square of the sine of the  same angle.
   By considering the preceding phenomena  it will  appear,
that whenever the  plane of  polarization  of a polarized ray,
whether ordinary or extraordinary, coincides with or is parallel
to the principal section, the ray will be refracted ordinarily;
and whenever the plane of polarization is perpendicular  to
the principal section, it will be refracted extraordinarily.  In
all intermediate positions  it will suffer both kinds of refraction,
and will be  doubly refracted; the ordinary pencil being the
brightest if the plane of polarization is nearer the position ot
parallelism than that of perpendicularity, and the extraordi-
nary pencil the brightest  if the plane of polarization is nearer
the position of perpendicularity than  that  of parallelism.  At
equal distances from both these positions, the ordmary and ex-
traordinary images are equally bright. 
142
A  TREATISE  ON  OPTICS.        PART II.
   (100.) It does not appear from  the preceding experiments
that the polarization of the two pencils is the effect of any po-
larizing force resident in the Iceland spar, or of any change
produced upon the light.  The Iceland spar has  merely sepa-
rated the common light into its  two elements, according to a
different law, in  the  same manner as a prism  separates all
the  seven  colors  of the spectrum  from the  compound white
beam by its power of refracting these  elementary colors in
different degrees.  The  re-union  of the  two oppositely  po-
larized  pencils produces common light, in the same manner as
the re-union of all the seven colors produces white light.
   The  method of producing polarized light by double  refrac-
tion is of all others the best, as we  can procure by this means
from a given pencil of light a stronger polarized  beam than in
any other way.  Through a thickness of three inches of Ice-
land spar we can obtain two separate beams of polarized light
one third of an  inch in diameter; and  each  of these beams
contains half the  light of the original  beam, excepting the
small quantity of light lost  by reflexion and absorption.   By
sticking a  black wafer on the spar  opposite either of these
beams, we  can procure a polarized  beam with its plane of po-
larization either in the principal section or at right angles to
it.  In all experiments on this subject,  the reader should re-
collect that every beam of polarized light, whether it is pro-
duced by the ordinary or the  extraordinary refraction, or by
positive or negative crystals, has always the same properties,
provided the plane of its polarization  has the same direction.
                      CHAP. XIX *

      ON  THE POLARIZATION OF  LIGHT BY REFLEXION.

  (101.) In the year 1810, the celebrated French philosopher
M. Malus, while looking through  a prism of calcareous spar
at the light  of the setting sun reflected from the windows of
the Luxembourg palace in Paris, was led to the  curious dis-
covery,  that a beam  of light reflected from glass at an angle
of 56°,  or from water at an angle of 53°, possessed the very
same properties as one of the rays formed by a rhomb of cal-
careous spar; that is, that it was wholly  polarized, having its
plane of polarization coincident with or parallel to  the plane
of reflexion.
 * For the formulae relating to this chapter, see (in the College edition,)
Appendix of Am. ed., Chap. VI. 
CHAP. XIX.   POLARIZATION BY REFLEXION.        143

  This most curious and important fact, which he found to be
true when the light was reflected from  all other transparent
or opaque bodies, excepting metals, gave birth to all those dis-
coveries which have, in our own day, rendered this branch of
knowledge one of the most interesting, as well as one of the
most perfect, of the physical  sciences.
  In order to explain this and the  other discoveries of Malus,
let C D, fig. 87., be a tube of brass or wood, having at one end
of it a  plate of glass, A, not quicksilvered,  and capable of

                          Fig. 87.
turning round an  axis, so that it may form different angles
with the axis of the tube.  Let D G be a similar tube a little
smaller than the other, and carrying a similar plate of glass B.
If the  tube D G is pushed into C D, we may,  by turning  the
one or the other round, place the two glass plates in any po-
sition in relation to one another.
  Let  a beam  of light, R r,  from a candle or a hole in  the
window-shutter, fall upon the glass plate A, at an angle of 56°
45'; and  let the glass be so  placed that the reflected ray r s
may pass  along the  axis of the two tubes, and fall upon the
second plate of glass  B at the  point  s.  If the  ray r s falls
upon the second plate B at an angle of 56° 45' also, and if the
plane of reflexion from this plate, or the plane passing through
s E and s  r, is at right angles to the plane of reflexion from
the first plate, or the plane passing through r R, r s, the ray r s
will not suffer reflexion from B, or will be so faint  as to be
scarcely visible.  The very same thing will happen if r s is a
ray polarized by double refraction, and  having its plane of po-
larization  in the plane passing through r R, r s.   Here  then
we have  a  new property or test of polarized  light,—that it
will not suffer reflexion from  a plate of glass B, when incident
at an angle of 56° 45', and when the plane of incidence or re-
flexion is  at right angles  to  the plane of polarization of the
ray.   If we now turn round  the tube  D G with the plate B, 
144            A TREATISE ON OPTICS.        PART II.

without moving the tube C D, the last reflected ray s E will
become brighter and brighter till the tube has been  turned
round 90°, when the plane of reflexion from B is coincident
with or parallel to that from A.   In this position the reflected
ray s E is brightest.  By continuing to turn the tube D G, the
ray s E becomes fainter and fainter, till, after being turned 90°
farther, the ray s E is faintest, or nearly vanishes, which  hap-
pens when the plane of reflexion from B  is perpendicular  to
that from A.   After a farther rotation of 90°, the ray s E will
recover its greatest brightness; and when, by a  still farther
rotation of 90°, the tube D G and plate B are brought back into
their first position, the  ray s E will again disappear.   These
effects may be arranged in a table, as follows:—
Inclination of the planes of the two reflexions, or the planes Kr 1 and r>£, or azimuth* of the plane r » K.	State of brightness of the image or ray t E reflected finm the second plate B.
At angles between 90° and 180° 180O............ At angles between 180° and 270° . 270°............ At angles between 270° and 300° . 360° or 0° ......... At angles between 0° and 90° . . !)(lo............	Scarcely visible The image grows brighter and brighter Brightest The image grows fainter and fainter Scarcely visible The i in age grows brighter and brighter Brightest The image grows fainter and fainter Scarcely visible
	
  If we now substitute in place of the ray r s one  of the po-
larized rays or beams formed by Iceland spar, so that its plane
of polarization is in the plane Rrs, it will  experience  the
very same changes as the ray R r does when polarized by re-
flexion from A at an angle of 56° 45'.  Hence it is manifest,
that a ray reflected at 56° 45' from glass has all the properties
of polarized light as produced by double refraction.
  (102.) In the  preceding observation?, the  rayRr is sup-
posed to be reflected only from the first surface of  the glass;
but Malus found  that the light reflected from  the second sur-
face of the glass  was polarized at the same time with that re-
flected from the first, although it obviously suffers reflexion at
a different angle, viz. at an angle equal to the angle of refrac-
tion at the first surface.
  The  angle of 56° 45',  at which light is polarized by re-
flexion from glass, is called its maximum polarizing angle, be-
cause the greatest quantity of light is polarized at that angle.
When the light was  reflected  at angles greater  or less than
56° 45',  Malus found that a portion of it only was polarized,
the remaining portion possessing all the properties of common
light.  The polarized portion diminished as the angle of inci-
dence receded on either side from 56° 45', and was nothing at
0°,  or a perpendicular incidence, and also nothing  at 90°, or
the most oblique  incidence. 
CHAP. XIX.   POLARIZATION BY REFLEXION.        145

  In continuing his experiments on this subject Malus found
that the angle of maximum polarization varied with different
bodies; and, after measuring it in  various substances, he con-
cluded  that it follows neither the order  of the  refractive
powers nor that of the dispersive powers, but that it is a prop-
erty of bodies independent of the other modes of action which
they exercise upon light.  After he had determined the angles
under which  complete polarization takes place in different
bodies, such as glass and water, he endeavored to ascertain the
angle at which it took place at their separating surfaces when
they  were put in contact.  In this  inquiry, however, he did
not succeed; and he remarks, " that  the  law according to
which this last angle depends on  the first  two remains to be
determined."
   If  a pencil  or beam of light reflected at  the maximum po-
larizing angle from glass  and other bodies were as completely
polarized as a pencil polarized by  double refraction, then the
two pencils would have been equally invisible when reflected
from the second plate, B, at the azimuths 90° and 270° ; but
this is not the case : the pencil polarized by double refraction
vanishes  entirely when it passes through a second rhomb, even
if it  is a beam of the sun's direct light; whereas the pencil
polarized by reflexion vanishes only if its light is faint, and if
the plates A and B have  a low dispersive power.   When the
 sun's light is used, there is a large quantity of unpolarized
 light and this unpolarized light is greatly increased when the
 plates A  and  B have a high dispersive power.  This curious
and most important fact was not observed by Malus.
   A very pleasing and instructive variation of the general ex-
 periment  shown in fig. 87.  occurred to me in examining this
 subject. If, when the plates of glass A and B have the position
 shown in the figure where the luminous body from which the ray
 s E proceeds  is invisible, we breathe gently upon the plate B,
 the ray s E will be  recovered, and  the luminous body from
 which it proceeds will be instantly visible.   The cause of this
 is obvious: a thin film of water is deposited upon the glass by
 breathing, and  as water polarizes light at  an angle of about
 53° 11', the glass B should have been inclined at an angle of
 53° 11' to the ray r s, in order to be incapable of reflecting the
 polarized ray ;* but as it is inclined 56° 45'  to the incident ray
 r s, it has the power of reflecting a portion of the ray r s.
   If the glass B is  now placed at an angle of 53° 11' to the
 ray r s, it will then  reflect a portion of  the polarized ray r s to
  * We neglect the consideration of the separating surface of the water
and glass, and suppose the glass B to be opaque.
                            N 
 146            A TREATISE ON OPTICS.         PART II.

 the eye at E;  but  if we breathe upon the glass B,  the re-
 flected light will disappear, because the reflecting surface is
 now water, and is placed at an angle of 53° 11', the polarizing
 angle for water.  If therefore we place two glass plates at B,
 the one inclined  56° 45', and the other 53° 11', to the beam
 r s, sufficiently large to fall  upon both, the luminous object
 will  be visible  in the  one but not in the other; but if we
 breathe upon the  two plates, we shall exhibit  the paradox of
 reviving an invisible image,  and extinguishing  a visible one
 by the same breath.  This experiment will be more striking
 if the ray r s is polarized by double refraction.

   On the Law  of the Polarization of Light by Reflexion.

   (103.) From  a  very extensive series of experiments  made
 to determine the maximum polarizing angles of various bodies,
 both  solid and fluid, I was led, in 1814, to the following simple
 law of the phenomena:—
   The index of refraction is  the tangent of the angle of po-
 larization.
   In  order to explain  this law, and to show  how to find the
 polarizing angle  for any body whose index  of refraction is
 known, let M N be the surface of any transparent body, such
 as water. From any point, r, draw r A perpendicular to M N,
        Fig. 88.         fig. 88., and round r as a centre de-
                       scribe a circle,  M A N D. From A
                       draw  A F, touching the circle at A,
                       and from any scale on which A r is 1
                       or 10 set off A F equal to 1-336 or
                       13-36, the index of refraction for water.
                       From F draw F r, which will be the
                       incident ray that will be polarized by
                       reflexion from the  water in the direc-
                       tion r S.  The  angle  A r R  will be
 53° 11', or the angle of maximum  polarization  for water.
 This  angle may be obtained more readily by looking for 1-336
 in the column  of natural tangents in a book of logarithms,
 and there will be found opposite to it  the corresponding angle
 of 53° 11'.   If we calculate  the angle of refraction T r D,
 corresponding to the angle of incidence A r R, or determine
 it by  projection, we shall find  it to be 36° 49'.
  From the  preceding  law we may draw the following con-
 clusions :—
  1. The maximum  polarizing angle, for all substances what-
 ever,  is the complement of the angle of refraction.  Thus, in
 water, the complement  of 36° 49' is 53° 11',  the polarizing
angle.
 
 CHAP.  XIX.   POLARIZATION  BY  REFLEXION.        147

   2.  At the polarizing angle, the sum of the angles of inci-
 dence and refraction is a right angle, or 90°.  Thus, in water,
 the angle of incidence  is 53° 11', and that of refraction 36°
 49', and their sum is 90°.
   3.  When a ray of light, R r, is polarized by reflexion, the re-
 flected ray, r S, forms a right angle with the refracted ray, r T.
   When light is reflected at the second surface of bodies, the
 law of polarization is as follows:—
   The index of refraction is the cotangent of the angle of
 polarization.
   In order to determine the angle in this case, let M N be the
 second surface of any body such as  water.   From r draw r A
        Fig. 89.        perpendicular  to  M N,  fig.  89.,  and
         j^         round r describe the circle M A N D.
    gr -r"^p±Jyi|'    From A draw A F, touching the circle
   /^^Mjjjjjj^     at A,  and  upon a scale in which r N is 1
   »===g^|jpl^d|i    take A F equal to -7485, that is to r.-^
 mJ     Jf^~~^[N' the reciprocal of the index of refraction,*
   I  ys\      J   and from F draw F r ;■ the ray R r will be
  t^         /    polarized when reflected in the direction
    ^^_L-^     r S.   The  maximum  polarizing angle
         11          ArR will be  36° 49',  exactly equal to
 the angle of refraction of the first surface.  Hence it follows,
   1.  That the polarizing angle at the second surface of bodies
 is equal to the complement  of the polarizing angle at  the first
 or to the angle  of refraction at the first surface.   The reason
 is, therefore, obvious why the portions of a beam  of  light re-
 flected at the first and second surfaces of a transparent parallel
 plate are  simultaneously polarized.
   2.  That the  angle which the reflected ray r S forms with
 the refraeted ray r T is a right angle.
   The laws of polarization now explained are applicable to
 the  separating  surfaces of two media of different refractive
 powers.  If the uppermost fluid is  wTater, and the undermost
 glass, then the index of refraction of their separating surface
 is equal to fiff^, to the greater index  divided by the lesser,
 which is 1-1415. By using this index it will be found that the
 polarizing angle is 48° 47'.
   When  the ray moves from the  less refractive substance
 into the greater,  as from water to  glass, as in the preceding
 case,  we must make use of  the law and the method above ex-
 plained for the first  surface of bodies; but when  the  ray
 moves from the greater refractive body into the less,  as from
oil of cassia to glass, we must use the law and method for the
second surface of bodies.
♦ The tangent of an angle to radius 1, is the reciprocal of the cotangent. 
148
A  TREATISE  ON  OPTICS.         PART II.
  If we lay a parallel  stratum of water upon glass whose
index of refraction is 1-508, the ray reflected from the refract-
ing surfaces will be  polarized when the  angle of incidence
upon the first surface of the water is 90°
  (104.) The preceding observations  are all applicable  to
white light,  or to the most luminous rays of the spectrum;
but, as every different color has a different index of refraction,
the law enables us to determine the angle of polarization for
every different color,  as in the following table,  where  it is
supposed that the most luminous ray of the  spectrum is the
mean one:—
	Index of tion.	Polarizing Angle.	Difference between the greatest and least Polarizing Angles.
C Red rays Water •? Mean rays / Violet rays I Red rays Plate Glass •? Mean rays f Violet rays ( Red rays Oil of Cassia •? Mean rays f Violet rays	1-330 1-336 1-342 1-515 1-525 1-535 1-597 1-642 1-687	53° 4'J 53 11 > 53 19 S 56 34 ) 56 45 } 56 55 \ 57 57 1 58 40 } 59 21 y	15' 21' 1°24'
  The  circumstance of the  different rays of the spectrum
being polarized at different angles, enables us to explain the
existence of unpolarized light at the  maximum polarizing
angle, or why the ray s E, in fig. 87., never wholly vanishes.
If we were to use red light, and set the two plates at angles of
56° 34', the polarizing angle of glass for red light, then the pen-
cil s E would vanish entirely.  But when  the light is white, and
the angle at which the  plates are set is 56° 45', or that which
belongs to mean or yellow rays, then it is only the yellow rays
that will vanish in the pencil s E.   A small portion of red and
a small portion of violet will be reflected, because the glasses
are not set at their  polarizing angles;  and  the mixture of
these two colors will produce a purple  color, which will be
that of the unpolarized light which remains in the pencil s E.
If we place  the plates  at the angle belonging to the red ray,
then the  red only will vanish, and  the color  of the unpo-
larized light will be bluish green.   If we place  the plates at
the angle corresponding with the blue light, then the blue
only will vanish, and the unpolarized light will be of a reddish
cast  In  oil of cassia, diamond, chromate of lead, realgar,
specular iron, and other highly dispersive substances, the color
of the unpolarized  light is extremely brilliant  and beautiful.
  Certain  doubly refracting crystals, such as  Iceland spar, 
CHAP. XIX.      PARTIAL POLARIZATION.             149

chromate of lead, Sic, have different polarizing angles on dif-
ferent surfaces, and in different directions on  the  same sur-
face ;  but there  is always one direction where the polariza-
tion is not affected by the doubly refracting force, or where
the tangent of the polarizing angle  is equal to the index of
ordinary refraction.

     On the partial Polarization of Light by Reflexion.

   (105.) If, in the apparatus in fig.  87, we make the ray R r
fall upon the plate A at an angle greater or less than 56° 45',
then  the ray s E will  not vanish entirely; but, as a consider-
able part of it will vanish like polarized light, Malus called it
partially polarized light, and considered it as  composed of a
portion of light perfectly polarized, and of another portion in
the state of common light  He found the quantity of polar-
ized light to diminish as the angle of incidence receded from
that of maximum polarization.
   M. Biot and M. Arago also maintained that partially polar-
ized light consisted partly of polarized and partly of common
light; and the latter announced that, at  regular angular  dis-
tances above and below the  maximum  polarizing angle,  the
reflected pencil  contained the  same proportion of polarized
light  In St. Gobin's glass he found that the same  proportion
of light was polarized at  an angle of incidence of 82° 48' as
at 24° 18'; in water he found that the same proportion was
polarized at 16° 12' as at 86° 31'; but he remarks,  "that  the
mathematical law which connects the value of the quantity of
polarized light with the angle of incidence and the refractive
power of the body has not yet been discovered."
  In  the  investigation of this  subject, I found that though
there was only one angle at which light could be completely
polarized by one  reflexion, yet  it might  be polarized at any
angle of incidence by a sufficient number of reflexions,  as
shown in the following Table.
BELOW THE POLARIZING ANGLE.		ABOVE THE POLARIZING ANGLE.	
No. of Refk-xjcms.	Angle at which the Light is polarized.	No. of Reflexions.	Angle at which the Light is polarized.
1 2 3 4 5 6 7 8	56° 45' 50 26 46 30 43 51 41 43 40 0 38 33 37 20	1 2 3 4 5 6 7 8	56° 45' 62 30 65 33 67 33 69 1 70 9 71 5 71 51
N2 
150            A TREATISE ON OPTICS.        PART II.

  In polarizing light  by successive reflexions, it is not neces-
sary that the  reflexions be  performed  at the same angle.
Some of them may be above and some  below the polarizing
angle, or all the reflexions  may be performed  at different
angles.
  From the  preceding facts it follows as  a necessary conse-
quence, that  partially polarized light, or light  reflected  at an
angle different from the polarizing angle, has suffered a physi-
cal change, which enables it to be more  easily polarized by a
subsequent reflexion.   The light, for example, which remains
unpolarized after five reflexions at 70°, in place of being com-
mon light, has suffered such a physical change that it is capa-
ble of being completely polarized by one reflexion more at 70°.
  This view of  the subject has been  rejected by M. Arago, as
incompatible with experiments and speculations  of his  own;
and, in estimating the value of the two opinions, Mr. Herschel
has rejected mine as the least probable.   It will be seen, how-
ever, from the following facts, that it is capable of the most
rigorous demonstration.
  It does not appear,  from the preceding inquiries, how a beam
of common light is converted into polarized light  by reflexion.
By a series of experiments made in 1829,1 have been able to
remove this difficulty.   It has been long known that a polar-
ized beam of light has its plane of polarization changed by re-
flexion from bodies.   If its  plane is inclined 45°  to the plane
of reflexion, its inclination will be diminished by a reflexion at
80°, still more by one at 70°, still more by  one at 60°; and at
the polarizing angle the plane of the polarized ray will be in
the plane of reflexion, the  inclination commencing again at
reflexions above the polarizing angle, and increasing till atO°,
or a perpendicular incidence, the inclination is again 45°.* I
now conceived a beam of common light, constituted as in fig.
81., to be incident on a reflecting surface, so that  the plane of
reflexion bisected  the angle of 90° which  the two planes of
polarization,  A B,  C D, formed with each other, as shown in
fig. 90.. No. 1., where M N is  the  plane of reflexion,  and
A B, C D the planes of polarization of the  beam of white
light, each inclined 45° to M N.   By a reflexion from  glass,
where the index of refraction is 1-525, at 80°, the inclination
of A B to M  N will be 33°  13',  as in No. 2., instead of 45°;
and in like manner the inclination of C D to M N will be 33°
13', in place  of 45°;  so that the inclination of A B to C D in
  * The rule for rinding the inclination is this:—Find the sum of the angles
of incidence and refraction, and also their difference; divide the cosine of
the former by the cosine of the latter, and the quotient will be the tangent
of the inclination required. 
CHAP. XIX.      PARTIAL POLARIZATION.
151
place of 90° is 66° 26', as in No. 2.  At an incidence of 65°
the inclination of A B to C I) will be 25° 36', as in No. 3.;
and at the polarizing angle of 56° 45' the planes A B, C D of
the two beams will be  parallel or coincident as in No. 4.  At
incidences below 56° 45' the planes will again open, and their

        No. 1.              Fig. 90.
         M     •     No. 2.         No. 3.         No. 4.






inclination will increase till at 0° of incidence  it is 90°, as in
No. 1., having been 25° 36' at an incidence of about 48° 15',
as in No. 3., and 66° 26' at  an  incidence of about 30°, as in
No. 2.
   In the process now described, we  see the manner in which
common light, as in No. 1., is converted into polarized light,
as in No. 4., by the action  of a reflecting surface.   Each  of
the two planes of its  component polarized beams is turned
round into a state of parallelism, so as to be a beam with only
one plane of polarization,  as in No. 4.; a  mode of polariza-
tion essentially different in its nature from that of double re-
fraction.  The numbers in  fig. 90. present us with beams  of
light in different stages of polarization from common light in
No. 1. to polarized light  in No. 4.  In No. 2. the beam has
made a certain approach  to polarization, having suffered a
physical change in the inclination ot its pianes; and  in No. 3.
it has made a nearer approach to it.   Hence we discover the
whole mystery of partial polarization, and we see that par-
tially polarized light  is light whose planes  of polarization
are  inclined at angles  less than 90°  and greater  than 0°.
The influence of successive reflexions is therefore obvious.  A
reflexion at 80° will turn  the  planes,  as in fig. 90., No. 2.;
another reflexion at 80° will bring them closer; a third still
closer; and so on: and though they never can by this process
be  brought  into a state of exact parallelism,  as in No. 4.
(which can only be done at the polarizing angle), yet they can
be brought infinitely near it, so that the beam will appear as
completely polarized as if  it had been reflected at the polar-
izing angle.   The correctness of my former experiments and
views is, therefore, demonstrated by the preceding analysis of
common light.
  It is manifest from these  views that partially polarized light
 
152             A TREATISE ON OPTICS.        TART II.
does not contain a single ray of completely polarized light;
and yet if we reflect it from the second plate B, in fig. H7,
at the polarizing  angle, a certain portion of it will disappear
as if it were polarized light, a result which led to the mistake
of Malus and others.  The light which thus disappears may
be called apparently polarized light; and I have  explained in
another place* how we may determine its quantity  at  any
angle of incidence, and for any refractive medium.  The fol-
lowing Table contains some of the results for glass, whose in-
dex of refraction  is 1-525.   The quantity of reflected  light is
calculated by a rule given by M. Fresnel.
Angles of Incidence.	Inclination of the Planes of Polarization, A B, C D, fie. 90.	Quantity of reflected Rays out of 1000.	--------------------- i Quantity of polarized Rays out of 1000.
0° 20 40 56 45' 70 80 85 90	90° 0' 80 26 47 22 0 0 37 41 66 26 78 24 90 0	43-23 43-41 49-10 79-5 162-67 391-7 616-28 1000-	0-7-22 33-25 79-5 129-8 156-6 123-75 0-
                       CHAR  XX.

 ON THE POLARIZATION OF LIGHT BV  ORDINARY REFRACTION.

   (106.) Although  it might have  been presumed that the
light refracted by bodies suffered some change, corresponding
to that which it receives from  reflexion, yet it was not until
1811 that it was discovered  that the refracted portion of the
beam contained  a portion of  polarized light.f
   To explain this property  of  light let R r, fig. 91.,  be a
beam of light incident at a great angle, between 80° and 90°,
on a horizontal  plate of glass, No. 1. ; a portion  of it will be
reflected at its  two surfaces, r and a, and the refracted beam
a is found to contain a small  portion of polarized light.
   If this beam  a falls  upon  a  second plate,  No. 2., parallel
to the first, it will suffer two  reflexions; and the  refracted
pencil 6 will  contain more  polarized light than  a.   In like
manner, by transmitting it through the plates Nos. 3, 4, 5, and

  *See  Phil. Transactions,  1830, p. 76., or Edinburgh Journal of Science,
New Series, No. V., p. 1C0.
  fThis discovery was made by independent observation by Malus, Biot,
and the author of this work. 
CHAP. XX.   POLARIZATION  BY  REFRACTION.
153
6., the last refracted pencil, fg, will be found to consist entirely,
so far as the eye can judge, of polarized light.  But, what is
very interesting, the beam fg is not polarized  in the plane of
refraction or reflexion, but in a plane  at right angles to it;
that is, its  plane of polarization is not  represented by A' B1

                          Fig. 91.
Us 81   as  is the ordinary ray in Iceland  spar, or as light
polarized  by reflexion,  but by C D' like the  extraordinary
ray in Iceland spar.   From a great number of experiments,
I found that the light of a wax candle at the distance of 10
or 12 feet was polarized at the following angles, by the fol-
lowing number of plates of crown glass.
No. of Plates of Crown Glass.	Observed Angles at which the Pencil is polarized.	[ No. of Plates of Crown Glass.	Observed Angles at which the Pencil is polarized.
8 12 16 21 24	79° 11' 74 0 69 4 63 21 60 8	27 31 35 41 1 47	57° 10' 53 28 50 5 45 35 41 41
  It follows from the above experiments, that if we divide the
number 41-84 by any number of crown glass plates, we  shall
have the tangent of the angle at which the beam is polarized
by that number.                              .  .
  Hence it is  obvious that the power of polarizing the re-
fracted light  increases with the angle of incidence, being no-
thing- or a  minimum at a perpendicular incidence, or 0°, and
the greatest  possible  or  a maximum at 90° of mcidence.  I
found  likewise, by various experiments, that the power of po-
larizing the light at any given angle increased with the re-
fractive  power of the  body, and consequently that  a smaller
number  of plates of a highly refracting  body was  necessary
than of a refracting body of low power, the angle of mcidence

b6Af MdiTBiot, and Arago considered  the beams a, b, &c.,
before they were completely polarized, as partially polarized, 
 154             A TREATISE ON  OPTICS.        PART II.

 and as consisting of a portion of polarized and a portion of
 unpolarized light; so, on the other hand, I concluded  from the
 following reasoning that the unpolarized light had suffered a
 physical change, which made it approach to the state of com-
 plete polarization.  For since sixteen plates  are required to
 polarize completely a beam of light incident at an  angle of
 69°, it  is clear that eight plates will not polarize the whole
 beam at the same angle, but will leave a portion unpolarized.
 Now, if this portion were absolutely unpolarized like common
 light, it would require to pass through other sixteen plates, at
 an angle of 69°, in order to be completely polarized; but the
 truth is, that  it requires to pass  through only eight plates to
 be completely polarized.  Hence I conclude  that the beam has
 been  nearly half polarized by the first  eight  plates,  and the
 polarization completed by the other eight.  This conclusion,
 though rejected by both the French  and English philosophers,
 is capable of rigid demonstration, as will appear from the fol-
 lowing observations.
   In order to determine the change which refraction produced
 in the  plane  of polarization of a polarized ray, I used prisms
 and plates of glass, plates of water, and a plate of a highly re-
 fractive metalline glass; and I found that a  refracting surface
 produced the greatest change at the  most oblique incidence, or
that of 90°;  and that the change gradually  diminished to a
 perpendicular incidence, or 0°, where  it was nothing. I found
 also that the greatest effect produced by  a single plate of glass
 was about 16° 39', at an angle of 86°; that it  was 3° 54' at an
 angle  of 55°, 1° 12' at an angle  of 35°, and 0° at an  angle
of 0°.*
   A beam of common light, therefore, constituted as in Jig.
92., No. 1.,  with each  of its planes A B, C D inclined 45° to
                    Fig. 92.
No. 1.     ,   No. 2.         No. 3.          No. 4.
the plane pf refraction, will have these planes opened 16° 39'
  * The rule for finding the inclination after a single refraction is as fol-
lows:—Find the difference between the angles of incidence and refraction,
and take the cosine of this difference.  This number will be the cotangent
of the inclination required ; and twice this inclination will be the inclina-
tion of A B to C D. 
CHAP. XX.    POLARIZATION  BY  REFRACTION.        155

each, by one plate of glass at an  incidence of 86°;  that  is,
their inclination,  in place of 90°,  will be 123° 18', as in No.
2.  By the  action of two  or  three plates more they will  be
opened wider, as in No. 3.; and by 7 or 8 plates they will  be
opened  to near 180°, or so that A B, C D nearly coincide, as
in No. 4., so as to form a  single polarized beam, whose plane
of polarization is perpendicular to  the plane of refraction.   I
have shown, in another place,* that these planes can never be
brought into mathematical coincidence by any number  of re-
fractions ; but they approach so near to it that the pencil is, to
all appearance, completely polarized with  lights of ordinary
strength.  All the light polarized by refraction  is only par-
tially polarized, and it has  the same properties  as that which
is partially  polarized by reflexion.  A certain  portion of the
light of a beam thus partially polarized, will disappear when
reflected  at the polarizing angle from the  plate B, fig. 87.;
and this quantity, which I  have elsewhere shown how to cal-
culate, is  given in the following table for a  single surface of
glass, whose index of refraction is  1-525.
Angle of Incidence.	Inclination of the Planes of Polarization A B, C D, fig. 93.	Quantity of transmitted Rays out of 1000.	Quantity of polarized Rays out of 1000.
0° 20 40 56° 45' 70 80 40' 85 90	90° 0' : 90 26 92 0 94 58 98 56 104 55 108 44 112 58	956-77 956-59 950-90 920-5 837-33 608-3 383-72 0	0 7-22 32-2 79-5 129-8 156-7 123-7 0
  Although the  quantity of light polarized by refraction, as
given  in  the last column of this Table, is calculated by a
formula essentially different from that by which the quantity
of light polarized by reflexion was calculated;  yet  it is  cu-
rious to see that the two quantities are precisely equal.  Hence
we obtain the following law:—
   When a ray of common light is reflected and refracted by
any surface, the quantity of light polarized by refraction is
exactly equal to that polarized by reflexion.
  This law is not at all applicable to plates, as it appeared to
be from the experiments of M. Arago.
  When  the preceding method of analysis  is applied to  the
light reflected by  the second surfaces of plates, we obtain the
following curious  law:—

  * See Phil. Transactions, 1830, p. 137., or Edinburgh Journal of Science,
New Series, No. VI.,  p. 218. 
156            A TREATISE ON OPTICS.        PART II.

  A pencil of light  reflected from the  second surfaces of
transparent plates, and reaching the eye after two refrac-
tions and an intermediate reflexion, contains at all angles of
incidence, from 0° to the maximum polarizing angle, a por-
tion of light polarized in the plane of reflexion.  Above the
polarizing angle, the  part of the pencil polarized in the
plane  of reflexion diminishes, till the incidence becomes 78°
7' in glass, rohen it disappears, and the whole pencil has the
character of common light.  Above this last angle the pencil
contains a quantity of light polarized perpendicularly to the
plane  of reflexion, which  increases to a maximum, and then
diminishes to nothing at 90°.*
  (107.)  As a bundle of glass plates acts upon light and po
larizes it as effectually as reflexion from  the  surface  of glass
at the polarizing angle, we may substitute a bundle  of glass
plates  in the apparatus, fig. 87, in place of the plates of glass
A, B.  Thus, if A {fig. 93.) is a bundle of glass plates which
Fig. 91
polarizes the transmitted ray s t, then, if the second bundle B
is placed as in the figure, with the planes of refraction of its
plates parallel to the planes  of refraction of the plates of A,
the ray s t will penetrate the second bundle; and if s t is in-
cident on B at the polarizing angle, not a ray of it will be re-
flected by the plates of B.   If B is now turned round its axis,
the transmitted  light v w will gradually diminish, and more
and more light will be reflected by  the plates of the  bundle,
till, after a rotation of 90°, the ray v w will disappear, and all
the light will be  reflected.   By continuing to turn round B,
the ray v w will re-appear, and reach its maximum brightness
at  180°, its minimum at 270°, and its maximum at 0°, after
having made one complete revolution.
  By this apparatus we may perform  the very same experi-
ments with refracted polarized light that we did with reflected
polarized light in the apparatus of fig. 87.
  We have now described  two methods of converting com-
mon light  into polarized light: 1st, By separating  by double
refraction the two oppositely polarized beams which constitute
common light;  and, 2dly, By turning round, by the action of
  * See Phil. Trans. 1830, p. 145.; or Edinburgh Journal of Science, No. VI.,
p. 234. New Series. 
CHAP. XXI.  COLORS  OP CRYSTALLIZED PLATES.     157

the reflecting and refracting forces, the planes of both these
beams till they coincide, and thus form light polarized in one
plane.  Another method still remains to be  noticed; namely,
to disperse or  absorb one of the oppositely polarized beams
which constitute  common light, and leave the other beam po-
larized in one plane. These effects  may be produced by agate
and tourmaline, &c.
   (108.) If we transmit a beam of common light through a
plate of agate, one of the oppositely polarized beams will be
converted into  a  nebulous light in one position, and the other
polarized beam in another position, so that one of the polar-
ized beams with a single plane of polarization is left.   The
same effect may  be produced by Iceland spar, arragonite, and
artificial salts prepared in a particular manner, to produce a
dispersion of one of the oppositely polarized beams.*
   When we transmit  common light through a thin plate  of
tourmaline, one of the oppositely polarized beams which con-
stitute common light is entirely absorbed in one position, and
the other in another position, one of them always remaining
with a single plane of polarization.
   Hence,  plates  of agate  and tourmaline  are of great use,
either in affording a beam of light polarized in one plane,  or
in dispersing and absorbing one of the pencils of a compound
beam, when we wish to analyze it or to examine the  color or
properties of one of the pencils seen  separately.
                       CHAP. XXI.

       ON THE COLORS  OF CRYSTALLIZED PLATES IN
                     POLARIZED  LIGHT.

  (109.) The splendid colors, and systems of colored rings,
produced by transmitting polarized light through transparent
bodies that possess double refraction, are undoubtedly the most
brilliant phenomena that can be  exhibited.   The  colors pro-
duced by these bodies  were  first discovered  by independent
observation, by M. Arago and the author of this volume ; and
they  have  been  studied with great success  by M. Biot and
other authors.
  In  order  to exhibit these phenomena, let  a polarizing ap-
paratus be prepared, similar in its nature to that in fig. 87.;
but without the tubes,as shown in fig. 94., where A is a plate

  * See Edinburgh Encyclopaedia, vol. xv. pp. 600, 601.; Phil.  Trans. 1819,
p. 146.
                            o 
158             A  TREATISE  ON  OPTICS.        FART II.

of glass which polarizes the ray R r, incident upon it at an
angle of 56° 45',  and reflects it polarized in the direction r s,
where it is received by a second plate of glass, B, whose plane
of reflexion is at right angles to  that  of the  plate A, and
which reflects it to the eye at O, at  an angle  of 56° 45'.  In

                          Fig. 94.
order that the polarized pencil r s may be sufficiently brilliant,
ten or  twelve plates of window glass, or, what is better
still, of thin  and well-annealed flint  glass,  should be substi-
tuted in place of the single plate A.   The  plate or plates at
A are called  the polarizing plates, because  their  only use is
to furnish us  with a broad and bright beam of polarized light.
The plate B  is called the analyzing plate, because its use is
to analyze,  or separate into its parts,  the  light  transmitted
through any body that may be placed between the eye and the
polarizing plate.
   If the beam of light R r proceeds from the sky, which will
answer well  enough for common purposes, then an eye placed
at O will see, in the direction O s, the part of the sky from
which the beam  R r proceeds.   But as r s will be polarized
light  if it is reflected at 56° 45' from A, almost none of it will
be reflected to the eye at O  from the plate B; that is, the  eye
at O will see, upon the part of the sky from which  R r pro-
ceeds, a black spot;  and when  it does not see this black spot,
it is  a  proof that the  plates A and B are  not placed at  the
proper inclinations to each other.   When a position is found,
either by moving A or B, or both, at which  the black spot is
darkest, the apparatus is properly adjusted.
   (110.) Having procured a thin film of sidphate of lime or
mica, between the 20th and the 60th of an inch thick, and
which may be split by a fine knife or lancet from  a  mass of
any of these minerals in a transparent state, expose it, as
shown at C E D F, so that the polarized beam r s may pass
through it perpendicularly.   If we now apply the eye at 0,
and look towards the  black spot in the direction O s, we  shall
see the surface of the plate  of sulphate of lime entirely  cov-
ered with the most brilliant colors.   If its  thickness is  per- 
CHAP. XXI.  COLORS  OF CRYSTALLIZED PLATES.     159

fectly uniform throughout, its tint will be perfectly uniform ;
but if it has different thicknesses, every different thickness
will display a different color—some  red, some green, some
blue, and some yellow, and all of the most brilliant  descrip-
tion.'   If we turn the film C E D F round, keeping it perpen-
dicular to the polarized beam, the colors will become less or
more bright without changing  their nature, and two lines,
CD E F at right angles will be found, so that when either of
them is in the plane of reflexion r s O, no colors whatever are
perceived, and the black spot will be seen as if the  sulphate
of lime had not been interposed,  or as  if a piece  of common
glass had been substituted for it.   It will also be observed, by
continuing the rotation of the sulphate of lime, that the colors
again  beo-in to  appear; and  reach  their greatest brightness
when  either of the lines G H, L K, which are inclined 45° to
C D, E F, are in the plane of reflexion rsO. The plane Rrs,
or the plane in which the light is polarized, is called the plane
of primitive polarization; the lines CD, E F,  the  neutral
axes; and G H, K L, the depolarizing axes, because they de-
polarize, or change the polarization of the polarized beam r s.
The brilliancy or intensity of the colors increases gradually,
from the position of no color, to that in which it is the most
brilliant.                             ,,   ,   „   , .   ^
   Let us now suppose the plate C E D F to be fixed in the po-
sition where it  gives the brightest color; namely, when G H
is perpendicular  to the plane of primitive polarization Rrs,
or parallel to the plane r s O, and let  the color be red.  Let
the analyzing plate B be made to  revolve  round the ray r s,
beginnino- its motion at 0°, and preserving always the same
 inclination to the  ray r s, viz.  56°  45'.  The brightest red
 beino- now visible at 0°, when the plate B begms to move from
 its position shown in the figure, its brightness will gradually
 diminish till B has turned round 45°, when  the red color will
 wholly disappear, and the black spot in the sky be seen.  Be-
 yond 45° a faint green will make its appearance, and will be-
 come brighter and brighter  till  it attains its greatest bright-
 ness at 90°.   Beyond 90° the green becomes paler  and paler
 till it disappears at 135°.  Here the  red again  appears, and
 reaches its maximum brightness at  180°.  The very same
 changes are repeated while the plate B passes from 180°
 round to its first position at 360° or  0°.  From this experi-
 ment it appears, that when the film C E D F alone revolves,
 only one  color is seen;  and when the plate B only  revolves,
 two colors are seen during each half of  its revolution
    If we repeat the preceding experiment with films ot diner-
 ent thicknesses, that give different colors, we shall find that the. 
160
A  TREATISE  ON  OPTICS.        PART n.
two colors  are  always complementary to each other, or to-
gether make white light
   (111.) In order to understand the cause of these beautiful
phenomena, let  the eye be placed between the film  and the
plate B, and it will be seen that the light transmitted through
the film is white, whatever be the position of the film.   The
separation of the colors is therefore produced, or the white
light is analyzed, by reflexion from the  plate B.  Now, sul-
phate of lime is a doubly refracting crystal;  and one of its
neutral axes, C D, is the section of a plane passing through its
axis, while E F  is the  section of a plane perpendicular to the
principal section. Let us now suppose either of these planes, for
example E F, to be placed, as in the figure, in the plane of po-
larization  R r s of the polarized light; then this ray will not be
doubled, but will pass into the ordinary ray of the crystallized
film; and falling upon B, it will not suffer reflexion.  In like
manner,  if  C D is brought into the plane Rrs,  it will  pass
entirely into the ordinary ray, which, falling upon B, will not
suffer reflexion.   In these two positions of the film, therefore,
it forms only a single image or beam ; and as the plane of po-
larization  of this image or beam is at right angles to the plane
of reflexion  from B, none of it is reflected to the eye at 0.
But in every other position of  the doubly refracting  film
C E D F,  it  forms two images  of different intensities, as may
be inferred from fig. 86.; and when either of the depolarizing
axes G H or K L is in the plane of primitive polarization, the
two images are of equal brightness, and are polarized in op-
posite planes; one  in the plane  of primitive polarization,  and
the other at right angles to it.   Now, one of these  images is
red, and the other green, for reasons which will be afterwards
explained; and as the green  is polarized in the plane of primi-
tive polarization Rrs, it does not  suffer reflexion from the
plate B; while the red, being polarized at right angles to that
plane, is reflected to the eye  at O, and is therefore alone seen.
For a similar reason, when B is turned round 90°, the red will
not suffer reflexion from it; while the green will suffer re-
flexion, and  be transmitted to the eye at O.  In this case the
plate  B analyzes the compound beam of white light trans-
mitted through the film of sulphate of lime, by reflecting the
half of it  which is polarized in  the plane  of  its reflexion, and
refusing to reflect the other half, which is polarized in an op-
posite plane.  If the two beams had been each white light, as
they are in thick plates of sulphate of lime, in place of seeing
two different colors during the  revolution of the plate B, the
reflected pencil s O would have undergone different variations
of brightness, according as the two oppositely polarized beams 
 CHAP. XXI.  COLORS OF CRYSTALLIZED PLATES.     161

 of white light were more or less reflected by it;  the positions
 of greatest brightness being those where the red and  green
 colors were the brightest, and the darkest points being those
 where no color was visible.
   (112.) The analysis of the white beam composed of two
 beams of red and green light, has obviously been effected by
 the power of the plate to reflect the one and to transmit or
 refract the other;  but the same beam may be analyzed  by va-
 rious other methods.  If we make it pass through a rhomb of
 calcareous spar sufficiently thick to separate by double refrac-
 tion the red from the green beam, we shall at the same time
 see both the colored beams, which we could not do in the for-
 mer case ; the one  forming the ordinary, and the other the ex-
 traordinary image.   Let us now remove the plate B, and sub-
 stitute for it a rhomb of calcareous spar, with its principal sec-
 tion in  the  plane  of reflexion r s O, or perpendicular to the
 plane of primitive polarization Rrs, and let the rhomb have
 a round aperture in the side farthest from the eye, and of such
 a size that the two images of the aperture, formed by double
 refraction,  may just touch  one  another.  Remove the film
 C E D F, and the eye placed behind the rhomb will see only
 the extraordinary  image of the aperture,  the  ordinary one
 having vanished. Replace the film, with its neutral axes as in
 the figure, parallel and perpendicular to the plane Rrs, and
 no effect will be produced ; but  if either of the  depolarizing
 axes  are brought into the plane Rrs, the  ordinary image  of
 the aperture will  be a brilliant red, and the extraordinary
 image a brilliant green; the double  refraction of the rhomb
 having  separated these  two differently colored and oppositely
 polarized beams.   By turning round the film, the colors will
 vary in brightness  ; but the same image will always have the
 same color.  If we  now keep the film fixed in the position
 that gives the finest colors, and move the rhomb of calcareous
 spar round, so that its principal section  shall make a complete
 revolution, we shall find that, after revolving 45°  from its first
 position, both images become white.  After revolving  90°, the
 ordinary image  that was formerly red is now green,  and the
 extraordinary image that was formerly green is now red. The
 two images become again white at 135°, 225°, and 315°; and
at 180°, the ordinary image is again red,  and the  extraordi-
nary one green; and at  270°, the ordinary image is green,
and the other red.
  If we use a  large circular aperture on  the  face of the
rhomb, the ordinary and extraordinary images O, E will over-
lap each other,  as  in fig. 95.; the overlapping parts at F G
being pure white light, and the parts at C and D having  the
                           02 
162
A  TREATISE ON OPTICS.       PART II.
colors above  described.  This experiment affords ocular de-
monstration that the  two colors at C  and D are comple-
mentary, and form white light.
  The analysis of the compound beam transmitted by the sul-
phate of lime may also be effected by a  plate of agate, or by

                          Fig. 95.

                         O     E
any of the  crystals, artificially prepared for the purpose of
dispersing  one of the component beams.   The agate being
placed between the eye and the film C E D F, it will disperse
into nebulous light the red beam, and enable the green one to
reach  the eye;  while in another position it will scatter the
green beam, and allow the red light to reach the eye.  With
a proper piece of agate this experiment is both beautiful and
instructive ; as the nebulous light, scattered round the bright
image, will be green when the distinct image is red, and red
when the distinct image is green.
   The analysis may also be effected by the absorption of towr-
maline and  other similar substances.  In one position the tour-
maline absorbs the green beam, and allows the red to pass;
while  in another position  it absorbs  the red, and suffers the
green to pass.  The yellow color of the tourmaline, however,
is a disadvantage.
   The analysis  may also he performed by a bundle of glass
plates, such as A or B, fig. 93.  In one position  such a bundle
will transmit all  the red, and reflect all the green; while in
another position it will transmit all the green, and reflect  all
the red,  in  the opposite manner, but  according to  the same
rules as the analyzing plate B, Jig. 94.
   (113.) In all these experiments the  thickness of the  sul-
phate  of lime has been  supposed such as to give a red and a
green tint; but if we take a film 0-00046  of an English inch
thick,  and  place  it at C E D F in fig. 94., it will produce no
colors at all, and the black spot in the sky will be seen, what-
ever be  the position of the film.   A film 000124 thick will
give the white of the first order in Newton's scale  of colors,
given  in p. 93;  and a  plate 0-01818 of an  inch thick,  and
all plates of greater thickness, will  give  a white composed
of all the colors.   Films or plates of intermediate thicknesses 
  CHAP. XXI.  COLORS OF CRYSTALLIZED PLATES.     163

  between 000124 and 0-01818 will give all the intermediate
  colors in Newton's Table between the white of the first order
  and the white arising from the mixture of all the colors. That
  is, the colors reflected to the eye at O will be those in column
  2d, while  the colors observed by turning round flie plate  B
  will be those in column 3d ;  the one set of colors correspond-
  ing to the reflected tints,  and the other to the transmitted tints
  of thin plates.  In order to determine the thickness of a  film
  of" sulphate of lime which gives any particular color in the
  Table, we  must have recourse to the numbers in the last col-
  umn  for glass, which has nearly the same refractive power as
  sulphate of lime. Suppose it is required to have the thickness
  which corresponds to the red of the first  spectrum or order  of
  colors.  The number in the column for glass,  opposite red, is
  5f; then,  since the white  of the first order is produced by a
  film 0-00124 of an inch thick, the number corresponding to
  which is 3| in the column for glass, we say, as 3f is to 54, so
  is 0-00124 to 0-00211, the thickness which will give the red
'  of the  first order.  In the same manner, by having tiie thick-
  ness of any film of this substance, we can determine the color
  which it will produce.
     Since the colors vary with the thickness of the plate, it is
  manifest that if we could form a wedge of sulphate of lime,
  with its thickness varying from 0-00124 to 0-01818 of an inch,
   we should observe at once all the colors in Newton's Table in
   parallel stripes.  An experiment of the same  kind may  be
  made in the following manner :—Take a plate of sulphate of
   lime, M N, fig. 96., whose  thickness  exceeds 0-01818 of an
Fig. 96.
inch.   Cement it  with isinglass on  a plate of glass;  and
placing it upon a fine lathe, turn out  of it with  a very sharp
tool a concave or hollow surface between A and B, turning it
so thin at the centre that it either begins to break or is on the
eve of breaking.   If the plate  M N is now placed  in water,
the water will after some time dissolve a small portion of its
substance, and polish the turned surface to a certain degree.
If the plate is now held at C E D F, fig. 94., we shall see all
the colors in Newton's Table in the form  of  concentric rings, 
164             A TREATISE ON OPTICS.        PART II.

as shown in the figure.  If the thickness diminishes rapidly,
the rings will be  closely packed together, but if the turned
surface is large, and the thickness diminishes slowly, the col-
ored bands will be broad.  In place of turning out the con-
cavity, it might be done better by grinding it out, by applying
a convex surface  of great radius,  and using the finest emery.
When the plate M N is thus prepared, we may give the most
perfect polish to the turned surface by cementing upon it a
plate of glass with Canada balsam. The balsam will dry, and
the plate may be preserved for any length of time.
  By the method  now described, the most beautiful patterns,
such as are produced in bank-notes, &c, may be turned upon
a plate of sulphate of lime 0-01818 of an inch thick, cemented
to glass.  All the grooves or lines that compose  the pattern
may be  turned  to different  depths,  so as to leave  different
thicknesses of the  mineral, and the grooves of different depths
will all appear as different colors, when the pattern is held in
the apparatus in fig. 94.  Colored drawings of figures and
landscapes may in like manner be executed, by scraping away
the mineral to the thickness that will  give the required colors;
or the effect may be  produced by an etching ground, and
using water  and  other fluid solvents of sulphate of lime to
reduce  the  mineral to the required thicknesses.  A cipher
may thus be executed upon  the  mineral; and if we cover
the surface  upon  which it is scratched, or cut, or dissolved,
with a balsam or fluid of exactly the same refractive power as
the sulphate, it will be absolutely illegible by common light,
and may be distinctly read in polarized  light, when placed at
C E D F in fig. 94.
  As  the colors produced in the preceding experiments vary
with the different thicknesses of the  body which  produces
them, it  is obvious that two films put-together, as they lie in
the crystal with similar lines  coincident or parallel, will pro-
duce a color corresponding to the sum of their thicknesses,
and not the color which arises from the mixture  of  the two
colors which they  produce separately.   Thus, if we take two
films of sulphate of lime,  one of  which gives the orange of
the  first order, whose number in the last column in Newton's
Table, p. 93., is 5», while the other gives the  red of the 2d
order, whose number is 111; then by adding these numbers, we
get 17, which corresponds in the Table to greenish yellow of the
3d order. But if the two plates are crossed, so that similar lines
in the one are at right angles to similar  lines in the other,
then the tint or color which they  produce will be that which
belongs to the difference of their thicknesses.   Thus, in the
present case,  the difference of the above numbers is 6|, which 
CHAP. XXI. COLORS OF CRYSTALLIZED PLATES.        165

corresponds in the Table  to  a reddish violet of the second
order.  If the plates which are thus crossed are equally thick,
and produce the  same colors,  they will destroy each oOier's
effects, and blackness will be produced; the difference of the
numbers in the Table being 0.   Upon this principle, we may
produce  colors by crossing plates of such a thickness as to
give  no  colors separately,  provided  the difference of 'their
thickness does not exceed  001818;  for  if the difference of
their thickness is greater than this, the tint will be white, and
beyond the limits of the Table.
  If the polarized  light  employed  in the preceding experi-
ments is  homogeneous, then the colors reflected from the plate
B will always be those of the homogeneous light  employed.
In red light, for example, the colors  or rather shades which
succeed eaeh other, with different thicknesses of the mineral,
will be red at one thickness, black at another, red at another,
and black at another, and so on with all the different colors.
  If we place the specimen shown in fig. 96. in violet light, the
rings  A B will be less  than in red light; and in intermediate
colors they will be of intermediate  magnitudes, exactly as in
the rings of thin plates formerly described. When white light
is used, all the different sets of rings are combined in the very
same  manner as we have already explained, in thin plates of
air, and  will form  by their combinations the various colored
rings in  Newton's Table.
                      CHAP.  XXH.

   ON THE SYSTEM OF COLORED RINGS IN CRYSTALS WITH
                         ONE AXIS.

   (114.) In all the preceding experiments the film C E D F
must be held at such a distance from the eye, or from the plate
B, that its surface may be distinctly seen, and in the apparatus
used by different philosophers this distance was considerable.
In the year 1813 I adopted another method, namely, that  of
bringing the film or crystal to be examined as close to the eye
as possible,  a very small  plate, B, not above one fourth  of  an
inch, being  interposed, as in fig. 94., between  the crystal and
the eye, to reflect the  light transmitted through the crystal.
By this means I discovered the systems of rings formed  along
the axes of crystals with one and  two axes,  which form the
most splendid phenomena in  optical science,  and which  by
their analysis have led philosophers to the  most important dis«
coveries. 
166            A TREATISE ON OPTICS.        PART II.
Fig. 97.
  I discovered them in ruby, emerald, topaz, ice, nitre, and a
great variety of other bodies, and  Dr. Wollaston afterwards
observed them hi Iceland spar.
  In order to observe the system of rings round a single axis
of double refraction, grind down the summits or obtuse angles
A X of a rhomb of Iceland spar, fig. 72., and replace them by
plane and polished surfaces perpendicular to the axis of double
refraction A X.  But as this is not  an easy operation without
the aid of a lapidary, I have adopted  the following  method,
which enables us to transmit light along the axis A X without
injuring  the rhomb.  Let C D E F, fig. 97, be Oie principal
                       section of  the rhomb; cement upon
                       its surfaces CD, FE, with  Canada
                       balsam,  two  prisms, D L K, F G H,
                       having  the  angles L D K, G F H
                       each equal to about 45°; and by let-
                       ting fall a ray of light perpendicu-
                       larly upon the face D L, it will pass
                     ^E along the axis A X, and emerge per-
                       pendicularly through  the face F G.
                       Let the rhomb thus prepared be held
in the polarized beam r s,fig. 94, so that r s may pass along
the axis A X, and let it  be held as near the plate B as possible.
When the eye is held very near to B, and looks along 0 s as
it were through the reflected image of the rhomb C E, it will
perceive along its axis A X a splendid  system of colored rings
resembling that shown mfig. 98., intersected by a rectangular
B^
black cross, A B C D, the arms of which meet at the centre
of the rings.   The colors in these rings are exactly the same
as those in Newton's Table  of colors, and consequently the 
CHAP. XXII. RINGS IN CRYSTALS WITH ONE AXIS.      167

same as the system of rings seen by reflexion from the plate
of air between the object  glasses.   If we turn the rhomb
round its axis, the  rings will suffer no change; but if we fix
the rhomb, or hold it  steadily, and turn  round the  plate B,
then,  in the azimuths 0°, 90°, 180°, and 270° of its revolution,
we shall see the same system of rings; but at the intermediate
azimuths of 45°, 135°, 225°, and 315°, we shall see another
system, like that in fig. 99., in which all the colors are  com-
plementary to those mfig. 98., being the same  as those seen
Fig. 99.
in the rings formed by transmission through the plate of air.
The superposition of these two systems of rings would repro-
duce white light.
  If, in place of the glass plate B, we substitute a  prism of
calcareous spar, that separates its two images greatly, or a
rhomb of great thickness, we shall see in the ordinary image
the first system of rings, and in  the  extraordinary image the
second system of complementary  rings,  when the  principal
section of the prism or rhomb is in the plane rsOas formerly
described.
  As the light which forms the first system of rings is polarized
in an opposite plane to  that which forms the second system,
we may disperse  the one system  by agate,  or absorb it by
tourmaline, and thus render the  other visible, the  first or the
second system being dispersed or absorbed according to the
position of the agate or the tourmaline.
  If we  split the  rhomb of  calcareous syar,fig. 97,  into  two
plates by the fissure M N, and examine the rings produced by
each plate separately, we shall find that the rings produced by
each plate are larger in diameter than those  produced by the
whole rhomb, and that the rings increase in size as the thick-
ness of the plate  diminishes.  It will also be found that the
circular area contained within any one ring is to the circular 
168
A  TREATISE  ON  OPTICS.        PART II.
area of any other ring, as the number in Newton's Table cor-
responding to the tint of the one ring is to Uie number cone-
sponding to the tint of the other.
   If we use homogeneous light we shall find that the rings
are smallest in violet light and largest in red light, and of in-
termediate sizes in the intermediate colors, consisting  always
of rings of  the color of the light employed, separated by black
rings.   In white light all the rings formed  by the seven dif-
ferent colors are combined, and  constitute the colored  system
above  described, according to the principles which were fully
explained in Chapter XII.
   (115.) All the other crystals which have one axis of double
refraction, give a similar  system of rings along their axis of
double refraction; but those produced by the positive crystals,
such as zircon, ice, Sic, though to the eye  they differ in no
respect from those of the negative crystals, yet possess dif-
ferent  properties.  If we take  a system  of rings  formed by
ice or  zircon, and combine it with a system of rings of the
very same  diameter  formed  by Iceland  spar,  we  shall  find
that the two systems destroy one another,  the one being nega-
tive and the other positive; an effect which might have been
expected from the opposite kinds of double refraction possessed
by these two crystals.
   If we combine two  plates of  negative crystals, such as Ice-
land spar and beryl, the system of rings  which they produce
will be such as would be formed by two plates of Iceland spar,
one of which is the plate employed, and the  other  a plate
which  gives rings of the same size as the  plate  of beryl.  But
if we  combine a plate of a negative crystal with a plate  of a
positive crystal, such as one of Iceland spar with  one of zir-
con or ice, the resulting system of rings, in place of arising
from the sum of their separate  actions, will arise from their
difference;  that is, it will be equal to the system produced by
a plate of Iceland spar whose thickness is equal to the differ-
ence of the thicknesses of the plate of Iceland spar employed,
and another plate of Iceland spar that would give rings of the
same size as those produced by the zircon or ice.
   These experiments of combining rings are not easily made,
unless  we employ crystals which have external faces  perpen-
dicular to the axis of double refraction, such as  the  variety of
Iceland spar called spath calcaire busee,  some of  the micas
with one axis, and well crystallized plates of ice, Sic   When
two such plates cannot be obtained, I have adjusted the axes
of the  two plates so as to coincide, by placing between them,
at their edges, two or three small pieces of soft wax, by press- 
 CHAP. XXII. RINGS IN CRYSTALS WITH ONE AXIS.      169

 ing which in different directions, we may produce a sufficiently
 accurate  coincidence of the  systems of rings to establish the
 preceding conclusions.
   If, when  two systems of rings are  thus combined, either
 both negative or both positive, or the one negative and tiie
 other positive, we interpose between the plates which produce
 them crystallized films of sulphate of lime or mica, we  shall
 produce the most beautiful changes in  the form and character
 of the  rings.  This  experiment I found  to be particularly
 splendid when  the film was placed between two plates of the
 spath calcaire basee of the same thickness,  and taken from the
 same crystal.  By fixing them  permanently with their faces
 parallel, and leaving a sufficient interval  between  them for
 the  introduction  of films of  crystals, I had an  apparatus by
 which the  most  splendid phenomena  were  produced.   The
 rings were  no longer symmetrical round their axis, but exhib-
 ited the most beautiful variety of forms during the rotation of
 the combined plates, all of which are easily deducible from the
 general laws of double refraction and polarization.
   The table of crystals that  have negative double  refraction
 shows the bodies that  have a negative  system of rings; and
 the table of positive crystals  indicates those that have a posi-
 tive system of rings.
   (116.) The following is the method which I have used for
 distinguishing whether any  system of rings is positive or
 negative.  Take a film of sulphate of lime, such as that shown
 at C E D F, fig. 94., and mark upon its surface the lines or
 neutral axes C D, E F as nearly as may be.   Fix this film by
 a little wax on the surface, L D or F G, fig. 97, of the rhomb
 which produces  the  negative  system of rings.  If the film
 produces alone the red of the second order, it will now, when
 combined with the rhomb,  obliterate part of the red ring of
 the second order, either in the  two quadrants A C, B D, fig.
 98., or in the other two, A D, C B.   Let it obliterate the red
 in A C, B D; then if the line C B,fig. 94, of the film crosses
 these two quadrants at right angles to the rings, it will be the
 principal axis of the sulphate of lime; but  if it crosses the
 other two quadrants,  then the  line E F, which  crosses the
 quadrants A C,  B D, will be the  principal axis of sulphate of
 lime, and it should be marked as such.  We shall  suppose,
 however,  that C D has been  proved to be the principal  axis.
 Then, if we wish to  examine whether any other system of
rings is positive or negative, we  have only  to cross the rings
with the axis C D,  by  interposing the film: and  if it obliter-
ates the red ring of the second order in the quadrant which it 
170            A TREATISE ON OPTICS.         PART II.

crosses, the system will be negative; but if it obliterates flie
6ame ring in the other two quadrants which it does  not  cross,
then the system will be positive.  It is of no consequence
what color the film polarizes, as it will always obliterate the
tint of the  same nature  in the system of rings  under exam-
ination.
  (117.) In order to  explain the  formation of the systems of
rings seen along the axes of crystals, we must consider the
two causes on which  they depend;  namely,  the thickness of
the crystal through which the polarized light passes, and the
inclination of the  polarized light to the axis of double refrac-
tion or the axis pf the rings.  We  have already shown how
the tint or color varies with  the thickness of the crystallized
body, and how, when we know the color for one thickness, we
may determine it for  all  other thicknesses, the inclination of
the ray to  the axis remaining always the same.   We have
now, therefore, only to consider the  effect of inclination to the
axis.   It is obvious that  along the axis of the crystal, where
the two black lines A B, C L>,fig. 98., cross each other,  there
is neither double refraction  nor  color.  When the polarized
ray is slightly inclined to the axis,  a  faint tint  appears, like
the blue in the first order of  Newton's scale; and as the  incli-
nation  gradually increases, all the colors in Newton's table are
produced in succession, from  the very black of the first  order
up to the reddish white of the seventh order.  Here, then, it
appears that an increase in the inclination of the polarized
light to the axis corresponds to an  increase of thickness; so
that if the  light always passed through the same thickness of
the mineral,  the different colors  of the  scale would  be pro-
duced by difference of inclination alone.  Now, it is found by
experiment, that in the same thickness of the mineral, the
numerical value of the  tints, or the numbers  opposite to the
tints in the last column of Newton's table, vary as the square
of the  sine of the inclination of the polarized ray to the axis.
Hence it follows, that at equal inclinations the same tint will
be produced;  and  consequently,  the similar  tints will  be at
equal distances from the  axis of' the rings, or  the lines of equal
tint or  rings will be circles whose centre is in the axis.   Let
us suppose that at an  inclination of 30° to the axis we observe
the blue of the second order, the numerical value of whose
tint is 9 in Newton's  table, and that we wish to know the tint
which  would be produced at  an inclination of 45°.   The sine
of 30° is -5, and its square -25.  The sine of 45° is  -7071, and
its square .5.  Then we say, as -25 is to 9, so  is -5 to 18, which
in the table  is the numerical value of the  red of the  third 
CHAP. XXII. RINGS IN CRYSTALS WITH ONE AXIS.     171

order.  If we suppose the thickness of the mineral to be in-
creased at the inclinations 30° and 45°, then the numerical
value of the tint would increase in the same proportion.
   It is obvious from what  has been  said, that the polarizing
force, or  that which produces the rings,  vanishes when the
double refraction vanishes, and increases and diminishes with
the double refraction,  and according to the  same law.  The
polarizing force, therefore, depends on the force of double re-
fraction;  and we  accordingly  find that crystals  with high
double refraction have the power of producing the same tint,
either at much less thicknesses, or at much less inclinations to
the axis.   In order  to compare the polarizing intensities of
different crystals, the best way is to compare the tints which
they  produce at right angles to the axis where the  force of
double refraction and polarization is a maximum, and with a
given thickness  of the mineral.   Thus,  in the  case given
above, we may find the tint at right angles to  the axis, by
taking the square of the sine of 90°, which is 1; so that we
have  the following proportion: as -25 is to 9, so is 1 to 36, the
value of the maximum tint of calcareous spar at right angles
to the axis, upon the supposition that a tint of the value of 9
was produced at an inclination of 30°.  If we have measured
the thickness of Iceland spar at which the tint 9 was produced,
we are prepared to  compare the polarizing intensity of Iceland
spar with that of any other mineral.  Thus, let us take a plate
of quartz, and let  us suppose  that  at  an inclination of 30°,
and with a thickness fifty-one times as great as  that of the
plate  of Iceland spar, it produces a yellow of the first order,
whose value is about 4.   Then  to find the tint at 90°, or at
right  angles  to the axis, we say, as the  square  of the  sine of
30°, or -25, is to 4,  so is the square of the sine of 90°, or 1, to
16, the tint at 90°, or the green of the third order.   Now the
polarizing power or intensity of the Iceland spar would have
been  to that  of the  quartz as 36 to 16, or 24. times as great, if
the thickness of the two minerals had been the same; but as
the thickness of the quartz was 51 times as great as that  of
the Iceland spar, the polarizing  intensity of the Iceland spar
will be 51 multiplied by 2\ times, or 115 times  as great as
that of quartz.   The intensities for various crystals have been
determined by several observers, but the following have been
given by Mr. Herschel:— 
172
A  TREATISE ON OPTICS.
PART II.
Polarizing Intensities of Crystals with One Axis.
--.— ---	Val'ie of	Thicknesses that produce
	highe.t Tint.	the same Tint.
Iceland spar -	35801	0-000028
Hydrate of strontia	1246	0-000802
Tourmaline -	851	0-001175
Hyposulphate of lime -	470	0-002129
Quartz......	312	0-003024
Apophyllite, 1st variety	109	0-009150
Camphor - - -	101	0-009856
	41	0-024170
Apophyllite, 2d variety	33	0-030374
----------3d variety	3	0-366620
  The above  measures are  suited to yellow  light,  and the
numbers in the second column show the proportions of the
thicknesses of the different substances that produce the same
tint.  The polarizing force of Iceland spar is so enormous at
right angles to the axis, that  it is almost  impracticable to pre-
pare a film of it sufficiently thin to exhibit the colors in New-
ton's table.
                      chap. xxni.

  ON  THE SYSTEMS OF COLORED  RINGS IN CRYSTALS WITH
                        TWO AXES.

  (118.) It was long believed that all crystals had only one
axis of double  refraction;  but, after I discovered the double
system of rings in topaz and other minerals, I found that these
minerals had two axes of double refraction as well as of polar-
ization, and that the possession of two axes characterized the
great body  of  crystals which are  either formed  by art, or
which occur in the  mineral kingdom.
  The double system of rings, or rather one of the sets of the
double system  of rings in  topaz, first presented itself to me
when I was looking along the axis of topaz, which reflected a
part of the light of the sky that happened to be polarized, so
that they were seen without the aid either of a polarizing or
an analyzing plate.  In this and some other minerals, however,
the axes of double refraction are so much inclined to one an-
other, that we cannot see the two systems of rings at once.  1
shall  therefore  proceed to explain them as exhibited by nitre,
in which I also discovered  them and examined many of their
properties. 
CHAP. XXIII. RINGS IN CRYSTALS WITH TWO AXES.    173
  Nitre, or saltpetre, is an artificial  substance which crystal-
lizes in six-sided prisms with angles of about 120°. It belongs
to the prismatic system of Mohs, and has therefore two axes
of double refraction along which a ray of light  is not divided
into two.  These axes are each inclined about 2^° to tiie axis
of the prism, and about 5° to each other. If, therefore, we cut
off a piece of a prism of nitre with a knife driven by a smart
blow from a hammer, and  polish two flat surfaces perpendicu-
lar to the axis of the prism, so as to leave a thickness of the
sixth  or eighth of an inch,  and then transmit  the polarized
light r s, fig. 94, along the axis of the prism, keeping the
crystal as near to the plate B as possible on one side, and the
eye as near it as possible on the other, we shall see the double
system of rings, A B, shown in fig. 100., when the plane pass-
ing through the two  axes of nitre is in the plane of primitive

            Fig. 100.                    Fig. 101.
polarization, or in the  plane  of reflexion rsO, fig. 94., and
the system shown in j%. 101. when the same plane is inclined
45° to either of these  planes.  In passing  from the state of
fig. 100. to that of fig. 101., the black lines assume the forms
shown in^s. 102. and 103.
  These systems of rings have, generally speaking,-the same
colors as those of thin  plates, or  as  those of the systems of
rings round one axis.   The orders of colors commence at the
                           P2
9 
174            A  TREATISE  ON  OPTICS.        PART II.

centres A and B of each system;  but at  a  certain distance,
which in fig. 100. corresponds to the  sixth ring, the rings, in
Fig. 102.                   Fig. 103.
place of returning and encircling each pole A and B, encir-
cle the two poles as an ellipse does its two foci.
  When we diminish the thickness of the plate of nitre, the
rings enlarge; the fifth ring will  then surround  both poles,
At a less thickness, the fourth ring will surround them, till at
last all the rings will surround both poles, and the system will
have a great resemblance to the system surrounding one axis,
The place of the  poles A, B never changes,  but the black
lines A B, C D become broad and  indefinite;  and the whole
system is distinguished from the  single system principally by
the oval appearance of the rings.
  If we increase the thickness of the nitre, the rings will di-
minish in size;  the colors will lose their resemblance to those
of Newton's scale; and the  tints do not  commence at the
poles A, B, but at virtual poles in their proximity.   The color
of the rings within the  two poles is red,  and without them
blue; and the great body of the rings is pink and green.
  As the same color exists  in  every part of the same curve,
the curves have been called isochromatic lines, or lines of
equal tint.   The lines or axes along which there is no double
refraction or polarization, and whose poles are A, B, fig. 100.,
have  been called  optical axes, or axes of no polarization, or
axes of compensation, or resultant axes;  because they have
been found not to be real axes, but lines along which the op- 
CHAP. XXIII. RINGS IN CRYSTALS WITH TWO AXES.    175
posite actions of other two real axes have been  compensated,
or destroy one another.
   (119.) In various crystallized bodies, such as nitre and ar-
ragonite, where the  inclination of the resultant  axes, A, B,
fig. 100., is small, the two  systems of rings may be  easily
seen at the same time; but when the inclination of the result-
ant axes is great, as in topaz,  sulphate of iron, Sic, we can
only see one of the systems of rings, which may be done most
advantageously by grinding and polishing two  parallel  faces
perpendicular to the  axis of the rings.  In mica and  topaz,
and various other crystals, the plane of most eminent cleavage
is equally inclined to  the two resultant axes;  so that in such
bodies the systems of rings may be readily found and  easily
exhibited.
   Let M N, for example, Jig-. 104., be a plate of topaz, cut or
split so as to have its face  perpendicular to tiie  axis of the
Fig. 105.
prism in which this body crystallizes.  If we place this plate,
fig. 104., in the apparatus fig. 94. so that the polarized ray rs,
                       fig.9A, passes along the line ABeE,
                       fig. 104., and if the eye receives this
                       ray when reflected from the analyzing
                       plate B, it will see in the direction of
                       that ray a system of oval rings, like
                       that in fig. 105.  In like manner, if
                       the polarized light is transmitted along
                       the line C B d D, the eye will see an-
                       other system perfectly similar to the
                       first.  The lines ABeE and C B <Z D
                       are, therefore, the  resultant axes of
                       topaz. The angle ABC will be found
                       equal to about 121° 16'; but if we
                       compute the inclination of the refract-
                       ed rays B d, B e,  we shall find it, or
                       the angle d B e, to be only 65°; which
                       is, therefore, the inclination of the op-
tical or resultant axes of topaz.
 
176
A  TREATISE ON OPTICS.        PART II.
  If we suppose tiie plate of nitre fixed in any of the positions
which give any of the  rings shown in figs. 100, 101, 102, or
103., then, if we turn round the plate B, we shall observe in
the azimuths of 90° and 270° a system of rings complement-
ary to each, in which  the black  cross in fig.  100. and the
black hyperbolic curves  in figs. 101. 103.  are white, all the
other dark parts light, and the red green, the green red, &c.
as in the single system of rings with one axis.
  In the preceding observations we have supposed the polari-
zation of the incident light, and the analysis of the transmitted
light, to  be necessary to the production of the rings; but in
certain cases they may be shown  by common light with the
analyzing plate, or by polarized light without the  analyzing
plate  B, and in some cases without "either the light being po-
larized or analyzed.  If in topaz, for example, fig. 104., we
allow common light to  fall in the direction A B, so as to  be
refracted along B e, one of the  resultant axes,  and subsequently
reflected at e from the second surface, and reaching the eye
at c, we shall see, after  reflexion  from the  analyzing plate,
the system of rings in fig. 105.; or if A B is polarized light,
the rings will be seen by the eye at c without  an  analyzing
plate. There are several other curious phenomena seen under
these circumstances, which I  have described  in  the Phil.
Transactions for 1814, p. 203. 211.
  I have found some crystals of nitre which  exhibit  their
rings without the use either of polarized light or an analyzing
plate; and Mr. Herschel has found the same  property in some
crystals of carbonate of potash.
  (120.) When the preceding phenomena are seen by polar-
ized homogeneous light, in place of white light, the rings are
bright curves, separated by dark intervals; the curves  having
always the color of the light employed.  In many crystals the
difference in the size of the rings seen in different colors is
not very great, and the  poles A, B of the two systems do not
greatly change their place; but Mr. Herschel found that there
were crystals, such as tartrate of potash and soda, in which
the variation in the size of the rings was  enormous, being
greatest in red, and least in  violet  light, and  in which the
distance  A B,figs. 100.101., or the inclination of the resultant
axes, varied from 56° in violet light to 76° in red, the inclina-
tion having intermediate values for intermediate colors, and
the centres of all the different systems lying in  the line A B.
When all these  systems of rings are combined, as they  are in
using white light, the system  of rings which they form is ex-
ceedingly irregular, the two oval centres, or  the halves of the
first order of colors, being drawn out with long spectra or 
CHAP. XXIII. RINGS IN CRYSTALS WITH TWO AXES.    177
tails of red, green, and violet  light,  and the ends of all the
other rings being red without the resultant axes, and blue
within.
  Mr. Herschel found other crystals in which the  rings are
smallest in red, and largest in blue  light, and in which the
inclination of the axes or A B is least in red, and greatest  in
violet light.
  In all crystals of this kind, the deviation of the tints, or the
colors of the rings seen in white light from Newton's Table
is very considerable, and may be calculated from the preceding
principles.   This deviation I found to be very great, even  in
crystals with one axis of double refraction and one system  of
rings, such as apophyllite where the rings have  scarcely any
other tints than a succession of greenish yellow, and reddish
purple ones.   By viewing  these rings in homogeneous light,
Mr. Herschel has found that the system is a negative one for
the rays at the one end of the spectrum, a positive one for the
rays at the other end of the spectrum, and that there are  no
rings at all in yellow light.
  A similar and equally curious anomaly  I  have  found  in
glauberite, which is a crystal which  has two axes of double
refraction, or two systems of rings for red light, and one nega-
tive system for violet light.
  (121). All the singularities of these phenomena disappear,
and may be rigorously calculated by supposing  the  resultant
axes of crystals where there are two, or the single axis where
there is one, with a system of rings deviating from Newton's
scale, as merely apparent  axes, or axes of compensation, pro-
duced by the opposite action of two or more rectangular axes,
the principal one  of which is the  line bisecting  the  angle
formed by the  two resultant axes.  Upon this principle, I have
shown that all the phenomena presented by such crystals may
be computed with as much accuracy as we can  compute the
 motions of the heavenly bodies.
   The method of doing this may be understood from the fol-
                      lowing observations.   Let A C B  D,
                      fig. 106.,  be a crystal with  two axes
                      turned into a sphere.  Let P, P be the
                      poles of the axes, O the point bisecting
                      them, and A B a line passing through
                    ■JD O,  and perpendicular  to C D, a line
                      passing through P, P.  Let us suppose
                      an  axis to pass through O, perpendicu-
                      lar to the plane A C B D, then we may
           B         account for all the phenomena of such
crystals, by supposing the  axis at O to be the principal one,
 
178
A  TREATISE  ON  OPTICS.        PART II.
and the other axis to be along either of the diameters A B or
C D.   If we take C D, then the axes O and C D must be both
of the same name, either both positive or both negative;  but
if we take A B, the axes  must be one positive and  the  other
negative;  or, what is perhaps the simplest supposition for il-
lustration, we shall suppose the two  rectangular axes which
produce all the phenomena to be A B, C D, either both positive
or both negative, leaving out the one at O.  Supposing A 0 B,
C P P D to be projections of great circles of the sphere, then
P, P are the points where the axis A B destroys the  effect of
the axis C D; that is, where the tints produced by each axis
must  be equal and opposite.  Now,  if we suppose  the arch
C P to be 60°, then, since A P is 90°, it follows that the axis
C D produces at 60° the same tint  that A B does at 90°, and
consequently the polarizing intensity of C D will be to that
of A B as the square of the sine of 90° is to the square of the
sine  of 60°, or as 1 to 0*75, or as 100 to 75.  The polarizing
force of each axis being thus determined, it is easy to find the
tint which will be produced by each axis separately at any
given inclination to the axis, by the method formerly explain-
ed.  Let E be any point on the surface of the sphere, and  let
the tints produced at that point be 9, or the blue of the second
order, by C D, and 16, or the green of the third order, by A B.
Let the inclination of the planes passing through A E, C E,
or the spherical  angle C E A be determined, then the tint at
the point E  will correspond to the diagonal of a  parallelogram
whose sides are  9 and 16, and whose angle is double the angle
C E A.  This law, which is general, and applies also to double
refraction, has been confirmed by Biot and Fresnel, the last
of whom has proved that it coincides rigorously with the law
deduced from the theory of waves.
  If the axes AB, CD are equal, it follows that they will
produce the same tint at equal inclinations;  that is, they will
compensate each other only at one point, viz.  O, and will pro-
duce round O a  system of colored  rings,  the  very same as if
O were a single axis of double refraction of an opposite name
to A B, C D.  If the axis A B has exactly the  same propor-
tional action that C D has upon each of the differently colored
rays, a compensation will  take place for each  color exactly at
O, the centre of the resultant systems of rings, and the colors
will be exactly those of Newton's scale. But if each axis  ex-
ercises a different proportional action upon the colored rays, a
compensation will take place at O for some  of the  rays (for
violet, for example), while the compensation for red will take
place on each side of O; consequently,  in 6uch a cage  the 
CHAP. XXIV. INTERFERENCE OF POLARIZED LIGHT.   179
crystal will have one axis for violet light, and two axes for red
light, like glauberite.
  The phenomena of apophyllite  may, in a similar manner,
be explained by two equal  negative  axes, A B, C D, and a
positive axis at O.
  According to this method of combining the  action of dif-
ferent rectangular axes, it follows that three equal and  rectan-
gular axes, either all positive or all negative, will destroy one
ahother at every point  of the  sphere, and thus produce the
very same effect as if the crystal had no double refraction and
polarization at all.   Upon this principle I have explained the
absence of double refraction in all  the crystals which form the
tessular system of Mohs, each of the primitive forms of which
has actually uiree similarly situated and rectangular axes.  If
one of these axes is not precisely equal to the other, and the
crystallization not  perfectly  uniform, traces of double refrac-
tion will appear, which is found to be the  case in muriate  of
soda, diamond, and other bodies of this class.
  (122.) The following table  contains the polarizing inten-
sities of some crystals with two axes, as given by Mr. Her-
schel :—

       Polarizing Intensities of  Crystals with Two Axes.
	Value of highest Tint.	Thicknesses that produce the same Tint.
	7400 1900 1307 521 249	0-000135 0-000526 0*000765 0-001920 0-004021
Anhydrite, inclination of axes 43° 48' Mica, inclination of axes 45° - - -Heulandite (white), inclination of) axes 54° 17'......$		
CHAP. XXIV.
INTERFERENCE OF POLARIZED  LIGHT.—ON THE  CAUSE OF THE
             COLORS OF CRYSTALLIZED BODIES.

  (123.) Having thus described  the principal  phenomena of
the colors produced by regularly  crystallized bodies  that pos-
sess one or two axes of double refraction, we shall proceed to
explain the cause of these remarkable phenomena.
  Dr. Young had the great merit of applying the doctrine of
interference to explain the colors produced by double refrac-
tion.  When a  pencil of light falls  upon a thin plate of a 
180
A  TREATISE ON OPTICS.       PART II.
doubly refracting crystal, it is separated into two, which move
through the plate with different  velocities, corresponding  to
the different indices of refraction for the ordinary and extra-
ordinary ray.   In calcareous  spar, the ordinary  ray moves
with greater velocity than the extraordinary one;  and there-
fore they ought to interfere with one another,  and in homo-
geneous light produce rings consisting of bright and dark cir-
cles round the  axis of double refraction.  According to this
doctrine, however, the rings ought to be produced  in common
as well as in polarized light; but  as this was not the case, Dr.
Young's ingenious hypothesis was long neglected.   The sub-
ject was at last taken up by Messrs. Fresnel and Arago, who
displayed great address in their  investigation of the subject,
and succeeded in showing how the production of the rings de-
pended on the polarization of the incident  pencil and its sub-
sequent analysis by a reflecting  plate  or  a doubly refracting
prism.
   The following are the laws of  the interference of polarized
light as discovered by MM. Fresnel and Arago:—
   1. When two rays polarized in the same plane interfere
with each other, they will produce by their interference fringes
of the very same kind as  if they were common light.
   This law may be proved by repeating the experiments on
the inflexion of light, mentioned in Chap.  XL, in polarized in
place of common light; and  it will  be found that the very
same fringes are produced in the one case  as in the other.
   2.  When two rays of  light are polarized at right angles
to each other, they produce no colored fringes in the same cir-
cumstances under which  two rays of common light would
produce them.   When the rays are polarized at angles inter-
mediate between 0° and  90°, they produce fringes of inter-
mediate brightness, the fringes being totally obliterated  at
90°, and recovering their greatest brightness at 0°, as  in
Law 1.
   In order to prove this law, MM. Fresnel and Arago adopted
several methods, the  simplest of which is the  following, em-
ployed by the  latter.  Having made two  fine slits in a thin
plate of copper, he placed the copper behind the focus F of a
lens, as in fig. 56., and received the  shadow of the copper
upon the screen C D, where the fringes produced by the inter-
ference of the rays passing through the two slits were visible.
In order, however, to observe  the fringes more accurately,  he
viewed them with an eye-glass,  as formerly described.  He
next prepared  a bundle of transparent plates,  like either  of
those shown at  A and B,fig. 93.,  made of fifteen thin films of
mica or plane glass, and he divided this bundle into two,  by 
CHAP. XXIV. INTERFERENCE OF POLARIZED LIGHT.  181

a sharp cutting instrument   At the  line of division these
bundles had as nearly as possible the same thickness, and they
were  capable of polarizing completely  light incident upon
them at an angle of 30°.   These bundles were then placed
before the slits so as to receive and transmit the rays from the
focus  F at an incidence of 30°, and through portions of the
mica  in each bundle that were very near to each other pre-
vious  to their separation.   The bundles were also fixed to re-
volving frames, so that, by turning either bundle round, their
planes of polarization could be  made either parallel or at right
angles to each other, or could be inclined at any intermediate
angle.  When the  bundles were placed  so as to polarize the
rays in parallel planes, the fringes were formed by the slits
exactly as when the bundles  were removed; but when the
rays were  polarized at 90°, or at right angles to each other,
the fringes wholly disappeared.  In all intermediate  positions
the fringes appeared with intermediate degrees of brightness.
  3.  Two rays originally polarized at right angles to each
other may be subsequently brought into the same plane of po-
larization, icithout acquiring the power of forming  fringes
by their interference.
  If, in the preceding experiment, a doubly refracting crystal
be placed between the eye and the copper slits, having  its
principal section inclined 45° to either of the planes of polari-
zation of the interfering  rays, each pencil will be separated
into two equal ones polarized in two rectangular planes, one
of which  planes is  the principal section itself.  Two systems
of fringes ought, therefore, to be produced; one system from
the interference of the ordinary ray from the right hand slit
with that of the ordinary ray from the left hand slit, and an-
other system from  the interference of the extraordinary ray
from the right hand slit  with the extraordinary ray from the
left hand slit; but no such fringes are produced.
  4.  Two rays polarized at right angles to each other, and
afterwards brought into  similar planes of polarization, pro-
duce fringes by their interference like rays of common light,
provided they belong to a pencil, the  whole of which was
originally polarized in the same plane.
  5.  In the phenomena of interference produced by rays that
have  suffered double refraction, a difference of half an undu-
lation must be allowed, as  one of the pencils is retarded  by
that quantity from some unknown cause.
  The  second of tihese laws  affords a direct explanation  of
the fact which perplexed Dr. Young, that no fringes are ob-
served when light is transmitted through a thin plate possess-
ing double refraction.  The two pencils  thus produced do not
                            Q 
182            A  TREATISE  ON  OPTICS.        TART II,
form fringes by their interference,  because they are polarized
in opposite planes.
  The production of the fringes by the action of doubly re-
fracting  crystals on polarized light may be thus  explained,
Let M N,fig. 107., be  a section of the  plate of sulphate of
          , 107.         lime, C E D F, fig. 94., and B the ana-
                        lyzing plate.  Let R r be a polarized
                    /j  ray incident upon the plate M N, and
                   fl   let O and E be the ordinary and ex-
          0     Jfflb    traordinary  rays  produced  by  the
          E    <|j/^    double refraction of the plate M N.
                        When the  plate M N is in such a po-
     N                 sition that  either of its neutral axes
C D, E F, fig. 94., are  in  the plane of primitive polarization
of the ray R r, fig. 107., then one of the pencils  will not suf-
fer reflexion by the plate B, and  consequently only  one of the
rays will be reflected.   Hence it is  obvious that no colors can
be produced by interference,  because there is  only one ray.
But  in every other  position of the  plate M N, the two rays,
O s, E s, will be reflected by the plate B; and being polarized
by the plate in the same plane, they will, by Law 1., interfere,
and produce a color or a fringe corresponding to the retardation
of one of the rays within the  plate, arising from the difference
of their velocities.  If we call d the interval  of retardation
within the plate M N, we must add to  it half an undulation
to get the real interval, as one of the rays passes  from the or-
dinary to the extraordinary  state.   If we now suppose the
plate B to make  a  revolution of 90°, M N remaining fixed,
then the ray E will be reduced to the ordinary state; and con-
sequently we  must subtract half an undulation  from d, the in-
terval of retardation within  the  plate, to have  the real differ-
ence of the intervals of retardation.  Hence the two intervals
of retardation will differ by  a whole undulation;  and conse-
quently the color produced when the plate B has been turned
round 90°, will be complementary  to that which is produced
when the plate B has the position shown in fig. 107.
  If we  suppose  the rays E and O to be received  upon  and
analyzed by a prism of Iceland  spar, we shall have-two or-
dinary rays interfering to form the  colors in one image,  and
two extraordinary rays interfering to produce the complement-
ary colors in the other image. 
CHAP. XXV. POLARIZING STRUCTURE OF ANALCIME.   183


                      CHAP. XXV.

       ON THE  POLARIZING STRUCTURE  OF ANALCIME.

  (124.) In a preceding chapter I have mentioned the very
remarkable double refraction which is possessed by analcime.
This mineral, which is also called  cubizite, has been regarded
by mineralogists as having the cube for its primitive form; but
if this were correct, it should have exhibited no double refrac-
tion.   Analcime has certainly no cleavage planes,  and it must
be regarded at present as forming in this respect  as great an
anomaly in crystallography as it does in optics by its  extra-
ordinary optical  phenomena.
  The most  common form of the  analcime is the  solid  called
the icositetrahedron, which is bounded by twenty-four equal
and similar trapezia; and  we may regard it as derived from
the cube, by cutting off each of  its angles by three planes
equally inclined to  the three faces which contain the  solid
angle. If we now conceive the cube to be dissected by planes
passing through all the twelve diagonals of its six faces, each
of these planes  will be  found to be a plane  of no double re-
fraction,  or polarization; that is, a ray of polarized light trans-
mitted in any direction whatever, provided  it is in  one of these
planes, will  exhibit none of the polarized  tints when the
crystal is placed in the  apparatus, fig. 94.   These planes of
no double refraction are shown by dark lines in figs. 108. and
          Fig. 108.           109.  If the  polarized ray is in-
                           cident in  any  direction which
                           is  out of these planes, it will
                           be divided into  two pencils, and
                           exhibit the  finest  tints,  all  of
                           which are related to the planes of
                           no double refraction.  The double
                           refraction is sufficiently great to
                           admit a distinct  separation of the
                           images  when the  incident  ray
                           passes  through any pair of the
                           four planes which are  adjacent to
                           the  three axes of the  solid, or of
                           the cube from which it is derived.
                           The least refracted image is the
                           extraordinary  one;  and  conse-
                           quently the  double refracuon is
                          negative  in relation to the  axes
                          to which the doubly refracted ray
                           is perpendicular.
 
184
A  TREATISE  ON OPTICS.        PART II.
  In all other doubly refracting crystals, each particle has the
same  force  of double refraction;  but in the  analcime,  the
double refraction of each particle varies with the square of its
distance from the planes already described.
  The beautiful distribution of the tints shown in figs. 108.
and 109. cannot, of course, be  exhibited to the eye at once,
but are  deduced  by transmitting polarized light in every
direction through the mineral.
  In  several of the  crystals,  the  tints rise to  the third  and
fourth order; but when  the crystals  are very small, the  tints
do not exceed the white of the first order.   The tints are ex-
actly  those of Newton's scale, which indicates that they  are
not the result of opposite and dissimilar actions. In figs. 108.
and 109. the tints are represented by the  faint shaded lines
having their origin from the planes where  the double  refrac-
tion disappears.
  The preceding property of analcime is a simple and easily
applied mineralogical character, which would identify the most
shapeless fragment of the mineral.
  The abbe Hauy first  observed in this mineral  its property
of yielding no electricity by friction,  and derived the name of
analcime from  its want of this  property.  When we consider
that the  crystal is intersected by numerous planes, in which
the ether does not exist at all, or has  its properties  neutralized
by opposite actions, we may ascribe to this cause the difficulty
with which friction  decomposes the  natural quantity of elec-
tricity residing in the mineral.
                      CHAP. XXVI.

                ON CIRCULAR  POLARIZATION.

  (125). In all crystals with one axis there is neither double
refraction nor polarization along the axis; and this is indicated
in the system of rings, by the disappearance of all light in the
centre of the  rings at  the  intersection of the black cross.
When we examine, however, the system of rings produced
by a plate of rock crystal whose faces are perpendicular to the
axis, we find that the  black cross is obliterated within the
inner ring,  which is occupied with  a  uniform tint of red,
green, or blue,  according to the thickness of the plate.   This
effect  will  be  seen in fig. 110.  M. Arago first observed
these colors in 1811. He found that when they were analyzed
by a prism of Iceland spar, the two images had complementary 
CHAP. XXVI.   ON CIRCULAR POLARIZATION.         185

colors, and that the  colors changed, descending in Newton's
      Fig. no.      scale as the prism revolved; so that  if
                    the  color of  the  extraordinary  image
                    was red, it became in succession orange,
                    yellow, green, and violet.  From this
                    result he concluded, that the differently
                    colored rays had been  polarized in dif-
                    ferent planes, by passing along the axis
                    of the rock  crystal.  In this state of the
                    subject, it was taken up by M. Biot, who
                    investigated it with much sagacity and
                    success.
  Let C E D F be the plate of quartz, fig. 94., along whose
axis a polarized ray, r s, is  transmitted.  When  the eye is
placed at O, above the analyzing plate fixed as in the figure,
it will see, for example, a circular red space in the  centre  of
the rings.  If we  turn the quartz round  its  axis,  no change
whatever takes place; but if we  turn the plate B from right
to left, through an angle of 100° for example,  we shall observe
the red change to orange, yellow, green, and violet, the latter
having  a dark  purple  tinge.   If  we now cut from the same
prism of rock crystal another plate of twice the thickness, and
place it in  the apparatus, the plate B remaining where it was
left, we shall find that its tint is different  from that of the
former  plate; but by  turning the plate  B 100° farther, we
shall again bring the tint  to its least brightness, viz., a sombre
violet  By a plate thrice as thick, the least brightness will
be obtained by turning the plate B 100° farther, and so on, till,
when the thickness is  very great, the plate B may have made
several complete revolutions.  Now, it might happen that a
thickness had been taken, so that the rotation of B which pro-
duced the  sombre violet was 360°, or terminated in the point
0°, from which it set out which would  have perplexed the
observer, if he  had not made the succession of experiments
which we  have mentioned.
  This phenomenon will be better understood, by supposing
that we take a plate of quartz ^th of an inch thick, and use
the  different homogeneous rays of the spectrum  in succession.
Beginning with red, we  shall find that the  red light in the
centre  of  the rings has its maximum brightness  when the
plate B is  at 0° of azimuth, as in fig. 94.  If we turn B from
right to left, the red tint  will gradually decrease, and after a
rotation of 17^° the red tint will wholly vanish, having reach-
ed  its  minimum.   With a  plate ^ths  thick,  the red will
vanish at  35°, every  additional  thickness of the  25th of  an
inch requiring an additional rotation of 17^°.  If the light is
 
186
A  TREATISE  ON  OPTICS.        TART II.
violet, the same thickness, viz., ^jth of an inch, will require a
rotation of 41° to make it vanish,  every additional 25th of
an inch of thickness requiring a rotation of 41° more.
  (126.) The rotations for different colors corresponding to 1
millimetre, or Jjth of an inch of quartz, are as follows:—
Homogeneous Ray.	Rotation.	Homogeneous Ray.	Rotation.
Limit of red and orange -Mean orange..... Limit of orange and yellow Mean yellow..... Limit of yellow and green-Mean green - - -	17° 30' 19 00 20 29 21 24 22 19 24 00 25 40 27 51	Limit of green and blue -Mean blue..... Limit of blue and indigo Mean indigo .... Limit of indigo and violet Extreme violet - - -	30°03' 32 19 34 34 36 07 37 41 40 53 44 05
  Upon  trying  various  specimens of quartz, M. Biot found
that there were several in which the very same phenomena
were produced  by turning the  plate B from  left to right.
Hence, in reference to  this property, quartz may be divided
into right-handed and left-handed quartz.
  From  these interesting  facts it  follows, that,  in  passing
along the axis of quartz, polarized light comports itself, at its
egress from the crystal, as if its planes of polarization revolved
in the direction of a spiral within the crystal, in some speci-
mens from right to left, and in others from left to right. " To
conceive this distinction," says Mr. Herschel, " let the reader
take a common cork-screw, and holding it with the  head to-
wards him, let him turn it in the usual  manner as if to pene-
trate a cork.  The head will then turn the same way as the
plane of polarization of a ray, in its progress from the spec-
tator through a right-handed crystal, may be conceived to do.
If the thread  of the cork-screw were reversed,  or were what
is termed a left-handed  thread, then the motion of the head
as the instrument advances would represent that of the plane
of polarization in a left-handed specimen of rock crystal."
  From the opposite characters of these two varieties of quartz,
it follows, that if we combine a plate of right-handed with a
plate of left-handed quartz, the result of the combination will
be that of a plate of the thickest of the  two,  whose thickness
is equal to the difference of the  two thicknesses.   Thus,  if a
plate Jjth of an inch thick of right-handed quartz is combined
with a plate ^ths thick of left-handed quartz, the same colors
will be  produced as if we used a plate  ^ths of an inch
thick of left-handed quartz. When the thicknesses are equal,
the plates of course destroy each other's effects,  and  the  sys-
tem of rings with the black cross will be distinctly seen.
  (127.)  In examining the phenomena of circular polarization, 
CHAP. XXVI.   ON CIRCULAR POLARIZATION.          187

in the amethyst, I found that it possessed the power in the
same specimen of turning the planes  of polarization both
from right to left and from left to right, and that it actually
consisted of alternate strata of right and left-handed quartz,
whose planes were parallel to the axis of double refraction of
the prism.  When we cut a plate perpendicular to the axis of
the prism, we therefore cut  across these  strata, as shown in
fig. in., which  exhibits sections of the strata which occur
                    opposite the three alternate faces  of the
                    six-sided prism.   The shaded  lines are
                    those which turn the planes of  polariza-
                    tion from right  to left, while the inter-
                    mediate unshaded ones and the three un-
                    shaded  sectors  turn  them from left to
                    right.   These strata are not  united to-
                    gether like the parts of certain composite
                    crystals, whose  dissimilar  faces  are
                    brought into mechanical contact; for the
                    right and left-handed strata destroy each
other at the middle line between each stratum, and each stra-
tum has its maximum polarizing force in its  middle line, the
force diminishing gradually to the lines of junction.
   In some specimens of amethyst the thickness of these strata
is so minute, that the  action of the right-handed stratum ex-
tends nearly to the central line of the left-handed stratum,
and vice versa, so as nearly to destroy each other; and  hence
in  such specimens we see  the  system of colored rings with
the black cross almost entirely uninfluenced by the tmts of
circular polarization.   A vein of amethyst, therefore, J^th of
an inch thick in  the direction of the axis, may be so thm in  a
direction perpendicular to the axis that the arc of rotation for
the red ray may be 0° ; and we shall have the  curious phe-
nomenon of  a plate which  polarizes circularly only the most
refrangible  rays of the spectrum.   By a greater degree of
thinness in the strata, the plate would be incapable of polar-
izing  circularly  the yellow ray; and by a greater thinness
still, there  would be no action  on  the violet light  These
feeble  actions, however, might he rendered  visible  at great
thicknesses of the mineral.
   We may therefore conclude  that  the  axes of rotation  in
amethyst vary from 0° to each of the numbers in the preceding
table, according  to the thickness of the strata.
   The coloring  matter of the amethyst I have found to be
curiously distributed in reference to these views; but I must
refer to the original memoir for farther information.*    _

              * Edinburgh Transactions, vol. ix. p. 139.
 
188             A  TREATISE ON OPTICS.        PART  II.

  M. Biot maintained that this remarkable property of quartz
resided in its ultimate particles, and accompanied  them in all
their combinations.   I have found, however, that it is not pos-
sessed by opal, tabasheer, and other silicious bodies, and that
it disappears in melted quartz.  Mr. Herschel also found that
it does not exist in a solution of silica in potash.
  Fie 112    Hitherto no connexion  could be traced between
   ^a.   '  the right  and left-handed  structure  in  quartz, and
          the crystalline form of the specimens which  possess-
          ed these  properties.  Mr.  Herschel,  however, dis-
          covered that  the  plagiedral quartz which contains
          unsymmetrical faces, x x x,fig. 112., turns the planes
          of polarization in the same direction  in  which tiiese
          faces lean round the summits Axx, axx.

    &            Circular Polarization in Fluids.
  (128.)  The  remarkable property of polarizing  light circu-
larly occurs in a feeble degree in certain  fluids, in which it
was discovered by  M. Biot and Dr. Seebeck.   Mr. Herschel
has found it in camphor in a solid  state, and I have discovered
it in certain specimens of unannealed glass.   If we take a
tube six or  seven inches long, and fill it with oil of turpentine,
and place it in the apparatus, fig. 94., so  that  polarized light
transmitted through the oil may be reflected to the eye from
the plate B, we shall observe the complementary  colors and a
distinct rotation of the plane of polarization from right to left.
Other fluids have the property of turning the planes of polari-
zation from left to  right,  as shown in the following table,
which contains the results of M. Biot's experiments.

          Crystals which turn the Planes from Right to Left.
Rock crystal..........
Oil of turpentine.........
Solution of 1753 parts of artificial camphor
  in 17359 of alcohol......
Essential oil of laurel.
------------ turpentine.
Crystals which turn the Planes from Left to Right.
	Arc of Rotation for every 25th of an inch in Thickness.	Relative Thick-nesses that produce the same Effect.
Essential oil of lemons....... Concentrated syrup (from sugar) - - - -	18°25' 0 26 0 33	1 38 4i
Arc of Rotation
 for every 25th
 of an inch in
 Thickness.
18°25'
 0 16
 0 01
Relative Thick-
 nesses that
 produce the
 same Effect.
 1
68$ 
CHAP. XXVI.   ON CIRCULAR POLARIZATION.          189

  In examining these phenomena, M. Fresnel discovered that
in quartz they  were produced by  the interference of two
pencils formed by double refraction along the axis of the quartz.
He succeeded in  separating these two pencils, which differ
both from  common  and polarized light.   They  differ from
polarized light, because when either of them is doubled by a
doubly refracting crystal, the pencil or image never vanishes
during the  revolution of the crystal.   They differ from com-
mon light, because when they suffer two total reflexions from
glass, at an angle of about 54°, the one will emerge polarized
in a plane inclined 45° to the right, and the other in a plane
45° to the  left,  of the plane of total  reflexion.   M. Fresnel
has also discovered  the following properties  of a circularly
polarized ray:—When it is transmitted through a  thin doubly
refracting plate parallel to its axis,  it is  divided into two
pencils with  complementary colors; and these colors will be
an exact quarter of a tint, or an order  of colors, either higher
or lower in Newton's scale, than the  color which the same
crystallized plate  would have given by polarized light.  M.
Fresnel also proved that a circularly polarized  ray,  when
transmitted along the axis of rock  crystal, will not exhibit the
complementary colors when analyzed.
  (129.) In the prosecution  of this curious subject, M. Fresnel
discovered the following method of producing a ray possessing
all the above properties, and therefore  exactly similar to one
of the pencils produced by circular double refraction.  Let
ABC D,fig. 113., be a parallelopiped of crown glass, whose
index of refraction is 1-510, and whose angles A B C, A D C
are each 54^°.  If a common  polarized ray, R r, is  incident
      Fig 113      perpendicularly  upon  A B, and emerges
            'n.   perpendicularly from  C D, after having
             I    suffered two  total reflexions at E and F, at
      A^^jjfB angles of 54£°; and if these reflexions are
       M^mlrE  Perf°rmed hi a plane  inclined 45° to  the
          Hy    plane of polarization of the ray, the emer-
         ^m      gent ray F G will have all the properties
        jay       of a circularly polarized ray, resembling in
  Fr  ___f       every respect one  of  those produced by
■r\mU=Mc       double refraction along  the axis  of rock
    |              crystal.   But as  this circularly polarized
   G-             ray may  be restored to  a single plane of
polarization,  inclined 45° to the plane of reflexion,  by two
total reflexions at 54A_°, it follows, and  I have verified the re-
sult  by observation, that if the parallelopiped A B C D  is
sufficiently long, the pencil will emerge circularly polarized, 
 190             A  TREATISE  ON  OPTICS.        PART II.

 at 2, 6,10, 14, 18 reflexions, and  polarized in a single plane
 after 4, 8, 12, 16, 20 reflexions.
   M. Fresnel  proved  that the  ray R r  would emerge at G,
 circularly polarized by three total  reflexions at 69° 12', and
 four total reflexions at 74° 42'.  Hence, according to the pre-
 ceding reasoning, Uie ray will  be circularly polarized by 9,
 15, 21, 27, &c. reflexions at 69° 12', and restored to common
 polarized light at 6, 12,  18, and  24  reflexions  at the same
 angle; and it will  be circularly  polarized by 12, 20,  28, 36,
 &c. reflexions at 74° 42', and be restored to common polar-
 ized light by  8, 16,  24, 32, &c.  reflexions.
   I have found that circular polarization can be produced by
 2\, 1\, 12Jj-, &c. reflexions,  or any other number which is a
 multiple of 2\; for though we cannot see  the ray in the mid-
 dle of  a reflexion, yet we can show it  when it is restored to a
 single  plane of polarization, at 5, 10, 15 reflexions.*   When
 we use homogeneous light, we find  that the  angle at  which
 circular polarization is produced is different for the differently
 colored rays; and hence these different rays cannot be restored
 to a single plane of polarization at the  same angle of reflexion.
 Complementary colors will therefore  be produced, such as I
described long ago,  and  which, I  believe, have  not been ob-
served by any other person. \   These colors are essentially
different from those of common polarized light, and will be
 understood when  we come to explain those of elliptical  polar-
 ization.
                      CHAP. XXVII.

    ON ELLIPTICAL POLARIZATION, AND  ON THE ACTION OP
                    METALS  UPON LIGHT.

                On Elliptical Polarization.

  (130.) The action of metals upon light has always present-
ed a troublesome anomaly to  the philosopher.   Malus at first
announced that they  produced no effect whatever;  but he
afterwards found that the difference between transparent and
metallic bodies consisted in  this,—that the former reflect all
the light which they polarize in one plane, and  refract all the
light which they polarize in an opposite plane; while metallic
bodies reflect what they polarize in both planes.  Before I was
* See Phil. Transactions, 1830, p. 301.
t See Phil. Transactions, 1830, p. 309. 325. 
CHAP. XXVII.   ELLIPTICAL POLARIZATION.           191

acquainted with any of the experiments of Malus, I had found*
that light was modified by the action of metallic bodies;  and
that, in all the metals  which I tried, a great portion of light
was polarized in the plane of incidence.  In February, 1815,
I discovered the curious property possessed by silver and gold
and other metals, of dividing polarized rays into their comple-
mentary colors by successive  reflexions: but I was misled by
some of the results into the belief, that a reflexion from a
metallic surface had the same effect as a certain thickness of
a crystallized  body;  and that the polarized tints varied with
the angle of incidence, and rose to higher orders, by increasing
the number of reflexions.   M. Biot, in  repeating my experi-
ments, and in an elaborate investigation of the phenomena,!
was misled  by the same causes, and has given a lengthened
detail of experiments,  formulae, and speculations, in which all
the real phenomena  are obscured and confounded.  Although
I had my full share in this  rash generalization, yet I never
viewed it as  a correct expression  of the  phenomena, and I
have repeatedly returned to the subject  with the most anxious
desire of surmounting its difficulties.  In this attempt I have
succeeded ; and I have been enabled to refer all the phenomena
of the action of metals to a new species of polarization, which
I have called elliptical polarization, and which unites the two
classes of phenomena which constitute circular and rectilineal
polarization.
  (131.) In the action of metals upon common light it is easy
to recognize  the  fact announced  by Malus, that the light
which they reflect  is polarized in different planes.   I have
found  that the pencil polarized in the  plane of reflexion is
always more intense than that polarized  in the  perpendicular
plane. The difference between these pencils is least in silver,
and greatest in galena, and  consequently the  latter polarizes
more light in  the plane of reflexion than silver.  The following
table shows the effect  which takes place with other metals:—

  Order in which the Metals polarize most Light in the Plane of
                          Reflexion.
Galena.
Lead.
Grey cobalt.
Arsenical cobalt.
Iron pyrites.
Antimony.
Steel.
Zinc.
Speculum metal.
Platinum.
Bismuth.
Mercury.
Copper.
Tin plate.
Brass.
Grain tin.
Jewellers' gold.
Fine gold.
Common silver.
Pure silver.
Total  reflexion
 from glass.
* Treatise on JVew Philos. Instruments, p. 347. and Preface.
t Traiti de Physique, torn. iv. p. 579. 600. 
192
A  TREATISE  ON  OPTICS.        PART II.
   By  increasing the number of reflexions,  the whole of the
 incident light may be  polarized in  the plane of reflexion.
 Eight reflexions from plates of steel, between 60° and H()°
 polarize the whole light of a wax candle ten feet distant.  An
 increased  number  of reflexions [above 36] is  necessary to do
 this with  pure  silver;  and in total reflexions from glass,
 where the circular polarization begins, and  where  the two
 pencils are equal, the effect cannot be produced by any number
 of reflexions.
   In order to examine  the action of metals  upon polarized
 light  we must provide a  pair of plates  of each metal, flatly
 ground and highly polished, and each at least H  inch  long
 and half an inch broad.  These parallel plates should be fixed
 upon a goniometer, or other divided  instrument, so that one
 of the plates can be made to approach to or recede from  the
 other, and  so that their surfaces can receive the polarized ray
 at different angles of incidence.  In place of giving the plates
 a motion of rotation round the polarized ray, I have  found it
 better to give the  plane of polarization  of the ray a motion
 round the plates, so that the planes of reflexion and of polari-
 zation may be set at any  required angle.  The ray reflected
 from the plates one or more  times is then analyzed, either by
 a plate of glass or a rhomb of Iceland spar.
   When the  plane of reflexion from  the plates is either par-
 allel or perpendicular to the plane of primitive polarization,
 the reflected light will receive no  peculiar modification, ex-
 cepting what arises from their property of polarizing a portion
 of light in the plane of reflexion.   But in every other position
 of the  plane  of  reflexion, and at every angle of  incidence,
 and  after any number of reflexions, the pencil will have re-
 ceived particular modifications, which we  shall  proceed  to
 explain.  One of these,  however, is so beautiful and striking,
 as to arrest our immediate attention.  When  the  plates are
 silver  or gold, the  most  brilliant complementary  colors are
 seen in the ordinary and extraordinary images,  changing with
 the angle of incidence and the number of reflexions.  These
 colors  are  most brilliant when the plane of  reflexion is in-
 clined 45°  to the plane  of incidence, and they vanish when
the inclination is 0° or 90°.  All the other metals in the table,
p.  191, give analogous colors;  but  they are  most brilliant  in
silver, and  diminish in brilliancy from  silver to galena.
  In order  to investigate  the cause of these phenomena, let
us suppose  steel plates to be used, and the plane of the polar-
ized ray to be inclined 45° to the  plane of reflexion.  At an
mcidence of 75° the light has suffered some  physical  change, 
CHAP. XXVII.   ELLIPTICAL POLARIZATION.          193
 which is a maximum at that angle.   It is not polarized light,
 because it does not vanish during the revolution of the ana-
 lyzing plate.   It is neither partially polarized light nor com-
 mon light;  because, when we reflect it a second time at 75°,
 it is restored to light polarized in one plane.   If we transmit
 the light reflected from steel at 75° along the axis of Iceland
 spar, the  system of rings shown in fig. 98. is changed into the
 system shown in fig. 114., as if a thin film of a crystallized
        Fig, in.        body which  polarizes the blue of the
                       first order had crossed the system.  If
                       we substitute for the calcareous spar
                       films of sulphate of lime which give
                       different tints, we shall find that these
                       tints are increased in  value by a quan-
                       tity nearly equal to a  quarter of a tint,
                       according as the metallic action coin-
                       cides with or opposes  that of  the crys-
                       tal.  It was on  the authority  of this
                       experiment  that I was led to believe
 that metals acted like crystallized plates.   And when I found
 that the  colors were better developed and more pure after
 successive reflexions, I rashly  concluded, as  M. Biot also did
 after  me, that each successive reflexion  corresponded to an
 additional thickness of  the film.  In order to prove the error
 of this opinion, let us  transmit the light reflected 2, 4, 6, 8
 times from steel at 75°  along the axis of Iceland spar, and we
 shall  find that the system of rings is perfect, and  that  the
 whole of the light is polarized in one plane ; a result absolutely
 incompatible with the supposition of the tints rising with the
 number of reflexions.   At 1, 3, 5, 7, 9,11 reflexions,  the light
 when transmitted along the axis of Iceland spar will produce
 an effect  equal to nearly a quarter of a tint, beyond  which it
 never rises.
   I now conceived that light reflected 1, 3, 5, 7,  9 times from
 steel at 75° resembled circularly polarized light.  In circularly
 polarized  light produced by two total reflexions from glass, the
 ray originally polarized -f 45° to the plane of reflexion  is, by
 the two reflexions at the same angle, restored to light polarized
 — 45° to  the plane of reflexion; whereas in steel, a ray polar-
 ized -f 45°, and reflected once from  steel at 75°, is restored by
 another reflexion at 75° to light polarized —17°.
   With different metals the same  effect is produced, but the
 inclination of the plane of polarization of the restored ray is.
different, as the following table shows:—
                            R 
194
A  TREATISE  ON OPTICS.
PART II.
Total Reflexions.		Inclination of restored Ray.	— Total Reflexions.		Inclinstioa of restored Ray.
From glass - ■ Pure silver -Common silver ■ Fine gold - -Jewellers' gold Grain tin - -Brass - - -Tin Plate - -Copper - - -Mercury - -Platinum - -		45° 0' 39 48 36 0 35 0 33 0 33 0 32 0 31 0 29 0 26 0 22 0	Bismuth - - - -Speculum metal Iron pyrites - -Antimony - -Arsenical cobalt Galena - - -Specular iron,		21° c 21 0 1910 17 0 14 0 16 15 13 0 12 30 11 0 2 0 0 0
  In total reflexions, or in circular polarization, the circularly
polarized ray is restored to a single plane by the same number
of reflexions and at the same angle at which it received cir-
cular polarization, whatever be the inclination of the plane of
the second pair of reflexions to the plane of the first pair; but
in metallic polarization,  the angle at which the second re-
flexion restores the ray to a single plane of polarization varies
with the inclination of the plane of the second reflexion to the
plane of the first reflexion.   In the case of total reflexions,
this angle varies as the  radii of a circle ; that is, it is always
the same.   In the case  of metallic polarization, it varies as
the radii of an ellipse.  Thus, when the plane of the polarized
ray is inclined 45° to the plane of primitive polarization, the
ray reflected once at 75° will be restored to polarized light at
an incidence of 75°; but when the two planes are parallel to
one another, the restoration takes place at 80°; and when they
are perpendicular, at 70°; and at  intermediate  angles, at in-
termediate inclinations.   For these reasons, I have called this
kind of polarization elliptic polarization.
  We have already seen that light polarized + 45° is ellipti-
cally polarized  by 1, 3, 5, 7 reflexions from steel at 75°, and
restored to a single plane of polarization by 2, 4, 6,8 reflexions
at the same angle; and we have  stated that the ray restored
by  two reflexions  has its  plane  of polarization brought into
the state of—17°.  The following are the inclinations of this
plane to the plane of reflexion,  by different numbers of re-
flexions from steel and silver:—
No. of Re-flexions.	Inclination of the Plane of the polarized Ray.		No. of Re-flexions.	------.------------- Inclination of the Plane of the polarized Ray.	
	Steel.	Silver.		Steel.	Silver.
2 4 6 8	—17° 0' + 5 22 — 138 + 030	— 38° 15' + 31 52 — 26 6 + 21 7	10 12 18 36	— 0°9' + 03 — 0 0 + 00	— 16° 56' +13 30 — 6 42 + 0 47
 
CHAP. XXVII.  ELLIPTICAL POLARIZATION.           195
  These results explain in the clearest manner why common
light is polarized by steel after eight reflexions, and by silver
not till after thirty-six  reflexions.  Common light consists of
two pencils,  one polarized + 45°, and the other — 45°; and
steel brings  these planes of polarization into the plane of re-
flexion after eight reflexions, while silver requires more than
thirty-six reflexions to do this.
  (132.) The angles  at which elliptical polarization is pro-
duced  by one reflexion  may be considered as the maximum
polarizing  angles  of the metal,  and  their tangents may be
considered as the indices of refraction of the different metals,
as shown in the following table :—
Angle of Maximum Polarization.		Index of Refraction
78° 30'		4-915
78	27	4-893
78	10	4-773
77	30	4-511
76	56	4-309
76	0	4011
75	25	3-844
75	0	3-732
74	50	3-689
73	0	3-271
72	30	3-172
70	50	2-879
70	45	\ 2-864
  Elliptical polarization may be produced by a sufficient num-
ber of reflexions at any given  angle, either above or below
the maximum polarizing angle, as shown in the following table
for  Steel:—
Number of Reflexions at which Elliptical Polariza-tion ie produced.	Number of Reflexions at . which the Pencil is re-stored to a single Plane.	Observed Angle of Incidence.
3 9 15 &c. 24 74 12J. &c. 2 6 10 k im n&c. 13 5 &c. 1444 7£&e. 2 6 10 &c. 24 7J 121 &c. 3 9 15 &c.	6 12 18 &c. 5 10 15 &c. 4 8 12 &c. 3 6 9 &c. 2 4 6 &c. 3 6 9 &c. 4 8 12 &c. 5 10 15 &c. 6 12 18 &c.	86° 0' 84 0 82 20 79 0 75 0 67 40 60 20 56 25 52 20
  When the number of reflexions is an *integer, it is easily
understood how an elliptically polarized ray begins to retrace
its course, and to recover  its state of polarization in a single
plane, by the same number of reflexions by which  it lost it;
but it is interesting to observe, when the number of reflexions
Name of Metal.
Grain tin -  -  -  -
Mercury  -  -  -  -
Galena   ....
Iron pyrites -  -  -
Grey cobalt -  -  -
Speculum metal
Ahtimony melted  -
Steel.....
Bismuth  -  -  -  -
Pure silver  -  -  -
Zinc.....
Tin plate hammered
Jewellers' gold -  - 
 196             A TREATISE  ON  OPTICS.        FART II.

 is H, 24, or any other mixed number, that the ray must have
 acquired its elliptical polarization  in the middle of the second
 and third reflexion ; that is, when it had reached its greatest
 depth within the metallic surface  it  then begins to resume its
 state of polarization in a single plane, and  recovers it at the
 end of 3, 5, and 7, reflexions.  A very remarkable effect takes
 place when one reflexion is made on one  side of the  maxi-
 mum polarizing angle, and one on the other side.   A ray that
 has received partial elliptical  polarization by one reflexion at
 85° does not acquire more elliptic polarization  by a reflexion
 at 54°, but it retraces its course and  recovers its state of single
 polarization.
   By a method which it would  be out  of place  to  explain
 here, I have determined the number of points  of restoration
 which can  occur at different angles of incidence from 0° to
90°, for any number of reflexions;  and  I  have represented
them in fig. 115., where the arches  I, I., II, II., &c. represent
the quadrant  of incidence,  for  one, two, Sic reflexions; C
                         Fig. 115.

                VI
                 V
                IV
     »
                III
                 II
                  I
                     I  II III  IV V  VI*
being the point of 0°,and  B that of 90° of incidence.  In the
quadrant, I, I. there is no point of restoration.  In II, II. there
is only one  point or node  of restoration, viz. at 73° in silver.
In III, III. there are two points of restoration, because  a  ray
elliptically polarized by one and a half reflexion will be re-
stored  by thiee reflexions at 63° 43' beneath the maximum
polarizing angle, and at 79° 40' above that angle. It may also
be shown that for IV. reflexions there are 3 points of restora-
tion, for V. reflexions 4 points; and for VI. reflexions 5 points,
as shown in the figure.  The loops or double curves are drawn
to represent the  intensity of the elliptic polarization which
has its minimum at 1,2,3, &c, and its maximum  in the middle
of the  unshaded parts.  If we  now use homogeneous  light,
we shall find that the loops have different sizes in the different
colored rays, and  that their minima and  maxima  are different.
 
CHAP. XXVII.   ELLIPTICAL POLARIZATION.          197

Hence, in the Vlth quadrant, C B for example, there will be
6 loops of all the  different colors, viz. CI; 1, 2;  2, 3; 3, 4,
&c.; overlapping one another, and producing by their mixture
those beautiful complementary colors which have already been
mentioned.   For a more full account of this curious branch
of the subject of polarization, I must refer  the reader to the
Philosophical Transactions, 1830;  or to the Edinburgh
Journal of Science, Nos. VII. and VIII. new series, April, 1831.
                     CHAP.  XXVIII.

 ON THE  POLARIZING STRUCTURE PROnUCED  BY HEAT, COLD,
        COMPRESSION, DILATATION, AND INDURATION.

  The various phenomena of double refraction, and  the  sys-
tems of polarized rings with one and two axes of double re-
fraction, and with planes of no double refraction, may be  pro-
duced either transiently or  permanently, in glass and other
substances, by heat and cold, rapid cooling,  compression  and
dilatation, and induration.

         1.  Transient Influence of  Heat and Cold.

  (1.) Cylinders of glass with one positive  axis of double
                        refraction.
  (133.) If we take a cylinder of glass,  from half an inch to
an inch in diameter, or upwards,  and about half an inch or
more in thickness, and transmit heat from its circumference to
its centre, it will exhibit when exposed to polarized light, in
the apparatus, fig. 94., a system of rings with a black cross,
exactly similar to those  in fig 98.; and the complementary
system  shown  in fig. 99. will appear by turning round the
plate B 90°.   In this case we  must  hold the cylinder at the
distance of 8 or 10 inches from the eye, when the rings  will
appear as it were in the inside of the glass.   If we cover up
any portion of the surface of the glass cylinder, we shall  hide
a corresponding portion of the rings, so that  the cylinder has
its  single  axis of double refraction  fixed in the  axis of its
figure, and not in every possible direction parallel to that axis
as in crystals.
  By crossing the rings with a plate of sulphate of lime, as
formerly explained, we shall find that it depresses the tints in
the two quadrants which the axis of the plate crosses; and
                         R2 
198
A  TREATISE ON OPTICS.        PART II.
 consequently that the system of rings is negative, like that of
 calcareous spar.
   As soon as the  heat  reaches the axis of the cylinder, flie
 rings  begin to lose  their brightness, and when the heat is
 uniformly diffused through the glass, they disappear entirely.

    (2.) Cylinders of glass with a negative axis of double
                         refraction.
   (134.)  If a similar cylinder of glass is heated  uniformly in
 boiling oil, or otherwise brought to a considerably high  tem-
 perature, and is made to cool  rapidly by surrounding its cir-
 cumference with  a  good conductor, it will exhibit a similar
 system of rings, which will  all vanish when  the glass is uni-
 formly cold.   By  crossing these  rings with  sulphate of lime,
 they will be found to be positive, like those of ice and zircon;
 or the same thing may be proved by combining this system of
 rings with the preceding system, when they will be found to
 destroy one another.
   In both these systems of rings, the numerical value of the
 tint or color at any one  point varies as  the square of the dis-
 tance of that point from  the axis.   By  placing thin  films of
 sulphate of lime between two  of these systems of rings,  very
 beautiful systems  may be produced.

 (3.) Oval plates of glass with two axes of double refraction.
   (135.)  If we take an oval  plate A B D C, fig.  116.,  and
                perform with  it the two preceding experi-
                ments,  we shall find that it has in both cases
                two axes of double refraction, the principal
                axis passing through O, being negative when
                it is heated at its circumference, and positive
                when cooled at its circumference. The curves
                A B, CD,  correspond  to the black  ones in
       q     "  fig. 101., and the distance mn to the  inclina-
 tion of the resultant  axes.   The effect  shown in fig. 116. is
 that which is produced by inclining m n 45° to the plane of
 primitive polarization; but when m n is  in the plane of prim-
 itive polarization,  or perpendicular to it,  the curves A B, C D,
 will form a black cross,  as mfig.  100.
   In all the preceding experiments, the  heat  and cold might
 have been introduced and conveyed through the glass from
 each extremity of  the axis of the cylinder or plate.   In  this
 case the phenomena would have been exactly the same,  but
 the axes  that were  formerly negative  will now be positive,
and vice versa.
 
CHAP. XXVIII.   POLARIZATION BY HEAT.
199
         (4.) Cubes of glass with double refraction.
  (136.) When the shape of the glass is that of a  cube, the
rings have the form shown mfig. 117. and when it  is a paral-
lelopiped  with  its length about three times its breadth, the
rings have the form shown mfig. 118. the curves of equal tint
near the angles being circles, as shown in both the figures.

 (5.) Rectangular plates of glass with planes of no double
                        refraction.

  (137.) If a well annealed rectangular plate of glass, E F D C,
is placed with its lower edge C D on a piece of iron  A B D C
fig. 119., nearly red hot, and the two together are placed in the

                          Fig'.119.
apparatus, fig. 94., so that C D may  be inclined 45° to the
plane of primitive polarization, and  that polarized light may
reach the eye at O from every part of the glass, we shall ob-
serve the following phenomena.  The instant that the heat
enters the surface C D, fringes of brilliant colors will be seen
parallel to C D, and almost at  the same time before the heat
has reached the upper surface E F, or even  the central line
a b, similar  fringes will  appear at E F.  Colors at first faint
blue,  and then  white, yellow, orange, Sic, all  spring  up at
a b; and these central colors will be divided from those at the
edges by two dark lines, M N, O P, in which  there is neither
double refraction  nor polarization.  These lines  correspond
with the black curves in fig. 101. and fig. 116., and the struc-
ture between  M N  and O P is negative, like that of cal-
careous spar; while the structures without M N and O  P are
positive, like  those of zircon.  The tints thus developed are
those of Newton's scale, and are compounded of the different 
200
A  TREATISE  ON  OPTICS.
PART  II.
Fig. 120.
sets of tints that would be given in  each of the homogeneous
rays of the spectrum.
  In these plates there is obviously an infinite number of axes
in the planes passing through M N, O P, and all the  tints, as
well as the  double refraction, can be calculated by the very
same laws as in regular crystals, mutatis mutandis.
  If the plate E F D C is heated equally all round, the fringes
are produced with more regularity  and quickness; and if the
plate,  first heated in  oil or otherwise, is cooled  equally  all
round, it will develope the same fringes, but the central ones
at a b will in this last case be positive, and the outer ones at
E F and C D negative.
  Similar effects to those above described may be produced in
similar plates of rock salt, obsidian, fluor spar, copal, and other
solids that have not the doubly refracting structure.
  A series of splendid phenomena are produced by crossing simi-
lar or dissimilar plates of glass when their fringes are developed.
When similar plates of glass, or those in which the fringes are
produced by heat, as in fig. 119., are crossed, the curves or lines
of equal tint at the square of intersection, ABCD, fig. 120.,
                        will he hyperbolas.  The tint at the
                        centre will be the difference  of the
                        central tints of each of the two plates,
                        and the tints of  the succeeding hy-
                        perbolas will  rise  gradually in the
                        scale above that central tint.   If the
                        tints  produced  by each  plate are
                        precisely the same, and the plates of
                        the same shape, the central tints will
                        destroy  each  other,  the  hyperbolas
                        will be equilateral ones, and  the tints
                        will gradually rise from the zero  of
                        Newton's scale.
  When dissimilar plates are crossed, as in fig. 121., viz. one
in which the fringes are produced by heat with one in which
they'are produced by cold, the lines of equal tint in the square
of intersection ABCD (fig. 121.),  will be ellipses. The tints
in the centre will be equal to  the sum of the  separate tints,
and the tints formed by the combination of the external fringes
will be equal to their difference.  If the plates and their tints
are perfectly equal, the lines of equal tint will be circles. The
beauty of these  combinations can be understood only from col-
ored drawings.   When the  plates are  combined lengthwise,
they add to or subtract  from each other's  eftect, according as
similar or dissimilar fringes are  opposed to one another.
 
CHAP. XXVIII.   POLARIZATION BY HEAT.
201
Fig. 121.
  (6.) Spheres of glass, <$cc. with an infinite number of axes
                    of double refraction.

   (138.) If we place a sphere of glass in a glass trough of hot
 oil, and observe the system of rings, while the heat is passing
 to the centre of the sphere, we shalf find it to be a regular
 system,  exactly like that in fig. 98.; and  it will suffer no
 change by turning the sphere in any direction.  Hence  the
 sphere has an  infinite number of positive axes of double re-
 fraction, or one along each of its diameters.
   If a very hot sphere of glass is placed in a glass  trough of
 cold oil, a similar system will be produced, but  the  axes will
 all be negative.

 (7.)  Spheroids of glass with one  axis of double refraction
   along the axis of revolution and two axes along the equa-
   torial diameters.

   (139.) If we place an oblate spheroid in a glass  trough of
 hot oil, we shall  find that it  has one axis of positive  double
 refraction along its shorter axis, or  that of revolution;  but if
 we  transmit the polarized light along any  of  its equatorial
 diameters, we shall find that it has two axes of double refrac-
 tion, the black curves appearing as  in fig.  116. when the axis
 of revolution is inclined 45° to the plane of primitive  polari-
 zation, and changing into a cross when  the axis is parallel or
perpendicular to the plane of  primitive polarization.
  The very same phenomena  will be  exhibited with a prolate
spheroid, only the black cross opens in a different plane when
 the two axes are developed.
  Opposite systems of rings will be developed in both these
cases, if hot spheroids are plunged in cold oil. 
202             A TREATISE ON OPTIC S.        PART II.

   The  reason of using oil is to enable the polarized  light to
pass through the splieres or spheroids without refraction. The
oil should  have a refractive power as near as possible to  that
of the glass.
   A number of very curious phenomena arise  from heating
and cooling  glass tubes, or cylinders, along their axes; the
most singular variations taking  place according as the heat
and cold are applied to the circumference, or to the axis, or to
both.

         (8.) Influence of heat on regular crystals.
   (140.) The influence  of uniform heat and  cold on regular
crystals is  very remarkable. M. Fresnel found that heat dilates
sulphate of  lime less in  the direction of its  principal axis
than in a direction perpendicular to it; and professor Mitscher-
lich has found that Iceland spar  is dilated by heat in the di-
rection of  its axis of  double refraction, while in all directions
at right angles to this axis it contracts; so that there  must be
some intermediate direction in which there is neither contrac-
tion  nor dilatation.   Heat brings the rhomb of Iceland spar
nearer to the cube, and diminishes its double refraction.
   In applying heat to sulphate of lime, professor Mitscherlich
found that the two resultant axes (P, P, fig. 106.) gradually
approach as  the  heat increases, till they unite at O, and form
a single axis.  By a  still farther  increase  of heat they open
out on each side  towards A and B.  A very curious fact of an
analogous  kind I have found in glauberite, which has one axis
of double  refraction  for violet, and two axes  for red light
With  a heat below  that of boiling water, the two resultant
axes (P, P, fig. 106.) unite at O, and, by a slight  increase of
heat, the resultant axes again open  out, one in the direction
O A, and the other in the direction O B.  By  applying cold,
the single  axis for violet light at  O opened out into two at P
and P.  At a certain temperature the violet axis also opened
out into two, in the plane A B.

     2.  On the permanent Influence of sudden Cooling.
   (141.) In March, 1814, I found that glass melted and sud-
denly cooled, such as prince Rupert's drops, possessed a per-
manent doubly refracting structure ;*  and in December, 1814,
Dr. Seebeck published an account of analogous  experiments
with cubes of glass.   Cylinders,  plates, cubes, spheres,  and
spheroids of  glass, with a permanent doubly refracting  struc-
* Letter to Sir Joseph Banks, April 8. 1814.  Phil, trans. 1814. 
CHAP. XXVIII. POLARIZATION BY SUDDEN COOLING.   203
ture, may be formed by bringing the glass to a red heat, and
cooling it rapidly at its circumference, or at its edges.  As
these solid bodies often lose their  shape in  the process,  the
symmetry of their structure is affected, and the  system  of
rings or fringes injured ;  so that the phenomena are not pro-
duced so perfectly as during the transient influence of heat
and  cold.  It is often  necessary, too, to grind and  polish the
surfaces  afresh:  an operation during which the solids are
often broken,  in consequence of the state  of constraint in
which the particles are held.
   An endless  variety of the  most beautiful optical  figures
may be produced by cooling the glass upon metallic patterns
(metals being the  best conductors) applied symmetrically to
each surface of the glass, or symmetrically round its circum-
ference.   The  heat may be thus drawn off from the glass in
lines of any form or direction, so as to give any variety what-
ever to its structure, and, consequently,  to the  optical  figure
which it produces when exposed to polarized light.
   (142.)  In all doubly refracting  crystals the form  of the
rings is independent  of  the  external  shape of the crystal;
but  in glass solids that  have received the doubly refracting
structure, either transiently or permanently, from heat, the
rings depend entirely on the  external shape of the solid.  If,
in fig. 119., we divide the rectangular plate E F D C into two
equal parts  through the  line a b, each half of the plate  will
have the same  structure as the whole,  viz. a negative and two
positive  structures, separated by two dark neutral lines.  In
like manner, if we cut a  piece of a tube of glass, by a notch,
through  its circumference to  its centre,  or if we alter the
shape of cylindrical plates and spheres, &c, by  grinding them
into different external figures, we  produce a complete change
upon the optical figures which they had previously exhibited.

     3. On the Influence of Compression and Dilatation.
   (143.) If we could compress and dilate the various solids
above mentioned with the same uniformity with which we can
heat and cool  them, we should produce the  same doubly re-
fracting  structures which  have been described, compression
and dilatation always producing opposite structures.
   The influence of compression and  dilatation may  be  well
exhibited by taking a strip of glass, A B D C, fig. 122., and
bendino- it by the force of the hands.  When it is held in the
apparatus,./^. 94., with its edge A B inclined 45° to the plane
of primitive polarization, the whole thickness of the glass wil
be covered  with colored fringes, consisting of  a negative set 
204             A  TREATISE  ON  OPTICS.        TART II.

separated from a positive set by the dark neutral  line M N.
The fringes on the convex side A B are negative, and those

                          Fig. 122.
on the concave side positive.   As the bending force increases,
the tints increase in number; and  as  it diminishes, they di-
minish in number, disappearing entirely when  the  plate of
glass recovers its shape.   The tints, which are those of New-
ton's scale, vary with  their' distances from M N ; and when
two such plates as that shown  in fig.  122. cross each other,
they produce in the square of intersection rectilineal fringes
parallel to the diagonal of the square which joins  the  angles
where the two concave and the  two convex sides of the plates
meet.
  When a plate of bent  glass is  made to cross a  plate crys-
tallized by heat, and suddenly cooled, the fringes in the square
of intersection are parabolas, whose vertex will be towards
the convex side of the bent plate,  if the principal axis of the
other plate is positive, but towards the concave side,  if that
axis is negative.
  The effects of compression and dilatation may be most dis-
tinctly  seen  by pressing or dilating plates or  cylinders of
calves'-feet jelly or soft isinglass.
  By  the  application of compressing  and dilating forces,  I
have been able  to alter the doubly refracting structure of
regularly crystallized bodies in every direction,  increasing or
diminishing their tints according to the direction in which the
forces were applied.*
  The  most remarkable influence of pressure, however, is
that which it produces on a mixture of resin and  white wax.
In all the cases hitherto mentioned of the artificial production
of double refraction, the phenomena are related to the shape
of the mass in which the change is induced: but I have been
able to communicate to the compound above mentioned  a
double refraction,  similar to that which exists in the particles
of crystals.   The  compressed mass has a single axis of double
refraction in  every parallel  direction,  and the  colored rings
are produced by  the inclination of the refracted  ray to the
axis according to the same law as in regular crystals.  If we
See Edinburgh Transactions, vol. viii. p. 2dl. 
CHAP. XXVIII.  POLARIZATION BY INDURATION.      205

remove the compressed film, any portion of it will be found to
have one axis of double refraction like portions of a  film  of
any crystal with one axis.   The  important deductions which
this experiment authorizes will be noticed at the conclusion
of this part of the work.

            4.  On the  Influence of Induration.

   (144.) In 1814 I had occasion to make some experiments on
the influence of induration in communicating double refraction
to soft solids.  When isinglass is dried in a glass trough of a
circular form, it exhibits a system of tints with the black cross
exactly like negative crystals with one axis.   When a thin
cylindrical plate of isinglass is indurated at its circumference,
it produces a system of rings with one positive axis.  If the
trough in the first of these experiments and the  plate in the
second are  oval, two axes of double refraction will be ex-
hibited.
   When jelly placed in rectangular  troughs of glass is grad-
ually indurated, we have a  positive and a negative structure
developed,  and  these are separated by a black neutral  line.
If the bottom of the trough is taken  out, so as to allow the
induration to go on at two parallel surfaces,  the same  fringes
are produced as in a rectangular  plate of  glass heated in oil,
and subsequently cooled.
   Spheres and  spheroids of jelly may be made by proper  in-
duration to produce the  same effects as spheres and spheroids
of glass when  heated  or cooled.  The lenses of almost  all
animals  possess the  doubly refracting structure.  In some
there is only one structure, which is generally positive.  In
others there are two structures, a positive and a negative one;
and in many there are three structures,  a  negative between
two positive, and a positive between two negative structures.
In some instances we have two structures of the same name
together.   By the process of induration we may remove en-
tirely the natural structure of the lens, especially when it is
spherical or spheroidal,  and superinduce the structure arising
from  induration. I have now before me a spheroidal lens of
the boneto fish, with one beautiful system of rings along the
axis of the spheroid, and  two systems along  the equatorial
diameters.   I have also several  indurated  lenses of the cod,
that display in the finest  manner  their doubly  refracting
structure.
                               S 
206
A  TREATISE  ON  OPTICS.        PART II.
                      CHAP. XXIX.

    PHENOMENA OF COMPOSITE OR TESSELATED CRYSTALS.

   (145.) In all regularly formed doubly refracting crystals, the
separation of the two  images, the size of the rings, and the
value of the tints, are exactly the same in all parallel direc-
tions.   If  two  crystals,  however, have grown  together with
their axes inclined to one another, and if we cut a plate out
of these united crystals  so that the eye cannot distinguish it
from a plate cut out of a single crystal, the exposure of such a
crystal  to  polarized light will instantly detect its composite
nature, and will  exhibit to the eye the very  line of junction.
This will  be obvious upon considering that the polarized ray
has different inclinations to the axis of each crystal, and  will
therefore produce different tints at these different inclinations.
Hence the examination of a body in  polarized light furnishes
us with a new  method of discovering structures which can-
not  be  detected by  the  microscope, or any  other method of
observation.
   A very fine  example of this is exhibited in  the bipyramidal
sulphate of potash, which Count Bournon and  other crystal-
lographers  regarded as one simple  crystal, whose primitive
form was the bipyramidal dodecahedron, like the crystal shown
in fig. 112.  But by cutting a plate perpendicular to the  axis
of the pyramid, and exposing  it  to polarized  light, I found  it
to be composed of several crystals, all united  so as to form the
regular figure  above represented.  The crystal has two axes
of double refraction, and the plane passing through  the  two
axes of one, is  inclined 60° to the plane passing through the
two axes of each of the other two.  So that when we incline
the plate, each  of the three combined crystals displays different
colors.  I have  found many remarkable structures of this kind
in the mineral  kingdom, and  among  artificial salts;  but  two
of these are so  interesting as to merit particular notice.
  (146.) The apophyllite from Faroe  generally crystallizes in
right-angled square prisms, and splits with great facility  into
plates by planes perpendicular to the axis of the prism.  If
we remove with a sharp knife the uppermost  slice, or the un-
dermost, it will  be found to have one axis of double refraction,
and to give the  single  system  of rings shown in fig. 98.  If
we remove other  slices in  the same  manner,  we shall  find
that when exposed to polarized light they exhibit  the curious
tesselated  structure  shown  in fig. 123.  The  outer case,
M O N P, consists of a number of parallel veins or plates. In 
Fig. 124.
Fig. 125.
CHAP. XXIX. COMPOSITE OR TESSELATED CRYSTALS.   207

the centre is a small lozenge, abed, with one axis of double
                         refraction,  and  round it are four
                      Q crystals, A, B, C, D, with two axes
                         of double refraction, the plane pass-
                         ing through the axes  of A and D
                         being  perpendicular  to  the  plane
                         passing through the axes of B and
                         C; and the former plane being in
                         the direction M N, and the latter in
                         the direction O P.
                            When the polarized light is trans-
                         mitted through the faces of certain
prisms, the beautiful tesselated figure shown in fig. 124 is ex-
hibited, all the differently shaded parts shining with the most
                splendid  colors.  As the prism  has  every-
                where the same thickness, it is obvious that
                the doubly refracting force varies in  different
                parts of the crystal; but this variation takes
                place in such a symmetrical  manner in rela-
                tion to the sides and ends of the prism, as to
                set at defiance all the
                recognized  laws  of
                crystallography.
                  With  the view of
                observing the form of
                the lines of equal co-
                lor,  I  immersed the
                crystal  in  oil,  and
                transmitted the polar-
                ized light in a direc-
                tion parallel to a di-
                agonal of the prism;
                the effect then exhib-
                ited is shown in fig.
                125., where ABCD
                is  the  crystal;  A C,
                and  B D,  its  edges,
where the  thickness is nothing, and
m n the edge through which  the  di-
agonal of the prism  passes.  Now, it
is obvious, that if this had been a regu-
lar crystal, the lines of equal tint or of
equal double refraction would  have
been all straight lines parallel to AC C
or B D; but in the apophyllite they present the most  singular
irregularities, all of  which are, however, symmetrically re-
06 
208            A  TREATISE  ON  OPTICS.        PART II.
lated to certain fixed points within the crystal.  In the middle
of the crystal, half way between m and n, there are on\y five
fringes or orders of  colors; at points equi-distant from this
there are six fringes, the sixth returning into itself in the
form of an oval.   At other two equidistant points near m and
n, the 3d, 4th, and 5th fringes are singularly serrated, and the
6th  and 7th fringes return into themselves in the form of a
square; beyond this, near m and n, there are only four fringes,
in consequence of the fifth returning into itself.
   (147.) A composite  structure of a  very different kind, but
extremely interesting  from the effects which it produces,  is
"exhibited in many crystals of Iceland spar, which are inter-
sected by parallel films  or veins of various thicknesses, as
shown in fig. 126.  These thin veins or  strata are perpendic-
                 ular to  the short diagonals E F, G H of the
                 faces of the rhomb, and parallel to the edges
                 E G, F H.  When we  look  perpendicularly
             ^F through  the  faces A E B F, D G C H, the
                 light will not pass through any of the pianes
                 ebeg, ABCD,  afhd, and consequently
                 we shall only see two  images of any object
             ^H. just as if the planes were not there.  But  if
                 we look through any of the other  two pair
       3>        of parallel faces, we shall observe the  two
common images at their usual distance; and at a much greater
distance, two secondary images, one on each side of the com-
mon images. In some cases there axe four, and in other cases
six, secondary images, arranged in two lines; one line being
on each side of the common images, and perpendicular to the
line joining their centres.  When the interrupting planes are
numerous, and especially when  they are also found  perpen-
dicular to  the short diagonals of the other two faces of the
rhomb  that meet  at  B,  the  obtuse  summit,  the secondary
images are extremely numerous, and sometimes arranged  in
pyramidal heaps of singular beauty, vanishing, and reappear-
ing, and changing their color and the intensity of their light,
by every inclination of the plate.  If the light of the luminous
object is polarized, the phenomena admit of  still greater va-
riations.  When the strata or veins are thick, the images are
not colored, but have  merely at their edges the  colors of re-
fracted light.
   Malus considered these  phenomena as produced by fissures
or cracks within the  crystal,  and he regarded the colors  as
those of thin plates of  air or space; but I have found that they
are produced by veins or twin  crystals firmly  united together
so as  to resist separation  more powerfully than the  natural
 
CHAP. XXIX. COMPOSITE OR TESSELATED CRYSTALS.   209

cleavage planes, and I have found this both crystallographically,
by measuring the angles  of the veins,  and optically, by  ob-
serving the system of rings seen through the veins alone.
  This  composite structure will be understood from fig. 127,
where A B D C is the principal section of a rhomb of Iceland

                         Fig. 127.
■D
"11
                         3ST
 spar whose axis is A D.   The form and position of one of the
 intersecting veins or rhomboidal plates, is shown atMmNn,
 but greatly thicker than it actually is; the angles Am M, and
 D n N, being 141° 44'.  A ray of common light R 6, incident
 on the face A C at  b, will be refracted in  the lines bc,bd.
 These  rays entering the vein M m N n, ate and d, will be
"again refracted doubly; but as the vein is so thin as to produce
 the complementary colors of polarized light by the interference
 of the two pencils which compose each of the pencils ce, df,
 these colors  will depend on the thickness of the vein  M N,
 and on the inclination of the ray to the axis of the plate M N.
 These double pencils  will emerge  from the vein at e,f, and
 will be refracted again  as in the figure into the pencils e m,
 e n,fo,fp;  the colors of en, fo, being complementary to
 those of e m, fp.   That the multiplication  and color of the
 images are owing to the causes now explained may be proved
 ocularly, as  I have done, by dividing rhombs  of calcareous
 spar, and inserting between them, or in grooves cut in a single
 plate of  calcareous spar, a thin film of sulphate of lime or
 mica.  In this way all the phenomena of the natural compound
 crystal may be reproduced in the artificial one, and we may
 give great variety to the phenomena by inserting thin films in
 different azimuths round the polarized pencils b c, bd, and at
 different inclinations to the axis of double refraction.
   The compound crystal shown in fig. 127. is  in  reality a
 natural polarizing apparatus. The part of the rhomb A m N C,
 polarizes the  incident liglit R b.   The  vein M N is the thin
 crystallized vein whose  colors are  to be examined; and  the
 part B M n D, is the analyzing rhomb.
   Various other minerals and artificial crystals are intersected
                           S2
 
210             A TREATISE ON OPTICS.        PART II.

with  analogous veins,  and  produce analogous  phenomena
There are several composite crystals which exhibit remarkable
peculiarities  of structure, and  display curious optical  phe-
nomena  by polarized light.  The  Brazilian topaz is one of
those which is worthy of notice, and whose properties 1  have
explained by colored drawings, in  the second  volume of the
Cambridge Transactions.
   For a  full  account  of the properties of composite  crystals,
and of the multiplication  of images by  the crystals of cal-
careous spar  that are intersected by veins, we refer the reader
to the Edinburgh Transactions, vol. ix. p. 317., and the Phil.
 Trans.,  1815, p. 270.;  or to the  Edinburgh  Encyclopedia,
art. Optics.
                      CHAP. XXX

  ON  THE DICHROISM, OR DOUBLE COLOR, OF BODIES; AND
          THE ABSORPTION OF POLARIZED LIGHT.

  (148.)  If a crystallized body has a different color in different
directions when  common  light  is  transmitted  through  its
substance, it is said to possess dichroism, which signifies two
colors.  Dr. Wollaston observed this property long ago in  the
muriate of palladium and potash, which appeared of a deep
red color along the axis, and of a vivid green in a transverse
direction; and M. Cordier observed the same change of color
in a mineral  called iolite, to which Haiiy gave the  name of
dichroite.  Mr.  Herschel  has observed a similar fact in a
variety of sub-oxysulphate of iron, which is of a deep blood
red color along the axis, and  of  a light green color perpen-
dicular to the axis.  In examining this class of phenomena, I
have found that they depend on the  absorption of light being
regulated by  the inclination of the incident ray to the axis of
double refraction, and  on a difference of color in  the  two
pencils formed by double refraction.
  In a rhomb of yellow Iceland spar, the extraordinary image
was of an orange yellow color, while the ordinary image  was
yellowish white along the axis.   The  color and  intensity of
the two pencils were the same, and the difference of color and
intensity increased with the inclination  to the  axis.  When
the two images overlapped each  other, their  combined color
was the same at all angles with the  axis, and this  color  was
that of the mineral.   If we expose the  rhomb  to  polarized
light, its color will be orange yellow in the position where the
ordinary image vanishes, and  yellowish white in the position
where the extraordinary image vanishes.  The crystals in the
i 
CHAr. XXX.  ABSORPTION OF POLARIZED LIGHT.     211

following Table possess the same properties, the  ordinary and
extraordinary  images having the  colors  opposite  to  their
names:—
Colors of the two Images in Crystals with ONE AXIS.
Name, of Crystals.	Principal Section In Plane of Polarization.	Principal Section, perpendicular to Plane of Polarization.
Zircon.	Brownish white.	Deeper brown.
Sapphire.	Yellowish green.	Blue.
Ruby.	Pale yellow.	Bright pink.
Emerald.	Yellowish green.	Bluish green.
Emerald.	Bluish green.	Yellowish green.
Beryl, blue.	Bluish white.	Blue.
Beryl, green.	Whitish.	Bluish green.
Beryl, yellowish ) green. \	Pale yellow.	Pale green.
Rock crystal, near-1 ly transparent \	Whitish.	Faint brown.
Rock crystal, yellow.	Yellowish white.	Yellow.
Amethyst	Blue.	Pink.
Amethyst.	Greyish white. Reddish yellow.	Ruby red,
Amethyst		Bluish green.
Tourmaline.	Greenish white.	Bluish green.
Rubellite.	Reddish white.	Faint red.
Idocrase.	Yellow.	Green.
Mellite.	Yellow.	Bluish white.
Apatite lilac.	Bluish.	Reddish.
Apatite olive.	Bluish green.	Yellowish green.
Phosphate of lead	Bright green.	Orange yellow.
Iceland spar.	Orange yellow.	Yellowish white.
Octohedrile.	Whitish brown.	Yellowish brown.
  (149.) When the crystals  have two axes of double refrac-
tion, the absorption of the incident rays produces a variety of
phenomena, at and near the two resultant axes.  These  phe-
nomena  are finely displayed in iolite.  This  ipineral, which
crystallizes in six and  twelve-sided prisms, is of a deep blue
color when seen along the  axis, and of a brownish yellow
when  seen in a direction perpendicular to  the axis of the
prism.  When we look along the resultant axes which are
inclined 62° 50' to one  another, we see a system  of  rings
which are pretty distinct when the plate is thin; but when it
is thick, and when the  plane passing through the axes is in
the plane of primitive polarization, branches of blue and white
light are seen to diverge in the form of a cross from the centre
of the system  of rings.   This curious effect is shown in fig.
12^., where P, P', are the centres of the two systems of rings,
O the principal negative axis of the crystal, ana C D the plane
passing through the  axes.   The  blue  branches,  which  are
shaded in the figure, are tipped with purple at their  summits
P, P',  and  are separated by whitish light in some specimens, 
212             A TREATISE ON OPTICS.        TART II.

       Fig. 123.       and by bluish light in others.   From P
         _&.          and P' to O, the white or yellowish light
  #  becomes more and more blue,  and at 0
                     it is quite blue; while from P and 1" to
                     C and  D it becomes more and more
                     yellow, and at C and D it is quite yel-
                     low, the yellow being  almost equally
                     bright in the plane A C  B D, perpendic-
                     ular to the principal axis O.  When the
         B           plane C D is perpendicular to the plane
of  primitive polarization, the poles P,  P' are  marked with
patches of white or yellowish light but everywhere else  the
light is a deep blue.
  When examined by common light, we find that the ordinary
image is brownish yellow at C and  D, and the  extraordinary
one faint blue;  the former acquiring some blue rays, and  the
latter some yellow ones from C to D, and from A to B where
there is still a great difference  in the  color of the images.
The yellow image becomes fainter from A to P and P', and
from B to P and P',  where it changes into  blue, the feeble
blue image being gradually reinforced by other blue rays till
the intensity of the two blue images  is  nearly equal.   The
faint blue image increases in intensity from C to P, and from
D to P', and the yellow one acquires an accession of blue liglit,
and becomes  bluish white from P and P' to O; the ordinary
image is whitish, and the other  a deep blue ; but the white-
ness gradually diminishes towards O, where the two images
are almost equally blue.   The following  table will show that
this property exists in many other crystals:—
Colors of the two Images in Crystals with two Axes,
Names of Crystals.	Plane of Axis in Plane of • Polarization.	Plane of Axia perpendicular to Plane of Polarization.
Topaz blue. , ---- green. ---- greenish blue ---- pink. ---- pink yellow. ---- yellow. Sulphate of haryta. ------yellowish ) purple. i J ------yellow. -----orange yellow Cyanite. Dichroite. Cyniophane. Epidote olive green. ------whitish green Mira.	White. White. Reddish grey. Pink. Pink. Yellowish white. Lemon yellow. Lemon yellow. Gamboge yellow. White. Blue. Yellowish white. Brown. Pink white. Reddish brown.	Blue. Green. Blue. White. Yellow. Orange. Purple. Yellowish white. Yellowish white. Blue. Yellowish white. Yellowish. Sap green. Yellowish while. Reddish vshite.
 
CHAP. XXX.  ABSORPTION OF POLARIZED LIGHT.       213
  The following table shows the color of the images in crys-
tals with two axes which have not been examined.
Names of Crystals.	Axis of Prism in Plane of Polarization.	Axis of Prism perpendicular to Plane of Polarization.
Mica.	Blood red.	Pale greenish yellow.
Acetate of copper.	Blue.	Greenish yellow.
Muriate of copper.*	Greenish white.	Blue.
Olivine.	Bluish green.	Greenish yellow.
Sphene.	Yellow.	Bluish. '
Nitrate of copper.	Bluish white.	Blue.
Chromate of lead.	Orange.	Blood red.
Staurotide.	Brownish red.	Yellowish white.
Augite.	Blood red.	Bright green.
Anhydrite.	Bright pink.	Pale yellow.
Axinite.	Reddish white.	Yellowish white.
Diallage.	Brownish white.	White.
Sulphur.	Yellow.	Deeper yellow.
Sulphate of strontia.	Blue.	Bluish white.
--------- cobalt.	Pink.	Brick red.
Olivine.	Brown.	Brownish white.
  In the last nine crystals in the preceding table, the tints are
not given in relation to any fixed line.
  The following list contains the colors of the two pencils, in
crystals, whose number of axes is not yet known.
Phosphate of iron.
Actynolite.
Precious opal.
Serpentine.
Asbestos.
Blue carbonate of
  copper.          }
Octohedrite (one axis.)
Chloride of gold and J
  sodium.           j
------------- and 3
ammonium.
potassium.
and
Fine blue.t
Green.
Yellow.
Dark green.
Greenish.
Violet blue.

Whitish brown.

Lemon yellow.'

Lemon yellow.

Lemon yellow.
Bluish white.
Greenish white.
Lighter yellow.
Lighter green.  ,
Yellowish.
Greenish blue.
Yellowish brown.
Deep orange."] ^

Deep orange. > % °

Deep orange. J *%
   (150.)  By the application of heat to certain crystals, I have
been able to produce  a permanent difference  in the color of
the two pencils formed by double refraction.  This experiment
may be made most easily on Brazilian topaz.   In one of these
topazes, in which one of the pencils was yellow and the other
pink, I found that a red heat acted more powerfully upon the
extraordinary than upon the ordinary pencil, discharging the
yellow color entirely from the one, and producing only a slight
 * The colors are given in relation to the short diagonal of its rhomboidal
base.
 t When the axis of the prism is in the plane of polarization. 
214
A  TREATISE  ON  OPTICS.
change upon the pink tint of the other.  When the topaz was
hot, it was perfectly colorless, and, during the process of cool-
ing,  it gradually acquired a pink tint, which  could  not  be
modified or renewed by  the most intense heat.  In various
topazes, the color of whose two pencils was exactly the same,
heat discharges more of  the color from one  pencil than the
other, and thus gives them the power  of absorbing light in
reference to the axes of double refraction.


        General Observations on Double  Refraction.

   (151.) The various facts which have  been  explained in the
preceding chapters, enable us to form very plausible opinions
respecting the  origin  and nature of  the doubly  refracting
structure.   The  particles  of  bodies reduced  to  a state  of
fluidity by heat, and prevented by the same cause from com-
bining into a solid body, exhibit no double refraction; and, in
like manner, the particles of  crystallized bodies,  including
metals when existing in a state of solution, exhibit no double
refraction.  As soon, however, as cooling in the one case, and
evaporation in the other, permits the particles to combine in
virtue of their mutual  affinities, these  particles have,  subse-
quent to the action of the forces by which they combine, ac-
quired the doubly refracting  structure.   This effect may  be
accounted for in two ways; either by supposing that the par-
ticles  have originally  a  doubly refracting structure, or that
they have no trace of such a structure.   On  the first of these
suppositions, we must ascribe the disappearance of the double
refraction in the fluid mass, and, in the solution, to the opposite
action of the particles, which must have had an axis in every
possible direction;  but as no double refraction  is visible, it is
more philosophical to suppose that none exists in the particles.
On the second supposition, then, that  the particles have  no
doubly refracting structure, it is easily understood how it may
be produced by the compression of any  two particles brought
together by attraction; for each particle will have an axis of
double refraction in  the direction  of the line joining their
centres, as if they had  been compressed by an external force.
By following out this idea, which I have done elsewhere,* 1
have shown how the various phenomena may be explained  by
the different attractive  forces of three rectangular axes, which
may produce a single negative axis, a single positive axis, or

  * Phil. Transactions, 1829, or Edinburgh Journal of Science, new series.
vol. vi. p. 328-337. 
CHAP. XXXI.     UNUSUAL REFRACTION.             215

two axes, either  both positive or both negative, or the one
positive and the other negative.  The influence of heat,  in
changing the  intensity of the two axes of sulphate of lime,
and in removing one of the axes, or in creating a new one,
admits of an easy explanation on these principles.
                    PART  III.

ON THE APPLICATION OF OPTICAL PRINCIPLES TO THE
      EXPLANATION OF NATURAL PHENOMENA.
                     CHAP. XXXI.
                 ON UNUSUAL REFRACTION.

  (152.) The atmosphere in which we live  is a transparent
mass of air possessing the property of refracting light.  We
learn from the barometer that its density gradually diminishes
as we rise in  the atmosphere, and, as we know from direct
experiment that the refractive power of air increases with its
density, it follows, that the refractive power of the atmosphere
is greatest at  the earth's surface, and gradually diminishes
till  the  air becomes so rare as scarcely to be able to pro-
duce any effect upon liglit.  When a ray of light falls ob-
liquely upon a medium thus varying in  density, in place of
being bent at once  out of its  direction,  it will be gradually
more and more bent during its passage  through it so as to
move in a curve line, in the same manner as if the medium
had consisted of an infinite number of strata of different re-
fractive powers.  In order to explain this, let E, fig. 129., be
                         Fig. 129.
 
 216            A TREATISE  ON OPTICS.        PART III.

 the earth, surrounded with an atmosphere ABCD, consisting
 of four concentric strata  of  different densities and different
 refractive powers.  The  index of refraction  for air at the
 earth's surface being 1-000,294, let us suppose that the index
 of the other three strata is  1-000,200, 1-000,120, 1 •()()( ),0.")0.
 Let B E D be the horizon, and  let a ray S n, proceeding from
 the sun  under the horizon,  fall  on the outer stratum at n,
 whose index of refraction is 1-000,050.   Drawing the per-
 pendicular E n m,  find by the rule formerly  given the angle
 of refraction, E n a, corresponding to the angle of incidence
 S n m.   When the ray n a falls on the second stratum at a,
 whose index  of refraction  is 1-000,120, we  may in like
 manner,  by  drawing a perpendicular E ap, find the refracted
 ray a b.  In the same way, the refracted rays b c and c d may
 be found.  The same  ray S n will therefore  have been re-
 fracted in a polygonal line nab cd, and  as it reaches the eye
 in the direction c d, the sun will be seen in the direction dcS',
 elevated above the horizon, by the refraction of the atmosphere,
 when it is still below it.   In  like manner it  might be shown
 that the sun appears above the horizon by refraction, when he
 is actually below it at sunset.
   Although  the rays of light move in straight  lines in vacuo
 and in all media of uniform density, yet, on the surface of the
 globe, the rays  proceeding from a distant  object, must neces-
 sarily move in a curve line, because they must pass through
 portions of  air of different densities and refractive powers.
 Hence it follows that,  excepting  in a vertical line, no object,
 whether it is a star  or planet  beyond our atmosphere, or is
 actually within it, is seen in  its real place.
   Excepting in astronomical  and  trigonometrical observations,
 where the greatest accuracy is necessary,  this refraction of the
 atmosphere does not occasion any inconvenience.   But since
 the density of the  air  and its refractive  power vary greatly
 when heated or cooled, great local heats or local colds will
 produce great changes of refractive power, and give  rise to
 optical phenomena of a very interesting kind.  Such phenom-
 ena have  received  the  name  of unusual refraction, and they
 are sometimes of such an extraordinary nature  as to resemble
 more the effects of magic than the results of  natural causes.
  (153.) The elevation of coasts,  mountains,  and ships, when
 seen over the surface of the sea, has long  been observed and
 known  by the  name of  looming.  Mr.  Huddart described
several cases of this kind, but  particularly  the very interesting
one of an inverted image of a ship seen beneath the real ship.
Dr. Vince observed  at Ramsgate a ship, whose  topmasts only
were seen above the horizon; but he at  the same time ob-
i 
CHAP. XXXI.     UNUSUAL REFRACTION.
217
Fig. 130.
served, in the field of the telescope through which he  was
looking, two images of the complete ship in the air, both di-
rectly above the ship, the uppermost of the two being erect
and the other inverted.  He then directed his telescope to
another ship whose hull was just in the horizon, and he ob-
served a complete  inverted image of it; the mainmast of
                  which just touched the mainmast of the
                  ship itself.  The first of these two  phe-
                  nomena is shown in fig. 130. in which A
                  is the real ship, and B, C the images  seen
                  by unusual refraction.  Upon  looking at
                  another  ship,  Dr. Vince  saw  inverted
                  images of some of its  parts which  sud-
                  denly appeared and vanished, " first ap-
                  pearing," says he, " below,  and running
                  up very rapidly, showing more or less of
                  the masts at different times as  they broke
                  out,  resembling in the swiftness of their
                  breaking out the shooting of a beam of the
                  aurora borealis." As the ship continued to
                  descend, more of the image gradually ap-
                  peared, till the image of  the  whole  ship
                  was at last completed, with the mainmasts
                  in contact.  When  the ship descended still
lower, the image receded from the ship, but no second image
was seen.  Dr. Vince observed another case, shown in fig.
Fig. 131.
131.,  in  which  the  sea was  distinctly
seen  between the ships B, C.  As the
ship A came above the horizon, the image
C gradually disappeared, and during this
time the image B descended, but the ship
did not  seem so near the horizon as  to
bring the mainmasts together.   The two
images were visible when the whole ship
was beneath the  horizon.
  Captain  Scoresby,  when navigating
the Greenland seas, observed several very
interesting cases of  unusual refraction.
On the 28th of June, 1S20, he  saw from
the mast-head eighteen sail of ships  at
the distance of about twelve miles. One
of them was drawn out or lengthened,
in a vertical direction; another was con-
tracted in the same direction; one  had an
inverted  image  immediately above it;
and other two had two distinct inverted 
218            A  TREATISE ON OPTICS.       PART III.

images above them, accompanied with two images of the strata
of ice.  In  1822, Captain  Scoresby  recognized his father's
ship, the Fame, by its inverted image in the air, although  the
ship itself was below the horizon.  He afterwards found that
the ship was seventeen miles beyond the horizon, and its dis-
tance thirty miles.   In all these cases, the image was directly
above the object; but on the 17th of September, 1818, MM.
Jurine and Soret observed a case  of unusual refraction, where
the image was on one side of  the object.  A bark about 4000
toises distant was seen approaching Geneva by the left bank
of the lake,  and at  the  same moment  there was seen above
the water an image of the sails,  which, in place of following
the direction of the bark, receded from it, and seemed to  ap-
proach Geneva by the right bank of the lake ; the image sail-
ing from east to west, while the bark was sailing from north
to south.  The image was of the same size as the object when
it first receded from the bark,  but it grew less and  less as it
receded, and was only one-half  that of the bark when the
phenomenon ceased.
  While the French army was marching through the sandy
deserts of Lower Egypt, they saw various phenomena of un-
usual refraction, to  which  they  gave  the name of mirage.
When the surface of the sand was  heated by the  sun,  the
land seemed to be terminated at a certain distance by a general
inundation.   The villages situated upon eminences  appeared
to be so many islands in the middle of a great lake, and under
each village there was an inverted image of it.  As  the army
approached  the boundary of the apparent inundation,  the
imaginary lake withdrew, and  the same  illusion  appeared
round the next village.   M. Monge, who has  described these
appearances  in the Memoires sur VEgypte, ascribes them  to
reflexion from a reflecting surface, which he supposes to take
place between two strata of air of different densities.
   One of the most remarkable cases of mirage was observed
by Dr. Vince.   A spectator at Ramsgate sees  the tops of  the
four turrets of Dover Castle  over a hill between Ramsgate
and Dover.  Dr. Vince, however, on the 6th of August, li-i06,
at seven p. m., saw the whole of Dover Castle, as  if it had
been brought over and placed  on the Ramsgate side of the hill.
The image of it was so strong that the hill itself was not seen
through the image.
   The celebrated fata morgana,  which is seen in the straits
of Messina, and which for many centuries astonished the vul-
gar and perplexed philosophers, is obviously a phenomenon of
this kind.   A spectator on an  eminence in the city of Reggio,
with his back to the sun and his face to the sea, and when die 
CHAP. XXXI.     UNUSUAL REFRACTION.             -<5l»
rising  sun shines  from that  point  whence its  incident ray
forms an angle of about 45° on the  sea of Reggio, sees upon
the water numberless series of pilasters,  arches, castles well
delineated, regular columns, lofty towers, superb palaces with
balconies and windows, villages and trees, plains with herds
and flocks, armies of men on foot and on horseback, all passing
rapidly in succession on the surface of the  sea.  These same
objects are, in particular states of the atmosphere, seen in the
air thouo-h less vividly; and when the  air is hazy, they are
seen on the surface of the sea, vividly colored, or frmged with
all the prismatic colors.
   (154.) That the phenomena above described are generally
produced by refraction through strata of air of  different den-
sities  may be proved by various experiments.   In order to
illustrate this, Dr. Wollaston poured into a square phial {fig.
132.) a small quantity of clear syrup, and above this he poured
       „.  ,~>       an equal quantity of water, which grad-
                   ually combined with the syrup, as seen at
                   A.  The word Syrup upon a card held
                   behind the bottle appeared erect when
                   seen through the pure syrup, but inverted,
                   as represented in the figure, when seen
                    through the mixture of water and syrup.
                    Dr. Wollaston then put nearly the same
                    quantity of rectified spirit of wine above
                    the water, as in the same figure at B, and
                    he saw the appearance there represented,
                    the true place of the word  Spirit, and
                    the inverted and erect images below.
                       Analogous phenomena may be seen by
 looking at objects over the surface of a hot poker,  or along
 the surface of a wall or painted board heated by the sun.
   The late Mr. H. Blackadder has described some phenomena
 both of vertical and  lateral mirage as seen  at King George's
 Bastion, Leith, which are very instructive.  The extensive
 bulwark, of which this bastion forms the  central part, is formed
 of huge blocks of cut  sandstone, and from this to the eastern
 end the phenomena are best seen.   To  the east of the tower
 the bulwark  is extended in a straight line to the distance of
 500 feet.   It is eight feet high towards the land, with a foot-
 way  about two feet broad,  and three feet from  the ground.
 The parapet is three feet  wide at top, and is slightly inclined
 towards the sea.
    When  the weather is favorable, the  top of the parapet re-
 sembles a mirror, or rather a sheet of ice; and if in this state
 another person stands or walks upon it, an observer at a little
 
220            A  TREATISE  ON  OPTICS.       PART III.

distance will see an inverted image of the  person under him.
If, while standing on the footway another person  stands on it
also, but  at  some distance, with his face turned  towards the
sea, his image will appear opposite  to him,  giving the appear-
ance of two persons talking or saluting each other.   If, again,
when standing on the footway, and looking in a direction from
the tower, another person crosses the eastern extremity of the
bulwark,  passing through the  water-gate,  either to or from
the sea,  there is produced the  appearance of  two persons
moving  in  opposite directions, constituting what  has  been
termed a lateral mirage:  first one  is seen  moving past, and
then the other in an opposite direction, with some interval be-
tween them.  In looking  over the parapet, distant objects are
seen variously modified;  the  mountains (in Fife) being con-
verted into immense bridges; and  on going to the  eastward
extremity of the bulwark, and directing the eye towards the
tower, the latter  appears  curiously modified, part of it  being
as it were cut off and brought down, so as to form another
small and elegant  tower in the form of certain sepulchral
monuments.  At other times it bears an exact resemblance to
an ancient altar, the fire  of which  seems to burn with great
intensity.*
  (155.)  In order to explain as clearly as possible  how the
erect and inverted image of a ship is produced as in fig. 131.,
let S P (Jig. 133.)  be  a ship  in the horizon, seen  at  E by

                          Fig.133.
      S'
means of rays S E, P E passing in  straight lines through a
track of air of uniform density lying  between the ship and the
eye.  If the air is  more  rare at c than at a, which it may be
from the coldness of the sea below a, its refractive power will
* Edinburgh Journal of Science, No. V. p. 13 
CHAP. XXXI.     UNUSUAL REFRACTION.
221
be less at c than at a.  In this case, rays S d, P c, which,
under ordinary circumstances,  never could  have reached the
eye at E, will  be bent into curve lines P c, S d; and if the
variation of density  is such that  the  uppermost of these rays
S d crosses the other at any point x, then S d will be under-
most, and will enter the eye E as if  it came from the lower
end of the object.  If E p, E s, are  tangents to these curves
or rays, at the  point where they enter the eye, the part S of
the ship will be seen in the direction E s, and the part P in
the direction E p; that is, the  image s p will be  inverted.  In
like manner, other  rays, S n, P m, may be  bent into curves
S n E, P m E, which  do not  cross one another, so  that the
tangent E s' to the curve or ray S n will still  be  uppermost
and the tangent Ep' undermost.  Hence the observer at E
will see an erect image of the ship at s' p' above the inverted
image s p, as in fig. 131.  It is quite clear that the state of
the air may be such as to  exhibit only one of these images,
and that these appearances may be all seen when the real ship
is beneath the  horizon.
  In  one of captain  Scoresby's  observations  we have seen
that the ship was drawn out, or magnified, in a vertical direc-
tion, while another  ship was  contracted or diminished in the
same direction.  If  a  cause should exist, which is quite pos-
sible, which elongated  the  ship horizontally at the same time
that it elongated it vertically, the  effect would be similar to
that of a convex lens, and  the ship would  appear magnified,
and might be recognized at a distance far beyond the limits of
unassisted  vision.   This very case  seems  to  have occurred.
On the 26th July, 1798, at  Hastings, at five p. m. Mr. Latham
saw the  French coast, which  is about 40 or 50 miles distant,
as distinctly as through the best glasses.  The  sailors and fish-
ermen could not at  first be  persuaded of the reality of the ap-
pearance ;  but as the  cliffs gradually appeared more elevated,
they were so convinced that they pointed out and named to Mr.
Latham the different places which they had been accustomed
to visit: such as the bay, the windmill at Boulogne, St Vallery,
and  other places on the coast of Picardy.   All these places
appeared to them as if they were sailing at a small distance
into the  harbor.  From the eastern  cliff or hill, Mr. Latham
saw at once Dungeness, Dover cliffs,  and  the French coast, all
the way from Calais, Boulogne, on to St. Vallery, and, as some
of the fishermen affirmed, as far as Dieppe.  The day was ex-
tremely  hot, without  a breath of wind, and objects at some
distance appeared greatly magnified.
  This class of phenomena may  be well illustrated, as I have
                           T2 
222
A  TREATISE ON OPTICS.       PART III.
 elsewhere* suggested, by holding a mass of heated iron above
 a considerable thickness of water,  placed  in  a glass trough,
 with plates of parallel  glass.  By  withdrawing  the  heated
 iron, the gradation  of density increasing downwards, will be
 accompanied by a decrease of density from the surface, and
 through such a medium the phenomena of the mirage may be
 seen.
   (156.) That some of  the phenomena ascribed  to unusual
 refraction are  owing to unusual reflexion, arising  from differ-
 ence of density, cannot, we think, admit  of a doubt   If an
 observer beyond the earth's atmosphere at S, fig. 129., were
 to look at one composed of strata of different refractive powers
 as shown in the figure, it is obvious that the light of the sun
 would be reflected at its passage through the boundary of each
 stratum, and the same would happen if the variation of re-
 fractive power were perfectly gradual.  Well described cases
 of this  kind are wanting to enable us to state  the laws of the
 phenomena;  but the following  fact,  as  described by Dr.
 Buchan, is so distinct, as to  leave no doubt respecting its ori-
 gin.  " Walking on  the cliff," says he, " about a mile  to the
 east of Brighton, on the morning  of the 18th of  November,
 1804, while watching the rising of the sun, I turned my eyes
 directly towards the sea  just as the solar disc emerged from
 the surface of the water, and saw the face  of the cliff on
 which I was standing represented precisely opposite to me at
 some distance on the ocean.  Calling the attention  of my
 companion to this appearance, we soon also  discovered our
 own figures standing on the summit of the opposite apparent
 cliff, as well as the representation of a windmill near at hand.
 The reflected images were most distinct precisely  opposite to
 where we stood, and the false cliff seemed to fade  away, and
 to draw  near  to the real  one, in  proportion as  it receded
 towards the west. This phenomenon lasted about ten minutes,
 till the sun  had risen nearly his own diameter above the sea.
 The whole then seemed to be elevated into the air, and suc-
 cessively disappeared, like the drawing up of a drop scene in
a theatre.  The surface of the sea  was covered with a dense
fog of many yards  in height, and  which  gradually receded
before the rays of the sun.  The sun's light fell upon the cliff
at an incidence of about 73° from the perpendicular."
* Edinburgh Encyclopaedia, art. Heat. 
 CHAP. XXXII       ON THE RAINBOW.               223


                      CHAP. XXXII.

                    ON THE  RAINBOW.*

   (157.)  The rainbow is, as every person knows, a luminous
 arch extending  across the region of the sky opposite to the
 sun.  Under very favorable circumstances, two bows are seen,
 the inner and the  outer, or the primary and flie secondary,
 and within the  primary rainbow, and in  contact with it  and
 without the secondary one, there have  been seen supernu-
 merary bows.
   The primary or inner rainbow, which is  commonly seen
 alone, is part of a circle whose radius is 42°.  It consists of
 seven differently colored bows, viz. violet, which is the  inner-
 most, indigo, blue, green, yellow, orange, and red, which  is
 the  outermost.   These colors  have  the same  proportional
 breadth as the spaces in the prismatic spectrum.  This bow is,
 therefore,  only  an  infinite number of prismatic spectra, ar-
 ranged in the circumference of a circle; and it would be easy,
 by a circular arrangement of prisms, or by covering up all the
 central part of a large lens, to produce a small arch of exactly
 the same colors.  All that we require, therefore, to form a
 rainbow, is a great number of transparent bodies capable of
 forming a great number of prismatic spectra from the light of
 the sun.
   As the  rainbow is never seen, unless when rain is actually
 falling between the spectator  and the sky opposite to the sun,
 we are led to believe that the transparent bodies required are
 drops of rain which we know to be small spheres. If we look
 into a globe of glass or water  held above the  head,  and oppo-
 site to the sun, we  shall actually see  a prismatic spectrum re-
 flected, from  the farther side of the globe.  In this  spectrum
 the violet rays will be innermost, and the spectrum vertical.
 If we hold the globe horizontal on a level with the eye, so as
 to see the sun's light reflected in a horizontal plane, we shall
 see a horizontal  spectrum with the violet rays innermost.   In
 like manner, if we  hold a globe in a position intermediate be-
 tween these two, so as to see the sun's light reflected in a
 plane inclined 45° to the horizon, we shall perceive a spec-
 trum  inclined 45° to the  horizon  with the violet innermost.
 Now, since in a shower of rain there are drops in all positions
relative to the eye,  the eye will receive spectra inclined at all
angles to the horizon, so that when combined they  will form
the large circular spectrum which constitutes the rainbow.
* In the College edition, see Appendix of Am.ed. chap. vii. 
224            A  TREATISE ON OPTICS.       PART III.

  To explain this more clearly, let E, F,fig. 134., be drops of
rain exposed to the sun's  rays, incident upon them  in the

                          Fig. 134.
directions R E, R F;  out of the whole beam  of light which
falls upon the drop,  those rays which pass through or near the
axis of the  drop will  be refracted  to  a  focus behind  it, but
those which fall on the upper side of the drop will be refracted,
the red rays least,  and the violet most, and will fall upon the
back of the drop with an obliquity such that many of them
will be reflected, as shown in the figure.   These rays will be
again refracted, and will meet the  eye at O, which will per-
ceive a spectrum or prismatic  image of the sun, with the red
space uppermost, and the violet undermost.  If the sun, the
eye, and the drops E,  F, are all in the same vertical plane, the
spectrum produced  by E, F will form the  colors at the very
summit of the bow as in the  figure.   Let us now suppose a
drop to be near the horizon, so that the eye, the drop, and the
sun, are in a plane inclined to the horizon; a ray of the sun's
light will be reflected in the same manner as at E, F, with
this difference only,  that the plane  of  reflexion will be in-
clined  to  the horizon, and will  form  part  of the bow distant
from the summit.  Hence, it is manifest,  that the drops of rain
above the line joining the eye and the upper part of the rain-
bow, and in the plane passing through the eye and the sun,
will form  the  upper part of the bow; and the drops to the
right and  left hand of the observer, and without the line join-
ing the eye and  the  lowest part of the bow, will form the
lowest  part of the  bow  on each  hand.   Not a single drop,
therefore, between the eye and  the space within the  bow is
concerned in its production: so that, if a shower were to fall
regularly from a cloud,  the rainbow  would appear before a
single  drop of rain had reached the ground.
  If we compute the  inclination of the red ray and the violet 
CHAP. XXXII.       ON THE RAINBOW.
225
ray to the incident rays R E, R F, we shall find it to be 42° 2'
for the red, and 40° 17' for the violet, so  that the breadth of
the rainbow will be the difference of those numbers, or 1° 45',
or nearly three times and a half the sun's diameter.  These
results coincide so accurately with observation, as to leave no
doubt that the primary rainbow is produced by two refractions
and one intermediate reflexion of the rays that fall on the
upper sides of the drops of rain.
  It is obvious that the red and violet rays will suffer a second
reflexion at the points where they are represented  as quitting
the drop, but these reflected rays will go up into the sky, and
cannot possibly reach the eye at O.  But  though this is the
case with rays that enter the upper side of the drop as at E F,
or the side farthest from  the eye, yet those which enter it on
the under side, or the side nearest the eye, may  after two re-
flexions reach the eye, as shown in the drops H,  G, where the
rays R, R enter the drops below.   The red and  violet rays
will be refracted in different directions, and after being twice
reflected will be finally refracted to the eye at O; the  violet
forming the upper part, the red the under part of the spectrum.
If we now compute the  inclination of these  rays to the inci-
dent rays R, R, we  shall find them to be 50° 57' for  the red
ray, and 54° 7' for the violet ray; the difference of which or
3° 10' will be the breadth  of  the bow, and the distance be-
tween the bows will be 8° 55'.*  Hence it  is clear that a
secondary bow will  be  formed exterior  to the  primary bow,
and with its colors  reversed,  in consequence of their  being
produced by two reflexions and two refractions.  The breadth
of the secondary how is nearly twice as great as  that  of the
primary one, and its colors must be much fainter, because  it
consists of light that has suffered two reflexions in place of one.
   (139.) Sir Isaac Newton found the semi-diameter of the in-
terior bow to be 42°, its breadth 2° 10', and its distance from
the outer bow 8° 30'; numbers which agree so  well with the
calculated results as to leave no doubt of the  truth of the ex-
planation which has been given.  But if any farther evidence
were wanted, it may be found in the fact, which I observed in
1812,  that the  light of both the rainbows is wholly polarized
in planes passing through the eye and the radii of the arch.
This result demonstrates that the bows are formed by reflexion
at or near the polarizing angle,  from  the surface of a trans-
parent body.  The production of  artificial  rainbows by the
spray of a waterfall, or  by a  shower of drops scattered by a
mop, or forced out of a syringe,  is another proof of the pre-

      * No correction for the sun's apparent diameter, is here made. 
226 *           A TREATISE  ON  OPTICS.       PART III.

ceding explanation.  Lunar rainbows are sometimes seen, but
the colors are faint and scarcely perceptible.  In 1*14,1 saw,
at Berne, a fog-bow, which  resembled a  nebulous  arch, in
which the colors were invisible.
  (159.) On the 5th of July, 1828,1 observed three supernu-
merary  bows  within the primary  bow, each  consisting of
green and red arches, and in contact with the violet arch of
the primary bow.  On the outside of the outer or secondary
bow I saw distinctly a red arch, and beyond it a very faint
green one, constituting a supernumerary bow, analogous to
those within the primary  rainbow.
  Dr. Halley has shown that the rainbow formed by three re-
flexions within  the drops will encircle the sun Itself, at the
distance of 40° 20', and that  the rainbow formed by four re-
flexions will likewise encircle him at the distance of 45° 33'.
The rainbows formed by five reflexions will be partly covered
by the secondary  bow.  The light  which forms these three
bows is  obviously  too faint to  make any impression on our
organs, and these rainbows have therefore never been observed.
  Many peculiar rainbows have been seen and described.  On
the 10th August, 1665, a  faint rainbow was seen at Chartres,
crossing the primary rainbow at its vertex.  It was formed by
reflexion from the river.
  On the 6th August 1698, Dr. Halley, when walking on the
walls of Chester, observed a remarkable rainbow, shown in
fig. 135., where A B C is the  primary bow, D H E the  second-
ary one, and A F H G C the new bow intersecting the  second-
ary bow D H E, and dividing it nearly into three parts.  Dr.
Halley observed the points  F, G to rise,  and the arch F G
gradually to contract, till at length the two arches F H G and
F G coincided, so that the secondary iris for a great space lost
its colors, and appeared like a white arch at the top.  The new
bow, A H C, had its colors in the same order as  the  primary 
CHAP. XXXIII.   ON HAIOS AND PARHELIA.
227
one ABC, and consequently  the reverse of the secondary
bow;  and on this account the two opposite spectra at G and F
counteracted each other, and produced whiteness.  The sun
at this time shone on the river Dee, which was unruffled, and
Dr. Halley found that the  bow A H C was only that part of
the circle of the primary bow that would have been under the
castle bent upwards by reflexion from the  river.  A third
rainbow  seen between the two common ones, and not con-
centric with them,  is described  in  Rozier's Journal,  and is
doubtless the same phenomenon as that observed by Dr. Halley.
Red  rainbows, distorted  rainbows, and  inverted rainbows on
the grass, have been seen.  The latter are formed by the drops
of rain suspended on the spiders' webs in the fields.
                     CHAP. XXXIII.

      ON HALOS, CORONjE, PARHELIA, AND PARASELENE.
   (160.) When the sun  and moon  are seen in  a clear sky,
 they exhibit their luminous discs without any change of color,
 and without any attendant  phenomena.  In other conditions
 of the atmosphere, the two luminaries  not only experience a
 change of color, but are surrounded with a variety of luminous
 circles of various sizes and forms.   When the air is charged
 with dry exhalations,  the sun is sometimes as red as blood.
 When seen through watery vapors, he is shorn of his beams,
 but preserves his disc white and colorless;  while, in another
 state  of the sky, I have seen  the  sun of  the most brilliant
 salmon color.  When light fleecy clouds pass over the sun and
 moon, they are  often encircled with one, two, three, or even
 more,  colored rings,  like those of thin  plates;  and in cold
 weather, when particles of ice are floating in  the higher re-
 gions, the two luminaries are frequently surrounded with the
 most complicated phenomena, consisting of concentric circles,
 circles passing through their discs, segments  of circles,  and
 mock suns or moons, formed at the points where these circles
 intersect each other.
   The name halo is given indiscriminately to  these phenom-
 ena, whether they are seen round the sun or the moon.  They
are called parhelia when seen round the sun, and paraselene
when seen round the moon.
   The small halos seen round the sun and moon in fine wea-
ther, when they are partially covered with light fleecy clouds,
have been also called corona?.   They are very common round 
228
A  TREATISE ON OPTICS.      PART III.
the sun, though, from the overpowering brightness of his rays,
they are best seen when he is observed by reflexion from the
surface of still water.  In June, 1692, Sir Isaac  Newton ob-
served, by reflexion in a vessel of standing water, three rings
of color round the sun, like three little rainbows.   The colors
of the first or innermost were blue next the sun, red without,
and white in the middle between the blue and red. The colors
of the second ring were purple and blue  within, and pale red
without, and green  in the middle.  The colors  of the third
ring were pale blue  within and pale red without.  The colors
and diameters  of the rings are more particularly given as fol-
lows :—
1st Ring   -      Blue, white, red      -    Diameter, 5° to 6°.


3d  Ring   -      Pale blue, pale red   -    Diameter, 12°.
   On the 19th  February, 1664, Sir Isaac Newton saw a halo
round the moon, of two rings, as follows:—
1st Ring  -   White,  bluish green, yellow, red  -  Diameter, 3°.
2d  Ring  -   Blue, green, red......Diameter, 5i°
   Sir Isaac considers these rings as formed by the light pass-
ing through very small drops of water, in the same manner as
the colors of thick plates.   On the supposition that the glob-
ules of water are the 500th of an inch in diameter, he finds
that the diameters of the rings should be as follows:—
       1st Red  ring.......Diameter, 7J°
       2d Red  ring.......Diameter, 10£°
       3d Red  ring.......Diameter, 12° 33'
   The rings will increase in size as the  globules become less,
and diminish if the globules become larger.
   The halos round  the  sun and  moon, which have  excited
most notice, are those which are about 47° and 94° in diame-
ter.  In order to form a correct idea of them, we shall give
accurate descriptions of two; one a parhelion, and the other a
paraselene.
  The following is the original  account  of a parhelion, seen
by  Scheiner in 1630:—
  (161.) " The diameter of the circle M  Q, N next to the sun,
was about 45°,  and that of the circle O R P was about 95° 20';
they were colored like the primary rainbow;  but the red was
next the sun, and the other colors  in  the usual  order.  The
breadths of all'the  arches were equal  to one another, and
about a third part less than the diameter  of the sun,  as repre-
sented mfig. 136.; though  I cannot say but the whitish circle 
CHAP. XXXIII.  ON HALOS AND PARHELIA.           229

O G P, parallel to the horizon, was  rather broader than the
rest.  The two parhelia M, N were lively enough, but the

                         Fig. 136.
 other two at O and P were not so brisk. M and N had a pur-
 ple redness next the sun, and were white in the opposite parts.
 O and P were all over white.  They all differed in their du-
 rations ; for P, which shone but seldom and  but faintly, van-
 ished first of all, being covered by a collection of pretty thick
 clouds. The parhelion O continued constant for a great while,
 though it was but faint. The two lateral parhelia M, N were
 seen  constantly for  three  hours  together.  M was in a lan-
 guishing state, and  died first, after several  struggles, but N
 continued an hour after it at least  Though  I did not see the
 last end of  it, yet I was sure it was the only one that accom-
 panied the  true  sun for a long  time,  having escaped  those
 clouds and  vapors which extinguished the rest.  However, it
 vanished at last upon the  fall of some small showers.   This
 phenomenon was observed to last 4^ hours at least, and  since
 it appeared  in perfection when I first saw it, I am persuaded
 its whole duration might be above five hours.
  " The parhelia Q, R were situated in a vertical  plane  pass-
 ing through the eye  at F, and the sun at G, in which vertical
 the arches  H R C, O R P either  crossed or touched one an-
 other.  These parhelia were  sometimes brighter,  sometimes
 fainter than the  rest, but were not so  perfect in their  shape
 and whitish color.  They varied their magnitude and color ac-
cording to the different temperature of the  sun's  light at G,
and the matter that  received it  at Q and R; and therefore
                           U 
230            A TREATISE ON  OPTICS.        PART III.

their light and color were almost always fluctuating, and con-
tinued, as it were, in a perpetual conflict.  I took  particular
notice that they appeared almost the first and last of the par-
helia, excepting that of N.
  " The arches which composed the small  halo M N next to
the sun, seemed to the eye to compose a single circumference,
but it was confused, and had unequal breadths;  nor did it con-
stantly continue like itself,  but was  perpetually fluctuating.
But in  reality it consisted  of  the arches expressed in  the
figure, as I accurately observed for this very purpose.* These
arches cut each other in a point at  Q,  and there they formed a
parhelion; the~parhelia M, N shining from the common inter-
sections of the inner halo, and the  whitish circle O N M P."
  (162.)  Hevelius observed at Dantzic, on the 30th of March,
1660, at  one A. M.,  the paraselene shown in fig. 137.  The
moon A  was seen  surrounded  by an  entire  whitish  circle
Fig. 137.
B C D E, in which there were  two mock moons at B and D;
one at each side of the moon, consisting of various colors, and
shooting out very long and whitish beams by fits.  At  about
two o'clock a larger circle surrounded the lesser, and reached
to the horizon.   The tops of both these circles were  touched
by colored  arches, like inverted rainbows.  The inferior arch
at C was a portion of a large circle,  and the superior at F a
portion of a lesser.  This phenomenon  lasted nearly three hours.
The outward great circle vanished first.  Then the larger in-
verted arch at C, and then the  lesser; and last of all the inner
  * The four intersecting circles which form this inner halo are described
from four centres, one at each angle of a small square. 
>A
 CHAP. XXXIII.   ON IIALOS AND PARHELIA.           231

 circle B C D E disappeared. The diameter of this inner circle,
 and also of the superior arch, was 45°, and that of the exterior
 circle and inferior arch was 90°.
   On another occasion Hevelius observed a large white rec-
 tangular cross passing through the disc of the moon, the moon
 being in the intersection of the cross, and encircled with a
 halo exactly like the inner one in the preceding figure.
   (163.) The frequent occurrence of the halos of 47° and 94°
 in cold  weather, and especially in the northern regions of the
 globe, led to the belief that they must be formed  by crystals
 of ice and snow floating in the air.  Descartes  supposed that
 they were produced by refraction, through flat stars of pellucid
 ice;  and Huygens, who investigated the subject both experi-
 mentally and theoretically, has  published an elaborate  theory
 of halos, in which he assumes the  existence  of particles of
 hail,  some of which are globular and others cylindrical, with
 an opaque nucleus or kernel having  a certain proportion to the
 whole.   He supposes  these cylinders to be kept in a vertical
 position, by ascending currents of air  or vapor, and to have
 their axes at all possible inclinations to the horizon, when they
 are dispersed by the wind or any other causes.  He considers
 these cylinders to have been  at first a globular collection of
 the softest and purest particles of snow, to the bottom of which
 other particles  adhere,  the  ascending currents  preventing
 them from adhering to the sides; they will, therefore, assume
 a cylindrical shape.  Huygens then supposes that the outer
 part of  the cylinders may be melted by the heat of the sun, a
 small cylinder remaining unmelted in the centre, and that if
 the melted part is again frozen, it may have sufficient  trans-
 parency to refract and reflect  the rays of the sun in a regular
 manner.  By means of this apparatus, the existence of which
 is not impossible, Huygens has given a beautiful solution of
 almost all the difficulties which have been encountered in ex-
 plaining the origin of halos.
   Sir Isaac Newton regarded the halo of 45° as produced by
 a different cause from  the small prismatic coronse; and he was
 of opinion  that it arose  from refraction " from some sort of
 hail or snow floating in the air in a horizontal posture, the re-
 fracting angle being about 58° or 60°."
   When we consider, however, the  great variety of crystal-
 line  forms which water assumes in freezing; that these crys-
 tals really exist in a transparent state in the atmosphere, in
the form of crystals of ice, which actually prick the skin hke
needles ; and that simple and compound crystals of snow, of
every conceivable variety of shape, are often  falling through
the atmosphere, and sometimes melting in passing through its 
^
232             A TREATISE ON OPTICS.       PART III.

lower and warmer strata, we do not require any hypothetical
cylinders to account for the principal phenomena of halos.
  Mariotte, Young, Cavendish, and others, have agreed  in
ascribing  the halo of 45° or 46° in diameter, to refraction
through prisms of ice, with refracting angles of 60° floating
in the air, and having their refracting angles in all directions.
The crystals of hoar-frost have actually such angles, and if we
compute the deviation of the refracted rays of the sun or moon
incident upon such a prism, with the  index of refraction for
ice,  taken at 1-31, we shall find  it to be 21° 50', the double
of which is 43° 40'.  In order to explain  the larger  halo, Dr.
Young supposes that the rays which have been  once refracted
by the prism may fall on other  prisms, and the effect then be
doubled by a second refraction, so as  to produce  a deviation
of 90°.  This, however, is by no means  probable, and Dr.
Young has candidly acknowledged  the " great apparent prob-
ability" of Mr. Cavendish's suggestion, that the external halo
may be produced by the refraction of  the rectangular termi-
nations of the crystals.   With an index  of refraction of 131,
this would give a deviation of 45° 44', or  a diameter of 01°
28', and the mean of several accurate measures is 91° 40',  a
very remarkable coincidence.
  The existence of prisms with such rectangular terminations
is still  hypothetical; but I have removed the difficulty  on this
point, by observing in the hoar-frost upon stones, leaves, and
wood,  regular quadrangular crystals of ice, both  simple and
compound.
  Although halos are generally  represented as circles, witli
the sun or moon  in their centres, yet their apparent form is
commonly an irregular oval, wider below than above, the sun
being nearer their upper than their  lower extremity.  Dr.
Smith has shown that this is an optical deception, arising from
the apparent figure of the sky, and he  estimates that when
the circle touches the horizon, its apparent vertical diameter
is divided  by the moon, in the proportion  of about 2 to 3 or 4;
and is to the horizontal diameter drawn through the moon as 4
to 3, nearly.
  With the view of ascertaining  if any of the halos  are form-
ed by reflexion, I  have examined  them with doubly refracting
prisms, and have  found that  the light which forms them has
not suffered reflexion.
  The production of halos may be illustrated  experimentally
by crystallizing various salts upon plates  of glass, and looking
through the plates at the sun or a candle. When the crystals
are granular and properly formed, they will produce the finest
effects.  A few drops of a saturated  solution of alum, for ex-
i 
CHAP. XXXIII.  ON HALOS AND PARHELIA.
233
ample, spread over a plate of glass so as to crystallize quickly,
will cover it with an imperfect crust,  consisting of flat  octa-
hedral crystals,  scarcely visible to the eye.  When the ob-
server, with his eye placed close behind the smooth side of the
glass  plate, looks through it at a luminous body, he will per-
ceive three fine halos  at  different distances,  encircling the
source of light.   The interior halo, which is the whitest of
the three, is formed  by  the refraction of the rays through a
pair of faces in  the  crystals that  are least  inclined to  each
other. The second halo,  which is blue without and red within,
with all the prismatic colors, is formed by a pair of more  in-
clined faces; and the third halo, which is large and brilliantly
colored, from the increased refraction and dispersion, is formed
by the most inclined faces.   As each crystal of alum has three
pairs of each of these included prisms, and as these refracting
faces will have  every possible direction to  the  horizon, it is
easy to understand how the halos are completed and equally
luminous throughout.  When the  crystals have the property
of double  refraction, and when their axis is  perpendicular to
the plates, more  beautiful combinations will be produced.
  (164.)  Among the luminous phenomena of the atmosphere,
we may here notice that of converging and diverging  solar
beams.  The phenomenon of diverging beams, represented in
fig. 138., is of frequent  occurrence in summer, and when the
sun is near the horizon ; and arises from a portion of the  sun's

                          Fig. 138.
 
234            A TREATISE ON OPTICS.       PART III.

shown in fig. 139., where the rays converge to a point A, as
far below the horizon M N as the  sun is above it.   This phe-
nomenon is always seen opposite to the sun, and generally at

                         Fig. 139.
the same time with the phenomenon of diverging beams, as if
another sun, diametrically opposite to the real one, were below
the horizon at A, and throwing out his divergent beams.  In a
phenomenon  of this kind  which I saw in 1824, the eastern
portion of the horizon where it appeared was occupied with a
black cloud,  which seems to be necessary as a ground,  for
rendering visible such feeble radiations.  A few minutes after
the phenomenon was first seen,  the  converging lines  were
black, or  very dark;  an effect which seems to have  arisen
from the luminous  beams having become broad and of unequal
intensity, so that the eye took up, as it were, the dark spaces
between the beams more readily than the luminous  beams
themselves.
  This phenomenon is  entirely one  of perspective.   Let us
suppose beams inclined to one another like the meridians of a
globe to diverge from the sun, as these meridians diverge from
the north  pole of the  globe, and  let us suppose that planes
pass through  all these meridians, and through the line joining
the observer and the sun, or their common intersection.  An
eye, therefore, placed in that line, or in the common intersec-
tion of all the fifteen planes, will  see the fifteen beams con-
verging to a point opposite the sun, just as an eye in the axis
of a globe would see all the fifteen  meridians of the  globe
converge to its south pole.   If we suppose the axis of a globe
or of an armillary sphere to  be directed to the centres  of the
diverging  and converging  beams, and a plane to pass through 
CHAP. XXXIV.  COLORS OF NATURAL BODIES.          235

the globe parallel to the horizon, it would  cut off the  meri-
dians so as to exhibit the precise appearances in fig. 138. and
fig.  139.; with this difference only, that there would be fifteen
beams in the diverging system in the place of the number
shown in fig. 139.
                        CHAP. XXXIV.

               ON THE COLORS  OF NATURAL BODIES.

      (165.)  There is no branch  of the application of optical
    science which possesses  a greater interest than that  which
    proposes to determine the cause of the colors of natural bodies.
    Sir Isaac Newton was  the first who entered into an elaborate
    investigation of this difficult subject;  but though his specula-
    tions are  marked with the peculiar genius of their author, yet
    they will not stand a rigorous examination under the lights of
    modern science.
      That the colors of material nature are not the result of any
f    quality inherent in the  colored body has been incontrovertibly
i    proved by Sir Isaac.   He found that  all bodies, of whatever
'    color, exhibit that color only when they are placed in white
    light  In homogeneous red  light they appeared  red, in violet
    light violet, and so on; their  colors being always best displayed
    when placed in their own daylight colors.   A red wafer, for
    example,  appears red in the  white light of day, because it re-
    flects red light more copiously than any of the other colors.
    If we place a red wafer in yellow light, it can no longer ap-
    pear  red,  because there is not a particle of red light in the
    yellow light which it could reflect   It reflects, however, a
    portion of yellow light, because there is some yellow in the
    red which it does reflect.  If the red wafer had reflected no-
    thing but pure homogeneous red light and not reflected white
    light from its  outer  surface, which all colored  bodies do,  it
    would  in that case have appeared  absolutely black  when
    placed in  yellow light.  The colors, therefore, of bodies arise
    from  their property of reflecting or transmitting to  the eye
    certain rays of* white light, while they stifle or stop the re-
    maining rays.  To this point  the Newtonian theory is  support-
    ed  by infallible experiments;  but the principal  part of the
    theory, which has for its  object  to determine the manner in
    which particular rays are stopped, while others are reflected
    or transmitted, is not so well founded. 
236             A  TREATISE  ON  OPTICS.       PART III.

  As Sir Tsaac has stated the principles of his theory with the
greatest clearness, we shall give them in his own words.
  "1st Those superficies of transparent  bodies reflect  the
greatest quantity of light which have  the greatest refracting
power; that is, which separate media  that differ most in their
refracting power.   And in the confines of equally refracting
media there is no reflexion.
  "2d, The least parts of almost all natural bodies are in
some measure transparent; and the  opacity  of these bodies
arises from the multitude of reflexions caused in their internal
parts.
  " 3d, Between the parts of opaque  and colored bodies  are
many spaces,  either empty, or  replenished with mediums of
other  densities;  as  water between  the  tinging corpuscles
wherewith  any  liquor  is  impregnated;  air between   the
aqueous globules that constitute clouds or mists;  and for  the
most part spaces, void of both air and water, but yet  perhaps
not  wholly  void  of all substance,  between the parts of all
bodies.
  " 4th, The  parts of bodies and their interstices must not be
less  than of some definite bigness, to render them opaque and
colored.
  " 5th, The  transparent parts of bodies, according to their
several sizes, reflect rays of one color, and  transmit  those of
another, on  the same grounds that  thin  plates or bubbles do
reflect or transmit these rays;  and this I take to be the ground
of all their colors."
  " 6th, The  parts of bodies on which their colors depend are
denser than the medium which pervades their interstices.
  "7th, The  bigness of the component parts of natural bodies
may be conjectured by their colors."
  Upon these principles  Sir Isaac  has endeavored to explain
the phenomena of transparency, black and  white opacity, and
coZor.  He  regards  the transparency of water, salt, glass,
stones, and such like  substances, as arising from the smallness
of their particles, and the intervals between them ; for though
he considers them to be as full of pores or  intervals  between
the  particles as other bodies are, yet he reckons  the  particles
and their intervals to be too small to cause  reflexion at their
common surfaces.   Hence it follows, from the table in page
93, that the  particles of air and  their intervals cannot exceed
the  half of a millionth part of an inch ; the particles of water
the  fth of a millionth, and those of glass the jd of a millionth;
because at these thicknesses the light reflected is nothing, or
the  very black of the first order. The opacity of bodies, such
as that of white paper, linen, &c.,  is ascribed by Newton to a
A 
 CHAP.  XXXIV.  COLORS OF NATURAL BODIES.          237

 greater size of the particles and their  intervals, viz. such  a
 size as to reflect the white,  which is a mixture  of the colors
 of the  different orders.  Hence in air  they must exceed 77
 millionths of an inch,  in water 57 millionths, and in glass 50
 millionths.
   In like  manner all the different colors in Newton's table
 are supposed to be produced  when the particles  and their in-
 tervals have an intermediate size  between  that  which pro-
 duces  transparency and that which produces white opacity.
 If a film of mica, for example, of an uniform blue color, is cut
 into the smallest pieces of  the  same thickness, every piece
 will keep its color, and a heap of such pieces will constitute a
 mass of the same color.
   So far the Newtonian theory is plausible; but in attempting
 to explain black opacity, such as that of coal  and other bodies
 absolutely impervious to light, it seems  to fail  entirely.   To
 produce blackness, " the particles must be less  than any of
 those which  exhibit color. For at all greater  sizes there is too
 much light reflected to constitute this color; but if they be
 supposed a little less than is  requisite to reflect the white and
 very faint blue of the first order,  they will reflect so very little
 light as to appear intensely black."   That such bodies will be
 black when seen by reflexion  is evident;  but what becomes
 of all  the transmitted light 1 This question seems to have
 perplexed  Sir Isaac.  The answer  to it is, " it  may perhaps
 be variously refracted  to and fro  within the body,  until it
 happens to be stifled and lost; by which means it will appear
 intensely black."
  In this theory, therefore,  transparency and blackness are
 supposed to be produced by the very same constitution of the
 body;  and a refraction to and fro is assumed to  extinguish
 the transmitted  light in the  one  case, while in the other such
 a refraction is entirely excluded.
  In  the  production of colors of every kind, it is assumed
 that the complementary color, or generally  one half of the
 light, is lost  by  repeated reflexions.  Now, as reflexion only
 changes the direction of light, we should expect that the light
 thus scattered would show itself in some form or other; but
 though many accurate experiments have been made to discover
 it, it has never yet been seen.
  For  these  and  other  reasons,*! which it would be out of
place here to enumerate, I consider the Newtonian tjieory of

  * See a more detailed examination of the theory in my Life of Sir Isaac
Newton.
  f For an account of Sir David Brewster's outline of a new theory of the
colors of natural bodies, see Note VII. of Am. ed. 
 238             A TREATISE ON OPTICS.       PART III.

 colors as applicable only to a small class of phenomena, while
 it leaves unexplained the colors of  fluids and  transparent
 solids, and all the beautiful hues of the vegetable kingdom. In
 numerous experiments on the colors  of leaves, and  on the
 juices expressed from them, I have never been able to  see the
 complementary color which disappears, and  I  have almost in-
 variably found that the transmitted and  the reflected tint is
 the same.   Whenever there was an  appearance of two  tints,
 I have found it to arise from there being two differently color-
 ed juices existing  in different sides  of the  leaf.   The New-
 tonian theory is, we doubt not, applicable to the colors of the
 wings of insects, the feathers of birds, the scales of fishes, the
 oxidated films on metal and glass, and certain  opalescences.
   The colors of vegetable life and those of various kinds of
 solids arise, we are persuaded, from a specific attraction which
 the particles of these bodies exercise over the differently col-
 ored rays of light. It is by the light of the sun  that the  colored
 juices of plants are elaborated, that the colors of bodies are
 changed, and that many chemical combinations and decompo-
 sitions are effected.  It is not easy to allow that  such effects
 can be produced by the mere vibration of an ethereal medium;
 and we are forced,  by this class of facts, to  reason as if  light
 was material.   When a portion of light enters a body,  and is
 never again seen, we are entitled to say that it is detained by
 some power exerted over the light by the particles of the body.
 That it is attracted by the particles seems extremely probable,
 and  that it enters  into combination with them, and  produces
 various chemical and physical effects, cannot well be doubted;
 and without knowing the manner in which this combination
 takes place, we may say that  the light is absorbed, which is
 an accurate expression of the fact.
  Now,  in  the case of water, glass,  and  other transparent
 bodies,  the  light which  enters their substance has a  certain
 small portion of its particles absorbed, and the  greater part of
 it which escapes from absorption,  and is transmitted, comes
 out colorless, because the  particles  have absorbed a  propor-
 tional quantity  of all  the  different rays which compose white
 light, or, what is the same thing, the body has absorbed white
 light.
  In all colored solids and fluids  in which the  transmitted
light  has a specific  color, the  particles of the body have ab-
 sorbed all the rays which  constitute the complementary color,
detaining  sometimes all the rays of a certain  definite  refran-
gibility, a portion of the rays of other refrangibilities, and al-
lowing other rays to escape entirely from absorption ; all the
rays thus stopped will  form by their  union a particular com-
A 
 CHAP. XXXIV. COLORS OF NATURAL BODIES.          239

 pound color, which will be exactly complementary to the color
 of the transmitted rays.
   In black bodies, such as coal, Sic, all the rays which enter
 their substance are  absorbed;  and hence we see the  reason
 why such bodies are more easily heated and inflamed by the
 action of the luminous rays.  The influence exercised by heat
 and cooling upon the absorptive power of bodies furnishes an
 additional support to the preceding views.
   (166.) Before concluding  this chapter, we may mention a
 few curious facts relative  to white opacity, black opacity, and
 color, as exhibited by some peculiar substances.
   1st, Tabasheer, whose  refractive power  is 1-111,  between
 air and water, is a silicious concretion found in the joints of
 the  bamboo.   The  finest varieties reflect a delicate  azure
 color, and transmit a straw-yellow  tint, which is complement-
 ary  to the azure.  When  it is slightly wetted with  a wet
 needle or pin, the wet spot instantly becomes milk white and
 opaque.  The application of a greater quantity of water re-
 stores its transparency.
   2dly, The cameleon mineral is a solid substance made by
 heating the pure oxide of manganese with potash.  When it is
 dissolved in a little warm water, the solution changes its color
 from green to blue and purple, the last descending in the order
 of the rings, as if the particles  became smaller.
  3dly, A mixture of oil of sweet almonds  with soap and sul-
 phuric acid is, according to M. Claubry,  first yellow, then
 orange, red, and violet.  In  passing from the orange  to the
 red,  the mixture appears almost black.
  4thly,  If, in place of oil of almonds, in the preceding ex-
 periment, we  employ the  oily liquid obtained  from alcohol
 heated with chlorine, the colors of the mixture will be pale
 yellow, orange, black, red, violet, and beautiful blue.
  5thly,  Tincture of turnsole,  after having been a consider-
 able time shut up in a bottle, has an orange color; but when
 the bottle is opened and the fluid shaken, it becomes in a few
 minutes red, and then violet-blue.
  6thly,  A solution  of hoematine in water containing some
drops of  acetic acid is a greenish  yellow.   When introduced
 into  a tube containing mercury, and heated  by surrounding  it
 with  a hot iron,  it  assumes the  various  colors of yellow,
orange, red, and purple, and returns gradually to its primitive
tint.
  7thly,  Several of  the metallic oxides exhibit a temporary
change of color by heat, and resume  their  original color by
cooling.  M.  Chevreul observed,  that when indigo, spread
upon paper, is volatilized,  its color passes into a  very brilliant 
240            A  TREATISE  ON OPTICS.       PART III.

poppy-red.   The yellow phosphate of lead grows green when
hot.
  8thly,  One of the most remarkable facts, however,  is that
discovered by M. Thenard. lie found that phosphorus, purified
by repeated distillations, though naturally of a whitish yellow
color when allowed to cool slowly, becomes absolutely black
when thrown melted  into cold  water.  Upon  touching some
little globules that still remained yellow and  liquid  when he
was repeating this experiment, M. Biot  found that Uiey in-
stantly became solid and black.
                     CHAP. XXXV.

                 ON THE EYE AND VISION.
   An account of the structure and functions of the human
eye, that masterpiece of divine mechanism, forms an interest-
ing branch of applied optics.  This noble organ,  by means of
which we acquire so large a portion of our knowledge of the
material universe, is represented in figs. 140. and 141., the
former being a front and external view of it, and the latter a
section of it through all its humors.
  The human eye is of a spherical form, with a slight pro-
jection in front.   The eyeball or globe of the eye consists of
four coats or membranes,  which have received the names of
the sclerotic coat, the choroid coat, the cornea, and the retina;
and these coats inclose three humors,—the aqueous humor,
the vitreous humor,  and the crystalline humor,  the last of
which has the form of a lens.  The sclerotic coat, aaaa, or
the outermost is a strong and tough membrane, to which arc
attached all the muscles which give motion to  the eyeball,
Fig. 140.
 
CHAP. XXXV.    DESCRIPTION OF THE EYE.           241

and it constitutes the white  of the eye, a a, fig. 140.  The
cornea, b b, is the clear and transparent coat which forms the
Fig. 141.
 front of the eyeball, and is the  first optical surface at which
 the  rays of light are  refracted.  It is firmly  united  to  the
 sclerotic coat, filling up, as it were, a circular aperture in its
 front.  The cornea is an  exceedingly tough membrane, of
 equal thickness throughout, and composed of several  firmly
 adhering layers, capable of opposing great resistance  to  ex-
 ternal injury.  The choroid coat is a delicate membrane lining
 the inner surface of the sclerotic, and covered on  its  inner
 surface with a black pigment.   Immediately within  this pig-
 ment, and close to it, lies the retina, r r r, which is the inner-
 most coat of all.  It is a delicate reticulated membrane, formed
 by the expansion of the optic nerve, O O, which enters the eye
 at a point about ^ of an inch from the axis on the  side next
 the nose.  At the extremity of the axis of the eye,  in  a line
 passing through the centre of the cornea, and  perpendicular
 to its surface, there is a small  hole  with a  yellow margin,
 called the foramen centrale, which, notwithstanding its name,
 is not a real opening, but only a transparent spot, free from
 the soft pulpy matter of which the retina is composed.
   In looking through the cornea  from without, we perceive a
 flat circular membrane, ef, fig. 141., or within, b b, fig. 140.,
 which is grey, blue, or black, and divides the  anterior of the
 eye into two very unequal parts.   In the centre of it there is
 a  circular opening, d, called the pupil, which widens or ex-
 pands when a small portion of light enters the eye, and closes
 or contracts when a great quantity of light enters.   The two
 parts into which the iris divides the eye are called the anterior
and the posterior chambers.  The anterior chamber, which is
anterior to the iris, ef, contains the aqueous humor;  and the
posterior chamber, which is posterior to the iris, contains the
crystalline and vitreous humors, the last of which fills a great
portion of the eyeball.
                            V 
242             A TREATISE  ON  OPTICS.       PART III.

   The crystalline lens, c c,fig. 141., is a more solid substance
than either the aqueous or the vitreous humor. It is suspended
hi a transparent bag or capsule by the ciliary processes, g g,
which are attached to every part of the  margin or circumfer-
ence of the capsule.   This lens is  more  convex behind than
before; the radius of its anterior surface being 0-30 of an inch,
and that  of  its  posterior surface 0 22 of an  inch.  The  lens
increases in  density from its  circumference to its centre, and
possesses the doubly refracting structure.  It consists of con-
centric coats, and these are  again  composed of fibres.  The
vitreous humor, V V, is contained in a capsule, which is sup
posed to be divided into several compartments.
   The total  length of the eye from O to b is about 0-91 of an
inch; the principal focal distance of the lens, c c, is 1-73; and
the range of the moving eyeball, or the diameter of the field
of distinct vision, is 110°.   The field of  vision is 50° above a
horizontal line and 70° below it or altogether 120° in a ver-
tical plane.  It is 60° inwards and 90° outwards, or altogether
in a horizontal plane  150°.
   I have  found the following to be the refractive powers of
the different humors of the eye;  the ray of light being inci-
dent upon them from air:—
  Aqueous               Crystalline Lens.             Vitreous
  Humor.          Surfrce.    Centre.      Mean.          HvmOT.
  1-3366.         1-3767.   1-3990.   1-3839.         1-3394.
   But as  the rays refracted by the aqueous  humor pass  into
the crystalline, and those from the crystalline into the vitreous
humor, the indices of refraction  of the separating surface of
each of these humors will be:—
   From aqueous humor to outer coat of the crystalline    10300
   From do. to crystalline, using the mean index   -  -   1-0353
   From crystalline outer coat to vitreous.....0-9729
   From   do.   to   do.  using the mean index -  -  -   0-9679
   As the  cornea and  crystalline lens must act upon the rays
of light which fall upon the eye exactly like a convex  lens,
inverted images of external objects will  be formed upon the
retina rrr in precisely the same manner as if the retina were
a piece of white paper in the focus of a single lens placed at
d.  There is this difference, however, between the two cases,
that in the eye the spherical aberration is corrected by means
of the variation in the density of the crystalline  lens, which,
having a greater refractive power near the centre of its mass,
refracts the central rays to  the same point as the rays which
pass through it near its circumference c c.  No  provision,
however,  is made in the human eye for the correction of color, 
CHAP. XXXV.     ON THE SEAT OF VISION.
243
 because  the. deviation of the  differently colored  rays  is too
 small to produce indistinctness of vision.   If we shut up all
 the  pupil excepting a portion  of its edge, or look past the
 finger held  near the eye, till the finger almost hides a narrow
 line of white light we shall see a distinct prismatic spectrum
 of this line  containing all the different colors; an effect which
 could not take  place if the eye were achromatic,
   That  an  inverted image of external  objects is formed on
 the  retina has been often proved, and may be  ocularly demon-
 strated by taking the eye of an ox, and paring away with a
 sharp instrument the sclerotic coat till it becomes thin enough
 to see the image through it  Beyond this point optical science
 cannot carry us.  In what manner the retina conveys  to the
 brain the impressions which it receives from the rays of light
 we know not, and perhaps never shall know.

          On the Phenomena and Laws of Vision

   (167.)  1. On the seat of vision.—The retina, from its deli-
 cate structure, and its proximity  to the vitreous  humor, had
 always been regarded as the seat of vision, or the  surface on
 which the refracted rays were  converged to their foci, for the
 purpose  of  conveying the  impression to the brain, till M.
 Mariotte made the  curious discovery that the base of the optic
 nerve, or the  circular section of it at O, fig. 141., was in-
 capable of conveying to the brain the impression of distinct
 vision.
   He  found that when the image of any external object fell
 upon the base of the optic nerve,  it instantly disappeared.  In
 order to  prove  this, we have only to place upon the wall, at
 the  height of the eye, three wafers, two feet distant from each
 other.   Shutting one eye, stand opposite to the middle wafer,
 and while looking  at the outside wafer on the same hand as
 the  shut  eye, retire gradually from the wall till the middle
 wafer  disappears.   This  will happen at  about five times the
 distance of the wafers, or ten feet from the wall; and when
 the middle wafer vanishes, the two outer ones will be distinctly
 seen.  If candles are substituted for wafers, the middle candle
 will not  disappear,  but it will become a cloudy mass of light.
 If the  wafers are placed upon a colored wall, the  spot occu-
 pied by the  wafer will be covered  by the color of the wall, as
 if the wafer itself had been removed.  According to Daniel
 Bernoulli, the  part of the optic  nerve insensible  to distinct
 impressions  occupies about the seventh part  of the diameter
of the eye, or about the eighth  of an inch.
  This unfitness of the  base of the optic nerve  for giving 
244            A TREATISE  ON  OPTICS.       TART III.

distinct vision,  induced Mariotte to believe that the choroid
coat, which lies immediately below the  retina, performs the
functions  ascribed to  the retina;  for where  there was no
choroid coat there was no distinct vision.   The opacity of the
choroid coat and the transparency of the retina, which render-
ed it an unfit ground for the reception of images, were argu-
ments  in favor of this opinion. Comparative anatomy furnishes
us with another argument, perhaps even more conclusive than
any of those urged by Mariotte.   In the eye of the sepia
loligo, or cuttle-fish, an opaque  membranous pigment is inter-
posed between the  retina and the vitreous humor;* so that, if
the retina is essential to vision,  the impressions of Oie image
on this black membrane must  be conveyed to the retina by
the vibrations of the membrane in front of it.  Now, 6ince
the human retina is transparent, it will not prevent the images
of objects from being  formed on the choroid coat;  and the
vibrations  which they  excite in this membrane, being  com-
municated to the retina, will be conveyed to the brain.  These
views are  strengthened by another fact  of some interest.   I
have observed in young persons, that the choroid coat (which
is generally supposed to be black, and to grow fainter by age,)
reflects a brilliant  crimson color, like that  of dogs and other
animals.   Hence, if the retina  is affected by rays which pass
through it, this  crimson light which must necessarily be trans-
mitted  by it ought  to excite the sensation of crimson, which I
find not to be the case.
  A French writer,  M. Lehot,  has  recently written a  work,
endeavoring to  prove that the seat  of vision is in the vitreous
humor; and that, in place of seeing a flat  picture of the ob-
ject, we actually see an image of three dimensions,  viz. with
length, breadth, and thickness.  To produce this  effect, he
supposes that the retina sends out a number of small nervous
filaments, which extend into the vitreous humor, and convey
to the brain the impressions of all parts of the image.  If this
theory  were true, the eye  would not require to adjust itself to
different distances; and we besides  know for certain, that the
eye cannot see  with equal distinctness two points of  an object
at different distances, when it sees one of them perfectly.  M.
Lehot might  indeed reply to the first of these objections, that
the nervous filaments  may not extend far enough into the
vitreous humor  to render adjustment unnecessary; but  if we
admit this,  we would be admitting  an imperfection  of work-
manship, in so far as the Creator would then be employing two
Dr. Knox, Edinb. Journal of Science, No. VI. p. 199.
J 
CHAP. XXXV.   LAW OF VISIBLE DIRECTION.          245

kinds of mechanism  to  produce an effect which could have
been easily produced by either of them separately.
  As difficulties still attach to every opinion respecting  the
seat of vision, we shall still  adhere to  the  usual expression
used by all optical writers, viz.  that the  images of objects are
painted on the retina.
  (168.)   2. On the  law of visible direction.—When a  ray
of light falls upon the retina,  and gives us vision of the point
of an object from which it proceeds, it becomes an interesting
question to determine in what direction the object will be seen,
reckoning from the point where it falls  upon the retina.  In
fig. 142., let F be a point of the retina on which the image of
a point of a distant object is formed by means of the crystalline
Fig. 142.
lens, supposed to be at L L.  Now, the rays which form the
image of the point at F fall upon the retina in all possible di-
rections from L F to L F, and we know that the  point F is
seen in the direction F C R.  In the same manner, the  points
//' are seen somewhere  in  the directions /' S,/T.  These
lines F R,/' S,fT,  which may be called  the lines of visible
direction, may either be those which pass through the centre
C of the  lens  L L,  or, in the case of the  eye,  through the
centre of  a lens equivalent to all the refractions employed in
producing the image; or  it  may be the  resultant of all the
directions within the angles L F L, LfL; or it may be a line
perpendicular to the  retina at F,f'f.   In order to determine
this point, let us look over the top of a card at the point of
the object whose image is at F till the edge of the card  is just
about to hide it, or, what is the same thing, let us obstruct all
the rays that pass through the pupil excepting the upper ones,
R L, R C; we shall then find that the point whose image is at F,
is seen in the same direction as when it was seen  by all the
rays L F,  C F,  L F.   If we look beneath the card in a similar
manner, so as to see the object by the lower rays, R L F,  R C F
                            V2 
246             A TREATISE  ON  OPTICS.       PART III.

we shall sec it in the same direction.   Hence it is manifest
that the line of visible direction does not depend on the direc-
tion of the ray, but is always perpendicular to the retina. This
important truth in the physiology of vision may  be  proved  in
another way.  If we look at the sun  over the top of a card,
as before, so as to impress the eye with a permanent spectrum
by  means of rays  L F falling obliquely  on the retina, this
spectrum will be seen along the axis of vision F C.  In like
manner, if we press  the  eyeball at any part where the retina
is, we shall see the luminous impression which is produced, in
a direction perpendicular to the  point of pressure;  and if we
make the pressure with the head of a  pin,  so as to press either
obliquely or perpendicularly, we shall  find that  the luminous
spot has the same direction.
  Now, as the interior eyeball is as nearly  as possible a perfect
sphere, lines perpendicular to the surface of the retina must
all  pass  through  one single point, namely,  the  centre of its
spherical surface.  This one point may be called  the centre of
visible direction,  because every point of a visible object will
be seen in the direction of a line drawn from this  centre  to
the visible point.  When we move the eyeball by means of its
own muscles through its whole range  of 110°, every point of
an object within the area of the visible field  either of distinct
or indistinct vision remains absolutely fixed, and this arises
from the  immobility  of the centre of" visible direction,  and,
consequently, of the lines of visible direction joining  that
centre and every  point in the visible field.  Had  the centre of
visible direction been out of the centre of the  eyeball, this
perfect stability of vision  could not have existed.  If we press
the  eye  with the finger, we alter the spherical form of the
surface of the retina; we consequently alter the direction of
lines perpendicular  to it, and also the centre where these lines
meet; so  that the directions  of visible objects should be
changed by pressure,  as we find them to be.
  (169.) 3.  On the  cause  of erect vision from an inverted
image.—As the refractions  which take place at the surface of
the cornea, and at the surfaces of the crystalline lens, act ex-
actly like those in a  convex lens in  forming behind it an in-
verted image of an  erect object; and as we know from direct
experiment that an  inverted image is formed on  the retina, it
has been long a problem among  the learned, to determine how
an inverted  image produces an erect object.  It would be a
waste of time to give even an outline  of the different opinions
which have been  entertained on this subject;  but there is one
so extraordinary as to merit notice. According to this opinion,
all infants see objects upside down, and it is only by comparing 
CHAP. XXXV.    CAUSE OF ERECT VISION.
247
 the erroneous information acquired by vision with the accurate
 information acquired by touch, that the young learn to  see
 objects in an erect position!  To refute such an opinion would
 be an insult to the intelligent reader.  The establishment of
 the  true cause of erect vision necessarily overturns all erro-
 neous hypotheses.
   The law of visible direction above explained, and deduced
 from direct experiment, removes at once every difficulty that
 besets the subject.  The lines of visible direction necessarily
 cross each other at the centre of visible direction, so that those
 from the lower part of the image go to the upper part of the
 object, and those from the upper part of the image to the lower
 part of the object.  Hence, in fig. 142. the  visible  direction
 of the point/', formed by rays coming from the upper end S
 of the object, will be/' C S, and  the visible direction of the
 point/, formed  by rays coming from the lower end T of the
 object, will be/CT; so  that an  inverted image necessarily
 produces an erect object.
   This conclusion may be illustrated in another way.  If we
 hold  up against the sun the  erect figure of a man, cut out of
 a piece of black  paper,  and look  at it steadily for a little
 while; if we then shut both eyes, we shall see an erect spec-
 trum of the man when the  figure  of the paper is erect,  and
 an inverted spectrum of him when the figure is held in an
 inverted position.   In this case, there are  no rays proceeding
 from the object to the retina after the eye is shut, and therefore
 the object is seen  in the positions above mentioned, in virtue
 of the lines of visible direction being in all cases perpendicular
 to the impressed part of the retina.
   (170.)  4.  On the law of distinct vision.—When  the  eye
 is directed  to any point of a landscape, it sees with perfect
 distinctness only that point of it which  is directly in the axis
 of the eye, or the  image of which falls upon the central  hole
 of the retina.   But, though we do not see any point but  the
 one with that distinctness which is necessary to  examine it,
 we still see the other parts of the landscape with sufficient
 distinctness to enable us to enjoy its general  effect.   The  ex-
 treme mobility of  the eye, however, and the duration of the
 impressions made upon  the retina, make up for this apparent
 defect, and  enable  us to see the landscape as  perfectly as if
 every part of it were seen with equal distinctness.
   The indistinctness of vision for all objects situated out of
 the axis of the eye increases with  their distances from  that
axis; so that we are not entitled to ascribe the distinctness of
vision  in  the axis  to the  circumstance of the image being
formed on the central hole of the retina, where  there  is no 
248             A TREATISE ON OPTICS.       PART III.

nervous matter;  for if this were the case, there would be a
precise boundary between distinct and indistinct vision, or the
retina would be found  to grow thicker and thicker as it  re-
ceded from the central hole, which is not the case.
   In  making some experiments on the indistinctness of vision
at a distance from  the axis of the eye, I was led to observe a
very  remarkable peculiarity of oblique vision.  If we shut one
eye, and direct the other to any fixed point, such as the head
of" a pin, we shall see indistinctly all  other  objects  within the
sphere of vision.   Let one of these  objects thus seen indis-
tinctly be a strip of white paper, or a pen lying upon a green
cloth. Then, after a short  time, the strip of paper,  or the pen,
will disappear altogether, as if it were entirely removed, the
impression of the green cloth upon  the surrounding parts of
the eye extending itself over the part of the retina which the
image of the pen  occupied.  In  a  short  time the vanished
image will reappear, and again  vanish.  When both eyes are
open,  the very same effect  takes place, but not so readily as
with one eye.  If the object seen  indistinctly is a black stripe
on a white ground, it will vanish in a similar manner.  When
the object seen obliquely is luminous, such as a candle, it will
never vanish entirely,  unless its light is much weakened  by
being placed at a great distance, but  it swells  and  contracts,
and is encircled  with a nebulous halo; so  that the luminous
impressions  must extend themselves  to adjacent  parts of the
retina which are not influenced by the light itself.
  If, when two  candles are  placed  at the distance of about
eight  or ten feet from the eye, and  about a foot  from each
other, we view the one directly and the other indirectly, the
indirect image will swell, as we have already mentioned, and
will be surrounded with  a bright ring of  yellow light, while
the bright light  within the ring will have  a pale  blue color.
If the candles are viewed through a prism, the red  and green
light of the  indirect image will vanish, and there will be left
only a large mass of yellow terminated with a portion of blue
light.   In making this experiment, and  looking  steadily and
directly at one of the  prismatic  images of the candles, 1 was
surprised to find that  the  red and green  rays began  to dis-
appear, leaving  only yellow and a small portion of blue; and
when the eye was kept immovably fixed on the same point of
the image, the yellow light became almost  pure white, so that
the prismatic image was converted into an elongated  image
of white light.
  If the strip of white paper which is seen indirectly with
both eyes is placed so near  the eye as to be seen double, the
rays which proceed from it no longer fall upon corresponding
M 
CHAP. XXXV.   LAW OF DISTINCT VISION.             249

points of the retina, and the two images do not vanish instan-
taneously.   But when the one  begins to disappear, the  other
begins soon after it, so that they sometimes appear to be ex-
tinguished at the same time.
   From these results it appears that oblique or indirect vision
is inferior to direct vision, not only in distinctness, but from
its inability to preserve a sustained vision  of objects;  but
though thus defective, it possesses  a superiority over direct
vision in giving us  more perfect vision of minute objects, such
as small stars, which cannot be seen by direct vision.   This
curious fact has been noticed by Mr. Herschel and  Sir James
South, and  some of the French astronomers.   " A rather sin-
gular method," say Messrs. Herschel and  South, "of obtaining
a view, and even a rough measure, of the angles of stars of
the last degree of faintness, has often been resorted to, viz. to
direct the  eye to  another part of the field.  In  this way, a
faint star, in the neighborhood of a large one, will often be-
come very conspicuous; so as to bear a certain illumination,
which will yet totally disappear, as if suddenly blotted  out,
when the eye is turned full upon it, and so on, appearing and
disappearing alternately as often as you please.  The lateral
portions of the retina, less fatigued by strong lights, and  less
exhausted by perpetual attention, are probably more sensible
to faint impressions than the central ones'; which may  serve
to account for this phenomenon."
   The following  explanation of  this curious phenomenon
seems to me more  satisfactory:—A luminous  point seen by
direct vision, or a sharp  line of light viewed steadily for a
considerable time,  throws the retina  into a state of agitation
highly unfavorable to distinct vision.  If  we look through the
teeth of a fine comb held  close to the eye, or even through a
single aperture of the same narrowness,  at a sheet of illumi-
nated white paper, or even at the sky, the paper  or the sky
will appear to be covered  with an infinite number of broken
serpentine lines, parallel to the aperture,  and in constant  mo-
tion ; and as the aperture is turned round, these parallel undu-
lations will also turn round.  These black and white lines are
obviously undulations on the  retina, which is sensible to the
impressions of light in one phase of the undulation, and insen-
sible to it in another phase.  An analogous effect is produced
by looking stedfastly, and for a considerable time, on the par-
allel  lines which represent the sea  in certain  maps.  These
lines will  break into portions of serpentine lines,  and all the
prismatic tints  will be seen included between the broken cur-
vilinear portions.   A sharp point or line  of light is therefore 
250             A  TREATISE  ON  OPTICS.       PART III.

unable to keep up a continued vision of itself upon the retina
when seen directly.
  Now, in the case of indirect vision, we have already  peen
that a luminous object does not vanish, but is seen indistinctly,
and  produces an  enlarged image on the retina,  beside that
which is produced by the defect of convergency in  the pencils.
Hence, a star seen indirectly,  will affect a larger portion of
the retina from these two causes,  and, losing its sharpness,
will be more distinct.   It is a curious circumstance, too, that
in the experiment with the two candles mentioned above, the
candles seen  indirectly  frequently appear  more  intensely
bright than the candle seen directly.
  (171).  5.  On the insensibility  of the eye to direct impres-
sions of faint light.—The insensibility of the  retina to  indi-
rect impressions of objects ordinarily illuminated, has a sin-
gular counterpart in its insensibility to the direct impression
of very faint light  If we fix the eye steadily on  objects in a
dark room that are illuminated with the faintest gleam of
light, it will be soon thrown into a state of painful agitation;
the objects will appear and disappear according as the retina
has recovered or lost its sensibility.
  These affections are no doubt the source  of many  optical
deceptions which have been ascribed to a supernatural  origin.
In a dark night, when objects are feebly  illuminated, their
disappearance and reappearance must seem very extraordinary
to a person whose fear or curiosity calls forth all his powers of
observation.  This defect  of  the  eye must have  been often
noticed by the sportsman in attempting to mark, upon the mo-
notonous heaths,  the particular spots where moor-game had
alighted.  Availing himself of the slightest difference of tint
in the adjacent heaths, he endeavors to keep his eye steadily
upon it as he advances ; but whenever the contrast of illumi-
nation  is feeble, he almost always loses sight of his mark, or
if the retina does take it up a second time, it is only to lose it
again.*
  (172.)  6.  On the duration of impressions of light on the
retina.—Every person must have  observed that  the  effect of
light  upon  the eye continues  for  some time.   During the
twinkling of the eye, or the  rapid  closing of the eyelids for
the purpose of diffusing the lubricating fluid over the cornea,
we never lose sight of the objects we are viewing.  In like
manner, when we whirl a burning  stick with a rapid motion,
its burning extremity will produce a complete circle of light,
* See the Edinburgh Journal of Science, No. VI. p. 2?8.
J 
CHAP. XXXV. SINGLE VISION WITH TWO EYES.        251

although that extremity can only be in one part of the circle
at the same instant.
  The most instructive experiment, however, on this subject,
and one which it requires a good  deal  of practice  to make
well, is to look for a short time at the window at the end of a
long apartment, and then quickly direct the  eye to  the  dark
wall.   In general, the ordinary observer will see a picture of
the window, in which the dark bars are white and the white
panes dark; but the practised observer, who makes the observ-
ation with great promptness, will see  an accurate  representa-
tion of the window with dark bars and bright panes; but this
representation is  instantly succeeded by the complementary
picture, in which the bars are bright and  the  panes dark.   M.
D'Arcy found that the light of a live coal, moving at the dis-
tance of  165  feet, maintained  its  impression on the retina
during the seventh part of a second.*
  (173.) 7. On the cause of single vision with two eyes.—
Although an image of  every visible object is formed on the
retina of each eye, yet  when the two eyes are capable of di-
recting their axes to any given object, it always appears single.
There is no doubt that in one sense, we really see two objects,
but these objects appear as one, in consequence of the one oc-
cupying exactly the same place as  the other.  Single vision
with two eyes, or with  any number of eyes, if we had them,
is the  necessary consequence of the  law of  visible direction.
By  the action of  the  external  muscles  of  the  eyeballs, the
axes of each eye can be directed to any point of space  at a
greater distance than 4 or 6 inches.  If we look, for  example,
at an aperture in a window-shutter,  we know that an image
of it is formed in each eye; but as the line of  visible direc-
tion from any point in the one image meets the line of visible
direction from the same point in the other image, each point
will be seen as one point, and, consequently, the whole aper-
ture seen by one  eye will coincide with or  cover the whole
aperture seen by the other.  If the axes of both eyes are di-
rected to a point beyond the window,  or to a  point within the
room, the aperture will then  appear double, because  the  lines
of visible direction from the same points in each image do not
meet at the aperture.   If the muscles of either of the eyes is
unable to direct the two axes of the  eyes to the same point,
the object will in that case also appear double.  This inability
of one eye to follow the motions of  the other  is frequently the
cause of squinting, as the eye which is, as it  were, left behind
necessarily looks in a different direction from the other.   The
* For a farther illustration, see Note VIII. of Am. ed. 
252             A TREATISE ON OPTICS.       PART III.

same effect is often produced by the  imperfect vision of one
eye, in consequence of which the good eye  only is  used.
Hence the imperfect eye will gradually lose the power of fol-
lowing the motions of the other,  and will therefore look in a
different direction.  The disease of squinting  may be  often
easily cured.
  (174.)  8.  On  the  accommodation of the eye  to different
distances.—When the  eye sees  objects distinctly at a  great
distance, it is unable, without some change, to see objects dis-
tinctly at  any less distance.  This will be readily seen by
looking between  the  fingers at a distant object.   When the
distant object is seen distinctly, the fingers will be seen  indis-
tinctly ; and, if we look at the fingers so as to see them dis-
tinctly, the distant object will be quite  indistinct.   The most
distinguished philosophers have  maintained different opinions
respecting the method by which  the eye adjusts  itself to dif-
ferent distances.   Some have ascribed it to the mere enlarge-
ment and diminution of the pupil; some to  the elongation of
the eye, by which the retina is removed from  the crystalline
lens; some to the motion of the  crystalline lens; and others
to a change in the convexity of  the lens, on the  supposition
that it consists of muscular fibres.  I have ascertained, by
direct experiment, that a variation in the aperture of the pupil,
produced  artificially, is incapable of producing  adjustment,
and as an elongation of the eye would alter the curvature of
the  retina, and consequently the centre of visible direction,
and produce a change of place in the image, we consider this
hypothesis as quite untenable.
  In order to discover the cause of the adjustment, I made a
series of experiments, from which the following inferences may
be drawn:—
  1st, The contraction of the pupil, which  necessarily  takes
place when the eye is adjusted to  near objects, does not pro-
duce distinct vision by the diminution of the aperture, but by
some other action which necessarily accompanies  it.
  2dly, That the eye  adjusts itself to near objects by two
actions; one of which is voluntary, depending wholly on the
will, and the other involuntary, depending on the stimulus of
light falling on the retina.
  3dly, That when the voluntary power of adjustment fails,
the adjustment may still be effected by the  involuntary stimu-
lus  of light.
  Reasoning from these inferences, and other results of ex-
periment,  it  seems difficult to avoid  the conclusion that the
power of adjustment depends on the mechanism  which con-
tracts and dilates the pupil; and as this adjustment is  inde- 
CHAP. XXXV. LONG AND SHORT-SIGHTEDNESS.        253

pendent of the variation of its aperture, it must be effected by
the parts in immediate contact with the base of the  iris.  By
considering the various ways in which the mechanism at the
base of the iris may produce the adjustment, it appears to be
almost certain that the lens is removed  from the retina by the
contraction of the pupil.*
   (175.)  9.  On the cause of longsightedness and shortsight-
edness.—Between the ages of 30  and  50,  the eyes of most
persons begin to experience  a remarkable  change, which
generally shows itself in a difficulty of  reading  small type or
ill-printed  books, particularly by candlelight.  This  defect of
sight,  which is  called longsightedness, because  objects are
seen best at a distance, arises from a change  in the state of
the crystalline lens, by which its density and refractive power,
as well as  its form, are altered.  It frequently begins at the
margin of  the lens, and takes  several months to go round it,
and it is often accompanied  with a partial  separation of the
lamina? and  even of  the fibres  of the  lens.  " If the  human
eye," as I have  elsewhere remarked, " is not managed  with
peculiar care at this period, the change in the condition of the
lens often runs into cataract, or terminates  in a derangement
of fibres, which, though not indicated by white  opacity, occa-
sions imperfections of vision that are  often  mistaken  for
amaurosis and other diseases. A skilful oculist, who thoroughly
understands the  structure of the eye, and all its optical func-
tions, would have no difficulty, by means of nice experiments,
in detecting  the very portion of the lens where this change
has taken place; in determining the nature and magnitude of
the change which is  going on; in applying the proper reme-
dies for stopping its progress;  and in ascertaining whether it
has advanced to such a state that aid  can be obtained from
convex or concave lenses.  In such cases, lenses are often re-
sorted  to  before the  crystalline lens has suffered a uniform
change of figure or of density, and the use  of them cannot
fail to aggravate the very evils which they are intended to
remedy.   In diseases of the lens, where the  separation of
fibres is confined to small spots,  and is yet of such magnitude
as to give separate colored images of  a  luminous object, or
irregular halos of light, it is often necessary to limit  the aper-
ture of the spectacles, so as to allow the vision to be performed
by the good part of the crystalline lens."
  This defect of the eye, when it is not accompanied  with
disease, may be  completely remedied by a convex lens, which
  * For a fuller account of these experiments, see Edinburgh Journal of
Science, No. I. p. 77.
                            w 
254             A  TREATISE  ON  OPTICS.       PART III.

makes up for the flatness  and diminished refractive power of
the crystalline, and  enables the eye to converge the pencils
flowing from near objects to distinct foci on the retina.
   Shortsightedness shows itself in an inability to see at a dis-
tance ; and those who experience this defect bring minute ob-
jects very near the eye in order to see  them distinctly.  The
rays from remote objects are in this case converged to foci be-
fore they reach the  retina, and therefore  the  picture on the
retina is indistinct.  This imperfection often appears in early
life, and arises from an increase of density  in the central partt)
of the crystalline  lens.   By using a suitable concave lens the
convergency of the  rays  is delayed, so that a distinct  image
can be formed on the retina.
                     CHAP. XXXVI.

      ON ACCIDENTAL  COLORS AND COLORED SHADOWS.

  (176.) When the eye has been strongly  impressed with
any particular species of colored light and when in this state
it looks at a sheet  of white paper, the paper does not appear
to it white, or of the color  with which the eye was impressed,
but of a different color, which is said to be the accidental color
of the color with which the eye was impressed.   If we place,
for example, a bright red wafer upon a sheet of white  paper,
and fix the eye steadily upon a mark  in  the centre of it, then
if we turn the eye upon the white paper we shall see a cir-
cular spot of bluish green light, of the same size as the wafer.
This color, which is called  the  accidental color  of red, will
gradually fade away. The bluish green image of the wafer is
called an ocular spectrum, because it is impressed on the eye,
and may be carried about with it for a short time.
  If we make the preceding experiment with differently col-
ored wafers, we shall obtain ocular spectra whose colors vary
with the color of the wafer employed, as in the following table.
Color of the Wafer	Accidental Color, or Color of the Ocular Spectrum.
Red.	Bluish green.
Orange.	Blue.
Yellow.	Indigo. Reddish violet.
Green.	
Blue.	Orange red.
Indigo.	Orange yellow.
Violet.	Yellow green.
Black.	White.
White	Black.
 
CHAP. XXXVI.    ON ACCIDENTAL COLORS.
255
  In order to find the accidental color of any color in the spec-
trum, take half the length of the spectrum in a pair of com-
passes, and setting one foot in the color whose accidental color
is  required,  the other will  fall upon  the accidental color.
Hence the law of accidental  colors derived from observation
may be thus  stated:—The accidental color of any color in a
prismatic spectrum, is that color which in the same spectrum
is distant from the first color half the length of the  spectrum;
or, if we arrange all the colors of any prismatic spectrum in
a circle, in their due  proportions, the accidental color of any
particular color  will be the color exactly opposite that par-
ticular color.  Hence  the two colors have been called  opposite
colors.
  If the primitive  color, or that which impresses the eye,  is
reduced  to the  same degree of intensity as the  accidental
color, we shall  find  that the one is the  complement of the
other, or what the other wants to make  it white  light; that
is, the primitive and  the accidental colors will, when reduced
to the same degree of intensity which they have in the spec-
trum, and when mixed  together, make white light.  On this
account  accidental colors have  been called complementary
colors.
  With the aid  of these facts, the theory of accidental colors
will be readily understood.  When the eye has been  for some
time fixed on the  red wafer, the part of  the retina occupied
by the red image is strongly excited, or, as it were, deadened
by  its continued  action.  The sensibility to  red  light will
therefore be diminished;  and, consequently, when  the eye  is
turned from the red wafer to the white paper, the deadened
portion of the retina will be insensible to the red rays- which
form part of the white light from the paper, and consequently
will see  the paper of that color which arises from all  the rays
in the white light of  the paper but the red; that is,  of a bluish
green color, which is therefore the true complementary  color
of  the  red.   When a black  wafer is  placed  on  a white
ground, the circular portion of the  retina, on which the black
image falls, in  place of being deadened, is refreshed,  as  it
were, by the absence of light, while all the surrounding parts
of the retina, being excited by the white light  of  the paper,
will be deadened by its continued action.   Hence, when the
eye is directed to the white paper, it will see a white circle
corresponding to the  black image on the  retina; so  that the
accidental color of black is white.  For the same reason,  if a
white wafer is placed on a black ground, and viewed stedfastly
for  some time,  the eye will afterwards see a black  circular
space; so that the accidental color of white is black. 
256            A  TREATISE  ON OPTICS.       PART III.

   Such are the phenomena of accidental colors when  weak
light is employed; but when the eye is impressed  powerfully
with a bright white light, the phenomena have quite a different
character.   The first person who made this experiment with
any care was Sir Isaac Newton, who sent an  account of the
results to Mr. Locke, but they were not published till 1H29.*
Many years before 1691, Sir Isaac, having shut his  left eye,
directed the right one to the image of the sun reflected from
a  looking-glass.  In order to  see the  impression  which was
made, he turned  his eye to a dark corner of his room, when
he observed a bright spot made by the sun,  encircled by rings
of colors.  This " phantom of  light and colors," as he calls it,
gradually vanished;  but whenever he thought of it, it return-
ed, and became as lively and vivid as at first.   He rashly re-
peated the experiment three times, and his eye was impressed
to such a degree, " that whenever I looked  upon the clouds,
or  a book, or a bright object, I saw upon it  a round bright
spot of light like the sun; and, which is still stranger, though
I looked upon the sun with my right eye only, and not with
my left, yet my fancy began to make an impression on my left
eye as well as  upon my right;  for if I shut my right eye, or
looked upon a book or the clouds with my left eye, I could see
the spectrum of the sun almost as plain as with my right eye."
The effect of this experiment  was such, tfiat Sir Isaac durst
neither write nor read, but was obliged to shut himself com-
pletely up in a dark chamber for three days together, and by
keeping in  the dark, and employing  his mind about  other
things, he began, in about three or four days,  to recover the
use of his eyes.  In these experiments,  Sir Isaac's attention
was more taken up with the metaphysical than with the op-
tical results of them, so that he has not described  either the
colors which he saw, or the changes which they underwent.
   Experiments of a similar kind were made  by M.  iEpinus.
When the sun  was near the horizon, he fixed his eye steadily
on  the solar disc for 15 seconds.  Upon  shutting his eye he
saw an irregular pale sulphur yellow image of the  sun, encir-
cled with a faint red border.   As soon as he opened his eye
upon a white ground, the image  of the sun  was a brownish
red, and its surrounding border sky blue.  With his eye again
shut, the image of the sun became green with a red border,
different from the last.  Turning his eye again upon a white
ground, the sun's image was more red, and its border a brighter
sky blue.  When the eye was  shut, the green spectrum be-
came a greenish sky blue, and then a fine sky  blue, with the
* In Lord King's Life of Locke. 
CHAP. XXXVI. IMPRESSIONS ON THE RETINA.         257
border growing a finer red;  and when  the  eye was open, the
spectrum became a finer red, and its border a finer blue.  M.
^Epinus noticed, that when his eye was fixed upon the white
ground, the image of the sun frequently disappeared, returned,
and disappeared again.
   About the year 1808, I was led to repeat the preceding ex-
periments  of  ^Epinus;  but, instead of  looking  at the sun
when of a dingy color, I took advantage of a fine  summer's
day, when the sun was near the meridian,  and I formed upon
a white ground  a  brilliant image of his  disc by the concave
speculum  of a reflecting telescope.   Tying up my right eye,
I viewed this luminous disc  with my left eye through a tube,
and when the retina was highly excited, I turned my left eye
to a white  ground, and observed the following spectra by al-
ternately opening and shutting it:—
           Spectra with left eye open.            Spectra with left eye shut.
    1. Pink surrounded with green.          Green.
    2. Orange mixed with pink.             Blue.
    3. Yellowish brown.                    Bluish pink.
    4. Yellow.
    5. Pure red.                           Sky blue
    6. Orange.                             Indigo.

   Upon uncovering my right eye,  and turning it to a white
ground, I was surprised to observe that it also gave  a  colored
spectrum,  exactly the reverse  of the  first spectrum,  which
was pink with a green border.  The reverse spectrum was  a
green with a pinkish border.  This experiment was repeated
three times, and always with the same result; so that it would
appear that the  impression of the solar image was  conveyed
by the optic nerve from the left to  the right eye.   Sir Isaac
Newton supposed that it was his fancy  that  transferred the
image from his left to his right eye; but we  are disposed to
think that in his experiment no transference took place, be-
cause  the  spectrum  which  he  saw with both eyes was the
same, whereas in my experiment it  was the reverse one. We
cannot however speak decidedly on this point, as  Sir Isaac
did not observe  that the spectra with the eye shut  were the
reverse of those seen with the eye open.  If a spectrum is
strongly formed  on one eye,  it is a very difficult matter to de-
termine on which eye it is formed, and it  would be impossible
to  do this if the  spectrum was the  same when the eye was
open and shut.
  The phenomena of accidental colors are often  finely seen
when the eye has not been  strongly impressed with any par-
ticular colored object.  It was long ago observed by  M. Meus-
                           W2 
258             A  TREATISE  ON  OPTICS.       PART III

nier, that when the sun shone through a hole a quarter of an
inch in diameter in a red curtain, the image of the luminous
spot was green.  In like manner, every person must have ob-
served in a brightly painted room, illuminated by the sun, that
the parts of any white object on which  the  colored light does
not fall, exhibit the complementary colors.  In  order to see
this class of phenomena, I have found  the  following method
the simplest and the best.   Having  lighted two candles, hold
before one of them a piece of colored glass, suppose bright red,
and remove the other candle to such a  distance that the  two
shadows of any body formed upon a piece of white paper may
be equally dark.  In this case one of the shadows will be red,
and the other green.   With blue glass, one of them will be
blue, and the other orange  yellow;  the one being invariably
the accidental color of the other.  The very same effect may
be produced in daylight by two holes in a window-shutter; the
one being covered with a colored glass, and the other trans-
mitting the white light of the  sky.   Accidental colors may
also be seen by looking at the image of a candle, or any white
object  seen  by reflexion from a plate  or  surface of colored
glass sufficiently thin to throw back  its  color from the second
Burface.   In this  case the  reflected image  will always  have
the complementary color of the glass.   The same  effect may
be seen in looking at the image of a candle  reflected from the
water in a blue finger-glass; the image of  the candle is  yel-
lowish : but the effect is not so decided in this  case, as the
retina is not sufficiently impressed with the  blue light of the
glass.
  These phenomena  are obviously different from those which
are produced by colored wafers; because in the present case
the accidental color is seen by a portion of the retina which
is not affected, or deadened as it were, by the primitive color.
A new theory of accidental colors  is  therefore requisite, to
embrace this class of facts.
  As in acoustics, where every fundamental sound is actually
accompanied with its harmonic sound, so in the impressions of
light, the sensation of one color  is accompanied  by a weaker
sensation of its accidental or harmonic color.* When we look
at the red wafer, we are at the same time, with the same por-
tion of the retina, seeing green;  but being much fainter, it
seems  only to dilute the red, and make it, as it were, whiter,
by the combination of the  two sensations.  When the eye
looks from the wafer to the  white  paper, the permanent sen-
  * The term harmonic has been applied to accidental colors; because the
primitive and its accidental color harmonize with each other in painting.
J 
CnAP. XXXVI.   INSENSIBILITY TO COLORS.
259
 sation of the accidental color remains,  and we see a green
 image.   The duration of the  primitive impression is  only a
 fraction of a second, as we have already shown ; but the dura-
 tion  of the  harmonic  impression  continues for a time propor-
 tional  to the strength of the impression.   In order to apply
 these views to the second class  of facts, we must have re-
 course to another principle;  namely, that  when the whole or
 a great part of the retina has the sensation of any primitive
 color, a portion of the retina protected from the impression of
 the color is  actually thrown into  that state which gives the
 accidental or  harmonic  color.  By the vibrations  probably
 communicated from the surrounding portions, the influence of
 the direct or primitive color is not propagated to parts free
 from its action, excepting in  the particular case of oblique
 vision formerly mentioned.   When  the  eye, therefore, looks
 at the white spot of solar light seen in the  middle of the  red
 light of the  curtain, the whole of the retina, except the por-
 tion  occupied by  tiie image  of the white spot, is in the state
 of seeing every thing green; and as  the  vibrations  which
 constitute this state spread over the portions of the  retina
 upon which no red light falls, it will, of course, see the white
 circular spot green.
   (177.) A very remarkable phenomenon of accidental  colors,
 in which the eye  is not excited by any primitive color, was
 observed by Mr. Smith, surgeon in Fochabers.  If we hold a
 narrow strip  of white  paper  vertically, about a foot from  the
 eye, and fix both eyes upon an object at  some distance beyond
 it, then if we allow the light of the sun, or the light of a can-
 dle, to act strongly upon the right eye, without affecting  the
 left, which may be easily protected from its influence, the left
 hand strip of paper will be seen of a bright green color, and
 the right hand strip of a red color.  If the strip of paper is
 sufficiently broad to make the two images overlap each other,
 the overlapping parts will be perfectly white, and free from
 color; which proves that the red and green are complementary.
 When  equally luminous candles are held near each eye, the
 two strips of paper will be white.  If when the candle is held
 near the right eye, and the strips of paper are seen red and
green, then on bringing the  candle  suddenly to the left eye,
the left hand image of the  paper will  gradually  change to
green, and the right hand image to red.
   (178.) A singular affection of the retina,  in reference to
colors, is shown in  the inability of  some eyes to distinguish
certain  colors of the spectrum.  The persons who experience
this defect have their eyes generally in a sound state, and are
capable of performing all the most delicate functions of vision. 
260            A TREATISE ON  OPTICS.        PART III.

Mr. Harris, a shoemaker at Allonby, was unable from his in-
fancy to distinguish the  cherries of a  cherry-tree from its
leaves, in so far as color was concerned.   Two of his brothers
were equally  defective in this respect,  and  always mistook
orange for grass green, and light green for  yellow.  Harris
himself could only distinguish black and  white.  Mr. Scott,
who describes his own case in the Philosophical Transactions,
mistook pink for a pale blue, and  a full red for a full green.
  All kinds of yellows and  blues, except  sky blue, he could
discern with great nicety.   His father,  his  maternal  uncle,
one of his  sisters, and  her two sons, had all the same defect.
  A tailor at  Plymouth,  whose case is described by  Mr.
Harvey, regarded the solar spectrum as consisting only of yel-
low and light blue; and  he could distinguish with certainty
only yellow, white, and green.  He regarded indigo and Prus-
sian blue as black.
  Mr. R. Tucker describes the colors of the spectrum as fol-
lows :—
  Red   mistaken  for   Brown.
  Orange   ....   Green.
  Yellow   sometimes   Orange.
  Green    ....   Orange. |
  A gentleman in the  prime of life, whose case I  had occasion
to examine, saw only two colors  in the spectrum,  viz. yellow
and blue.   When the  middle  of  the red  space was absorbed
by a blue glass, he saw the  black space, with what he called
the yellow, on each side of it.  This defect in the  perception
of color was experienced by the late Mr. Dugald Stewart,
who could  not perceive any difference in the color of the scar-
let  fruit of the Siberian crab and  that  of its  leaves.  Mr.
Dalton is unable to distinguish blue from pink by daylight a"d
in the solar spectrum the red is scarcely visible, the rest  of it
appearing  to consist of two  colors.   Mr. Troughton has the
same defect, and is capable of fully appreciating only blue and
yellow colors; and when he names colors, the names of blue
and yellow correspond to the  more and  less refrangible rays,
all those which belong to the former exciting the sensation of
blueness, and those which belong to the latter the sensation of
yellowness.
  In almost all these cases, the different prismatic  colors  have
the power  of exciting  the sensation of light and giving a dis-
tinct vision of objects, excepting in  the case of Mr. Dalton,
who is said to be scarcely able to see the red extremity of the
spectrum.
  Mr. Dalton has endeavored to explain this peculiarity of
Blue  sometimes  Pink.
Indigo   ...  Purple.
Violet    ...  Purple.
J 
CHAP. XXXVII. PLANE AND CURVED MIRRORS.        261

vision by supposing that in his own case the vitreous humor is
blue,  and, therefore, absorbs a great portion  of the red rays
and other least refrangible rays ;  but this opinion is, we think,
not well founded.  Mr. Herschel attributes this state of vision
to a defect in the sensorium, by which  it is rendered incapable
of appreciating exactly  those differences between  rays on
which their color depends.*
PART IV.
         ON  OPTICAL INSTRUMENTS.

   All the optical instruments now in use have, with the ex-
ception of the burning mirrors of Archimedes, been  invented
by modern philosophers and opticians.  The principles upon
which most of them have been constructed have already been
explained, in the preceding chapters, and we shall therefore
confine ourselves, as much as possible, to a general account of
their construction and properties.


                    CHAP. XXXVII.

             ON PLANE AND CURVED MIRRORS.

   (179.) One of the simplest optical instruments is the single
plane  mirror, or looking-glass, which consists  of a plate of
glass with parallel surfaces, one of which is covered with tin-
foil and quicksilver.  The glass performs no other part in this
kind of plane mirror than that of holding and giving a polished
surface to the thin bright film of metal which is extended over
it  If the surfaces of  the  plate of glass are  not parallel, we
shall see two, three, and four  images of all  luminous objects
seen obliquely; but even when the surfaces are  parallel, two
images of an  object are formed,  one reflected from  the first
surface of glass, and the other from  the posterior surface of
metal;  and the distance of these images  will  increase with
the thickness of the  glass.  The image  reflected from the
glass is, however, very faint compared with the other; so that
for ordinary purposes  a plane  glass  mirror is sufficiently ac-
curate ; but when  a plane  mirror forms a part  of an optical
  * For the theory recently advanced by Sir David Brewster to explain
these cases, see Note IX. of Am. ed. 
262             A TREATISE ON OPTICS.        PART IV.

instrument where accuracy of vision is required, it must be
made of steel, or silver, or of a  mixture of copper and tin;
and in this case  it  is called  a speculum.  The formation of
images by mirrors and specula  has  been fully  described in
Chap. II.                                   J

                       Kaleidoscope.
   (180.) When two plane mirrors are combined in a particu-
lar manner, and  placed in a particular position relative to an
object, or series  of objects, and the  eye, they constitute the
kaleidoscope,  or instrument  for  creating  and  exhibiting
beautiful forms.  If A C, B C, for example, be sections of two
plane mirrors, and M N an object placed between them or in
      Fig. 143.      front  of each, the mirror A C will  form
                  behind it an image m n of the object M N,
                   in the manner shown  in fig. 16.  In like
                   manner,  the  mirror  B C will form an
                   image M' N' behind it.  But, as we have
                   formerly  shown, these images may be con-
                   sidered as new objects, and  therefore the
                   mirror A C will form  behind it  an  image,
                   M" N", of the object or image M' N, and
B C will form behind it an image, m' n', of the object or image
m n.  In like manner it will be found that m" n" will  be the
image of the object or  image M" N", formed  by B C, and of
the object or image m! n', formed  by A C.   Hence m" n" will
actually  consist of two images overlapping each  other and
forming one, provided the angle A C B is exactly 60°, or the
sixth part of a circumference  of 360°.  In this  case all the
six images  (two of the six  forming only one,  m"n",) will,
along with the original  object, M N, form a perfect equilateral
triangle.  The object,  M N, is drawn perpendicular to the
mirror B C,  in consequence of which M N and M' N' form
one straight line; but if M N is moved, all the  images will
move, and  the figure of all  the  images combined  will form
another figure of perfect  regularity,  and  exhibiting the most
beautiful variations, all of which may be drawn by the methods
already described.  In reference  to the multiplication and ar-
rangement of the images, this is the principle of the kaleido-
scope; but the principle of symmetry,  which is essential to the
instrument, depends on the position of the object and the eye.
This principle will be understood  from fig. 144., where ACE
and B C E represent  the two mirrors inclined  at  an angle
A C B, and having C  E for their  line of junction, or common
intersection.  If the object is placed at a distance, as at M N,
then there is no position of the eye at or  above E which will
 
CHAP. XXXVII.   ON THE KALEIDOSCOPE.
263
give a symmetrical arrangement of the six images shown in
fig. 143.; for the corresponding parts of the one will never

                         Fig. 144
 join the corresponding parts of the other.   As the object is
 brought nearer and nearer, the symmetry  increases, and is
 most complete when the object  M N is quite close to A B C,
 the ends of the reflectors.  But  even here it will not be per-
 fect, unless the eye is placed as near as possible to E, the line
 of junction of the reflectors.  The following,  therefore, are the
 three conditions of symmetry in the kaleidoscope:—
   1. That the reflectors should  be placed at an angle which
 is an even or an odd aliquot part of a circle, when  the object
 is regular and similarly situated with respect to both the mir-
 rors ; or an even aliquot part of a  circle, when the object is
 irregular.
   2. That out of an infinite number of positions for the object
 both within and without the reflectors, there is only one posi-
 tion where perfect symmetry can be obtained, namely, by
 placing the object in contact with the ends of the reflectors, or
 between them.
   3. That out of an infinite number of positions for the situa-
 tion of the eye, there is  only one where the  symmetry is per-
 fect, namely, as near as  possible  to the angular point, so that
 the whole of the circular field can be distinctly seen; and this
 point is the only one at which the  uniformity of the reflected
 light is greatest.
   In order to give variety to the  figures formed by the instru-
 ment, the objects,  consisting of pieces of colored glass, twisted
 glass of various curvatures, &c,  are placed in a narrow cell
 between two circular pieces of glass, leaving them  just room
 to move about, while this cell  is turned  round by the hand.
 The pictures thus presented to the eye are beyond all descrip-
 tion splendid and beautiful;  an endless variety of symmetrical
 combinations presenting  themselves to view,  and never again
 recurring with the same form and color.
   For the purpose of extending the power of the instrument
and introducing into symmetrical  pictures  external objects, 
264
A  TREATISE ON  OPTICS.       PART IV.
whether animate or inanimate, I applied a convex  lens, L L,
fig. 141., by means of which  an inverted image of a distant
object, M N, may be formed at the very extremity of the mir-
rors, and therefore  brought into a position of greater sym-
metry  than can be effected in any other way. In this construc-
tion the lens is placed  in  one tube and  the reflectors  in an-
other ; so that by pulling out or pushing in the tube next the
eye, the images of objects at any distance  can be  formed at
the place of symmetry.  In this way, flowers,  trees, animals,
pictures, busts, may be introduced  into symmetrical combina-
tions.   When the distance E B is less than that at which the
eye sees objects distinctly, it is necessary to place a convex
lens at E, to give distinct vision of the objects in the picture.
See my Treatise on the Kaleidoscope.

                  Plane burning Mirrors
   (181.) A combination of plane burning mirrors forms a pow-
erful burning instrument;  and it is highly probable  that it was
with such  a combination that Archimedes destroyed the ships
of Marcellus.  Athanasius Kircher, who first  proved the effi-
cacy of a union of plane mirrors, went with  his pupil Scheiner
to Syracuse, to examine the position of the hostile  fleet; and
they were both satisfied that the ships of Marcellus could not
have been more than thirty paces distant from Archimedes.
   Buffon constructed a burning apparatus upon this principle,
which may be easily explained. If we reflect the light of the
sun upon  one  cheek by a small piece of plane looking-glass,
we shall experience a sensation of heat less than if the direct
light of the sun fell upon  it.   If with the other hand we re-
flect the sun's light upon the  same cheek with another piece
of mirror, the warmth will be increased, and so on, till with
five or six pieces we can no longer endure  the heat.   Buffon
combined 168 pieces of mirror, 6 inches by 8, so that he could,
by a little mechanism  connected with each, cause them to
reflect the light of the sun upon one  spot   Those pieces of
glass were selected which gave the smallest image of the sun
at 250 feet.
   The following were the effects produced  by different num-
bers of these mirrors:—
                           EiTcct produced.
                   Small combustibles inflamed.
                   Beech plank burned.
                   Tarred beech plank inflamed.
                   Pewter flask 61b. weight melted.
                   Tarred and sulphured plank set on fire.
No. of	Distance of
Mirrora.	Object.
12	20 feet
21	20
40	66
45	20
98	126
 
CHAP. XXXVII. CONVEX AND CONCAVE MIRRORS.      265
                    Plank covered with wool set on fire.
                    Some thin pieces of silver melted.
                    Tarred fir plank set on fire.
                    Beech plank sulphured inflamed violently.
                    Tarred plank smoked violently.
                    Chips of fir deal sulphured and mixed with
                     charcoal set on fire.
                    Plates of silver  melted.
   As it is difficult to adjust the mirrors while the sun changes
 his  place, M. Peyrard proposes to produce great  effects by
 mounting each mirror in a separate frame, carrying a tele-
 scope, by means of which one person can direct  the reflected
 rays to the object which  is to be burned.  He conceives that
 with 590 glasses,  about 20 inches in diameter, he could reduce
 a fleet to ashes  at the distance of a quarter  of a league,  and
 with glasses of  double  that size at the distance of half a
 league.
   Plane glass mirrors have been combined permanently  into a
 parabolic form, for the purpose of burning objects placed in
 the  focus of the  parabola,  by the sun's rays; and the  same
 combination  has been used, and is still in use, for lighthouse
 reflectors, the light being placed in the focus of the parabola.

                Convex and Concave Mirrors.
   (182.)  The general properties of convex and concave mir-
 rors have been already described in Chap. II.  Convex mirrors
 are used  principally as household ornaments, and are charac-
 terized by their  property of forming erect and diminished
 images of all objects placed before them, and these images ap-
 pear to be situated behind the mirror.
   Concave mirrors are  distinguished by their  property  of
 forming in front of them, and in the air, inverted images  of
 erect objects, or erect images of inverted  objects, placed at
 some distance beyond their principal  focus.   If a fine trans-
 parent cloud of blue smoke is raised,  by means of a chafing-
 dish, around  the focus of a large concave mirror, the image of
 any  highly illuminated object will  be depicted, in the middle
 of it, with great beauty.  A skull concealed from the observer
 is sometimes used, to surprise the ignorant;  and when a dish
 of fruit has been  depicted  in a similar manner, a  spectator,
 stretching out his  hand to seize it, is met with the image of a
drawn dagger, which has been quickly substituted for the fruit
at the other conjugate focus of the mirror
No. of Mirrors.	Distance of Object.
112	138
117	20
128	150
148	150
154	150
154	250
224	40
 
266            A TREATISE  ON  OPTICS.        PART IV.

  Concave mirrors have been  used as lighthouse reflectors,
and as burning instruments.  When used in lighthouses, they
are formed  of plates of copper plated with silver, and they
are hammered into a parabolic  form,  and then polished with
the hand.   A lamp placed  in the  focus  of the  parabola will
have its divergent light thrown, after reflexion, into something
like a parallel beam, which will retain its intensity at a great
distance.
  When concave mirrors are used for burning, they are gene-
rally made spherical, and regularly ground and  polished upon
a tool, like the specula used in telescopes.  The  most cele-
brated of these were made  by M. Villele, of Lyons, who exe-
cuted five large ones.   One of the best  of them, which con-
sisted of copper and tin, was very nearly four feet in diameter,
and its focal length thirty-eight inches.   It  melted a piece  of
Pompey's pillar in fifty seconds, a silver sixpence in seven
seconds and a half, a halfpenny in sixteen  seconds,  cast-iron
in sixteen seconds, slate in  three seconds, and thin tile in four
seconds.

                    Cylindrical Mirrors
  (183.)  All  objects seen by reflexion in a cylindrical mirror
are necessarily  distorted.   If an observer looks into such a
mirror with its axis standing vertically, he will see the image
of his  head of the same length as the original, because the
surface of the mirror is a straight  line in a vertical direction.
The breadth of the face will be greatly contracted in a hori-
zontal direction, because the surface is  very convex in that

                           Fig. 145.
j 
CHAP. XXXVIII.       ON SPECTACLES.
267
direction, and  in intermediate directions the head will have
intermediate breadths.  If the axis of the mirror is held hori-
zontally, the face will be as broad as life, and exceedingly
short.  If a picture or portrait M N is laid  down horizontally
before the mirror A B, fig. 145.,  the reflected image of it will
be highly distorted; but the picture may be drawn  distorted
according to  regular laws, so that  its image shall have  the
most correct proportions.
  Cylindrical  mirrors, which are now very uncommon, used
to be made for this purpose, and were  accompanied with a
series of distorted figures, which, when seen by the eye, have
neither shape nor meaning, but when laid down before a cylin-
drical mirror,  the reflected image  of them has the most per-
fect proportions.   This effect is shown in fig. 145., where
M N is a distorted figure, whose image in the mirror A B has
the appearance of a regular portrait.
                    chap, xxxvin.

             ON SINGLE AND COMPOUND LENSES

   Spectacles and reading glasses are among the simplest and
most useful of optical  instruments.  In order to enable a per-
son who has imperfect vision to see 'small objects distinctly,
when they are not far from the eye, such as small manuscript,
or small type, a convex lens of  very short focus must be  used
both by those who are  long and short sighted.
   When a short-sighted  person, who cannot see well at a dis-
tance, wishes to have distinct vision at any particular distance,
he must use a concave lens, whose focal length will be found
thus,—Multiply the distance at which he sees  objects most
distinctly by the distance at which he wishes to see them dis-
tinctly with  a concave lens, and  divide this product by  the
difference of the above distances.
   A long-sighted person, who cannot see near objects distinctly,
must use a convex lens, whose focal  length  is found by  the
preceding rule.
   In choosing spectacles, however, the  best way is to select,
out of a number,  those which  are found to answer best  the
purposes for which they are particularly intended.
   Dr. Wollaston introduced a new kind of spectacles, called
periscopic, from their property  of  giving a wider field of  dis-
tinct vision than the common ones. The  lenses used for this
purpose, as shown  at  H and L Jig. 19.,  are meniscuses, in 
268             A TREATISE ON OPTICS.       PART IV
which the convexity predominates,  for long-sighted persons,
and  concavo-convex lenses, in which  the  concavity predomi-
nates, for  short-sighted persons.   Periscopic  spectacles de-
cidedly give more  imperfect vision than common spectacles,
because they increase  both the aberration of figure  and of
color; but they may be of use  in a  crowded city, in warning
us of the oblique approach of objects.


             Burning and Illuminating Lenses.

   (184.) Convex lenses possess peculiar advantages for con-
centrating the sun's rays, and  for conveying to an immense
distance a  condensed  and parallel beam of light.  M. Buffon
found that  a convex lens, with a long focal length, was prefer-
able to one of a short focal length for fusing metals by the
concentration of the sun's rays.   A lens,  for  example, 32
inches in diameter and  6 inches in focal length, with  the di-
ameter  of  its focus  8  lines,  melted  copper in less  than a
minute; while a small  lens 32 lines  in diameter, with a focal
length of 6 lines, and its focus § of a line, was scarcely capa-
ble of heating copper.
   The most perfect burning lens ever constructed was exe-
cuted by Mr. Parker, of Fleet  Street, at an expense of 700/.
It was  made of flint glass, was three  feet in diameter,  and
weighed 212 pounds.   It was 3£ inches thick at the centre;
the focal distance was 6 feet 8 inches, and  the diameter of the
image of the sun in its focus  one inch.   The rays refracted
by the lens were received on a second lens, in whose focus the
objects  to be fused were placed.  This second lens had an ex-
posed diameter of 13 inches; its  central thickness was If of
an inch; the length of its focus was 29 inches.  The diameter
of the  focal image was § of  an  inch.   Its  weight was 21
pounds.  The combined focal  length of the two lenses was 5
feet 3 inches, and the diameter of the focal image \ an inch.
By means of this powerful burning lens, platinum, gold, silver,
copper,  tin, quartz, agate, jasper, flint, topaz, garnet, asbestos,
&c. were melted in a few seconds.
  Various causes have prevented philosophers from construct-
ing burning lenses of greater  magnitude than that made by
Mr. Parker.  The impossibility of procuring pure  flint glass
tolerably free from veins and impurities for a large solid lens;
the trouble  and expense of casting it into  a  lenticular  form
without flaws and  impurities; the great increase  of  central
thickness which becomes necessary by increasing the diameter
of the lens ; the enormous  obstruction that is thus opposed to
the transmission of the solar  rays,  arid the increased aber- 
CHAP. XXXVIII.   ON POLYZONAL LENSES.             269
ration which dissipates the rays at the focal point, are insuper-
able  obstacles to the  construction of solid  lenses of any con-
siderable size.
  (185.) In order to improve a solid lens formed of one piece
of glass, whose section isAmpBEDA, Buffon  proposed to
cut out all the glass  left white in the figure, viz. the portions
between mp,fig. 146., and n o, and between n o and the left
 F-  146  hand surface of D E. A lens thus constructed would
   %s'   '  be incomparably superior to the solid one AmpBE
    •**■    DA; but such a process we  conceive to be imprac-
      (j  ticable on a large scale, from the extreme difficulty
      i   of polishing the  surfaces Am,  B p, C n,  F o, and
      P* the left hand surface of D E; and even if it were
          practicable, the greatest imperfections in the glass
          might happen to occur in the parts which are left.
      |E   In  order to  remove  these  imperfections,  and to
      fF construct lenses of any size, I proposed, in 1811, to
     J    build them up of separate zones or rings, each of
     Jt    which rings was again to  be  composed of separate
segments, as shown in the front view of the lens in fig. 147.
This lens is composed of one central  lens, A B C D, corre-
sponding with its section D E in fig. 146., of a  middle ring
                        G E LI corresponding to C D E F in,
                        fig. 146., and  consisting of five seg-
                         ments ; and another ring, N P R T,
                         corresponding to A C F B, and con-
                         sisting of eight segments.
                           The preceding construction obvi-
                      -]x ously puts it in our power to execute
                         these  compound lenses, to which I
                         have given the name  of polyzonal
                         lenses, of pure flint  glass  free from
                         veins;  but it possesses another great
                         advantage, namely,  that of enabling
 us to correct, very nearly, the spherical aberration, by making
 the  foci of each zone coincide.
   One of these lenses was  constructed,  under my direction,
 for  the Commissioners of Northern Lighthouses, by  Messrs.
 W. and P. Gilbert.  It  was made of pure  flint  glass,  was
 three feet in diameter, and consisted of many zones and  seg-
 ments.   Lenses of this kind have been  made in  France of
 crown  glass, and have  been introduced into the principal
 French lighthouses; a purpose  to which they are infinitely
 better  adapted than  the  best constructed parabolic reflectors
 made of metal.
   A polyzonal  lens of at least four feet in diameter  will be
                            X Z
 
270
A  TREATISE ON OPTICS.        PART IV.
speedily executed as a burning-glass, and will, no doubt, be
the most powerful  ever  made.  The means of executing it
have been,  to a considerable degree, supplied by the scientific
liberality of Mr. Swinton and Mr. Calder, and other gentlemen
of Calcutta.
                      CHAP. XXXIX.

*             ON SIMPLE AND COMPOUND  PRISMS.

                      Prismatic Lenses.

   (186.) The general properties of the prism  in  refracting
 and decomposing light have  already been  explained; but its
 application as an optical instrument, or as an important part of
 optical instruments, remains to be described.
   A rectangular prism, ABC, fig. 148., was first applied by
 bir Isaac Newton as a plane mirror for reflecting to a side the
 rays  which form the image in reflecting telescopes.   The
 angles  B AC, BCA, being each 45°, and B a right angle,
 rays falling on the face A B will be reflected by the back sur-
 face H  C as if it Were a plane  metallic mirror; for whatever
 be the refraction which they suffer at  their entrance into the
 face A B, they will suffer an equal and opposite one at the
 face BC.  The great value  of such a mirror is,  that as the
 incident rays  fall upon A C at an angle greater than that at
 which total reflexion  commences, they will all suffer total re-
flexion, and not a ray will be lost; whereas in  the best me-
tallic speculum nearly half the light  is lost.  A portion of
light, however, is lost by reflexion at  the  two surfaces  A B,
«   T     ATSma11 PTtlon ^ the absorption of the glass itself.
bir Isaac Newton also proposed the convex prism, shown at 
CHAP. XXXIX.    ON PRISMATIC LENSES.
271
D E F, the faces D F, F E being ground convex.  An analo-
gous prism, called the meniscus prism, and shown at G H I,
has been used  by M. Chevalier, of Paris, for the camera  ob-
scura.   It differs only from Newton's in the second face, I H,
being concave in place of  convex.
  On account of the difficult execution of these prisms, I have
proposed to use a hemispherical lens, L M N, the two convex
surfaces of  which are ground at the  same time.  When a
longer focus is required, a  concave lens, R Q, of a longer focus
than the hemisphere P R Q, may he  placed or cemented on
its  lower surface, and  if the concave lens is formed out of a
substance of a different dispersive power, it may be  made  to
correct the color of the convex lens.
  A single prism is used with peculiar advantage for inverting
pencils of light, or for obtaining an erect image from pencils
that would give an inverted one.  This effect is shown in fig.
149., where A B C is a rectangular prism,  and R R' R" a par-
allel pencil of light, which, after being refracted at the points
1, 2, 3, of the face A B, and  reflected at the points a, b, c, of
the base B C, will be again refracted at the points 1,2, 3, of
the face A C, and move on in parallel lines, 3 r", 2 r', 1 r;  the
ray R I, that was uppermost,  being now undermost, as at 1 r.

             Compound and Variable Prisms.
   (187.) The great difficulty of obtaining  glass  sufficiently
pure for a prism of any size,  has rendered it extremely diffi-
cult to procure good ones;  and they have therefore not been
introduced as they would otherwise have been into optical in-
struments.  The principle upon which polyzonal lenses  are
constructed is  equally applicable  to prisms.  A  prism con-
structed like AD,fig. 150., if properly executed, would have
exactly the same properties  as A B C, and  would be incom-
parably superior to it, from the  light passing through such a
small thickness of glass. It would obviously be difficult to exe-
cute such a prism as A D out of a single piece of glass, though 
272
A  TREATISE  ON OPTICS.       TART IV.
it is quite practicable; but there is no difficulty in combining
six small prisms all cut out of one prismatic rod, and therefore

                          Fig. 150.
                            JL
necessarily similar.   The summit of the  rod  should have a
flat narrow face parallel to its  base, which would  be easily
done if the prismatic  rod were cut out of a plate of thick par-
allel glass.  The separate prisms being cemented to one an-
other,  as  in the  figure, will form a compound prism, which
will he superior to the common prism for all purposes in which
it acts solely by refraction.
  (188.) A compound prism of a different kind, and  having a
variable angle, was proposed by Boscovich, as shown in Jig.
151., where AB C is a hemispherical convex lens, moving in
a concave lens, DEC, of the same curvature.  By turning
one of the lenses round upon the other, the  inclination of the
faces A B, D E, or A B, C E, may be  made to vary from 0° to
above 90°.
  (189.)  As this apparatus is both troublesome to execute and
difficult to use, I have employed an  entirely different principle
for the construction of a variable prism, and have used it to a
great extent in numerous experiments on the dispersive pow-
ers of  bodies.  If we produce a vertical line of light by nearly
closing the  window-shutters, and view the line with a flint
glass prism whose refracting angle is 60°, the edge of the re-
fracting angle being held vertical, or  parallel  to the line of 
CHAP. XXXIX.    ON PRISMATIC LENSES.
273
light, the luminous line will be  seen as a brightly  colored
spectrum, and any small portion of it will resemble almost ex-
actly the solar spectrum.   If we  now turn the prism in the
plane of one of its refracting faces, so that the inclination of
the edge to the line of light increases gradually from 0° up to
90° when it is perpendicular to the line of light, the spectrum
will gradually grow less and less colored, exactly as if it were
formed by a prism of a less and less refracting angle, till at an
inclination of 90° not a trace of color is  left.  By this simple
process,  therefore, namely, by using a line of light instead of
a circular disc, we have produced the very same  effect as if
the refracting angle of the prism had been varied from 90°
down to  0°.
  (190.)  Let it now be required to determine the relative dis-
persive  powers  of  flint  glass  and  crown  glass.   Place the
crown glass prism so as to produce the largest spectrum from
the line  of white light, and let the refracting angle of the
prism be  40°.   Then  place the flint glass  prism  between it
and the eye, and turn it round, as before described, till it cor-
rects the color produced by the crown glass prism, or till the
line  of light is perfectly colorless.   The inclination of the
edge of the flint glass prism to the line of light being known,
we can easily find, by a simple formula the  angle  of a prism
of flint glass which corrects  the  color  of a prism  of crown
glass  with  a refracting angle of 40°.   See my Treatise on
New Philosophical  Instruments, p. 291.

                    Multiplying Glass.
  (191.)  This lens is more amusing than useful, and is intend-
ed to give a number of images of the  same  object.   Though
it has the circular  form of a lens, it is nothing more than a

                          Fig. 152
^»
99�9420 
274
A  TREATISE ON OPTICS.       PART IV.
number of prisms formed by grinding various flat faces on the
convex surface of a plano-convex glass, as shown in fig. 152.,
where A B is the section of a multiplying glass in which only
three of the planes are seen.   A direct  image  of the object
C will be seen through the face G H, by the eye  at E; an-
other image will  be seen at D, by the refraction of the face
H B, and a third  at F, by the  refraction  of the  face A G, an
image being  seen through every plane face that is cut upon
the  lens.   The image at C will be  colorless, and all those
formed by planes inclined to A B will be colored in proportion
to the angles which the planes form with A B.
  Natural multiplying glasses  may  be  found  among trans-
parent minerals which are crossed with veins oppositely crys-
tallized, even though they are  ground into plates with parallel
faces.  In some specimens of  Iceland spar  more than a hun-
dred finely colored  images may be seen at once.  The theory
of such multiplying glasses has already been explained in
Chap. XXIX.
                       CHAP.  XL.

      ON THE CAMERA OBSCURA, MAGIC LANTERN, AND
                      CAMERA LUCIDA.

  (192.) The camera obscura, or dark chamber, is the name
of an amusing and  useful optical instrument, invented by the
celebrated Baptista Porta.  In  its original state  it is nothing
more than a dark room with an opening in the window-shutter,
in which is placed a convex lens of one or more feet focal
length.   If  a sheet of white  paper is held perpendicularly be-
hind the  lens,  and passing through its focus, there will be
painted  upon it an accurate picture of all  the  objects seen
from the window, in which the trees and clouds will appear to
move in the wind,  and all living objects to display the same
movements and gestures which they exhibit to the eye.  The
perfect resemblance of this picture to nature astonishes and
delights every person, however often  they may have seen it.
The image is of course inverted,  but if we look over  the top
of the paper it will  be seen as if it  were erect.   The ground
on which the picture is received should be hollow, and  part of
a sphere whose radius is the focal distance of the convex lens.
It is customary, therefore, to make it of the whitest plaster of
Paris, with as smooth and accurate a surface as possible.
  In order to exhibit the picture to several spectators at once, 
CHAP. XL.      ON THE CAMERA OBSCURA.            275

and  to enable any person to copy it, it  is desirable that the
image should be formed upon a horizontal table.  This  may
be done by means of a metallic mirror, placed at an angle of
45° to the refracted rays, which will reflect the picture upon
the white ground  lying horizontally;  or,  as in the portable
camera obscura, it  may be reflected upwards by the mirror,
and received on the lower side of a plate of ground glass,  with
its rough side  uppermost,  upon which the picture may be
copied with a fine sharp-pointed pencil.
  A  very convenient  portable camera obscura for drawing
landscapes or other objects is shown in fig. 153., where A B
is a meniscus lens, with its concave side uppermost, and the
          Fig. 153.          radius of its convex surface being
                          to the radius of its  concave sur-
                          face as 5 to 8, and  C D a plane
                          metallic speculum inclined at an
                          angle of 45° to the horizon, so as
                          to reflect the landscape downwards
                          through  the  lens  A  B.    The
                          draughtsman introduces his  head
                          through an opening in one  side,
                          and his hand with the  pencil
                          through another opening, made in
                          such a manner as to allow no  light
                          to fall upon the picture which is
                          exhibited on the  paper at  E F.
                          The tube  containing the  mirror
                          and lens can be turned round by a
rod within, and the inclination of the mirror changed, so  as to
introduce objects in any part  of the horizon.,
  When the camera is intended for public  exhibition,  it  con-
sists  of the  same parts similarly arranged; but they  are in
this case  placed on the top of a building, and  the  rotation of
the mirror, and its motion in  a vertical  plane, are effected by
turning two rods within the reach of the spectator, so that he
can introduce any object into the picture from all points of the
compass and at all distances.  The picture is received on a
table, whose surface is made of stucco,  and  of the same radius
as the lens, and this surface is made to  rise and fall  to accom-
modate it to the change of focus produced by objects  at dif-
ferent distances.  A camera obscura which throws the  image
down upon  a horizontal surface may be made  without any
mirror,  by using any of the lenticular  prisms D E F,  G H I,
M L N, when the objects are  extremely near, and P R Q,,fig.
148.   The convex surfaces of these prisms  converge the rays
which are reflected to their focus by  the  flat faces D E, G H,
 
276            A  TREATISE  ON  OPTICS.        PART IV.

L N, and P Q; tiiese lenticular prisms may be formed by ce-
menting plano-convex or concave lenses on the faces A B, B C
of the rectangular  prism A B C, or the convex lens  may be
placed near to A B.
  If we wish to form an erect image on a vertical plane, the
prism ABC, fig. 148., may be placed in front of the convex
lens, or immediately behind it.  The same effect might be
produced by three reflexions from three mirrors or specula
  I have found that a peculiarly brilliant effect is given to the
images formed  in the camera obscura when they are received
upon the silvered back of a looking-glass, smoothed by grind-
ing it with a flat and soft hone.  In the portable camera ob-
scura I find that a film of skimmed milk, dried upon a plate of
glass, is superior to ground glass for the reception of images.
   A modification of the camera obscura, called the megascope,
is intended for taking magnified drawings of small objects
placed near the lens.  In this case, the distance of the image
behind the lens is greater than the distance of the  object be-
fore it.  By altering the distance of the object, the size of the
image may be reduced or enlarged.  The  hemispherical lens
L M N,fig. 148., is particularly adapted for the megascope.


                      Magic Lantern.
   (193.) The  magic  lantern, an  invention  of Kircher,  is
shown in fig. 154., where L is a lamp  with a powerful Argand
burner, placed  in a dark  lantern.  On one side of the lantern

                          Fig. 154.
is a concave mirror M N, the vertex of which is opposite to
the centre of the flame, which is placed in its focus.  In the
opposite side of the lantern is fixed a tube A B, containing a
hemispherical  illuminating lens A, and a convex lens B; b&
tween A and B the diameter of the tube is increased for the 
 CHAP. XL.       ON THE CAMERA LTJCIDA.            277

 purpose of allowing sliders to be introduced through the slit
 C D.  These sliders contain 4 or 5 pictures, each painted and
 highly  colored  with transparent varnishes, and, by  sliding
 them through C D, any of the subjects may be introduced into
 the axis of  the tube and between the two  lenses A, B.  The
 light of the lamp L, increased by the light reflected from the
 mirror falling upon the lens A, is concentrated by it upon the
 picture in the slider; and this  picture, being in one of .the
 conjugate foci of the lens B, an  enlarged image of it will  be
 painted on a white  cloth, or on  a screen of white paper,  E,
 standing  or suspended perpendicularly.  The distance of the
 lens B from the object or the slider may be increased or dimin-
 ished by pulling out or pushing in the tube B, so that a distinct
 picture of the object  may be formed  of any  size and  at any
 distance from B,  within moderate limits. If the screen E F is
 made of fine semi-transparent  silver paper, or  fine  muslin
 properly prepared, the image may be distinctly seen by a spec-
 tator on the other side of the screen.
   (194.)  The phantasmagoria is nothing more than a magic
 lantern,  in  which the images are received on a transparent
 screen,  which is fixed in view of the spectator.  The magic
 lantern, mounted upon wheels, is made to recede from or ap-
 proach to the screen; the consequence of which is, that the
 picture on the screen expands to a gigantic size,  or  contracts
 into an invisible object or mere luminous spot.  The lens B is
 made to recede from  the slider in C D when the lantern ap-
 proaches  the  screen, and to  approach to it when the lantern
 recedes from the  screen,  in order that the picture  upon the
 screen may always be distinct.   This  may be accomplished,
 according to Dr. Young, by jointed rods or levers,  connected
 with the  screen,  which pull out  or push in  the  tube B; but
 we are of opinion that the required effect may be much more
 elegantly and efficaciously produced by the simplest piece  of
 mechanism  connected with the wheels.


                      Camera Lucida.

  (195.) This instrument, invented by Dr. Wollaston in 1807,
 has  come into very general use for drawing landscapes, de-
 lineating  objects of natural history, and copying and reducing
 drawings.
  Dr. Wollaston's form  of the  instrument is shown in  fig.
 155., where A B C D is a glass prism, the angle BAD being
90°, A D C  67^°, and D C B 135°.  The rays proceeding from
any object, MN,  after being reflected by the faces  DC,  CB
to the eye, E, placed above the angle B, the observer will see 
278            A TREATISE ON OPTICS.        PART  IV.
an image mnof the object M N projected upon  a piece of
paper at m n.   If the eye is now brought down close to the
                          Fig. 155.
                E
             B l_^   A
                                             M
angle B, so that it at the same time sees into the prism with
one-half of the pupil, and past the angle B with the other half,
it will obtain distinct vision of the image m n, and also see the
paper  and the  point of the  pencil.  The draughtsman has,
therefore, only to trace the  outline of the image upon the
paper, the image being seen  with  half of the pupil, and the
paper and pencil with the other half.
  Many persons  have acquired  the art of using this instru-
ment with great facility, while others have entirely failed. In
examining the  causes of this  failure, professor Amici, of Mo-
dena, succeeded in removing them,  and has proposed various
forms of the  instrument free from  the defects of Dr. Wollas-
ton's.*  The one which M. Amici thinks the best is shown in
                          Fig. 156.
  * An account of these various forms will be found in the Edinburgh
Journal of Science, No. V. p. 157. 
CHAP. XLI.      ON SINGLE MICROSCOPES.            279
fig. 156., where A B C D is a piece  of thick parallel glass,
F G H C a metallic mirror, whose  face, F G, is highly polish-
ed, and inclined 45° to B C.  Rays from an object, M N, after
passing through the glass ABCD, are reflected from F G,
and afterwards from the face B C of the glass plate to the eye
at E,  by which the object, M N, is seen at m n, where the
paper is placed.  The pencil and the paper are readily seen
through the plane glass ABCD.   In order to make the two
faces of the glass, AD, B C,  perfectly  parallel,  M. Amici
forms  a triangular prism of glass, and cuts it through the
middle; he then joins the two prisms or halves, A D C, C A B,
so as to form a parallel plate, and by slightly turning round
the prisms, he can easily find  the position in which the two
faces are perfectly parallel.
                       CHAP.  XLI.

                     ON MICROSCOPES.
  A microscope is an optical instrument for magnifying and
examining minute objects.  Jansen and Drebell are supposed
to have separately invented the single microscope, and Fon-
tana and Galileo seem to have been the first who constructed
the instrument in its compound form.

                    Single Microscope.
  (196.) The single microscope is nothing more than a lens
or sphere of any transparent  substance, in the focus of which
minute objects are placed.   The rays which issue from each
point of the object are refracted by the lens into parallel rays,
which, entering the eye placed immediately behind the lens,
afford  distinct vision  of the object.  The magnifying power
of all such microscopes is  equal to the distance at which we
could examine the object most distinctly, divided by the focal
length of the lens or sphere.   If this distance is 5 inches,
which  it does not exceed in good eyes when they examine mi-
nute objects, then the magnifying power of each lens will be
as follows:—
Focal length in	Linear magnifying	Superficial magnifying
Inches.	power.	power.
5	1	1
1	5	25
i Tff i lAA	50	2500
	500	250000
 
1
280            A  TREATISE  ON  OPTICS.       PART IV.

  The linear magnifying power is the  number of times an
object is magnified  in length, and the  superficial magnifying
power is the number of times that it is magnified in surface.
If the object is a small square,  then a  lens of one inch focus
will magnify the side  of the square 5  times, and its area or
surface 25 times.
  The best single microscopes  are minute  lenses ground and
polished on a concave  tool; but as the perfect  execution of
these requires considerable skill, small spheres have been often
constructed as substitutes.  Dr. Hooke  executed  these spheres
in the following manner:  having drawn out  a thin strip of
window-glass into threads by the flame of a lamp, he held one
of these threads with its extremity in or near the flame, till it
ran into a globule.   The globule was  then cut off and  placed
above a small  aperture, so that none of  the rays which it
transmitted passed through the  part where it was joined to the
thread of glass.  He  sometimes ground off the end  of the
thread, and polished that part of the sphere.  Father di Torre
of Naples improved these globules by  placing them  in small
cavities in a piece of calcined tripoli, and remelting them
with the blowpipe; the consequence of which was, that they
assumed a perfectly spherical form.  Mr. Butterfield executed
similar spheres by taking upon the wetted point of a  needle
some finely pounded glass, and  melting it by a spirit lamp into
a globule.  Lf the part next the needle was not melted, the
globule was removed from the  needle  and taken up with the
wetted needle  on its  round side, and  again presented  to the
flame till  it was a perfect sphere. M. Sivright, of Meggetland,
has made lenses by putting pieces of glass in small  round
apertures  between the 10th and 20th of an inch, made  in pla-
tinum leaf.   They  were then melted by the blowpipe,  so that
the lenses were made and set at the same time.
  Mr. Stephen Gray made globules for microscopes by  insert-
ing drops  of water  in small apertures.  I have made  them in
the same  way with oils and varnishes; but the finest of all
single microscopes may be executed by forming minute plano-
convex lenses upon glass with different fluids.   I have also
formed excellent microscopes by using the spherical crystal-
line lenses of minnows and other small fish, and taking care
that the axis of the lens is the axis  of vision, or that the ob-
server looks through the lens in the  same manner that the fish
did*
  The most perfect single microscopes  ever executed of solid
substances are those made of the gems, such as garnet, ruby,
See Edinburgh Journal of Science, No. III. p. 98. 
CHAP. XLI.      ON SINGLE MICROSCOPES.
281
sapphire, and diamond.  The advantages of such lenses I first
pointed  out  in  my  Treatise on  Philosophical Instruments;
and two lenses, one of ruby and another of garnet, were  exe-
cuted for me by Mr. Peter Hill, optician in Edinburgh. These
lenses performed admirably, in consequence of their producing,
with  surfaces  of inferior  curvature, the same magnifying
power as a glass lens; and the distinctness of the image was
increased by their  absorbing the  extreme blue rays of the
spectrum.  Mr. Pritchard, of London, has carried this branch
of the art to the highest perfection, and has executed lenses
of sapphire and diamond of great power and perfection of
workmanship.
  When the diamond can be procured perfectly homogeneous
and free from double refraction, it may be wrought into a lens
of the highest excellence ; but the sapphire, which has double
refraction, is less fitted for this purpose.  Garnet is decidedly
the best material for single lenses, as it has no double refrac-
tion, and may  be procured, with a little attention,  perfectly
pure  and homogeneous.   I have now in my  possession  two
garnet microscopes, executed by Mr. Adie, which far surpass
every solid lens I have seen.  Their focal  length is  between
the 30th and the 50th of an inch.  Mr. Veitch, of Inchbonny,
has likewise executed some admirable garnet lenses  out of a
Greenland specimen of that mineral given to me by Sir Charles
Giesecke.
  (197.) A  single  microscope, which occurred to me some
years ago, is shown in fig. 157., and consists in a new method
of using a hemispherical lens so as to  obtain from  it twice
Fig. 157.
the magnifying power which it possesses when used in the
common way.   If A B C is a hemispherical lens, rays issuing
from any object, R,  will be refracted at the first surface A C,
and, after total reflexion at  the plane surface B C, will  be
again refracted at the second surface A B, and emerge in par- 
282            A TREATISE ON OPTICS.        PART IV.

allel directions def, exactly in the same  manner as if they
had  not  been reflected at the points  a, b, c, but had passed
through  the other half B A' C of a perfect  sphere A B A' C.
The object at R will therefore be magnified in the  same man-
ner, and will be seen with the same distinctness as if it had
been seen through a sphere of glass A B A' C.   We obtain,
consequently, by  this contrivance, all the  advantages of a
spherical lens, which we believe never has been executed by
grinding.  The periscopic  principle, which will presently
be mentioned, may be communicated to this  catoptric lens, as
it may be  called,  by merely grinding  off the angles B C, or
rough  grinding an annular  space  on the  plane  surface B C.
The confusion arising from the oblique refractions will thus
be prevented, and the pencils from every part of the object
will fall  symmetrically upon the  lens, and be symmetrically
refracted.
  Before I had thought of  this lens, Dr. Wollaston had pro-
posed a method of improving lenses, which  is shown in fig,
     Fig.lSd.     158.  He  introduced  between two plano-
                  convex lenses of equal size  and  radius, a
                  plate of metal with a circular aperture equal
                  to ^th of the focal length, and when the
                  aperture was well centered, he  found that
                  the visible  field was 20°  in diameter.   In
                  this compound lens the oblique pencils pass,
                  like the central ones, at right angles to the
                  surface.   If we compare this lens with the
                  catoptric one above described, we shall see
                  that the effect which is produced in the one
case with two  spherical and  two plane surfaces,  all ground
separately, is produced in the other case by one spherical  and
one plane surface.
  (198.)  The idea of Dr. Wollaston  may,  however, be im-
proved in other ways, by filling up the central aperture  with
Fig. 159.
J 
CHAP. XLI.      COMPOUND MICROSCOPES.
283
a cement of the same refractive power as the lenses, or, what
is far  better, by taking a sphere of glass and grinding away
the equatorial parts, so as  to limit  the central  aperture, as
shown in fig. 159.;  a construction which, when  executed in
garnet, and used in homogeneous light, we conceive to be the
most perfect of all lenses, either, for single microscopes, or for
the object lenses of compound ones.
  When a single  microscope is used for opaque objects, the
lens is placed within a concave silver  speculum, which con-
centrates  parallel or converging rays upon the face of the ob-
ject next the eye.

                 Compound Microscopes.
  (199.) When a microscope consists of two or  more lenses
or specula, one of which forms an enlarged image of objects,
while the  rest magnify that image,  it  is called  a compound
microscope.  The lenses, and the progress of the rays through
them  in such an instrument, are shown in fig. 160., where
A B is the object glass, and C D the eye glass.   An object,
                         Fig. 160.
M N, placed a little farther from A B than its principal focus,
will have an enlarged image of itself formed at m n in an in-
verted position.  If this enlarged image is in the focus of an-
other lens, C D, placed nearer the eye  than  in the figure, it
will be again magnified, as if m n were an object  The mag-
nifying effect of the lens A B is found by dividing the distance
of the image m n from the lens A B by the distance  of the
object from the same lens; and  the magnifying effect  of the
eye glass C D is found by the rule for single microscopes; and
these two  numbers being multiplied  together,  will  be  the
magnifying power of the compound microscope. Thus, if M A
is 4th of an inch,  An, 5 inches, and C n £ an inch, (m n being
supposed in the focus of C D,) the effect of the lens A B will
be 20, and that of C D 10, and the whole power 200. A larger
lens  than any of the other two,  called  the field  glass,  and
shown at E F, is generally placed between A B and the image 
284             A TREATISE ON OPTICS.        PART IV.

m n, for the purpose of enlarging the field of view.   It has
the effect of diminishing the magnifying  power of the instru-
ment by forming a smaller image at v u, which is  magnified
by CD.
  The ingenuity of philosophers and of artists has been nearly
exhausted in devising the best forms of object  glasses and of
eye glasses for  the compound microscope.  ^Mr. Coddington
has recommended four lenses to be employed in the eye piece
of compound  microscopes, as shown in fig. 161.;  and along
with these he uses, as an object glass, the sphere excavated at

                          Fig. 161.
the equator, as in fig. 159., for the purpose of reducing the
aberration and dispersion.  " With a sphere," says he, "prop-
erly cut away at the centre, so as to reduce the aberration and
dispersion to insensible quantities, which  may be done most
completely and most  easily, as I have found in practice, the
whole image is perfectly distinct, whatever extent of it be
taken; and the radius of curvature of it is  no  less than the
focal length, so that the one difficulty is entirely removed, and
the other at least diminished to  one-half.  Besides all this,
another advantage appears in  practice to attend this construc-
tion, which I did not anticipate, and for which I cannot now
at all account.   I have  stated  that when a pencil of rays is
admitted into the eye, which, having passed  without deviation
through a lens,  is bent by the  eye, the vision is never free
from  the  colored  fringes produced by excentrical dispersion.
Now, with the sphere I certainly do  not perceive this defect,
and I therefore  conceive that if it were possible to make  the
spherical glass on a very minute scale, it would  be the most
perfect simple microscope, except,  perhaps, Dr. Wollaston's
doublet. * * *  Now,  the sphere has  this advantage, that it is
more peculiarly fitted for  the object glass of a compound in-
strument, since it gives a perfectly distinct  image of any re-
quired extent, and  that, when combined with  a proper  eye
piece, it  may without difficulty be employed for opaque ob-
 jects."*  The difficulty of making  the spherical glass on a
* Cambridge Transactions, 1830. 
CHAP. XLI.      COMPOUND MICROSCOPES.
285
Fig. 162.
very minute scale, which Mr. Coddington here mentions,  and
which is by no means insurmountable, is, I conceive, entirely
removed by substituting a hemisphere, as shown in fig. 157.,
and contracting the aperture in the manner there mentioned.
  Dr. Wollaston's microscopic doublet shown in fig. 162., con-
                    sists of two plano-convex  lenses m, n,
                    with their  plane sides  turned towards
                    the  object.   Their focal lengths are as
                    one  to three, and their distance from lT4o
                    to 1^ inch, the least  convex being next
                    the  eye.  The tube is about  six inches
                    long, having at its lower end, C D, a cir-
                    cular perforation about -^ of an inch in
                    diameter; through which fight radiating
                    from R is reflected by a plane mirror ab
                    below it.  At the upper end of the tube
                    is a plano-convex lens A B, about f of an
                    inch focus, with  its  plane side next the
                    observer, the object of which is' to form
                    a distinct image of the circular perfora-
                    tion, at e, at the distance of about T8T of
                    an inch from A B. With this instrument,
                    Dr.  Wollaston saw the finest striae  and
                    serratures upon the scales of the lepisma
                    and podura, and upon  the scales of a
                    gnat's wing.
                      (200.) Double and triple achromatic
                    lenses have been recently much used for
                 "B. the  object  glasses of microscopes,  and
                    two or  three of them have  been com-
                    bined;  but  though  they  perform well
                    they are very expensive, and by no means
superior to other  instruments that are properly constructed.*
The power of using homogeneous light, indeed, renders them
in a great measure unnecessary, especially as we can employ
either of Mr. Herschel's  double lenses shown in figs. 43. and
44., which are entirely free from spherical aberration.  One of
these, fig. 44., has been  executed 4_ of an inch focus, with an
aperture of -J^ of an inch; and Mr. Pritchard, to whom it be-
longs, informs us that it brings out all the test objects, and ex-
hibits opaque ones with facility.
   In applying the compound microscope to the examination of
objects of natural history, I have recommended the immersion
of the object in a fluid, for  the purpose of expanding it and
* See Edinburgh Journal of Science, No. VUI. new series, p. 244. 
286            A  TREATISE  ON  OPTICS.        PART IV.

giving its minute parts their proper position  and appearance.
In order to render this method perfect, it is proper to immerse
the anterior surface of the object glass in the  same fluid which
holds the object; and if we use a  fluid of greater dispersive
power than the object glass,  and  accommodate the interior
surface to the difference of their dispersive powers, the object
glass may be made  perfectly achromatic.  The superiority of
such an instrument in viewing animalculse and the molecules
of bodies noticed by Mr. Brown, does not require to be pointed
out.
                On Reflecting Microscopes.
   (201.) The simplest of all reflecting microscopes is a con-
cave mirror, in which the face of the observer is always mag-
nified when its focus is more remote than the observer.  When
the mirror is very concave, a small object m n, fig. 14., will
have a magnified picture of it formed at M N; and when this
picture is viewed by the  eye, we have a single reflecting mi-
croscope, which magnifies as many times as  the distance A n
of the object from the mirror is contained in the distance A M
of the image.
   But if,  instead of  viewing  M N with the naked eye, we
magnify it with a lens, we convert the simple reflecting mi-
croscope into a compound reflecting microscope, composed of
a mirror and a lens.  This microscope was first proposed by
Sir Isaac Newton;  and after being  long in disuse has been re-
vived in an improved form by Professor Amici of Modena. He
made  use of a concave ellipsoidal  reflector,  whose focal dis-
tance was 2r4ff inches. The image is formed in the other focus
of the ellipse, and this image is magnified by  a single or double
eye piece, eight inches from the reflector.   As it  is imprac-
ticable to illuminate  the  object m n when situated as  in fig.
14., professor Amici placed it without  the  tube or  below the
line BN, and introduced it into the speculum A B by reflexion
from  a small  plane speculum  placed  between  m n and A B,
and having its diameter about half that of A  B.
   Dr. Goring, to whom microscopes of all kinds owe so many
improvements, has greatly improved this instrument. He uses
a small plane speculum less  than  4, of the  diameter  of the
concave speculum, and employs the following specula of very
short focal distances:—
        Focal distance in inches
            1-5
            10
            0-6
            0-3
0-6
0-3
0-3
0-3 
CHAP. XLI.     MICROSCOPIC OBSERVATIONS.
287
  That ingenious artist Mr. Cuthbert, who executed these im-
provements, has more  recently,  under Dr. Goring's direction,
finished truly elliptical specula,  whose aperture is equal to
their  focal length.   This he has done with  specula  having
half an inch focus and half an inch aperture, and three tenths
of an inch focus and three tenths of an inch aperture.   Dr.
Goring assures us that this microscope exhibited a set of lon-
gitudinal lines on the scales of the podura in addition to the
two sets of diagonal ones previously discovered,  and two seta
of diagonal lines on the scales of the cabbage butterfly in ad-
dition to the longitudinal ones with the cross stripe,  hitherto
observed.*

                     On Test Objects.
   (202.) Dr. Goring has the merit of having introduced the
use of test objects,  or objects whose  texture or markings re-
quired a certain excellence in the microscope to be well  seen.
A few of these are shown in fig. 163. as given by Mr. Pritch-
ard.  A is the wing of the menelaus, B and C the hair of the

                          Fig. 163
bat,  and D and E the hair of the mouse.  The most difficult
of all the test objects are those in the scales of the podura and
the cabbage butterfly mentioned above.

            Rules for microscopic Observations.
  (203.)  1. The eye should be protected from all extraneous
light, and should not receive any of the light which proceeds
from the illuminating centre, excepting what is transmitted
through or reflected from the object.
  2.  Delicate observations should not be made when the fluid
which lubricates the cornea is in a viscid state.
  3.  The best position for microscopical observations is when
* See Edinburgh Journal of Science, No. IV. new series, p. 321. 
288            A TREATISE ON OPTICS.       PART  IV.

the observer is lying horizontally on his back.   This arises
from the perfect stability of his head,  and from  the equality
of the lubricating film of fluid which covers the cornea.  The
worst of all positions is that in which we look downwards ver-
tically.
  4. If we stand  straight up and look horizontally,  parallel
markings or lines will be seen most perfectly when their di-
rection is vertical; viz. the direction in which the lubricating
fluid descends over the cornea.
  5. Every part of the object should be excluded, except that
which is under immediate observation.
  6. The light which illuminates the object should have a
very small diameter.   In the day-time it should be a single
hole in  the window-shutter of a darkened room, and at night
an aperture placed before an Argand lamp.
  7. In all cases,  particularly when high powers are used,
the natural diameter  of  the illuminating light should be di-
minished, and its intensity increased, by optical contrivances.
  8. In every case of microscopical observations, homogeneous
yellow light, procured from a monochromatic lamp, should be
employed.  Homogeneous red light may be obtained by colored
glasses.*

                     Solar Microscope.
  (204.) The solar microscope is nothing more than a magic
lantern, the light of the  sun  being  used  instead  of that of a
lamp.  The tube A B, fig. 154., is inserted in a hole in the
window-shutter, and the sun's light reflected into it by a long
plane piece of looking-glass, which the observer can turn
round to keep the light in the tube  as the sun moves through
the heavens.
  Living objects, or objects of natural history, are put upon a
glass slider, or stuck  on  the point of a needle, and introduced
into the opening C D, so as to be illuminated by the sun's rays
concentrated by the lens A.  An enlarged and brilliant image
of the object will then be formed on the screen E F.
  Those who wish to see the various external forms of micro-
scopes of all kinds, and  the different modes of putting them
up, are referred to the article Microscope, in the Edinburgh
Encyclopaedia, vol. xiv. p. 215—233.  In the latest work on
the microscope, viz. Dr. Goring and  Mr.  Pritchard's " Micro-
scopical Illustrations," London,  1830,  the  reader  will  find
much valuable and interesting information.
* See the article Microscope, Edinburgh Encyclopedia, vol. xiv. p. 22*. 
CHAP. XLII.    ASTRONOMICAL TELESCOPE.           289


                      CHAP. XLII.

       ON  REFRACTING AND REFLECTING TELESCOPES.

                 Astronomical Telescope.

  (205.) That the  telescope was invented in the thirteenth
century, and perfectly known to Roger Bacon, and that it was
used in England by Leonard  and Thomas Digges before the
time of Jansen or Galileo, can scarcely admit of a doubt  The
principle of the refracting telescope, and the method of com-
puting its magnifying power, have been already explained.
We  shall therefore proceed  to describe the different forms
which it successively assumed.
  The astronomical 'telescope is represented in fig. 164.  It
consists of two convex lenses A B, C D, the former of which

                         Fig. 164.
 is called the object glass, from being next the object M N, and
 the latter the eye glass, from its being next the eye E.  The
 object glass is a lens with a long focal distance; and  the  eye
 glass is one of a short focal distance.  An inverted image m n
 of any distant object M N is formed in the focus of the object
 glass A B ;  and this image is magnified by the eye glass C D,
 in  whose anterior focus it  is placed.   By tracing the  rays
 through the two lenses, it will be seen that they enter the eye
 E parallel.  If the object M N is near the observer, the image
 mn will be  found  at  a  greater distance from AB;  and  the
 eye glass C D must be drawn out from A B to obtain distinct
 vision of the image m n.  Hence it is usual to fix the object
 glass A B at the  end of a tube longer than  its focal distance,
 and to place  the eye glass C D in a small tube, called the  eye
 tube, which will slide out of, and into, the larger tube, for  the
 purpose of adjusting it to objects at different distances.  The
 magnifying power of this telescope is equal to the focal length
 of the object glass divided by the focal length of the eye glass.
  Telescopes of this  construction  were  made  by  Campani
 Divini and Huygens, of the enormous length of 120 and 136
feet; and  it was with instruments 12 and 24 feet  long  that
Huygens discovered the ring and the fourth satellite of Saturn. 
290             A  TREATISE  ON  OPTICS.       PART IV

In order to use object glasses of such great focal lengths with-
out the encumbrance of tubes, Huygens  placed  the  object
glass in a short tube at the top of a very  long pole, so that
the tube could be turned in every possible direction upon a
ball and socket by  means of a string, and brought into the
same line with another short  tube containing the eye glass,
which  he held in his hand.
   As these telescopes were liable  to  all  the imperfections
arising from the aberration of refrangibility and that of spher-
ical figure, they could not show objects distinctly when the
aperture of the object glass was great; and on this account
their magnifying power was limited.  Huygens found that the
following were the proper proportions:—
Focal length of toe	Aprrtu	re of the	Focal length of the	Magnifying power.
object glass.	obji-c	glass.	eye gloss.	
1ft,	0-545	inches.	0-605	20
3	0-94		104	33£
5	1-21		1-33	44
10	1-71		1-88	02
50	3-84		4-20	140
100	5-40		5-95	197
120	5-90		6-52	216
  In the astronomical telescope, the object, M N, is always seen
inverted.
                   Terrestrial  Telescope.
  (206.) In order to accommodate this telescope to  land ob-
jects which require to be seen  erect, the instrument is con-
structed as in fig. 165., which is  the same  as the preceding
one, with the addition of two lenses E F, G H, which  have the

                          Fig. 165.
same focal length as C D, and are placed at distances equal to
double their common focal length.  If the focal lengths are not
equal,  the distance of any two of them  must be equal  to the
sum of their focal lengths.  In this telescope the progress of
the rays is exactly the same as in the astronomical one, as far
as L, where the two pencils of parallel rays C L, D L cross in
the anterior focus L of tfie second eye glass E F.  These rays
falling on E F form in its principal focus an erect image, m' n', 
CHAP. XLII.     GREGORIAN TELESCOPE.             291

which is seen erect by the third eye glass G H, as the rays
diverging from m' and n' in the  focus of G H enter the eye
in parallel pencils at E'. The  magnifying power of this tele-
scope is the same as that of the former when  the eye glasses
are equal.

                    Galilean Telescope.

  (207.) This telescope,  which  is the one  used by Galileo,
differs in nothing from the astronomical telescope, excepting
in a concave eye glass C D, fig.  166. being substituted for the
convex  one.  The  concave lens C D is  placed between the

                         Fig. 166.
image m n and  the object glass, so that  the image is in the
principal focus of the concave lens.  The pencils of rays
ABn, AB»! fall uponC D, converging to its principal focus,
and will therefore  be refracted into parallel lines, which will
enter the eye at E, and give distinct vision of the object. The
magnifying power  of this telescope is  found  by the same rule
as that for the astronomical telescope:  it gives a smaller and
less agreeable field of view than the  astronomical telescope,
but it has the advantage of showing the object erect, and of
giving more distinct vision of it.

             Gregorian Reflecting Telescope.
  (208.) Father Zucchius seems to have been  the first person
who magnified  objects by means of a lens and a concave spec-
ulum ; but there is no evidence that he constructed a reflecting
telescope with a small speculum.
  James Gregory was the first who described the construction
of this  instrument, but he does not seem to  have executed
one;  and  the honor of doing this with his own  hands was  re-
served for Sir Isaac Newton.
  The Gregorian  telescope is shown in fig. 167., where A B
is a concave metallic speculum  with a hole in its centre.  For
very  remote  objects the curve of the  speculum should be a
parabola.  For nearer ones it should  be an ellipse in whose
farther focus is the  object, and in whose nearer focus is the
image; and in  both these  cases the speculum would  be free
from  spherical  aberration.  But, as these curves  cannot be 
292
A TREATISE  ON OPTICS.       PART IV.
communicated with certainty to specula, opticians are satisfied
with giving to them a correct spherical figure.  In front of the

                         Fig. 167.
large speculum is placed a small  concave one, C D, which
can be moved nearer to and farther from the large speculum
by means of the screw W at the side of the tube.  This spec-
ulum should have its curvature  elliptical, though  it is gene-
rally made spherical.   An eye-piece consisting of two convex
lenses, E, F, placed at a distance equal to half the sum of their
focal lengths, is screwed into the tube immediately behind the
great speculum A B, and permanently fixed  in that position.
If rays M A, N B, issuing nearly parallel from the extremities
M and N of a distant object, fall upon the speculum A B, they
will  form an inverted  image of it  at m n, as more distinctly
shown in fig. 14.
  If this image m n is farther from the small  speculum C D
than its principal focus, an inverted image of  it, m' n', or an
erect image of the real object, since m n is itself an inverted
one,  will be formed somewhere between E  and F, the  rays
passing through the opening in the speculum.  This image
ml n' might have been viewed and magnified  by a convex eye
glass at F, but it is  preferable to receive the  converging rays
upon a lens E called the field glass, which hastens their con-
vergence, and forms the image of m n in the  focus of the lens
F, by which they are magnified; or, what is the same thing,
the pencils diverging from the image m! n' are  refracted by F,
so as to enter the eye parallel, and  give distinct vision of the
image.   If the object M N is brought nearer the speculum
A B, the image of it, m n, will recede from A B and approach
to C  D;  and, consequently, the  other image m' n' in the  con-
jugate focus of C D will recede from its place m'n', and cease
to be seen distinctly.  In order to restore it to its place m' n',
we have only to turn  the screw W, so as to remove C D
farther from A B, and  consequently farther from m n, which
will  cause the image m' n'  to appear perfectly  distinct as be- 
CHAP. XLII.    CASSEGRAINIAN TELESCOPE.          293
fore.  The magnifying power of this telescope may be found
by the following rule:—
  Multiply the focal distance of the great speculum by the
distance of the small mirror from the image next the eye, as
formed  in the anterior focus of the convex eye glass, and mul-
tiply also the focal distance of the small speculum by the focal
distance of the eye glass.   The quotient arising from dividing
the former product by the latter will be the magnifying power.
   This  rule supposes the eye-piece to consist of a single lens.
   The  following table, showing the focal lengths, apertures,
powers, and prices of some of Short's telescopes, will exhibit
the great  superiority  of  reflecting telescopes  to refracting
ones:—			
Focal lengths in feet	Aperture In inches.	Magnifying powers.	Price in guineas
1	30	35 to 100	14
2	4-5	90 300	35
3	6-3	100 400	75
4	7-6	120 500	100
7	12-2	200 800	300
12	180	300 1200	800
                 Cassegrainian Telescope.
  (209.) The  Cassegrainian  telescope, proposed by M. Cas-
segrain, a Frenchman, differs from the Gregorian only in hav-
ing its small speculum C D,fig. 168., convex instead of con-
cave. The speculum is therefore placed before the image m n

                          Fig. 168.
of the object M N, and an image of M N will be formed at
m' n' between E and F as in the Gregorian  instrument.  The
advantage of this form is, that the  telescope is shorter than
the Gregorian  by more  than twice the focal  length of the
small speculum; and it  is generally admitted that it gives
more light, and a distincter image, in consequence of the con-
vex speculum correcting the aberration of the concave one.
                           Z2 
294            A TREATISE ON OPTICS.        PART IV.

                  Newtonian Telescope.

  (210.) The  Newtonian  telescope, which may be regarded
as an improvement upon the Gregorian one, is represented in
fig. 169., where A B is a concave speculum, and m n the in-
verted image which it forms of the object from which the rays

                         Fig. 169.
                                           ■E
M, N proceed.  As it is impossible to introduce the eye into
the tube  to view this image  without  obstructing the light
which comes from the object, a small plane speculum C D, in-
clined 45° to the  axis of the large speculum, and of an oval
form, its axes being to one another as 7 to 5, is placed between
the speculum and the image m n, in order to reflect it to a side
at m' n', so that we can magnify it with  an eye glass E, which
causes the rays to enter the eye parallel.   The small  mirror
is fixed upon a slender arm, connected with a slide, by which
the mirror may be made to approach to or recede from  the
large speculum A B, according as the image m n approaches
to or  recedes from it.  This adjustment might also be effected
by moving the eye lens E to or from the small speculum. The
magnifying power of this telescope is equal to the focal  length
of the great speculum divided by that of the eye glass.
   As about half of the light is lost in metallic reflexions, Sir
Isaac Newton  proposed to substitute, in place of the metallic
speculum, a rectangular prism ABC, fig.  148., in which the
light suffers total  reflexion.  For this purpose, however, the
glass  requires to be perfectly colorless  and free from veins,
and hence such a prism has  rarely been used.  Sir Isaac
also proposed to make the two faces of  the prism convex, as
DEF, fig. 148., and  by  placing  it between  the image m n
and the object, he not only erected the image, but was enabled
to vary the magnifying power of the telescope.   The original
telescope,  constructed by Sir Isaac's own hands,  is  preserved
in the library of the Royal Society.
   The following  table shows the dimensions of Newtonian 
CHAP. XLII.     NEWTONIAN TELESCOPE.
295
telescopes, which we have computed by taking a fine telescope
made by Hawksbee as a standard:—
al length of great speculum.	Aperture of speculum.	Focal length of eye glass.	Magnifying power.
1ft.	2-23 inches.	0-129 inches.	93
2	3-79	0152	158
3	514	0-168	214
4	6-36	0-181	265
6	8-64	0-200	360
12	14-50	0-238	604
24	24-41	0-283	1017
  (211.)  On account of the great loss of light in metallic re-
flexions, which, according to the accurate experiments of Mr.
R. Potter, amounts to 45 rays in every 100, at an incidence of
45°,* and the imperfections of reflexion, which even with per-
fect surfaces make the rays  stray five or six times more than
the same imperfections in refracting surfaces, I have proposed
to construct the Newtonian  telescope, as shown in fig. 170.,
where A B is the concave speculum, m n the  image of the

                         Fig. 170.
object M N, and C D an achromatic prism, which refracts the
image m n into an oblique  position, so that it can  be viewed
by the eye at E through a magnifying lens.  Nothing more is
required by the prism than  to turn the  rays as much aside as
will enable the observer to see the image without obstructing
the rays from  the object  M N.   As the prisms of crown  and
flint  glass which compose the achromatic prism may be ce-
mented  by a substance of intermediate  refractive  power, no
more light will be lost than what is reflected at the two sur-
faces.
  In place of setting the small speculum, C D, of the  New-
tonian telescope, fig. 169., at 45°, to the incident rays, I have
proposed to place it much more obliquely, so as  to reflect  the
image m n, fig. 170., out of the way of the observer, and no
farther.   This would of course require a plane speculum, C D,

      * Edinburgh Journal of Science, No. VI., new series, p. 283. 
296            A  TREATISE  ON  OPTICS.       PART  IV.

of much greater length; but the greater obliquity of the re-
flexion  would more than compensate  for this inconvenience.
It might be advisable, indeed, to use a small speculum of dark
glass, of a high refractive power, which at great incidences
reflects as much light as metals,  and which is capable of
being brought to a much finer surface.  The fine surfaces of
some crystals, such as  ruby silver, oxide of tin, or diamond,
might be used.
   A Newtonian reflector, without an eye glass, may be made
by using a reflecting glass prism, with one or both of its sur-
faces concave, when the prism is placed between the image
m n and the great speculum, so as  to reflect the  rays parallel
to the eye.  The magnifying power will be equal to the focal
length  of the great speculum,  divided by  the  radius of  the
concave surface of  the prism if both the surfaces are concave,
and of  equal  concavity, or by twice  the radius, if only  one
surface is concave.

             Sir William HerscheVs Telescope.

   (212.) The fine Gregorian telescopes executed  by Short
were so superior to any other reflectors, that the Newtonian
form of the instrument fell  into disuse.   It was revived, how-
ever, by Sir W. Herschel, whose labors form the most brilliant
epoch in optical science.  With an ardor never before exhibit-
ed, he  constructed no  fewer than 200 seven feet Newtonian
reflectors, 150 ten feet,  and 80 twenty feet  in  focal  length.
But his zeal did not stop here.  Under the munificent  patron-
age of George III., he began, in 1785, to construct a telescope
forty feet long, and on the 27th of August, 17H9, the day on
which it was completed, he discovered with it the sixth satel-
lite of  Saturn.
   The  great  speculum had a diameter of 49£ inches, but its
concave surface was only 48 inches.   Its thickness  was about
3£ inches, and its  weight when cast was 2118  lbs.  Its focal
length  was forty feet, and  the  length of the sheet  iron tube
which contained it was 39 feet 6 inches, and its breadth 4 feet
10 inches.  By using small convex lenses, Dr. Herschel  was
enabled to apply a power of 6450 to the fixed stars, but a very
much lower power was in general  used.
   In this telescope the observer sat at the mouth of the tube,
and observed by what is called  the front view,  with his back
to the  object, without using a plane speculum,  the eye  lens
being applied directly  to magnify the image formed by the
great speculum.   In order to prevent  the head,  Sic from ob- 
CHAP. XLIII.  ON ACHROMATIC TELESCOPES.         297
structing too much of the incident light, the image was formed
out of the axis of the speculum, and  must,  therefore, have
been slightly distorted.
  As the frame of this instrument was exposed to the weather,
it had greatly decayed.  It was, therefore, taken down, and
another telescope, of 20 feet focus, with a speculum 18 inches
in diameter,  was erected  in its place,, in 1822, by J. F. W.
Herschel, Esq., with which many important observations have
been made.

                Mr. Ramage's Telescope.

  (213.) Mr. Ramage, of Aberdeen, has constructed various
Newtonian telescopes, of great lengths and high powers. The
largest  instrument at  present in use in this country, and we
believe  in Europe, was constructed by him, and erected at the
Royal Observatory of Greenwich in 1820.   The great specu-
lum has a focal length of 25 feet, and a diameter of 15 inches.
The image is formed out of the axis of the  speculum, which
is inclined so as to throw it just to the side of the tube, where
the observer can view it without obstructing the incident rays.
The tube is a 12-sided prism of deal, and when the instrument
is not in use it is lowered into a box, and covered with canvas.
The apparatus for moving and  directing the telescope  is ex-
tremely simple, and displays much ingenuity.
                     CHAP. XLIII.

               ON ACHROMATIC  TELESCOPES.

  (214.) The principle of the achromatic telescope has been
briefly explained in Chap. VII., and we have there shown how
a convex lens, combined with a concave lens of a longer focus,
and having a higher refractive and dispersive power, may pro-
duce refraction without color, and consequently form an image
free from the primary prismatic colors. It has been demonstrated
mathematically, and the reader  may convince himself of its
truth by actually tracing the rays through the lenses, that a
convex and a concave lens will form an achromatic combina-
tion, or will give a colorless  image, when their  focal lengths
are in the same proportion as their dispersive powers.  That
is, if the dispersive powers of crown and flint  glass are as 0-60
to 1, or 6 to 10; then an achromatic object glass could be
formed by combining a convex crown glass lens of 6, or 60, or 
298
A  TREATISE  ON  OPTICS.        PART IV.
.AC
600  inches  with a concave flint glass lens of 10, or 100, or
1000 inches in focal length.
  But  though such a combination would form an image free
from color, it would not be free from spherical aberration, which
can only be removed by giving a proper proportion to the cur-
vatures of the first and  last surface, or the two outer surfaces
of the  compound lens.  Mr. Herschel has found that a double
object  glass will be nearly free from  aberration, provided  the
radius  of the exterior surface of the crown  lens be 6-72, and
of the  flint 14-20, the  focal length of the combination  being
10-00,  and the radii of the  interior surfaces being computed
from these data by the formula? given in elementary works on
optics, so as to make the focal lengths of the two glasses in
  Fig. 171.  the direct ratio of their dispersive powers.  This
           combination is shown  in fig. 171., where A B is
           the convex lens of crown  glass, placed on the out-
           side  towards the object, and C D the concavo-con-
           vex  lens of flint glass placed  towards  the eye.
           The two inside surfaces that come in contact  are
           so nearly of the same curvature that they may be
           ground on the same tool, and united together by a
           cement to prevent the loss of light at the two sur-
           faces.
              In the double  achromatic object glasses con-
           structed previous to  the publication of Mr. Her-
           schel's investigations, the surface of the concave
            lens  next the eye was, we believe, always con-
            cave.
   Triple achromatic object glasses consist of three lenses A B,
   Fig. 172.   C D, E F, fig. 172., A B and E F being convex
             lenses of crown glass, and C D a double concave
             lens of flint glass.
               The object  of using three lenses  was  to ob-
             tain a better correction  of the spherical aberra-
             tion; but  the greater complexity of their con-
             struction,  the greater risk of imperfect centering,
             or  of the axes of the three lenses not being m
             the same straight line, together with the loss of
             light at six surfaces, have been considered as
             more than compensating their advantages;  and
             they have accordingly fallen into disuse.
               The following were  the radii of two  triple
             achromatic  object  glasses,  as  constructed by
  B*>  E   Dollond:-
 
CHAP. XLIII.  ON ACHROMATIC TELESCOPES.         299
                  A B, or first Crown Lens.
        FIRST OBJECT GLASS.               SECOND OBJECT GLASS.
Radii of first surface,  -  -  -  28 inches......28
------  second surface, --40........35-5

                    C D, or Flint Lens.
Radii of first surface,  -  -   -  20-9........21-1
------  second surface, --28......    -  25-75

                 E F, or second Crown Lens.
Radii of first surface,  -  -   -  28-4-  -  -.....28
------  second surface, -   -  28-4........28
Focal length of the compound
  lens,.......46 inches......46-3

  In consequence of the great difficulty of obtaining  flint
glass free from veins and imperfections, the largest achromatic
object glasses constructed in England did not greatly exceed
4 or 5 inches in diameter.   The neglect  into which this im-
portant  branch of our national manufactures was allowed to
fall by the ignorance and  supineness of the British government,
stimulated foreigners to rival us in the manufacture of achro-
matic telescopes.   M. Guinand  of Brenetz, in Switzerland,
and M. Fraunhofer, of  Munich, successively devoted  their
minds to the subject of making large lenses of flint glass, and
both of them succeeded.   Before  his death,  M. Fraunhofer
executed two telescopes  with achromatic object glasses of 9^
inches,  and 12 inches in diameter;  and he informed me that
he  would undertake to  execute one 18 inches in diameter.
The first of these object  glasses was for the magnificent achro-
matic telescope ordered  by  the emperor of Russia, for the ob-
servatory at Dorpat  The object glass was a double one, and
its  focal  length was 25  feet; it was mounted  on a  metallic
stand which weighed 5000 Russian pounds.  The telescope
could be moved by the slightest force in any direction, all  the
movable parts being balanced by counter weights. It had four
eye glasses, the lowest of which magnified 175, and the  high-
est 700 times.  Its price  was 1300Z., but it was liberally given
at prime cost, or 950Z. The object glass, 12 inches in diameter,
was made for the king of Bavaria, at the  price of 2720Z.;  but
as it was not perfectly complete at the time of Fraunhofer s
death, we do not  know that it is at present in use.  In the
hands of that able observer, Professor Struve, the telescope of
Dorpat has  already made  many important discoveries in as-
tronomy.
  A French optician, we  believe, M. Lerebours, has  more 
300            A TREATISE  ON  OPTICS.       PART  IV.
recently executed two achromatic object glasses of glass made
by Guinand.   One  of them is nearly 12 inches in diameter,
and another above 13 inches. The first of these object glasses
was mounted as a telescope at Uie Royal Observatory of Paris;
and the French government had expended 500/. in the pur-
chase of a stand for it, but had not  the liberality to purchase
the object glass, itself. Sir James South, our liberal  and active
countryman, saw the  value of the two object glasses, and ac-
quired them for his  observatory at Kensington.


                ON ACHROMATIC EYEPIECES.
  (215.) Achromatic  eyepieces when one lens only  is wanted,
may be composed of two or three lenses exactly on the same
principles as object glasses.  Such eyepieces, however, are
never  used, because  the  color can  be corrected in a superior
manner, by a proper arrangement of single lenses of the same
kind of glass.   This arrangement is shown in fig. 173., where
A B and C D are two plano-convex lenses, A B being the one
              B
 next the object glass, and C D the one next the eye, a ray of
 white light R A, proceeding from the achromatic object glass,
 will be refracted by A B at A, so that the red ray A r crosses
 the axis at r, and the violet ray A v at v. But these rays being
 intercepted by the second lens C D at the points m, n, at dif-
 ferent distances from the axis, will suffer different degrees of
 refraction.   The red  ray m r suffering a greater  refraction
 than  the violet one n v, notwithstanding its inferior refran-
 gibility, so that the two rays will emerge parallel from the
 lens C D (and therefore be colorless) as shown at m r', m v'.
  When these two lenses are made of crown glass, they  must
 be placed  at a distance equal to half the sum  of their  focal
 lengths, or, what is more accurate, their distance must be
 equal to half the  sum  of the focal distance,  of the  eye  glass
 C D, and the distance at which the field glass A B would form
an image of the object glass of the telescope.  This eyepiece
is called the negative eyepiece.   The stop or diaphragm  must
be placed half-way between the two lenses.  The focal length
of an equivalent  lens, or one that has the same magnifying 
CHAP. XLIII.  ON ACHROMATIC TELESCOPES.
301
Fig. 174.
power  as the  eyepiece, is equal to twice the product of the
focal lengths of the two lenses divided by the sum of the same
numbers.
  An eyepiece nearly achromatic, called Ramsderts Eyepiece,
and much used in transit instruments and telescopes with mi-
crometers,  is  shown in fig. 174., where A B,  CD, are two
                             plano-convex lenses with their
                             convex  sides  inwards.   They
                             have the same focal length, and
                             are  placed at a  distance from
                             each other, equal to two-thirds
                             of the  focal  length of  either.
                             The focal  length of an equiva-
                             lent lens is equal to three-fourths
                             the focal length of either lens.
                             The use of this  eyepiece is  to
give a  flat field, or a distinct view of a system of wires placed
at M N.  This eyepiece is not quite achromatic, and it might
be rendered more so by increasing the distance  of the lenses;
but as this  would require the wires  at M N to be brought
nearer A B, any particles of dust or imperfections in the lens
A B would be  seen magnified by the lens C D.
  The erecting achromatic eyepiece  now in universal use  in
all achromatic telescopes for land objects is shown in fig. 175.
It consists of four lenses, A, C, D, B, placed as in the  figure.
                          Fig. 175.
Mr. Coddington has shown, that if the focal lengths, reckoning
from A, are as the numbers 3, 4, 4 and  3, and the distances
between them on the same scale  4, 6,  and 5-2,  the radii,
reckoning from the outer surface of A, should be thus:—
           First surface    27
           Second surface
First surface
Second surface
First surface
 jcond surface
First surface
Second surface
11)
E > nearly plano-convex-
a meniscus.
ano-convex.
21 i nearly pi:

ni > Double convex.

  2A 
302             A TREATISE ON OPTICS.        PART IV.

   The magnifying power of this eyepiece, as usually  made,
differs little from what would  be produced by using the first
or  fourth lens  alone.  I have shown, that the  magnifying
power of this eyepiece  may be  increased or diminished  by
varying the distance between C and D, which even in common
eyepieces of this kind may be done, as A and C are  placed in
one tube A C, and D and B in another tube  D B, so that the
latter can be drawn out  of the general  tube.  In fig. 17">., I
have shown the eyepiece constructed in this way, and capable
of having its two parts separated by a screw nut E,  and rack.
This contrivance for obtaining a variable  magnifying  power,
and consequently of separating optically a pair of wires fixed
before the eye glass, I communicated to  Mr. Carey in ls(>;5,
and had one  of the instruments constructed by Mr. Adie  in
1806.  It is fully described  in  my Treatise on Philosophical
Instruments, and has been more recently brought out as a new
invention by Dr. Kitchener, under the name of the Pancratic
Eye Tube.

                     Prism Telescope.

   (216.)  In 1812, I showed  that colorless refraction may be
produced by combining two prisms of the same substance, and
the experiments which led to this result were published in my
Treatise on New Philosophical  Instruments in lHl.'i.   The
practical purposes to  which this  singular principle seemed to
be applicable were the construction of an achromatic  telescope
with lenses of the same glass, and the construction of a
Teinoscope, for extending or altering the lineal proportions of
objects.
   If we take a prism, and hold its refracting edge downwards
and horizontal, so as to see through it one of the panes of glass
in a window,  there will  be  found a position, namely, that in
which the rays enter the prism and emerge from it  at  equal
angles, as in fig. 20., where the square pane of glass is of its
natural  size.   If we  turn the refracting edge  towards the
window, the pane will be extended or magnified in its length
or vertical direction, while its breadth remains the same.  If
we now take the same prism and hold its refracting edge ver-
tically, we  shall find,  by  the  same process, that  the pane of
glass is extended or magnified in  breadth.  If two such prisms,
therefore, are combined in these positions, so as to magnify the
same both in length and breadth, we  have a telescope com-
posed of two  prisms,  but unfortunately the objects are all
highly fringed  with the  prismatic colors.   We may correct 
CHAP. XLIII.   ON ACHROMATIC TELESCOPES.         303

these colors in three ways: 1st, We may make the prisms of
a kind of glass which obstructs all the rays but  those of one
homogeneous color;  or, we may use a piece of the same glass
to absorb the other rays when two common glass prisms are
used: 2d, We may use achromatic prisms in place of common
prisms: or, 3d, What is best  of all  for common  purposes,  we
may place  other two prisms  exactly similar,  but in reverse
positions, or they may be placed as shown in fig. 176, which
represents the prism telescope; A B and A C being two prisms

                         Fig. 176.
 of the same kind of glass, and of the same refracting  angles
 with their planes of refraction vertical, and E D, E F other
 two perfectly similar prisms,  similarly placed,  but with their
 planes of refraction horizontal.  A ray of light, M a, from  an
 object, M, enters the first prism, E F, at a, emerges from the
 second  prism, E D, at 6, enters the third prism, A C at c
 emerges from the fourth prism, A B, at d,  and enters the eye
 at O.  The object, M, is extended or magnified horizontally
 by each  of the two prisms, E F, E D, and vertically by each
 of the two prisms, A B, A C; objects are  magnified by look-
 ing through the prisms.
   This instrument was made in Scotland by the writer of this
 Treatise, under the name of a  Teinoscope,  and also  by Dr
 Blair, before it was proposed or executed  by Professor Amici
 of Modena.   Dr. Blair's model is now before me, being com-
 posed of four prisms of plate  glass with refracting angles of
 about 15°.   It was presented to me two years ago by his son •
 but as no account of it was ever published, Mr. Blair could not
 determine the date of its construction.
  In constructing this  instrument, the perfect equality  of the
four prisms is not necessary.  It will be sufficient if A  B  and
D E  are equal,  and A C and E F, as the color of the  one
prism can be made to correct that of the other by a change in
its position.  For the same reason it is not necessary that they
be all made of the same kind of glass.
999 
304             A  TREATISE ON OPTICS.       PART IV.
      Achromatic Opera Glasses with Single Lenses.
  (217.) M. d'Alembert has long ago shown that an achromatic
telescope may be constructed with a single object glass and a
single eye glass of different refractive and dispersive powers.
To effect this, tiie eye glass must be concave, and be made of
glass of a much higher dispersive power than that of which
the object glass is made; but the proposal was quite Utopian
at the time it was suggested, as substances with a sufficient
difference of dispersive power were not then known.  Even
now, the principle can  be applied only to opera glasses.
   If we use an object glass of very low dispersive  power, the
refraction  of the violet rays may be corrected  by a concave
eye lens of a high dispersive power, as will be seen  by the
following table.
          Object glass                Eye gloss           Magnifying
           made of                 made of             power.
        Crown glass           Flint glass           1J
        Water                 Oil of cassia         2
        Rock crystal           Flint glass           2
        Rock crystal           Oil of anisesccd      3
        Crown glass           Oil of cassia         3
        Rock crystal           Oil of cassia         6
   Although all the rays are made to enter the eye parallel in
these combinations,  yet the correction  of  color is not satis-
factory.

            Mr. Barlow's Achromatic Telescope.
   (218.) In the year  1813 I discovered the remarkable dis-
persive power  of sulphuret of carbon, having  found  that  it
" exceeds all  fluid bodies in refractive power, surpassing even
flint glass, topaz, and tourmaline; and that in dispersive power
it exceeds every fluid  substance except oil of cassia, holding
an intermediate place between phosphorus and balsam of tola
  * * *  Although oil of cassia surpasses the sulphuret  of car-
bon in  its power of dispersion, yet, from the yellow color with
which it is tinged, it is greatly inferior to the latter as  an op-
tical fluid, unless in cases where a very thin concave  lens  is
required.  The extreme volatility of the sulphuret is undoubt-
edly a disadvantage; but  as this volatility may  be restrained,
we have no hesitation in considering the sulphuret of  carbon
as a fluid  of great value in optical researches,  and which
may be of incalculable service in the construction of optiml
instruments."*   This  anticipation has  been  realized by Mr.

  * On the Optical Properties of Sulphuret of Carbon, in Edinburgh Tram.
vol. viii. p. 285.  Feb. 7. 1814. 
CHAP. XLIII.  ON ACHROMATIC TELESCOPES.          805

Barlow, who has employed sulphuret of carbon as a substitute
for flint glass, in correcting the dispersion of the convex lens.
It had been proposed, and the experiment even tried, to place
the concave lens between the convex one and its focus, for the
purpose of correcting the dispersion of the convex  lens, with
a lens of less diameter, but Mr. Barlow has the  merit of hav-
ing first carried this into  effect
   The telescope which he has made on this principle, consists
of a single object lens of plate glass, 7-8 inches in clear aper-
ture, with a focal length  of 78 inches.   At the distance of 40
inches from this lens was placed a concave lens of sulphuret
of carbon, with a focal length of 59-8 inches, so that parallel
rays falling on  the convex plate  lens, and converging to its
focus, would, when refracted by the fluid concave lens, have
their focus at the distance of 104 inches from the fluid lens,
and  144 inches, or 12 feet, from the plate glass lens.  The
fluid is contained between two meniscus  cheeks, and a glass
ring, so that the radius of the concave fluid lens is 144 inches
towards the eye, and 56-4 towards the object  lens.  The fluid
is put in at a high temperature, and the contraction which it
experiences in cooling is said to keep every thing perfectly
tight.  No decomposition of the fluid has yet been observed.
The great secondary spectrum which I found to exist  in  sul-
phuret of carbon is approximately corrected by the distance of
the fluid lens from the object glass; but we are persuaded that
it is  not free from  secondary color.  Mr. Coddington remarks,
that  the general course of an oblique pencil is  bent outward
by the fluid lens, and the violet rays more than the red, so as
to produce indistinctness; but we are not  aware  that  this
defect was observed in the instrument.  The tube of the tele-
scope is 11 feet, and the  eyepieces one foot. "The telescope,"
says Mr. Barlow, " bears a power of 700 on the closest double
stars in South's and Herschel's catalogue, although the field
is not  then so  bright as  I could desire.  Venus  is beautifully
white and well denned with a power of 120, but shows some
color with 360.   Saturn, with  the  120 power,  is a very bril-
liant object, the double ring and belts being well and satisfac-
torily defined, and with  the 360 power it is  still very fine."
Mr.  Barlow remarks,  also, that the telescope is not so com-
petent to the opening of the close  stars, as it is powerful in
bringing to light the more minute luminous points.

      Achromatic Solar  Telescopes with single  Lenses.
  (219.) An  achromatic  telescope for viewing the sun or any
highly luminous object may be constructed by using a single
                          2A2 
 306             A TREATISE ON OPTICS.        PAHT IV.

 object glass of plate glass;  and by making any one of the eye
 glasses out of a piece of glass which transmits only homoge-
 neous light: or the same thing may be effected by a piece of
 plane glass of the same color; but this introduces the errors
 of other two surfaces.  In such  a  construction  it would be
 preferable to absorb all the rays but the red;  and there are
 various substances by which this may be readily effected.  The
 object glass of this telescope, though thus rendered monochro-
 matic, will still be liable to spherical aberration.  But if the
 radii of the lens are properly adjusted, the excess of solar light
 will  permit  us to diminish the aperture, so as to render the
 spherical aberration almost imperceptible.   Such a telescope,
 when made  of a great length, would, we  are  persuaded, be
 equal to any instrument that has yet been directed to the sun.
 If we could  obtain a solid or a fluid which would absorb all
 the other rays  of the spectrum but the yellow, with as little
 loss as there is in red glasses, a telescope of the preceding
 construction  would answer for day objects, and for all the  pur-
 poses of astronomy.  If the art of giving lenses a hyperbolic
 form  shall be brought to  perfection, which we have  no doubt
 will  yet be done, the  spherical aberration would disappear;
 and a telescope upon this principle would be the most perfect
 of all instruments.
   Even by using red light only, a great improvement might
 be effected in the common  telescopes  for day objects and for
 astronomical purposes.  If the red rays,  for example, form  ^th
 of white light, we have only to increase the area of the aper-
 ture  10 times to make  up completely for this defect of light
 The  spherical aberration is, no doubt greatly increased also:
 but if we  consider that, when compared to the aberration of
 color, it is only as 1 to 1200, we can afford to  increase it in
 order to gain  so  great an advantage.  Common  telescopes,
 indeed, may be considerably  improved  by applying colored
 glasses, which  absorb only the extreme rays of the spectrum,
 even  though they do not produce  an achromatic  or  homoge-
 neous image.
   These observations are made for the benefit of those  who
 cannot afford expensive instruments, but who may yet wish to
 devote themselves to astronomical  observations, with the ordi-
 nary instruments which they may  happen to possess.

 On the Improvement of imperfectly achromatic Telescopes.

   (220.) There are  many achromatic  telescopes of consider-
able size, in which the flint  lens either  over corrects or under
corrects the colors of the crown glass lens.  This defect may
J 
CHAP. XLIII.   ON ACHROMATIC TELESCOPES.          307

be easily removed by altering slightly the curvature of one or
other of the lenses.  But all achromatic telescopes whatever,
when made of crown and flint glass, exhibit the secondary
colors, viz. the wine-colored and the green fringes.   These
colors are not very strong; and in many,  if not in  all cases,
we may destroy them by absorption through glasses that will
not weaken greatly the intensity of the light.  The glasses
requisite for this purpose must be found by actual experiment;
as the secondary tints, though generally of the colors we have
mentioned, are variously composed, according to the nature of
the glass of which the two lenses are made. 
' SI 
                                                          309
            APPENDIX OF THE AUTHOR,

                         CONTAINING

TABLES OF REFRACTIVE AND DISPERSIVE POWERS, &c

                   OF DIFFERENT MEDIA.
                     TABLE I.
              (Referred to from Page 30.)

Table of the Refractive Powers of Solid and Fluid Bodies.
1-961
1-830
1-815
1-779
2-028
1-794
              (1$    Index of Refraction.
Realgar artificial ......... 2-549
Octohedrite.............. 2-500
Diamond ................ 2439
Nitrite of lead............ 2-322
Blende.................. 2-260
Phosphorus .............. 2-224
Sulphur melted  .......... 2-148
Zircon ................
Glass—lead 2 parts, flint
  1 part                 '.
Garnet .................. 1-815
Ruby  .................
Glass—lead 3 parts, flint
  1 part
Sapphire ..............
Spinelle................. 1-764
Cinnamon stone .......... 1-759
Sulphuret of carbon....... 1-768
Oil of cassia.............. 1641
Balsam of Tolu ..........  1628
Guaiacum  ...............  1-619
Oil of aniseseed ..........  1-601
Quartz  ..................  I-548
Rock salt................  1-557
Sugar melted ............  1-554
Canada balsam...........  1549
            jg  ~  Index of Refraetion.
Amber..................  1-547
Plate glass, from 1-514 to ...  1-542
Crown glass, from 1-525 to ..  1-534
Oil of cloves.............  1-535
Balsam capivi............  1-528
Gum arabic ..............1-502
Oil of beech nut..........  1-500
Castor oil................  1-490
Cajeputoil  ..............  1-483
Oil of turpentine  .........  1'475
Oil of olives .............  1470
Alum ...................  1457
Fluor Spar...............  1434
Sulphuric acid............  1-434
Nitric acid...............  1-410
Muriatic acid  ............  1-410
Alcohol  .................  1372
Cryolite.................  1349
Water...................  1336
Ice  .....................  1309
Fluids in minerals 1-294 to .  1131
Tabasheer ...............  Mil
Ether  expanded to thrice )  j.q^
   its volume...........  S
Air...................1000294
Table of the Refractive Powers of Gases.
Index of Refractk
Vapor  of sulphuret  of?
  carbon  ............)
Phosgene gas..........
Cyanogen.............
Chlorine..............
Olefiant gas...........
Sulphurous acid .......
Sulphuretted hydrogen .
Nitrous oxide .........
Hydrocyanic acid......
Muriatic acid .........
1001530
1001159
1-000834
1-000772
1-000678
1-000665
1-000644
1-000503
1000451
1000449
              - ' •  Index of Refraction.
Carbonic acid......... 1000449
Carburetted hydrogen .. 1-000443
Ammonia............. 1-000385
Carbonic oxide........ 1-000340
Nitrous gas ........... 1-000303
Azote  ................ 1-000300
Atmospheric air ....... 1-000294
Oxygen............... 1-000272
Hydrogen............. 1-000138
Vacuum.............. 1-000000 
310
A  TREATISE  OIN OPTICS.
                  TABLE II.

            (Referred to from Pa<re 31.)

Table of the Absolute Itifrurtiir I'otrtrs of Bodies.
Tabasheer.............
Cryolite ...............
Fluor spar .............
Oxygen................
Sulphate of baryta......
Sulphurous acid gas.....
Nitrous gas ............
Air  ...................
Carbonic acid  ..........
Azote  .................
Chlorine...............
Nitrous oxide  ..........
Phosgene..............
Selenite ...............
Carbonic oxide.........
Quartz  ................
Glass..................
Muriatic acid  ..........
Sulphuric acid .........
Calcareous spar.........
Alum  .................
Borax  .................
' lleirm-tiun.
,  00976
  0-2742
  0-3126
.  0-3799
  0-382'.)
  0-4455
  0-4491
  0-4528
  0-4537
  0-4734
  0-4813
  0-5078
  0-5188
  0-5386
  0-5387
  0-5415
  0-5436
.  0-5514
  0-6124
  0-6424
  0-6570
  0-6716
        ~            Index of Refrortiot.
 Nitre ..................0-7079
 Rainwater  .............0-7845
 Flint glass.............. 0-7980
 Cyanogen............... 0-802!
 Sulphuretted hydrogen ... 0-8419
 Vapor  of sulphuret of  ) fio7.n
   carbon..............J v'°'u
 Ammonia............... 10032
 Alcohol rectified......... 1-0121
 Camphor ...............1-255]
 Olive oil................ 1-2G07
 Amber .................l-3fi54
 Octohedrite  .............l-381ti
 Sulphuret of carbon .....1-4200
 Diamond ............... 1-4566
 Realgar  ................ l-fififfi
 Ambergris.............. 1-70(10
 Oil of cassia*............ 1 -70.3-1
 Sulphur  ................2-2000
 Phosphorus  .............2-HH57
.Hydrogen...............30953
                               No. I.

                      (Referred to from Page 72.)

  In order to convey 1o the reader some idea of the variety of dispersive
powers which exist in solid and fluid bodies, I have given the following
table, selected from a much larger one, founded on observations wliicli
I made in 1811 and 1812.+
  The first column contains the difference of the indices of refraction
for the extreme  red and violet rays, or the part of the whole refraction
to which the dispersion is equal; and the second column contains the
dispersive power.

              Table of the Dispersive Powers of Bodies.
                                               _.    . '   Diff. of Inilicei
                                               Di»r,er»iTe     f R,fr,rl,n„
                                                Pow"-    for extreme Hay.
    Oilof cassia ...........................  0139      O-OW
    Sulphur after fusion.....................  0130      0-149
    Phosphorus  ............................ 0-128       0-156
    Sulphuret of carbon  .................... 0-115       0-077
    Balsam of Tolu  ........................ 0103       0-065
    Balsam of Peru  ........................ 0-093       0-058

       * See Edinburgh Journal of Science, No. XX. p.  308.
       f See my  Treatise on New Philosophical Instrument*, p. 315 
TABLE OF
Barbadoes aloes..........
Oil of bitter almonds.....
Oil of aniseseed .........
Acetate of lead melted ...
Balsam of Styrax........
Guaiacum ..............
Oil of cumin............
Oil of tobacco...........
Gum ammoniac .........
Oil of Barbadoes tar......
Oil of cloves............
Oil of sassafras ... .>......
Rosin  ..................
Oil of sweet fennel seeds .
Oil of spearmint.........
Rock salt...............
Caoutchouc.............
Oil of pimento...........
Flint glass..............
Oil of angelica..........
Oil of thyme............
Oil of caraway seeds.....
Flint glass ..............
Gum thus...............
Oil of juniper ...........
Nitric acid..............
Canada balsam..........
Cajeput oil..............
Oil of rhodium  ..........
Oil of poppy.............
Zircon, greatest ref.  ....
Muriatic acid ...........
Gum copal..............
Nut oil.................
Oil of turpentine.........
Feldspar................
Balsam capivi.........
Amber.................
Calcareous spar — greatest
Oil of rape-seed.........
Diamond  ...............
Oil of olives  ............
Gum mastic.............
Oil of rue...............
Beryl ..................
Ether..................
Selenite................
Alum..................
Castor oil...............
Crown glass, green  ......
Gum arabic.............
Winer..................
Citric acid..........---
Glass of Borax  ..........
E POWEItS.          311
Dispersive	Dili, of Indieea of Refraction
towir.	for extreme Rays.
0-085	0-058
0-079	0-048
0077	0044
0069	0040
0-067	0-039
0-066	0041
0-065	0033
0064	0035
0063	0037
0062	0032
0-062	0033
0060	0-032
0-057	0-032
0-055	0-028
0-054	0026
0-053	0029
0052	0028
0-052	0.020
0-052	0-026
0051	0-025
0-050	0024
0-049	0.024
0-048	0029
0-048	0-028
0047	0-022
0.045	0-019
0-045	0021
0044	0-021
0-044	0-022
0044	0-022
0044	0045
0043	0-016
0043	0-024
0-043	0022
0042	0-020
0042	0-022
0041	0-021
0041	0-023
0040	0027
0-040	0019
0-038	0-056
0038	0018
0-038	0-022
0037	0016
0037	0022
0037	0-012
0037	0020
0036	0017
0036	0018
0-036	0020
0036	0-018
0035	0012
0-035	0019
0034	0018
 
312              A TREATISE  ON OPTICS.

                                             Dl.per.lv. '  l>"l f{ li»llr«i
                                              power.    "' "•''•"""n
                                                   '.for extreme Ra,t.
    Garnet................................ 0-034      0-018
    Chrysolite ............................. 0-033      0022
    Crownglass............................ 0033      0018
    Oil of wine............................ 0032      0-012
    Glass of phosphorus..................... 0031      0-017
    Plateglass............................. 0032      0-017
    Sulphuric acid......................... 0031      0-014
    Tartaric acid........................... 0030      0016
    Nitre, least ref.......................... 0030      0-009
    Borax.................................0-030      0-014
    Alcohol  ............................... 0029      0011
    Sulphate of baryta  ................____0029      0-011
    Rock crystal ........................... 0026      0014
    Borax glass (1 bor. 2 silex)................ 0026      0014
    Blue sapphire.......................... 0026      0021
    Bluishtopaz ........................... 0-025      0016
    Chrysoberyl............................ 0025      0019
    Blue topaz............................. 0-024      0016
    Sulphate of strontia..................... 0-024      0015
    Prussicacid............................  0-027     0008
    Fluor spar............................. 0-022      0010
    Cryolite...............................  0022     0007
                             No. II.

                     (Referred to from Page 73.)

  The following table contains the results of several experiments which
I made in the manner described in pp. 72, 73. The bodies at the top of
the table have the least action upon green light, and those at the boiiom
of it the greatest  The relative  position of some of the substances is
empirical; but, by referring to the original  experiments in my  Treatise
on New Philosophical Instruments,  p. 354., it will be  seen whether or
not the  relative action of any  two bodies upon  green light has been
determined.
Table of Transparent Bodies, in the order in which Otey exercise the
                 least action upon Green Liglit.
Oil of cassia.
Sidphur.
Sulphuret of carbon.
Balsam of Tolu.
Oil of bitter almonds.
Oil of aniseseed.
Oil of cumin.
Oil of sassafras.
Oil of sweet fennel seeds.
Oil of cloves.
Canada balsam.
Oil of turpentine.
Oil of poppy.
Oil of spearmint
Oil of caraway seeds.
Oil of nutmeg.
Oil of peppermint
Oil of castor.
Gum copal.
Diamond.
Nitrate of potash.
Nut oil.
Balsam of capivi.
Oil of rhodium.
Flint glass.
Zircon. 
TABLE OF ACTION OF MEDIA ON GREEN LIGHT, &C.  313
Table of Transparent Bodies, <yc.—continued.
Oil of olives.
Calcareous spar.
Rock salt.
Gum juniper.
Oil of almonds.
Crown glass.
Gum arabic.
Alcohol.
Ether.
Glass of borax.
Selenite.
Beryl.
Topaz.
Fluor spar.
Citric acid.
Acetic acid.
Muriatic acid.
Nitric acid.
Rock crystal.
Ice.
Water.
Phosphorous acid.
Sulphuric acid.
                          No.  III.

                  (Referred to from Page 80.)

Table of the Indices of Refraction of several Glasses and Fluids
Refracting Media.	Qrav.	Indices of Refraction for the Seven Rays in the Spectrum marked 1 in fig. 55. with the fallowing Letters.						
		B Bed ray.	C Red ray.	D Orange.	E	F Blue.	o	H Violet.
Water---- Solution of { Potash ) Oil of Tur- \ pentine 5 Crown Glass Crown Glass Flint Glass Flint Glass	1-000 1-416 0-885 2-535 2-756 3-723 3-512	1-330935 1-399629 1-470496 1-523S32 1-554774 1-627749 1-602042	1-331712 1-400515 1-471530 1-526849 1-555933 1-629681 1-603800	1-333577 1-402805 1-474434 1-529587 1-559075 1-635036 1-608494	1-335851 1-405632 1-478353 1-533005 1-563150 1-642024 1-614532	1-337818 1-408082 1-481736 1-536052 1-566741 1-648266 1-620042	1-341293 1-412579 1-488198 1-541657 1-573535 1-660285 1-630772	1-344177 1-416368 1-493874 1-546566 1-579470 1-671062 1-640373
2B 
 
NOTES
                 BY

THE AMERICAN EDITOR.
                             No. I.

                    (Referred to from article 32.)

  If the remark of Mr. Herschel be admitted, the consequence may be
drawn in relation to all the simple gases, except oxygen, that their ab-
solute refractive powers will be expressed by the  square of the index
of refraction, diminished by unity: for in  them, the specific gravity is
directly proportional  to the weight of the atom.   The  same remark
applies to the vapors of simple bodies, and to many compound gases.
  If the specific gravity, and weight of the atom, of hydrogen be called
unity, the specific gravity of nitrogen, chlorine? &c. will be expressed
by the weight of the atom of each: hence the square of the index of
refraction diminished by unity, will be, by the process directed in article
32, multiplied and divided by the same quantity.
  The inflammable substance  hydrogen,  instead of presenting a high
intrinsic refractive power, would occupy a low place on the scale, while
chlorine would rank high upon it   This consequence was observed by
Mr. Herschel himself.

                             No. II.

                  (Referred to from article 66, page 67.)
  This remark in relation to the absorptive power of water, though true
for moderate thicknesses, in relation to the colored  rays of the spec-
trum, appears, by a recent discovery of Signor Melloni, not to be true
in regard to the heating rays.  An account of the interesting experi-
ments which have established this  fact, will  be more in place, in
connexion with the article which, treats  of the heating power of the
spectrum.

                             No. in.

                (Referred to from article 66, page 70.)

  This interesting analysis of the solar spectrum, by Sir David Brewster,
will, probably, have its value to the reader increased by a brief state-
ment of the experiments from which the results, given in pages 69 and
70, were deduced.   The matter of this note is taken from a paper, by
Sir David Brewster,  in the Edinburgh Journal of Science for October,
1831. *
  1. The first position is that, " red, yellow, and blue light exist at every
point of the solar spectrum."   The eye gives evidence of the existence
of red light, in the red, orange, and violet spaces, which, together, con-
stitute more than half the length of the spectrum.   If the blue and
* And Edinburgh Trans. Vol. XII. Part. I. 
316               A TREATISE  ON OPTICS.
indigo spaces be transmitted through olive  oil,  the  light becomes of a
violet tint, rendering evident the red,  previously existing in Ihe blue
and indigo, by absorbing rays which had neutralized it.   In the yellow
and green spaces the existence of red is proved by showing thai while
light may be detected in them.
   Yellow light is recognized by the eye in rather more than one-fiflh
of the length of the spectrum, namely in the orange, yellow, and green
spaces.   It may be  proved to be present in the blue and indigo spaces
by many experiments, among  which is the  ono already described, in
which a violet tint is developed, by passing the spaces ihrough olivo
oil:  the tint absorbed by the oil cannot be  red, because violet, reddish
blue, is made to appear by the transmission; it cannot be blue, for blue
taken from  blue will not leave violet; it is, then, yellow, which mixed
with the red and blue had formed white light, at this part of the spec-
trum. Farther,  the spectrum examined through  a deep blue glass, show
green in the blue  space, and through a transparent wafer of gelatine,
produces a whitish band in the same space.  Yellow is shown to exist
in the red space  by examining it through a prism of port wine, the re-
fracting angle of which is 90°,  and the whole of the red space assumes
a yellowish tint by the absorption of the blue rays, by certain thicknesses
of pitch, balsam of  Peru, &c. In the violet space, owing to the extreme
facility with which that color is absorbed, and the extreme faintness of
the rays, yellow light has not yet been detected.
   Blue light is perceptible  to the eye through more than two-thirds of
the length of the spectrum, that is in the green, blue, indigo, and violet
spaces.   The absorptive powers of pitch, balsam of Peru  &c, show
green light  extending considerably within the red  space; and the blue
is farther proved to be spread throughout that space by the yellow tinge
which it assumes, when viewed through the media already alluded to;
a tint which could only result from the absorption of blue rays.
   2.  White light exists, at every point of the spectrum,  and may be in-
sulated by absorbing the excess of the colored rays at any point.
   By a particular  thickness of smalt-blue  glass, the yellow space, the
brightest of the spectrum, becomes greenish white, ana, with a different
blue, reddish white.  A mixture of red ink and sulphate of copper re-
duces the yellow space to nearly a white, the tint being slightly red
when the ink is in excess,  and green  when there is too much of the
solution of  sulphate of copper. By particular methods, not described,
Sir David Brewster states  that he has succeeded  in insulating white
light in both the  orange and green spaces.
   The curious property possessed by this white light of not being de-
oomposable by refraction, is  a powerful support of the new theory of
the spectrum.
   By a principle of absorption  applied  to the heating rays of the solar
spectrum, and which will be described  in a subsequent note, the exist-
ence of a spectrum  of heating rays, exceeding in length the three colored
spectra, is proved, bringing a new analogy to bear upon this question.


                              No. IV.

                 (Referred to from article 71, page 82.)

   Comparative experiments on the heat in different parts of the solar
spectrum, require the most delicate instruments.   The  new  branch of
science, thermo-magnetism, furnished Signor Melloni with a much more
sensible means of measuring temperature, than the common  thermom-
A 
NOTES BY  THE AMERICAN  EDITOR.       317
eter, namely, by the magnetic currents developed by heat, in a battery
composed of bars of bismuth and antimony.
  By the aid of this instrument, he found  that the heat accompanying
the violet space of the spectrum, from a crown glass prism, was not at all
absorbed by pure water, while a small portion of the heat in the indigo
was absorbed, a greater portion of  that in the blue, and  so on through
the colored spaces and into the invisible heating rays beyond the spec-
trum, the extreme rays of which were entirely absorbed.
  The relative degrees of heat in the  spectrum formed by a  crown
glass prism, which, however, it must  be recollected, has absorbed the
rays unequally, is represented by the annexed diagram, in which the
 ordinates, 2 v, 5 i, &c., of the upper curve represent, nearly, the relative
 temperatures;  the lengths of the ordinates, and of course the relative
 degrees of heat, being expressed by the numbers written above them: thus
 the amount of heat in the middle of the space v, the violet space, compared
 with that in the middle of the blue space b, is as 2 to 9; with the middle
 of the red space r, as 2 to 32. The points marked 25,12, &c. beyond the
 colored space r, correspond to the bands, in the spectrum, having,  re-
 spectively, the  same temperatures as the middle of the yellow, of the
 green, of the blue, &c.   By passing the spectrum through a thickness
 of less than a twelfth of an inch of water, contained between plates
 of thin glass with parallel surfaces and free from defects, the heating
 powers of the several rays became as represented in the lower curve;
 none of the heat accompanying the violet rays having been  absorbed^
 a little of that  accompanying the blue, and so on increasing as the  re-
 frangibility diminished, until in the  band, 2, 0, having  the same tem-
 perature as the violet, all the heating rays were absorbed.  Different
 media stopped  the heating rays in different degrees, those of higher
 refractive powers permitting them to pas more readily than those of
 lower powers.
  Although this subject cannot be considered as fully developed, we
 are able to understand by it, why Seebeck found the  point of greatest
 heat to vary, according to  the  material of the prism used to form the
 spectrum, being in the red when a prism of crown glass was used, and
 in the yellow when w ater was the refracting material.   Melloni  found
 the greatest heat in the  orange after passing the spectrum formed  by
 the crown glass prism  through water, as  appears by the diagram, in
 which the greatest heat  in the red and yellow are 20, and in the orange
is 21. Seebeck found the point of greatest heat in the yellow, when the
prism contained water;  a sufficiently  near coincidence with the ob-
servation of Melloni, if we consider that in the experiments of Secbeck
the rays were exposed to absorption by the glass forming the hollow
prism in which the water was contained, and in those of Signor
                              2B2 
318
A  TREATISE  ON  OPTICS.
Melloni to the absorption by the crowTi glass prism first, and then to
that by two plates of glass and the water contained between them.
  The power of  transmitting  the heating ra\s without absorption
being greater as the  refractive  power is greater, according to the  law
before referred to, we should expect that in flint glass the greatest heat
would lie farthest  from the violet  end of the spectrum; in plate  and
crown glass, that it should  be at a  less distance from that end ;  in sul-
phuric acid and oil of turpentine still less; in alcohol and water yet
nearer to  the violet end:  and these deductions we find, by consulting
the table on page 82, to be correct.  Minute differences, which could not
be detected by the instruments used  by Seebeck and others, in the points
of greatest heat as given by that table, will probably hereafter appear.
  Tho experiments discussed in this note authorize the adtlition of n
fourth spectrum to the three colored spectra  represented  in fig. 51.,
namely, a heating  spectrum containing rays which are less  refrangible
than the  extreme  red rays  of the  spectrum.
                              No. V.

                     (Referred to from page 116.)

  The undulatory hypothesis represents in so simple a manner the phe-
nomena to which Dr. Young applied his principle of interference, thru
I have been induced to refer to it here, with a view to a general expla-
nation of the hypothesis.  The reader will be better satisfied if he lake
up the subject, as briefly referred to in the 84th article of the text, be-
fore entering upon the account to be given in this note.  As stated in
that article, the hypothesis of undulations supposes all space, the planet-
ary spaces as well as  the interstices between the particles of bodies,
to be occupied by an elastic medium,  or ether, which is put in a state
of vibratory motion by luminous bodies,  and in which impulses nre
propagated according  to the same  mechanical laws, as the impulses
which, communicated to air, produce sound.
  If we suppose  a luminous point surrounded  by this elastic medium,
ihe particles immediately about the point  have a vibratory  motion im-
pressed upon  them, or a motion to and fro ; this they communicate lo
the adjacent particles, and thus a wave is formed, which spreads about
the point as a centre, just as the waves formed by a stone, thrown inlo
still water, spread around the point at which the stone struck the sur-
face. As these waves would communicate to a floating body which they
might meet, an impulse in a direction radiating from the point where
they originated, so luminous waves striking the  retina,  give the sensa-
tion of light in a similar direction.
  In the annexed diagram, let AB, No. 1, represent one of the directions
in which the impulse given by a luminous body, is propagated; we shall
find, according to the hypothesis, along that line particles of the elastic
medium, or ether, having all rates of motion, from rest, or when the mo-
tion is nothing, to the greatest rapidity of the vibration; and in the two
opposite directions, from A towards B, and from B towards A. For example,
let the particles at A, D, and C, be at rest, then if from A to D we rind
particles moving towards B,  their velocities, or rates of motion, will be
found increasing between A and a!, mid-way from A to D, and then de-
creasing between a' and D; between D and C the vibration will be in the
contrary direction, namely, from B towards A; and the  velocities afirr 
             NOTES BY THE AMERICAN EDITOR.        319

increasing to V, will diminish to C. The distance between A and C includes
all velocities from nothing to the greatest, and in the two opposite direc-
tions A B and B A; the same would he true of C B, if made equal to AC:
this distance A C, or C B, is called the length of a wave, and it is this which
in red light is 256 ten millionths, and in violet light 171 ten millionths,
of an inch (see  page 119, text.)  To represent  to the eye the velocities
of the different particles of ether between A and D, the  curve A cf D
is described, in  which the  ordinates, or lines  in the same direction as
a'a, represent these velocities,  as, for example, a! a the velocity at a';
the particles between D and C vibrating  in  a contrary direction, the
curve D 6 C, representing their  velocities, is traced on the side of the
line A B opposite to A a D.  In the same way  the velocities between C
and E, and between E and B, are shown by similar curves, on opposite
sides of A B.   Let No. 2. represent a system of waves  in which the
lengths  M P and P O are equal to A D and  D C, and let the motion
between M and P coincide  in direction with that between A and D, the
curves of velocities MmPandAaD coinciding, and the motion be-
tween P and O with that between D and C, the curves P p C and D 6 C
coinciding; then it is plain  that  the motion of the particles between OQ,
C E, and Q N,  E B, &c. will be the same, or  that the undulations will
coincide throughout; one undulation will, therefore,  add to the effect
of the  other, and  the light will be the united light produced by the
two undulations.  The same will be true whether the point M coincides
with A, with C, or with B, &c.; that is, whether the lengths of the paths
of the two rays A B and M N are exactly equal, or differ by one, two, or
more undulations.  If the rays,  instead of moving in the same direction,
meet under a small angle, the remarks will  still apply;  and the first
result,  stated on page 115, text, is in accordance with the hypothesis,
under consideration, namely, that bright spots, illuminated by the  sum
of the  two lights, will be formed when the differences in the lengths
of the  paths of the rays, are d, (A C,) 2 d, (A B,) 3 d, &c.
  Next let the curves described in  No. 3 represent the velocities in
another system of undulations, S Vand V U being equal to A D and BC
in No. 1, the particles of the ether between S and V, as shown by the
curve being in a state of vibration from T towards S, those between V
and U from S towards T, and so on. If the ray S T in No. 3 were brought
to coincide  with  A B in  No.  1, the  point S being  placed at A,  the
particles between S and V in No. 3 moving in opposite directions lrom 
320               A TREATISE  ON OPTICS.
those between A andD, and those between V and U in opposite direc
tions from those  between D and C, and their velocities being supposed
equal, their motions would destroy each other, and the wave would be
destroyed, or darkness would result.  The path S T differs from A B by
the distance S V, or half an undulation. The same would be the result
if No. 3 began one undulation to the left hand of S, or two or more un-
dulations, that is if the path S T, No. 3, differed from A B, No. 1, by one
and a half, two and a half,  &c. undulations.  As the same consequences
would follow, if the rays S T and A B met under a small angle, we infer
(page 115,  text) that when the difference in the lengths of the paths,
of the two pencils of rays, is ^d (AT)), lid (A E), 2£d,&c., "instead of
adding to one another's intensity, they destroy each other and produce
a dark spot"


                             No. VI.

                     (Referred to from page 120.)

   Professor Hare has observed, in relation to the  translucency of gold
leaf;—"Gold  leaf transmits  a greenish light, but it is questionable,
if it be truly translucent.   Placed on glass, and viewed by transmitted
light, it appears like a retina. It is erroneously spoken of as a continuous
superficies."  The  nature  of the process by which gold is reduced to
leaves, strengthens this conclusion.
   On examining gold leaf-by the solar microscope, I find  in it innumer-
able rents, and  also various gradations of thickness; the rents have
their edges colored, a blue fringe appearing on one side, and a reddish
brown on the  opposite side; the  thickest parts transmit no light, and
through  the very thin parts a delicate green light is transmitted.  The
surface thus exhibited is very beautiful, The rents are  visible to the
nalied eye, when the leaf is very strongly illuminated.
                             No. VIL

                      (Referred to from page 237.)

   The following classification of colored bodies is alluded to in the text,
 as given by Sir David Brewster, in the life of Newton.  The colors of
 each of the classes require, in his view, to be explained upon different
 principles.
   1. "Transparent colored fluids, transparent colored gems, transparent
 colored glasses, colored powders, and the colors of the leaves and flowers
 of plants."
  The colors of these bodies are derived from the absorption of particu-
 lar colored rays:  thus, water at great depths appears red by transmitted
 light, owing to the  absorption of the blue and yellow rays which with
 the red constituted  white light;  certain thicknesses of smalt-blue glass
 appear intensely blue by transmitted light, while at greater thicknesses
 the glass appears red.  In the case of opaque and colored bodies, as in
 the leaves of plants, we are to suppose  certain rays to  be absorbed by
 the indefinitely  thin film  through which the rays reflected  from any
 surface maybe supposed to pass, the complementary tint being reflected;
 as all the incident light is not reflected, the transmitted tint will be com-
 plementary to the colors absorbed, and thus the body will appear of the
same color by both reflected and transmitted light.  Colored powders 
NOTES BY THE AMERICAN EDITOR.         321
would seem not always to belong to this class, since many of them
change their hues with a change in the size of the particles.
  2. " Oxidations on metals, colors of Labrador feldspar, colors of precious
and hydrophanous opal and other opalescences, the colors of {he feathers
of birds, of the wings of insects, and of the scales of fishes."
  To these the Newtonian theory is strictly applicable.
  3. " Superficial colors, as those  of mother-of-pearl and striated sur-
faces."
  4. " Opalescences and colors in composite crystals  having double re-
fraction."
  5. " Colors from the absorption  of common and polarized light, by
doubly refracting crystals."
  6. " Colors at the surfaces of media of different dispersive powers."
  7. " Colors at  the surfaces of media in which the reflecting forces
extend to different distances, or follow different laws."
                            No.  VIII.

                     (Referred to from page 251.)

  The philosophic toy called by its inventor, Dr. Paris, the thaumatropc,
or wonder-turner, illustrates very perfectly the fact of the duration of
impressions on the retina.
  On one side of the card, represented in the diagram, is drawn a chariot
and horses, and on the opposite side the charioteer; on causing the card
to revolve by turning the strings C and D between the thumb and fore-
finger of each hand, the charioteer appears in the act of driving the
chariot, as in the figure.  It requires but little skill to give to the card,
exactly the motion, which shall perfectly unite  the two objects on the
opposite sides into one picture, and yet not render it confused by the
rapidity of the turning.  Many amusing illustrations accompany the toy;
for example, Harlequin and Columbine are painted upon opposite sides,
and by a turn of the card are seen to join in a dance: royalty, stripped
of its robes, occupies one side of the card, and the robes the opposite,
the robes are donned by a turn:  a potter seated at his wheel moulding
the unformed clay, occupies one side of the card, and an urn is grasped
by an arm on the reverse; on turning, the urn appears  grasped by the
potter's arm, the foot of the vase being yet unfinished. 
1
322               A TREATISE  ON OPTICS.
                             No. IX.

                     (Referred to from page 261.)

  Referring to his new analysis of the solar spectrum, Sir David Brews-
ter advances the following  hypothesis to account  for  certain of the
cases just detailed.*
  "By means of this analysis we are now able to explain the phe-
nomenon observed by  those who are insensible to particular colors.
(Edin. Journ. Sc. No. XIX. old series.  No. IX, new  series.)  The eyes
of such persons are blind to  red light; and when we abstract all the
red rays from a spectrum constituted as alreadyt described, there will be
left two colors,  blue and yellow, the only colors which are recognized by
those who have  this defect of vision. To such eyes, light is always
seen in the red space; but this arises from the eye being  sensible to the
yellow and blue rays, which are mixed with the red light.
  Hence blue light will be seen  in the place of the violet,  and a greenish
yellow will appear in the orange and red spaces, or, which is the same
thing, the spectrum will consist only of the  yellow and the blue spectra,
The physiological fact, and the optical principle, are therefore in perfect
accordance; and while the  latter gives a  precise explanation of the
former, the former  yields to  the latter a new and an unexpected sup-
port"
  The details of the cases  referred to, fully sustain this  conclusion.
There are other circumstances  connected with them, and with others
described in the text, page 260, not unworthy of notice.
  In  the second of the cases described in the Journal of Science, No.
XIX, although the individual never failed  to detect a full blue or a
full yellow, he seems to have had very imperfect ideas of those colors
when presented in a state of mixture; green, as such, he did not know,
and when blue was diluted with yellow, forming what  to a good eye
would appear yellowish green, the blue tint escaped him, and the mix-
ture appeared yellow.   In like  manner, his discrimination of yellow,
when mixed with blue,  was very  defective; he called grass green
yellow, and yet yellowish green appeared  to be " yellow with a good
deal  of blue in it."  This remark may serve to explain  why the same
white seen at different times, appeared to him to vary in  its tint, at one
time being white, at another " white with a dash of yellow and blue,"
at another " white  with yellow and  blue in it." When requested to
arrange colors so as to produce the strongest contrasts, he divided them
into two classes, to  one  of which he gave the name  of blue, and to the
other yellow.  In these contrasts he invariably placed white among the
blues, and was never perplexed, as in the preceding examinations, when
tasking himself as to the precise shades.  That white should be classed
by him as blue, appears consistent with  the other observations,  foi
being blind to red light  the tint  of white should be that which appears
when red is removed from the spectrum  or  a bluish green, which tint
he saw as a blue.
  In examining other cases we shall find reason to be satisfied that this
blindness extends to light of other colors than  red, and that in those
cases also there is a want of discrimination between shades in mixtures
of the colors to which the eye is sensible. The Plymouth tailor, whose
  * Edinburgh Trans. Vol. XII. Part.I., or Edin. Journalof Science, No. X
new series.
  t See text, p. 60. 
NOTES BY  THE  AMERICAN  EDITOR.        323
case is described by Mr. Harvey (see page 260, text), seems not to have
been entirely blind to red light, and to have been in a measure blind
to blue; thus  the  prismatic spectrum appeared to consist entirely of
yellow, and light blue; the  red, orange,  and yellow spaces appearing
as if red had  been withdrawn from them, while the full blue, the in-
digo and violet were  light  blue, and  dark blue and indigo stuffs ap-
peared to be black, and crimson was either blue or black.  A dark green
he  regarded as brown, by which, since  he was blind to red light, he
must  have meant a shade  of black, and hght green  as orange, by
which,  for the reason just stated, he meant a variety of yellow.  The
blue in  both these mixtures escaped his perception.
  An extreme case seems to have occurred in the vision of Mr. Harris,
of Allonby,  who, according to the statement of Mr. Huddart, could
only distinguish black from white, or was entirely blind to colors.
  It is much to be desired, for  the  elucidation of this curious subject,
that more well  examined  cases were on record.  The colors of the
spectrum afford rigid  tests, not to  be  found in colored stuffs;  and by,
such tests only, the minutia of peculiarities of vision can be satisfac-
torily determined. 
\
iijBXA 
              APPENDIX


                      TO



        SIR  DAVID BREWSTER'S



 TREATISE  ON  OPTICS;



INTENDED TO ADAPT THE WORK TO USE, AS A TEXT-BOOK,
       IN THE COLLEGES OP THE UNITED STATES.


               BY A. D. BACHE,
      PROFESSOR OF NAT. PHILOS. AND CHEMISTRY IN THE
             UNIVERSITY OF PENNSYLVANIA. 
  Entered according to the act of congress, in the year 1833, by Carey,
Lea, & Blanchard, in the clerk's office of the district court of the
eastern district of Pennsylvania.
STEREOTYPED BY J. HOWE. 
                                  iii
ADVERTISEMENT.
  The object of the following Appendix is to place in
the hands of the students of our Colleges, a text-book
which will furnish  them with  some of the  analytical
methods of the most recent writers, upon the elements of
Optics.
  The work of Sir David Brewster is from the pen of a
master, and presents, in a popular form, the results which
have flowed from experiment and from theory, applied to
the investigation of the different branches of Optics.  The
Appendix merely aims at supplying  to the student the
mode of determining the results given in the text, more
particularly in what relates to Reflexion  and Refraction.
  It  may not be amiss to state, that I do not  present, to
the notice of instructers, an untried course, but that most
of the propositions in the following pages have entered
into the  mathematical portion of the course, taught to
the Senior Class of the University of Pennsylvania.
  The works in  which the full  development of these
subjects may be found, and which have been consulted
in the composition  of the Appendix, are Coddington's
Optics, Coddington on Reflexion and Refraction, Lloyd's
Treatise on Light  and Vision.  The  more advanced
student will find the subject treated by the most general
methods in Herschel's Treatise on Light.
                                    A. D. BACHE.
    Philadelphia, March, 1833. 
.J 
CONTENTS OF THE  APPENDIX

                  BY THE

    THE  AMERICAN EDITOR.
                      REFLEXION.

                        CHAPTER I.

         REFLEXION BY SPHERICAL AND PLANE MIRRORS.
Irt.                                                      Page
 1. Direct and oblique pencil defined........................   9
 2. Reflexion of a small direct pencil by a Spherical Mirror.—
     Case 1. Diverging Rays falling upon a Concave Mirror ...  ib.
 3. Case 2. Converging Rays upon a Concave Mirror..........  10
 4. Comparison of the Formulae for the two Cases.............  ib.
 5. Case 3. Diverging Rays upon a Convex Mirror............  11
 6. Case 4. Converging Rays upon a Convex Mirror  ..........  ib.
 7. Comparison of Formulae.—General Formula for the Reflected
     Pencil  ..............................................  12
 8. Rule.—The sum of the vergencies of incident and reflected
     Pencils constant......................................  ib.
 9. Application of the general Formula for the reflected Pencil,
     to the Plane Mirror...................................  13
10. Parallel Rays falling upon a Plane Mirror.................  ib.
11,12.  Diverging and Converging Rays......................  ib.
13. Application of the general Formula to Reflexion by a Concave
     Mirror..............................................  ib.
14. Principal Focal distance found...........................  14
15. Case of diverging Rays; Reflected Pencil  considered for dif-
     ferent values of the distance of the Radiant Point........  ib.
16. Converging Rays upon a Concave Mirror.—Rule for  Focal
     length  ..............................................  16
17. Reflexion of a small direct  Pencil by a Convex Mirror......  ib.
18,19,20. Parallel, diverging, and Converging Rays............  ib
21,22.  Incident and reflected Pencils referred to the centre of the
     Mirror..............................................  1?
23,24.  Distance of the principal Focus from centre, a mean propor-
     tional between the Distance of the Radiant Point and Focus,
     of any small Pencil, from the same centre...............  18
26. Cases of the oblique Pencil  .........................: • • •  19
27,28.  Reflexion of a small oblique Pencil which meets a Mirror
     near its vertex..............................• • •---••  ib-
29. Focus of reflected Pencil in primary plane.  General i ormula  a)
30. Geometrical interpretation  of general Formula.—Rule......  ib.
31. Focus of reflected Pencil in secondary plane.—Rule---....  21
32. Reflexion of a small oblique Pencil which crosses the Axis of
     a Mirror before meeting its Surface.....................  22
                             A2 
VI              CONTENTS OF THE APPENDIX.


                          CHAPTER II.

               FORMATION OF IMAGES BY REFLEXION.

Art.                                                        Page
 33. Image of a Plane, formed by a Concave Mirror............  23
 34. Curvature of Image at Vertex............................  24
 35. Change in the form of Image, by change  in distance of object
      from  Mirror .........................................  ib.
 36. Hypothesis that Image and Mirror are similarly curved......  26
 37. Image formed from Section..............................  ib.
 38. Case of Convex Mirror.—Case of Plane Mirror............  ib.


                       REFRACTION.

                         CHAPTER m.

                REFRACTION BY PRISMS AND  LENSES.

 39,40.  Refraction at a plane Surface.—Law expressed.—Angle of
      deviation, general Formula............................  26
 41. Case of total Reflexion .................................  27
 42. Formulas for Reflexion deduced from those for Refraction ....  ib.
 43. Refraction by a Prism, general Formula...................  28
 44,45.  Case in which the angle  of  Incidence  is  small.—Other
      cases  in which the general Formula is simplified .,.......  ib-
 46. Case of Rays incident perpendicularly upon  the first Surface
      of a Prism...........................................  29
 47. Case in which the angles of Incidence  and emergence are
      equal...............................................  ib.
 48. Proof that, in this  last case, the deviation is a minimum.....  30
 49. Refraction  by  Lenses.—Suppositions made  in first approxi-
      mations.—Corrections................................   31
 50. A small Pencil of Rays refracted by a spherical Surface.—
      Formulae............................................  ib.
 51. Signs of the algebraic quantities'in the general Formula ....  32
 52. Refraction of a direct Pencil by a Lens.—Approximate re-
      lations ..............................................  33
 53,54.  Formula, neglecting thickness of Lens.—Rule.—Cases in
      practice.............................................  34
 55. Mode of applying  Correction for thickness in any case......  35
 56. Refraction by a plate with parallel Surfaces...............  ib.
 57. Particular case of  thick plate............................  36
 58,59, 60. Parallel, diverging, and converging Rap falling upon a
      thick  Plate..........................................  ib.
 61. Refraction by a thin double Convex Lens.................  37
 62, 63.  Parallel Kays.  Principal focal  distance................  38
 64,65.  Diverging Rap.—Rules,  for  focal distance, in different
      forms  of double Convex Lens, investigated..............  39
 66. Converging Rays.—Rules for Focal Length................  40
 67,68.  Correction for thickness applied to Case of parallel Rap
      falling upon a double Convex Lens.—Nature of results ...  41
 69. Principal Focal Length of a Refracting Sphere deduced from
      the general Formula..................................  42
 70. Refraction by a thin plano-convex Lens...................  43 
CONTENTS OF THE APPENDIX.            vti
Art.                                                         Page
 71. Plane Side turned to Incident Light.—No correction for thick-
      ness  ................................................  44
 72,73. Convex side to Incident light.—Correction for thickness ..  ib.
 74. Refraction by a double Concave Lens....................  45
 75, 76, 77. Cases of parallel, diverging, and converging Rap.—
      Effect of thickness in a practical  point of view..........  46
 78. Plano-concave Lens.—Concave side to incident Rays.—Plane
      side to incident Rays..................................  47
 79. Refraction by a Meniscus.—Power  of the Meniscus with Con-
      vex side to incident Rays..............................  48
 80,81. Meniscus  with Concavity towards Incident Light.—Focal
      length..............■................................  ib.
 82. Concavo-convex Lens..................................  ib.
 83. Spherical Surfaces of the same Curvature  ................  49
                        CHAPTER IV.

             FORMATION OF IMAGES BY REFRACTION.

84. Formation of Images by  Lenses.—Property of Centre of a
     Lens................................................  50
85. Centres of different kinds of Lenses found.................  51
86. Mode of using this Point in the formation of Images........  ib.
87. Image of a plane, by a Convex Lens  .....................  52
88. Object considered as a portion of a circular arc ............  53
89. Image by a Concave Lens  ..............................  54
90. Theory of the magnifying power of a Lens used as a Telescope  ib.
                         CHAPTER V.

         SPHERICAL ABERRATION OF MIRRORS AND LENSES.

 91. Spherical aberration of Mirrors.—Longitudinal aberration de-
      fined.—Lateral aberration  .............................  55
 92,93. Formula for longitudinal abermtion.—Approximate rela-
      tions introduced......................................  ib.
 94. Aberration for parallel Rap.............................  58
 95. Lateral Aberration.....................................  ib.
 96. Physical Focus of a Mirror, or circle of least Aberration, de-
      termined in position and magnitude.....................  ib.
 97. Introduction to Aberration of Lenses.—Aberration produced
      by a single spherical Refracting Surface.................  61
 98. Approximate values introduced into general Formula.......  63
 99. Aberration by a Convex Surface.—Cases of no Aberration ..  64
100. Spherical Aberration of Mirrors found from Formula for Aber-
      ration by a Single refracting Surface....................  ib-
101. Spherical Aberration of a Lens..........................  65
102,103. Case of parallel Rays.—Double Convex Lens..........  66
104. Spherical Aberration of an equi-convex Lens of glass......  68
105. Case of Lens of which the Radii of the Surfaces are in the
      ratio of two to five............................W'-°
106. Aberration by plano-convex Lens.—Plane side towards Inci-
      dent Light.—Convex side towards Incident Light.........  69
107. Lens of least Spherical Aberration with a given power.....  lb 
1
Viii             CONTENTS OF THE APPENDIX.
Art.                                                         PtiRc
108.  Equations of Curves producing Surfaces of accurate Refrac-
      tion.................................................  71
109.  Convex Surface of accurate Refraction for parallel Rays pass-
      ing from a rarer to a denser Medium.—Lens formed......  72
110.  Concave Surface of accurate Refraction for parallel Rap pass-
      ing from a denser to a rarer Medium.—Lens corresponding
      to this Case..........................................  73
112.  Surfaces of accurate Reflexion determined................  74
113.  Caustic for parallel Rays  falling upon a  concave  spherical
      Mirror..............................................  ib.
     For Rap diverging from the Extremity of a Diameter......  75
114. Equation of caustic Curve for any reflecting Curve.........  76
115. Example.—Caustic by Reflexion from a portion  of a Logarith-
      mic Spiral...........................................  78


                         CHAPTER VI.

         DOUBLE REFRACTION AND  POLARIZATION OF LIGHT.

117. Formula for extraordinary Refraction in the Plane of Section
      of a doubly refracting' Crystal.........................  79
118. Law of extraordinary Refraction  in Plane of principal Section  80
     In Plane perpendicular to principal Section...............  ib.
119. Formula  for relative Brightness of Images, after transmission
      through a second doubly refracting Crystal...............  ib.
120. Law of Polarization of Light by Reflexion at  the Maximum
      Polarizing Angle.—Consequence of the Law.............  81
121. Relation of polarized Ray to refracted Ray................  ib.
122. Polarization at the second Surface of a Plate..............  82
123. Law of partial Polarization of Light, by Reflexion..........  ib.
124. Effect of successive Reflexions of Common  Light,  at Angles
       differing  from the Maximum Polarizing  Angle...........  83
125. Law of Polarization  of Light by ordinary  Refraction.......  ib.
126. Polarization of Light  by Refraction and Reflexion  at the two
       Surfaces  of a Plate...................................  ib.
     Consequences of the Law,...............................  84


                          CHAPTER VII.

                         ON THE RAINBOW.

127. To determine the Deviation, produced by two Refractions and
      any number  of Reflexions, by a sphere, when the deviation
      is a Minimum.......................................  85
128. Theory of the  Primary Rainbow.........................  86
129. Theory of the  Secondary Rainbow.......................  87 
APPENDIX.
REFLEXION.
                         CHAP. I.

     REFLEXION BY SPHERICAL AND PLANE MIRRORS*

  (1.) In considering the cases of reflexion from spherical or plane
6urfaccs, two divisions will be made : in the first, the  axis of the
incident pencil will be supposed perpendicular to the surface of the
mirror, in the second, oblique to it; in  the  first case the pencil is
termed direct, in the second, oblique.
(2.) Prop. I. To determine the form given by reflexion to a small
  direct pencil of light, proceeding from a point in the axis of a
  mirror.
  Case 1. In Jig. 8., p. 18 of the  text, let A represent  the radiant
point of a pencil of diverging rays, F its focus, and C ihe centre
of a concave mirror.   Call AD = u, FD = v, and CD = r.  The
angle of  reflexion FMC being equal  to the  angle of  incidence
AMC, the line CM bisects the vertical angle of the triangle AMF:
whence (Legendre's Geom., Book III.,  Art. 201., or Euclid, VI. 3.)
                 AC : AM :: FC : FM, or
                      AC  _  FC_
                      AM  "  FM
  The pencil being, by supposition, very small, the point Mis very
near to D, hence for the approximate value of AM we  may take
AD, and  for that  of  FM, FD.  The equation just found becomes

                        AC __ FC_
                        'AD ~~ FD
  By the notation adopted above AC = AD — CD — u — r, and
FC = CD__FD — r — v.   We have therefore

                    !inr = rjzi,  or
                      U         V
                     r      r    ,    ,
                1-----=-----1, whence
                     u      v
  * Throughout the Appendix, the student is supposed to be acquainted
with the corresponding chapters in the body of the work. 
10                 CONCAVE MirvRORS.         APPENDIX.

dividing by r and transposing
                  112         , ,
                 ---------= —.....(a)
                  u      v      r
   (3.*) Case 2. We next proceed to the case in which a converging
pencil falls upon a concave mirror.  Fig. 10., p. 20 of the text.
   AM,  AN representing the two extreme rays of the pencil, con-
verging to A', the imaginary  radiant point, MF and NF are the
reflected rays.  The radius CM therefore bisects the angle AMF,
the outward angle of the triangle A'MF, and (Euc. VI. A.)
                       AC  _ FC_

                       AM ~ FM '
but for AM and FM we may substitute A'D and FD, their ap.
proximate values, whence
                       AC_ _ FC_

                       A'D ~ FD
   Using the notation before employed, A'D == u, FD = v, and
DC =  r, whence  AC  = u -4- r, and FC = r — «; these values
substituted in the equation just found give,
                     u 4- r     r — v
                    —E—- = -----, or,
                        u         v

                    1  + — = — — 1 and
                         u     v

                _-L + _l=A.....o
                    u      v      r
   (4.) Comparing equation (b) with (a) we find that it differs from

it only  in the sign of — which is  positive in (a) and negative in
                     u
(b); both these cases may, therefore, be represented  by the same
equation, if we agree to give the positive sign to the distance of
the radiant point for diverging rays, negative for converging rays;
that is,  if we consider  the distance («) positive, when the  radiant
point is in front  of the mirror, negative when  it is behind the
mirror.
   If, then, i.« represents the distance of the radiant point from the

mirror,____will  denote the degree of divergency or convergency


of the incident rays. In like manner, — will represent  the con-
                                     v
vergency of the reflected rays.   From equations (a) and (b) 
 CHAP. I.                REFLEXION.                      \\

 where  r  is a constant quantity;  a result which may be thus ex-
 pressed:  the divergency (or convergency) of the incident rays to-
 gether  with the  convergency of the rejlected rays is  a constant
 quantity for the same  mirror.  The  curvature of the  mirror  is

 measured by —, the reciprocal of the radius.
              r
   (5.) Case 3. Diverging rays falling upon a convex mirror. Fis. 12..
 p. 21, text.                                              s    '
   The  line CM, bisects the outward  angle of the triangle AMF
 whence (Euc. VI. A.)                                         '
                        AC _   FC

                        AM ~  ~FM '
 substituting their approximate values for AM and FM,

                          AC  _ FC
                          AD  ~~FD~'
 and by the notation adopted in the cases already considered,
                   u M r     r — v    ,
                   —!—  :=-----,  whence
                     u        v
               _1______1_  =      _2_
                u       v          r

   On comparing this equation, in  which we have made -—  posi-
                                                      u
 tive as  it  corresponds to a real  radiant, with (a), we perceive that
                  1         2
 the signs of both---and ---are different  From the figure we
                  v         r
 observe that v corresponds  to an  imaginary focus, and that the ra-
 dius is  now behind the mirror.   Equation (a) may, then, be used
 to represent this case  if the sign of the radius be changed; the re-
 sulting negative value of the focal distance corresponds to a focus
 behind the mirror.
   (6.) Case 4. Converging rays  falling upon a convex mirror. The
 formula for this  case may be deduced from Jig. 12., if B M and
 B N be made to represent the incident, and MA, NA the reflected
 rays.  We should have by  proceeding as in the last case,

                       FC  _   AC_

                      FM  ~  ~AM  '
                        FC      AC_

                       ~FD  ~~  AD  '
and since  FD --= u, F being the imaginary radiant point, and
AD = v,  A being the focus; FC = r — u, and AC = r -f- »,
whence 
12   CONVEX MIRRORS.--GENERAL FORMULA.  APPENDIX.
                     r — u     r 4- v
                     ——_ —    ~__, or
                       U         V

              -Jl + J_=_A.....(d)
                  u       v          r
the first term being made negative  to correspond to the case of
converging rays.   This formula differs from  (c)  in the  sign of

—, which was negative in (c), and  in that of _ .  The figure
v                                              u
shows that v in this latter case corresponds to a real focus, while
in the former —v denoted the distance  to  an imaginary focus.

The change  of sign in  —  conforms  to  the remarks  made in
                          u
article (4.)
   (7.)  Comparing  the four equations (a) (b) (c) and (d), we  per-
ceive that the formula

                i + ^-  = A.....a)
                 u       v       r
may be made to include them all, by attributing to u, and r, respec-
tively, the positive  sign when the radiant point or the centre is in
front of the mirror, the negative sign when either of these points
is behind the mirror, and by considering the positive value of v as
corresponding to a focus in front of the mirror, its negative value
to one behind it.

   (8.)  We  might  have commenced by giving to the student this
conventional mode of considering the  quantities used  in the an-
alysis, and then have deduced the general equation by reference to
a single diagram : we have preferred in the outset to show him that
the variations in the algebraic signs are not arbitrary, but required
by the geometrical relations of the quantities.

   From the formula

                i + -L=A.....a,
                 u       v       r
we deduce the general rule, that the sum of the vergencies* of tkt
incident and reflected pencils is  a constant quantity.

  (9.) Having obtained an equation (1) expressing the relation be-
tween the distances of the radiant point and focus of a small pencil,
by means of the radius of the mirror, we shall proceed to interpret
it in its application to different kinds of mirrors, and under different
circumstances of the incident  pencil.
  * This convenient term, expressing, as the case may be, either divergency
or convergency, is proposed and used by Lloyd in his Treatise on Light and
Vision. 
CHAP. I.     REFLEXION.--PLANE  MIRRORS.           13
Prop. II. To determine the form given to a small pencil of rays by
  reflexion from a plane mirror.
  In the plane mirror the  radius is infinite, or  — = o, whence

from (1),

               — +  — = o  .....(2), or,
                u       v
                          V = — w .
  The focus and  radiant  point are  at  equal distances from  the
mirror, but on opposite sides of it.
  (10.)  If parallel rays fall upon the mirror u = oo, and v = — oo,
or the reflected rays are parallel.   This corresponds to the  case
represented in fig. 4., p. 15, text.
  (11.)  If the rays diverge before reflexion, the formula
                        1           1
shows that they will be equally divergent after reflexion; and
                          v = — u
that the focus is as far behind the mirror as the radiant point is in
front of it.   Fig. 5., p. 15, text.
  (12.) For converging lays (fig. 6., p. 16, text,) u takes the nega-
tive sign, and (2) becomes

           ____-—|-----= o.....(3), whence,
                           v = u.
  The sign of v being positive the focus is real, its distance  v in
front of the mirror is equal to  the distance of the imagmary ra-
diant point behind it.
(13.) Prop. III.  Reflexion of a small pencil of light by a concave
  mirror.
  The formula which applies to this case is,

                1 +  1=1.....CD-
                 u       v       r
  By transposition

              ! -  •  — _ i_ = 2"~r, or,
               v  ' '  r     u        ur
                                ur
                             2« —r 
                                                               \

14                CONCAVE  MIRRORS.          APPENDIX

  This value of » gives the rule on page ID of the  text.

  (14.) When the rays are parallel — = o.   And

                        1       2
                           ""  2
   The  sign  of v being positive shows that the focus is in front of

the mirror, and its value  _ ,  that the  focal distance is  half the

radius.   This  is represented in fig. 7., p. 17,  of the text, where
FD =  i CD.
   The  focus of parallel rays is called the principal focus.  As it
serves as a point of comparison for the foci of other rays, we shall
                                                   2
represent its distance by a symbol, /.   Placing for  —  its  value
                                                   r

— ,  formula (1) becomes
/

                ±+i-= ±.....(4).
                 u      V      f
   (15.)  The  next  case to be considered is that of diverging rays.
In this, « is  positive and the formula is

             ----f- --- = —.....(4),  whence,
              u       v       f

                      .1  -   _L     1
                       V       f       u

   As long as  u > f, or the point A in  fig. 8., p. 17, is farther

than the  principal focus from D, we  have ---<;___, and _
                                              u     f        f
     1   .       . .
------is a positive quantity, or the  rays  converge  after re

flexion.   Since---------<*---,  we  have _! <- _L , and
                 /      "     /              »     /
» > /, or the focus is fartlier from  the mirror than the  principal
focus.

   If we suppose, besides, that  A is beyond the centre C, we have

u >  r, and---<; — , whence-------— , or E-_____— > -
             u      r             f      n    , ,  r      u    r

------. or >---» and  .— > — , or « < r, the focal distance
    r          r          v     r
is less than the radius. 
 CHAP. I.                 REFLEXION.                      15

   We  infer,  then, that rays  diverging from a point beyond the
 Centre  of the mirror, are reflected to a point between the principal
 focus and the centre.


   As--- diminishes in value,  — evidently must increase, or,


 as u increases, v diminishes, and vice versa: that is, as the radiant
 point recedes from the mirror  the  focus  approaches it, and vice
 versa.  When the radiant point becomes infinitely distant, the rays
 are parallel, and their focus is the principal focus.

   We have seen that as long as the radiant point  is farther from
 the mirror than its centre (that is, u > r) the focus cannot coincide
 with that centre (or v < r) which it approaches.  If u = r, or the
 radiant point coincides with the  centre, then

          1112      11
        ---=  —--------=----------=  — ,or,
          v      f       r       r       r       r

                             v = r,
 and the rays arc reflected to the  centre.

   Let the radiant point now pass the centre towards the principal

 focus, that is, let u < r, and at  the same time u> f,  or — > -
                                                         u    r

 and — <- — . Since —  <;  — ,---------is still a positive
      u    f           u     f    f      u

 quantity; the rays  are, therefore,  still  brought to a  focus: but

 since   — > — ,__________or------.___<; — , whence
        u      r     f      u        r      u      r

 - <; — and v >■ r; that is, the focus lies beyond the centre.
 v    r

   When the radiant point coincides with the principal focus u =f,

 whence,  — =  — , and — =---------=  o, or v = oo ;
          u      f         v      f      u
 the rays are rendered parallel by reflexion.

   If we suppose the radiant point  to approach still nearer to the

 mirror,  so that a</,we have — > — , whence  —.------is
                                «     /            /      «

 negative;  that is, ___  has  a negative sign,  or  the   focus  is
                    v
 imaginary,  the  rays diverging  after reflexion.   The divergency
of the  reflected  rays  is less than that  of the  incident  rays,

for J_  = _(1_____L\  ,  and  J_____1 < -1, the  di-
     V         V u      f /          u      f      u

vergency before reflexion. 
16                  CONVEX  MIRRORS.          APPENDIX.

   (16.)  For the case of converging rays falling upon a concave mir-
ror (fig. 9., p. 19, text,) we make, in formula (4), v negative; whence,

                1       1      1         /rv
               ---------=  -v-.....(5), or,
                v       u      f

                     i=  >  +-L.
                      v      f       u
   The  signs of the quantities on the right-hand side  of this equa-

tion being both positive,   — is always positive.  Converging rays
                         v
are, therefore, always brought to a focus.   Moreover,  since___L i
                                                        u   f
     111
> —  , — > — , or the convergency is greater after, than be-
     u    v     u

fore, reflexion.   Since-----f-  — >. — , — > — and v </,
                       »      /     /    v     f
whence the focus of converging rays is nearer the mirror than tho
principal focus.
                                   1              2
   Substituting, in equation (5) for ___, its value___, and trans-

        1                         f
posing  — , we have
        u
                     _L .. J|.  _L
                      v      r       u
whence the value of v is
                                 ur
                         v =  —------,
                              2m -f r
agreeing with the rule on p. 20 of the text.

(17.) Prop. III. Reflexion of a small pencil of rays  by a convex
   mirror.
   In this case, r, the radius of the mirror, takes the negative sign,
and  equation (1) becomes

              -1  + —  = - —.....(6).
               u        v         r

   (18.)  For parallel rays (fig. 11., p. 21, text,) — = o,  whence
                                             u
1        2               r
-=------, OTV = —-■
v        r               2
   The focus  is behind the mirror,  and at the distance, from the
vertex, of half the radius.
   If we call the principal focal distance /, the formula for the re-
flexion by a convex  mirror becomes

              ±  +  ± =  -±.....m.
               u       v         f 
CHAP. I.                REFLEXION.                     17

  (19.) For diverging rays (Jig. 12., p. 21,) wc have

        _!.=      >—L = _(>  +JL\,
          V          f       u         \f       u J

and therefore  — is always negative;  such rays diverge  after re-
              v
flexion, and since — -L. --->--- , or___>__ , their di-
                  f      u     u       v      u
vergency is increased by the reflexion.

  (20.) Figure 12. will, as has  already been stated, represent the
case of converging rays falling upon a convex mirror, if we sup-
pose BM, and BN to represent  the  incident rays, meeting in the
imaginary radiant point F.  To express this case analytically, u
must be made negative, and the formula is
             -1__1=__L.....(8), or,
              v      u         f

                    _! — -1   _L
                     v       u      f
  The position of the focus, as shown by this equation, passes
through variations, corresponding to those in the case of diverging
rays falling upon a concave mirror (Art. 15.).

  (21.) It is sometimes convenient to refer the distance of the ra-
diant point and focus, to the centre of the mirror, instead of to the
vertex.  Formula (1) may be readily transformed into one which
shall refer to the centre.

Prop. IV. To determine the relation  of the distance of the focus and
  radiant point, of a small pencil  of rays, from the centre of a
  mirror.
  By fig. 8., p. 18, text,  it appears that AD = AC + CD, and
FD = CD — CF.  Calling AC = u' and FC = v', CD, as be-
fore, = r, AD = u, and FD  — v; we have u  = u' -4- r,  and
v = r — »'.
  Substituting these  values of u and v  in the equation
            1        1       2
          _i_ _|-----— --- ...... (1), it becomes
      ---— — 1=1----------
      r — v                u 4- r
B2 
18              REFERENCE TO CENTRE.       APPENDIX.

  Bringing to a common denominator the quantities on both sides
of the equation, and reducing, we have,
r	— v'	«' +	r
r	— v'	_ «' + u'	r
	v'		
	• 1 =	= 1 + -	r
	1	2	
or,
                                          (9).
                 our
  (22.) This equation  is the  same in  form with  the one found
when  the distances were estimated from the vertex, except that
the sign of u' is negative,  the distances v' and  u' being  now
reckoned in opposite directions.
  Equation (9) might have been deduced directly from the relation
of the lines AM, FM,AC, CF, Jig. 8., in the triangle AMF.  The
solution of the question by  that method, would have  been more
simple.

  (23.) If we substitute for  — ,  in equation (9), its value - , we

have

                 ±-\=i......<«*
                  »      w      /
  From this equation, may  be deduced the  relation expressed in
the following proposition.

(24.) Prop. V.  The distance of the principal focus of a mirror from
  the centre, is a mean proportional between the distances of the
  radiant point and focus of any small pencil, from the principal
  focus.
  Equation (10) gives,
              1        1   ,"    1      / 4- a'
             ■—T =----(- — = J ~— , or,
              »       /        «'        /a'
/-»' = /■
r/ =  —-----, whence,
                            f + u'     / +  «'
                   /-»':/::/:/+ a',
in which proportion /— v' represents FO (Jig. 8., p. 18, text,) or
CO — CF, and u'  + /, AO or AC + CO.
  (25.)  The subject of reflexion at curved surfaces in general, will
be treated briefly in a subsequent chapter.  The properties of the 
CHAP. I.
REFLEXION.
19
surfaces formed by the revolution of the parabola and ellipse about
their axes, are readily understood,  from very simple geometrical
considerations, and gain nothing by being presented analytically.
They will, however, be referred to in another part of this Appendix.
  (26.) We pass next to the reflexion  of a small oblique pencil by
a spherical  mirror, on which subject  two  propositions  will be
given; in the first will be considered the case in which the axis
of the pencil does  not cross that of the mirror, and the pencil falls
upon the mirror near to the vertex, and in the second, the case in
which the axis of  the pencil  crosses that of the mirror.
(27.) Prop. VI. A  small pencil having its radiant point out  of the
  axis of a mirror, meets the surface near the vertex, required the
  form of the reflected pencil.
   Let LMN represent a section  of the  mirror, made by a plane
 passing through the lines MR and MC, or through the axis of the
 pencil and the centre of the mirror.  RL is an extreme ray of the
 pencil incident very near to M,  LF is the  corresponding reflected
 ray, meeting  the  reflected ray  MF, which corresponds to the axis
 of the pencil,  in the point F.  F  is the focus of the pencil LRN,
 in the plane of the section RMC.
   (28.) To determine  the focus of rays which meet the  mirror in
 a plane perpendicular  to  RMC; suppose  a plane to pass through
 RM at right  angles to that of the figure, this plane will cut from
 the pencil, RLN, two rays, which  reflected will meet the axis, MF,
 of the reflected pencil, in the focus required.  If, now, a plane be
 passed through one of the incident rays, just described, and the
 corresponding reflected ray, it will pass through  the centre of the
 mirror; the line RC, joining the radiant point and centre, will be,
 therefore, its intersection with the plane RMC containing the axis
 of the incident and of the reflected pencil; and the point, F!, in
 which RC produced meets MF, will be the point in which the sup-
posed plane meets MF, that is, the focus of the reflected  pencil.
  The  focus,  of the reflected pencil, in any plane between RMC
and the one at right angles to it,*  will be found between F and F'.

  *Coddington  terms the former of these planes the  primary  plane, the
latter, the secondary plane. 
I
20                  OBLIQVE  PENCIL.          APPENDIX.

   (29.)  To determine, nnnlytically, the position  of the point F;
draw from the vertex M, MX and MZ perpendicular, respectively,
to RL and LF; also from C, CP and CQ perpendicular, reflec-
tively, to RM and MF.  As the arc LM is very small, it. may be con-
Bidered a straight line perpendicular to the radius  CI.; whence the
incident and reflected rays making equal angles with CL, also make
equal angles with  LM, and the triangles LMX and  LMZ arc
similar. But they have the side LM common, hence they are equal,
and MX is equal to MZ.  The two triangles CPM and  CQM  being
also  equal,  CP is equal to CQ.   And since CT  and  CS may be
considered as perpendicular to RL and LF, they  may be taken ns
equal. Whence PT— QS. By the similar triangles IIP Tand KMX,

                   RP  : RM  :: PT :  MX;
and  by the similar triangles FMZ and FQS,
                   QS : MZ :: FQ : FM;
whence, since PT = QS, and MX = MZ,
              RP : RM ::  FQ : FM.....(e).
   Let RM = u,MF = v, CM = r, and the angle RMC = <p .
   Then,

        RP = RM— MP = RM—CM . cos RMC, or,
                   RP = u — r cos <p; and
        FQ = MQ — MF=MC.  cos CMF — MF, or,
                     FQ = r.  cos <p —  v.
   By substituting, in the proportion  (e) above, for RP, and FQ,
the values just found, and for RM, and FM, u, and v, we have
           u — r.  cos <p : u  ::  r. cos ft — »  :  »,  or,
        u —  r. cos 0    r. cos  ft — ».    tv • v  t
        ------------ =------------.  Dividing by  r,
             u                 v                   J
               1     cos <p  ___  COS </>     1      j
                r       m         »       r
             cos0    cos ft  _  _2_        .jj
              v        u        r
whence,
                                ur
                            cos 0
  (30.)  To interpret equation (11) geometrically, we should  ob-

serve that the ratio of MX to RM, that is, I ^E\\ , measures  the
                                        \RMI
divergency of the incident pencil  LRN. But  in the triangle MXL,
MX = ML  . cos LMX,  and since the angles XMR and LMC 
CHAP. I.               REFLEXION.                      21
are nearly right angles, LMX is nearly equal to RMC, or MA' =
ML. cos ft.    —  therefore varies with  cos *  ,  which, conse-
              RM                        u
quently  measures  the  vergency of the  incident rays.  In  liKe
manner, cos ^ expresses the vergency of the reflected pencil.
           v
  We infer, then, from (11), that the  sum of the vergencies of the
incident and reflected pencils is a constant quantity.
   If ft = o, or the  rays proceed from  a point in the axis, cos
ft = l, and formula (11) becomes
                     i-  + -L = ±.
                     u      v      r
agreeing with equation (1).
   If the rays are parallel,----  = o, and
                           u
                       cos ft      2
                      ---l = — , or,
                        v        r
                       v = J— . cos ft.

   (31.)  To find the position of the point F', for the plane perpen-
dicular to RMC, we have, in the triangle  RMC,
            MC : MR ::  sin MRC  : sin MCR, or,
calling the angle RCD, 0, whence MRC = 0 — ft,
                  r : m : sin (9 — ft) :  sin 0,
                      r__sin (9 — ft)
                      u         sin d
and, by substituting for sin (0 — ft) its value,
               r   _ sin 0 cos ft — cos 0. sin ft ^
              "*«                sin 0
   If MF' be  called »',  we may deduce from the triangle CMF',
by a method similar to  that just used,
                    r      sin (9 + <t>)   or
                    --- — -----:---n--- '  '
                    e'         sm a
               r   _ sin 6. cos ft 4- cos 0. sin ft
              ~i7                sin 0
   Adding together the values found  for  —, and  —T , and re-

ducing, we find,
                    r       r      e\
                  ___  4-  —- = 2  . cos ft , or,
                    u       v 
22                  OBLIQUE PENCIL.            APPENDIX.

              1 + 1 = 2" cos *.....(12).
                u      v         r

  Since  ---1  represents the vergency of the incident rays, and

!!2L? that of the reflected rays, it is evident, from (12), that the

sum of these quantities, for the incident  and reflected rays, in tho
secondary plane, is not constant for the same mirror, but depends
upon the angle, ft, of inclination  of the  axis of the pencil, to the
axis of the mirror.

(32.) Prop. VII.  A  small oblique pencil crosses  the axis before
  meeting the mirror, required the form of  the reflected pencil.

                             Fig.B.
   In  the  figure, R is  the  radiant point, RL tho axis of a small
 pencil, of which one of the extreme rays is RN; MC is the axis
 of the mirror, which is  cut by RL at the point D.   F is the focus
 of the reflected pencil  in the plane RMC.  F' (found as in the
 last proposition) is  the  focus in the plane perpendicular to RMC.
 In this proposition must be found,  besides the distances LF and
 LF', the point Y, at which LF cuts the axis, and the relation of the
 angle  L YM to the angle LDM, which latter angle is given.
   Call, as in article (29.), RL = u, LF = v, LF' = »', CL = r\
 also let MF = z, MD = b.
   Since CL bisects  the angle DLY, we have,
                    YL  : DL ::  YC : DC,
or, taking for  YL and DL, the distances YM and DM, which  are
nearly equal to them,
                YM : DM:: YC : DC, that is,
                  z : b :: r — z  : b — r,  or,
                     r — z     b — r 
CHAP. II.
    REFLEXION.


+ -1-=-"
    o       r
23
• (13).
  An equation from which z may be determined.
  The approximate value of the angle L YM may be obtained by
supposing that the tangents  of L YM and LDM are to each other
as the distances YM and DM, or,
tan F
YM
DM
                   tan D     DM     b

  The distances LF and LF'  will be given, as before, by the
equations
            cos ft     cos ft__2
             u        v        r
                      1      2
-E +
cos ft
(11),  and

... (12).
                         CHAP. II.

          FORMATION OF IMAGES BY REFLEXION.

   33.) In discussing the  subject of the formation of images by
reflexion, the surface of the mirror will be assumed to be spherical,
and the most useful case will be solved; namely, that of a plane
perpendicular to the axis  of the mirror.   The  objects, of which
images are to be formed by mirrors, when not plane, are generally
of irregular forms, and  the images can be found, only, by points.

Prop. VIII. To determine  the image of a plane, perpendicular to
  the axis of a mirror.
                           Fig. C.
  In the figure, let MN be the intersection of the plane constituting
the object, with a plane passing through the axis of" the mirror. It is
evident that if a small pencil of rays be supposed to proceed from
each point of MN, and  the several foci of the pencils be deter- 
24
IMAGES BY MIRRORS.        APPENDIX.
mined,  the assemblage  of the points  thus found will give the
image.  From M and N, through C, the centre of the mirror, draw
MB and NA,  the  axes of two pencils of rays from M and N, and
upon which, therefore, the foci  of the pencils will be found.  Let
the focus of the pencil, of which MB is the axis, be at m.

   By our former notation, (art. 21.), MC = u', Cm = v', and CB
= r. Call CEt the distance of the object from  the centre  of the
mirror,  d; and the angle MCE, 8:  then, since

                    CE = CM.  cos MCE,
                      d = u'. cos 0, or,
                                 d
                         a  = ----7, ■
                               cos 0

   The general formula, for the relation  of the distance of the ra-
diant point, and of the focus, from the centre, was, art. (23.),


                     v      f      u'

and, substituting the value of u', just found,
                1^=   1  +co,0.....
                »       f       d
  The polar equation of a conic section, the focus being the pole,
is (Young's Analyt. Geom., Chap. V.)

                   r =  A
1 -t- e . cos o>
1           e . cos ui
            r      A(l— e2)     .4(1 —e2)

This equation will be identical with (14), if we suppose — =
                                                      r
      1           1         e           1      ,     a
                                 =  — , and co = 0.
v'  4(1 —e2)      f    4(1 —e2)      d
  The image of the line MN is, therefore, a portion of a conic sec-
tion, of which C, the centre of the mirror, is one of the foci.

  (34.)  Since / = A (1 — e2) =  A (l _ il\  = ?1, (Analyt
                                   \     A-/     A
Geom.) the semi-parameter of the conic section, and / is constant,
it follows that the radius of curvature at the vertex of the conic
section, which is  equal to the semi-parameter, is independent of
the distance of the object from the centre of the mirror.
  (35.) We proceed to examine the variation in the figure of the
conic section just found, when the  distance of the object from the
mirror varies. 
4(1 —e2)
CHAP. II.               REFLEXION.                      25

  First, let the object be infinitely distant  Then, — =  o, and
                                                 d

            =z o,  that is, e = o, and the  image is  a portion of a

circle, corresponding to  the equation __ = E_ ,  or, v' = S— .
                                    v'       r             2
The radius of the circle is half that of the mirror.
  As the  object is brought nearer to the mirror,  d  diminishes, or

— increases; but, __ ==  ----f---, and,  since A (1 — e2) ==/,
d                 d      4(1 —e2)

or is constant,  e  must vary as---, and, therefore, increases.  As
                             d
                  11          -        '    1
long as d > /, or — <; ---,
                  d      f    A (I — e2)   A (1 — e2)
 < 1, and the image is a portion of an ellipse.
   When d =f, e = 1, and the curve is  a parabola.  For d -</,
 e >• 1, and the curve becomes a branch of a hyperbola.
   If d = o, or the object  passes through the centre of the mirror,
 _ = oo , and e is infinite, or the  image is a straight line, coin-
 d
 riding with the object.
   When the object passes  the centre, towards  the  mirror, d be-
 comes negative, and the equation (14) changes, if we reckon 0 from
 GC, to
               1 = 1 _ 22ll.....(15).
                v'      f      d
   This  equation gives a  hyperbola  while d </, a parabola when
 d = /, an ellipse when d >f.
   Each of these cases presents curious circumstances.   For ex-
 ample, in the case, d </, if a point be taken in the object, so that
 u' = /, that is,---- = /, the equation for the focal distance of
               cos 0
 the pencil proceeding from that point, is
                       1
                      — =o, v' = co;
                       v
 the image is infinitely  remote  from the mirror.   If we suppose the
 object to be sufficiently extended to cut the mirror, the point com-
mon to the object and mirror is its own image, and for that point
vl = r, and v' = r ; between the points for which u' = r and u' =
/, the  distance of the image has varied  from r to  infinity, and,
therefore, that portion  of the object which is between these limits,
 has a virtual  image,  the  part of each branch of which, between
 v' = r and the vertex,  is wanting.
               C 
26           REFRACTION AT  A SURFACE.     APPENDIX

  The part of the object between u' = d and u' = f, has its image
beyond the  centre; it is the branch of a hyperbola conjugate to
the first.
  (36.)  If the section  of the object be an arc concentric with tho
mirror, u' is constant, and
is constant, or the image is also a circular arc concentric with the
mirror.   In this case,  the relative magnitudes  of the object and
image are as their distances from the centre of  the mirror.
  (37.) We have considered a section  of the object, of the image,
and  of the mirror, made by a plane passing through the axis  of
the mirror; if these sections be supposed to revolve about the com-
mon axis, the section of the object will generate a plane, and that
of the mirror, and  of the image,  surfaces of revolution correspond-
ing to the sections.
  (38.) The case of a convex mirror is embraced by equation (14),
if r be made  negative.
                                     2
  For the plane mirror, r = oo,  and  — = o, whence,
                                     r

                   1 =  ^-S.....(16),
                    v       a
and the image is similar to the object.
REFRACTION.
                         CHAP.  in.

            REFRACTION BY PRISMS AND LENSES.

  (39.)  The most simple case of the refraction of light, is that in
which it takes place at a plane surface.  The perpendicular bein<*
drawn, the refracted ray is connected with the incident, by the law
(p. 29, textj) that the sine of the angle of refraction bears a constant
ratio, for a given medium, to the sine  of the angle  of incidence.
To  represent this law analytically, suppose a ray passing from a
rarer to a^denser medium, call the angle of incidence ft, the angle
of refraction ft', and let the sign  of incidence be to that of refrac-
tion, as m is to 1, when m will represent the index of refraction of
the denser medium,  that of the rarer medium being unity;  we
have
                  sin ft  : sin ft' :: m :  1, or,
                sin ft = m . sin ft'.....(17). 
CHAP. III.
REFRACTION.
27
  When the ray passes from a denser to a rarer medium, ft' repre-
sents the angle of incidence,  and ft that of refraction.
  If the angles ft and ft'  be very small, they may be taken instead
of their sines, in which case,
                          ft = m . ft'.
  (40.)  The difference between the angles of incidence and refrac-
tion is termed the deviation of the ray, for a single surface.  When
the angles are very small, we have, for the deviation,
               ft — ft' = mft' — ft' = (m — 1) ft',

                     ,   m — 1
               ft — ft' =-------ft.....(18).
                            m
  The deviation, therefore, when the angle of incidence is small,
bears a constant ratio to that angle.

  (41.)  The case of the total reflexion of a ray moving in a denser
medium, and arriving at  the separating  surface of the denser and
of a rarer medium, (pp. 34, 35, text), is comprehended in equa-
tion (17).  The ray passing from a denser to a rarer medium, if ft'
be taken to represent the angle of incidence, and ft  that of refrac-
tion, m will remain greater than unity, the angles being, as before,
connected by the equation
                 m . sin ft' = sin ft.....(17).

  In this equation, since sin ft  can never exceed unity, m sin ft'

cannot exceed unity, whence sin ft' cannot exceed — , (in which
                                                m
case, m . sin ft' = 1,) or,  the  equation cannot be satisfied if sin ft'

> — .   The ray  then  ceases to be refracted; it is wholly re-
   m
fleeted.
  The angle, at which, light, passing through a denser medium,
and  meeting  the separating surface  of  the denser  and  of  a
rarer medium, ceases to be  refracted, is found  from the equation
sin ft' = ___.  If the denser medium be glass, and the rarer, air,
          m
sin ft' = 1  =  1, whence,  ft' = 41° 48'.
          m       3
  (42.)  The phenomena of reflexion might be derived, analytically,
from those of refraction,  by considering that the angle of reflexion
is measured with the same  perpendicular as that of refraction, or,
is the supplement of that  measured by ft', (see text, fig. 22., p. 35),
and that the angles of incidence and reflexion are equal, but on
opposite  sides of the perpendicular; so that, in the case of re-
flexion,  sin ft  = — sin ft', or, m = — 1. 
                                                                 \

28             REFRACTION BY A PRI3M.      APPENDIX.

(43.) Prop. IX.  To determine  the course of a single ray, or of a
  small pencil of rays, refracted  by a prism.   (Fig. 20.,  p. 32,
  text.)
  Let  the angle,  HRM, of incidence, upon the first  surface,  be
called ft; H'RN, the corresponding angle of refraction,  ft'; H'R'N,
the angle  of incidence on the second surface, »//; n'R'b', the angle
of emergence from the  prism, \p; the angle, A, of the  prism, a.
The angles, RAR', ARR', and AR'R, together,  are equal to two
right angles :  but ARR' = 90° — ft',  and AR'R — 90° — <//
whence,
             a 4- (90 —ft')  4- (90 — i//) = 180, or,
                   a = ft' 4.  <//.....(19).

  The  angle of total deviation, DEH, is  equal  to the  sum of the
partial deviations, ERR' and ER'R, that is, calling the devia-
tion S,
                  6= ft — ft' 4- ip — <//, or,
                  6 = ft 4- <p — (ft' 4- .//'), or,
                  <5 = ft 4- </, — a.....(20).

  This value of the  deviation contains the given angles ft and a,
and the angle i//, which may be found by means  of the  relations of
0» $'» $'* and $, as given by the following equations:
                  sin ft = m . sin ft'.....(17),
                     0/ = a — ft'.......(19),
                 sin d/ = wt. sin \\>'.....(I?')-

   (44.) If we suppose the angles very small, we  have, from (17),

                       ft = wift', whence,
                      ft —ft'= (m—l)ft'.
  Also, from (17'),
                           xp  = Wli//,
                   </. — <// = (m — 1) $',  and
         a=ft —ft' + ./, — ^'=(m — l)(ft'4- f)\
but (19) gives
                     ft' 4- <// = c,  whence,
                  « = (m — 1). a.....(21),
a value depending only upon the refractive power of the material
of the prism, and upon its refracting angle.

  (45.)  There are two other cases in which the deviation, as given
by  equations (20), (17),  (17'), and (19),  assumes a somewhat
simple form.  These we shall  consider, in order. 
CHAP. III.             REFRACTION.                     29
(46.) Prop. X.  To find the deviation of a ray,  or of a small pencil
  of rays, incident perpendicularly on  one of the surfaces  of a
  prism.
  When the incidence upon the first surface is perpendicular, ft =
o, and ft' = o, and equation (20) becomes,
                         i = if — a,
and (19),
                       t// = a, whence,
                 sin tf' = m . sin if' = m . sin a;
but from the value just found for S, we have,
                     xp = a 4- S, whence,
              sin (a 4- S) = m . sin a.....(22):
from which  equation, 5 becomes known when a and m are given;
or,
                         _ sin (a 4- S)
                      m = ---:---■---i
                                sin a
may be found  by determining <5 and a.   This is  one  method  of
determining the refractive power of a substance. Another method
is furnished by the next proposition.
(47.) Prop. XL  To determine the deviation of a ray, or of a small
  pencil of rays, when the angles of incidence  and emergence are
  equal.   (Fig. 20., p. 32, text.)
  By the condition of the question ft = if, and since, from (17),
                  .,     sin ft    ,  ■   .,     sin ii
             sm ft  = —- , and sin if  =  ---1,
                         m                   m
                  sin ft' = 6in if', or, ft' = xp'.
  Equation (19), gives,
                    2ft = a, or, ft' ■=  —- ,
and from (20), by making ft = if,
                     _          ,      a  4-  6
                 S = 2ft — a, and ft =  —±— •

  Substituting these values for ft and ft' in (17), it  becomes
         sin —-  (a 4. 5) = m . sin — a.....(23);

an equation from which S may be determined when a and m are
given.
  By transmitting light through a prism so that ft = if, we have
a method of measuring the refractive  power of the  substance  of
which it  is composed, for,
         C2 
30 DETERMINATION OF REFRACTIVE POWER. APPENDIX.
              sin J (a 4- S) = m . sin h a,  whence,
                m = 8in * (a +J).....(24).
                          sin  4 a
   The equality of the angles of incidence and emergence may bo
ascertained, by measurement, with the instrument which has been
devised for this purpose,  or by the  use of the proposition which
follows.
(48.)  Prop. XII.  The angles of incidence and emergence, of a rau
  passing through  a  prism,  are  equal, when the  deviation is a
   minimum.
   The proposition requires the deviation to be a minimum.  Wo
therefore find its value, differentiate it, and put the differential co.
efficient equal to zero.  The value  of the deviation, from equation
(20), is,
                       s = ft 4- ^' —  ai
the differential of which, a being constant, is,
                    dS = d<p 4- d^,'whence,
                       dS  _ 1     d4>
                       d<p           dip
   Equation (19), is
                          a = ft' +  ^',
by the differentiation of which, we obtain
                         d^l  _
                         dip'  ~
  From equation (17), by a similar process, we have,
                  d . sin ft = m . d  sin ft', or,
                cos ft . d<p = 771 . cos ft'. dip',  or,
                     dip = m . °2ll .  dip'.
                               cos ft
  In like manner, we obtain, by differentiating (17'),
                     j , i        COS -d/   , ,,
                    d-b = m .------.  a\l/.
                              cos -dy
  Dividing the second of these equations by the first,
             d-dy     cos ft . cos -d/   d\J/
             -—- =  -----;------- •  —-, or, since
             oft      cos ft . cos -y>   aft'

                        dip'        *'
                  tf^__     cos ft . cos -4/
                  dft        cos ft'. cos -4* 
CHAP. III.
REFRACTION
31
  Substituting this value for

above, it becomes,
                  M  _
d^h
the value  of----found
cos ft . cos -d/
                              cos ft'. cos -^

  This  value  found for  the  differential coefficient of I,  is to be
made equal to  zero, whence,

                cos ft . cos ^'
=  1
(25);
                cos ft' . cos ^

an equation which is satisfied  if ft = ■>!>, or the angles of incidence
and emergence are equal.  The method of using this proposition
is described in art. 35, pp. 33, 34, of the text.

                   Refraction  by Lenses.

  (49.) In discussing the  subject of refraction by lenses, we  shall
consider the refracting surfaces as  spherical.   The pencil of inci-
dent rays will be, first, assumed as indefinitely small, and the thick-
ness of the lens neglected ; next, the correction for thickness will
be introduced, and, lastly, under the title of Aberration of Lenses,
the case of a pencil of any magnitude will be considered.   Refrac-
tion  through spheres and parallel plane surfaces, will be deduced
from a consideration of the general case.

(50.) Prop. XIII.  To determine the course of an indefinitely small
  pencil of rays, passing from a rarer to a denser medium, through
  a spherical surface.
                            Fig.V.
  Let E be  the  radiant point of a small pencil, of which EV is
the axis, and ER the  extreme ray; the ray ER, passing into the
denser medium, will be refracted towards the perpendicular, CR,
making the angle mRM less than ERC, and in such a proportion
that  sin ERC  : sin mRM :: m : 1, m representing the index of
refraction of the  denser medium, that of the rarer medium being
unity. 
32
SINGLE  SPHERICAL  SURFACE.    APPENDIX.
  Call EV= u, FV=u',CV= r.  In  the triangle, ERC, wc
have,
                      EC _  sin ERC
                      ER ~  sin ECR '
and in FRC,
                      FC _  sin FRC
                      FR ~~  sin FCR '
  Dividing the first of these equations by the second,

               EC   FR    sin ERC     ,   .
               ---.  -i~ = _______, and since
               ER  FC    sin FRC
           sin ERC = m . sin mRM = m . sin FRC,

                        EC  FR
                        --•-- =  m.
                        ER  FC

  The pencil of rays  being very small, the point R is very near to
 V, and for ER and  FR we  may take  their  approximate values
EV and  FV, whence,  EC = EV — CV =  u — r, and FC =
FV—CV=u'—r.   Substituting these values  of EC and FC,
and the values of ER and  FR for those lines in the equation just
found, it becomes
m, or,
-----; and, dividing by r,
  1       mm       ,   .        ...
---=--------,  or, by transposition,
(26).
  (51.)  This formula may be adapted to all cases of a small pencil
incident upon a spherical surface, by a conventional mode of con-
sidering the algebraic signs, of the different quantities involved in
it.  Let us, as in reflexion, consider v, u', and r,  essentially posi-
tive when the radiant point, the focus, and centre, arc, respectively,
in front of the lens, negative in the contrary case.
  The positive sign of u' will, then, correspond to an imaginary,
or virtual, focus ; and the negative  sign, to a real focus.  This  it
is important  to the  student to recollect,  since the reverse lias been
the case in reflexion.
  It would hardly be useful  to discuss the variations of formula
(20), by changes in the quantities concerned in it, since we must
consider, in refraction by lenses, the  action of two surfaces. 
CHAP. III.
REFRACTION.
33
(52.) Prop. XIV.  To determine the course of a small pencil of rays
  falling upon a lens; the radiant point of the pencil being in the
  axis of the lens.
                            Fig. E.
  Let ER, as before, be  the  extreme ray of the pencil, EV the
axis of the pencil and of the lens, RM the ray refracted at the first
surface, CR the  radius  of that surface, M the point in which RM
meets the second  surface, CM the radius  of the  second surface
drawn  to  the point M.  Producing MR until it meets the axis, F
is the virtual focus  of rays  refracted by the first surface.  Since
the refraction  at M is from a denser to a rarer  medium, the ray
is refracted from the perpendicular, taking the direction MN, which,
prolonged  until it intersects the axis, gives F' for the virtual  focus
of the pencil refracted by  the lens.

  Keeping the notation of the  last proposition EV= u, FV= u\
CV=  r, and the equation  for refraction at the first surface is, (26),
               u'      u         r
   The equation for the refraction at the second surface may be in-
 ferred from (2G),'or may be obtained directly, by proceeding in the
 triangles FMC and F'MC, as was done, in the last proposition,
 in ERC and FRC.   Thus,
         FC _ sin FMC  and  F'C^     sin  F'MC
         ~FM ~~ sin FC'M'      FM ~~  sin  F'C'M*

 and, by division,
               FC   F'M _ sin FMC _    1
               ~FM' F'C ~ sin F'MC ~~  rn
  Call VV, the thickness of the lens,-*; F'V, v; C'V, r';  then
FV = u' -j- t, FC = u' 4- t — r', and F'C = u — jE  These
values being substituted, in the equation just  found, instead of FM,
F'M, &c, we have,

             »' + *-*'.  _2_  =  1,  whence,
               u' 4- t     v — tr1      m 
34             LENS.--GENERAL  FORMULA.     APPENDIX
— r'     ™   M' 4- t — r'
   — — m .          --- .  or,
u' 4- t
a  + «/
                1 —
                                         +
dividing by r', and collecting the terms,
                   1        m >      1 — i
                   v      w'4- t       r'
m  being greater than  unity, the sign of the second  member is
really negative, and  as it will be  convenient to show this, we
change the form of that member, and the equation becomes,
            1        m           m — 1       ,nn\
            v      u' 4- t          r'
  This  is the  general equation between u', v, r', t, and m, which,
combined with  (26)> will determine v in terms of u, r, r', t, and m,
all of which are given quantities.

  (53.)  If the  thickness of the lens  is  so  small that it may be
neglected, equation (27) becomes,

          ______—  =______, but, from (26),
           v       u'           r1

                ™=I+ m~1 ,  whence,
                u'      u        r
            1       1      m — 1        m —1__
       J----1 = (m -1)  (1----1\.....(28).'
        v       u               \ r       r f

  Since —  represents  the  divergency, or  convergency,  of tho
          u'
incident pencil, and — that of the refracted pencil, we deduce,
                     v
from (28), that the difference of the vergencies of the refracted and
incident pencils is a constant quantity for the same lens.
  This formula applies to the different cases of the incident pencil
and lens, as in the single surface (art. 51), if we consider the dis-
tances of the  radiant point, focus,  and centre, positive when in
front of the lens, and negative in the contrary case.  The same
remark applies to the general equations (26) and (27).

  (54.)  In most  of the cases which occur in practice, the thick-
ness of the  lens may be neglected, and, therefore, equation  (28) is
applicable to them; in all cases this equation determines  an ap-
proximate value of the focal distance, to which a correction  for tho
thickness  of the lens may be, conveniently, applied. 
CHAP. III.             REFRACTION.                      35


  (55.) To obtain this correction for thickness, expand _!
                                                     u' 4- t
in equation (27), into a series, by division,

         m        m      mt  .   mt2 —  c
       —-----  = —.---------\- —-  _(_ &.C.
       w 4-  '      a      u-     u3   '

                   =  JL__  JUL (!_*!  +&c\
                       a      u  \u     u'2         /

   If the thickness of the lens is not very great compared with the

distance of the point F, the powers of 1  , higher than the first,
                                      u'
may be neglected, and we have,

                     m       m     mt

                   u' 4-  £ ~  ~u'     u72 '


   Substituting  this value  for ——— in (27), it  becomes,
                            u  4- t

               1       m_  ,    mt          m — 1

               v       u'       u'2  ~. ZZ    r'  '

or substituting for --- its valua from (26),
                   u

       1       1   _ m — 1      mt          tn — 1

       v       u         r        u'2  ~         p   '

by transposing  and collecting the  terms,



   v      u               \ r      r1  J      u'2

in which we perceive the approximate value of  1  given  by
                                                  v

equation (28), and a correction----!E_ for the thickness of the
                                  u'2
lens.

(56.) Prop. XV. To determine the form of a small pencil refracted
  by a medium, bounded by parallel planes.


  For plane surfaces, r and r1 are infinite, whence ---= o, and
                                                  r
1


  Equation (28) becomes,


        J_____L = o,  or,  -L —  L.....(30). 
36        REFRACTION BY  THICK  PLATES.    APPENDIX.

  The vergency of the refracted pencil  is the same with that of
the incident pencil, when the plate is indefinitely thin.
  (57.)  Applying the correction from equation (29), we obtain,
                      1      1       mt
 but, from (26), by making — = o,
                            r

                 ___= — , whence,  u' = mu;
                 u'      u

   Substituting this value of a' in that last given, for___,
                                                    v

                   — ■   —     -1  or
                     v      u       mu2

               l=l(l__l\.....(31).
                v       u  \      mu /
   Equation  (31) contains the theory of refracting plates of con-
 siderable thickness.

   (58.) If the incident rays are parallel, — — o, and___= o,
                               *        u             v
 or, v = oo, or parallel rays are unchanged by the refraction (fig.
 23, p. 36, text.)

   (59.) The value  of — for diverging rays, given by (31), is posi-
                      v
 tive, zero, or negative, according as we have

              1 >----, 1 = —,1< —• that is,
                   mu      mu       mu
 as t < mu, t = mu, t > mu, in which m is greater than unity.

 For ordinary cases of relation  between u,  and t, (t < mu) — is
                                                          v
positive, and therefore the focus imaginary, or the rays still diverge
after refraction.  As  u is measured from the first surface, and v
from  the  second, the effect of  refraction in bringing the object
nearer to, or removing it farther from the plate, is not expressed by
the relation of v and u, but by that of v — t, and u.  To ascertain
the effect of refraction in this point of view, we take the value of
v in (31), or,
mu — t 
 CHAP. III.              REFRACTION.                    37*

 this, by dividing and neglecting the  powers of t above the first,
 gives

                     p = u 4- —, whence
                                TO


             — « = „  + *(JL_l).....(32).



 In which since m > 1, — < i, and  —----1  is  subtractive,
                          TO             m

 or, v — t <  «; the point of divergence, therefore, is brought


 nearer the first surface by the distance, t (1___LY


   In a glass plate, m = -, and 1 — 1 = 1    The  point  of
                        2            to     3

 divergence is,  therefore,  brought nearer the first surface  by one
 third the thickness of the plate.

                    4             ii
   For water, to = -5- , and 1------=____
                    3            to      4

   (60.)  If the incident rays  converge,  u is negative, whence
 from (31),


            ~ =  - — (1+—).....C33)'
             v          u  \     mu/


 in which — is always negative, and, therefore, the rays still con-

 verge.

   Proceeding to find the value of v — *, as before, we have

                       mu2                     t

                      mu 4- t     ~          to     '


                v — t = — u — til----V


          „ _ t = _ (u + t (1 -  1)).....(34,)


from  which it appears that the focus is farther from the first surface

than  the  imaginary radiant point, by the distance til------1.


The reverse of the result for diverging rays.


(61.)  Prop.  XVI. To  determine the form of a small pencil of rays
   refracted by a double convex lens.

  We will  first consider the case in which the thickness of the
lens may be neglected.  To this, equation (28) will be adapted by
                                D 
38
DOUBLE  CONVEX  LENS.        APPENDIX.
making r, the radius of the first surface, negative, since its centre
is turned from incident light.  This gives
»      a                 \r      r' /
(35),
since — represents the vergency of the refracted pencil, and__
       0                                                    u

that of the incident pencil, and---------is negative,m,r,r',being

constant, for the same lens, we infer that the divergency destroyed,
or convergency produced, by this lens, is a constant quantity.

   (62.) For parallel rays,— = o, and


          1=_(to-1)(1+1).....(36).

In  the refracting media  of which lenses are made, m > 1, and

this value of— is negative: hence the focus lies behind the lens.
             0                                               '
and is real.  For the distance of the principal focus we have by
taking the value of v from (36),
                         1        rr'

             ^-■m^l-T+7.....^
                            3           1
If the lens is of glass, m = —- , and--------= 2, whence,
                            2        m — 1
                                2rr'
                       " —-----,—r *
                               r 4-  r
 corresponding to the rule given in the text (page 42).
   If the glass is equally convex, r = r', and
                       _____2r3  _
                      v~    "27  ~~r-
 The principal focal distance is equal  to the radius of the surfaces
 of the lens.

  (63.) The principal focal distance  may  serve  as  a convenient
term of comparison for the focal distances of diverging and con-
verging rays.  Denoting it by/, we have the value off given  by

 (37), and of — by (36), substituting in (35) 1 for its equal; and
                                           /
transposing, we have, 
CHAP. III.             REFRACTION.                     39
  (64.) Diverging rays.   Equation (38) applies to this case.
  From  that equation it appears that for the  same lens,  the
vergency of the refracted rays I — \  is less than the divergency
of the incident rays  f — \ by  a constant quantity  (—-- ,\ de-
pending upon the index of refraction of the material of the lens,
and upon the curvature of its  surfaces.
   If u >/ (flg. 29., p. 43), — < —- , and ---is negative, or
                             u     f         v
the rays  are brought to a focus.  The reciprocal of the focal dis-
tance is, from (38),

                    v         \f      u J
   Since  1_____1 < 1 , -- < 1 ,  or u > /, and the focus
         /      a     /    v      f
 is farther from the lens than the principal focus.  As the radiant
 point approaches the lens, — increases, and, of course, ---, or
 2.__L , diminishes, or v increases: that is, as the radiant point
 /    u
 approaches the lens, the focus recedes, and vice versa.
   WhenU = 2/, 1 = -(f ~^) = -if 'and° =
 — 2/.  The focus is as far from the lens as the radiant point.
   If the rays proceed from a point as far from the lens as the prin-
 cipal focus  (as from O', fig. 29.), a =/, and __ = o; the re-
 fracted rays are parallel.
   The radiant point being still supposed to approach the lens, we
 ,          <• „_  1  .   1 •         _£_, or 1, is then positive,
 have a </, or — > _ ,  —------7^ '    »
                 u     j     «     /       "
 and the rays are no longer brought to a focus.
   (65.) Equation (35) will give  the value of the  focal distance for
 diverging rays: to determine this,  transpose  _-  and bring the
 terms on the right-hand side of the equation to a  common denomi-
 nator, whence,
              1      rr'— u (m — 1) (r + r')  Rnd 
40               DOUBLE CONVEX LENS.       APPENDIX.
                  _          urr'
                    rr' — u (m — 1) (r 4- r')'
or changing the sign to correspond  to a real focus, to which we
have found the rays to be brought as long asa>/,
                                urr'
               v = — ——------------------; •
                      u (to — 1)  (r 4- r) — rr
                          3                    1
   For a glass lens, m = — , and  m — 1 = — ; whence,

                                            •(40).
                     u (r 4- r') — 2rr/
  This value of v gives the rule found in art. 45, of the text. The
arithmetical operation there directed is changed for the subtraction
                                                       2rr'
of u (r 4- r') from 2rr', when 2rr' p- u (r 4- r'), or u <       ,
                                                     r+r'
or u </; the algebraic expression shows by its change of sign, in
that case, that the focus is imaginary.
  If r = r', or  the lens is equally convex, (40)  becomes
                              2wra
2ur — 2r2
      ur
, or,
 agreeing with the rule just referred to,
   (66.) For converging rays falling upon a double convex lens, we
 make  —  , in equations (35) and (38), negative, whence,
        u

 V = -[v+(m-I'(T-+^)].....<41>'™d

             J-=-(-L+   ').....(42).
              V        \ U      f  f
  The sign of — being always negative, whatever be  the rela-
                v
tion  of u and /, the focus is always real.  Since __ 4. ___ >.
1                                               "     f
_, the  convergency of the rays is increased by refraction.
u
                                                         q
  Taking the value of v from (41), and making, in it, to = —,
as was done in  the case of diverging rays in the last article, wo
find for a glass lens, 
CHAP. III.              REFRACTION.                    41
                           2arr/
                   u (r 4- r1) 4- 2rr'
For a glass lens equally convex, we have,
                          ur
(43).
                          u 4- r
  These values for the focal distance give the rules on p. 44, of the
text
  (67). In what precedes, we have neglected the thickness  of the
lens, and next proceed to  show how a correction, for the effect of
the thickness, may be introduced,
Prop. XVII. To show the  method of applying, to the approximate
  focal length found for a double convex lens,  a correction for the
  effect of the thickness.
  As an example, let us take the case  of parallel rays falling upon
the lens.   Equation (29) is applied to  this case by making r nega-
tive, whence,
   l=l_(m__1)(l+-l)-^  .....(45).
    v       u             \ r       r  /     u2
  And for parallel rays, for which — = o,
                                  u

     L=-(m-l)(±-+±7)-%  .....(46).
      v                 \  r       r /    m 3
   Equation  (26) adapted to the case of a convex surface, gives,
                    m      1     to — 1
and for parallel rays,
                  m         to — 1
,  whence,
                      to_ __ (m — l)a
                      u'2       mr2
  Substituting this value  of J!L- in (46), we have,
                            M 3
      1                  / 1       1  \    (m —l)2*  „.

            1__     _ 1____(to — l)3*.....(47)#
            v   ~   ~ f       mr*
  To determine  from this equation the correction  to be applied to
the focal length,  v, we reduce  the terms of the second member to
a common denominator, whence,
        D2 
42  CORRECTION FOR THICKNESS.---SPHERE.  APPENDIX.

              1 = _  mr2 + (m-\)2ft  ^
               0               mr'~f

                               mr2f
                 mr2 4- (m — I)2ft

             f i
(m — \)2fH
„ _ (_/) = „  .  / =------!eej^----- 4- / =
               ^ J       mr*  + (to — l)2ft ^
    mr2 4- (to — l)2/t
  Dividing, and neglecting the terms  containing powers  of (
higher than the first,

              v+f=^-Yf2t.....(48),
                            mr*
the correction which is to be applied to the focal distance obtained
by equation (37).
  When the lens is equiconvex and of glass, we find (art 62) that
/ = — r, to which a correction,
» +
      (l)3.r2f _
J  —    3  ,2   —  B li
 is to be applied.  The sign of the correction is contrary to that
 of the focal distance, and the effect is therefore subtractive.  The
 corrected focal length is
                       v = — r 4- £r.
   (68.) The method which has just been  shown  gives, at last,
 only an approximate value of the  focal distance, which, however,
 is sufficiently accurate for  all  cases in which the thickness docs
 not bear a considerable relation to  the focal distance.  In the case
 of a sphere used as  a lens, the thickness is too considerable to use
 the method of correction already exhibited.

   (69.) Prop. XVIII.  To find the focal  length of a sphere for
 parallel rays.

  The supposition that the rays are parallel simplifies the question,

 without deducting much from its utility.  Since — =  o, equations

 (26) and (27) become, by making r negative,
                  m       to — 1
                 77 ■=-----—.......(49), and

             1        TO        TO -- 1
            —  =   -r—:-------;—.....(50).
             v      u 4-  t        r          v  '
For the sphere t = 2r, r = r'; and (50) gives
             1        m        to  — 1  ,
            —  =   ,  ,  n----------, but from (49),
             v     u  4-  2r       r                ' 
CHAP. III.            REFRACTION.                     43

                          mr
                          , whence
                   to — x

u' 4- 2r = 2r
« —1
mr      2mr — 2r — wir
                         to — 1          m _ l

              u' + 2r = mr~2r = r{m-2)
                           to — 1      m — 1
 Substituting this value of u' 4. 2r in (50),
                _1_ _ TO (TO — 1)    to --1
                v      r(m — 2)      r    ' °r

 _1_ _ TO (TO — 1) — (OT — 1) (TO—2)  __ (TO —1) (TO —TO 4- 2)
  v              r(m — 2)            ~       r(mZ^j     '
 and
                    1      2 (m — 1)   ,
                   ~T =  T7^—57' whence
                    b     r (to — 2)
                        c _ r (to — 2)
                           ~ 2 (to — 1)'
 The value just found is the distance of the focus  from the second
 surface; call / the distance from the centre, then
                ,              r(m — 2)
               / = v — r = -E----^ _ r
                              2 (ot — 1)
 or, bringing to a common denominator and reducing,


                '"tf==nr.....<51>-
 The rule on page 40 of the text,  is given  by the value of/ just
 found.
   If the refracting sphere beofta6asAeer,TO = 1.11145,and/= —5r,
 of course FQ (fig. 26, text,) = — 4r.  If the  sphere be of water,
 m = 1.3358  and / = — 2r nearly, or FQ (fig. 26)  = — r.   For a
 sphere of glass, to = 1.5, / = — \\r, and FQ  = — \r.   For a
 sphere of zircon, to = 2, / = — r, and FQ = o.
  (70.) Returning to the discussion of the formula for the refracted
 pencil when  the  lens is indefinitely thin, we take up  the case next
 in order.

  Prop. XIX.  To determine the form  of a  small  pencil after re-
 fraction by a plano-convex lens.

  As  in other discussions, the refractive power of the substance
of the lens is assumed to exceed that of the  medium traversed by
the incident pencil, or m > 1.
  The question  obviously includes two cases; in  the one, the 
V
44                PLANO-CONVEX  LENS.        APPENDIX.

plane side is turned towards incident light, in the other the curved
side is thus directed.
   (71.) First:  when  the plane  side  is turned to incident  rays,

— = o,  whence from (28),

                J__ l = _!EZl2.....(52).
                 v     a          r'
From this we infer,  that the divergency destroyed, or convergency
 produced, by this lens, is a constant quantity,  as  in  the double
convex  lens (art. 61), but the effect is less than in that lens by

m — 1  , the power of the first surface; it is  not necessary, there-
    r
 fore  to carry out the discussion of the properties of this lens. There
 will' be no correction for thickness, for parallel rays, no refraction
 being produced by the first surface.  This is shown by the analysis,
            ±                                    TO
 the term ^ (in 29) vanishing, since from  (26), -y = o.
          u2                                   "

    The principal focal distance given by making — = o in (52),

 and inverting, is
                        /=__!--.
                        J ~     m — l

    For a glass lens,
                            / = -2r'
    (72.) Second:  when the convex side is  turned to incident light,

  __ — o, and r is negative; from (28),

                 1_1 = _'!L=_1.....(53).
                 v     u           r
    The effect is, as in  the  first  case, to destroy divergency in the
  incident  pencil, or to produce,  or increase, convergency; and it we
  suppose r =  r', that is, the same lens to be used in both cases, the
  effect produced is the same.
     For the principal focal distance,


                         * ~ ~  to —1 '
   and for a glass lens,
                            /=-2r.
     (73.) The thickness of the  lens  produces in this case an effect
   on the principal focal length, since the  rays refracted  by the first
   surface fall obliquely upon the second. 
CHAP. III.            REFRACTION.                     45
  To introduce the correction into the value of the  principal focal
iistance, we recur  to equations (26) and (27); making,  in these,

r negative, —T = o, and — = o, we obtain.
    &       r'           u                '

                5=-^.....(54),
                u           r
                   1        m
                   v     u' 4- t
The value of u' from (54), is
                              mr
  (55).


, and
                             m —1
              mr                /   r      t \
a' 4- t «=-------- 4- t =a — m (----------I, but from (55),
            to — 1'            \to — 1    OT/
                             u' 4- «
                               OT
and substituting for u' 4- * the value just found,

              o ~----^-  +1.....(56).
                     TO--1      TO
The correction, therefore, shortens the approximate focal length of

the lens by —th part of its thickness; if the lens be of glass,

                    v = — 2r 4- § t, or,
                    r = — (2r — It).

(74.) Prop. XX. To determine the  form of  a small pencil, after
  refraction by a double concave lens.
  In this form the radius of the first surface is positive, that of the
second negative.  When the thickness of the lens may be neglect-
ed, we have from (28),
E_E  = (m_D(± + E)
 v      u             \  r ■    r /
(57),
and where an approximate value of the thickness may be  used,
from (29,)

    1_1 = (ot-1)(1+A)-^.....(58),
     v      u              \ r     r /    u2
in which u' is determined from equation (26), or 
46              DOUBLE  CONCAVE LENS.       ArrENDIX.

  (75.) When the incident rays are parallel, (fig. 31, p. 44, text,)
(57) gives
± = (m-i,(E  + -L)
or calling / the principal focal distance, and determining it from
the equation just given,

              f = -------------—.....(60).
              J    (to - 1)  (r + r')
This value being positive, the focus is imaginary, and at a distance
expressed by the product of the radii  divided  by the  index of
refraction, less one, into  the  sum  of  the  radii.  The  rule cor-
responds to that for a double convex lens; in fact, equations (60)
and (37), differ only in their sign.

   (76.)  Diverging rays.   In this case — is  positive, and, there-
                                     u
fore, as long as to > 1,  the value  of 0, from equation  (57), will
always be positive  and the focus imaginary; since
i. = -i  + 0._i,(J. + 4)
 0      u              \ r      r /
(57),
it appears that —> — , or the divergency of the rays is increos-

ed by the refraction, (fig. 32, text.)
  In a glass lens,
0     u+2\r+r'/'
2urr'
                       2r/ 4- m (r 4- r') '
whence the rule on page 45 of the text.

  (77.)  When the rays converge, — is negative, and (57) becomes

    1=_1+(OT_1)(1 + 1).....(61).

  The pencil still converges, is rendered parallel, or diverges, ac-

cording to the relation between — and (m — 1) I-----(- —rY or


its equal —- .   If — < —T or u > /, — is positive, and the
         /        a      /             v     ^

rays still converge; if — = ——, or u = /, -—  = 0, and the
                      u       j              v 
  CHAP.  III.               REFRACTION.                    47


  refracted rays are parallel; if — ^ __ 0r u < f, —  is  nera-
                               a    /         J   v        6
  tive, and the rays are brought to a focus. This real focus is as far

  behind the lens, as the virtual focus of parallel rays is in front of

  •t -e        f      1      2   „           11          1
  it, if m =  -E_, 0r--- —  — ; for then----- 4-  _ =____L,
             2      u      f               a  T  /          / '

  and 0 = — /:  the  distance exceeds that just named if u > -1
                                         j                   2,


  when we  shall  have  —  < — and----- 4. 1 <;___L , 0r

  1        1                                    /       /
  — <----7- and v  > — /:  the reverse will of course be true if





   If, as was first supposed, u > f or — <—-, though  the rays

 still converge after refraction, they converge less than before it, for

     1   ,   1     l


   We shall not introduce the correction for thickness, as it would
 be determined by the same  method with that for the double convex
 lens.  Practically the  thickness of double  concave lenses is of little
 importance, since it is least at the central parts.

 (78.)  Prop. XXI.  To determine the form of a small pencil of rays,
   after refraction by  a plano-concave lens.

   First.   When the concave side is turned to incident rays,  1 =
     ,  .     . .                                           ir1
 0, and r is positive; equation (28) gives

               1      1      (to —1)
              —--------=  i-------.....(62).
               v      u         r

   The divergency produced by this  lens is, therefore, less  than

 that produced by a double concave lens, by ^m ~ ^, the  effect of
                                            r1
 the second surface.

   Second.   When the plane side is towards incident light   Then

 — = 0, and r' is negative, whence,

               l__l_  = (m-l).....

               v      u          r

agreeing  with the  expression found above, if the lens is the same
in each case, or r  = /.

  (79.) The next form to be considered is the meniscus. 
48     MENISCUS.---CONCAVO-CONVEX  LENS.   APPENDIX.

Prop. XXII.  To determine the form of a small pencil of light after
  refraction by a meniscus.

  When the convex side of the meniscus is turned towards inci-
dent light, the  signs of both r and  r' are negative.  The general
formula (28)  gives

      l_l=_(m_1)(l_lr).....(64),
       0      u                  \ r      r /

in which, by the nature of this lens, r' > r, or ---<; — .
                                              tr1      r

   From this relation of —  and  —  it follows, that--------
                          r'        r                  r     r*
is a positive quantity, and  therefore the sign of the right hand side
of this equation is negative.   The  equation corresponds to that for
the double convex lens (35), but the divergency destroyed by the me-

niscus is (m — 1)1--------r \ , while that by the double convex


lens was (m — 1) I __ 4. __\ .  The power of the meniscus

is the difference betioeen the powers of its two surfaces.

   (80.) When the concavity of the meniscus is turned to incident

light, r and r' are positive, and r >■ r', or  — ^ — .
                                          r      r1
   Equation (28) applies directly to this case, and

      1-1= _(ot-1)(1-1).....(65).
       0      u                  \  r      r  /

   Since___^ ,__. ,_________is a positive quantity, hence _
          r   "  r'     r'     r                              v

______is  negative.  Equations (65) and  (64)  are  identical, the
     u
surface which first received the  incident rays in the case of (64),
being now the  second surface.

  (81.)  For tho focus of parallel rays, we have, from (64),
                        1           /
             / =----—r ■  -t^~.....(66).
                      ot — 1    r — r

  (82.)  The formulas just found for the meniscus apply to the con-
cavo-convex lens, recollecting that when the convexity is turned to
incident light r > r', and the reverse, r' >  r, when the concavity
is thus turned. 
CHAP. III.
REFRACTION.
49
  For the first of these cases we have (64),

     ------= -(ot-1)(1--1\.....(64),
      v      u                  \ r      r /

which, since---< —_ , will be more expressive if written
             r      r

       !_!=(„_!) (i._I).....(67).
        v      u               \ r      r /
  The second member of this equation is positive, and by referring
to the  case of the double concave lens (art. 74), we shall find that
the convergency destroyed by the concavo-convex lens is the differ-
ence  of the effects of its two surfaces, while in the double concave
lens  it was the sum of the same effects.
   It is obvious that turning the  concave  side  of this lens to inci-
dent light does not alter the  effect of the lens, as was shown in the
case  of the meniscus.
   The virtual focal length of the concavo-convex lens for parallel
rays, when the convex  side of the lens  is turned to the incident
pencil, is

               / =  —1 •  -^—,.....(68).
                     m —  l   r— r

   (83). For two spherical surfaces of the same curvature, we have
r =  r', and (28) gives

                       I     1  =
                        0      u
   The effect  is that of a plane glass.
E 
50
IMAGES BY LENSES.         APPENDIX.
                         CHAP. IV.

          FORMATION OF IMAGES BY REFRACTION.

  (84.) The subject of the formation of images by lenses becomei
simple, by introducing the consideration of the ray which passes
through the two surfaces of the lens, at points where the tangents
are parallel.

Prop. XXIII.  All the rays which suffer no deviation by refraction,
  by a lens, pass through a single point.

                            Fig.F.
   In the figure, let RL be a ray, refracted  by the first surface of
the lens MN into LL', and finally emerging in the direction L'R,
parallel to RL. Produce LL' until it intersects the axis of the lens
in O.  Since, by hypothesis, the tangents at the points L and 11
are parallel, the radii CL and GL1 are also parallel, and the tri-
angles COL and COL' are similar.   Whence,

                 CO : CO  ;: CL' : CL, or,

                    CO = CO . — , and
                                CL

           C'C= CO — CO=CO. (— — \\ .

  Calling CL = r, CL' =  r', the thickness of the  lens = t, and
CO = u', we have
 C'C= C'V—CV=CV— VV — CV=r' — t — r, and 
CHAP. rv.
REFRACTION.
51
               r>-r-t = u'.(1_A,  or,


               u'=(/_r-r).(_J_\, ^d


                a' = r — t. -JL-.....(69).
                             r— r
  This value of CO is  made up of quantities, constant for the
same lens ; from which we infer, that all the rays which experience
no deviation in passing through a lens, would, if produced after the
first refraction, meet in a single point in the axis of the lens.
  This point is called the centre of the lens.

  (85.) The distance of the centre of a lens from the vertex  of the
first surface is found, readily, from equation (69); for, since VO =
CV — CO = r — u', we have, by taking the  value of u' from
(69) and calling r — u', x.

             x = r — u'=t___L_.....(70).
                               r'— r
  The distance from the centre of a lens  to the  vertex of its first
surface, is equal to the thickness of the lens, multiplied by the ra-
dius of that  surface, and divided by the difference  of the radii of
the two surfaces.
  In the double convex lens, r is negative and / positive, whence,
                                  rt
                                r' 4- r
   The sign of (x) VO shows that, in this case, it lies on the right-

 hand side of the vertex.   Since ------  is a  fraction, x < t, the
                               r' 4- r
 centre is therefore between the two surfaces.

   In the equiconvex lens  r' = r, and
                           _       *
                        x          _.

   The centre is midway between the vertices.  It is from this cir-
 cumstance that the point, which we have defined, derives its name.
 The same remarks apply to the double concave lens, since for that
 lens r is positive  and r' negative, whence,
                                  rt
                       x =----.----,
                                r 4- r
the same expression which we have above.

   (86.)  To use the  position of the centre  of the lens, in de-
termining the image formed by an object, we observe, that one 
52    IMAGES BY A  DOUBLE CONVEX LENS.    APPENDIX
ray  of  the pencil,  which  proceeds from every  point  of the
object to the lens, passes through this centre.  This ray is called
the principal ray or  axis of the pencil, and when the lens  is thin
may be regarded as suffering no refraction.  It docs not fall per-
pendicularly, nor nearly so, upon the surface which it meets, and,
therefore, in strictness, the refraction of an oblique pencil should
be investigated and  applied to this case. Approximate results may,
however, be obtained, by taking the  focal distance already deter-
mined for a direct pencil;  this distance being found, for the pencil
proceeding from each point of the object, we have a series of points,
the assemblage of which  constitutes  the image.  An application
of this  method is given in the following proposition.

(87.)  Prop. XXIV.  The object, of which the image by a convex lens
   is required, is a plane perpendicular to the axis of the lens.
                             Fig.G.
  Let AB represent a section of the object, MM that of a double
convex lens, PC a line drawn from any point in the object through
the centre, C, of the lens; this line may be regarded as the axis
of a pencil of rays  proceeding from P, and may, farther, be con-
sidered to suffer no  refraction.  Call a the distance DC; u, PC;
and 0 the angle DCP: we have from the triangle DCP,
                     a = u . cos 0, whence
                          1     cos 0
but from equation (38), article 63,
                 J_____l_    _L_
                  0  "" a      /

or substituting for — its value just found,
                   u
(38),
cos 0
(71).
The polar equation of a conic section referred to the focus. 
CHAP. IV.
REFRACTION.
53
  From this equation, inferences might be drawn similar to those,
found in the chapter on the formation of images by mirrors.
  (88.) When the object subtends a small angle, we may consider
its section as a circular  arc; the image will be, also, a circular arc,
since if u is constant (38), v will be so; and the arcs will, evidently,
be similar.   If the distance of the object and image, respectively,
from the centre of the  lens, be called a and  v, their magnitudes
d and d', we  shall have
                     d!    v
                    i = -a-.....<72>;
0 being assumed very small, equation (71) gives, making cos 0 = 1,
                       1      f—a
                      —  =  J—r-, or
                       v        of

                        0 = -j----:

this value of  r> substituted in (72), gives

                   d'       f
(73).
                   d     f—a
  As long as a > /, / — a is negative, and the image is real. To
show the results of this case more clearly, put equation (73) under
the form
                       d'  _       f
d        a—f
      d!_
     T
If a > 2/, a —f > f and -j- is a fraction, or the image is less
than the object.
                  d'
  When a = 2/,  —— = 1, the image is equal to the object.

  For a <; 2/, a—/</, and the image is  larger than  the
object.

  The object still  approaching the lens when a = /, — =  oo,

and no image is formed.
  Next, if a </, equation (73) gives for

                       -       f
                        d     f — a '
a positive value, and the image is now a virtual one, on the same
side of the lens with the object.  It is greater than the object until
/ — a = /, or a = o, that is, until the object touches the lens.
  Since the rays  cross at  the  centre of the lens, it is evident
that when the object and image  are on the same side of the lens,
or the image is virtual, the latter is erect; when  on different sides,
       E2 
54                MAGNIFYING POWER.         APPENDIX.
                           d'
it is inverted.  The Bign of ——, therefore, determines whether the

image will be erect or inverted with respect to the object, the posi-
tive  sign corresponding to an erect image,  the negative  to an
inverted one.
  (89.)  For the double concave lens from (57), article 76,

                    l=l + l
                     v      u     f
Whence we derive, by following the samo  process  as for the con-
vex lens,
                    1      cos 0     1      ,
                   — =------r-  —- , and
                    0       a      f
                        d     f+a
an expression which is always positive, and a fraction : hence the
image formed by a concave lens is  on the same sido of the lens
with the object erect, and less than the object.
  We have spoken only of sections of the object, image, and lens;
the remarks made in the chapter on the formation of images by
mirrors, (Chap. II., art. 37,) apply equally to this case.
  (90.) Having found an expression  for tho ratio of tho linear
magnitudes of an object and its image formed by a double convex
lens, if we would view this image at the distance of distinct vision,
(pp. 48  and 49, text,) the apparent  magnitude  will be increased,
in the ratio of the distance of the object from the eye, to the limit
of distinct vision.   Let <5' express the former distance, &  the latter,
the magnifying power of a convex  lens used as above described,
will be expressed by


               6    d           6    f — a
where / is the focal length, and a the distance of the object from
the lens. 
CHAP. V.
SPHERICAL  ABERRATION.
55
                         CHAP. V.

     SPHERICAL ABERRATION OF MIRRORS AND LENSES.

  ^I'l1!, the tGXt' P¥eS 57 and 58'the fact is stated that the rays
which fall upon a spherical mirror, at a  distance from the axis,
are not converged to the same point, with those nearer to the axis.
Ihis is illustrated m the annexed figure, in which LM, LM are the
extreme rays of a pencil diverging from L, and F' is the point on
the axis at which the reflected rays  MF', MF' meet; LM', LM'
are two rays meeting the mirror near to its vertex D, the focus of
the reflected  rays M'F and M'F being at F.

                           Fig. H.
   FF' is the aberration in length, or longitudinal aberration, of
 the reflected pencil, and if from F a perpendicular to the axis be
 drawn meeting the reflected ray MF' in /, FI will  be  half the
 aberration in breadth, or lateral aberration of the same pencil.

 (92.) Prop. XXV.  To determine the aberration of a pencil of rays
   reflected by a spherical mirror.
 First : To find the amount of the longitudinal aberration.
   With the  centre L  and radius LM describe an arc cutting the
 axis of the mirror in E. According to the usual notation LD = u,
 CD = r; to distinguish  between DF'  and DF,  call DF' = v'
 and DF = v, and let MN = y, the semi-breadth of the portion of
the mirror occupied by the pencil.
  The relation  of the segments CF' and LC to the sides F'M and
LM in the triangle LMF', gives  (as in Prop. I.),
                        F'C     LC
F'M    LM 
56   SPHERICAL ABERRATION OF MIRRORS.   APPENDIX.

   But  LM =LE= LD — ED, and  ED = ND — NE,  or,
since ND is the versed sine of the arc MD to the radius CD,

                      ND = yl;
                             2r
                        MN2
for the same reason NE = _^, or, using for LM its approxi-

mate value LD,
                         v2
                  NE = J— , whence,
                         2u

              ED=yin_i_\,!md
                     2  \r     u )


                         2  V r     u )


                  "L1""^" \r    T/J
   Taking the  reciprocal of this value of LE, and neglecting the
terms containing the powers of the sine, y, divided by the diameter

2k, after JL_ we  have,
         2u

       1 = J- = 1 \l  + *L /!_ 1Y| .
       LM    LE      it L     2u  V r    u /J

   By a similar mode of proceeding, using the  versed sines A7/
and ND,  instead of  ND and NE, we should obtain


      FM     F'H     v' L     20' V  0'      r )\

   But F'C = r — 0', and LC = u — r; and substituting the
values of F'C, F'M,  LC and LM in  the ratio  found in the be-
ginning of this article,

           EEZ^ri_jEl(l_l)l  =
             0   L    20  \ 0'     r f \

           = iiziETi4-i(i_iy|.
                m   L    2u \ r      u 1 \
  Dividing by r and performing the multiplications by the quanti-
ties outside of the vinculum, in each member  of the equation just
found, 
CHAP. V.       SPHERICAL ABERRATION.             57

  We have thus a general relation between v' and « in terms of
the radius and semi-aperture of the mirror.

  (93.) If in (75) we use for 1----1 , in the multiplier of Jl. ,
                           0      r                    2v'
its approximate value,  derived  from equation (1), art. (7), namely,

             11       1     1       Kf
            ---— — =--------, we obtain
             0      r       r     u

            J___L _1_  (1___iy =
             0'     r      2d' \ r     m /

           = E_i+|f.(±__LV,or,
              r      u      2u  \ r      u /

E_ J_ = E_JL +  sl(± + JA(JL_ !)'.
 0     r      r      tt      2\m       0  / \  r      u /

farther, by using for (-----[-  —| the value given by (1),
                   V u     v' /

                    1 4-  1= —,
                     v      u      r
and substituting this in the equation last found,

    111=  1_1 4-  JE!(i-iy\ or,
     0'     r       r     a      r   \ r     up

    i = i_i + j/_/i_iy.....(76).
     »'     r      u       r  \ r     a /

  In this equation, which gives the value of —T , corresponding

to a point of incidence distant from the vertex, we find the recip-
rocal of the approximate focal  length, obtained when the rays were

supposed  to meet the mirror near the vertex,  namely,-----

1, and a correction for aberration.   This correction contains
 u

 V2 ,  a quantity proportional to the versed sine of the semi-angle
 r
of the pencil, and therefore depending upon this  angle, or the semi-
aperture of the mirror; and also r, and u, the radius of the mirror
and distance of the radiant point.  If these  latter quantities are
constant the aberration is a function of the  semi-aperture of the
mirror   The correction  for aberration is additive,  showing  that
the  reciprocal of the focal length, for rays distant from the vertex,
is greater than the reciprocal focal length of those near the vertex,
or that the point F' is  nearer to the mirror than F. 
53        CIRCLE OF LEAST ABERRATION.      APPENDIX.
(94.) For parallel rays — =  o, and from equation (76),
                       u

              —7 =  — 4- —— , whence,
               0       r      rl
                             2r2  + y* '
 performing the division, and neglecting the powers of y higher
 than the second,

                  v' =  _L -  J!L.....(77).
                         2     4r        ^  ;

 The correction for aberration is, therefore, — — , or---?—, or is
                                           4r         8/
 subtractive, and equal to the square of the semi-aperture of the
 mirror divided by eight times  the principal focal length.
   (95.) Second : To find the lateral aberration  of the extreme ray.
   The value of FI, which measures  the lateral aberration of the
 extreme ray, may be obtained as follows.  In the similar triangles
 F'FI and F'MN,

            ^L=^L,a,dFI=FF.M..
           FF'     F'N                    F'N
                           MN
 To approximate to the ratio —;— , F'D may be taken instead  of

 F'N, and the value of the aberration is
                               MN
                  FI= FF'- —.....(78),
                               F'D       K  '
 in which all the terms are known when FF' has been determined.
   (96.) We propose in  this article to determine the position and
 magnitude of the physical focus of a mirror, or of the circle which
 includes all the rays of a reflected pencil, when they are spread
 over the least space.

 Prop. XXVI.  To determine the position and magnitude of the cir-
  cle of  least  aberration, in a pencil of rays reflected by a concave
  mirror.
  In the figure let LM be the extreme ray of a pencil, incident
 upon the mirror, MF' the corresponding reflected ray, F the focus
 of rays very near  the  vertex.  Farther,  let LP  be any incident
 ray,  in the  lower portion  of the pencil; PR the corresponding re-
 flected ray intersecting MF' produced in c : draw cb perpendicular
 to the axis, from the point c. If we suppose the arc DP very small,
the reflected ray PR will coincide very nearly  with  the axis, and
the distance cb will be indefinitely small; as the arc DP increases, 
CHAP. V.        SPHERICAL ABERRATION.              59
the reflected rays being removed farther from the axis, cb, at first,
increases ; it afterwards diminishes as the point of intersection of
PR with the axis approaches to F', and when DP = DM, PR
coincides with PF', and cb vanishes.  Between the  case, then, in
which DP is very small and DP = DM, there is  a position of
the incident  ray LP, for which the reflected ray PR  gives a maxi-
mum value for cb  When cb is a maximum, all the rays of the

                            Fig-. I.
S
reflected pencil pass through the circle  of which that line is the
radius, which is, thus, the physical  focus  of the  mirror.   The
question resolves itself into determining the values  of F'b and cb
when the latter is a maximum.

  Call MN, y; PT,y'; DF,v;  DF', v'; FF', the longitudinal
aberration, a; F'b =  x;  be = z.   Since (art. 93)  the aberration
of a ray is proportional to the square  of the distance of its point
of incidence, from the axis of the mirror,
FR : FF':: PT*	: MN2 : : y'2 : y<	, or
FF'—FR : FF'	::y2-.yf2:y2,	whence
F'R =	af~y'\ y2	
from the similar triangles	F'bc and F'NM,	
6c : bF':	: MN: F'N, ox	
bc =	IF.™. F'N	
And from the similar triangles Rbc and RTP
                   Rb:bc::RT: TP, or

                       Rb = bc. ~;
                                 TP
substituting for be in this expression the value found above, 
60        CIRCLE OF LEAST ABERRATION.     APPENDIX.

                              MN  RT
and by the notation,
Rb = bF.
          F'N  TP

 Rb=x   y   RT
                            FN   y"
  If we approximate, by considering RT and F'N to be equal,

                    Rb =a x . 2-r-, and

       FR = Rb + bF'= x . 4- + x = x . SLEfcl.

Equating this value of FR with the one before found,
y + y' _	= a.	-—-z— , whence	
y		y2	
	V	y2 — y'2	
x = a			, or
	y*	y + tf	
               x = a.^.(y-y')..'...(19).

As we have supposed the ray LM to remain fixed, and LP to take
different positions, and have found,

                    z — oc — or . ^^ ,

be (or z) will be a maximum when bF' (or x) is a maximum; but
from (79), a; is a  maximum,  since a and y are constant, when
y> (y — y') is a maximum, or when
                   t/(y-y') = y'2,ot
V     2
In that case, from (79),
            * = -^- =4-.....(80), and,
                  Ay2       4
                _     MN  _  _o_   MN
              z ~ x '  p>N      4- •  pN  •

  If a perpendicular, FI, be drawn from the focus F, of rays in-
cident near the vertex, to the axis, meeting the extreme ray MF'
in I, by article (95),
                     MN      FI
                    ~F'N ~'   FF''

whence the value of z, or __ . ——--, becomes
                        4    F'N
                         FT
                         4 
CHAP. V.         SPHERICAL ABERRATION.
61
  From these values, (80) and (81), of x and z, it appears, that the
distance of the  circle of least  aberration, from the focus of rays
near the vertex, is three fourths of the longitudinal aberration of the
extreme ray, and that the radius of the -same circle is one  fourth
of the lateral aberration of the extreme ray.

               Spherical Aberration of Lenses.

  (97.) In this investigation we begin by determining the aberra-
tion produced by a single surface.  We shall assume  the light to
pass from a rarer into a denser medium, as when it enters  a  lens
through its first surface.

Prop. XXVII.  To determine the aberration produced  by a  single
  refracting surface.
                           Fig. K.
-R.         V        C
  Let the ray RL fall upon the spherical surface LV &t any point
L, and be refracted into the direction LM.  Continue LM until it
intersects the axis  of  the surface, at  F.  Draw the radius CL.
Call to  the ratio of the sine of incidence to that of refraction, in
the passage of the  ray from the rarer to the denser medium, the
sine of refraction being unity; then, by proceeding as in article
(50), Chap. III., we find
                      RC  __     FC_
                     [RL  ~   ' FL '

  From the  centres R and F with the radii RL and FL, respec-
tively,  describe the arcs LS and LT cutting the axis in S and T;
SV will be  the difference between RV and RL, and TV that
between FV and FL.  If the perpendicular LN'be let fall, from
L, upon the axis RV, SV = NV- NS,  and TV = NV- NT.
As in the notation of Chap. Ill, let R V = u, FV = a ,CV= r,
and call LN,  y.  NS is the versed sine of the arc LS, Nl ot L,i,
and JVFof LV; and if for the chord of each of those arcs we
substitute, as an approximate value, the sine, we have
               F
�73�135�336662 
62       SINGLE SPHERICAL SURFACE.     APPENDIX.


                  ns = JLL,
                       2RL

                 NT=JLL,
                       2FL

                 NV= JLL ,
                       2C V

or substituting for RL and FL, the approximate values R V and
FV,

  NS= JLL,  NT=JLL ,  NV = JL , whence,
         2«          2u'          2r

      SV = NV— NS= ll (1____I\ , and
                       2  V r     u )

        TV=NV—NT = JL(1___1\ .
                         2  \ r    u' /

  From these values of SV and  TV we obtain,

    RL — RV—SV = u — JLL(A____I\, and
                         2  V r    u )

     FL = FV— TV=u' — JLL(2___1\ .
                          2  V r    m'/

 Taking the reciprocals of RL  and FL, that is dividing unity by
the values just found for those lines, and neglecting the terms which
involve the quotients of the powers of y- by those of u, after the

first term, / y  \ , we have



      _i=iri4-i(i-iyyi|,
       RL     uLa\r    a/2 J


      i_ = iri4i(i-i\-^l.
       FL     u'  L     a' V r     a' / 2 J

 From the figure we have RC = RV—CV=u — r, and
FC = FV— CV = u'— r, and the equation for the relation of
RC, RL, FC and FL, becomes
 La\r"~u/2j

zi.h + ±(±-±\yl-]
<i   L     u  \ r    a / 2 J
.  Performing the divisions by u and u', indicated by the terms of
the equation, and dividing both sides of the equation by r,



     \r    u~)l     u \~r     u~)~2~l = 
CHAP. V.       SPHERICAL ABERRATION.           63


    = -(-r-^)[1 + v(f-4)fl-
performing the multiplications required by the expressions,

         i___I 4- ifl-lV-lf =
         r     u     u \ r     u /  2

   = m (1  _ _1)  + ^  (1 _ IVl! , whence,


          _ =----r- (m—1)----1_  y



   [7(T-v)"-4-(7--4-n.....(82)-

  In this value of---, the first two terms  correspond to the
               u'
value  found, (26) art. 50, on the supposition that the pencil is
small, and the third contains the correction for the aberration, pro-
duced by the single spherical surface.
  (98.) The expression just found, may be simplified by substi-
tuting in the terms of the second member for «', its approximate
value from equation (26),  art. 50.  From that equation

          —- =----(- (m — 1) — , whence,
          u     u            r

         1=1(1 + (m-1) —) , and
         u     to  \ u            r /

   !_l = l_i(!+o»-i)!yor,
    r    «'     r     to  \ u           r /

         1__1=  Wi_l\,and
         r     a'     to \ r     a /

         (---Y=-(--±Y-
         \ r     a'/    m2 \ r     u /
  In order to reduce, with greater convenience, the coefficient of

iL. in (82) to its simplest form,  call that coefficient k, then sub-

stituting in it the approximate values  of —— and /-------r j ,

just obtained, we have,

      *=(!+(«- Dl). 1(1-I)2
          \u          r /  m- \r    u /


                 u \r    u / 
64          CASES OF NO ABERRATION.      APPENDIX.

  *= (1 _ J-y*. [±(1+(-»-!, 1)_ El,
       \r     «/   L«J\u           r I    u\
      t_/j     i\2  fl-m2     fLzill,
          V r     a/La         rjm2

      fc = (l_lVYl-^-±i\   m~1
           \r     a/\r      u   /    m3  '
whence (82) becomes
             to  __   1     to—1     to — 1
             »'  ~~  "a"      r        ot2

     •(f-^-d-^'-f.....<*
  (99.) When the surface is convex, r is negative, and (83) takes
the form,
             ml     ot — 1     m — 1
             a'     u       r        »na

    ■(f+^-Cr+£-)'•£.....«">
  And for converging rays, w being negative,
            ot_     1    m — 1     to — 1
            u'        wr        TO3
       (1_!!L+I).(1_1)3.^.....(85).
       \r       u  /  \r     u/   2
  Tho term in (85) which contains the correction for aberration, will
vanish if either of the factors composing it  should be equal to zero.
  First, let
 1     TO4-1       .i   OT 4- 1     1
---------JZ__= o, then —EE—  — — ,
or,
                  1 : r :: to 4- 1 : m.
  There is, therefore, no aberration  for converging rays, falling
upon a convex spherical surface, when the distance of the radiant
point is a fourth proportional to 1, r, and m 4- 1.  From whence
the result on page 56, of the text, is easily deduced.
  Next, let
               ________=  o, and u = r,
                r     «
or the incident rays converge to the centre of the spherical surface.

  (100.) In art. 42, it was remarked that making m = — 1 in
the formula for refraction, the  cases would  represent the cor-
responding ones in reflexion.  Making to = — 1 in (83) we have,
A 
CHAP. V.        SPHERICAL ABERRATION.              65

      -l=i__-!_-l/l_!\a  li  or
         a      a      r     r  \r      u / '  2  '   '

           1= —-1  4-  l!(l_l\3
            a'      r      u      r  \r     m / '
a result which agrees with equation (76), art. 93.

(101.) Prop. XXVIII. To determine the aberration in a pencil of
  rays, after refraction by a spherical lens.

                           Fig. L.
  R being the radiant point of a pencil of rays  falling upon the
lens L VV'L', let RL be the extreme ray of the pencil, and R' the
virtual focus of the extreme rays, after refraction by the first sur-
face  of the lens.  If now we suppose a pencil to proceed from R',
considered as in the denser medium, the extreme ray of this pencil,
RL', will be refracted into the  direction, L'M, which, if continued
backward to F, will give the virtual focus of the extreme rays.
  As before, represent R V by u, R' V by u', and CL by r; farther,
let R V = 0', FV = 0, CL' = r', and VV = t.
  By the preceding proposition (equation 83,) we have
              to      1   ,ot — 1   ,   to — 1
+ ^i +
Yl_"±iV  (l_i\2.
 \r       a   /   \r     u /
(83).
  The case of the second  surface will correspond  to that of the
first, if we consider F the  radiant point, and R! the virtual focus;
o must be written in (83), for u,  v' for u', and /for r; we then
obtain
               TO       1    .  TO —1      TO--1
              —r =-----r ---;----r ——r,—
/ l      to + 1\  /_i_     J_\2  yj_
VET"       ~ )  V r    ' v  / '  2
or,
F2 
66    SPHERICAL ABERRATION OF LENSES.   APPENDIX.

               1_ _ m_    OT —1   ot — 1
               0     v'    ~p~     m2


     i-v-H1)-^-^.....™>

but 0' = u' 4- t, whence J!L = _ m   , performing the division
                      v'     u' + t
and neglecting the powers of < above the first,
                   to __  OT     OTt
                   0' ~"  «'  ~~ IT2 '

   We  may farther approximate to this value of — , by substi-
                                           0'
tuting  for —__ , in the second member of the equation,  its ap-
          u2
proximate value, from (26), art 50, namely,

          1= 1(1+JILZ1V; whence,
          u'2     m2 \  u        r  J

            OT  __  TO     '/*   1  m--1\3-
            0'      «'    to V «      r  /

in which the  value  of —;-, from (83), being written,


     Jl =  l 4-mE=l_I(l + !^zIV +
      0'      m        r     to\u       r/.
         ot —1 /J___m-t-l\  /J_    1_\a _y»_
           OT2\r      a  /  V r     a/   2

   By substituting for — , in equation (86), its value just  found,
                   0'
we have

1=  1 4-(m-l)(!_-I)_i(! 4- ^V +
 0     m           Vr     r  /   m\u       r   /
  to —1  T/J___ot 4- 1\ /_1_  _ I\2_ /J___m + l\
    m2   LV r      u  )\r  " u)    Xr1      v   )


             (v-vYlf.....<8"'
  We see, in  this formula, first, the two terms which denote the
reciprocal of the focal distance of an indefinitely small pencil j
second,  the  correction for thickness; and, in  tho last term, the
correction for aberration.
  (102.) The general formula, (87), becomes less complex, and
gives results of considerable practical importance, when applied to
the  case of parallel rays. 
CHAP. V.      SPHERICAL ABERRATION.         67

Prop. XXIX. The incident rays being parallel, to determine the
 aberration of the pencil after refraction by a spherical lens.

 In this case,--- = o, and (87) becomes,
           u

   1 = (to-1) (1  _ 1\ _ _L . ("»- !)2 4.
    0          V r    r' /    ot     r2



 The correction for thickness, contained in the second term, has
already been separately considered, articles 55, 67, &c.; we  may
therefore leave it out of the question here, making in (88) t  = o.
Farther, to approximate to the value of v, we may substitute for

.__in the second member of the same equation, the approximate
v
value __ , or (m — 1) (------—- j , obtained by making —
    /         \  r    f /               u
= o in (28), art. 53.
 We have, then, from (88),
      l  _ j_   ot —l r l___/J___ot 4-1\
      »    7"+TO2Lr3    Kr1      f )


            •(4-7-n-f-.
or, taking the value of v, dividing by the denominator thus found,
and neglecting the powers  off higher than the second,
          .   to —  i r i    /1   «»+1\
      B=/-—L7-S--V-——)


              \-~T) J  "2"'
the aberration in length, therefore, is represented by
             m — l r 1    / 1   to + 1\
       a = --m-2-[-^-Y7---—)


        ■^-m-^.....<89>-
 (103.) To apply the formula just obtained, to a double convex
lens r and / (art. 62,) must be made negative, whence

      a  = _^ir_-i-(i + ^-1)
             to3 L   r3   \r'     f /
(wn--^- 
68        ABERRATION OF GLASS LENSES.     APPENDIX.


          a = !EE^ri+/I_ +  !M!\
                 ma  L ra ^ V  r'       /   /


           •(W)']'^.....«■

  This value of the aberration having the positive sign, while the
approximate focal length has the negative sign, its effect on tho
focal length, for rays not near the vertex, is subtractive ; showing
that the  focus of such rays  is nearer the lens, than the focus of
rays incident near the vertex.

  (104.)  For an equi-convex, glass lens, r = r, m =----, and /

= r, disregarding the sign, since / has already been made nega-
tive in (90); and from (90),

          2rl  ,   / 1   ,   5\   4l   y2r3
    a = — I----U  I-----U  ---1 . — I .  1---., or,
          9  Lr3  T  \r   T  2r/  r* \     2

           0=l[1 +  (l + 4).4].A


                    a  =  1  ll
                    0  ~  3  '  r

  If we  suppose the beam of light to occupy the whole aperture

of the lens, y becomes the semi-breadth, and y3 = — •  2r


nearly, or y3 = rt, and t =  ^—', writmg t for -^— in the value
                         r               r
of a, just found,

                        a = lit,

the result stated in paragraph 3, page 53, of the text.
(105.) If to — .1 , and r : r':: 2 : 5, or r' =	—r, we have 2
from (36),	
/ V r ^ r'f 10	— , and r
, 10 / =----r. J 7	
  Substituting these values of m, r',  and / in (90), recollecting
that / has been already made negative in that equation, and that
now its value is to be placed there without regard  to the sign, it
gives,

          . = iri+  (i 4-i-i)
               9  Lr'      \5r      2    lOr / 
CHAP. V.        SPHERICAL ABERRATION.            69

               (2_     7  y~\   103r3   y*_
               5r     10r) J '  T3" '  2/  '


      9L^V5^20^\5^10/j73    /
                      _   7    y2
                       ~~~e~'T"
  This is the case of an unequally convex lens, in which the more
convex side is turned to incident light.

  (106.)  In  the plano-convex lens,  if the plane  side be turned

towards parallel rays, — = o, and / = 2r'; if the material be
                    r
           3
glass, m =  ___, and from (90) we obtain







                 a = 4 . 5 J^I — 4 . 5 r.
                          /
  The result given in paragraph 1, page 53, of the text

  In the same lens, with the convex side turned to parallel rays,

l=o and  f = 2r, whence from (90), r and / having already
 r1
been made negative,


           °=l[' + 44]8-f-

                0 = 1.17JL_ =  1.17.*.
                          f
  The result stated in paragraph 2, page 53, of the text

(107.) Prop. XXX. To determine the ratio of the radii of the sur-
  faces of a  double convex lens, which shall produce the least aber-
  ration, with a given focal length and aperture.

  To solve this question we must determine the ratio of r and r,
when a is a  minimum, / and y being constant.

  Differentiating the value  of a given in (90), considering rand
r' as variable, and disregarding the constant multipliers, we obtain,
after changing all the signs, 
70        LENS OF LEAST ABERRATION.      APPENDIX.







            + (W)"-£.....(9,)-

  But from equation (36), art. 62, we have


           j-=_(m_l)(l+Ir)1orI


  —  + —~ =-------r • —f- , whence, by differentiating,


                  dr   ,   rfr'
                 -----r- ---- = ", or,
                  r3  ^  r>3

                    dr'        dr
                              dr1
  Substituting in (91) this value of---, and dividing by dr
                              r'2


             dr " ~r~*    \V ^   J~)



        " \T + T/ "^ ~ w + T/ "H *
which, by the question, is equal to zero.  Multiplying by r3 we
obtain

       ^•(E +  ^MW)



            -(-^+t)i=°.....(92)-
  From equation (36), disregarding the sign off,

              1 -     !     _i    J_
               r     to — 1  /    r1

  Substituting this value  in (92), and arranging the terms

  _3_        6             3       _2___2(ot 4.2)

  r'2    .(to —1)// + (m —l)2/2   r'2      fr>

         2(to +  1)____1_     _2_    1

            /'       r'3     /r'   f2   -0'Or,

           _ (   6    4- 2to 4- 6\ . 1 4.
              Vot —1 ^        /  //  ^ 
 CHAP. V.        SPHERICAL ABERRATION.               71

 and by transposition and multiplication,

                (—*-  4  2to  + 6\ .  1 =
                \ot —1             /    r'

         =  (j~W-^-")-j-.....(93).

   If the lens is of glass, or to — 1
                                  2  '
                 21       6        ,21
                —r == -p-7  or r' — _ . f.
                 r       f              6   J'
 but  from  (36)

   1=  1___I_  2      6     1       12     i      A
   r"   /      r> ~ ~f     -n'T^T'T'

                         r —  1   f.
                               12  '*'*
   Comparing together the values obtained for r and r\
                        r : r' :: 1  :  6.
   This lens is known to opticians as the crossed lens.  With the

 more convex  side turned to parallel rays the aberration is 11. -fl
                                                      14   / '
 which is less  than that  for the plano-convex lens with the  convex
 side turned to parallel rays.
   (108.) It would carry us beyond the limits of this Appendix, to
 go into the investigation of the aberration of combined lenses.
  Before leaving this subject we purpose  to show a method by
which the  surfaces which refract rays accurately to a point, may
be determined.

Prop. XXXI.  To determine the curvature of the surface of a me-
  dium, so that rays passing into it, from a rarer medium, may be
  refracted to a point.
                           Fis: M.
  As we have found a concave surface to give only a virtual focus,
we proceed, at once, to examine the case in which the surface of
the denser medium is convex.   Let R be the radiant point, RL a
ray meeting the surface at L and refracted to F: let L' be a point
farther from the vertex  V than L, RL being the incident  and 
72        LENSES WITHOUT ABERRATION.      APPENOIX
L'F the refracted ray for tins  ]K)inL   Draw the perpendiculars
L'R! and LS upon the incident and refracted rays RI1 and L'F,
respectively.   LR' will  be nearly equal  to  the increment of the
incident ray,  and LS to the  decrement of the  refracted ray, in
passing from the point L to L'. Call RL, u', and LF, —  ?>'. Then
if L' be supposed very near to L, LR = du', and LS = dv'.
  In the triangle L'LR', _J_  = Cos. RLL' =  sin. incidence,
                         LL
and  in  L'LS,
whence,
LL'          1
                LS       cos. SLL'      sin. refraction
                  LR'__sin. incidence
                  LS       sin. refraction
                         du'
                       ----= to, or,
                         dv'
                  du' — mdv' — o.....(94),
the differential equation of the curve which, by a revolution about
the axis RF, will produce  the surface required.  To integrate, let
RV=u, and FV = — 0, the complete integral of (94) will be
               u' — u = ot (0' — 0).....(95).
  (109.)  If the incident rays be parallel, u' — u =  FM, fig. N.
                           Ffc-. N.
                  V M
If we put FM =4 — a;, (95) becomes
             A — x = to (v — 0), whence,
                      ,     A — x
              0 — 0' _ -1 4-  ±.....(96).
                         TO      TO
  The equation for the distance of any point in an ellipse from
the farther focus is, (Young's Analyt. Geom. art. 47, p. 72),
                        v = A 4- ex,
in which e < 1; with this (96) agrees in form, and will be identical if
            1            c      ,  ,    A      .
            .—  = e  = .—  , and 0_____=  A.
            m          A              m 
 CHAP. V.       LENSES WITHOUT ABERRATION.         73

   Substituting for  to  in the second of these  equations, its value
 from the first,
                  v' — c = 4, or v' = A 4- c.

   We  find, then,  that an ellipsoid of which  the semi-transverse
 axis is to  the excentricity as the index of refraction is to unity

 (— =  to 1 will refract parallel rays, accurately,  to the farther

 focus.

   If a lens be formed,  of which  the first surface is a portion of
 the ellipsoid just determined, the second surface should be (art. 99.)
 a portion of a sphere,  having the farther focus of the ellipsoid as
 its centre (fig. 38, text).

   (110.) Equation (95) may be applied to the case in which the inci-
 dent pencil  passes from a denser to a rarer medium, through a con.
 cave surface.   Then FL, FL', fig. M, would represent the incident
 rays, and LR, L'R the refracted rays,  and the  ratio of the sine of
 incidence to  the sine  of refraction  would be  represented by the

 fraction ---; substituting  this for to in (95)  we have
          m

               a' — a  —  1 («' — v).....(97).
                           OT
   For the case of parallel rays, (fig. 40., p.  55, text,) by proceed-
 ing as in the last article, making u' — u = 4  — x,
                   v' — v = to (4 — x),  and   -.
                     v = v' — to4 4- mx;
 an equation of the  same  form with that before obtained, and re-
 presenting the distance of a point in a  conic section from the
 farther focus;  in it

               to ^= e  =  _£_ , and v' — ot4  = 4.
                           4
   Since to >- 1,  e > 1, and the equation belongs to a hyperbola,
 (Young's  Analyt. Geom.,  article 79,  p.  104,) the  equation of
 which is
                   v' "= 4 4- to4 =  4 4- c.

   If, then, we  form a  lens with  the first surface plane, and the
 second that of a hyperboloid of which the excentricity is to the semi-
 transverse as the index of refraction, of the material of the lens,
 is to unity,  parallel rays, incident perpendicularly upon the first
 surface  of the lens, will  be refracted to the farther focus of the
hyperboloid  which forms the second  surface  (fig. 40, text).
  (111.) The  cases  in  which  the aberration of converging  rays
upon a spherical surface is zero, (art. 99,) are contained in (95); it
is unnecessary, however, to discuss it farther.
          G 
 74         CAUSTIC FOR PARALLEL RAYS.      APPENDIX.

   (112.)  The  forms of mirrors without aberration may also be
 inferred  from the equations just  discussed.  The convex mirror
 will be given by making m = — 1 in (94), whence,
                 du' 4- dv = o, and integrating
                  u'— t/(-») = C.....(98).
   By this property we recognize the hyperbola, the distances u' and
 - 0' being those  of the point, from the two  loci.
   For a concave mirror, u' and 0' have the  same sign, in equation
 (94), and
                       du' 4- dv'= 0, or,
                    u' 4-  0' = C.....(99).
   The mirror is an ellipsoid, the radiant point  coinciding with one
 focus, and the rays being collected at the opposite focus.
   If one focus remove to an infinite distance, the ellipsoid becomes
 a paraboloid, into the focus of which the  rays which have  been
 supposed parallel are collected.

                     Caustics by Reflexion.

   (113.) It is not intended to enter fully into this  subject  in  rcla-
 tion to both reflexion and refraction, but to confine the discussion
 to examples of the  caustics produced by reflexion.
   The formula for  the oblique pencil, art. 29, &c, gives, in certain
 cases, an elegant and easy method of determining the form  of a
 section  of the  caustic surface,  produced  by reflexion  from a
 spherical mirror.

 Prop. XXXII. To determine the form of the caustic produced by
   the reflexion of  a  pencil of rays from a spherical mirror, when
   the rays are parallel; and  also when the radiant point is at a
   diameter's distance from the vertex of the mirror.
   First.  When the radiant point is infinitely distant, or  the  rays
 parallel.
   LDM representing a section of the mirror, let RL be a ray  inci-
 dent upon it and reflected into LB; then, the focus of a small
 pencil meeting the mirror near to L will be'the point F found from
 the value  of 0 in the equation which concludes art. 30, namely,
                             r        .
                       0  =  __. cos. <p.
                             2
  To construct this value  of v; let fall from C, CP perpendicular
to the reflected ray LB, then
            LP = LC . cos. <p = r.  cos. ^, whence,
                               LP
                                2 
CHAP. V.
CAUSTICS  BY REFLEXION.
75
  Bisect LC in E, and LP in F, then LFE is a right angle; the
point F, of the caustic,  is,  therefore, in  the  circumference of a

circle of which LE = — LC is the diameter.  This being true

of each point in the curve, the caustic curve is an epicycloid, the

                            Fig. O.
^        \   )   \
\\
■W-
r-
!\
diameter of the generating circle of which is equal to the radius of
the base; this latter being half the radius of the mirror. This curve
and the circle LDM revolving about DC, as an axis, would gene-
rate, respectively, the surface of the caustic and that of the mirror.
  Second.  Let  the  radiant point be at the extremity of the di-
ameter of the mirror.
                            Fig.V.
  The ray RL is reflected into LB.  To find the position of F
upon LB, we recur to equation (11), art. M. 
76  GENERAL EQUATION  OF CAUSTIC CURVE.  APPENDIX
                  -L +  .L=_L_.
                   a       0      r .  cos. <p
  Drawing CP perpendicular to RL,
                   LP = r . cos. <p, whence,

           1  4- 1  — _JL —   4   —  1
            «      0  "  Li'  ~  flZ      ~u~
                        1       3
                       — =  — ,  or,
                        0       M
                      „_ _a_    _iB_
                            3        3
                      pi
  If C£ be made = ----  , the perpendicular from E to LB will

intersect it in the focus F.   The locus of theso  foci  is, therefore,
an epicycloid, of which  the diameter  of the  generating circle is
to the radius of the base as two to one. This curve is the cardioid.
  (114.) In  considering  the subject in a  more general point of
view, we may determine the equation  of the  curve of section of
the caustic, the position of the radiant point and section of the
mirror being given.

PRor. XXXIII. To determine the equation of the curve which is the
  section  of a caustic formed by  a curved mirror.  The section
  being made  by a plane passing through the axis of the mirror.
  We  refer the curve to polar co-ordinates, the radiant point being
the pole.
                            Fig. Q.
  Let B and B' be two points very near each other upon the curve
which is a section of the mirror; let C be the centre of the oscu-
lating circle to  the  curve at either of these points, so that the
portion of the circle  and curve nearly coincide between B and B'.
RB, RB' representing two incident rays, BF, B'F are the re- 
CHAP. V.        CAUSTICS BY REFLEXION.             77

fleeted rays.  Call RB = u, BF = v, the angle RBC = ^, RBC
= ip', the perpendicular RP, upon the tangent BP, = p,  the ra-
dius CB = CB' = r.  Join FR and let fall the perpendicular RQ,
upon BF.   F being a point in the caustic,  FR is the  radius
vector of that point and RQ a perpendicular  upon the tangent;
call RF, «',  and RQ, p'. An equation between u' and p' will be
that of the caustic curve.

  In the acute angled triangle RFB, since the segment BQ =
RB . cos. RBQ,

        u'2 = u2 4- 03 — 2uv . cos. 2<p____. (100);
and in the right angled triangle RBQ
                p'=u. sin. 2ip.....(101).

  To eliminate cos. 2ip and sin. 2<p, we proceed as follows.  Since
RP and  CB are perpendicular to BP, they are parallel, and the
angle PRB = RBC = <p, and

                      cos. <p = -I— .
                                u

  But by trigonometry,
                  cos. 2<p = 2 cos.2 <p — 1,

and by substituting for cos. <p the value just given,
                           2p


Wo have also, by trigonometry,
cos. 20 = -~£-----1.
           u'~
,2<p = 'fl — cos.2 2ip, or,
-*=/£
4p"
                          u  v      a-
  Substituting these  values of cos. 2<p and sin 2<p in (100), and
(101), respectively, we obtain
          u>2 _ U2 + „2 _ 2^0 (?L---1\ , and
           /=*/>-■£.....(Ke!)-
The value of a'2 may be written under the more simple form,
          u.2 = {u + v)2-^.....(103).
     G2 
78              LOGARITHMIC SPIRAL.         APPENDIX.

  The relation between « and 0 given by equation (11), art. 29, or

                   ! 4-  i=     2
                                     COS. (p
may be applied to this case by taking r to represent the radius of
the osculating circle, which is,
                           __  udu
                                dp
  Substituting this value for r,
     1,1           2              2dp
    —  4- — =------------=-----------,  or, since
     a       v      udu             udu . cos. <p
                          cos. <p
dp
cos. 0 =: J_
— 4-   — =  —— 7  whence,
 u       0      pdu
  1      2udp — pdu      ,
  0          pudu
 _      pudu
(104).
                     2udp — pdu

  If this value  of 0  be substituted in equation (103), we shall
obtain a new equation, which, in conjunction with (102), will give
the relation of u' and  p' in terms of u, p, du, and dp. The relation
of the last four quantities mentioned will be given by the equation
of the reflecting curve  and  by  its differential; eliminating these
quantities, there will  result a single equation between u' and p',
the equation of the caustic curve.
  (115.) To give an example of  this method of proceeding, let the
reflecting curve be any portion of a logarithmic spiral, of which
the equation is,
                           p = mu.

  The general value  of 0 (104),  is first to be applied to this par-
ticular case.
  Differentiating the  equation of the curve,
               dp =  mdu, whence (104) becomes
              pudu              pu           pu
       2mudu — pdu      2mu — p      2p — p
This value of 0 substituted in equation (103), gives

         m'9 — \u2 _  4p2u —  Aun- _ 4p2,  or,
                          v
        u'2 = in2 — Am"u2 = 4m2 (1 — m2),  and 
CHAP. VI.          DOUBLE REFRACTION.               79

                    u' = 2m  V 1 —OT2.
  Also, from (102),

        p' = 2mu JT-™£ = 2mu yrr^r

or, since we have just found
                    2« Vl — ot2 = u'
                        p' = mu'.
  The section of the caustic surface  is, therefore, a logarithmic
spiral differing only in position  from the reflecting curve.
                        CHAP. VI.

ON THE DOUBLE REFRACTION AND POLARIZATION OF LIGHT.

  (116.) Although it does not enter into the design of this Appen-
dix to show the method of deducing, from theoretical considera-
tions, any of the general laws of Optics, I have  thought that it
may  assist the student  to give the formulas to which these consid-
erations lead, or which  have been deduced  from experiment, in
certain particular cases, discussed in the text.  The formula, or
general law, once remembered, the details of the  phenomena flow
naturally from it and the  memory is not tasked to recollect indi-
vidual results.

                Double Refraction of Light.

  (117.) The formula which represents the  law  of extraordinary
refraction in doubly refracting crystals, becomes, when  the inci-
dent  ray is in a plane passing through the axis of the crystals,
       m'2 = m2 — (to2 — to'2) . sin. 2 <p.....(105),
in which ml is the index of refraction of the  extraordinary ray, to
that of the ordinary ray, and <p the inclination to the axis.  In the
spheroids constructed in the text (figs. 77. and 79.), to  give the
index of refraction of the extraordinary ray, if the axis which co-
incides with that of the rhomb be called b, and that perpendicular to

the same axis a, then by the construction a — — , and 6 = — ,
                                           m'           m
whence (105) becomes
              ,o     1     /  1     1 \   •   o
            m 2 =------I---------1 . sin.  j<a .
                    b~     \b2     a2)

  As long as m > m', or —. > —, that is a > b,— I----------|
                        b    a                \b2     a2 / 
80     INCIDENCE ON A SECOND CRYSTAL.     APPENDIX.

(fig. 77), will be negative, whence the  term crystals with a nega-
tive axis  which  apphes to this class.  When  a < b (fig. 79),

— <- — or ot' > m, and__(-----.___\ becomes additive, and
 o    a                    \b2    a3 /
the crystals are said to be crystals with a positive axis of double
refraction.

  (118.)  In the plane of principal section the tangents of the angles
of extraordinary and of ordinary refraction are in a constant ratio
to each other.   In the plane perpendicular to this,  the law of the
sines applies equally to the  extraordinary and to the ordinary ray,
but the value of the constant quantity is different for the two rays.
These are the only two cases, in which the extraordinarily refracted
ray is contained in the plane of incidence.             '"*-  < ,

   (119.)  When light which has been polarized by double refraction,
in the plane of principal section of a  crystal  Iceland spar (figs.
84. and  85., text),  passes  through a second crystal, the relative
brightness of each  image,  supposing  that no hght is lost by re-
flexion or absorption, may be expressed by the following formula;
in which Oo, Ee, Oe, and Eo represent the images formed as de-
scribed on page 140  of the text, a is the angle which the plane of
principal section  of the second rhomb  makes with the same plane
in tho first, and 4 is the brightness of the incident ray.

            Oo = ~ A. cos.3 a — Ee.....(106).


            Oe — —  A. sin.2 a — Eo.....(107).

   The sum of the brightness of the four images,

     Oo  4-  Ee 4- Oe  4- Eo = A (cos.2 a  4- sin.3 o) = A.

   From  the foregoing formulae (106. and 107.) we may trace the
changes of brightness in the several images, as described in pages
146, 141, of the text (fig. 86.)

   When the principal sections are parallel, a ±= 0, cos. a = 1, and
sin. a = 0, therefore

          Oo — Ee= 14          Oe — Eo — 0.
                       2

   By turning the lower crystal, a assumes a finite value and the
images Oe, Eo appear.  As a increases, sin. a increases and cos. a
diminishes; Oe and Eo, therefore, increase in brightness, and Oo,
Ee decrease.  When a — 45°, cos. a  = sin. a, and the four im-
ages are  equally bright.   The  angle a increasing farther, Oo and
Ee become  more and more faint, and disappear when a = 90°; at 
f

    CHAP. VI.        FOLARIZATION OF LIGHT.             81

    this angle Oe — Eo = — A.   The rotation of the lower crystal
                           2
    being continued beyond 90°, cos. a  takes  the negative sign and
    increases  negatively,  while  sin. a again diminishes ; when a =
    180°, cos. a = — 1, sin. a = 0, and Oe, Eo again disappear.  At
    this angle the two images Oo, Ee coalesce,  the  two extraordinary
    refractions taking place in opposite directions.

                 Polarization of Light by Reflexion.

      (120.)  When hght  has been polarized by reflexion from a Sur-
    face, upon which  it falls at  the maximum polarizing angle,  the
    following empyrical formula, determined by Malus, will represent
    the intensity  of  the  light reflected from  another surface, upon
    which the pencil is incident at the polarizing angle: (fig. 87, page
    143, text.)
                      T*= 4. cos.2 a.....(108),

    in which I is the intensity of the reflected light, A that of the inci-
    dent hght, and a the angle between the plane of mcidence and that
    of the second reflexion, or  the azimuth of the plane of the second re-
    flexion.  When a = 0, or 180°, J is  a  maximum, and when a =
    90°, or 270°, 1=0, and no light is reflected.
      As a consequence of this law, a beam of  common light, as far as
    brightness  is  concerned,  may  be represented  by two beams  of
    polarized light, having their planes of polarization at right angles
    to each other: for, the angle between the planes of polarization and
    of reflexion of the one being  called a, that of the other will  be
    90° — a, and from (108) we shall have, for the brightness of the
    two reflected pencils,
                             I =■ A. cos.2 a
                  I' = A. cos.3 (90 — a) = A. sin. ,Ja;
    whence,
                 / 4. I' = A (cos.2 a 4- sin.2 a) = 4,
    the sum of the intensities, of the two supposed pencils, remaining
    the same whatever be the angle a, which is characteristic  of com-
    mon light.
       Equation (108) applies  to the case of light polarized by refrac-
    tion, and incident upon a reflecting  surface at  the angle  of com-
    plete polarization, a being the angle between the plane of polariza-
    tion of the incident ray  and the plane  of  reflexion.

       (121.) The law, deduced by Sir David Brewster, as expressing
    the relation between the phenomena of  refraction and polarization
    by reflexion, when hght falls upon the first  surface of a body, is
                        tan. P  = to.....(109).
     P being the polarizing angle, and m the index of refraction of the
     material used. 
82        POLARIZATION BY REFLEXION.      APPENDIX.

  From this formula, if we suppose the light to be incident at the
polarizing angle, and call R  the angle of refraction at this inci-
dence,

       tan. P = sin,P ; but tan. P =  sin' P , whence
                 sin. R                cos. P

                 sin. R = cos. P,.....(110),

or the maximum polarizing angle is the complement of the cor-
responding angle of refraction, and the  reflected ray is  perpen-
dicular to the refracted  ray.

   (122.)  If the light which has passed through  the first surface
fall upon a second, parallel to the first, the angle of incidence upon
the first surface being P, that on the second is R, and R = 90 —
P (110);  whence,

   tan. R = cot. P:  but cot. P =----- —  — , and therefore,"
                                tan. P     m

                         tan. R = 1
                                  OT
or the tangent of the incidence  upon the second surface is the
index of the refraction from the denser to the rarer medium.  R is,
therefore, the angle of polarization for the second surface, and the
light reflected from that surface, as well as that from the first, will
be polarized.

    Law of Partial Polarization of Light by Reflexion.

   (123.) Sir David Brewster  has verified by an extensive series of
experiments  a law, which is due to Fresnel, by which the effect
of any  number of reflexions, on  the  inclination  of the planes of
polarization of a beam of light, may be determined.  The  effect
of a single reflexion at an angle differing from the polarizing angle,
is given by the equation
                .   .  „   cos. (i 4- i')       ,„,.
            tan. <p = tan. x----i_x—I.....(HI),
                           cos. (i — i')

in which formula, i is the angle of incidence, i' the corresponding
angle of refraction, x the primitive  inclination of the plane of
polarization of the polarized ray to the plane of reflexion, and <p the
inclination of the same planes after reflexion. The angle i — i' is
evidently the deviation produced by refraction,  and i  4.  i' is the
supplement of the angle between the refracted and reflected rays.
When x = 45°, the case considered in the text, p. 150, tan. x = 1,
 and
               .    ,    cos. (i 4- i')       /1in.
               tan. <P =----E_E—'......(112).
                        cos. (i — i') 
CHAP. VI.
POLARIZATION OF LIGHT.
83
  (124.)  The effect of successive reflexions of a pencil of common
light, or in which x = 45°, may be deduced from the first equation
for the value of tan. <p (111); for if 0 represent the inclination of
the plane of  polarization to that of reflexion after  n reflexions,
tan. 0 = tan." <p, x and  <p  preserving the  same relations to each
other after any number of reflexions; whence,
                       cos. (i  4- i')n         ni9\
              tan. 6 = ----  ■    ,—.....(lld)-
                       cos. (i  — i )D
  Since tan. 0 = tan.n <P, and by the  supposition tan. <p is not zero,
it appears that although partially polarized light  may have its
planes brought indefinitely near to parallelism, by increasing the
number of reflexions, yet tan. 0, and therefore 0, cannot  become
absolutely equal to zero by any number of reflexions.
   The formula for the quantity of the apparently polarized light
could not, advantageously, be introduced in this place.*

       Polarization of Light  by ordinary Refraction.
   (125.)  From an examination of the effect produced by  a single
surface upon the two planes of polarization in the beam of common
light,  Sir David Brewster inferred, that it depended upon the angle
of deviation of the ray, and was represented by the formula,
                cot. <p = cos. (i — i').....(H4).
 in which <p is the inclination of the planes  of polarization  to the
 plane of the  refraction, and i and i' the angles of incidence and
 refraction of the ray. When i — i' = 0° or i = 90°, cos. (i — i)
 = 1  and cot. <p = l,or<p= 45°,  and no change is produced in
 the  inclination.   When i — i' = 90°, cos. i — i' = 0, and cot. <p
 = 0,  or <p = 90°.
   When the light is not  common  light, or  light  in  which the
 planes of polarization are inclined 45° to the plane of refraction,
 if x be taken to represent the inclination of the planes of  polariza-
 tion of the beam to the plane  of refraction,
             cot <p = cot. x. cos. (i — i').....(H5).
   If  the light fall upon a second surface, parallel to the first, x for
 that surface is the value of <p  found  for the first and if 0  be called
 the inclination after n refractions,
        cot. 0 =  cot." <p  = cot." x. cos.n (i — i').....(116).
   When cot x = 1, that is, in the case of common hght,
                cot. 0 = cos."  (i — i').....(H7).
   (126 ) Bv combining this formula with that for the partial polar-
 ization by reflexion, we can readily obtain the effect produced upon
 h>ht which should reach the  eye, after two refractions, at the first
 surface of a plate,  and an intermediate reflexion at the  second
 surface.________________________________—______________—
              * Sir D. Brewster, in Phil. Trans. (Lon.) 1830. 
84        POLARIZATION BY  REFRACTION.      AITENDIX

  Let xp represent the  inclination of the  plane of polarization to
that of refraction, after refraction by the first surface of the plate;
4>' the inclination produced  by the reflexion at the second surface ;
and xp" that produced by the  second refraction at the first surface,
as the ray emerges.  Calling, as before, i the angle of incidence
on the first surface, i' that  of refraction,  and  x the inclination of
the planes of polarization of the  incident hght to the  plane of inci-
dence ;  then from (115),
                         ..    ...                  tan. x
    cot 9 = cot x . cos. (i — t),  or  tan. <p =----------— •
                                                cos. (i —»)

   From formula (111), for partial polarization by reflexion,
    t    a/    *      cos. (i 4-  i'             cos. (i 4- i')
    tan. p = tan. <p _____v  ~___. — tan. x-----1 ~   '
                     cos. (i — i')             cos. (i — i')2

   Equation (115), applied to the second surface of the  plate, gives
                 cot. xp" = cot. <p'. cos. (i — i');
whence, by substituting  for cot <p' the reciprocal of  the value just
found for tan. xp',
               .  .„        1      cos. (i — i')3
             cot. <p" —-------.  -----1-----E_,  or,
                         tan. x     cos. (i 4-  i)

           cot. 9" = cot. x . ______]_   .... (n8).
                              cos. (i  4. i)

   For common light, in which  x =  45°,
                       cot  xp  =  cos. (i — i'),
                         .      cos. (i 4- i')
                    tan. xp' = ----EET—L.,
                               cos. (i — i')2

              cot <p" = cos-(* —{')3  .....(H9).
                         cos. (i  4- i')
   If, in this latter case,
                   cos. (i — i')3 = cos. (i 4-  i'),
                    cot. xp" = 1, or <p" = 45°,
and the light polarized by the first refraction  and the intermediate
reflexion, will be restored by the refraction at emerging, to the «late
of common light.  The above equation will be satisfied in  glass of
which  sln,t  = ot = 1.525,  at 78° 7'.
        sin.  i'

   If cos. (i — i')3 > cos. (i 4- i'), which would occur by dimin-
ishing i, cot. <p" >■ 1, and <p" < 45°.                       •'  -E

   If cos.  (i — i')?. = cos. (i 4- V) tan. <p'= 1, ^=45°,orthe'light
polarized by the first refraction is restored to common light|by the
reflexion.  When refracted at the second surface, since, 
CHAP.  VII.        THEORY OF THE RAINBOW.            85

  22±p_W = cos. (j_0 .  cos.(i-i')2     cog> (i_0
   cos. (i 4. t')                   cos, (i _j_ j/)
                     cot 0"= cos. (i—i'),
or the light is repolarized at the second refraction, and the effect of
the plate is  that of a single surface.*
   CHAP. VII.

OF THE RAINBOW.
  (127.)  To explain the theory of the rainbow, we begin by the
following proposition.

  Prop. XXXIV. 4 ray of light enters a refracting sphere, is re-
flected any number of times, and emerges ; to determine the devia-
tion when it is a maximum, or minimum.

                            Fig.R.
   Let RL be a ray of light, meeting the refracting sphere LMNP
 at L, and refracted into LM;  LM meeting the  second surface of
 the sphere at M, is in part reflected into MN, which farther suffers
 reflexion at NP, taking the  direction NP;  that part of NP which
 is not reflected, passes out of the sphere, being  refracted into the
 direction PF. By the law of reflexion the angles CML, CMN, &c,
 are all equal to CLM the angle of refraction at the first surface;
 the angle of emergence IPF is, therefore, equal to the angle of
 incidence RLK.   Call the angle of incidence <p, that of refraction
 <p'; the angle of deviation of the refracted ray LM, or the angle
HLM = <p — xp'; the angle of deviation at emergence, or the angle
 NPG.= <p — <P';  and the sum of the deviations is 2 (9 —9E) The
deviation produced by the first reflexion, or EMN = 180 — LMN
= 180 — 20', and at each succeeding reflexion a new deviation of
equal amount is produced; the total deviation, therefore, after n re-

       * Memoir by Sir D. Brewster, in Phil. Trans. (Lon.) 1830
            H 
  86               PRIMARY RAINBOW.           CHAP.  VII.
  flexions is 180n —2n0'.  The sum of tho effects produced by both
  refraction and reflection is, calling S the total deviation,
                S = 180« — 2n0'  4. 2 (0 — xp',) or
           S = 180n + 20 — 2 (n 4. 1) 0'.....(120).
  In which equation <p and  xp' are connected  by the equation, (17,
  urt.  ojj)
                  sin. 0 = to. sin.  0'.....(17).
    When 5 is a maximum or minimum, d5 = o, and by differen-
  tiating (120,) considering xp and 0'  as variable,
                  2d<p — 2 (n + 1) dxp' = o, or
                       d<p  = (n 4- 1) d<p'.
    By differentiating (17),
                    d. sin. <p —m. d sin. 0', or
                   cos. 0 . dip = ot. cos. 0'. d<p',
  in which substituting the value just found for dxp,
             (n 4- 1) cos. xp .dxp' = m cos. xp' dxp', and
                  (n 4- 1)  cos. 0 = m . cos. 0'.
   To combine this with  (17), square both equations and add, we
  have
     sin. 20 4- (n 4- l)2. cos.3 <p = ot2. (sin.3  xp' 4. cos.2 0');
  but,  sin.2 0=1 — cos.2  0, and sin.2 0' 4. cos.a 0' =  1, whence

               cos.3 0 ( (n + l)a— l) 4-  1  = m2

    cos.30=   »'-*    =  ™2-l =   ot3-1
              (J14-I)3—1     w24-2w     w(w4-2)

               cos. 0 =  Jm2~\   .....(121).
                         v  n(» + 2)            ;
   (128.) The primary rainbow is formed by two refractions  and
 one reflexion of the sun's hght by drops of rain, as shown in Us
 134, page 224 of the text.                                     s'
   Producing the incident and emergent  rays  RF and Oq, until
 they  meet at q; (120) gives  by making n  = 1,
                    3 =  180 4- 20 — 40'.
   The angle q (RqO) is the  supplement of the deviation  6, whence
                 q = 40' — 20.....(122),
and since q increases as 5 diminishes, 5 is a maximum when 6 is a
minimum.   Near the maximum  value, q will  change  less for a
given change of incidence than at other values; and near this maxi- 
CHAP. VII.      THEORY OF THE RAINBOW.            87
 mum, therefore, the incident pencil will emerge most copiously, and
 affect the eye most strongly. When S is a minimum, we have from
 the preceding article, by making n = 1 in equation (121),

              cos.0= y^f^.....(123);

 in which, if for to, the index of refraction of water for the dif-
 ferently colored rays be substituted, the angle of incidence will be
 found at which each color is most copiously transmitted to the eye.
 The angle q will be given at the  same time by equation (122), and
 by the relation,
                sin. 0 = to . sin. xp'.....(17).
  To determine the limits of the value  of q for the differently
 colored rays, we take the value of to for the least and most refran-
                                    108
 gible of those rays: for the red  m =---, and for the violet m

    109
 =----These values substituted for m, in equation (123), we
     81
 obtain from (123), for the red rays,
  cos. 0 = .5092, 0 = 59° 21',  and  sin.  0 = .8603, whence from
 (17) sin. 0' = .6452 and <p' = 40° 11'; therefore from (122), q =
 160° 44' — 118° 42' = 42° 2'.
  For the violet rays the same equations give,
 cos. 0 = . 5199, 0 = 58° 41J', and sin. 0 = .8543; whence sin.
 9' = .6352, and 0' = 39° 25',  and i = 157° 40' — 117° 23' =
 40° 17'.
  The breadth of the bow is measured by the angle qOxjf = OnR —
 q' = q — q1 = 42° 2'— 40° 17' = 1° 45'.  This  supposes the
 rays to flow from a point.  The angle q being greater than q", the
 hne Oq is above Oq1, and the red is the highest color in the bow.
  (129.) The secondary rainbow, shown in the same figure of the
 text, is formed by two reflexions and two refractions: it corresponds
to the case of m = 2 in the formula for the deviation.  From this
 formula (120)
                   S = 360 4- 20  — 60';
but the angle GqO between the incident and emergent rays is the
excess of the angle of deviation above two right angles, whence
              q = 180 4-  20 — 60'.....(124).
  By the same reasoning which was used in the preceding article,
it may be shown, that the different colors will be transmitted most
copiously, at incidences given by equation (121), in which n = 2, and
m is the index of refraction corresponding to the colored rays of
which  the incidence is sought  From (121)
-v^F>
cos. 0 = J-----1.....(125). 
88
SECONDARY RAINBOW.        APPENDIX
  The relation of 0 and xp' is given by
                sin. 0 = m . sin. xp'.....(17).
  The limits of the value of q will be found  by placing for to in
(125), and (17), the index  of refraction  for  tho  least and  most
  r    -ui              108     ,      109
refrangible rays, or ot =---- and m =----.
     s     J           81            81
  By proceeding  as in the last article, we have for the red rays:
cos. 0 = .3118, 0 = 71° 494', sin. <p = .9501, sin. 0' = .  7126,
9' = 45° 27', and 5 = 50° 57'.
  For  the violet rays: cos. 0 = . 3184, xp = 71° 27J, sin. <p' =
.7046, 0' = 14° 48', whence j = 54° 7'.
  The angle, q1, for the violet rays, being  greater than the cor-
responding angle for the  red, the  violet is higher than the red, in
the bow; the colors are therefore  inverted in relation to those of
the primary.   The angle q'Oq = q1 — 5 = 3° 10'.
  The angular distance between the bows, qOq = 50° 57' — 42'
2'=  8° 55'.
  (130.) The breadth of the bows,  and of the space between  them,
having been measured on the supposition that  the rays flow from
a point, correction must be made for the apparent diameter of the
solar disc, which  is about 32'.   On  this account the breadth  of
each bow is  increased by 32', so that the primary  is 2° 17'  in
breadth, and the secondary 3° 42'.  The breadth of the space be-
tween the two bows is, thus, diminished by 32', and is 8° 23'. The
angle, q, for the highest red of the primary bow will be (42° 2' 4.
16') 42° 18';  whence, if the sun is more than  42° 18' above the
horizon, the primary bow is not seen; the corresponding limit for
the secondary bow is 54° 23'.
  (131.) A portion  of the light which enters any drop of rain, is
lost  at each  reflexion: for, by art 41, in order that total reflexion
shall take place at the separating surface of the denser and of the
rarer medium, the  relation  to sin. 0'= 1, or > 1,  must  subsist;
but from  the  investigation it appears, that sin. xp is always less
than unity, and that the  condition necessary to total "reflexion is
never satisfied.  The colors of the secondary bow  are therefore
fainter than those of the primary.
  The method of investigating the theory of the bows formed by
three or more reflexions  combined with two  refractions, must be
obvious from what  has been said in relation  to  the primary and
secondary bows. 
INDEX.*
                A.
Aberration, spherical, of lenses and
  mirrors, 51;  longitudinal; lateral,
  52. Appendix, 55.  By a single re-
  fracting surface, App. 61.  By a lens,
  App. 65. Values of, for different
  lenses, 56; App. 68. Chromatic, 74.
jEpinus, M., his experiments on ac-
  cidental colors, 256.
Air, the absorptive power  of, 120.
Amici, professor, of  Modena,  pro-
  poses various forms of the camera
  lucida, free from the defects of Dr.
  Wollaston's, 278.  Revives the re-
  flecting microscope in an improved
  form, 286.
Analcime, the  polarizing  structure
  of this mineral;  derives its name
  from its property of not yielding
  electricity by friction, 183.
Arago,  M., the colors produced by
  crystallized  plates  in  polarized
  light, studied with success by Biot
  and other authors, first discovered
  by, 157.
Archimedes, the manner  in which
  he  is  supposed to have  destroyed
  the ships of Marcellus, 264.
Atmosphere, the refractive power of
  the, 215.
Axis,  of lenses, 32.
                B.

Barlocci, professor, his experiments:
  finds that the armed natural load-
  stone had its power nearly doubled
  by twenty-four hours' exposure to
  the  strong light of the sun, 85.
Barlow, his achromatic  telescope,
  304.
Barton,  John, produces  color  by
  grooved surfaces, and  communi-
  cates these  colors by pressure to
  various substances, 106.
Batsha, the tides of, explained by
  Newton and Halley, 119.
Baumgartner, M., his experiments :
  discovers that a  steel wire, some
  parts of which were polished, the
  other parts without lustre, becomes
  magnetic by exposure to the white
  light of the sun; a north pole ap-
  pearing at each polished part, and
  a south pole appearing at each un-
  polished part;  obtains eight poles
  on a wire eight inches long, 84.
Beams, solar, diverging, 233; and
  converging, 234.
Berard, M., his experiments on the
  heating power of the spectrum, 82.
Berzelius, M., his experiments on ab-
  sorption, 123.
Biot and Arago  on polarized light,
  149.
Blackadder, some phenomena both
  of vertical and lateral mirage,
  seen by him at King George's Bas-
  tion, Leith, 219.
Blair, Dr., his achromatic lens, 77.
  Constructs a prism telescope, 303.
Bodies, absorptive power of; the na-
  ture of the power by which they
  absorb light, not yet ascertained;
  all colored transparent  bodies do
  not absorb the colors proportion-
  ally, 120. Absorb heating rays un-
  equally, 316.   Natural, the colors
  of, 235 and 320.
Bovista lycoperdon, the seed of, 101.
Boyle, his observations on the colors
  of thin plates, 90.
Brereton, lord, his observations on
  the colors of thin films, 90.
Buchan, Dr., case of nnusual reflex-
  ion, 222.
Buffon, constructs a burning appa-
  ratus ; the principle of it explained,
  264.
Camera obscura, an optical instru-
  ment, invented by the celebrated
  Baptista Porta, 274.
Camera lucida, invented by Dr. Wol-
  laston, 277.
Cameleon mineral, 239.
Carbon, sulphuret of, of great use in
  optical researches; employed  as a
  substitute for flint glass by Mr.
  Barlow, 305.
Carpa,  M.,  and M. Ridolfi, repeat
  Dr. Moricliini's  experiment with
  success, 84.
  * The Appendix rfferred to is that of the American editor, unless when
the contrary is expressly stated. 
)EX.
90                           ini

Cassia, oll'of, 31—79.
Catoptrics, 13. App. 9.
Caustics formed by  reflexion,  58.
  App. 74. Formed by refraction, 62.
Charcoal the most absorptive of all
  bodies, 120.
Chevalier, M., of Paris, makes use
  of a meniscus prism for  the  ca-
  mera obscura, 271.
Christie, Mr;, of Woolwich, his  ex-
  periment confirmed by those of M.
  Barlocci and M. Zantedeschi, 84.
Coddington, Mr., his observations on
  the compound microscope, 284.
Colors,   accidental,   and   colored
  shadows, 254.  Phenomena of,  il-
  lustrated by various experiments,
  259.
Compression and  dilatation,  their
  optical influence, 203.
Crossed lens, App. 69.
Crystals with one axis of double re-
  fraction, 128.  Whether  mineral
  bodies or chemical substances have
  two axes of double refraction, 133.
  A  list of the primitive  forms of,
  according  to  Haiiy,  134.  With
  one axis; system of colored rings
  in, 172.  The influence of uniform
  heat and cold on, 202. Composite
  exhibited in the bipyramidal sul-
  phate of potash, 206.  In apophyl-
  lite, ib.  In Iceland spar, 208. The
  multiplication of images  by the
  crystals of calcareous spar with
  one axis, 210.  Different colors of
  the two images produced  by dou-
  ble refraction in crystals with one
  axis, 211.
Cubes of glass with double refrac-
  tion, 199.
Curves, caustic, formed by reflexion
  and refraction, 58-  App. 74.
Cylinders of glass with one positive
  axis  of  double  refraction,  197.
  With a negative axis of double
  refraction,  198.

                D.
D'Alembert,304.
Davy, Sir Humphry, repeats Berard's
  experiments on the heating power
  of the spectrum in Italy and  at
  Geneva; the result of these ex-
  periments a confirmation of those
  of Dr. Herschel, 82.
De Chaulnes, duke, 98.
Descartes, 54.
Deviation, angle of, 33. App. 27.
Diamond, 31.
Dichroism, or the double color  of
  bodies, 210.
Dioptrics, 26. App. 26.
Dispersion, irrationality of, 73.
Dispersive powers, table of, Author's
  App. 310.
Di Torre, father, of Naples,  his im-
  provement on Dr. Hooku's spheres
  for microscopes, 280.
Dollond,  Mr.,  the  achromatic tele-
  scope brought to a high degree of
  perfection by, 76.

               E.
Ellipsoid, 54. App. 72.
Englefield, Sir Henry, 81.
Eriometer,  an instrument proposn]
  by Dr. Young, a description of it;
  and the manner in which  it is  to
  be used, 101.
Eye, the  human; the structure and
  functions of, 240.   The refractive
  powers of humors of, 242.  The in-
  sensibility of, to direct impressions
  of faint light; duration of the im-
  pressions  of light on the retina,
  250 and 321.    The cause of sin-
  gle vision with two eyes, 251.  The
  accommodation  of,  to  diflerent
  distances, proved  by various  ex-
  periments, 252.  Long-sightedness
  and short-sightedness accounted
  for, 253.  Insensibility to particu-
  lar colors, 259 anil 322.
Eye-pieces, achromatic, Ramsden's;
  in universal use in all achromatic
  telescopes for land objects, 301.

               F.
Faraday, Mr., his observations on
  glass tinged purple with  manga-
  nese; its absorptive power altered
  by  the transmission of the solar
  rays, 124.
Fata Morgana, seen in the straits of
  Messina, accounted for, 218.
Fibres, minute, colors of, 101.
Fits,  the theory  of, superseded  by
  the doctrine of interference, 111.
Fluids, circular polarization in, dis-
  covered by M. Biot and Dr. See-
  beck, 188.
Focal point, 18.
Foci, conjugate, 18. App. 15.
Focus, principal, for parallel rays, 17
  rules for finding the principal; for
  convex lenses,  41. App. 38 ;  for
  mirrors, 17. App. 14. Distance from
  centre, a mean proportional, &c,
  App. 18. Physical, of mirrors, App.
  59.
Fraunhofer, M., of Munich, his ob-
  servations on the lines in the spec-
  trum, 78; perceives similar bands
  in the light  of planets,  and fixed
  stars, 79.   Illuminating  power of
  the spectrum, 80.
Fresnel, M.,explainsthe phenomena 
INDEX.
91
  of inflexion or diffraction of light,
  86.  Formula for  polarization of
  light by reflexions,  not at  maxi-
  mum polarizing angle, App. 82. His
  experiment on the interference of
  polarized light, 180. Discoveries
  of, on circular polarization, 189.

                G.
Glasses, plane, 31. Multiplying, 273.
  Refraction of light through plane,
  30.  App. 35.  Achromatic  opera,
  with single lenses, 304.
Gordon, the duchess of, 91.
Goring, Dr., his improvements in all
  kinds of microscopes;  introduces
  the use of test objects, 287; a work
  published by him  and Mr. Pritch-
  ard on the microscope, 281.
Gray, Mr. Stephen, 280.
Gregory, James,  the  first who de-
  scribed the construction of  the re-
  flecting telescope, 291.
Grimaldi, his discovery  of the in-
  flexion or diffraction of light, 86.

                 H.
Hall, Mr., inventor  of  the  achro-
  matic telescope, 76.
Halley, Dr., his observations on the
  rainbow, 226.
Halos, 227. The colors of, described;
  the origin  of, and  how produced,
  232.
Hare, Dr., observation on translu-
  cency of gold leaf, 320.
Haiiy, the abbe, discovers the want
  of electricity by friction in anal-
  cime, 184.
Heat, the influence  of,  on the ab-
  sorbing power of colored  media ;
  analogous phenomena in mineral
  bodies, 123.  Heat  and cold, tran-
  sient influence of,  197.
 Herschel,  Mr., his discovery of an-
  other pair of prismatic images  in
   thin plates of mother-of-pearl, 105.
   The principal  data of the  undula-
   tory theory given by, 119.  The
   results of many authors  on the
   subject  of colored flames, given
   by, 124.  His  discovery that  in
   crystals with  two axes the axes
   change their position  according to
   the  color of the light employed,
   134.
 Herschel, Sir W., his experiment of
   the heating power of the spectrum
   confirmed by Sir Henry Englefield,
   81. Ink applied by him for obtain-
   ing a white image of the sun, 121.
   Constructs  a telescope,  40 feet
   long, with which he discovers the
   sixth satellite of Saturn, 296.
Hevelius, his observations on a pa-
  raselene, 230.
Home, Sir Everard, his description
  of the pearl, 105.
Hooke, Dr., constructs small spheres
  for microscopes, 280.
Huygens, his discovery of the law of
  double refraction in crystals, 131;
  determines the extraordinary re-
  fraction  of  any  point of  the
  sphere, 132. Publishes an elaborate
  history of halos, 231; his  discov-
  ery of the ring and the fourth sat-
  ellite of Saturn, 289.
Huddart, Mr., several cases described
  by him of unusual refraction, 216.

                I.
Iceland spar;  of what composed;
  found in  almost all countries, 1-26.
Image by mirrors, curvature at ver-
  tex of, App. 24.  Change of form
  by change  of distance  of object,
  App. 24; formed from section, App.
  26; by a convex lens, App. 5-2; by
  a concave lens, App. 54.
Incidence,  angle of,  equal to angle
  of emergence, 14. App. 29.  Plane
  of, 14.
Induration, the influence of, 205.
Ink, diluted, absorbs all the colored
  rays of the sun in equal  propor-
  tion ; applied by Sir William Her-
  schel, as a darkening  substance,
  for obtaining a white image  of
  the sun, 121.
 Interference, the law of, 115.
 Iolite, properties of, 211.
 Iris  ornaments invented by John
   Barton, Esq. 107.

                J.
 Jansen, his  invention of the single
   microscope, 279.
 Jurine and Soret, observation of an
   unusual refraction, 218.

                K.
 Kaleidoscope,  formation and  prin-
   ciple of the, 262.
 Kircher, the inventor of the magic
   lantern, 276.
 Kitchener's, Dr., pancratic eye-tube,
   302.
                 L.
 Landriani, 81.
 Lantern,  magic, invented  by Kir-
   cher, 276.
 Latham, Mr., 221.
 Lerebours,  M., has lately executed
    two  achromatic   object-glasses,
    which  are  in Sir James South's
    observatory at Kensington, 300 
92
INDEX.
Lehot, his work on the seat of vis-
  ion, 244.
Le Maire, 51.
Lens,  spherical,  concavo-convex,
  double-convex, planoconvex, dou-
  ble-concave, plano-concave, 31; of
  least aberration, App. 69.
Lens, a plano-convex, the principal
  focus of,  42.  App. 43. Plano-con-
  cave, refraction by, App. 47.
---- achromatic, 76.
Lenses, the formation of images by,
  App. 50. Their magnifying power,
  40. Convex and concave, 207. App.
  37, 45, and 54.  Burning and  illu-
  minating, 268.
Lenses, polyzonal,  constructed for
  the Commissioners of  Northern
  Lighthouses; introduced into the
  principal French lighthouses,  269.
Light,  the  velocity with  which it
  moves; moves in straight lines, 12.
  Falling upon any  surface, the an-
  gle of its reflexion equal to the
  angle of its incidence, 14. The to-
  tal reflexion of, 34.  App. 27.  Re-
  fraction of,  through curved  sur-
  faces, 37.   Refraction of, through
  spheres, 38. App. 42.  Refraction of,
  through concave and convex  sur-
  faces, 40. App. 31.  Refraction of,
  through convex lenses, 41.  App.
  37. Refraction of, through concave
  lenses, 44. App. 45. Refraction of,
  through meniscus and  concavo-
  convex lenses, 45. App. 48.  On the
  colors and decomposition of white
  light, the composition of, discover-
  ed by Sir I. Newton, 63.  Different
  refrangibilities of the rays of; re-
  composition of white light, 65. De-
  composition of,  by absorption, 67,
  and 315.  The inflexion or diffrac-
  tion of, 86.  Several curious prop-
  erties of,  107. The interference of,
  111.  The absorption of,  120. A
  new method proposed of analyzing
  white light, 124. Double refraction
  of, first discovered in Iceland spar,
  126. Polarization of, by double re-
  fraction,  138. Partial polarization
  of, by reflexion, 149. App. 82; and
  by ordinary refraction, 152.  App.
  83.  Polarized,  the colors of crys-
  tallized plates in, 182.  The action
  of metals upon ; absorptive powers
  of, 210.
Loadstone, various experiments by
  professor  Barlocci  and  Zante-
  deschi on the magnetizing power
  of light on, 85.
               M.

Magnetism, experiments illustrative
  of, as developed by light, 85.
Malus, M., discovers  the polariza-
  tion  of light by  reflexion,  112.
  Law of, App. 81.
Mariotte, his curious discovery that
  the base  of the  optic  nerve was
  incapable  of  conveying  to  the
  brain  the impression of distinct
  vision, 243.   Theory  of  vision
  proved by comparative anatomy,
  244.
Megascope,  a modification  of the
  camera obscura, 276.
Melloni,  Signor, observations on ab-
  sorption of heating rays of spec-
  trum, 316.
Meniscus, 32.  Its effect on parallel
  rays;  its effect on diverging rays,
  45. App. 48.
Microscope, single, 51. The magni-
  fying  power of; invented by Jan-
  sen and  Drebell, 279.   Made  of
  garnet, diamond, ruby, and sap-
  phire, 280.
-----------reflecting,  51.  First
  proposed by Sir Isaac Newton ; re-
  vived in an improved form by pro-
  fessor Amici of Modena, 286.
-----------compound, 283.
-----------solar, 2rt<
Microscopic observations, rules for,
  287.
Mirage, a name given to  certain ef-
  fects of unusual refraction, by the
  French army, while   marching
  through the sandy deserts of Lower
  Egypt, 218.
Mirrors,  13.   Images formed  by, 22.
  App. 23.   Concave,  formation of
  images by, 23. App. 24.   The prop-
  erties  by  which they are distin-
  guished, 265;  used  as  lighthouse
  reflectors, and as burning instru-
  ments, 206. Convex, formation of
  images by, 21.  App. 26.   Plane,
  formation of images by, 2.3.  App.
  26.  Spherical aberration  of,  57.
  App. 55.   Plane  and curved, 261.
  Plane burning, the effect produced
  by a number of these, 264.  Plane,
  reflexion by, App. 13.
Mohs, prismatic system  of, 173.
Morichini, Dr., his experiment on the
  magnetizing  power of the solar
  rays, 83; magnetizes several  nee-
  dles in the  presence  of Sir H.
  Davy, professor Play fair, and other
  English philosophers, 84. 
INDEX.
93
Mother-of-pearl, the principal colors
  of, obtained from the shell of the
  pearl oyster; long employed in the
  arts;  the  manner  in which its
  colors may be observed, 102.

                N.

Newton, Sir Isaac, his discovery of
  the composition of white light, 63.
  Recomposes white light, 65.   In-
  flexion or diffraction of  light de-
  scribed by, 86.  His method of pro-
  ducing a thin  plate of air;  com-
  pares  the colors seen by reflexion
  with those seen by transmission,
  92.  His observations on the colors
  of thick plates produced by  con-
  cave glass mirrors, 97.  His theory
  of the colors of natural bodies, 236.
  His experiment of accidental co-
  lors, 256. Was the first who applied
  a rectangular prism in reflecting
  telescopes,270. Executes a reflecting
  telescope with his own hands, 291.
Non-luminous bodies, 11.

                O.

Objects, test, the use of, introduced
  by Dr.' Goring, 287.
Oblique pencil, cases of, App. 19; re-
  flexion by mirror, App. 19;  which
  crosses axis of mirror, App. 22.
Opacity, 239.
Optical  figures,  beautiful,  how pro-
  duced, 203.
Optics, physical, 63.
Oxides,  metallic, exhibit  a  tempora-
  ry change of color by heat, 239.

                 P.

Parabola, 58.
Parker, Mr., of Fleet-street, executes
  the most perfect burning lens ever
  constructed, 268.
Pencil,  direct and oblique, defined,
  App. 9.
Phantasmagoria, 277.
Plates,  thin, colors of, 90. Of air,
  water,  and glass, 93.   Thick, first
  observed  and described  by  Sir  I.
  Newton as produced  by concave
  mirrors, 97; a method by which
  the colors may be best seen and
  their  theory best studied, 99.  Re-
  fraction through, App. 36.
Polarization  by reflexion, law of,
  App.  81.   Relative to refraction,
   App. 82.   At second surface of a
   plate, App. 82.  Partial  by reflex-
   ion. App. 82.  By ordinary refrac-
   tion, App. 83.  Circular; this sub-
  ject  studied with much sagacity
  and success, by M. Biot, 185. Ellip-
  tical, 190.
Porta,  Baptista, his invention of the
  camera obscura, 274.
Primary  plane, App. 19.
Principal focal distance, 17.
Prisms, refraction of light through,
  32. App. 28.
Prisms, crown glass, and diamond,
  the dispersive powers of, compared,
  71.
Prism, meniscus, used for the came-
  ra obscura, by  M. Chevalier of
  Paris, 271.
Prisms, compound and variable, 271.

               Q.
Quartz, 91.

               R.

Radiant point, 18.
Rainbow, primary and  secondary,
  223.  App. 86.   The light of both
  wholly polarized in the planes of
  the radii of the arch, 225.
Ramage,  Mr.,  of  Aberdeen,  con-
  structs various  Newtonian tele-
  scopes ;  the  largest  of these is
  erected at  the Royal Observatory
  of Greenwich, 297.
Rays, reflexion of parallel, 15. App.
  14.   Reflexion of  diverging,  15.
  App. 14.  Reflexion of converging,
  16.  App. 16.  Reflexion of, from
  concave mirrors, 16. App  13.   Re-
  flexion of, from convex mirrors, 20.
  App. 16.   Incident; refracted, 28.
  Focus of parallel; diverging; con-
  verging,  for sphere, 39.  App. 42.
  Parallel, 41.  Rule for finding the
  focus  of parallel for a glass  un-
  equally convex, 42. App. 38. Rule
  for finding the focus of a convex
  lens for diverging rays, 43. App. 39.
  Rule for finding  the focus of con-
  verging, 40. App. 44. Rule for find-
  ing the focus of a concave lens for
  diverging, 45. App. 46. Solar rays,
  the  magnetizing power of,  83.
  Falling  perpendicularly  on  the
  surface of a prism, App. 29.
Rectangular plates of glass with no
  double refraction, 199.
Reflexion  by specula and mirrors;
  angle of, 13.  Plane of. 14. Fits of,
  111.  Total, 34. App. 27.  Formula;
  for, deduced from those of refrac-
  tion, App. 27. Surfaces of accurate,
  App. 71.
Refraction, angle of, 28;  index of, 30.
  App. 28. Through prisms, 31. App.
  28.  And lenses, 31. App. 31.  The
  composition of white light discov- 
94                           i xi

  ered  by, 03.  Polarization by, 152.
  App. 83.  Unusual, 215.  Surfaces
  of accurate, App. 71.
Refraction, double, the law of, as it
  exists in Iceland spar, 120.  Law
  of, App. 79.  Communicated to
  bodies by heat, rapid cooling, pres-
  sure, and induration, 135.  Gene-
  ral observations on, 214.
Refractive power, tables of. Author's
  App. 310.  Absolute, 310, 315.
Ricss and Moser, MM., their experi-
  ments  on the magnetizing power
  of solar rays, 85.
Rochon, 81.
Scheele, the celebrated, 82.
Scheiner, the original account of a
  parhelion seen by, 228.
Scoresby, captain, in navigating the
  Greenland seas, observed several
  cases of unusual refraction, 217.
Secondary  plane,  focal  length  in,
  found, App. 21.
Seebeck, M., his experiments on the
  heating power of the spectrum, 82.
  And on the chemical influence of,
  83.   Published an  account of ex-
  periments with cubes and  glass
  cylinders, 202.
Self-luminous bodies, 11.
Senebier, 81.
Solar  spectrum consists of  three
  colored spectra of equal lengths,—
  red, yellow, and blue, 68.
Somerville, Mrs., her experiments;
  produces magnetism in  needles,
  which were  entirely  free  from
  magnetism  before, by the  solar
  rays; her experiments repeated by
  M.  Baumgartner, 84.
Spectrum,  the,  78.  Properties  of;
  the existence of fixed lines in, 78.
  The  illuminating  power of,  80.
  The heating power of, 81.  Chem-
  ical influence of, 82.
Spectacles, periscopic, invented by
  Dr. Wollaston, 267.
Specula, plane, concave, and con-
  vex, 13.
Sphere, rule for finding the focus of,
  40. App. 42.
------ of  glass, with a  number  of
  axes of double refraction, 201.
Spherical  surfaces,  of same curva-
  ture, refraction by, 49.
Spheroids, glass, with  one axis of
  double refraction, 201.
 Substances with circular double re-
  fraction, 136.
 Sulphuric  acid, 72.
Surfaces, grooved, the production of
  color bv; and of the cemmunica-
  bility of these colors  to  various
  substances,  104;  applied  to the
  arts by John Barton, EBq., 106.
Tabasheer, the refractive power of,
  2.19.
Talbot, Mr., his experiments on the
  colors of thin plates, 97.  His ob-
  servations on films of blown glass,
  100.
Teinoscope, 303.
Telescope, reflecting, 50.  Astronom-
  ical refracting, 51.  Achromatic,
  one of the greatest inventions of
  the last  century,  pronounced by
  Newton  to be  hopeless;  accom-
  plished soon after Ins death, by Mr.
  Hall; brought to a high degree of
  perfection by Mr. Dollond, 70.  Ter-
  restrial, 290. Galilean, 291.  Gre-
  gorian reflecting, 291.   A  rule  to
  find the magnifying power of, 293.
  Cassegrainian, 293. Newtonian,
  an improvement on the Gregorian
  one, 294.  Sir William Herschel's,
  296.  Mr. Ramage's, 297.  Achro-
  matic solar, with si ngle lenses, 305.
  Imperfectly Achromatic;  the im-
  provement of, 306.
Thenard,  the first  who observed
  blackness  produced on phospho-
  rus, 124.
Thickness, correction for, App.  35.
Topaz, Brazilian, 210.
Transmission, fits of, 111.


                U.

Undulations, theory of, 116 and 318.
  Great progress of, in modern times;
  the doctrine of interference in ac-
  cordance with, 118.
Vergency, term used by Lloyd, App.12
Villele,  M.,  of Lyons, burning in-
  struments made by, 266.
Vince, Dr., his observations on un-
  usual refraction, 216.  And  on  a
  most  remarkable case  of mirage,
  218.
Vision, the  seat  of, 243.   Erect, the
  cause of, from an inverted imase,
  246.  Distinct, the law of, 247.  Ob-
  lique, 248.
Visible direction, the law of, 245;
  the centre of, 246. 
INDEX.
95
                w.

Water, the absorptive power of, 120.
   For heating rays, 317.
Wollaston, Dr.,  his  discoveries on
  the  chemical effects of light on
  gum guaiacum, 83. His invention
  of the  periscopic spectacles,  207.
  Of the  camera  lujjida, 277.  Dou-
  blet, 284.    Refraction   through
  strata of air of different densities
  proved  by, 219.
Wunsch,  M-, his  observations  on
  alcohol and oil of turpentine,  82.
               Y.

Young, Dr., his invention of the in
  strument called the eriometer, 101.
  His illustration of the undulatory
  theory drawn from the spring and
  neap tides, 118.  His experiments
  on the interference of the rays of
  light, 114.
               Z.
Zantedeschi, M., his observations on
  oxidated magnets and those which
  are  not  oxidated;  repeats Mr.
  Christie's  experiments on needles
  vibrating in the sun's light, 85.
THE END. 

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