C y » r ft A 7k 
rLLLfcK 1 Ui^ 
ON THE PERCEPTION OF SMALL 
DIFFERENCES ' Publications of the University of Pennsylvania. 
PHILOSOPHICAL SERIES. 
EDITED BY 
GEORGE STUART ^FULLERTON, 
Professor of Philosophy in the University of Pennsylvania, 
AND 
JAMES McKEEN CATTELL, 
Professor of Experimental Psychology in Columbia College. 
No. 2. 
May, 1892. 
On the Perception of Small Differences 
With Special Reference to the Extent, Force 
and Time of Movement. 
by 
THE EDITORS. 
PHILADELPHIA 
UNIVERSITY OF PENNSYLVANIA PRESS 
PUBLISHERS CONTENTS. 
INTRODUCTION. 
PAGE. 
Sec. i. The Least Difference which can be Perceived, . 9 
2. Method of the Just Noticeable Difference, . . 10 
3. Method of Right and Wrong Cases, . . .12 
" 4. Method of Average Error, . . . . . 18 
5. Method of Estimated Amount of Difference, . 19 
6. The Laws of Weber and of Fechner, . . .21 
7. Interpretations of Weber's Law, ... 22 
8. The Relation between the Error of Observation 
and the Magnitude of the Stimulus, . . 23 
9. The Perception of Bodily Movement, . . 26 
" 10. The Importance of Uniform Conditions, . . 28 
" 11. Concluding Remarks, 29 
PART I. 
Sec. 12. Introductory, ....... 30 
" 13. Apparatus and Methods, 32 
" 14. Experiments by the Method of the Just Notice- 
able Difference, 35 
" 15. Experiments by the Method of Estimated 
Amount of Difference, 42 
" 16. Experiments by the Method of Average Error, 47 
" 17. Later Experiments by the Method of Average 
Error, 53 
" 18. Experiments by the Method of Right and Wrong 
Cases, 59 
ON THE EXTENT OF MOVEMENT. 
5 6 
PART II. 
ON THE FORCE OF MOVEMENT. 
PAGE. 
Sec. 19. Introductory, 65 
" 20. Apparatus and Methods, . .... 66 
" 21. Experiments by the Method of the Just Notice- 
able Difference, 70 
" 22. Discussion of the Method and Results, . . 74 
" 23. Reduction of the Method to that of Right and 
Wrong Cases, 77 
" 24. The Variable Error of the Just Noticeable Differ- 
ence, 80 
" 25. Further Results, 81 
" 26. Experiments by the Method of Average Error, . 83 
" 27. Further Experiments by the Method of Average 
Error, 87 
" 28. Analysis of the Error into an Error of Perception 
and an Error of Movement, . . . .91 
" 29. The Distribution of Errors, 95 
" 30. The Summation of Errors, 97 
" 31. Experiments by the Method of Estimated Amount 
of Difference, 99 
PART III. 
ON THE TIME OF MOVEMENT. 
Sec. 32. Introductory, ....... 103 
" 33. Apparatus and Methods, 103 
" 34. Results of Experiments, 106 
" 35. The Constant Error, . . . . . .109 
" 36. Analysis of the Error into an Error of Per- 
ception and an Error of Movement, . . 111 
" 37. The Distribution of Errors, . . . . 113 
" 38. The Summation of Errors, 114 
" 39. The Maximum Rate of Movement, . . .114 7 
PART IV. 
ON LIFTED WEIGHTS. 
Sec. 40. Introductory, .116 
" 41. Apparatus and Methods, 118 
" 42. Results of Experiments, 120 
" 43. The Amount of Difference and the Degree of 
Confidence, 124 
" 44. The Constant Error, 127 
" 45. Conditions affecting the Perception of Lifted 
Weights, 129 
PAGE. 
PART V. 
ON LIGHTS. 
Sec. 46. Introductory, 134 
" 47. Apparatus and Methods, 135 
" 48. Results of Experiments, 138 
" 49. The Amount of Difference and the Degree of 
Confidence, 142 
" 50. The Constant Error, 145 
" 51. Memory for Weights and Lights, . . . 147 
CONCLUSION. 
Sec. 52. Psychophysical Methods, 150 
" 53. The Error of Observation and the Magnitude of 
the Stimulus, 152 
" 54. The Extent of Movement, 154 
" 55. The Force of Movement, 155 
" 56. The Time of Movement, 157 
" 57. Lifted Weights, 158 
" 58. Lights, 158 ON THE 
Perception of Small Differences 
WITH SPECIAL REFERENCE TO THE EXTENT, 
FORCE AND TIME OF MOVEMENT. 
INTRODUCTION. 
Section i. The Least Difference which can be Perceived.-In- 
struments discover differences which the unaided sense cannot 
detect. Sensations may appear alike to common observation, 
while the stimuli differ by amounts which may be measured 
with the help of rules, clocks, balances, thermometers and other 
instruments. We thus find a psychological problem which ad- 
mits of scientific study. How great a difference must there be 
between two magnitudes in order that it may be perceived, and 
how does this least noticeable difference vary with varying cir- 
cumstances ? The relation between the intensity of a stimulus 
and the least noticeable difference is the only problem which 
has hitherto received much attention. But the intensity of the 
stimulus is only one of a number of factors affecting the error 
of observation. The time and space relations of the stimulus 
are not less essential than its intensity; and the variations of 
different observers, and of the same observer in different con- 
ditions, deserve study. The determination of the relation of one 
magnitude to another makes of psychology an exact science, 
whereas the study of personal differences may find practical ap-- 
plications in medicine and education. 
9 10 
We shall first review the methods which have been used in 
studying the perception of small differences, as these throw light 
on the nature of the mental process. 
Sec. 2. Method of the Just Noticeable Difference.-If it be 
wished to determine the least difference between two stimuli 
which can be noticed, the simplest and most natural way would 
seem to be to make stimuli so different that the difference can 
just be noticed. This was indeed the method used by Weber1 
and by Fechner2 in their first experiments on discrimination. 
It has been elaborated and praised by Wundt3 and others, but 
has not escaped criticism.4 Our own experiments have led us 
to the discovery of serious practical and theoretical objections 
to this method. These cannot be fully discussed in this para- 
graph. It is, however, evident that if it be left to the observer 
to say when he can just notice a difference, different observers 
may attach different meanings to the words. Lotze5 and others 
adopt the curious supposition that stimuli seem exactly alike so 
long as the difference is less than a certain amount, whereas, 
when the difference is made greater than this amount, it becomes 
1 Ernst Heinrich Weber, Annotationes de Pulsu, Resorptione, Auditu etTactu, Lipsiae, 
1834; Der Tastsinn und das Gemeingefuhl, in Wagner's Handworterbuch der Physiologic, 
III. 2, Braunschweig, 1846 ; Annotationes Anatomicae et Physiologicse, Lipsiae, 1851. 
2 Gustav Theodor Fechner, Zend-Avesta, Leipzig, 1851; Elemente der Psycho- 
physik, Leipzig, i860; In Sachen der Psychophysik, Leipzig, 1877; Revision der 
Hauptpunkte der Psychophysik, Leipzig, 1882. 
3 Wilhelm Wundt, Grundziige der Physiologischen Psychologic, Dritte umg. Aufl., 
Leipzig, 1887; Preceding editions 1880 and 1874; Das Weber'sche Gesetz und die 
Methode der Minimalanderungen, Lipsiae [1882]; Ueber die Methode der Minimal- 
anderungen, Philosophische Studien, Leipzig, 1883, I. 556. 
* Georg Elias Muller, Zur Grundlegung der Psychophysik, Berlin, 1879. Joseph 
Jastrow, a Critique of Psychophysical Methods, Am. Journal of Psychology, 1888, I. 
271. Both of these writers, however, think that the method can be used in a correct 
manner-namely, by gradually increasing or decreasing a stimulus until the observer 
notice the direction of the change. This may, in some cases, be an improvement in tech- 
nique, but does not seem to us to alter the fundamental assumptions of the method. 
6 Hermann Lotze, Metaphysik, Leipzig, 1879 (English translation, Oxford, 1884), § 259, 
"Ein Intervall bleibt, durch welches hindurch der Reiz sich erfolglos verstarkt, um erst 
mit dem erreichten Endwerthe einen merklichen Unterschied der Empfindung her- 
vorzubringen." 11 
suddenly apparent. This is by no means the case. The clear- 
ness with which a difference is distinguished varies gradually 
from complete doubt to complete certainty. The variation is 
continuous, and no point can be taken and called the "just 
noticeable difference," and kept constant for different observers, 
or even for the same observer at different times. If complete 
certainty be taken as the standard, a difference in the stimuli 
will be required much greater than that which can ordinarily be 
distinguished, and the standard will be found to differ greatly 
with different observers, measuring, if anything, rather their 
character than their fineness of sensation. The clearness of 
discrimination is a continuous function of the amount of differ- 
ence between the stimuli, but even with a fixed amount of differ- 
ence it does not remain constant. It depends on many varia- 
bles, so that a difference which may be distinctly perceived at 
one time will be indistinguishable at another. It has always 
been assumed that these variations may be eliminated by using 
the average of a large number of observations, but as they affect 
each single observation, they make it extremely difficult for a 
conscientious observer to come to any decision. 
The only means by which scientific results can be obtained 
with the method of the just noticeable difference would seem 
to be to keep the observer in ignorance of the real relations of 
the stimuli, and see to it that when he said he could just notice 
a difference, the difference was of the sort he thought it to be. 
If in a long series of trials the observer make no mistake with a 
given difference, we may, indeed, assume that he can really dis- 
tinguish this difference. But we should also conclude that he 
might notice a smaller difference, and have no basis for com- 
paring different observers, or the same observer under different 
circumstances. If we make the difference smaller until the 
observer be sometimes wrong in his judgment, the method is 12 
reduced to that of right and wrong cases which we have next to 
consider. 
Sec. 3. Method of Right and Wrong Cases'-If two stimuli 
betaken so nearly alike that an observer cannot always distinguish 
the difference between them with clearness and certainty, and a 
series of trials be made, he will sometimes notice the difference 
distinctly, at other times vaguely or not at all, at still other 
times a difference will apparently be observed contrary to the 
real relation of the stimuli. Owing to the complex physical, 
physiological and psychological antecedents of perception, the 
same stimulus is not always accompanied by the same sensation. 
There is a normal error of observation; it has been treated 
by mathematicians and students of the physical sciences with 
a view to elimination, but it can be studied with advantage 
by the psychologist. Experience has led to the conclusion that 
small errors of observation are more common than large ones, 
and that an error is as likely to be positive as negative. The 
theory of probability further assumes that an error of observa- 
tion is the algebraic sum of a very large number of small errors 
due to independent causes. Based on these assumptions, the 
probability of an error of any size has been determined by math- 
ematical analysis, giving the well-known probability curve and 
its equation :2 
y = ce~ hi 
What happens when an observer tries to distinguish a small 
difference may be illustrated by a diagrammatic figure (Fig. i). 
Let the two magnitudes be proportional to the lines xo, XO, 
1 This method seems to have been first used by Hegelmayer working under Vier- 
ordt's direction (Vierordt's Archiv, XI. 844). Fechner (op. cit^, however, deserves the 
chief credit for its elaboration. See, also, Muller (op. cit^. 
2 This paper is not written by mathematicians, nor for mathematicians. We give the 
commonly accepted formulae, but our results may be understood by those who are not 
familiar with their significance. The theory of errors of observation is treated without 
mathematics by Jevons, Venn and Galton, and with the aid of comparatively simple 
mathematics by Airy and Merriman. 13 
their difference would then be PO. Now, when the observer 
perceives the first stimulus xo, the error of observation will 
cause him to perceive it as sometimes smaller, sometimes larger 
than xo. The distribution of his errors is represented by the 
curve vyu. That is, the probability of any error, say po, is pro- 
portional to the corresponding ordinatepq, or the number of errors 
in one direction not greater than po is proportional to the por- 
Fig. i-The Distribution of Errors of Perception. 
tion of the area of the curve above this abscissa, poyq. The 
shape of the curve depends on h and c in the equation, and 
these depend on the average precision of observation. Thus if 
the curve vyu represent the precision of one observer, the pre- 
cision of a better observer would be represented by another 
bell-shaped curve taller and narrower than vyu. 
This curve may be drawn if any one abscissa and its ordinate 
be known. An abscissa of such a size that half the errors are 14 
smaller and half larger (in the figure this would be po) may 
conveniently be used as a measure of the accuracy of discrimi- 
nation. This abscissa or error is called the probable error.1 
The error of observation cannot, however, be directly deter- 
mined. After the stimulus xo has been perceived, a second (say 
greater) stimulus, XO, is presented to the observer. Supposing 
the perception of this stimulus to be subject to the same error 
of observation as the first,2 the distribution of errors is repre- 
sented by the curve VYU. Now, a consideration of these two 
curves will show that the first stimulus might be perceived as 
xv, and the second as XU, in which case the difference would 
be perceived with distinctness and certainty. But more fre- 
quently the first stimulus would be perceived as larger than xv, 
say as xp, while the second stimulus might be perceived as XM ; 
in this case the observer might distinguish the difference, but it 
would seem smaller than before, and would be noticed with less 
clearness and confidence. It is further possible that the two 
stimuli might be perceived respectively as xo and XP, in which 
case they would seem alike. Lastly, the first stimulus might 
be perceived as xu, while the second stimulus might be perceived 
as XV, in which case the objectively greater stimulus would ap- 
pear the less. If the observer could define the amount of differ- 
ence which he perceives in a series of trials, his curve of pre- 
cision could be immediately drawn, but he can only do this 
vaguely.3 He may, however, be required to say which of the 
i The mean error, the error of mean square, and the modulus are all used to measure 
the error of observation. These quantities are related theoretically. If the probable 
error be i.oo, the mean error is 1.18, the error of mean square 1.48 and the modulus 2.10. 
2 It would usually be different, but to an inconsiderable degree. (See Muller's criticism, 
op. cit., p. 21 seq.). 
3 We indeed required the observer to give the degree of confidence which he felt in 
his judgment, and noted it by the letters a, b, c, d-a. plan proposed by Peirce and Jastrow 
(C. S. Peirce and J. Jastrow, On Small Differences of Sensation, National Academy of 
Sciences, HI. [1884] ). The great variation in the use of these letters (say " a " practi- 
cally sure) by different observers shows how little reliance can be placed on their judg- 
ment of the just noticeable difference, Method I. 15 
two stimuli seems the greater, and the number of times he is 
right and the number of times he is wrong may be recorded.1 
If, with the same difference, two observers be right in their 
decisions the same percentage of times, we conclude that their 
accuracy of discrimination is the same. If one observer be 
right a larger percentage of times than another, the difference 
in the stimuli can be decreased for the better observer, until he 
be right the same percentage of times, and we regard the ac- 
curacy of discrimination of the two observers as proportional 
to the difference which they can discriminate equally well. In 
the same manner the accuracy of the same observer under 
varying conditions (for example, when both magnitudes are 
greater, or when the time between their presentation is in- 
creased) may be determined. 
It would be tedious to experiment until a difference were found 
for which the discrimination of different observers should be 
alike, nor is this necessary. It is evident that the percentage of 
right cases will increase as the difference is taken greater, and 
the relation between the two can be determined by mathematical 
analysis. It follows the formula developed by Gauss for deter- 
mining the relative area of the probability curve above any 
abscissa. Adapted to the percentage of right cases it is 
= 1 + -I Ce-^dt 
n 2 j/jt I 
J 0 
in which P.E. is the probable error and A the amount of differ- 
1 As we have seen, the stimuli might appear alike, and most experimenters have re- 
corded right, wrong and doubtful cases. Strictly speaking, however, the two stimuli 
would never appear exactly alike (except once in an indefinitely large number of trials), 
and it is theoretically better and practically more convenient to require the observer to 
come to some decision. " Doubtful cases" are analogous to the just noticeable differ- 
ence, depending on the judgment (or caprice) of the observer. The assumption that 
there is no difference in sensation when the observer is doubtful as to its nature has led 
Fechner, Muller, Wundt and others into many difficulties. 16 
ence.1 The values of this integral given in works on probability 
are the converse of what is needed for the present purpose. We, 
therefore, give a table in which the entries and arguments have 
A 
been reversed. This table gives the values of p p for every 
percentage of right cases above fifty.2 
TABLE I. 
Table for Determining the Probable Error from the Percentage of Right 
Cases and Amount of Difference. 
1.90 
i-99 
2.08 
2.19 
2.31 
2-44 
2.60 
2-79 
3-°5 
3-45 
* 
O ci re rf tn\o r\00 Os 
o^ososososososos^os 
u 
i 
1.25 
i-3° 
1-36 
1.41 
'•47 
i-54 
1.60 
1.67 
i-74 
1.82 
O ci re rf inO r^OO Os 
0O0O0O0O0O000O0O00O0 
<1 
.78 
.82 
.86 
•9i 
•95 
1.00 
1.05 
I.IO 
1.14 
1.20 
K 
W. 
I 
70 
71 
72 
73 
74 
75 
76 
77 
78 
79 
r 
•38 
.41 
•45 
•49 
•53 
•57 
.61 
•65 
.69 
•74 
* 
O *- ci rt ie\O r^oo Os 
sO'O'O'O'O'O'O'OsO'O 
I 
.00 
.04 
.07 
.11 
•15 
•19 
.22 
.26 
•3° 
•34 
5° 
5i 
52 
53 
54 
55 
56 
57 
58 
59 
The values of in this table are inversely proportional 
to the probable error of an observer, and measure directly his 
accuracy of discrimination. Thus if two weights be used weigh- 
ing 100 and 108 gm., and one observer can correctly distinguish 
the difference ninety-one per cent, of the time, while another can 
distinguish it only seventy-five per cent, of the time, the ac- 
curacy of the former is about twice as great as that of the latter, or 
the observer who could distinguish a difference of 8 gm. ninety- 
1 We had in view the testing of this formula by experiment; it had, however, been con- 
firmed by Fechner, by Peirce and Jastrow and during the course of our experiments by 
Higier (Experimentelle Priifung der psychophysischen Methoden im Bereiche des 
Raumsinnes der Netzhaut. Dorpat, 1890. Also Philos. Stud., VII. 232,1891). 
2 In case the percentage of right cases be less than fifty, which can only occur when 
there is a constant error larger than A or when the number of experiments is limited, use 
the percentage of wrong cases, and take negative. 
J7.£L. 17 
one per cent, of the time could distinguish a difference of 4 gm. 
seventy-five per cent, of the time. Thus whatever the difference 
between the stimuli and whatever the percentage of right cases, 
the difference with which an observer would be right seventy- 
five per cent, of the time can be discovered. 
The equation given above can, of course, only be solved with 
the aid of the integral calculus, but its meaning may be under- 
stood by regarding Fig. 1. Thus if the difference between 
the stimuli be PO, the number of times that XO will seem 
larger than xo or XP will be to the number of times that it will 
seem smaller, as the area PUNQ is to the area P VQ, but in this 
case the area PUNQ is to the area P VQ as 3 : 1 ; consequently, 
when the difference (A) is equal to the probable error QP.E.\ the 
observer will be right in his decision seventy-five per cent, of 
the time.1 
We trust the considerations brought forward in this section 
will throw light on the real nature of the just noticeable differ- 
ence. It seems to us that it is a difference so large that the 
error of observation will not often cause the observer to err as 
to its nature. We learn by experience what difference we can 
usually distinguish correctly, and regard this as the just notice- 
able difference. But what point shall be called the just notice- 
able difference depends on the character and experience of the 
observer. Some people are sure they are right when they have 
but little to guide them, whereas others are slow in coming to a 
decision. If this view of the just noticeable difference be cor- 
1 While the probable error seems the best standard for comparison, it is better in 
making experiments to choose a difference with which the observer will be right about 
eighty-four per cent, of the time, the error of mean square. This is an abscissa about half 
again as large as PO (Fig. i), where it will be seen there is an inflection point in the 
curve, and it approaches the horizontal axis most rapidly. Here an alteration in the 
error makes the greatest difference in the size of the ordinate, and a reliable result is 
reached with fewest experiments. Near Y and near K, on the other hand, the curve is 
nearly parallel to the horizontal axis, and very large (near 100) or very small (near 50) 
percentages of right cases are the most unfavorable for calculating the probable error. 18 
rect, it cannot be regarded as a fixed unit fit for measuring the 
intensity of sensation, and preceding theories on this subject 
must be radically altered. 
Sec. 4. Method of Average Error.-If a stimulus be pre- 
sented to an observer, and he be required to make a second stim- 
ulus as nearly as possible equal to it, the error of observation 
which we have been considering will lead him to make it some- 
times greater, sometimes less. If a series of trials be made and 
the average error taken, the observer's accuracy of discrimina- 
tion is directly measured. As (apart from constant errors) the 
real errors are known, the average error directly obtained is 
proportional to the probable error (1.000: 0.845), and results ob- 
tained by this method may be readily compared with those obtained 
by the method of right and wrong cases. By the latter method, 
however, we only determine the number of errors in one direction 
greater than a given size, and many experiments must be made 
before we can proceed to calculate the probable error, whereas 
by the method of average error the amount and direction of 
error are in each case measured, and a reliable result is obtained 
more directly and quickly. A single experiment by the method 
of average error indicates the observer's accuracy of discrimina- 
tion, and half-a-dozen give a result sufficiently reliable for clini- 
cal and anthropometric purposes. This method is therefore to 
be preferred whenever it can be used. If, however, the observer 
be required to make a stimulus, such as a weight or light, appar- 
ently equal to another, he can only do so with repeated trials, 
and it is difficult to keep the conditions constant. In such a 
case it is better to use a fixed difference and the method of right 
and wrong cases. With movements, on the other hand, the 
method of average error can be used to great advantage. The 
observer is required to make one movement and then another 
movement as nearly as possible equal to it, the error in extent, 19 
force or time being measured by suitable apparatus. In this 
case, however, the error is complex, being partly an error of per- 
ception, and partly an error in adjusting the second movement. 
We do not see how the error of adjustment can be eliminated 
otherwise than by using the method of right and wrong cases. 
The error of an observer is apt to consist of two compo- 
nents-one a variable error dependent on the normal variation 
in perception, which we have been considering-the other a 
constant error due to overestimating or underestimating the 
second stimulus as compared with the first. With the method 
of average error the constant error is directly determined by the 
experiments, and it may be subtracted algebraically from each 
error, and thus the variable error obtained.1 The precision of 
observation has usually been measured by the variable error, 
but the constant error also measures an important error in per- 
ception. 
Sec. 5. Method of Estimated Amorint of Difference.-When 
experiments are made with stimuli of different degrees of inten- 
sity, a stronger and a weaker stimulus may be presented to the 
observer, and he may be required to fix on a third stimulus 
which appears to him midway between the two. This has been 
called the method of mean gradation.2 An observer may further 
be required to double3 a given stimulus, also to halve it, etc. We 
regard this method as distinct from the three preceding methods, 
which rest on the error of observation.4 It is true, the error of 
1 The theoretical relation of the variable to the constant error is determined by values 
of P. E. and A in the probability integral. See, below, the method of eliminating and de- 
termining the constant error, when the method of right and wrong cases is used. 
3 This method has long been used by astronomers in estimating the apparent magni- 
tudes of the stars. It is said to have been first used for psychological purposes by Pla- 
teau (Bull, d 1'Acad. de Belgique, XXXIII. 376, 1872). 
Apparently first done by Merkel, Philos. Stud., IV. 541; V. 245, 499, 1888-9. 
' The method of just noticeable difference has been used in such a form that the differ- 
ences should appear not just noticeable, but equal. In this case the method comes under 
the present head, and is subject to the same criticism. 20 
observation is measured in this method, but, in addition, the ob- 
server attempts to estimate definite quantitative differences in 
sensation. If an observer can, in fact, estimate quantitative 
amounts of difference in sensation, apart from association with 
known quantitative differences in the stimuli, a relation between 
mental and physical intensity can be determined. The writers, 
however, agree in finding that they cannot estimate such quanti- 
tative differences in sensation in a satisfactory manner. We 
can indeed say when one weight seems approximately double 
another, but this is doubtless because we have often lifted first 
one volume, and then two, and the like. But we cannot say when 
one sound seems twice as loud, or one day twice as hot as an- 
other. We have made experiments to see how nearly different 
observers would agree in adjusting one shade of light midway 
between two others, and have found hesitation in coming to a 
decision and great divergence of opinion. Most men will think 
that a just king is happier than a tyrant, but few will agree with 
Plato in considering him 729 times as happy. 
An important distinction in the senses should be kept in mind. 
In estimating the quantitative relations of weights, for example, 
we should expect an approach to the true relations. We know 
when one weight is twice as heavy as another, because two ob- 
jects of the same sort are twice as heavy as one, and from fre- 
quent use of the balance. We are also interested in the objective 
relations of weights. Two pounds of sugar cost twice as much 
as one, and go twice as far. Sounds and lights, however, are 
rarely doubled, and can be measured only with difficulty, if at 
all. We, therefore, have no association with the real magnitudes 
of the stimuli. On the other hand, we see the same object, 
sometimes in a bright light, sometimes in a dim light; we hear 
the same noise, sometimes near by, sometimes at a distance. We 
may, therefore, come to regard relative differences as equal dif- 
ferences. 21 
Sec. 6. The Laws of Weber and of Fechner.-The methods 
we have been considering have been used chiefly in studying 
the relation between the least noticeable difference and the in- 
tensity of the stimulus. Weber1 observed that the least differ- 
ence between two stimuli which could be noticed was not the 
same for stimuli of different magnitudes, but a proportional 
part of the stimulus. When an observer could just distinguish 
the difference between I and 1.05 kg., he could not distinguish 
any difference between 4 and 4.05 kg., but could only distinguish 
the difference when it was about four times as- great-i.e., about 
.2 kg. 
Since Weber published this result, an amount of experiment 
and theorizing has been devoted to this subject, which has prob- 
ably never been surpassed in the history of any science. It is 
commonly claimed that the experiments confirm Weber's gen- 
eralization. At all events, the difference which can just be 
noticed is usually found to become greater as the stimulus is 
taken greater. Weber's law can be expressed by the equation 
N = C -AtL 
in which TV is the least difference which can be noticed, and 
the increase in the stimulus S which causes this differ- 
ence. No objection can be made to this statement of Weber's 
law so long as the meaning of N be left an open question. 
We ourselves consider it a physical quantity, dependent on the 
error of observation, and this indeed, if Weber's law hold, 
1 Op. cit. As long ago as 1730 Daniel Bernoulli (cf.Todhunter, History of the Theory 
of Probability, Cambridge and London, 1865) maintained that the value of money is 
relative, so that (to use the terms of Laplace) the fortune morale is proportional to the 
logarithm of the fortune physique. The general principle is expressed by Hobbes and 
Spinoza, and is implicit very early, as in the institution of Mosaic tithes. Bouguer 
(Traite d'optique, etc., Paris, 1760) seems to have been the first to observe that the just 
noticeable difference (between lights) is proportional to the magnitude of the stimulus. 
Lambert (Neues Organon, Leipsic, 1764, II. 268-9) seems to have made the same obser- 
vation independently, as his least noticeable difference (35) differs from that of Bouguer (B't). 22 
would increase in direct proportion to the stimulus. Fechner, fol- 
lowed by many others, regards N as an equal increase in sensa- 
tion, always the same whatever value maybe given to S. If Wbe 
such a unit fit for measuring the intensity of sensation, it fol- 
lows directly from Weber's law that the sensation increases in 
arithmetic ratio as the stimulus increases in geometric ratio. 
Fechner further assumes that the difference between no sensa- 
tion and the least sensation which can be noticed is a mental 
quantity equal to the other N's, whence follows the equation : 
E = C log. S 
that is, the sensation is equal to the logarithm of the stimulus 
multiplied by a constant. This equation, which leads to nega- 
tive sensations and other difficulties, has been discussed at 
great length by many writers.1 
Sec. 7. Interpretations of Weber's Law.-According to 
Wundt and other writers, there are three possible interpreta- 
tions of Weber's law. These are respectively psychophysical, 
physiological and psychological. The first is adopted by Fech- 
ner, who holds that his development of Weber's law expresses 
an ultimate relation between mind and matter. Such a func- 
1 Some of the more important works not hitherto mentioned in this paper are as 
follows: 
H. Aubert, Physiologic der Netzhaut, Breslau 1865. 
H. Helmholtz, Handbuch der Physiologischen Optik, Leipzig 1867. Second edition 
in course of publication. 
J. J. Muller, Sitz. d. Sachs. Ges., Leipzig 1870. 
J. Delboeuf, Etude psychopbysique, Bruxelles, 1873, and later publications. 
F. Brentano, Psychologic, etc., Leipzig 1874. 
E. Hering, Ueber Fechner's psychophysisches Gesetz, Sitz. d. Wiener Acad. 1875. 
P. Langer, Die Grundlagen der Psychophysik, Jena 1876. 
F. A. Muller, Eine Untersuchung etc., Marburg 1882. 
J. v. Kries, Ueber die Messung etc., Viertelj. f. wissensch. Philos., VI. Leipzig 1882. 
P. Tannery, Revue Philos., Paris 1884. 
A. Elsas, Ueber die Psychophysik, Marburg 1886. 
A. Kohler, Ueber die hauptsachlichsten Versuche etc.; Philos. Stud., Leipzig 1886. 
A. Grotenfelt, Das Weber'sche Gesetz etc., Helsingfors 1888. 
H. Miinsterberg, Beitrage etc., Freiburg i. B. 1889. 23 
tional relation is, of course, only conceivable on the assumption 
that the sensation is measurable. If this be the case, however, 
a direct proportion would seem far more natural, and should 
only be rejected under stress of facts which cannot otherwise 
be explained. The physiological interpretation of Weber's law 
is based on the supposition that the sensation is, indeed, directly 
proportional to the brain changes immediately correlated with 
it, but that these increase as the logarithm of the physical 
stimulus. It has been argued that this may be due to resistance,1 
radiation2 or fatigue8 in the nervous system. Owing to our 
complete ignorance of the minute physiology of the nervous 
system, such hypotheses may be made more easily than proved 
or disproved. The inertia of the nervous system may account 
for a very weak stimulus not being perceived, and its exhaustion 
or destruction for a very strong stimulus failing to call forth a 
corresponding sensation, but so long as there is no proof it must 
be regarded as unlikely that brain changes should vary as the 
logarithm of the stimulus. A psychological explanation of 
Weber's law is favored by Wundt, who argues4 that it is a case 
of the "relativity of feeling." This principle may account for 
certain effects of contrast, etc., but scarcely seems relevant to 
the supposed relation between intensity of sensation and inten- 
sity of stimulus. Indeed, Wundt seems to think that the physi- 
ological explanation may be combined with his. 
Sec. 8. The Relation between the Error of Observation and the 
Magnitude of the Stimulus.-As may be gathered from what has 
preceded, the writers cannot accept any of these explanations of 
Weber's law. All the experiments made by the first three 
1 J. Bernstein, Untersuchungen fiber den Erregungsvorgang im Nerven- und Mus- 
kel-Systeme, Heidelberg 1871. 
2 James Ward, An Attempt to interpret Fechner's Law, Mind I. 464, London 1876. 
3 Herm. Ebbinghaus, Ueber den Grund der Abweichungen von dem Weber'schen 
Gesetzbei Lichtempfindungen, Pfliiger's Archiv, XLV. 113, 1889. 
4 Phys. Psy., I. 377 seq. 24 
methods which we have described seem to us to determine the 
error of observation under varying circumstances, and not to 
measure at all the quantity of sensation. The fourth method, 
it is true, seeks to estimate quantitative relations in sensation, 
but we find that this cannot be done apart from association with 
known quantitative relations of the stimuli. Our own experi- 
ments by this method, to be described below, show that in such 
cases we tend to estimate the intensity of sensation as directly 
proportional to the intensity of the stimulus; consequently, in 
so far as any deduction concerning quantitative relations in sen- 
sation can be made from such estimation, the sensation increases 
as the stimulus and not as its logarithm. The experiments 
which we have made by the first three methods (and incidentally 
by the fourth) show that the error of observation usually in- 
creases as the stimulus is taken greater, but more slowly. We 
therefore believe that Weber's law does not hold for the 
perception of movement. We believe, however, that the 
error of observation tends to be related to the magnitude 
of the stimulus in a simple manner; namely, that the error 
of observation (and the so-called least noticeable differ- 
ence) is proportional to the square root of the stimulus. We 
believe, further, that we can explain this relation in a sat- 
isfactory manner. Suppose time to be taken as the phys- 
ical magnitude. If one second be estimated, a certain error 
of observation will occur. If the experiment be repeated, an 
error on the average of the same size will be made. Now, if 
the two seconds be estimated continuously, each second com- 
posing the interval may be regarded as subject to its own error. 
If these errors were always in the same direction, the error in 
estimating two seconds would be twice as great as the error in 
estimating one second, and Weber's law would hold. In half 
the cases, however, the two errors would be in opposite direc- 25 
tions and would partly counterbalance each other, and the 
average error in estimating two seconds would, consequently, 
be smaller than twice the average error for a single second. It 
would, indeed, according to the theory of probability (confirmed 
by our own experiments), be the average error for one second 
multiplied by the square root of two. Or, generally, the alge- 
braic sum of a number of errors is the average error multiplied 
by the square root of the number. There seems to be reason 
for maintaining that this deduction should hold for the estima- 
tion of any physical magnitude composed of equal units, although 
this view has not, we think, been hitherto proposed. 
As the result of our own experiments we find that the error 
of observation varies more nearly as the square root of the mag- 
nitude than as the magnitude. An exact correspondence with 
such a law can indeed only be looked for under the simple con- 
ditions assumed by the theory of probability. In actual percep- 
tion the fractions of the magnitude, physically equal, would sel- 
dom or never be subject to exactly the same errors of observa- 
tion. Every general law is subject to secondary laws, and the 
greater our knowledge the more do these secondary laws in- 
crease. The planets do not, in fact, move in ellipses, but by 
very complex paths ; the volume of a gas does not, in fact, vary 
inversely as the pressure, but in a very complex manner. It is 
usually a token of our ignorance rather than of our knowledge 
when the complex phenomena of nature seem to be related by 
a simple mathematical formula. The error of observation would, 
in fact, seldom or never increase as the square root of the mag- 
nitude, but the summation of errors would seem to account in 
an entirely satisfactory manner for the usual increase of the 
error of observation as the magnitude is taken larger. We, 
therefore, substitute for Weber's law the following: The error of 
observation tends to increase as the square root of the magnitude, 26 
the increase being subject to variation, whose amount and cause 
must be determined for each special case.1 
Sec. 9. The Perception of Bodily Movement.-The larger 
part of the experiments described in this paper are concerned 
with the perception of small differences in movements of the 
body. We choose this class of sensations, partly because it 
offers peculiar advantages for studying the perception of small 
differences, and partly because more knowledge is needed con- 
cerning the perception of movement. Movements are conveni- 
ent for such study, because the observer himself causes the 
stimulus which he perceives, while at the same time it can be 
measured more exactly than light or sound. The complexity of 
the perception of movement is so considerable that there is 
much difference of opinion as to its nature, and for its study ex- 
act experiments are needed. 
Common observation does not tell us what nervous and mus- 
cular mechanism is involved in movement, nor what sensory ap- 
paratus is used in its perception. In this paper we shall lay stress 
on the fact that movements differ in their force, in their extent 
and in their time, and we shall try to show how accurately these 
different factors are discriminated, and how far they are inter- 
dependent. This analysis of movement seems to us natural, as 
it is in accordance with the magnitudes, energy, space and time, 
on which physical science is based. But we have still to inquire 
how we come to perceive the force, extent and time of our 
movements. The perception of movement was formerly identi- 
fied with touch, and passive sensations from the skin undoubt- 
edly give us knowledge concerning bodily movements. These 
1 One of the writers (G. S. F.) gives but a qualified assent to the subject-matter of this 
section and its application elsewhere in this paper, on the ground that mathematicians 
are not agreed as to the soundness of the theory upon which the law is based, and also 
on the ground that the errors in question may not be independent errors. He regards, 
however, the results obtained by the writers as sufficiently in accord with the law to jus- 
tify them in holding it tentatively, and subiect to criticism. 27 
sensations may be of two sorts-one due to pressure at the place 
where force is applied, the other due to contraction, relaxation, 
etc., of the skin of the moving member and its connections. In 
addition to these sensations from the skin, sensory nerves end- 
ing in the muscles1 and about the joints2 have been discovered, 
and these doubtless play some part in the perception of 
movement.3 Then in all movements, and especially in such as 
are violent and powerful, many parts of the body are concerned. 
The whole position and balance of the body may be altered, and 
vital processes, respiration, circulation, etc., are affected. In 
addition to these sensations from the skin and inner body, im- 
portant knowledge concerning the movement may be obtained 
from the other senses, particularly from sight. 
It is well known that besides the numerous sensory impressions 
resulting from the movement, an immediate consciousness of the 
motor impulse calling forth the movement has been considered 
an essential factor in its perception. Wundt,4 Bain5 and others 
lay great stress on this "feeling of innervation," holding it to be 
an important, if not the only ground for belief in matter and 
space. On the other hand, Bastian,6 Ferrier,7 James8 and others 
1 C. Sachs, Archiv f Anat. u. Phys., 1874, pp. 175, 491, 645. Sachs' microscopical re- 
sults have, however, been disputed. See Mays, Zeitsch. f. Biol., XX. 
2 Rauber, Vater'sche Korper der Bander und Periostnerven und ihre Beziebung zum 
sogenannten Muskelsinn, Munchen, 1865. See William James, The Principles of Psy- 
chology, New York, 1890, II. 189, seq. 
3 Wundt (Phys. Psy., I. 464) writes : " Die Annahme eines specifischen Muskelsinnes 
wurde zuerst, wie es scheint, von Ch. Bell aufgestellt." The distinction was probably 
known to Aristotle, and is plainly made by Scaliger (1557). See Hamilton's edition of 
Reid's Works, Edinburgh 1846, p. 867. 
* Beitiage zur Theorie der Sinneswahrnehmung, Leipzig 1862 ; Vorlesungen uber die 
Menschen- und Thierseele, Leipzig 1863; Phys. Psy. 
5 Senses and Intellect, 3d ed., 1868. The author's views were gradually elaborated in 
the preceding editions, 1855 and 1864. 
6 On the Muscular Sense, Brit. Med. Journ., 1869; The Brain as an Organ of Mind, 
appendix, 3d ed., London, 1882; Brain, 1887, with discussion. 
7 The Functions of the Brain, 1st ed., London, 1876; 2d ed., 1886, Ch. X. 
8 The Feeling of Effort, Anniv. Mem. of the Boston Soc. of Nat. Hist., 1880, Psy., Ch. 
XXVI. 28 
argue that there is no consciousness accompanying the outgoing 
current. The extended discussion concerning the sense of in- 
nervation may be partly confused by misunderstanding. It is 
evident that we know what movement we purpose to make, but 
it seems equally evident that we cannot thus know what move- 
ment we carry out. We may will to lift a heavy box, and dis- 
cover from the unexpected movement resulting that the box 
is empty. Our perception of the movement we have carried 
out may, however, be affected by our knowledge of the move- 
ment we have willed. The box which unexpectedly turns out 
to be empty seems to fly up of itself. 
Sec. io. The Importance of Uniform Conditions.-Before 
proceeding to describe our experiments, some mention of 
special methods should be made. One of the most important 
considerations is the need of keeping all the conditions con- 
stant, except the variable to be investigated. This is such an 
evident precaution in all exact experiments that it would seem 
needless to mention it, were it not that it has been neglected in 
many psychological investigations. The resulting confusion 
has been very great. If experiments are being made on the 
variation of the error of observation, with the magnitude of 
the stimulus, care must be taken to keep the time of stimula- 
tion, the interval between the stimuli, the place and it may be 
the area of stimulation, the condition of the observer, etc., 
constant. This has often been neglected, especially when the 
methods of just noticeable difference and of average error have 
been used. But the greatest confusion has been due to the 
previous knowledge of the observer. If he know the results 
of preceding experiments, he may seek to distinguish apparent 
differences in sensation, or he may seek to make his judgment 
objectively correct, and the difference in result will be great. 
If the observer know beforehand the real relations of the 29 
stimuli in each case, he will find that this will affect the appar- 
ent relations of the sensations, and almost any result can be 
obtained. We regard, therefore, all experiments made by a 
single observer on himself as provisional. Lastly, many psycho- 
logical researches have but little value because the author has re- 
jected experiments and series which seemed wrong to him. Under 
such circumstances even the most conscientious observer may 
obtain results conforming better to his theory than to the truth. 
Sec. ii. Concluding Remarks.-The experiments published 
in this paper were mostly carried out in the psychological 
laboratory of the University of Pennsylvania during the years 
1889, 1890 and 1891. The observers who took part in the ex- 
periments are designated by letters : F. and C, the writers; R. and 
J., respectively, their wives ; K., also a woman, a student of psy- 
chology and biology; D., N and 5., advanced students of psy- 
chology ; L. and S., advanced students of biology. The calcula- 
tions were made by the observers. Most of the apparatus 
used was given to the University by Dr. Weir Mitchell, Mr. H. 
H. Houston and Mrs. Matthew Baird. Some part was obtained 
through an appropriation from the Bache Fund, and is the prop- 
erty of the National Academy of Sciences. The new apparatus 
was made under our direction by Clay & Torbensen, Camden, 
N. J. The drawings of apparatus were made by Miss "Wash- 
ington, University of Pennsylvania, and the curves were drawn 
by F, C. and J. PART I. 
ON THE EXTENT OF MOVEMENT. 
Sec. i 2. Introductory.-Until quite recently but little exper- 
imental work had been done on the discrimination of the ex- 
tent of movements. Wundt1 gives the results of an investiga- 
tion concerned with movements of the eyes, and considers 
them in harmony with Weber's law. Numerous experiments2 
have been made on the perception of small differences in the 
length of lines ; and as the eye in most cases has not been care- 
fully fixed, the movement was doubtless an important factor in 
the judgment. But it cannot be separated from the influence 
of the size of the image. Several researches have been pub- 
lished comparing movements made with the right and left 
hands. Thus Hall and Hartwell,8 Loeb4 and Bloch5 have studied 
movements of the arm with a view to comparing those made 
with the right side of the body and those made with the left. 
Jastrow6 has investigated the perception of space by dispa- 
rate senses, and in so doing has made use of various move- 
ments, but he is concerned rather with the relations which 
different senses, or different aspects of the same sense, hold to 
1 Phys. Psy , Bd. II. s. 116 ff. 
2 The most recent by Higier(o/. «V.),who gives references to preceding investigations. 
3 Bilateral Asymmetry of Function, Mind, Vol. IX. pp. 93-109. 
4 J Loeb, Untersuchungen etc., Pfliiger's Archiv, 1887 and 1890. 
6 A. M. Bloch, Experience etc., Rev. Scientifique, 1890. 
•The Perception of Space by Disparate Senses, Mind, Vol. XI. pp. 539-54. 
30 31 
each other than with the accuracy with which the extent of 
movements can be discriminated, in itself considered.1 
There are peculiar difficulties connected with the study of 
ocular movements, and it has seemed desirable to the writers to 
choose for investigation movements of another class. In judg- 
ing by the eye of the distance between two points, or of the 
length of a line, active sensations of movement may contribute 
much to the result ; but the passive sensations arising from the 
extent, the position or the motion of the image on the retina, or 
in binocular vision, the difference in character of the retinal 
images, may contribute no unimportant factor. In free move- 
ments made by the arm this difficulty is obviated. They are, 
to be sure, complex, and in so far not wholly satisfactory; but 
they are at least less unsatisfactory than ocular movements, 
and offer the special advantage that they enable us to compare 
directly the judgment of the extent of a movement with the 
judgment of its force and time. 
Our experiments on the extent of movement were under- 
taken with a view to : 
(l) The investigation of psychophysic law in the sphere of 
bodily movements. 
(2) A comparison of the results obtained by the use of the 
four psychophysical methods. 
(3) The determination of the effects of practice. 
(4) The determination of the significance of the confidence 
of the experimentee in relation to objective correctness. 
(5) The investigation of constant errors. 
i No less than four inaugural dissertations have recently been published bearing more 
or less directly on the subject. Those by Cremer and by Ostermann, Wurzburg, and by 
Falk, Dorpat, we have not seen. Ueber Bewegungsempfindungen, by Edmund Burke 
Delabarre, Freiburg i. B., 1891, we have received since our paper has been written. This 
dissertation contains interesting experiments on movements of the arm, but the relation 
of the variable error to the magnitude of the movement is not considered. 32 
Sec. 13. Apparattis and Methods.-The larger part of the 
experiments on the discrimination of the extent of movements 
were made with very simple apparatus. A scale, graduated to 
millimeters, was fastened along the edge of an ordinary table. 
At one end of the scale, to the left of the experimentee as he sat 
facing the table, was a fixed upright which served as point of 
departure for the movement. A movable upright, which could 
be placed at any desired point on the scale, determined the ex- 
tent of the movement chosen as standard of comparison. To 
allow for the width of the finger used in making the movements, 
the two uprights were set on flat bases, with bevelled edges pro- 
jecting in such a way that, when these edges were in contact at 
the o point of the scale, there was just room for the finger be- 
tween the uprights above. A seconds pendulum, swinging above 
the table, gave the time for each movement, and the interval 
between the movements to be compared. The experimentee 
sat directly opposite the 10 cm. mark on the scale, with the 
fixed upright to his left at the o point. His body was about 20 
cm. back from the edge of the table. The movement was made 
with the index finger of the right hand, the arm swinging freely 
and being used as a whole in the movement. The motion was 
from left to right, and when in motion the finger was from 2 to 
6 cm. above the table. Each movement occupied, as nearly as 
possible, one second, and one second was allowed for recovering 
position between the two movements to be compared in each 
experiment. A screen prevented the experimentee from seeing 
the movements made. The recorder sat to the right of the ex- 
perimentee and regulated the standard movement in each exper- 
iment by placing in position the movable upright. The ex- 
perimentee then moved his finger over the space between the 
two uprights, and, having thus gained a standard, made a second 
movement to which the first served as guide. The point at 33 
which the finger rested upon the scale at the end of this second 
movement was recorded, the finger-nail being cut to a point as a 
help to the recorder. The experimentee knew nothing of the 
results of the experiments until after the whole number was 
completed. The danger of his unconsciously allowing for con- 
stant errors, and vitiating the results, was thus avoided. 
With this apparatus, and in the manner described, 4400 ex- 
periments were made, and all four of the so-called psychophys- 
ical methods were employed. The method of just noticeable 
difference was used in the attempt to mark off a distance just 
greater and one just less than 500 mm. The method of esti- 
mated amount of difference was used in the attempt to halve 
500 mm., to double 300 mm., and to find the mean between 300 
and 700 mm. The method of average error was used in the 
attempt to measure off successively 100, 300, 500 and 700 mm. 
Finally, the method of right and wrong cases was used in the 
attempt to decide whether the two uprights were nearer together 
or farther apart in the second of each pair of movements, their 
distances being as 500:510 mm., or as 510:500 mm. 
We have here ten different kinds of experiments; or, since 
the experiments by the method of right and wrong cases may 
count double, the second' movement in each experiment being 
half the time longer and half the time shorter than the first, we 
may say that we have eleven kinds. A complete series, includ- 
ing ten experiments of each kind, or in all 110 experiments, was 
made at each sitting. Sometimes one and sometimes two series 
were made in a day. The ten experiments of each kind (ex- 
cept those by the method of right and wrong cases) were 
made before passing to the next, and the eleven different kinds 
were arranged in the series in the order in which they are given 
above. Every second series was made in reversed order. Fur- 
ther details of these experiments will be given later, when each 34 
class will be discussed by itself. All these experiments were 
made upon F., the recorder being M., who is not otherwise rep- 
resented in this paper. 
After these experiments were made, it occurred to the writers 
that there was a possible source of error in the method of get- 
ting the standard movement in most of the experiments. In all 
the experiments, the first movement was determined by two 
uprights, while the second movement, except in the experiments 
by the method of right and wrong cases, was made with the 
movable upright removed. It would seem to be quite possible 
Fig. 2.-Apparatus for Measuring the Extent of Movement. 
that the force expended in the touch against the upright limiting 
the first movement would have made the movement appreciably 
greater if the limit had not been there, and that a second move- 
ment, made in imitation of the first, but without such an inde- 
pendent limit, would have a tendency to be too great. Eight 
hundred later experiments were made by the method of average 
error, upon two subjects, and care was taken to overcome this 
difficulty. The apparatus used in making them was more elab- 
orate than that employed before, and was constructed partly 
with a view to discriminating between the force and the extent 35 
of movements, and modifying either factor independently of 
the other. 
This apparatus consists of a brass plate one meter long, grad- 
uated to millimeters, and grooved for the wheels of a small brass 
carriage. Along the scale is a wire, carrying an indicator (A) 
which is moved by a bar attached to the carriage. To the left 
of the experimentee, as he sits facing the apparatus, are two 
brass pins, which stop the carriage at the o point of the scale. 
Between the front and back wheels of the carriage, and parallel 
with the track, is a ring (7?) into which is to be inserted the 
finger used in moving the carriage. Attached to the carriage is 
a cord ( W} which runs over a pulley to the left of the experi- 
mentee, and admits of being weighted to any desired extent. 
Here, as before, the movement is from left to right. The indi- 
cator, which slides along the wire as the carriage moves, and 
remains at the farthest point reached, makes the readings much 
easier than in the apparatus used before, where the recorder had 
to catch the reading during the very brief time before the finger 
began the return movement. The carriage may be moved alone, 
or made to raise any weight attached to the cord. Thus one 
may study the effect of increasing or decreasing the amount of 
force demanded by movements of a given extent, and see to 
what degree judgments of distance are conditioned by the 
amount of force expended. So far, the writers have not found 
time to carry out such experiments. 
Sec. 14. Experiments by the Method of the Just Noticeable 
Difference.-The method of the just noticeable difference has 
been described. It assumes that the smallest difference be- 
tween two stimuli that can be consciously noted may be taken 
as a measure of the fineness of discrimination of any sense- 
the smaller the just noticeable difference, the greater the fine- 
ness of discrimination. But the use of this unit of measurement 36 
is, as we have seen, not without its difficulties. We do not find 
that all things consciously noted are noted with equal clearness. 
We are clearly conscious of some things, less clearly conscious 
of others, and still others seem to be on the border-line between 
consciousness, or at least what usually passes by that name, and 
the obscure region which has been designated as "sub-con- 
sciousness" or "latent thought." And, just as there maybe 
many degrees of objective difference below the level of the least 
degree that can be consciously noted as such by the unaided 
sense, so it is quite thinkable that there may be many corre- 
sponding degrees of sensation below the level of that which we 
are accustomed to regard as on the threshold of consciousness. 
Such sub-conscious mental modifications may differ from those 
commonly called conscious, simply as the fainter among these 
latter differ from the clearer. That such is the case, and that 
we have between the clearest consciousness and a complete 
cessation, if such there be, of all mental life a descending series, 
each member of which differs continuously from the one imme- 
diately preceding it, and a portion of which lies below what we 
commonly recognize as the threshold of consciousness, there 
seems good reason to believe. We do not have, then, two 
clearly marked divisions, the one lying below and the other 
above a well-defined threshold or ideal limit between sensations 
of quite different classes. Neither do we have a series begin- 
ning abruptly with what would be recognized as a "conscious" 
sensation. 
If this be the case the just noticeable difference would seem 
to be a unit arbitrarily taken. Since we can, by varying the ob- 
jective difference between two simuli, bring about a varying de- 
gree of clearness in the consciousness of difference, the mean- 
ing of the words "just noticeable" must necessarily be vague. 
A larger difference than the one settled upon would be noted 37 
with a clearer consciousness, and a smaller with a fainter, the 
differences between the cases being one of degree and not of 
kind. And although it is possible that the same observer may 
at different times give the words approximately the same mean- 
ing, there would appear to be no way at all of deciding that the 
just noticeable difference means the same thing in the case of 
different persons. 
Again, the same degree of difference in the stimuli may, by 
the same observer, be perceived as a difference much more clearly 
at one time than at another. Many causes concur to produce at 
times a heightened consciousness and at times a less vivid one. 
Suppose that the difference between two weights is so small as 
to be distinguished with some clearness at one moment, and not 
thus distinguished at the next. Is it to be called a noticeable 
difference ? If it be distinguished in nine cases out of ten, may 
we call it the jtist noticeable difference ? Probably in several of 
those instances a smaller difference would have been noticed. It 
is this fluctuation in the power of discrimination, due partly to 
nervous changes in the observer, that has caused a series of ex- 
periments to be regarded as indispensable in the determination 
of the least noticeable difference. 
The method of just noticeable difference may be applied in 
two somewhat different ways. Either two stimuli with a certain 
difference between them may be given the experimentee, and 
the difference increased or decreased until the observer considers 
it just noticeable-or a single stimulus may be given, and the 
experimentee required to fix upon another just noticeably greater 
or less. The just noticeable difference is here assumed to be 
the average obtained by combining the results of a number of 
experiments. In all the experiments given below as made by 
this method, the latter course was adopted. In this form, it will 
be seen that the method is analogous to that of average error; 38 
and as the numbers of right and of wrong cases were recorded, 
the three psychophysical methods have been combined. 
With the simple apparatus first described, 800 experiments 
were made by the method under discussion. Half of them were 
attempts to mark off on the scale a distance just greater, and half 
a distance just less than 500 mm. As details concerning the 
methods have already been given, it is only necessary to say that 
the recorder placed the movable upright at the 500 mm. mark 
on the scale, the experimentee moved his finger between the 
two uprights in the manner described, and then attempted, the 
movable upright having been taken away, to measure off a dis- 
tance just greater or just less, as the case might be. The re- 
sults of the experiments appear in the accompanying table. 
TABLE IL 
Extent of Movement-Method of Just Noticeable Difference-F. Experi- 
menter- 800 Experiments-VII. VIII. 1890. 
JUST GREATER THAN $00 mm. 
JUST LESS THAN $00 mm. 
EXT. 
V.E. 
V. 
! w. 
EXT. 
V.E. 
V. 
w. 
Av 
Var. . . . 
+ 36-6 
16.6 
II.4 
1.6 
5-7 
3 
- 24-7 
13* 
10. 
1.8 
5-4 
1.2 
6 
Av 
+ 60.1 
9.8 
5-o 
0 
- 4.8 
9-i 
5-i 
33 
Var. . . . 
8.2 
2-7 
1.2 
8.4 
^■3 
1.1 
Av 
Var. . . . 
+ 39-4 
5-0 
7-8 
1.6 
4-7 
0 
- 24.2 
7-2 
9- 
1.8 
5-3 
4 
Av 
Var. . . . 
+ 21.5 
^3 
8.9 
2. 
4-7 
5 
- 37-7 
8.6 
1.2 
5-i 
i-3 
0 
Av. Av. . . 
Av. V. . . 
+39.4 
9.2 
9.5 
2. 
5. 
1.1 
2 
-22.8 
9.9 
9.2 
1.8 
5.2 
1.2 
IO.7 
Ext. The extent of the second movement as compared with 500 mm. 
V.E. The average variable error made. 
V. The mean variation of the variable error. 
The percentage of wrong cases; i. e., attempts at greater which resulted in less, 
and vice versa. 
Av. The average extent, variable error, and variation of the variable error for 100 
experiments. 
Var. The mean variation of the results of the ten sets of ten experiments each from 
their average as given above opposite Av. 
Av. Av. The average of the four sets of figures opposite Av. above, giving the aver- 
ages for 400 experiments. 
Av. V. The average of the four sets of figures opposite Var. above. 39 
As has been said, ten experiments of each of the eleven kinds 
made with the first apparatus were made at one sitting. The 
results of each sucli set of ten experiments have been calculated, 
and then these results combined in sets of ten to get the aver- 
ages which appear in the four tables embodying the results of 
this investigation. The averages given in the tables conse- 
quently represent in each case 100 experiments. As 400 
experiments of each kind were made, there are four such sets 
of averages for each kind of experiment, and these are com- 
bined at the foot of the tables in figures which represent aver- 
ages for the whole 400 experiments. The averages are given in 
their chronological order, those at the top of each table rep- 
resenting the first hundred experiments made of the class indi- 
cated, those below them the second hundred, etc. The experi- 
ments made later with the second piece of apparatus were also 
made in sets of ten, and the results have been computed in the 
same way. 
To enter more into detail, let the first set of averages in the 
table just given and under just greater than 500 mm. be taken. 
The figures to the right of Av. and under the headings Ext., 
V.E., V. and % W. indicate that, in a hundred experiments, the 
average distance measured off as just greater than 500 mm. was 
536.6 mm.; the average variable error or mean variation of the 
individual distances from the average of each series of 10 experi- 
ments was 11.4 mm.; the mean variation of the variable error, 
or the average of the deviations of the variable error from its 
average, was 5.7 mm.; while three per cent, of the attempts to 
mark off a distance just greater than 500 mm. resulted in mark- 
ing off a distance actually less. 
The figures in italics below these averages have reference to 
the results of the separate sets of ten experiments each. For 
each set of ten experiments the average extent of the distance 40 
measured off, the variable error or mean variation of the 
single distances from this average, and the mean variation of 
this variable error, were computed. The averages described in 
ihe preceding paragraph were, of course, obtained by taking 
the average of ten sets of results thus obtained. The mean 
variation from their average of these results representing in 
each case ten series appears in the second row opposite Var. 
Thus, the mean variation of the average distance measured off, 
in the first ten sets of ten experiments, from the 536.6, the figures 
obtained by averaging these averages, was 16.6 mm. Similarly 
the mean variation of the variable error for the ten sets was 1.6, 
and the mean variation of the variation of this variable error for 
the ten sets 1.1. 
These explanations will apply, with a few obvious changes, to 
all the sets of figures on the table. Of course, where the en- 
deavor was to measure off a distance just less than 500 mm., 
and the minus sign is used under the heading Ext., it is meant 
that the distance measured off was as much less than 500 mm. 
as is indicated by the figures. For example, in the first block of 
averages under "just less than 500 mm.," the - 24.7 means that 
the average distance measured off in 100 experiments was 475.3 
mm. The significance of the averages at the bottom of the table 
is too plain to need comment. 
It appears, as a result of 400 experiments, that a movement, 
to be just noticeably greater than a previous movement of 500 
mm. must be 539.4 mm. long; and from the same number 
of experiments, it appears that to be just noticeably shorter a 
movement must be 477.2 mm. long. The difference between the 
attempt at just less and the standard movement (5 per cent, of 
the former) is thus considerably less than the difference between 
the standard movement and the attempt at just greater (8 per 
cent, of the standard). A glance at the table will show, however, 41 
that these results arc modified by a marked tendency to overrate 
the standard in the second group of 100 experiments of both 
kinds. In these experiments the attempts at just greater gave 
an average of 560.1 mm., and the attempts at just less one of 
495.2 mm., while 33 per cent, of the attempts at just less resulted 
in a movement actually greater than 500 mm. Still, in three out 
of the four groups of experiments of both kinds there appears to 
be a tendency to overrate the standard movement. 
What has been said of the highly variable character of the 
just noticeable difference may be well illustrated from the table. 
Even for groups of 100 experiments it varies, for the attempt at 
just greater, between 60.1 mm. and 21.5 mm., and for the attempt 
at just less between 37.7 mm. and 4.8 mm. For the smaller groups 
of ten experiments the variation was still more marked. In the 
attempts at just greater it varied between 76 mm. and 9 mm., 
and in the attempts at just less between 49 mm. and 1 mm. 
In striking contrast with these figures stands the very slight 
degree of fluctuation in the variable error and its mean varia- 
tion. These would seem to be a much better measure of dis- 
criminative power than the least noticeable difference itself, for 
they do not appear to be subject to the disturbing influence of 
constants. For example, the average increment necessary to 
produce the consciousness of just greater in the second group 
of 100 experiments was 60.1 mm., and in the fourth group only 
21.5 mm.; but the mean variations corresponding to these were 
respectively 9.8 and 8.9, and their variations in turn 5.0 and 4.7. 
Or, to take the second group of 100 experiments alone, as the 
tendency to overrate the stimulus was there most marked, while 
the increment under "just greater" was 60.1 mm., and the de- 
crease under "just less" only 4.8 mm., their variable error was 
respectively 9.8 and 9.1, and the mean variation of this 5.0 and 5.1. 
It is sufficiently evident that the just noticeable difference is of 42 
little value in psychophysical experiments. It varies within 
wide limits even for sets of 100 experiments, and the only ob- 
jective criterion of its significance-the percentage of right and 
wrong cases obtained-varies correspondingly. If the average 
of a large number of experiments be taken, the difference will 
be found to be too large, and the percentage of wrong cases too 
small to be satisfactory in experiments by the method of right 
and wrong cases. Where the proportion of right cases is so 
large, the results may easily be modified by accidental varia- 
tions. Moreover, as the percentage of right cases does not in- 
crease uniformly as the difference increases, the average per- 
centage of errors given in the table does not really correspond to 
the average just noticeable difference, but is somewhat too low. 
One. may, it is true, apply the probability integral to the sepa- 
rate averages for 100 experiments, and compute the percentage of 
errors corresponding to the average just noticeable difference, 
but even in this case the results would not be wholly trust- 
worthy, as the difference was highly variable even within the 
sets of 100 experiments. The variable error of the just notice- 
able difference furnishes a more satisfactory measure of dis- 
crimination, but there is no special advantage in taking the 
average variable error of this difference, as it is more regular 
when the method of average error is used. 
Sec. 15. Experiments by the Method of Estimated Amount 
of Difference.-The method of mean gradation favored by 
Wundt1 concerns itself only with the attempt to find a mean 
between two given stimuli (sensations ?). There appears, how- 
ever, no good reason for not using other relations as a measure 
of accuracy in discrimination. If one observer can fix more ac- 
curately than another upon a stimulus half as great, or twice as 
great, as a given one, his sense of the quantitative relations 
1 Phys. Psy., Bd. I. 8, g i. 43 
between stimuli must be finer. The method of estimated amount 
of difference would include all such estimates of the quanti- 
tative relations of differing stimuli. 
Twelve hundred experiments were made by the method of 
estimated amount of difference. Of these 400 were attempts to 
mark off on the scale the half of 500 mm., 400 were attempts to 
double 300 mm., and 400 were attempts to find the mean be- 
tween 300 and 700 mm. The details of the experiments were 
similar to those of the experiments by the method of just 
noticeable difference. In the attempt to halve 500 mm., the 
movable upright was placed by the recorder at the 500 mark on 
the scale, the experimentee moved his finger between the up- 
rights, and then tried to make a movement half as long. In the 
same way a stimulus of 300 mm. was given, and then an attempt 
was made to double the distance; and, in the third case, two 
stimuli of 300 and 700 mm. were given one second apart, and 
the attempt made to give their mean in a third movement. 
The results have been calculated and tabulated in precisely 
the same way as those of the experiments by the method of just 
noticeable difference, so that it is scarcely necessary to give a 
detailed description of the table. The four sets of averages for 
each kind of experiment represent in each case 100 experiments. 
Var. gives the mean variation from their average of the results 
obtained from the separate sets of ten experiments in each group 
of 100. Thus, to take the first 100 attempts to halve 500 mm., 
the figures on the first line in the table indicate that the average 
distance marked off was 305.8 mm., or 55.8 mm. too great; that 
the mean variation of the single distances from the average of 
each series of ten was 13.3, and the mean variation of the varia- 
tions 7.5. The - 41.2 to the right of these in the same line in- 
dicates that the average distance measured off in the attempt to 
double 300 mm. was 558.8 mm, or 41.2 mm. too short. The 44 
corresponding figures for experiments of the third kind show 
that the average of 100 attempts to find the mean between 300 
mm. and 700 mm. resulted in a distance of 517.4 mm., or 17.4 
mm. too great. The three sets of averages at the bottom of the 
table represent in each case 400 experiments. 
Extent of Movement-Method of Estimated Difference-F. Experimentee- 
T2oo Experiments-VII. VIII. 1890. 
TABLE III. 
halve 500 mm. 
double 300 mm. 
MEAN 300 AND 700 mm. 
EXT. 
V.E. 
v. 
EXT. 
V.E. 
v. 
EXT. 
V.E. 
v. 
AV; . . . 
+ 55-8 
13-3 
7-5 
- 41.2 
l6.6 
9.1 
4-17-4 
15-2 
8.2 
Var. . . 
^3 4 
2-5 
II 
234 
4.6 
23 
255 
3.8 
2.4 
Av. . . . 
+ 7i-9 
12.3 
6-7 
-18.1 
14-4 
7-7 
4- 36-3 
16.4 
9.2 
Var. . . 
9-9 
2 3 
21 
I2.7 
3* 
2-9 
8.7 
4-4 
2.8 
Av. . . . 
+ 5i-9 
II. 
6.6 
- 42.3 
"•5 
7-i 
4- 3-3 
14. 
8.9 
Var. . . 
10 8 
3-2 
2. 
l6.1 
2-3 
' 7 
13-9 
3-2 
2.3 
Av. . . . 
4- 41-3 
"•3 
5-9 
58.2 
131 
6.6 
- 7-4 
14.1 
8.6 
Var. . . 
6.9 
2-7 
2-7 
17.6 
2.1 
1.6 
11.6 
4-7 
2. 
Av. Av. . 
+55.2 
12. 
6.7 
- 39 9 
13.9 
7.6 
+ 12.4 
14.9 
8.7 
Av. V. . 
10.2 
2.7 
1.7 
17.4 
3. 
1.9 
12.4 
4. 
2.4 
As a result of 400 experiments of each kind, we find that the 
attempt to halve 500 mm. gave an average distance 55.2 mm. too 
long; the attempt to double 300 mm. a distance 39.9 mm. too 
short; and the attempt to find a mean between 300 and 700 mm. 
a distance 12.4 mm. too long. From the figures above these 
averages it is evident that the tendency toward the error they 
exhibit was fairly constant. In the attempts to halve 500 mm., 
not one of the forty sets of ten experiments gave an average that 
was not too great. In the attempts to double 300 mm., on the 
contrary, all but one set (1 mm. too great) gave an average 
that was too small. The experiments of the third kind were 
less constant. In three of the four groups of 100, the mean 
fixed upon was too great, but in only one of these groups 45 
did each of the ten sets of ten experiments give an average too 
great. The final averages are considerably modified by the 
marked tendency to overrate the standard observable in the 
second 100 experiments of each kind. This tendency has been 
pointed out in discussing the corresponding groups of the ex- 
periments made by the method of just noticeable difference. 
It is not easy to compare with each other the results obtained 
in the three kinds of experiments under discussion, for the ex- 
periments are quite different. It is, however, worthy of note 
that, whether we take the relation of the constant error made to 
the distance aimed at or to the stimulus given (in the experi- 
ments of the third class, to either of the stimuli), we find that 
the attempt to strike a mean between two stimuli results in a 
smaller error than the attempt to halve or to double a single 
stimulus. This is in harmony with the statement Wundt makes1 
in speaking of the measurement of mental intensities-namely, 
that the mean between two stimuli can be more accurately deter- 
mined than can a stimulus bearing some given relation to a 
single stimulus. The difference does not, however, appear to be 
so great as his words would lead one to expect. Moreover, it 
should be kept in mind that the mean between 300 and 700 mm. 
is 500 mm., that this was one of the stimuli most constantly used, 
and that the objective quantitative relations of the stimuli were 
known to the experimentee. Before the first ten experiments 
of the kind in question were made, a stimulus of 500 mm. had 
been given the experimentee twenty times, and a but slightly 
different stimulus of 510 mm. had been given ten times. As a 
result of the experiments discussed in this monograph, the writers 
have been led to believe that the memory for absolute stimuli is 
better than has been commonly thought. Some experiments to 
show the memory of the experimentee for the stimuli employed 
1 Phys. Psy , Bd. I. 8, § i. 46 
in these experiments on the extent of movement will be given in 
a later section. 
Furthermore, if we take, not the constant error, but the vari- 
able error or its variation, as a measure of the fineness of dis- 
crimination, what has been said of the accuracy with which a 
mean between two stimuli may be found does not hold. The 
400 attempts to halve 500 mm. gave a variable error of 12; the 
attempts to double 300 mm. a variable error of 13.9; while the 
attempts at the mean between 300 and 700 mm. gave a variable 
error greater than either, 14.9. The average variation of the 
variable error for the forty sets of ten experiments in each case 
(in the table, under the figures just quoted) was, too, least for 
the experiments of the first kind, and greatest for those of the 
third, being respectively 2.7, 3.0 and 4.0. 
The results of these experiments are all contrary to Fechner's 
law. If we assume that in such experiments we base our judg- 
ment on th£ quantitative relations of two sensations, and if, as 
Fechner maintained, the sensation increase as the logarithm 
of the stimulus, the attempt to halve 500 mm. should result 
in a distance less than 250, whereas it has given one of 305.2; 
the attempt to double 300 mm. ought to give a distance 
greater than 600, whereas it has given only 560.1; and the 
attempt to find the mean between 300 and 700 mm. should 
give a distance under 500, whereas the mean found is 512.4. 
That these results are not due simply to a constant ten- 
dency to over or underestimate the given stimuli, is evident 
from the fact that they are all contrary to Fechner's law, while 
any constant tendency to an over or underestimation of the 
stimuli would have the effect of making all the results greater 
or less together, and would not diminish the results of the ex- 
periments of the second kind, while increasing those of the first 
and third. 47 
It will be noticed that in these experiments by the method of 
estimated amount of difference, the variable error is, in pro- 
portion to the whole extent of the movement made, greater 
than in the case of the experiments by the method of just notice- 
able difference. The attempts at just greater than 500 mm. 
resulted in a distance of 539.4, with a variable error of 9.5, or 
.018, and the attempts at just less in a distance of 477.2, with a 
variable error of 8.7, or .018 of that movement. In the three 
kinds of experiments now under discussion the movements were 
respectively 305.2, 560.1 and 512.4 mm., and the variable errors 
corresponding to them .039, .025 and .029. The attempt to 
double 300 mm. resulted in a variable error smaller in relation 
to the whole extent of the movement made than was the variable 
error in the experiments of the first and third kinds. 
Sec. i6. Experiments by the Method of Average Error.-By 
the method of average error 1600 experiments were made with 
the apparatus first described, and 800 more were made later upon 
two different experimentees with the second piece of apparatus. 
Only the 1600 first made will be discussed in this section. 
They were of four different kinds, and consisted in attempts to 
measure off on the scale 100, 300, 500 and 700 mm. Thus, the 
recorder placed the movable upright at the 100 mm. mark 
on the scale, the experimentee moved his finger as described be- 
tween the uprights, and then, the movable upright having been 
taken off the scale, attempted to mark off a distance equal 
to the standard. The same thing was done for 300, 500 and 700 
mm. Four hundred experiments of each kind were made, and 
they were made, as were those described in the preceding sec- 
tions, in sets of ten. The results have been counted out and 
tabulated in the same way as those already given. The four sets 
of averages for each kind of experiment represent in each case 
100 experiments. The general averages at the bottom of the 48 
table give the results of the whole 400 experiments of each 
kind. The figures in italics refer to the results obtained for the 
separate sets of ten experiments. To take again as representa- 
tive a single block of figures: the +8.9, in the first line under 
100 mm., and opposite Av., signifies that the first hundred 
attempts to repeat a movement of 100 mm. resulted in an 
average movement of 108.9 mm., or one 8.9 mm. too long; the 
5.7 in the same line gives the mean variation of the single 
distances from the average of each series of ten; and the 3.5 
gives the mean variation from its average of this variation. In 
the second line, the 5.1, .6 and .6 give the mean variations of the 
results obtained for the ten separate sets of ten experiments 
from their respective averages as indicated in the line above. 
Extent of Movement-Method of Average Error-F. Experimentee-1600 
Experiments-VII. VIII. 1890. 
TABLE IV. 
ioo mm. 
300 mm. 
500 mm. 
700 
mm. 
EXT. 
V.E. 
V. 
EXT. 
V.E. 
V. 
EXT. 
V.E. 
V. 
EXT. 
V.E. 
N. 
Av. . . 
+ 8.9 
5-7 
3-5 
+ 
3-7 
8.8 
4.6 
4- .8 
10.8 
6.2 
4- 2.8 
10.4 
6.1 
Var. . . 
5 
.6 
.6 
g.6 
1.6 
1.0 
7-4 
1.8 
1.2 
10.8 
2.8 
i-7 
Av. . . 
4~ 18.4 
5-3 
3-1 
4-21-4 
9-5 
5-5 
4-27-6 
10.8 
6-3 
4-13-1 
7-7 
4-9 
Var. . . 
3 1 
1.1 
•5 
g.o 
z-9 
i-7 
2.8 
i-3 
4-7 
2.1 
13 
Av. . . 
+ H-7 
5-3 
2.9 
+ 
2.4 
7-3 
4.6 
4- 3-° 
8-3 
4-3 
- 6.4 
9-4 
5-7 
Var. . . 
2.g 
i-7 
•7 
8.6 
2.4 
1.6 
6.2 
2.2 
1.1 
7-8 
2.0 
1.2 
Av. . . 
+ 5-4 
4.8 
2.8 
- 
16.4 
9-5 
5-2 
-14.0 
8.2 
5-4 
- 28.9 
8. 
4.9 
Var. . . 
3-6 
.8 
.6 
g.6 
2.1 
.8 
8.6 
2.0 
i-4 
93 
2.0 
r-7 
Av. Av.. 
4- 11.8 
5.3 
3.1 
4- 2.8 
8.8 
5.0 
4- 4.3 
9.5 
5.5 
- 4.8 
8.9 
5.4 
Av. Var. 
3.7 
1. 
.6 
9.2 
2. 
6.9 
2.2 
1.2 
8.1 
.22 
15 
From the table it appears, in each case as a result of 400 ex- 
periments, that the attempts to mark off 100, 300, 500 and 700 
mm. gave an average of, respectively, 111.8, 302.8, 504.3 and 
695.2 mm. The tendency to overestimate the shortest distance, 
100 mm., was very marked. The greatest distance, 700 mm., 49 
was underestimated. There is a general agreement between 
these results and those of the experiments made with the 
second piece of apparatus, as will be pointed out in the next 
section; and it will be seen, when the experiments made on the 
force of movement are described, that a similar tendency is ap- 
parent. Evidently the constant error, as it stands here, cannot 
be taken alone as a measure of the fineness of discrimination in 
judging of the extent of movements. It is much the greatest 
in the case of the smallest movement. 
The variable error increases as the distance increases, until 
we come to the 700 mm. movement, when it falls somewhat. 
For the four movements it is respectively 5.3, 8.8,9.5 and 8.9; or 
Compared with the standard 
movement, 
For ioo mm. 300. 500. 700. 
.053 .029 .019 .013 
.O47 .029 .019 .013 
Compared with the move- 
ment actually made, 
Thus the variable error does not accord with Weber's law, but 
increases much more slowly than the stimulus. Its relation to 
the square root of the stimulus is for the four movements : 
For ioo mm. 300. 500. 700. 
.530 .513 .424 .336 
It increases, therefore, somewhat more slowly than the square 
root of the stimulus, for the first three movements, and falls 
considerably with the last. 
If this variable error be taken as a measure of discriminative 
power, it would seem that a distance of 500 mm. is not as ac- 
curately discriminated as one of 700. It should be taken into 
consideration that, in making such movements as those employed 
in these experiments, very different muscles are brought into 
play at different times, and the tension upon the skin and joints 
is much greater in the more extended movements. Placed as the 50 
experimentee was, before the 100 mm. mark on the scale, 
there was a distinct feeling of strain in reaching to the 700 mm. 
point, although the finger could readily be extended some 
distance beyond it. It is not inconceivable that the peculiar fac- 
tors entering into the consciousness of a movement of 700 mm. 
should make discrimination finer for such a movement than 
for one of 500 mm. A similar fall of the error with the last 
movement will be seen in the results of the later experiments 
to be described below. 
The amount of the variable error, of its variation, and of the 
constant error, for each of the four stimuli selected may be seen 
at a glance from the accompanying curves. 
Fig. 3. 
K.E. Curve representing the amount of the variable error for 100, 300, 500 and 700 mm. 
as determined for each point by 400 experiments by the method of average error. 
F. experimentee. 
V. Similar curve representing the mean variation of the above variable error. 
C.E. Similar curve representing the amount of the constant error. 
W's Law. A line representing the increase of the variable error according to Weber's 
law, the smallest magnitude being taken as norm. 
Vst. Curve representing the increase of the variable error as proportional to the square 
root of the stimulus. 
It is interesting to compare the variable error obtained in 
these experiments with that obtained in the experiments by the 
methods of just noticeable difference and estimated amount of 
difference, when the movement was approximately the same. In 51 
the experiments by the method of just noticeable difference, 
with movements of 477.2 and 539.4 mm., the error was .018; in 
those by the method of estimated amount of difference, with 
the movement of 512.4, the error was .029; in these experiments, 
with a movement of 504.3, the error was .019. Again, in the ex- 
periments by the method of estimated amount of difference, with 
a movement of 305.2, the error was .039 ; in these experiments, 
with a movement of 302.8, the error was .029. 
We find, therefore, that the variable error is greatest in exper- 
iments by the method of estimated amount of difference-a 
result one would naturally expect, in view of the difficulty of 
holding such relations in mind during the experiments. 
It may be well to insert here the record of a few attempts 
made to mark off from memory, and without previous stimulus 
being given, the distances used as stimuli in the experiments 
on the extent of movement. After twenty complete series of 
experiments had been made, that is, after one-half of the whole 
number, the experimentee tried to measure off ten times in suc- 
cession a distance of 100, then one of 300, of 500 and of 700 
mm. After forty series-the whole number of experiments- 
this was done again, but then done three times, making thirty 
attempts at each of the distances. The results of the forty 
trials at each of the four stimuli are as follows : 
TABLE V. 
100 mm. 
300 mm. 
500 mm. 
700 mm. 
EXT. 
V.E. 
EXT. 
V.E. 
EXT. 
V.E. 
EXT. 
V.E. 
After 20 . 
+ 34 
14 
4- 48 
21 
+ 37 
25 
+ II 
IO 
After 40 . 
+ 22 
IO 
" 7 
31 
- 5 
17 
- 10 
14 
-j- 20 
12 
+ 1 
13 
- 20 
13 
+ 8 
IS 
+ 19 
8 
- 9 
14 
- 29 
21 
- 16 
l6 
Av. . . . 
+ 24 
11 
+ 8 
20 
- 4 
19 
- 7 
14 52 
Thus forty attempts to measure off without previous stimulus 
100 mm. resulted in a distance 24 mm. too great; as many to 
measure off 300 gave a distance 8 mm. too great; the same 
number for 500 and 700 gave distances too short by, respect- 
ively, 4 and 7 mm. There is evidence of the tendency before 
remarked to overrate the shorter distance. The variable error 
is in every case greater than in the corresponding division of 
the experiments by the method of average error, where previous 
stimulus was given. The error falls, as it there does, with the 
last movement, indicating a greater accuracy of discrimination. 
These experiments show that our memory for absolute dis- 
tances is fairly accurate, and they are of some significance in 
connection with the experiments by the method of estimated 
amount of difference. The attempt to find a mean between 300 
and 700 mm. resulted in a constant error of +12.4; this mean 
was known by the experimentee to be 500; the attempt to mark 
off 500 mm. from memory gave a constant error of -4 only. It 
follows that the smallness of the constant error in the attempts to 
find the mean between 300 and 700 mm. proves little in regard to 
the accuracy with which a mean can, in general, be found between 
two stimuli. On the other hand, the fact that the variable error in 
these attempts to find a mean was .029 of the movement made, 
while the variable error in the memory-experiments was .038 of 
the movement made, indicates that the presence of the stimulus 
had the effect of heightening the accuracy of the judgments. 
The amount of the variable error, in the experiments by the 
method of average error, when previous stimulus was given, 
was, as has been said, 
Compared with the standard 
movement, 
For ioo mm. 300. 500. 700. 
.053 .029 .019 .013 
.047 .029 .019 .013 
Compared with the move- 
ment actually made, 53 
In these experiments it is 
Compared with the move- 
ment aimed at, 
For ioo mm. 300. 500. ' 700. 
.110 .067 .038 .020 
.089 .065 .038 .022 
Compared with the move- 
ment actually made, 
It is thus about double the error obtained with previous stim- 
ulus given. Its excess over this would seem to indicate the 
variable error of memory. 
The amount of this error, and of the constant error for each 
of the four distances, is represented in the accompanying curves. 
They represent the amount of the error as compared with the 
distance aimed at, and not as compared with the movement 
actually made. As they are drawn on the same scale as the 
former curves, their divergence from these may be readily seen. 
Fig. 4. 
Sec. 17. Later Experiments by the Method of Average Error. 
-The 800 experiments by the method of average error with 
the second piece of apparatus were made upon two subjects- 
400 upon each. The distances taken as stimuli were, as before, 
100, 300, 500 and 700 mm. The details of the experiments 
4 54 
were similar in the two cases, except that in the later ex- 
periments the stimulus was not determined by the two uprights, 
but was set by the experimentee himself. Before each set of 
ten experiments with the one stimulus, the experimentee was 
told to measure off the distance chosen as stimulus. He was 
then informed whether the distance he had measured off was 
too great or too small. Four more trials were allowed him, after 
each of which he was informed of the nature and amount of his 
error, and he was then required, in making the experiments, to 
determine the extent of the first movement by the standard thus 
held in memory. The second movement was an attempt to 
repeat the first. It will be seen that the conditions of the first 
and second movements were, in this case, similar, neither of 
them being terminated by a fixed limit. 55 
ioo mm. 
300 mm. 
500 mm. 
700 mm. 
EXT. I. 
EXT. II. 
AV. E. 
N. E. 
EXT. I. 
EXT. II. 
AV. E. 
V. E. 
EXT. 1. 
EXT. II. 
AV. E. 
V. E. 
EXT. 1. 
EXT. II. 
AV. E. 
V. E. 
K. 
Av 
IO5.9 
+ 3-8 
II.9 
11.4 
307-8 
+ 0.2 
21.8 
21.1 
506.3 
- 3-2 
25-8 
22.6 
692.9 
- O.7 
21.0 
16.9 
Var 
8.8 
5-6 
2.1 
2.0 
21.2 
10.0 
4.6 
JZ 
'5-6 
II.O 
7-4 
6-5 
^3-o 
'3'8 
5-9 
3-7 
D. 
Av 
101.0 
- 2.9 
17-3 
l6.2 
301.1 
- 14-3 
32-4 
26.1 
473-9 
- 24.1 
4i-3 
3i-3 
697-7 
- I9.7 
33-5 
25-7 
Var 
5-6 
43 
2.8 
3-7 
^4-7 
10.0 
5-r 
7-i 
^•9 
IO.I 
t>3 
53 
15-6 
'3-9 
10.2 
7-5 
Av. Av. . . 
103.4 
+ 0.4 
14.6 
13.8 
304.4 
- 7.0 
27.1 
23.6 
490.I 
- 13.6 
33.5 
26.9 
695.3 
- 10.2 
27.2 
21.3 
Av. Var. . . 
7.2 
4.9 
2.4 
2.8 
17.9 
10.0 
4.8 
6.1 
15.2 
10.5 
6.8 
6.0 
14.3 
13.8 
8.0 
5.6 
Extent of Movement-Method of Average Error-K. and D. Experimentees-800 Experiments-1890. 
TABLE VL 56 
These experiments were, unfortunately, not made on F., the 
subject in the earlier experiments, and one cannot be sure that 
the difference in the constant error is due to the difference in 
the conditions under which the experiments were made. Still, 
it is interesting to note that in these experiments there is a tend- 
ency to diminish the relative extent of the second movement, 
and it is not improbable that the absence of the upright, which, 
in the earlier experiments, limited the first movement, has some- 
thing to do with this. 
The tendency to underestimate the standard movement, as the 
distance increases, is here, as before, sufficiently marked. For 
the first three movements the constant error varies nearly as the 
movement. With the last movement it diminishes. 
Neither the average error nor the variable error increases 
directly as the extent of the movement, and hence they do not 
agree with Weber's law. For both subjects they increase more 
nearly as the square root of the distance for the first three move- 
ments, though the approximation is not very close. For both 
subjects these errors fall decidedly with the last movement. 
Whether, therefore, we consider the constant error, the vari- 
able error or the average error, we find that discrimination is 
finer for a movement of 700 mm. than for one of 500. This 
accords with the results of the earlier experiments, as far 
as the variable error is concerned. In those experiments, how- 
ever, the constant error does not follow the course it does here. 
It falls, then rises, then falls again to a negative error rather 
greater than the positive error at 500 mm. 
It will be seen that, though the results obtained with the two 
subjects are of the same general nature, all three errors are much 
greater in the case of the second subject. This would indicate 
a greater fineness of discrimination in the first, who is a woman. 
Whether women generally have a finer discrimination in judging 57 
of the extent of movements would have to be determined by- 
experimenting on a considerable number of persons. A com- 
parison of the variable error obtained in the experiments on K. 
with that obtained in the earlier experiments on F will show 
that F.'s error is smaller for all four points. F.'s constant error, 
however, is larger. 
Curves indicating the amount of the constant error, the aver- 
age error, and the variable error for K. and D., as determined by 
loo experiments for each point, as also curves giving the 
average for the two subjects, are subjoined. 
Fig. 5.-Curves for K. 58 
Fig. 6.-Curves for D. 59 
Fig. 7-Average for K. and D. 
The amount of the average and variable errors is thus, in 
relation to the distance aimed at: 
For ioo mm. 
300. 
500. 
700. 
Average error for K., 
.119 
•073 
.051 
.030 
" " D., 
.173 
.108 
.O83 
.O48 
Variable error for K., 
.114 
.070 
•045 
.024 
" " D., 
.162 
.087 
.O63 
•037 
If the error be compared with the distance actually measured 
off, the figures are but little changed. The abscisses in the 
curves indicate the distance aimed at. The relative amount of 
the constant error in figures might be misleading, as it passes 
from + to -. 
Sec. i 8. Experiments by the Method of Right and Wrong 
Cases.-By the method of right and wrong cases 800 experi- 60 
merits were made, the stimuli used being 500 and 510 mm. 
The method has been described and criticised in the introduc- 
tion, and it is not necessary to dwell upon it here. Most of the 
needful details concerning the experiments have already been 
given. Briefly stated, the experiments were made as follows : 
The movable upright was placed by the recorder at the 500 or 
the 510 mm. mark on the scale, as the case might be. The ex- 
perimentee then moved his finger between the uprights, and 
thus obtained the standard movement. As he recovered posi- 
tion for a second movement, the movable standard was shifted 
to the 510 or the 500 mm. mark. He again made a movement 
between the two uprights, and then gave a decision as to whether 
the second movement was longer or shorter than the first. In 
giving his decision he used the letters a, b or c to express his 
degree of confidence in the correctness of this judgment, a indi- 
cating that he was quite confident, b that he was fairly confident, 
and c that he was less confident. 
Twenty experiments by this method were made at a sitting; 
in ten of them the second stimulus was the greater, and in ten 
the less. Of course, the second stimulus was made greater or 
less at random, and care was taken to avoid following any order 
which could be surmised by the experimentee. The twenty 
experiments were all made, as were the sets of ten in the other 
kinds of experiments, before taking up another kind. The 
results of the experiments are given in the accompanying table. 61 
TABLE VII. 
Extent of Movement-Method of Right and Wrong Cases-500:510 mm.- 
F. Experimentee-800 Experiments-VIL VIII. 1890. 
2D THE GREATER. 
2D 
THE LESS. 
AVERAGE. 
R. 
w. 
D. 
R. 
w. 
D. 
R. 
w. 
D. 
i st Group 
76 
21 
3 
75 
22 
3 
75-5 
21.5 
3 
2d " 
67 
32 
1 
7i 
28 
1 
69 
30 
1 
3d " 
8l 
19 
0 
67 
33 
0 
74 
26 
0 
4th " 
76 
24 
0 
74 
26 
0 
75 
25 
0 
Average . * 
75 
24 
1 
71.8 
27.2 
1 
73-4 
25.6 
1 
of whole 
number of 
right cases when the 2 
d was greater . 
51-1 
44 44 
44 
wrong 
44 44 
4 
44 
46.9 
DEGREE 
OF CONFIDENCE. 
a. 
b. 
c. 
d. 
o]o of times used 
•5 
12.5 
86. 
1. 
44 44 
" when 
right . 
.... 
•7 
151 
84.2 
44 44 
44 
44 
wrong 
. . . . 
.0 
5.1 
94-9 
44 44 
right in the 
case of each degree . 
100 
97- 
71.7 
R, Right. 
IV, Wrong. 
D, Doubtful. 
First group representing 100 experiments of each sort, as does each of the other groups. 
Average, The average of 400 experiments. 
a, Quite confident. 
b, Fairly confident. 
c, Less confident. 
The 400 experiments in which the second movement was 
greater, and the 4C0 in which it was less, are each divided into 
four groups of 100 experiments, and the percentages, which are 
also the numbers of right, wrong and doubtful judgments, are 
given for each group, under R, W, and D. The groups are ar- 
ranged in chronological order. Thus, in the first 100 experi- 62 
merits made with the second movement greater, it was judged 
greater seventy-six times, less twenty-one times, and the experi- 
mentee was unable to come to a decision three times. In the 
first ioo with the second movement less, it was judged less 
seventy-five times, greater twenty-two times, and three times 
no decision was arrived at. Combining these two sets of figures 
in an average we get (in the table, to the right of the figures 
just given) 75.5 per cent, of right judgments, 21.5 per cent, of 
wrong ones, and three per cent, of cases in which the experi- 
mentee was in doubt. Below the averages for the groups of 
100 are the averages for the four groups of each kind. To the 
right of these is their average, which, consequently, represents 
800 experiments. 
The percentage of the whole number of right cases when 
the second movement was greater (51.1) is obtained, of course, 
by adding the two averages 75 and 71.8, and dividing 75 by their 
sum. The percentage of the whole number of wrong cases 
when the second movement was greater (46.9) is similarly ob- 
tained from the averages 24 and 27.2. 
As regards the degree of confidence expressed, the figures in 
the table are sufficiently clear. They have reference to the 
whole 800 experiments. Thus, the experimentee was quite 
confident he was right only 0.5 percent, of the time; he was 
fairly confident 12.5 per cent., less confident still 86.0 per cent., 
and wholly undecided 1.0 per cent. To pass to the second line, 
he was quite confident he was right 0.7 per cent, of the whole 
number of times in which he was right; he was fairly confi- 
dent 15.1 per cent., and less confident 84.2 per cent. The third 
line of figures so resembles the second as to need no comment. 
The fourth shows to what degree the subjective feeling of con- 
fidence corresponded with objective correctness in the judg 
ment. It indicates that when the experimentee was quite con- 63 
fident he was right, he was always right; when he was fairly 
confident, he was right 97 per cent, of the time; and when he 
was less confident, he was right 71.7 per cent, of the time. 
Thus it appears that with the two stimuli 500 and 510 mm., 
75 per cent, of the judgments were right when the second 
stimulus was the greater, and 71.8 per cent, were right when 
the second was the less. This would indicate that the first 
movement was slightly underrated on the whole. In the earlier 
experiments by the method of average error a standard move- 
ment of 500 mm. was slightly overrated. In the later experi- 
ments it was underrated. The second group of the experiments 
by the method of right and wrong cases differs from the other 
three in showing an inclination to overestimate the standard. 
It will be remembered that the corresponding groups in the ex- 
periments by each of the other three methods showed a similar 
tendency. The ten series of experiments represented by these 
groups were made within a period of seven days. During this 
time the standard movement seems to have been pretty con- 
stantly overrated. 
As in the criticism of the just noticeable difference as a 
standard of measurement it was objected that the significance of 
the words "just noticeable" is not easily determined, so with 
regard to the degrees of confidence a, b, and c, it may be ob- 
jected that the terms "quite confident," "fairly confident " and 
"less confident" are extremely vague. In a series of experi- 
ments with the one observer each of these terms may be as- 
sumed, perhaps, to have approximately the same meaning in 
different parts of the series; but the quantitative relations of 
the subjective feeling of confidence in the three cases remain 
very obscure, nor can it be assumed that they may be measured 
by the percentage of right cases corresponding to each degree 
of confidence. The fact that an observer is always right when 64 
he feels quite confident, and right ninety-seven per cent, of the 
times when he feels fairly confident, does not prove that the 
amount or intensity of his confidence in the two instances is as 
too to 97. We see, however, from the figures in the table that 
the observer was sure to be right when he felt confident enough 
to say a, nearly sure to be right when he was willing to say b, 
and much less likely to be right when moved to say c. 
The average variable error obtained with a stimulus of 500 
mm. in the experiments by the method of average error 
was 9.5. Now, if we eliminate in these experiments the con- 
stant error (.7), and calculate the average error (that cor- 
responding to 78 per cent, of right cases) by applying 
the probability integral to the figures in the above table, we 
obtain an error of 12.1. The average variable error in the ex- 
periments by the method of just noticeable difference was, for 
just greater than 500 mm., 9.5, and for just less 8.7. If, 
by the use of the probability integral, we calculate from the 
just noticeable difference (Table II.), the difference with which 
the observer should be right in his judgment seventy-eight per 
cent, of the time, we obtain a difference of 12.9, which agrees 
very well with the results obtained by the method of right and 
wrong cases. But it is evident that the several series vary 
more, and that the probable error of the result is greater. PART II. 
ON THE FORCE OF MOVEMENT. 
Sec. 19. Introductory.-The experiments hitherto made on 
the so-called " muscular sense," or " sense of innervation," 
have been carried out with weights. We also began with such 
experiments (which are described in Part IV.), but the great 
advantage of using a dynamometer soon occurred to us. With 
a dynamometer the several psychophysical methods may be 
used and compared, and the factors concerned in the perception 
of movement may be distinguished and studied separately. If 
lifted weights be used as stimuli, the only method which can be 
applied conveniently is that of right and wrong cases. It is 
not possible to make one weight equal to another, just heavier 
or twice as heavy, without repeated trials, and if regularity be 
observed the method of right and wrong cases must be intro- 
duced as a secondary help. The method of right and wrong 
cases is indeed admirably suited to the discrimination of lifted 
weights, as the magnitude of the stimulus can be measured 
without appreciable error. But this method, as we have seen, 
requires numerous experiments before conclusive results can be 
reached.1 
Further, it is not easy to analyze the factors concerned in the 
perception of lifted weights. We might expect the lighter 
1 Fechner heroically made 67072 trials with lifted weights, and yet considered these too 
few. Galton (Inquiries into Human Faculty, London, 1883, Appendix D) suggests that 
the influence of chance may be lessened by allowing the observer to arrange three 
weights in order. But the observer might distinguish the heaviest weight from the light- 
est and place the third between. 
65 66 
weight to be lifted higher and faster than the heavier, and the 
judgment of the observer might depend on the perception of 
the force, of the extent or of the time. Muller and Schumann1 
conclude that the observer's judgment is due chiefly to the time, 
and explain Weber's law by the principle of mechanics, accord- 
ing to which a proportional increase in force is required to cause 
an equal increase in velocity. Our experiments prove the in- 
correctness of this ingenious theory, as they show that the force 
of a movement can be judged better than its time, and that the 
judgment of time follows Weber's law more nearly than the 
judgment of force. 
Sec. 20. Apparatus and Methods.-The clinical dynamome- 
ters in use are too inaccurate for scientific experiment.2 We 
therefore constructed a special dynamometer for our work. This 
is shown in Fig. 8. 
Fig. 8.-Apparatus for Measuring the Force of Movement. 
A heavy spiral spring is enclosed in the brass cylinder (7? P) 
to which the handle (77) is attached by a bar. The bar runs on 
1 G E. Muller und Fr. Schumann. Ueber die psychologischen Grundlagen der Ver- 
gleichung gehobener Gewichte. Pfliiger's Archiv, XLV. 1889. 
2 We tested an oval (Collin) dynamometer and found the absolute readings to be 
worthless, while the average error of adjustment was very large. A reputable instrument- 
maker informed us that he did not know whether the reading on his dynamometers gave 
lbs. or kg. He made the springs, and, without testing them, copied the scale from a 
French dynamometer. The scientific value and practical usefulness of clinical observa- 
tions would be much increased by more accurate methods and instruments. 67 
double wheels almost without friction. When the handle is 
pulled out the amount of force applied is shown on the scale (T3), 
as in an ordinary spring balance. The pointer, however, not 
being attached to the bar, is only pushed forward, and stays at 
the point of the maximum pull. The recorder can thus take the 
exact reading before replacing the pointer. The scale registers up 
to 10 kg., and in addition a second scale (7?) is attached, register- 
ing 15 kg. further. By means of the bar, pivot and screw (at N) 
the spring can be set at any point up to 15 kg. In such a case, 
the observer must pull the set amount before the handle move, 
while the force of his pull beyond this amount is registered on 
the front scale. The advantage of a double scale is twofold. 
The maximum force of movement may be kept the same, while 
the extent and rate are altered, and the total extent of the 
movement may be made as small as desired.1 The instrument 
was tested carefully, and the errors of adjustment and of read- 
ing were small as compared with the quantity measured. The 
force of pull was varied from 2 to 16 kg., and the reading was 
taken to .01 kg.2 
The movement used in our experiments was a free pull with 
the arm. The observer sat on a high chair with his right 
shoulder opposite the instrument, which was fastened to an or- 
dinary table. The handle was conveniently grasped, always in 
1 In some clinical dynamometers the amount of movement is so great that the grasp with 
which the observer begins must be altered while the movement is being made; if, on the 
other hand, the amount of the movement be less, it is too small to be accurately registered. 
In our dynamometer there is an attachment which may be fastened to the extension (below 
H), if it be desired to use the grasp of the hand, or the pressure of the thumb and forefinger. 
The range (25 kg.) of this dynamometer is suitable for the latter movement, which we 
recommend for clinical observation, as it is less liable to accidental alterations of position 
than the grasp of hand. 
2 The error of reading would not increase as the magnitude of the stimulus is taken 
greater, and would form a larger part of the (apparent) error of observation when small 
than when great. It is, however, so small a part of the total error in these experiments 
that it may be disregarded. The error of the instrument and possible errors of calcula- 
tion are more serious complications, but we should not know how to allow for these. 68 
the same manner. A natural pull was made, the muscular 
mechanism being different for pulls of different force, but kept 
as nearly as possible the same for the same pull. A screen (at- 
tached in S) hid the scale and record-book from the observer. 
We had intended to give the observer the first or normal force 
of pull by an adjustable stop which can be attached to the exten- 
sion (below H), and removed before the second or judgment pull 
be made. We saw, however, that some part of the force of the 
first pull would be spent in striking the stop, which led us to 
adopt the method described in Part I., which we believe to be a 
great advantage in all psychological experiment in which it can 
be used. 
The observer was told to give a pull of (say) 2 kg. As might 
have been expected, his error in estimating a standard mag- 
nitude was usually very great. He was then told the direction 
and approximate amount of his error and allowed to try again. 
This was repeated until he had made five trials, by which time 
he could usually give the standard without great variation. A 
series of ten judgments was then made, the observer giving in 
each trial first the standard pull from memory, and then a pull 
as nearly as possible (say) equal to it. A series of five trials 
preceded each series of ten judgments, and if in the course of 
the series the first pulls varied greatly from the standard, the 
observer was told to make his pulls less or greater as the case 
might be. The observer was told not to attend particularly to 
the amount of the standard, but to concentrate his attention on 
the comparison; and the error of his observation, as discovered 
in the difference of the stimuli, was made the basis of the aver- 
ages. With this method the observer produces and perceives 
both stimuli under like conditions, and the comparison can be 
made to great advantage. 
The time conditions were made as constant as possible. A 69 
pendulum, giving seconds, swung in front of the observer, and 
after a little practice its movement could be followed automati- 
cally. The first or standard pull was made at the beginning of a 
second and lasted about one second, a second was allowed for 
recovery, and the judgment pull was made during the third 
second. The stronger pulls lasted, approximately, one second, 
while the weaker pulls were made in a little less time, the ob- 
server being allowed to make them in the manner which he 
found most convenient. He was expected to make his decision 
during the fourth second. If we had been able to continue our 
work, we should have measured these times by means of a 
chronoscope, so as to discover more exactly the relations between 
force, extent and rate. We believe that interesting results 
would be disclosed by such a research. 
When 2, 4 and 8 kg. were used as stimuli, the extent of the pull 
was proportional to the force. When a stimulus of 16 kg. was 
used, the spring was set at io, and the handle began to move when 
the force was io, and the distance it moved was proportional to the 
force beyond io. The handle moved 6.4 mm. for each kg. If 
time had permitted, we should have used dynamometers in which 
the extent would have been kept constant, and also those in 
which it would have varied otherwise than in direct proportion to 
the force. We believe the relations between these magnitudes 
to be an interesting subject for research. The present experi- 
ments on the force of movement are complicated by the varia- 
tion in extent. The observer's attention was concentrated on 
the force, and casual introspection indicated that the difference 
in force only was regarded. More careful introspection (taken 
in connection with the accuracy with which the extent of move- 
ments can be judged, as shown in Part I.) seemed, however, to 
indicate that the observer was helped more by the variation in 
extent than he had at first supposed. As the extent varied 
5 70 
directly with the force for 2, 4 and 8 kg., any general rela- 
tion (such as Weber's law) between the error of observation and 
the magnitude of the stimulus would not be affected. With 16 
kg., however, the extent was to the force as 6: 16. If the 
observer therefore judged partly by the extent, the error for 16 
kg. would be relatively smaller than if he judged only by the 
force. 
Most of the experiments (4000 judgments) on the force of 
movement were made by J. and C., who acted alternately as ob- 
server and recorder. They did not know any results of their 
own experiments until the whole series had been completed, and 
but little concerning the results of the other observer, as these 
were not calculated while the work was in progress. A series 
of experiments with the several magnitudes and by the different 
methods was made together, the order being altered in each 
series, so as to eliminate, as far as possible, the effects of fatigue 
and contrast. By the method of average error, 400 experiments 
each were made on A, D. and K., C. or F acting as recorder. 
Sec. 21. Experiments by the Method of the Just Noticeable 
Difference.-In these experiments a standard pull, 2, 4, 8 or 
16 kg., was fixed in mind by a series of five trials, and then a 
series of ten judgments was made. The observer gave the 
standard, and then a pull which appeared just greater (or less). 
We have already considered the difficulties in the way of 
making experiments by the method of the just noticeable 
difference. In these experiments the observer did not at- 
tempt to make the difference in the pulls so great that he was 
sure that he was right in the objective relation of the 
magnitudes, but so great that he felt tolerably confident in 
the correctness of his judgment. He would have guessed 
himself to have been right in his judgment about nine 
times out of ten, the confidence corresponding to that marked 71 
" b" in the method of right and wrong cases. If the observer 
made the difference greater than he had intended, or for some 
other reason felt sure that his judgment was correct, this ex- 
periment was marked if, on the other hand, he felt uncer- 
tain as to the correctness of his judgment, the experiment was 
marked " c." It would have been more accurate to have treated 
these different classes of judgment separately, but as " and 
" c" seldom occurred, we did not undertake the additional calcu- 
lation which this would have involved. 
The results of the experiments by this method are given in 
Table VIII., which is arranged in a manner analogous to those 
in Part I. 72 
Force of Movement-Method of the Just Noticeable Difference-J. and C -1600 Experiments-IX. 1890-I. 1891. 
2 KG. = 200. 
4 KG. = 409. 
8 KG. = 800. 
16 KG. = l600. 
I. 
V. 
V'. 
V. E. 
< w. 
i 
L 
v- 
n. 
V'. 
V. E. 
^w- 
I. 
V- 
n. 
vl- 
V.E. 
w. 
I. 
V. 
11. 
vl- 
V. E. 
51 w. 
ATTEMPT AT JUST GREATER. 
J- • • 
Var. . 
205 
8 
>5 
6 
+ 34 
7 
+ 65 
10 
17 
6 
I3-I 
4-2 
3 
389 
29 
20 
8 
+ 49 
23 
26 
8 
20. 
5-2 
7 
781 
32 
43 
24 
+ 55 
27 
48 
25 
35-8 
256 
12 
l6o8 
56 
5i 
25 
+ 56 
35 
47 
16 
53-5 
23-2 
27 
C. . . 
Var. . 
209 
7 
21 
7 
20 
7 
17-3 
43 
I 
389 
15 
3° 
8 
+ 84 
15 
34 
8 
33-5 
8-7 
3 
789 
31 
52 
10 
+ 91 
23 
48 
37-5 
83 
5 
1609 
32 
40 
IO 
+ 79 
20 
45 
9 
37-3 
5-2 
5 
ATTEMPT AT JUST LESS. 
J- • • 
Var. . 
210 
10 
14 
4 
- 20 
8 
13 
3 
18 
7 
10.8 
2.8 
20. 
3-8 
10 
400 
32 
24 
8 
- 3i 
26 
5 
i9-7 
7-2 
17 
778 
43 
34 
8 
- 72 
28 
36 
7 
29.2 
7-8 
8 
1617 
36 
40 
IO 
- 89 
24 
49 
IO 
39-6 
IO. 
9 
C. . . 
Var. . 
215 
18 
18 
- 36 
7 
383 
18 
3i 
6 
- 54 
27 
25 
9 
26.8 
7- 
11 
812 
39 
45 
IO 
-142 
36 
42 
38-9 
IO.Q 
5 
1646 
42 
49 
29 
-133 
28 
34 
16 
34-9 
4-7 
2 
Av.. . 
210 
17 
±39 
17 
15.3 
52 
390 
26 
±54 
28 
25. 
9.5 
790 
43 
± 90 
43 
35.3 
7.5 
1620 45 
1 
± 89 
44 
41.3 
10.7 
TABLE VIII. 73 
We think the table will be readily understood. The unit 
used in the tables and text is .01 kg. The results for the two 
observers, J. and C, and for "just greater" and "just less" are 
given separately. The basis of the figures opposite J. and C. is 
ten series, each containing ten judgments, while the average is 
the result of the 400 experiments with each magnitude. The 
figures in italics opposite Var. give the average variations of the 
ten series. This variation is very nearly proportional to the 
. -845 / 
probable error of the average, which is about ~~ or % (more 
nearly .282), the variation given in the tables. Under I. is given 
the average force of the first or standard pull, and under II. the 
average amount of difference between the two pulls, together 
with its direction-that is, the "just noticeable difference." 
Under V. and K1, respectively, are given the average variations 
of the pulls from the series to which they belong. These varia- 
tions are dependent on memory for the standard, and the ob- 
server was directed not to pay special attention to this. They 
are, therefore, more complex than the other errors, and less suit- 
able for studying the relation between the magnitude of the 
stimulus and the error of observation. Under V. E., the vari- 
able error, is given the average variation of the difference 
between the pulls. Thus, if in a series of ten experiments, the 
observer make the second pull on the average .5 kg. greater 
than the first, sometimes the difference will be greater than .5 
kg., sometimes less, sometimes even negative; the average 
variation or error of these differences is the variable error, 
V. E., in the tables. Under % JU is given the percentage 
(as there are 100 experiments, it is also the number) of wrong 
cases; that is, those cases in which the real relation of the 
stimuli was the reverse of that intended by the observer. Thus, 
in the first quarter of the first line of figures in the table, we see 
that the observer, y, in giving the standard pull 2 kg. =200 74 
from memory, made it on the average 205, or too great, with 
an average variation in each series of 15. In attempting to make 
the second pull just greater than the first, the observer made it 
on the average 34 greater, with an average variation in each 
series of 17. The just noticeable difference would, therefore, be 
34, or % of the stimulus. This is the only figure usually re- 
corded in experiments by this method. We, however, believe 
that the just noticeable difference only has an exact meaning 
when taken in connection with the number of errors, or the vari- 
able error. By regarding these quantities we are able to com- 
bine the several methods, and regard this as one of the chief 
results of our work. In the case under consideration Jis vari- 
able error was 13.1, and in the course of the 100 judgments the 
second pull was made less than the first three times. 
Sec. 22. Discussion of the Method and Results.-The relation 
of the just noticeable difference to the magnitude of the stimulus 
and its variation with the two observers, and according as just 
greater or just less was attempted, may be conveniently seen in 
the accompanying curves. From the just noticeable differences 
Fig. 9. 
Fig. io. 75 
shown in the table and curves, we should be compelled to con- 
elude that J's fineness of discrimination is nearly twice as great 
as C.'s, whereas the variable error and ratio of right to wrong 
cases in the same experiments give nearly the same fineness of 
discrimination for the two observers. In fact,y. was right 88% 
of the time, while C. was right 95%, which shows that the just 
noticeable difference meant something quite different to each 
of the observers. With unskilled observers we might expect 
still greater variation in choosing a just noticeable difference. 
The just noticeable difference seems, further, to have been 
given a different value, according as just less or just greater was 
attempted. It is evident that the corresponding curves for J. 
and C. have much the same shape, whereas the "just less " curves 
differ entirely in their shape from the "just greater" curves. 
For the two smaller magnitudes "just less" was taken smaller 
than "just greater," whereas the reverse holds with the two 
larger magnitudes. This seems to show a marked tendency to 
underestimate the second pull as compared with the first in the 
case of the smaller magnitudes, and to overestimate it in the 
case of the greater magnitudes. Such a tendency is also ap- 
parent in the experiments by the method of average error (see 
Table X.), but is not there so marked. 
If Weber's law obtained, all the curves representing the just 
noticeable difference in the figures would be straight lines be- 
ginning at the origin, and the average would follow the broken 
line. The just noticeable difference is, on the average, about 
one-eighth of the stimulus, but it grows rapidly smaller as the 
stimulus is taken greater. It is approximately 
Magnitude, 2 4 8 16 kg. 
Just noticeable difference, i f i A 
consequently, according to the method of just noticeable dif- 
ference, Weber's law does not obtain, even within narrow limits, 
for the force of movement. 76 
We have given our reasons for holding that the just noticeable 
difference is an arbitrary point without objective criterion. We 
think it cannot be used as a test for comparing different ob- 
servers. We see that it failed with J. and C. In Parts IV. and 
V. we shall find that some observers rarely think their judg- 
ment correct, unless it be in fact correct, while others are con- 
tinually sure that they notice a difference which is not present. 
If observers be arranged according to the difference they think 
they can perceive, an order exactly the reverse of their real fine- 
ness of discrimination will often be obtained. An observer 
may, indeed, keep the just noticeable difference fairly constant 
under the same circumstances, but this, we think, is because he 
has in mind a certain apparent amount of difference or degree 
of confidence. When the circumstances are altered, it is very 
difficult for the observer to give the same value to the just no- 
ticeable difference. Thus "just greater " for J. was 34 with 200, 
and 56 with 1600. We find, therefore, that the just noticeable 
difference was less than doubled when the stimuli were taken 
eight times as great. On regardingthe variable error, however, 
we see that for 200 it is less than half the just noticeable differ- 
ence, and we conclude that the observer could usually notice this 
difference, and on regarding the percentage of wrong cases, we 
see that it was, in fact, misjudged only three times out of 100 
trials. With 1600, on the other hand, the variable error is 
nearly as great as the just noticeable difference, and we must 
conclude that the observer would often fail to notice this differ- 
ence ; and on regarding the percentage of wrong cases, we see 
that it was, in fact, misjudged twenty-seven times out of 100. 
The just noticeable differences in the two cases do not mean the 
same thing, and cannot be used to measure the accuracy of dis- 
crimination ; still less can they be regarded as equal mental 
magnitudes. The fact that the just noticeable difference tends 77 
to increase more slowly than the variable error shows an incli- 
nation on the part of the observer to make the just noticeable 
difference, not an equally noticeable difference, but an equal 
objective difference. As in our other experiments with estimated 
amounts of difference, therefore, the observer tends to judge 
such magnitudes (doubtless through association) according to 
their real relations, and not according to the logarithms of these. 
Sec. 23. Reduction of the Method to that of Right and Wrong 
Cases.-In making a difference so great as to be just noticeable, 
the observer did not aim to make it so great as to be absolutely 
sure of the correctness of his judgment, but so great as to feel 
fairly confident of its correctness, expecting in a general way to 
be right about nine times out of ten. If an observer could, in 
fact, employ such a criterion, the just noticeable difference 
would measure his fineness of discrimination. We should in this 
case, however, have substituted the method of right and wrong 
cases for the method of just noticeable difference. In our ex- 
periments the observer did not know the result of his judg- 
ments, and could not tell how nearly he was coming to such a 
standard. The table shows that, in fact, the percentages of right 
cases varied from 73 to 99. Still, if the probability integral may 
be applied to these series, we can reduce them to a common 
standard. From the just noticeable difference and the percent- 
age of right cases actually obtained, we can calculate how large 
the just noticeable difference should have been made in order to 
give a certain constant percentage of right cases. We thus re- 
place the method of the just noticeable difference by the method 
of right and wrong cases. The probable errors (the differences 
with which the observer might expect to be right 75% of the 
time) are given in Table IX., and the relations are shown in 
Figs. 11 and 12. 78 
Probable Errors from the Just Noticeable Difference. 
TABLE IX. 
200 
400 
• 800 
1600 
JUST GREATER. 
J 
12.2 
22.4 
30.2 
61.5 
c 
lS-3 
30.1 
37-3 
32-4 
JUST LESS. 
J 
10-5 
22. 
34-6 
447 
c 
16.4 
29-7 
58. 
43-6 
Av 
14.3 
26. 
40. 
45.5 
Fig. ii. 
Fig. 12. 
The errors given in the table and shown in the curves are evi- 
dently more regular and accordant than those for the method of 
just noticeable difference. They also agree with the results 
obtained by other methods. They show that the fineness of 
discrimination of the two observers is nearly the same, but that 
J. is more accurate with the three smaller, C. with the largest 
magnitude. This result seems natural, as C.'s arm is much 
stronger than J.'s, but it is contrary to Weber's law. As the 
curves for J. and C, and it may be for " just greater " and "just 79 
less," follow different equations, an average is fictitious, but the 
average curve given in the figure is evidently quite regular. It 
shows no inclination to follow Weber's law. The error of ob- 
servation increases as the stimulus is taken greater, but much 
more slowly than in direct proportion to the stimulus. A curve 
drawn according to the square root of the magnitude is also 
shown in the figure. It follows fairly well the curve obtained by 
experiment. 
The just noticeable difference converted into the method of 
right and wrong cases in this manner is a satisfactory test of the 
accuracy of discrimination. It suffers, however, from the prac- 
tical drawback that a very large number of experiments must 
be made before a reliable result can be reached. Although one 
hundred judgments of each sort were made in the case before 
us, the value of at least one ordinate (C. just less 8 kg.) seems 
to be considerably influenced by chance. As has been explained, 
the method of right and wrong cases can be used most advan- 
tageously when the difference is such as to give about 84% 
of right cases. But this difference is smaller than most ob- 
servers would call just noticeable. When the observer is always 
right in his decision, our theory fails us entirely, for it re- 
quires us to say that the fineness of discrimination of such an 
observer is infinitely greater than that of one who is right 999 
times out of 1000. The theory is, indeed, saved by the assump- 
tion, that if the series were continued long enough, there would 
be a mistake, but we think the theory of probability cannot 
assign a probable time for such a mistake to occur. If several 
observers, even with varying differences between the stimuli, are 
always right in their judgments, we must regard their fineness of 
discrimination as alike and indefinitely great. The method of 
the just noticeable difference, in its original form, is thus reduced 
to an absurdity. 80 
Sec. 24. The Variable Error of the Just Noticeable Differ- 
ence.-We have seen that the method of right and wrong cases, at 
its best, requires many trials, whereas the variable error gives 
us a reliable result more quickly. In these experiments we have 
calculated the variable error of the just noticeable difference, 
and believe this to be a far better test than the just noticeable 
difference itself. The value of the variable error is given in the 
table and shown in the accompanying curves (Figs. 13 and 14). 
Fig. 13. 
Fig. 14. 
It is evident that the curves agree with those obtained inde- 
pendently by the method of right and wrong cases, and this 
proves conclusively the reliability, both of the methods and of 
the experiments. The average curves for the probable error 
and for the variable error are shown together in Fig. 16, below. 
The variable errors are smaller than the probable errors 
because they are calculated from series of io experiments, 
while the probable errors are from series of 100 experiments. 
The fact that the probable error of the final average is smaller 
for the variable error, and more especially that the variation of 
the separate series of the same sort is much less, shows that the 
variable error gives the more reliable result. We see that the 
accuracy of discrimination of the two observers is about the same, 
J. being the better for the three smallest magnitudes, and C. for 
the largest. The variable error for just less tends to be smaller 81 
than for just greater, which is natural, as the magnitude aimed 
at is smaller. The average curve gives no indication of follow- 
ing Weber's law, but does tend to follow the law of the square 
root of the magnitude. We conclude that the variable error of 
the just noticeable difference is a satisfactory test of the accu- 
racy of discrimination, but as the variable error, when just equal 
is attempted, is still smaller and more constant (Table X.), we 
see no advantage in using the just noticeable difference. 
Sec. 25. Further Results.-We have still to call attention to 
the average variation of the pulls from the average of the series 
to which they belong, as also the absolute value of the first pull. 
These have turned out far more constant than we had expected. 
They depend on memory for the stimulus, and show that during 
a series of ten trials a magnitude can be held in mind very well. 
The average variation of the pulls is slightly larger than the 
variable error of the just noticeable difference, and almost ex- 
actly proportional to it, so that the curves could scarcely be 
distinguished. This confirms independently the results obtained 
by the other methods, and shows that a greater dependence on 
memory did not alter the relation between the error of observa- 
tion and the magnitude of the stimulus. 
The average variations of the several series, given in italics in 
the table, should be noticed, as these show what reliance may be 
placed on the results, they being about four times the probable 
error. It is, however, a somewhat complex quantity, depending 
not only on the normal variation of perception for the different 
magnitudes and methods, but also on the condition of the 
observer from day to day, on practice, etc. 
Before leaving the comparison of methods, it may prove inter- 
esting to give curves indicating the confidence which may be 
placed on a few experiments. They show the results of the first 
ten experiments on J. "just greater," the general character of 
curves for Jis " just less " and for C. being the same, with great 82 
variation in the separate curves. In Fig. 16 the final results ob- 
tained by the three methods are shown together. 
Fig. 15. 
Fig. 16. 
A comparison of the curves in Fig. 15 with the average of one 
hundred experiments will show what reliance may be placed on 
ten trials. Weber and others have based their laws on fewer ex- 
periments than this. From the curve for the just noticeable dif- 
ference (Weber's method) the conclusion would follow that the 
just noticeable difference is nearly a constant magnitude whatever 
the stimulus may be. But the just noticeable difference here 
does not agree at all with the final result. It is about half the 
variable error and was mistaken about one-fourth of the time. 
That is, the observer called a difference noticeably greater that 
was half the time smaller than the probable error of observation. 
When the just noticeable difference is corrected by the propor- 
tion of right cases, an entirely different curve is obtained. It 
agrees very well with the final result for the first three magni- 
tudes, but fails with 16 kg. In this case there were five errors, 
and the probable error must be regarded as indefinitely great. 
In about half the series there were no errors, and the probable 
error must be regarded as indefinitely small; that is, in a 
series of ten experiments, the method of right and wrong cases 83 
is apt to fail. The variable error gives a curve agreeing fairly 
well with the final result; but as it is nearly a straight line, it 
would follow from it that Weber's law obtains. 
We finally conclude that the just noticeable difference can- 
not be used in comparing different observers, and cannot even 
be kept constant for the same observer under changed conditions. 
The method of right and wrong cases, including the application 
of the probability integral, gives correct results, but requires a 
large number of experiments (at least 100), and should be used 
with a difference smaller than "just noticeable." The method 
of average error gives the best results with a limited number of 
experiments (say 100), and a few (say 10) will serve for clinical 
and anthropometric purposes. But the average error of "just 
equal " is a better test than the average error of the just notice- 
able difference. 
Sec. 26. Experiments by the Method of Average Error.-Ex- 
periments by this method were made by J. and C. on the same 
days as the experiments treated in the preceding sections. After 
the standard had been set by five pulls, a series of ten judgments 
was made, in each of which the observer first gave the standard, 
and then a second pull as nearly as possible equal to the first. 
The standard was then again set, and a second series made, in 
order to obtain as many experiments as by the method of the 
just noticeable difference. After each pair of pulls had been 
made, the observer was required to decide or guess which of the 
two had been the greater, and give the confidence of his decision. 
This was, naturally, nearly always e, little or no confidence.1 
It will be noticed that we have thus combined the two funda- 
mental psychophysical methods. 
The table of results will be readily understood, as it is much 
like the foregoing. 
1 b only occurred twice for J. and once for C. In later experiments we have sub- 
divided c, so as to distinguish small confidence from none whatever. 84 
0 i 
II 
8 
: 
4 
KG = 4. 
8 KG 
. = 800. 
8 KG. - 1600. 
! 1 J 
vj "• |V'|AE 
V2. 
V E. 
I 
v. 
I 
J 1 S 
V* . A E V2. 
V.E I. V. 
II 
1 
V1. A.E V2. 
V.E 
I. 
v- 
II. V1. A.E. 
V«. V.E. 
J. I. . . 
212 
i l 1 
16 4- 4 17 15 
12-4 
428 
-9 
- 
13 31 3i J4 
21-9 808 52 
-25 
47 48 28 
4i-9 
1630 
38 
- 17 52 55 
29 48-9 
Var. . . 
7 
5 y 4 
3 
4.2 
3^ 
13 
14 17 5 
73 \ 3*\io 
15 
9 i4 6 
11.1 
32 
12 
ig ig 16 
8 14-9 
J. II. . 
204 
11 '+ 10 13 14 
7 
9-5 
385 
17 
+ 
17 18 2112 
14-4 805 39 
4- 2 
37 39j22 
35-6 
1597 
37 
- 19 44 47 
24 37-6 
Var. . . 
II 
J 7 4\ 5 
3 
2-9 
13 
3 
7 2 
34\ 3 
ib 
g.2 
38 
12 
20,72 8 
3 7-4 
Av. . . 
208 
13 + 7 15 H 
8 
10.9 
406 
23 
4- 
2 24 26 13 
18.1 806 45 
-11 
42 43 25 
38.7 
l6l3 
37 
l8 48 51 
26 43.2 
C.I . . 
218 
15 + 2 15 15 
9 
14-4 
404 
27 
- 
, 1 
6 23 27 17 
24.5 822 41 
- 46 
38 59 29 
37- 
1626 
40 
- 60 38 68 
35 41-8 
Var. . . 
19 
5 6 3 J 
2-4 
20 
9 
a? y 7 7 
7-7 2g 13 
72/ £ 
8.2 
3' 
8 
zz // 9 
7 6. 
C. II. . 
191 
18 + 10 17 17 
9 
i3-i 
385 
3° 
+ 
12 27 24 15 
21.4 839 46 
- 47 
38 58,27 
34-i 
1625 
39 
- 56 33 61 
27 32-6 
Var. . . 
IO 
5 5 J 6 
3 
43 
12 
8 
u 8 5 
3- 19 10 
18 
9 12 7 
6.1 
34 
7 
13 \ 8 13 
11 7.2 
Av. . . 
204 
16 + 616 16 
1 
9 
13-7 
394 
28 
4- 
3 25 25 16 
22.9 830 43 
- 46 
38 58 28 
I 
35-5 
1625 
39 
- 58 35 64 
3i 37-2 
Av. Av. 
206 
15 + 6 15 15 
8 
12.3 
400 26 
4- 
2 25 26 14 20.5 818 44 
- 29 40 51 26 
37.1 
1619 
38 
- 38 42 58 
29 40.2 
Force of Movement-Method of Average Error-J. and C.-1600 Experiments-IX. 1890-IL 1891. 
TABLE X. 85 
In this table, two series of the same sort, each containing 100 
experiments, are given for J. and C., the final average being thus 
the result of 400 experiments. In so far as Pull II. is much dif- 
ferent from I., we must regard this as due to a constant error. 
After the two pulls and their variations, we give the average 
error (A. E.) of the observer. If there were no constant error 
present, this is the only quantity we should have to consider. 
But, as has been explained, when there is a constant error, the 
average error is made up of two factors-this constant error and 
a variable error. Both of these factors are interesting, and de- 
serve separate study. In a series of ten experiments, indeed, the 
apparent constant error is complex, being partly a true constant 
error, due to overestimating or underestimating the force of the 
second pull as compared with the first, and partly due to the vari- 
able error. In calculating the variable error of the separate series, 
however, we have treated the apparent constant error as a true 
constant error, and subtracted it algebraically from each average 
error to obtain an apparent variable error, V. E., in the table. This 
was necessary, as the constant error might have been either in- 
creased or decreased by the normal variation of the variable error. 
The limited number of experiments in the series makes the 
variable error too small, but as the series throughout contained 
the same number of experiments the relative value is not af- 
fected. The variation of the average error ( K2) gives an addi- 
tional measure of the accuracy of discrimination. 
From the table and the curves (Figs. 17 and 18) showing the 
relations of the variable error, we see that these are much the 
same as for the variable error of the just noticeable difference. 
The two series of the same sort on each observer correspond 
almost exactly, the second series showing the improvement with 
practice, but not altering the shape of the curves. This im- 
provement was greater for J., which is doubtless explained by 86 
the fact that, though neither observer had previously made ex- 
periments of this sort, C. had had the more practice in psycho- 
Fig. 17. 
Fig. 18. 
logical experiments. This calls attention to a difficulty in all 
anthropometric experiments. The results depend as much on 
adaptation to the conditions of experiment as on differences in 
the senses and faculties. C's error of observation was the less 
with the greater, and J's with the smaller magnitudes, probably 
due to the greater strength of C's arm. Yet this is contrary to 
Weber's law. It is worth notice that the experiments on J. 
throughout depart less from Weber's law than those on C. In 
the present case J's curves follow Weber's law fairly well for 
the three smaller magnitudes. 
Curves for the constant errors of the first series are given 
below (Fig. 21). The constant errors show a tendency to slightly 
overestimate the second pull with the smaller magnitudes, and to 
considerably underestimate it with the greater. At first sight, 
one is tempted to account for the underestimation of the second 
pull with the large magnitudes by means of fatigue. If the 
observer were tired by the first pull, a second pull might seem 
as great when, in fact, less. But this theory would require J. 
(whose arm is the weaker) to have a larger constant error than 
C., whereas the contrary holds, and, besides, a series of ten pairs 
of pulls was made continuously, and fatigue should tell if any- 
where in the pairs becoming smaller, whereas the standards are 
overestimated. 87 
The average or crude error, being the resultant of the con- 
stant and variable errors, is, of course, larger than the variable 
error, but it is no less regular. It has usually been neglected, 
the variable error alone being used, but it seems to us that it 
measures a real error of discrimination. Thus C.'s variable error 
for the larger magnitudes is smaller than J's, whereas his con- 
stant error and his average error are larger. This indicates for 
C. perhaps greater practice in such experiments, more constant 
attention, etc., but it is a fact that J. could distinguish the smaller 
difference. It remains a question, therefore, which of the two 
quantities should be regarded as the measure of fineness of dis- 
crimination. 
As regards the other figures in the table, it is worth noticing 
that the variation of the average error is about one-half this error, 
while the variation of the pulls from the series to which they be- 
long is about equal to the average error for the smaller magni- 
tudes, but is less for the larger magnitudes, where there is a con- 
siderable constant error. Either of these quantities could thus be 
used to measure the error of observation, and we have in all six 
different quantities, any of which could be used for this purpose. 
In all cases the error becomes too small for Weber's law, as 
the magnitude is increased, which, we believe, is the conclusion 
of most careful experiments on the discrimination of small dif- 
ferences. The values follow fairly well the law of the square 
root of the magnitude, but the departure is greater than can be 
attributed to chance variations. 
Sec. 27. Fzirther Experiments by the Method of Average Error. 
-Experiments by this method were also made with three addi- 
tional observers. The results are given in the accompanying 
table, together with the first series made with J. and C. The 
variations of the series, and the variations of the average 
error have not been calculated, as these have been sufficiently 
illustrated. 88 
2 KG. = 200. 
4 KG. = 400. 
8 KG. = 800. 
16 KG. = 1600. 
I. 
11. 
A.E. 
V.E. 
I. 
11. 
A.E. 
V.E. 
1. 
11. 
A.E. 
V.E. 
I. 
11. 
A.E. 
V.E. 
K 
204 
+ 16 
32 
25.2 
396 
+ 6 
37 
32-6 
792 
-|~ 17 
6l 
55- ' 
1580 
+ 6 
59 
54-3 
Var 
31 
13 
8 
3-2 
37 
18 
8 
8.2 
34 
23 
9 
7-8 
62 
23 
II 
11.9 
D 
189 
26 
22.8 
411 
+ 17 
45 
41.2 
765 
+ 21 
54 
43-5 
1625 
4- 8 
49 
42.7 
Var 
20 
4 
6 
6. 
29 
12 
9 
9.8 
29 
'9 
8 
10.7 
35 
22 
6 
5-7 
F 
228 
+ 
18 
25 
18.3 
408 
+ 11 
3° 
28. 
794 
+ 8 
38 
37-3 
1642 
- 8 
44 
43-4 
Var 
30 
8 
7 
5-9 
47 
8 
4 
5-8 
38 
10 
9 
7-9 
60 
'4 
10 
II. 
J-I 
212 
+ 
4 
i5 
12.4 
428 
- 13 
3i 
21.9 
808 
- 25 
48 
4i-9 
1630 
- 17 
55 
48.9 
Var 
7 
8 
4.2 
3<> 
'7 
'7 
7-5 
32 
15 
14 
11.1 
32 
19 
16 
^4-9 
C. I 
2l8 
+ 
2 
IS 
14.4 
404 
- 6 
27 
24.2 
822 
- 46 
59 
37- 
1626 
- 60 
68 
41.8 
Var 
*9 
6 
3 
2-4 
20 
*3 
7 
7-7 
29 
29 
21 
8.2 
3i 
11 
9 
6. 
Av 
210 
+ 
II 
23 
18.6 
409 
+ 3 
34 
29.6 
796 
- 5 
52 
42.9 
1621 
- 14 
55 
46.2 
Force of Movement-Method of Average Error-5 Observers and 2000 Experiments-IX. 1890-III. 1891. 
TABLE XI. 89 
The relations of the variable error to the magnitude of the 
stimulus for W, D. and F. are shown in Fig. 19, and the averages 
for K., D., F., J. I. and C. I. (together with curves for Weber's 
law and the square root of the magnitude) are shown in Fig. 20. 
Fig. 19. 
Fig. 20. 
Thesurves for F and C. {C's curves are given in Fig. 17) cor- 
respond closely, and this was usually the case when the two 
observers made the same experiments. A similarity in pursuits 
and interests thus seems to lead to uniformity in the processes 
of perception and comparison. When results correspond so well 
as those of F, C. and J., and the probable error for each observer 
is so small, we may regard the results as established, and it 
would not be profitable to further increase the number of experi- 
ments, unless it were in order to study the effects of practice. 
The curves for K. and D. bear witness to a larger error and 
greater irregularity, but tend toward the same shape as the 
others. J. and K. are women, and their curves do not permit us 
to draw any inference as to greater fineness of discrimination in 
one of fhe sexes. 
In Fig. 20 the average curve for the several observers is given, 
together with a curve for Weber's law and for the square root of 
the magnitude. There is no doubt but that the average curve 
is regular and bends away from the vertical axis. Weber's law, 90 
consequently, does not obtain. The error of observation be- 
comes greater as the stimulus is taken greater, but more slowly. 
Unfortunately, we cannot compare 16 kg. with the other mag- 
nitudes, as the extent of the movement was even less than for 
8 kg. But in the experiments on extent of movement (Part I.) 
we saw that the error was actually smaller for 700 than for 500 
mm., and a similar inclination of the curve toward the horizontal 
axis might have occurred if movements of greater force than 
16 kg. had been used. 
The curve drawn according to the square root of the magni- 
tude is evidently more like the experimental curve than is that 
drawn according to Weber's law, and it would correspond better 
if the error for 16 kg. were corrected for the smaller extent of 
this movement. The curves for F and C. follow closely the 
theoretical curve, but the difference in the curves cannot be 
attributed to chance, and we may assume the presence of factors 
in addition to the summation of errors, for which alone the law 
of the square root of the magnitude accounts. Hypotheses to 
account for these secondary variations might easily be made, but 
it may be better to wait for further experiments, more particu- 
larly on magnitudes smaller than 2 and larger than 16 kg. 
The combined curve for the average error would follow almost 
exactly the same course as that for the variable error. The 
average error for the individual observers, however, differs con- 
siderably from their variable error. Thus C. has large con- 
stant errors with 8 and 16 kg., and his average error is, conse- 
quently, much larger than his variable error; with F, on the 
other hand, the constant error is small, and the average,error is 
not much reduced by its elimination. K. and D. had a positive 
constant error with all the magnitudes, but with a comparatively 
large probable error. All the observers had a positive constant 
error with the smallest magnitude. The relation of the constant 91 
errors to the magnitude of the stimulus can be studied most 
conveniently in the accompanying figures. 
Fig. 21. 
Fig. 22. 
Sec. 28. A nalysis of the Error into an Error of Perception and 
an Error of Movement.-In all our experiments on the perception 
of movement the observer first adjusted the two movements in 
the manner required, and afterward estimated the correctness 
with which this had been done. Thus, after the observer had 
made the force of his second movement as nearly as possible 
equal to that of his first, we did not consider the experiment 
complete, as has hitherto been the case, but required the observer 
to decide or guess which of the two movements had been the 
greater, in spite of his attempt to make them exactly alike, and 
to assign a degree of confidence to his decision. The degree of 
confidence was nearly always c, a and b occurring rarely except 
in the first few series with K. and D. 
The percentages of right cases for the several observers and 
magnitudes are given in the accompanying table. 92 
TABLE VIL 
Percentages of Right Cases in Judging ^he Force of Two Movements 
Made as Nearly as Possible Alike. 
SECOND LESS. 
SECOND GREATER. 
2 KG. 
4 
' 8 
16 
2 
4 
8 
16 
K 
82 
70 
70 
70 
57 
67 
67 
82 
D 
8l 
92 
82 
9i 
60 
55 
55 
57 
F 
71 
77 
47 
54 
69 
65 
69 
55 
J. I. . . . 
67 
49 
55 
56 
46 
62 
60 
J. II.. . . 
52 
60 
52 
62 
59 
59 
58 
5i 
c. I. . . - 
64 
64 
61 
55 
58 
46 
56 
77 
C. II. . . 
69 
7i 
56 
5i 
59 
55 
33 
58 
This table shows that the observers could often correctly dis- 
tinguish a difference, even when they had attempted to make 
the two movements exactly alike. We are thus led to believe 
that the error which we have hitherto been considering is com- 
plex, being partly due to an error of perception and partly 
to an error of movement.1 If the entire error, in the attempt to 
make the two movements alike, were due to an error of percep- 
tion, they would seem, when completed, exactly alike, and the 
observer's judgment would have been a mere guess, as likely to 
be wrong as right. If, on the other hand, the observer did not 
make the two movements apparently alike, he should perceive 
his error of movement as a difference, and this difference would 
give a percentage of right cases corresponding to its size. Fur- 
ther, the average variable error of perception, when used as 
the amount of difference in the method of right and wrong 
1 This was suggested by Muller (op. cit. 71 sq^. He says, " Es ist aber ganz unmog- 
lich, naher anzugeben, in welcher Weise diese beiden Factoren, die Triiglichkeit unseres 
Urtheils und die Unsicherheit der Hand, das Wahrscheinlichkeitsgesetz der Einstellungs- 
fehler bei Anwendung verschiedener Normaldistanzen beeinfliissen." Our experiments, 
however, accomplish this. 93 
cases, should theoretically give about 78 per cent, of right 
cases.1 If the error of movement gave this same percentage, 
we should conclude that the error of movement and the error 
of perception were equal. The table shows that the percent- 
age of right cases (apart from constant errors, which are elimi- 
nated as explained in Part IV.) was always less than 78 per 
cent.; we, therefore, know that in these experiments the error 
of movement was less than the error of perception, and by com- 
paring the value of (Table I.), which would give 78 per 
cent, of right cases, with those values of ^^ corresponding to 
the percentages of right cases in the table, we determine the 
size of the error of perception as compared with the error 
of movement. We already know the size of the complex vari- 
able error, and can thus readily calculate the actual values of the 
error of perception and the error of movement, the complex 
variable error being equal to the square root of the sum of the 
squares of the separate errors. 
The analysis of the variable error in judging the force of 
movement (Tables X. and XI.) gives the results contained in 
the accompanying table. 
1 So Fechner (Revision, 115), who gives the percentage as 78.4, and we used the 
nearest whole percentage in calculating the table. Fechner, however, seems to have made 
a slight mistake in his interpolation, the correct percentage being 78.7. The variable 
error, calculated from series of ten experiments, is, further, relatively smaller than the 
probable error, calculated from series of varying size. It should also be remembered 
that the difference varied in each experiment. But the number of experiments is too 
small to require great accuracy in calculation. 94 
TABLE XIII. 
Analysis of the Error into an Error of Perception and an Error of 
Movement. 
ERROR OF PERCEPTION. 
ERROR OF MOVEMENT. 
2 KG. 
8 
16 
2 
4 
8 
16 
K 
21 
28 
47 
40 
15 
17 
29 
37 
D 
18 
29 
40 
3° 
13 
29 
24 
3° 
F. . . . 
15 
23 
36 
43 
IO 
16 
ro 
6 
J. I. . . . 
11 
22 
40 
48 
4 
0 
9 
12 
J. II. . 
9 
14 
35 
36 
2 
5 
6 
8 
C. I. . . 
U 
24 
36 
37 
5 
4 
IO 
20 
C. II. 
12 
20 
34 
32 
6 
9 
0 
5 
Av. . . 
14 
23 
38 
38 
8 
II 
13 
17 
The averages show that the error in adjusting the movement 
was about half as large as the error in perceiving it, and that 
the variable error obtained by the method of average error is 
reduced by about one-eighth when the error of adjustment is 
eliminated. The error of perception increases more rapidly than 
the error of movement for the three smallest magnitudes, but is 
the same for 8 and 16 kg. Curves for the averages are given 
below (Fig. 49). 
It is to be noted that the observers who had the larger com- 
plex errors had the larger errors both of perception and of move- 
ment, but that in their case the error of movement is relatively 
larger, being nearly equal to the error of perception. Hence 
we may, perhaps, conclude that exactness in movement, as 
needed in manual work and games, can, by training, be improved 
more than accuracy of discrimination. The traits latest acquired 
by the race and individual are the most easily altered. So far 
as a conclusion can be drawn from five observers, it would seem 
that woman as compared with man is relatively more accurate 
in movement than in perception. 95 
Sec. 29. The Distribution of Errors.-An average cannot give 
as exact information as the separate quantities from which it is 
taken. It is like a common noun, which, for convenience, 
neglects individual differences. The arithmetical average to- 
gether with its probable error gives, indeed, for ordinary pur- 
poses, all the information needed ; but as we have in this paper 
under consideration the nature of the method of average error, 
some study of the distribution of the errors making up the 
average seems desirable. In so far as this distribution follows 
the exponential equation (curve of error, § 3), we can discover 
it from the averages. But it may be useful to compare the 
mathematical theory with actual experiment. We have, conse- 
quently, divided the errors made by the five observers (Tables X. 
and XI.) into fourteen classes in accordance with their size 
and direction. All the errors of 20 (.2 kg.) or smaller are 
placed together in two classes, one for positive, the other for 
negative errors; all the errors between 21 and 40 are placed in 
classes together, and so on up to 120, and in the seventh 
pair of classes all errors larger than 120 are placed. The few 
cases (in all 34) in which there was no error (i. e., the error 
was smaller than the reading of the instrument) are divided 
equally between and -. In place of a table, which would 
be somewhat cumbersome, we give curves, by means of which 
the distribution of errors for the several observers and mag- 
nitudes may be conveniently seen. There are 100 errors in 
each set, and the area of the curve may be regarded as unity. 
Owing to the limited number of errors, however, it is necessary 
to place them in classes, and a continuous curve is not, of 
course, obtained. The abscissae are proportional to the size of 
the mean error (or mid-error) of each class, and the ordinates 
to the number of errors in each class. Two sets were made on 
J. and C, the curve representing the second set being drawn in 
heavier lines. 96 
These curves show distinctly how far the distribution of 100 
errors follows the bell-shaped curve required by the theory of 
Figs. 23 to 42. 
probability. All the curves do, in fact, tend to approach the 
theoretical curve, and the agreement is best for the best ob- 
servers (A,y. and C), and for the stimulus (4 kg.) in which the 97 
relation of the ordinates to the abscissae is the most favorable 
for drawing the curve. If the number of errors were greater 
than 100, the curves would of course become more regular. 
For various reasons we are not prepared to introduce into this 
paper more mathematical discussion than necessary. But 
these curves might be treated in a manner which would compare 
the average and variable error, and the relation and relative ac- 
curacy of these compared with the probable error, the error of 
mean square and the modulus. The departure from the the- 
oretical curve could further be used to discuss and possibly 
to amend the assumptions made by Laplace and Gauss in their 
mathematical deductions. 
Many of the facts already discussed may be observed with 
advantage in these curves. Thus the taller and narrower curves 
show distinctly which of the observers were the more accurate, 
and which of the stimuli were the more exactly perceived. The 
position of the curve to the right or left of the perpendicular 
axis shows the amount and direction of constant errors. The 
two curves given for J. and C. show the accuracy with which 
the same series can be repeated and the improvement with 
practice. 
Sec. 30. The Summation of Errors.-Our theory to account 
for the increase of the error of observation with the size of 
the magnitude assumes that the algebraic sum of a number 
of errors increases as the square root of the number. It has 
seemed to us that it might be worth while to test the mathemat 
ical theory by experiment, as we are not aware of this having 
been directly done. We have therefore added in pairs algebrai- 
cally the variable errors (with 2 and 4 kg.) for F., J. and C. As 
there were ten errors in each series we thus obtained five com- 
plex errors, and took the average of these. According to theory, 
this average should be nearly the variable error, multiplied by 98 
j/2 = i.414, and should further be equal to the variable error 
made with a stimulus double the size. In the accompanying 
table we give the variable error ( K A), its value when multi- 
plied by the 3/2? the result of adding the errors algebraically in 
pairs (S.E.), which should theoretically be nearly the same as 
the preceding, and the variable error with a stimulus double the 
former, which should be the same as the two preceding, sup- 
posing the error of observation to be due entirely to the sum- 
mation of errors. 
TABLE XIV. 
2 KG. = 200. 
4 KG. = 400. 
V.E. 
X V2 
S.E. 
V.E 4 KG. 
V.E. 
X }2 
S.E. 
V.E. 8 KG. 
F. . . . 
18.3 
25-9 
23-4 
28. 
28. 
396 
34-i 
37-3 
J. I. . . 
12.4 
17-5 
15-8 
21.9 
21.9 
31.0 
33-3 
4i-9 
J. IL. . 
9-5 
13-4 
12.6 
I4.4 
14-4 
20-4 
18.7 
35-6 
C. I. . . 
14.4 
20.4 
20.1 
24-5 
24-5 
34-6 
34-4 
37- 
C. II. . 
i3-i 
18.5 
17- 
21.4 
21.4 
3°-3 
35-5 
34-i 
Av. . . 
13.5 
19.1 
17.8 
22. 
22. 
31.2 
31.2 
37.2 
Summation of Errors-iooo Experiments. 
From the figures given in the table we see that the summa- 
tion of errors corresponds closely to the variable error multi- 
plied by y/2. In eight sets out of ten it is, however, slightly 
smaller and in a tolerably constant ratio. At all events the 
mathematical theory, according to which combined errors in- 
crease as the square root of the number of observations, and 
the probable error of an average decreases as the square root of 
the number, is confirmed fairly well by these experiments. 
The error of observation when the stimulus was doubled was 
greater than either mathematical theory or actual summation of 
the errors admits. We consequently assume that the error of 99 
observation was increased by the summation of errors in the 
manner required by theory, but also by other factors entering 
into the execution of the movement. It would not be likely 
that the error made (say) in giving 2 kg. would be the same 
as in going on from 2 to 4 kg. Many factors-the magni- 
tudes most commonly compared in daily life, the muscular 
mechanism used, the strength of the observer, fatigue, etc.- 
would alter the error of observation with different magnitudes, 
and these must be empirically determined by experiment. But 
while these would sometimes increase and sometimes decrease 
the error of observation, the general tendency of the error to 
increase as the magnitude is taken larger is accounted for in a 
satisfactory manner by the summation of errors. 
Sec. 31. Experiments by the Method of Estimated Amount of 
Difference.-This method was used by J. and C. in conjunction 
with the experiments just described. The attempt was made to 
double 3 kg., to halve 12 kg., and to find a mean between 2 and 
4 kg., and 4 and 8 kg. respectively. The standards were set in 
the manner already described. When the method of mean 
gradation was used, two standards had to be determined and 
held in mind; after this had been done in five trials the observer 
gave one pull (the greater or less in alternate series), then the 
second, and thirdly a pull which seemed midway between the 
two. The error of each experiment was separately calculated. 
In halving 12 kg. the dynamometer was set at 4 kg., the extent 
of this pull being consequently proportional to the force of 8 
kg., when the stop was set at zero. The magnitudes to be esti- 
mated were chosen so as not to be the same as any of the stand- 
ards. The results of these experiments are given in the accom- 
panying table. 100 
TABLE XV. 
Force of Movement-Method of Estimated Amount of Difference-J. and 
C.-800 Experiments-IX. 1890-I. 1891. 
DOUBLE 3 KG. = 300. 
HALVE 12 KG. = 1200. 
I. 
V. 
11. 
V1. 
A.E. 
V3. 
V.E. 
I. 
V. 
II. 
V1. 
A.E. 
V3. 
V.E. 
J. - • 
326 
14 
26 
6 
- 10 
92 
48 
14 
72 
// 
37 
6 
56 
8 
1238 
33 
43 
IO 
+ 266 
50 
41 
9 
266 
50 
41 
II 
43 
*4 
c... 
312 
24 
24 
6 
-131 
39 
37 
8 
132 
39 
44 
12 
46 
IO 
1232 
38 
53 
17 
+ 239 
49 
37 
11 
239 
49 
34 
6 
34 
4 
Av. . 
319 
25 
- 70 
42 
102 40 
51 
1235 
48 
+252 
39 
252 
37 
38 
MEAN BETWEEN 2 AND 4 KG. 
MEAN BETWEEN 4 AND 8 KG. 
I. 
V. 
11. 
V1. 
hl 
ri 
A.E. 
V3. 
V.E. 
I. 
V. 
11. 
V1. 
in. 
V3. 
A.E. 
V3 
V.E. 
44 
16 
J-• • 
C.. . 
231 
II 
20 
428 
28 
28 
7 
- 7 
21 
27 
7 
36 
*3 
18 
6 
28 
IO 
424 
34 
3i 
9 
849 
33 
44 
'4 
- 44 
40 
39 
9 
69 
21 
32 
12 
236 
24 
■9 
7i 
4i7 
32 
3i 
4 
- 3 
3^ 
26 
6 
4i 
18 
6 
24 
9 
450 
38 
10 
858 
18 
5i 
J3 
- 54 
17 
48 
16 
*3 
16 
33 
12 
42 
8 
Av. . 
233 
19 422 
| 
29 
- 5 
26 
1 
38 
18 
26 
437 
34 
853 
47 
+ 49 
43 
66 
32 
43 
In attempting to double a given pull J. did so very nearly, 
whereas C. gave a pull scarcely more than half again as great as 
his first pull. We cannot explain this difference in the two 
observers, whose variable error here as elsewhere agrees very 
well, but consider it of interest as showing how difficult it is 
to make quantitative estimates of sensations. We, ourselves, 
think that such estimates depend on association with known 
quantitative relations of the stimuli, and it seems quite likely 
that these should vary greatly with different observers. This 101 
experiment of doubling a stimulus (sensation ?) would seem to 
reduce Fechner's law to an absurdity. If we measure the inten- 
sity of a sensation by estimating the number of just noticeable 
differences which it contains, and if the just noticeable difference 
be proportional to the magnitude of the stimulus, the difference 
between no sensation and the force of 3 kg. ought to seem 
immensely greater than the difference between 3 and 6 kg., and 
in attempting to double 3 kg. the observer ought to give a pull 
beyond the strength of any arm. But, in fact, both observers 
less than doubled the first magnitude. 
In attempting to halve 12 kg., the two observers agree 
tolerably well in their estimates, greatly overestimating the true 
half. This, too, is exactly contrary to the result required by 
Fechner's law. The experiment was complicated by the fact 
that the dynamometer was set at 4 kg., so that both observers 
more nearly halved the extent than the force of the movement. 
This indicates that throughout these experiments the observer 
may have based his judgment largely on the extent (and possibly 
the time) of the movement. 
The results of the two observers in using the method of mean 
gradation agree very well. The arithmetical mean is under- 
estimated, and so far this is the result required by Fechner's law. 
But the estimates do not give the geometric mean. 
The results of experiments by this method are considerably 
influenced by contrast, as has been found to be the case with 
lights.1 Thus when the order was 200-400-estimated mean, J's 
constant error was-28, and C.'s-50; whereas,when the order was 
400-200-estimated mean, J.'s constant error was +14, and C.'s 
-4-34. With 4 and 8 kg., a similar but less marked result was 
1 Alfr. Lehmann, Ueber die Anwendung der Methode der mittleren Abstufungen 
auf den Lichtsinn, Philos. Stud., III. 497, 1886. 
Hjahmar Neiglick, Zur Psychophysik des Lichtsinns, Philos. Stud., IV. 28, 1887. 102 
obtained for both observers. These, like many other effects 
of contrast, seem somewhat difficult to explain. We may, if we 
like, assume that the latter standard was somewhat better in mind 
than the former, and a greater objective difference between it 
and the judgment-pull seemed equal to a less difference between 
this judgment-pull and the more vaguely remembered first 
standard. 
The variable error in these experiments is more nearly alike 
for the two observers, and has a smaller probable error than the 
constant error of estimation, and is better suited for measuring 
the fineness of discrimination. But for ordinary purposes it 
cannot be recommended, as the variable error, when "equal" is 
attempted, is less complex and more regular. 
It will be remembered that experiments on extent of move- 
ment by this method also gave an irregular constant error, but 
that the half of a movement was also overestimated and the 
double underestimated. In that case, the mean tended to be 
overestimated. All our experiments, therefore, are contrary to 
the assumption that sensations are estimated as proportional to 
the logarithm of the stimulus. We believe that differences 
(they may be qualitative) in sensation, being associated with 
quantitative differences in the stimuli, are used to estimate the 
relations of these magnitudes, and our experiments show that 
such estimations are subject to large constant errors, but tend 
to discover the true objective relations of the extent and force 
of movements. PART III. 
ON THE TIME OF MOVEMENT. 
Sec. 32. Introductory.-Experiments on the time of move- 
ment have not, we think, hitherto been published. Numerous 
experiments have, indeed, been made on the estimation of inter- 
vals of time and on the accuracy with which a rhythm can be 
kept.1 These researches contradict each other concerning the 
validity of Weber's law, and in most other respects. Galton 
has recently recommmended the rate of movement as an anthro- 
pometric test, and devised an instrument for its measurement.2 
Our chief object was to determine the accuracy with which 
the time of movements can be discriminated, and to study the 
part played by rate as compared with extent and force in the 
perception of movement. We also give some experiments on 
the maximum rate of movement, which subject deserves study, 
if only on account of its possible pedagogical and clinical appli- 
cations. 
Sec. 33. Apparatus and Methods.-A detailed description of 
the apparatus needed to measure short intervals of time with 
exactness need not be given here. The arrangement which we 
consider the most convenient has been described elsewhere by 
i The most recent, by Herbert Nichols, " The Psychology of Time," Am. Jour, of Psy., 
Vol. III. 4, IV. i, 1891, who gives an excellent summary and criticism of preceding re- 
searches, twenty-one in number. 
2 Francis Galton, A New'Instrument for Measuring the Rate of Movement of the 
Various Limbs, Journal of the Anthropological Institute, 1890. 
103 104 
one of us.1 It consists of an electric chronoscope, with the pieces 
necessary to control and regulate it. The great accuracy of this 
apparatus is sufficiently shown by the fact that the average 
variation in the time of fifty blows (in five series) in one case 
was 1.6 thousandth of a second, which very small variation in- 
cludes the error of all the apparatus as well as the irregularity 
of the blow. In addition to the apparatus for measuring the 
time, we needed an instrument which would close an electric 
current when the movement was begun, and break it when the 
movement was ended. This is shown in Fig. 43. 
Fig. 43. 
An electric current passes through the contacts at C and C'. 
The observer sat in a convenient position in front of the instru- 
ment, and placed his right hand on H and against the bar A H C. 
When the hand was moved the spring drew up the bar and 
closed the contact. We give a special figure of this contact. 
We should like to call attention to it, as we think it might be 
useful in physical instruments. Contacts are usually closed by 
means of mercury, but owing to combustion and spilling it is 
1 Mind, XI. 1886; Phil. Stud., III., 1886. We have, however, introduced several im- 
provements. The chronoscope, as made by Peyer and Favarger, often fails to give the 
proper time. We have obviated this fault by replacing the string with which the instru- 
ment is started by a bar and key, by which means it can be started constantly and regu- 
larly, and by placing a clamp on the carriage of the tuning-fork, which prevents it from 
becoming loosened with the vibrations. A wooden house, with padding placed over the 
clock-work, deadens the noise and keeps out dust. We also constructed a new regulator, 
made by C. Krille, Leipzig. 105 
difficult to keep the mercury at a constant level. Besides, mer- 
cury is always troublesome in a laboratory. This contact is 
closed at an exact point by the platinum bar touching the plat- 
inum sunk into the inclined plane at C. The pressure can be 
regulated by the spring and screw at 5, and the contact is kept 
clean by the rubbing. The contact is never broken after it has 
once been made. There is no rebound, as the movement is 
stopped gradually by the increasing friction of the inclined plane. 
Fig. 44. 
When the hand has moved from H to R and touches the bar, 
the lever breaks the contact at C\ This bar is very light, and 
the most violent blow will not injure hand or apparatus. The 
electric current is thus closed while the hand moves, and the 
time of the movement in thousandths of a second is directly 
read from the chronoscope. The duration of either a quick 
blow or a slow movement can thus be measured conveniently 
and exactly. 106 
In our experiments on the discrimination of the time of 
movement, the extent was always 50 cm. The force of the blow 
therefore varied, being greater for the quicker movements. The 
observer could, consequently, use the perception of a difference 
in force to judge the time. We regret that we have not been 
able to carry out experiments which we had planned to discover 
how the judgment of the time of movement is affected by alter- 
ations in the extent and force. The time between the two move- 
ments was made as nearly as possible one second, but owing to 
the nature of the experiments this interval could not be given 
to the observer. 
We used as normal times 1, % and X sec., and the time of a 
blow made as quickly as possible. This blow required about % 
sec., so we had a geometric series of four magnitudes, as with 
the force of movement. The normal time was given to the ob- 
server in the same manner as in the experiments already de- 
scribed. He made a movement in what he supposed to be about 
(say) 1 sec., and was told the direction and amount of his error. 
After he had learned the approximate size of the standard by 
five such trials, a series of ten judgments was made. The ob- 
server made the normal movement, and then a second movement, 
as nearly as possible, in the same time. The four magnitudes 
were used at one sitting, the order being varied, so as to elimi- 
nate the effects of contrast and fatigue. 
Sec. 34. Results of Experiments.-By the method of average 
error 400 experiments each were made on K., D., F and C. The 
method of calculating and tabulating the results was the same 
as in the corresponding experiments on extent and force. The 
table will, therefore, be readily understood. 107 
MAXIMUM. 
% sec. = 250 a. 
% SEC. = 
500 a. 
I SEC. - IOOO (I 
n. 
A.E. 
V.E. 
i- 
11. 
A.E. 
V.E. 
I. 
n. 
A... 
V.E. 
II. 
AE. 
V.E. 
K 
188 
-13 
26 
23-7 
283 
+ 8 
37 
31-3 
496 
+ II 
61 
53-9 
878 
4- 45 
145 
133 
Var 
35 
'3 
^5 
I2.<) 
IO 
16 
6 
5-7 
38 
43 
21 
19-3 
"5 
65 
24 
27.6 
D 
136 
-16 
21 
14.2 
215 
- 4 
38 
36.7 
544 
+ 16 
96 
85- 
1020 
+ 41 
168 
139- 
Var 
9 
IO 
9 
3-8 
21 
IO 
IO 
"■7 
7i 
34 
28 
26.3 
99 
86 
43 
32-7 
c 
87 
- 5 
12 
IO. 
286 
+ 3 
28 
24-7 
480 
+ 27 
47 
35-i 
1055 
+ 36 
78 
66.7 
Var 
6 
6 
4 
2-4 
'3 
9 
5 
5-i 
3^ 
19 
8 
7-9 
109 
22 
15 
12.9 
F 
IIO 
- 7 
13 
9-7 
273 
- M 
24 
18.8 
492 
+ 2 
35 
34-i 
959 
+ 69 
88 
60.6 
Var 
6 
6 
5 
4-3 
14 
6 
6 
6 
34 
24 
7 
8-7 
87 
3i 
30 
75.6 
Av 
130 
- 10 
18 
14.4 
264 
- 2 
32 
27.9 
503 
+ 14 
60 
52.0 
978 
+ 48 
120 
99.8 
Time of Movement-Method of Average Error-4 Observers-1600 Experiments-XL 1890-III. 1891. 
TABLE XVI. 108 
We have in the table the time of the first movement (I.), the 
time of the second movement (II.) compared to the first (the 
constant error), the average error (A. E.) in attempting to make 
the two movements alike, and the variable error (V. E.) obtained 
by eliminating as far as possible the constant error from the 
average error. One-thousandth of a second (a) is used as the 
unit of time. 
The variable error has usually been made the measure of dis- 
crimination in experiments by the method of average error. We 
give in Figs. 45 and 46 curves showing the relation of the varia- 
ble error to the magnitude of the stimulus. The relations given 
Fig. 45. 
Fig. 46. 
in the table and shown in the figures prove that the error in 
judging the time of movement tended to vary as the time, al- 
though, as is usually the case, the error becomes too small as the 
magnitude is increased. The apparent close agreement of the 
average with Weber's law is, indeed, due to the irregularity of 
K's curve. The curves for the other observers depart about 109 
equally from Weber's law and from the square root of the 
magnitude.1 
Sec. 35. The Constant Error.-The average error is, of course, 
larger than the variable error, but would give nearly the same 
curves. It is the constant error which has received most atten- 
tion in experiments on rhythm and the sense of time, but dif- 
ferent investigations differ greatly in their results. We give 
curves showing the constant errors in our experiments (Figs. 47 
Fig. 47. 
Fig. 48. 
and 48). The curves for the several observers agree better than 
is usually the case with constant errors. The average curve is 
nearly a straight line crossing the horizontal axis just beyond 
one-fourth second. With one-half and one second, the second 
movement was made more slowly than the first. With the 
maximum blow the second movement was made more quickly 
than the first and also seemed easier and quicker. The accom- 
plishment of one blow thus makes one following it quicker and 
more powerful. 
The error made by an observer in judging a standard magni- 
tude is usually very great. Before making these experiments 
the observers were required to estimate one, one-half and one- 
fourth second. The averages of ten such estimations are given 
in the accompanying table. 
1 One of the writers (J. McK. C.) thinks that the comparatively large errors with the 
slower movements were due to the fact that the force of the blow was used in judging the 
quicker movements-force being judged more accurately than time. 110 
TABLE XVII. 
Estimation of a Standard Interval of Time-4 Observers-100 Eperiments. 
% SEC. 
2500". 
% SEC. 
= Joocr. 
I SEC. 
= looocr. 
TIME. 
V. 
TIME. 
V. 
TIME. 
V. 
K 
201 
10 
D 
171 
i5 
143 
6 
274 
20 
C 
39° 
63 
616 
"5 
901 
226 
F 
191 
19 
273 
27 
437 
55 
Av 
276 
32 
344 
49 
453 
78 
The table shows that one second was much underestimated, 
although all the observers had been using one second (given by 
a pendulum) as the time of movement in other experiments. 
The attempt to estimate longer intervals of time and other 
standard magnitudes gave very great errors, which shows how 
little reliance should be placed on such estimates-for example, 
in a court of justice. 
As we did not make other experiments on the time of move- 
ment by the method of estimated amount of difference, it is 
worth noting that in the present case when the observer at- 
tempted to give an interval half as great as another (the order 
of the series was one, one-half, and one-quarter second), the half 
was greatly overestimated. All through the experiments the 
same constant error was present: one-half second seemed to F. 
and C. less than the half of one second, and more than the 
double of one-fourth second. That is, the estimation of inter- 
vals of time gives a result contrary to Fechner's law. This in- 
crease in subjective estimation of equal objective increments 
may be related to the fact that the error of observation increases 
faster than the objective increment warrants, according to 
the summation of errors. Both facts are perhaps due to the 111 
greater part played by force in the quicker movements, and 
may be related to the commonly observed phenomenon that 
time passes more quickly when we are active than when we are 
passive. 
Sec. 36. Analysis of the Error into an Error of Perception 
and an Error of Movement.-As in the corresponding experi- 
ments on extent and force, the observer, after he had made the 
time of the two movements as nearly as possible alike, was re- 
quired to judge the success with which this had been done. 
The percentages of right cases for the several observers and mag- 
nitudes are given in the table. 
TABLE XVIII. 
Percentages of Right Cases in Judging the Time of Two Movements Made 
as Nearly as Possible Alike. 
SECOND LESS. 
SECOND GREATER. 
MAX. 
250. 
500. 
1000. 
MAX. 
250. 
500. 
1000. 
K 
IOO 
88 
83 
69 
5 
40 
51 
69 
D 
82 
76 
67 
47 
63 
64 
79 
82 
F 
82 
68 
86 
83 
32 
75 
5i 
61 
C 
67 
58 
57 
73 
20 
76 
62 
58 
The table shows that the movements could be judged much 
better than they could be made. The great excess of right cases 
with the maximum blow, when the second movement was the 
quicker, shows that the second' blow seemed far quicker and 
more powerful than the first. It was, in fact, quicker, but less 
so than it seemed. 
When the complex variable error given in Table XVI. is an- 
alyzed into an error of perception and an error of movement in 
the manner described (Sec. 28), we obtain the results given in 
Table XIX., and the accompanying curves (Fig. 50). 112 
Analysis of the Error into an Error of Perception and an Error of 
Movement. 
TABLE XIX. 
ERROR OF PERCEPTION. 
ERROR OF MOVEMENT. 
MAX. 
250. 
500. 
1000. 
MAX. 
250. 
500. 
1000. 
K 
22 
26 
46 
112 
IO 
16 
29 
72 
D 
I I 
3° 
66 
122 
9 
21 
53 
66 
F 
9 
i5 
28 
47 
3 
11 
20 
38 
C. 
IO 
21 
33 
58 
0 
12 
IO 
3i 
Av 
13 
23 
43 
85 
5 
15 
28 
52 
Fig. 50. 
Fig. 49. 
We thus see that both the error of perception and the error 
of movement increase nearly in direct proportion to the magni- 
tude of the stimulus, but somewhat more slowly. The error of 
perception is somewhat less than double the error of movement, 
and about four-fifths of the complex variable error. 
In Fig. 49 the corresponding curves for the force of move- 
ment are shown. In comparing the curves it must be noticed 
that the ordinates representing the error in force are twenty 
times the abscissae, whereas those representing the error in time 
are only ten times the abscissae. It is evident that force was 113 
judged relatively much better than time, especially for the larger 
magnitudes. 
Sec. 37. The Distribution of Error.-The errors in adjusting 
the time of movements were classified in the same manner as 
those on the force of movement (Sec. 29). The accompanying 
curves show clearly the distribution of small and large errors, 
and the variation for the several observers and magnitudes. The 
classes are 0-250-, 26-500-, etc. 
The curves are tolerably regular and approach the bell-shape 
required by the theory of probability. Small errors are evi- 
Figs. 51-66. 114 
dently more frequent than large ones. The taller and narrower 
curves of F. and C. bear witness to greater fineness of discrim- 
ination, and their more regular shape to smaller variation. 
Regularity cannot be expected with the larger magnitudes, as 
there were comparatively few small errors. 
Sec. 38. The Summation of Errors.-In exactly the same 
manner as in the experiments on the force of movement (Sec. 30), 
we added algebraically in pairs the variable errors of F. and C. 
with and % second. The results are given in the table. 
TABLE XX. 
% SEC. = 2500-. 
% SEC. = 5OO(T. 
V.E. 
X f 2. 
S.E. 
V.E. %. 
V .E. 
X 1 2. 
S.E. 
V.E. I. 
F. . . . 
18.8 
26.6 
29.2 
34-i 
34-i 
48.2 
50.3 
60.6 
c. . . . 
24-7 
34-9 
35-4 
35-i 
35-i 
49.6 
49.2 
66.7 
Av.. . . 
21.7 
30.7 
32.3 
34.6 
34.6 
48.9 
49.7 
63.6 
Summation of Errors-400 Experiments. 
It is evident that the results obtained by the summation of 
the errors correspond very closely with the variable error mul- 
tiplied by the square root of two. The variation is not larger 
than the probable error would lead us to expect, and the math- 
ematical theory is thus confirmed by the experiments. The 
variable error in these experiments on the time of movement 
increases, as we have seen, more rapidly than the mere summa- 
tion of errors would warrant. 
Sec. 39. The Maximum Rate of Movement.-We believe that 
the time of the quickest blow which can be made is a useful test 
for clinical and anthropometric purposes. Under constant con- 
ditions it is very regular, but differs considerably with different 
observers and under changed conditions. In ten series of ten 115 
experiments made on C. the average variation of a blow fifty 
cm. in extent was only 5a. The average time of the separate 
series, however, varied from 96 to 138(7, and this variation tells 
of some change in the condition of the observer. In certain 
diseases of the nervous system the rate of movement might be 
most useful as a test of progression or recovery, perhaps fully 
as valuable as the dynamometer test now used universally. The 
difference in the rate of movement with right and left hand (or 
foot) may also prove useful in the diagnosis of unilateral disease. 
The rate of movement for C. as the result of 120 experiments 
was exactly the same (1190-) for the two hands. C's reaction 
time with right and left hand is also the same, and this test may 
also be recommended for clinical experiment. For C., a blow with 
the right hand from right to left was a little slower (70- as the 
result of 120 experiments) than from left to right. 
In the experiments which we have described (Table XVI.) 
the maximum rate of movement for the four observers was 
respectively 87, no, 136 and 188a, and the results obtained 
from other observers fell within these limits. The time for 
women is decidedly longer than for men, which measures a well- 
known difference in the sexes, due doubtless to selection and 
heredity. 
In these experiments only the time of the entire movement 
was measured, its extent being always 50 cm. We also made 
a few experiments in which the extent was varied, in order to 
study the rate of a movement during its course. As the result 
of 200 experiments on C. we obtained the following times: 
Extent, 10 30 50 70 cm. 
Time, 13 63 104 1410- 
The motion was therefore nearly uniform at the rate of 5 M. 
per second. But at the beginning of the movement there was 
apparently a greater followed by a lesser rate. PART IV. 
ON LIFTED WEIGHTS. 
Sec. 40. Introductory.-Weber's generalization concerning the 
relation of the least observable difference in sensation to the mag- 
nitude of the stimulus was chiefly based on experiments with 
lifted weights.1 These experiments are of considerable historical 
interest, but they were too few and too inexact to contribute to 
our present knowledge. Fechner's research with lifted weights 
is well known, and has, perhaps, never been surpassed in extent 
and thoroughness by any investigation in any science. The ex- 
periments were made 1856-1859, and Fechner always intended 
to publish the details, but left this unaccomplished. Summaries 
are, however, contained in the Elemente and in the Revision. 
Fechner made 67072 experiments on himself by the method of 
right and wrong cases. The normal weight was varied between 
300 and 3000 grammes, and the "comparison weight " was from 
1.5 to 8 per cent, heavier. The experiments show that the per- 
centage of right cases (+% the doubtful cases) remained toler- 
ably constant, when the difference between the two weights was 
proportional to the normal weight. The experiments thus tend 
to support Weber's law. The percentage of right cases, how- 
ever, generally becomes less, as the normal weight is taken 
smaller, and this Fechner explains by supposing that the weight 
of the arm should be added to both of the weights. The most 
serious criticism to which Fechner's experiments are open. 
1 See references, Sec. 2. 117 
would seem to be that it is difficult, when the relations of the 
weights are known, to compare the sensations without taking the 
known differences of the stimuli into account. Fechner defends1 
this method, naively arguing that its objectivity is proved by the 
close correspondence of his experiments with the law in defence 
of which they were undertaken. We had intended to compare 
the results of experiments, when the subject knew the objective 
relations of the stimuli, with those in which he was ignorant of 
these. We have, however, only found that it is more difficult for us 
to come to a decision when we know the objective relations. The 
large number of doubtful cases recorded by Fechner (they are 
nearly one-seventh of the whole number) seems, indeed, to bear 
witness to a similar difficulty. 
Hering2 gives an account of some experiments on lifted 
weights made by Biedermann and Loewit at his suggestion. 
These seem to show that the least noticeable difference becomes 
greater as the stimulus is taken greater, but much more slowly 
than in direct proportion. The variation from Weber's law is 
thus in the same direction as in Fechner's experiments, but far 
greater. 
Merkel,8 in his extended psychophysical study, found Weber's 
law to hold for pressure made by the moving finger with great 
exactness, the least perceptible difference being of the stimu- 
lus. These experiments were made by Merkel on himself, and 
we judge them to be in need of independent confirmation. 
Pierce and Jastrow (op. cit.} found the probable error for press- 
ure, from which the muscular sense was not entirely excluded, 
to be about of the stimulus (250 gm.), and to be further dimin- 
ished by practice. 
1 Revision, p. 58. 
• Ueber Fechner's psychophysisches Gesetz, Sitzungsb. d. Wiener Akad. d. Wiss., Ill 
Abth.,1875, p. 33. 
3 Op. cit. 118 
Several observers have studied the perception of lifted weights 
in order to determine what part is due to the skin sensations, and 
what part to the muscular sensations. These experiments have 
been made on paralytic patients, and by stimulating the muscle 
electrically.1 They seem to show that the discrimination of 
lifted weights is not interfered with, though the skin be ataxic, 
or though the movement be caused by electric stimulation. But, 
as is so common in clinical experiments, cases are cited from 
which exactly opposite conclusions are drawn. 
The only research remaining for us to mention is the recent 
interesting paper by Muller and Schumann, already noted. In 
view of the effects of contrast, etc., they study the nature of 
the motor impulse, and conclude that the velocity with which a 
weight is lifted is the chief factor in the discrimination of the 
force of movement. 
Our experiments with lifted weights were made with a view 
to studying methods for determining the fineness of discrimina- 
tion and the factors entering into the perception of movement. 
They are not concerned with the relation between the magni- 
tude of the stimulus and the error of observation, and were 
mostly made earlier than the experiments in the foregoing Parts 
of this paper. 
Sec. 41. Apparatus and Methods.-When lifted weights are 
used as stimuli they can be easily arranged and accurately 
measured. We loaded wooden boxes so that they weighed (to 
teeeett) I00' io4' i0&' 112 and 116 gm. Care was taken to 
place the load in the centre of the box and to make the pack- 
ing firm. The boxes were cylinders about 6 cm. in diameter 
and 3 cm. high. The weights looked alike, and several sets 
were used, so that the observer might not by any chance learn 
them. In the later experiments paper caps were made, and 
1 References will be found in Wundt, James and Muller. 119 
shifted from weight to weight. But these precautions were 
scarcely necessary, as the observer did not look at the weights. 
The observer sat on an ordinary chair facing a low table. The 
boxes were placed, two at a time, in a line parallel to the edge 
of the table, but a little back and opposite the left shoulder. 
A seconds pendulum swung in front of the observer. 
The recorder placed the 100 gm. weight and one of the 
heavier weights for the observer. One was about i cm. to 
the right of the other, and this was always lifted first. The 
weight was lightly grasped on the side with the thumb and 
fingers, the observer being allowed to choose the position which 
seemed to him the most convenient. The weights were not 
raised perpendicularly, but carried along from left to right. We 
chose this movement because we thought it could be kept the 
most constant. The movements were made in time with the 
pendulum, and in the direction in which it swung. The first 
weight was lifted, as nearly as possible, for one second, during 
the next second the position was recovered, and, lastly, the 
second weight was lifted for one second. The time was easily 
kept, as the motion of the pendulum was followed almost auto- 
matically. It was our intention to have the extent of movement 
30 cm., but all the observers showed a tendency to make it 
greater, and we did not check this. A swing of the arm of 
about 40 cm. would thus seem to be most natural, as there is 
a most natural swing of the leg in walking, due to its acting as a 
pendulum. 
After the observer had lifted the two weights he was required 
to say whether the second weight seemed lighter or heavier than 
the first. He was also required to estimate what confidence he 
felt in his decision, a, b and c were used in the manner already 
described, a meaning sure of the correctness of the decision, b 
less certain, and c uncertain. If the observer could discern no 120 
difference whatever, he was allowed to say " doubtful " (d\ but 
was asked to use this as seldom as possible, and in our later ex- 
periments he was required to guess even when quite doubtful. 
Forty experiments were made in a series, each of the four differ- 
ences being used ten times, and the second weight being, in 
half the cases, the heavier, in half the lighter. The order was 
irregular, and could not have been guessed by the observer. 
Ten series were made with each observer, so that 100 trials were 
made with each difference. The observer was always ignorant 
of the real relations of the weights, and only the writers knew 
the results of previous experiments. 
Sec. 42. Results of Experiments.-The results of 4000 ex- 
periments made on nine observers are contained in the accom- 
panying table. The number of right, of wrong and of doubt- 
ful cases is given, for each amount of difference, and according 
to whether the second weight was the lighter or the heavier. 121 
The Discrimination of Lifted Weights Weighing about ioo gm.-Method of Right and Wrong Cases-9 Observers- 
4000 Experiments-II. 1890-I. 1891. 
SECOND WEIGHT THE LIGHTER. 
SECOND WEIGHT 
THE HEAVIER. 
ioo : 
104. 
108. 
112. 
116. 
104. 
108. 
112. 
116. 
R. W. D. 
R. W. D. 
R. W. D. 
R. W. D. 
R. W. D. 
R. W. D. 
R. W. D. 
R. W. D. 
R 
36 14 O 
35 i5 0 
45 5 0 
45 5 0 
28 22 O 
36 14 0 
4i 9 0 
45 5 0 
K 
25 25 O 
30 20 0 
40 10 0 
36 13 1 
40 IO O 
43 7 0 
46 4 0 
49 1 0 
G 
17 27 6 
29 15 6 
32 13 5 
37 11 2 
43 7 0 
47 1 2 
48 I I 
50 0 0 
L 
33 i7 0 
36 14 0 
45 5 0 
47 3 0 
37 13 0 
39 11 0 
49 1 0 
48 2 0 
S 
33 17 0 
37 11 2 
4i 9 0 
45 3 2 
33 17 0 
36 14 0 
47 2 1 
46 4 0 
N 
27 13 10 
33 12 5 
43 5 2 
45 4 1 
34 14 2 
44 2 4 
38 3 9 
49 0 1 
D. I. . . ■ 
3° 16 4 
36 10 4 
47 2 1 
50 0 0 
39 9 2 
39 i° 1 
49 1 0 
50 0 0 
D. II 
43 7 0 
34 14 2 
46 4 0 
47 3 0 
36 13 1 
42 80 
50 0 0 
50 0 0 
F 
33 16 1 
4i 6 3 
46 3 1 
47 3 0 
36 10 4 
45 2 3 
44 5 1 
48 1 1 
C 
38 11 1 
42 62 
45 4 1 
50 0 0 
35 13 2 
4i 6 3 
42 7 1 
50 0 0 
TABLE XXL 122 
As we have seen, the figures given in the table do not directly 
measure the fineness of discrimination. In the manner de- 
scribed in the Introduction we have calculated the probable 
errors of the several observers for the four amounts of differ- 
ence and give the results in Table XXII.1 
Probable Error with Lifted Weights. 
TABLE XXII. 
ioo : 
104. 
108. 
112. 
116. 
AV. 
v. 
R 
7-4 
9.8 
7-3 
8-3 
8.2 
.8 
K 
6.4 
8.1 
7-2 
8.2 
7.5 
•7 
G 
6-5 
5-i 
6.8 
7- 
6.3 
.6 
L 
5-i 
8. 
4.8 
6.6 
6.1 
1.2 
S 
6.8 
8.4 
6-3 
7-7 
7.3 
■7 
N 
6.1 
5-5 
7-3 
5-9 
6.2 
■5 
D. I 
4-5 
7-i 
4-3 
4.6 
5.1 
1.0 
D. II 
3-2 
7-i 
4-3 
5-6 
5. 
1.3 
F 
4-7 
4-3 
6. 
6.2 
5.3 
.8 
C 
4.1 
5-i 
6.8 
4.6 
5.1 
.8 
Av 
5.5 
6.8 
6.1 
6.5 
6.2 
.8 
V 
1.2 
1.5 
1. 
1.1 
.9 
This table shows clearly the probable error of the several 
observers-that is, the difference in grammes which each of 
them could expect to distinguish 75 times out of 100. On 
the average, weights can thus be discriminated when they are 
1 In this calculation the doubtful cases have been divided equally between " right and 
wrong." Much discussion has arisen concerning the disposition of doubtful cases, but 
we think the correct solution has not been suggested. We believe it is found in the 
proper application of the probability integral. However, when there are but few doubtful 
cases, this would not give a result appreciably different from an equal division. As has 
been said, we think it best to let the observer divide his doubtful cases by guessing, and 
this has been done in our later experiments. When the observer is always right in his 
judgment his fineness of discrimination has not been measured. We must assume that 
errors would occur if the series were continued long enough, and have somewhat arbi- 
trarily set 99 as the most probable percentage of right cases. 123 
about as 100:106 (or 110:116) gm.1 or when the difference 
is (say) of the stimulus. If we wish to know what 
difference can be distinguished 99 times out of 100 the figures 
in the table are to be multiplied by 3.45, or about 3%. Thus, 
on the average, the difference between 100 and 121 gm. can be 
distinguished 99 times out of 100. 
The table shows that the fineness of discrimination of the 
observers did not differ greatly, the probable error varying 
from 8.2 to 5 gm., or from about to of the stimulus. If 
the observers be classified we find that the average probable 
error for two women was 7.8, for two advanced students of 
biology 6.2, for three advanced students of psychology 6.2, and 
for the two writers 5.2. As 400 experiments were made with 
each observer the results must be considered accurate, but it 
should be remembered that in all anthropometric experiments 
the size of the error may be due to accommodation to the condi- 
tions of experiment, as well as to real differences in the senses 
or faculties. The smaller probable error of the writers (nearly 
the same for the two) is thus probably not so much due to finer 
senses as to more practice in careful observation. 
From this point of view as well as for other reasons it is in- 
teresting to note the improvement with practice. We have 
therefore divided the experiments into five sets, each containing 
eighty judgments, and give in the accompanying table the per- 
centages of right cases for the observers with increasing practice. 
1 It will be noticed that in these calculations the error of observation is assumed to be 
the same for the several weights. This assumption is not strictly correct, but cannot be 
avoided until the relation between the error of observation and the magnitude of the 
stimulus is known. 124 
Right Cases in Percentages with Increasing Practice. 
TABLE XXIII. 
I. 
II. 
in. 
IV. 
V. 
R 
78 
75 
70 
81 
85 
K 
78 
79 
74 
80 
74 
G < . . 
82 
82 
83 
68 
78 
L 
70 
89 
85 
9° 
83 
S 
83 
78 
80 
78 
83 
N 
84 
75 
84 
84 
83 
D. I 
84 
88 
81 
88 
92 
D. II 
86 
9° 
83 
88 
89 
F 
86 
85 
89 
81 
93 
C 
9° 
87 
86 
86 
88 
Av 
82 
83 
81 
82 
85 
The averages in this table show a slight improvement in ac- 
curacy of discrimination as the series were continued, but it is 
so little that it might be due to chance variations. Two sets 
were made on D., and the second was slightly the better. The 
different series on the same observer vary considerably, and 
eighty judgments by the method of right and wrong cases 
would, therefore, scarcely be sufficient as a test. Ten experi- 
ments by this method would be worthless. 
Sec. 43. The Amount of Difference and the Degree of Confi- 
dence.-In these experiments we used four amounts of differ- 
ence, our object being to combine the method of the just notice- 
able difference with the method of right and wrong cases, and 
to study the application of the probability integral. As regards 
this latter, our experiments agree very well with the mathemati- 
cal theory. The probable error is nearly the same, whether it 
be calculated from a small difference and a relatively small per- 
centage of right cases, or from a larger difference and a larger 
percentage of right cases. With the smallest difference the 
probable error is the least, but this is most likely due to chance 125 
variations. If the probable error became gradually smaller as 
the differences were made smaller, it would indicate either a 
larger probable error with the greater weights (which is doubt- 
less the case but to a small degree) or, if considerable, the pres- 
ence of factors other than the amount of difference, such as un- 
conscious recognition of the weights by touch, or an inference 
from the order of presentation. Below (in Sec. 49) we give a 
curve showing the percentages of right cases for the several dif- 
ferences required by the probability integral, and those obtained 
by actual experiment. 
By varying the amount of difference and making it in some 
cases so large that it was usually perceived correctly, we have 
combined the method of right and wrong cases with the method 
of the just noticeable difference. This shows clearly that the 
method of the just noticeable difference in its scientific form is 
simply a case of the method of right and wrong cases. We 
also see that it is not the most favorable amount of difference 
to choose for experiment; if the observer be always right, we 
have no satisfactory basis for comparison; and, further, a chance 
error, due, say, to complete distraction of the attention, affects 
the probable error more when the percentage of right cases is 
above 90 than when it is between 80 and 90. 
The method of the just noticeable difference, in its less scien- 
tific form, has also been combined with the method of right and 
wrong cases by recording the degree of confidence of the ob- 
server. If the just noticeable difference be the difference the 
observer is sure that he can notice, it is that difference with which 
a was given ; if it be the difference he feels some confidence that 
he can notice, it was that with which b was given. The number 
(which is also the percentage) of times a and b were given by 
the several observers is shown in the accompanying table.1 
1 c was given in the remaining cases of the 100, unless the observer were doubtful. 
The number of times this occurred is shown in Table XXI. 126 
The Degree of Confidence of the Observers with Varying Amounts of 
Difference. 
TABLE XXIV. 
ioo : 
104. 
108. 
III. 
116. 
AV. 
A. B. 
A. B. 
A. B- 
A. B. ' 
A. B. 
R 
10 45 
19 46 
25 46 
33 46 
22 46 
K 
41 0 
51 0 
62 I 
64 0 
54 0 
G 
47 37 
64 22 
71 15 
72 21 
63 24 
L. . . 
13 72 
22 69 
32 58 
39 53 
26 63 
S 
60 36 
51 45 
83 15 
81 16 
69 28 
N 
34 29 
40 28 
47 21 
62 26 
46 26 
D. I. . . 
14 61 
11 68 
28 64 
39 56 
23 62 
D. II. . . 
5 5° 
8 54 
24 59 
30 57 
17 55 
F 
3 3i 
17 29 
23 34 
37 34 
20 32 
C 
1 16 
2 28 
9 3i 
12 58 
6 33 
Av. . . . 
23 38 
• 28 39 
40 34 
47 37 
35 37 
- 
J 
- 
We see from the figures in the tablethat the number of times 
the observer felt confidence in the correctness of his decision 
usually increased as the amount of difference was taken greater. 
Compared with the percentages of right cases, the fictitious 
averages are: 
ioo : 104. 108. 112. 116. 
% of times right, 69 77 89 94 
% of times a and b, 61 67 74 84 
The confidence thus varies nearly as the percentage of right 
cases, and some reliance may, therefore, be placed on such in- 
trospection. We see, however, from the table, that different in- 
dividuals place very different meanings on the degree of confi- 
dence. Some observers are nearly always quite or fairly confi- 
dent, while others are seldom quite confident. If this table be 
compared with the preceding, it will, indeed, appear that those 
observers who felt the greatest degree of confidence in their 
judgment had the largest probable error, while those who were 127 
most seldom quite confident had the smallest probable error. It 
is, therefore, absurd to measure the fineness of discrimination of 
an observer by the difference which he thinks he can notice, as 
is done when the method of the just noticeable difference is 
used in its usual form. 
We give in Table XXV. the percentage of times the observer 
was right in his judgment when he used a, b and c respectively. 
TABLE XXV. 
Percentage of Times the Observers were Right with Each Degree of 
Confidence. 
R. 
K. 
G. 
L. 
s. 
N. 
D. I. 
D. II. 
F. 
c. 
AV. 
a . 
95 
89 
9° 
97 
86 
91 
96 
100 
IOO 
100 
94 
b 
79 
58 
81 
66 
78 
89 
91 
97 
98 
82 
c 
65 
63 
68 
65 
75 
72 
68 
74 
77 
81 
71 
We see from the table that an observer is more apt to be 
right than wrong, even when he feels very little confidence in 
the correctness of his decision. We also obtain a rough meas- 
ure of what reliance may be placed on the judgment of the ob- 
server. 
Sec. 44. The Constant Error.-Table XXL shows that most 
of the observers were the more often right in their judgment 
when the second weight was the heavier. The second weight 
seemed relatively heavier than the first, which increased the 
apparent difference when it was in fact the heavier, and de- 
creased it when it was in fact the lighter. This constant error 
becomes eliminated in calculating the probable error, for we 
have in the one series and in the other ft so that C 
disappears in the average value of from which the probable 
error is calculated. If, however, we subtract the one equation 
from the other, we can find the relation of C. to P. E., and, 128 
consequently, the value of the constant error. These values for 
the several observers and differences are given in the accom- 
panying table.1 
Constant Error in Distinguishing Weights Weighing about too gm.-Method 
of Right and Wrong Cases-9 Observers-4000 Experiments. 
TABLE XXVI. 
too : 
104. 
108. 
112. 
116. 
AV. ± 
R 
-2.4 
•4 
-2. 
0. 
1.2 
K 
4- 
5- 
3- 
8.6 
5-1 
G 
6-5 
5-3 
7- 
86 
6.8 
L 
•9 
1.1 
2.8 
3-8 
2.1 
S 
.1 
- .8 
3-4 
0. 
i-3 
N 
•7 
3-5 
- .8 
4 3 
2-3 
D. I 
1.6 
•5 
i-3 
0. 
.8 
D. II 
-1.1 
2.5 
3- 
3-2 
2.4 
F 
•9 
1.4 
- 1.1 
i-5 
1.2 
C 
- -5 
• 2 
-M 
0. 
•5 
Av. ± 
1.9 
2.1 
2.6 
3. 
2.4 
The table shows that the constant error varied more than the 
probable or variable error. For two of the observers the con- 
stant error is about as large as their probable error. The con- 
stant error for F. and C. is small (within the limits of chance 
variations), and this may possibly be owing to the observers 
having unconsciously made allowance for a constant error, as 
they knew the tendency of other observers to overestimate the 
second weight. The constant error of the average increases 
regularly as the difference is taken greater. As is usually the 
case with constant errors, the overestimation of the second 
weight in these experiments is difficult to explain. At first sight 
it might be attributed to fatigue ; but as many pairs of trials were 
Owing to the limited number of experiments, some part of the errors given in this 
table is due to the variable error. 129 
made in succession, fatigue could scarcely tell in a single pair. 
The overestimation of the second weight may possibly be due 
to the fact that it was the more recently perceived. But the 
various effects of contrast have not yet been explained. 
Sec. 45. Conditions Affecting the Perception of Lifted 
Weights.-In the experiments which we have been discussing 
the weights were lifted as described, and the movement was 
always made in the same manner. It would evidently be an 
advantage to alter the nature of the movement, and from the 
variation in the probable error attempt to learn the factors con- 
cerned in the perception of lifted weights. With this object in 
view we made the movements in various ways, and give the 
results in the accompanying table. Only 200 experiments of 
each sort were made, and the probable and constant errors can 
only be regarded as approximate. The weights used were 100 
and 108 gm., the probable and constant errors being calculated 
as described. The nature of the movement is given in the table, 
and will be described more exactly below. The series were made 
in the order in which they are given in the table, but the groups 
collected under Av. Av. were made simultaneously, and half of 
the experiments on lifting the weights up and down with one 
and with two hands were made at the close. 130 
TABLE XXVII. 
Probable and Constant Errors in Lifting Weights 100:108 gm.-J. and C.- 
3000 Experiments-I. 1891-I. 1892. 
NATURE OF THE MOVEMENT. 
PROBABLE ERROR. 
CONSTANT ERROR. 
J- 
c. 
AV. 
AV.AV 
J- 
c. 
AV. 
AV. AV. 
Right to left ca. 40 cm. . 
5-7 
5-i 
5-4 
5-4 
- -4 
.2 
- -3 
- -3 
Up and down, right w't first 
8.8 
7.6 
8.2 
- -4 
- i-5 
- -9 
Up and down, left w't first 
6-3 
4.8 
5-5 
6.9 
+ i-3 
4-3-0 
4-2.1 
4- -6 
Two hands, right first . . 
7-3 
9.8 
8-5 
~3-7 
- -4 
- 2.0 
Two hands, left first . . . 
11.1 
6.4 
8-7 
8.6 
+ 4-7 
4-3-i 
-r-3-9 
4- -9 
Up only, 16 cm 
8.6 
9.8 
9.2 
+ 2.8 
- i-3 
4- -7 
Down only, 16 cm. . . . 
5-i 
13-1 
9.1 
9-i 
- -7 
4- 1.0 
4- -i 
4- -4 
From wrist only 
7-5 
8.2 
7-8 
4- 2.2 
4-2.3 
4- 2.2 
From forearm only . . . 
6.2 
6.6 
6.4 
4~ 2- 
- i-7 
4- -i 
From shoulder only . . . 
10.3 
8.2 
9.2 
7-8 
4-1.8 
- 6.2 
-2.2 
4- -i 
First 4 cm., second 16 cm. 
4.8 
6.4 
5-6 
- -3 
4- -7 
4- -2 
First 16 cm., second 4 cm. 
7-8 
5-4 
6.6 
+ 5-6 
- 3-3 
4-1.1 
First sec., second 2 sec. 
5-6 
5-3 
5-4 
- i-7 
- 3-o 
-2.3 
First 2 sec., second % sec. 
5-4 
4.6 
5-o 
5-7 
- i-3 
.0 
- .6 
- -4 
Pressing the weights . . . 
10.9 
10.3 
10.6 
10.6 
~ i-4 
-1 .8 
-1.6 
- 1.6 
Averages 
7.4 
7.4 
7.4 
7.4 
4- .6 
- .6 
0 
0 
The average probable error for J. and C. is exactly the same- 
7.4 gm., or about of the stimulus. Throughout our experi- 
ments the results for E, J. and C. agree closely. The average 
constant error is small for both J. and C.-less than of the 
probable error. It is negative for C.-that is, the second weight 
seemed relatively slightly the lighter, as was the case before. 
It is positive for J., as was the case with eight of the ten ob- 
servers given in Table XXVI. 
In considering the probable errors for the different kinds of 
movements, it must be remembered that they (being the result 
of only 100 experiments with each observer) would vary consid- 
erably even were the movement always the same. However, 131 
some reliance may be placed on the figures given under Av. Av. 
The probable error was smaller, and was subject to smaller con- 
stant errors when the weights were lifted across 40 cm., than when 
they were lifted up 8 cm. and set down again. The time of the 
movement in each case was one second. As before noted, a 
movement of about 40 cm. seems the most natural. J. was not 
told how long to make the movement, but made it on the average 
42.6 cm. When the weights were lifted up and down the height 
was determined by loose horizontal silk threads, which marked 
the distance without offering resistance. The distance the 
weights were lifted by J. in series two and three was noted with- 
out the observer's knowledge. When the judgment was right 
the heavier weight was lifted on the average 7.9, the lighter 7.8 
cm.; when wrong the heavier weight was lifted 7.9, the lighter 
7.9 cm. The distance which the weights were lifted was, con- 
sequently, not proportional to the weight, and the observer can- 
not judge of the difference by the distance (or rate) as main- 
tained by Muller and Schumann {op. cit^). The probable error 
was increased by about | when the weights were lifted with 
different hands. It was considerably increased when the weight 
was lifted up only (16 cm.) or down only. It became larger 
when the movement was made from the shoulder only, the wrist 
and forearm not being bent. It seems, therefore, that the per- 
ception depends on sensations accompanying the movements of 
the wrist and forearm, but when either of these is present the 
error is not appreciably greater than when the three joints are 
used. When the rate of movement was altered so that the one 
weight was lifted four times as rapidly as the other, either by 
being lifted higher in the same time or the same distance more 
quickly, the probable error was not increased. This result was 
unexpected, and proves conclusively that we do not judge of 
difference in weights by the rate at which they are lifted. When 132 
the weights were pressed by the fingers with force sufficient to 
lift 2 kg., the probable error was considerably increased. The 
observer is, consequently, helped by the lightness of touch, 
which normally is made just sufficient to hold the weights. The 
sensation may be partly touch and partly a muscular sensation 
from the fingers. When two hands were used the weight lifted 
with the right hand seemed relatively the heavier. But the 
constant errors are throughout within the limits of probable 
chance variations. 
In all these experiments the observer's confidence in his de- 
cision was recorded. The confidence seemed to correspond 
exactly with the apparent amount of difference. Thus the ob- 
server said a if the difference seemed quite evident, b if the 
difference was apparent but not so great that he felt sure of the 
correctness of his judgment, c if he noticed but little difference 
and felt doubtful as to his correctness, and d if he could discern 
no difference, and felt that his judgment was a mere guess. The 
percentages of the times the several degrees of confidence were 
given were as follows : 
a. 
b. 
c. 
d. 
. . 10 
25 
46 
19 
c. . 
. . 0 
13 
60 
27 
a or b is the just noticeable difference, and d is the threshold 
of German psychologists. The just noticeable difference (and 
also the threshold) would make J. a much better observer than 
C. No weight, consequently, can be laid on such determinations. 
The percentages of times the observer was right in his judg- 
ment were as follows: 
a. 
d. 
J. . 
• • 93 
86 
74 
65 
c. . 
. . X 
97 
8i 
60 
It is interesting to note that when the decision of the observer 
seemed a mere guess, he was considerably more likely to be right 133 
than wrong. This bears witness to the part played by subcon- 
scious mental processes in our daily life. C.'s judgment was 
more accurate than J's, as he was right a larger percentage of 
times when he felt some confidence, and wrong a larger per- 
centage of times when he felt none. C. could notice no differ- 
ence when the apparent difference (assuming the theory of the 
distribution of errors to obtain in this case) was less than | the 
probable error, and could notice the difference with some con- 
fidence when the apparent difference was more than double the 
probable error. C. should by theory have been right in about 57 
per cent, of the cases when he gave d, and in 97 per cent, of the 
cases when he gave b. These are nearly the percentages of 
right cases actually obtained, whence it follows that an observer 
can judge apparent differences in sensation with considerable 
accuracy. PART V. 
ON LIGHTS. 
Sec. 46. Introductory.-Numerous experiments have been 
made with a view to testing Weber's law for the sense of sight. 
We need not give an abstract of these, as our own experiments 
do not bear directly on the subject. The early experiments by 
Bouguer, Lambert, Steinthal, Masson, Fechner and Volkmann 
seemed to show that the just noticeable difference is a propor- 
tional part of the stimulus. The conclusion of later experi- 
ments,1 however, is that Weber's law holds only approximately 
and for lights of medium intensity. 
In these experiments the figures obtained as expressing the 
accuracy of discrimination proper to the sense of sight place it 
very high. Sight has, accordingly, been described as the most 
accurate of the senses. But it is evident that in the experi- 
ments with shadows and with disks, as they have been carried 
out, the conditions are very different from those of experiments 
1 Volkmann, Physiol. Untersuch., etc., Leipzig, 1863. 
Aubert, op. cit. 
Helmholtz, op. cit. 
Delboeuf, op. cit. 
Camerer, Mon. Bl. f. Augenheil., 1877. 
Kraepelin, Phil. Stud., II. 1885. 
Neiglick, Phil. Stud., IV. 1887- 
Merkel, op. cit. 
Koenig und Brodhun, Sitz. d. Acad, zu Berlin, 1888. 
Muller-Lyer, Archiv f. [Anat, u.] Physiol., Sup. Bd., 1889. 
The two papers last mentioned are of special importance. The German physiological 
archives and proceedings are, unfortunately, not included in the libraries of the Univer- 
sity of Pennsylvania or Cblumbia College, and were not accessible to us while this 
paper was being written. Since going to press we have examined some papers in the 
Astor Library, and added references, but it is too late to give abstracts. 
134 135 
with lifted weights. In comparing a shadow with its back- 
ground, the observer receives both impressions simultaneously, 
and attention passes from one to the other and back again, thus 
receiving a complex series of impressions. So it is also with 
variegated disks. In lifting weights, on the contrary, the im- 
pressions are successive. This must also be the case in experi- 
menting on hearing and other senses. In our own experiments 
with lights we have tried to make the conditions similar to those 
of our experiments on weights. Our results, in connection 
with which the size of the illuminated area on the retina must, 
of course, be taken into account, do not give pre-eminence in 
discriminative power to the sense of sight as compared with the 
muscular sense. 
Sec. 47. Apparatus and Methods.-The apparatus used in 
these experiments was planned by the writers. It is represented 
in the accompanying cut. 
Fig. 67 136 
A lamp (Z), provided with a metal hood, runs on a wooden 
slide. At one end of the slide is an upright plank pierced by a 
hole. This aperture can be enlarged or diminished in size by 
pulling out or pushing in two brass rods (7? 7?1) which are attached 
to overlapping plates of metal so cut as to leave between their 
edges a square hole. A seconds pendulum (P) swings in front 
of this hole, and a sheet-iron screen (5), attached to the bar of 
the pendulum, covers the hole, when the pendulum is held up 
by the electro-magnet (JZ). There is in the hood of the lamp, 
opposite the flame, a small hole (15 mm.) with a hinged cap 
(K). As the pendulum swings freely, the light of the lamp 
shines for one second through the aperture in the upright plank, 
and then is for one second cut off by the screen on the pendu- 
lum. The pendulum, weighted by a heavy iron ball, can at any 
time be held up by the electro-magnet (JZ), the recorder manipu- 
lating the key (at 0. The hooded lamp (A) gives the recorder 
light to keep his record and to measure the movement of the lamp. 
In the following cut may be seen the positions of the observer, 
of the recorder, and of the several parts of the apparatus. 
Fig. 68. 
The observer (O), behind a wooden partition (S), looked with 
one eye through a tube fixed in the partition at a piece of white 137 
paper (P) on the opposite wall and distant 4 M. A diaphragm 
(7?) permitted him to see on this paper a circle 10 cm. in di- 
ameter. The lamp (L} illuminated this paper through the aper- 
ture in the upright plank, when the aperture was not covered 
by the screen on the pendulum. The lamp was 250 cm. distant 
from the paper when at its farthest point. The recorder (2?) 
moved the lamp to or from the paper, and regulated the swing- 
ing of the pendulum by means of the key. During the experi- 
ments the room was darkened by the use of double blinds at 
the windows, and every effort was made to prevent scattered 
light escaping into the room from the two lamps. 
The experiments were made as follows :-The observer having 
placed himself in position, the recorder gave him a signal to 
secure attention, and then, turning the key, allowed the pen- 
dulum to swing away from the magnet. The lamp1 being, say, 
at what we may call the normal distance (250 cm. from the 
paper), the observer was thus given the standard sensation 
of light. With the return swing of the pendulum, and while the 
light was cut off by the screen, the lamp was moved nearer to 
the paper by a certain distance marked on the slide, so that the 
second light seen by the observer should be more intense than 
the first in the ratio of 100: no, 120, 130 or 140. At its next 
return the pendulum was caught up by the magnet. Between 
the next two lights the movement might be from the 140 point 
to the 100 point, etc. Each light lasted one second, and there 
was an interval of one second between the two lights. The 
observer was required to state whether the second light in each 
pair seemed brighter or fainter than the first, and to express his 
degree of confidence by a, b or c. A record was made after 
each experiment. 
1 The lamp at the usual height was ten-candle power. To prevent the observer from 
gaining a familiarity with the absolute intensity of the normal light, the lamp was fre- 
quently turned up or down a little. These variations in intensity were not great. 138 
Forty experiments were made in each series, in twenty of 
which the second light was fainter, and in twenty brighter than 
the first. Five experiments of each of these kinds were made 
in the one series for each of the four differences chosen. The 
results have been computed and tabulated as in the experiments 
on lifted weights. Of course the different experiments in each 
series were made in an irregular order, and the observer had no 
clue to guide him except the objective difference in the lights. 
Only the writers knew the results of previous experiments. 
Sec. 48. Results of Experiments.-The results of 4000 ex- 
periments made on nine observers are contained in the accom- 
panying table. The number of right, wrong and doubtful cases 
is given for each amount of difference and according to whether 
the second light was the fainter or the brighter. 139 
SECOND LIGHT THE FAINTER. SECOND LIGHT THE BRIGHTER. 
ioo : 
IIO. 120. 130. 140. IIO. 120. 130. 140. 
R. W. D. R. W. D. R. W. D. R. W. D. R. W. D. R. W. D. ! R. W. D. R. W. D. 
R 
... 37 13 0 42 8 0 49 i 0 50 0 0 28 22 0 29 21 0 31 19 0 37 13 0 
K 
... 38 12 O 39 II 0 44 6 O 46 4 O 36 14 O 38 12 0 41 9 O 46 40 
G 
... 37 9 4 46 2 2 48 2 0 50 0 0 18 27 5 31 18 i 40 7 3 40 82 
I 
... 44 6 0 46 4 0 48 2 0 50 0 0 28 22 0 39 11 0 48 2 0 50 00 
S 
48 2 0 50 0 0 49 1 0 50 0 0 14 35 1 29 20 1 33 16 1 46 40 
N 
... 37 8 5 42 1 7 47 1 2 49 0 1 29 12 9 41 2 7 39 1 10 48 02 
I). I 
... 42 2 6 45 1 4 48 1 1 50 0 0 21 19 10 32 9 9 36 11 3 4° 73 
D. II 
• • • 45 5 0 5° 0 0 50 0 0 50 0 0 25 15 10 42 3 5 45 4 1 49 10 
F 
... 43 6 1 48 1 1 47 0 3 50 0 0 22 18 10 33 8 9 42 2 6 49 01 
C 
• •• 37 10 3 43 4 3 48 2 0 50 0 0 24 12 14 35 7 8 43 0 7 45 05 
The Discrimination of Lights-Method of Right and Wrong Cases-9 Observers-4000 Experiments. 
TABLE XXVIII. 140 
As the figures given in the table do not directly measure the 
fineness of discrimination of the observer, we have calculated, as 
in the experiments with lifted weights, the probable errors of the 
several observers for the four amounts of difference. The 
results are contained in Table XXIX.1 
TABLE XXIX. 
Probable Error with Lights. 
ioo : 
IIO. 
120. 
130- 
140. 
AV. 
V. 
R 
17.0 
22.5 
17.2 
18.0 
18.7 
1.9 
K 
10.4 
18.3 
i93 
19.2 
16.8 
3.2 
G 
24.6 
14-3 
15-0 
16.6 
17.6 
3.5 
L 
10.0 
12.4 
"•5 
11.6 
11.4 
0.7 
S 
11.2 
10.6 
16.2 
14-5 
13.1 
2.2 
N 
10.8 
14-5 
13.8 
11.6 
12.7 
1.5 
D. I 
10.1 
12.5 
15-8 
16.4 
13.7 
2.4 
D. IL ... 
8.8 
7.6 
11.0 
12.3 
9.9 
1.7 
F 
"■0 
10.6 
12.8 
11.6 
11.5 
0.7 
C 
12.9 
•M 
13-7 
13.6 
13.4 
0.3 
Av 
12.7 
13.7 
14.6 
14.5 
13.9 
I.8 
V 
3.3 
3.0 
2.1 
2.4 
2.3 
.8 
We have here the difference, given in hundredths, of the in- 
tensity of the light chosen as a standard, which each observer 
may oxpect to distinguish 75 times out of 100. On the average, 
lights can thus be distinguished when they are about as 100:114, 
or when the difference is about one-seventh of the stimulus. 
Reckoning upon this basis, we find that a difference, to be dis- 
tinguished 99 times out of 100, would have to be about 48, or 
nearly one-half the stimulus. It will be noticed that the differ- 
ence is here much greater than that obtained in the experiments 
with lifted weights, and our results would seem to contradict the 
1 As before, the doubtful caseshave been divided equally between the right and wrong 
cases. 141 
commonly accepted doctrine of the greater fineness of discrimi- 
nation of the sense of sight. It should, however, be borne in 
mind that in these experiments the retina was illuminated over 
a comparatively small area, and but one eye was used. Had the 
area illuminated been greater, and had the light been seen with 
both eyes, a lesser difference could probably have been perceived- 
The lights used, moreover, were comparatively faint. We have 
as yet made no experiments to determine the influence upon 
our results of such changes in the conditions of experiment. In 
one perhaps very important respect, the experiments on lights 
and those on lifted weights were alike. The two lights were 
seen, as the two weights were lifted, successively, and the time 
allowed each stimulus, as well as the time between each pair of 
stimuli, was accurately determined and kept constant. Any 
experiments designed to determine the relative degree of fine- 
ness of discrimination proper to sight and the other senses must 
observe such conditions if trustworthy results are to be obtained. 
The variation of the averages for the several observers from 
their average (2.3) shows that the difference in discriminative 
power was not great, though there is a greater disparity between 
the best and the worst series (9.9 and 18.7) than in the case of 
lifted weights. Classifying the observers as before, we find the 
average probable error for two women was 17.7; for two ad- 
vanced students of biology, 14.5 ; for three advanced students 
of psychology (the first series on D. only being used), 13.2; and 
for the two writers, 12.4. 
To show the effects of practice on accuracy of discrimination, 
the experiments have been divided into five sets, each contain- 
ing eighty judgments, and we give in the accompanying table 
the percentages of right cases for each set in the case of each 
observer. The second series of experiments made on D. was, 
of course, made after the first series was completed. In this- 142 
table, however, it is treated like the other series. It will be no- 
ticed that it is considerably better than the first series on D. 
TABLE XXX. 
I. 
n. 
in. 
IV. 
V. 
R 
82 
74 
72 
82 
69 
K 
82 
81 
74 
91 
81 
G 
59 
81 
80 
85 
82 
L 
86 
87 
84 
86 
90 
S 
77 
82 
79 
81 
79 
N 
84 
85 
80 
86 
80 
D. I 
72 
79 
74 
77 
85 
D. II 
87 
85 
87 
9i 
92 
F 
72 
84 
69 
9i 
97 
C 
81 
76 
74 
87 
87 
Av 
78 
81 
77 
86 
84 
Right Cases in Percentages with Increasing Practice. 
The table shows a slight improvement in accuracy of discrim- 
ination in the successive sets of eighty judgments, but it is very 
slight, and much stress cannot be put upon it. It might be due 
to accidental variations. 
Sec. 49. The Amount of Difference and the Degree of Confi- 
dence.-In these experiments, as in those with lifted weights, 
four amounts of difference were used. The probable error is, as 
the table shows, nearly the same whether calculated from the 
number of right cases with a lesser difference or the number of 
right cases with a greater. This verifies experimentally the 
mathematical theory of the relation of the number of right cases 
to the difference used. The probable error is, however, as 
before, somewhat smaller with the smaller differences, and this 
indicates the presence of some small disturbing factor. We 
give curves showing the results of our experiments on weights 
and lights, as compared with curves showing the relations re- 143 
quired by the probability integral. The agreement is close, and 
the probable error, whatever the difference and corresponding 
Fig. 69. 
percentage of right cases, may, consequently, be used as a 
measure of accuracy of discrimination. The curves show clearly 
that under the conditions employed by us weights can be dis- 
tinguished much better than lights. 
By making the difference in some cases so large that the 
observer was nearly always right (100:140), we have combined 
with the method of right and wrong cases that of the just notice- 
able difference. It is scarcely necessary to point out again that 
this last-mentioned method, when used in any satisfactory way, 
is simply a form of the method of right and wrong cases, and 
that the difference employed is not the best difference to use in 
experiments by the method of right and wrong cases. 
The observer was required in these experiments to use a, b 
and c, in the manner already described, to indicate the degree 
of his confidence in the judgment made. The accompanying 
table shows how many times a and b were given by the several 
observers.1 
1 The rest of the time the observer gave c or d. The d's are recorded in Table 
XXVIII. The figures in Table XXXI. are also percentages. 144 
The Degree of Confidence of the Observers with Varying Amounts of 
Difference. In 
TABLE XXXI. 
ioo: 
IIO. 
120. 
130- 
140. 
AV. 
a. b. 
a. b. 
a. b. 
a. b. 
a. b. 
R.. . . •. 
o 38 
5 42 
15 48 
14 58 
8 46 
K 
1 6 
2 6 
8 13 
12 l8 
6 11 
G 
50 34 
63 28 
71 15 
87 8 
68 19 
L 
6 44 
16 52 
35 55 
70 26 
32 44 
S 
68 28 
65 3° 
77 17 
79 19 
72 23 
N 
16 23 
28 34 
54 26 
69 20 
42 26 
D. I. . . 
11 33 
20 34 
33 48 
56 34 
3° 37 
D. II. . . 
3 26 
1 65 
15 65 
47 48 
16 51 
F 
5 i7 
9 36 
20 47 
50 35 
21 34 
C 
1 13 
4 23 
17 32 
28 35 
12 26 
Av. . . . 
16 25 
21 35 
34 37 
51 30 
31 32 
_ 
. 
It will be seen that the as and b's increase as the difference 
increases, and as the percentage of correct judgments increases. 
Compared with the percentage of right cases the averages are : 
ioo : 
IIO. 
120. 
130. 
140. 
Percentage of times right, 
69 
83 
89 
95 
" " a and b, 
40 
56 
71 
81 
The confidence of the observer is, hence, a fair measure of 
the correctness of his judgment, but it is evident that a and b 
have a widely different meaning in the case of the several 
observers. G. and S. used a much more freely than did the 
others, and yet their probable error is not correspondingly small. 
That they used these letters in a looser sense than did the 
others is clear from the following table, which gives the per- 
centage of times the several observers were right with each 
degree of confidence. S. was willing to say b when he was 
right only 62 per cent, of the time. He seems, indeed, to have 
made no real distinction between b and c. It is worth noting 145 
Percentage of Times the Several Observers were Right with Each Degree 
of Confidence. 
TABLE XXXII. 
R. 
K. 
G. 
L. 
s. 
N. 
D.I. 
D. II. 
F. 
c. 
AV. 
a. . . . 
94.1 
100 
80.8 
99-2 
87.2 
IOO 
95-8 
IOO 
100 
100 
95.7 
b. . . . 
81.2 
95-° 
70.7 
88.7 
61.7 
96.I 
86.3 
95-7 
94-i 
95-i 
86.5 
c. . . . 
66.7 
79.0 
48.6 
72-9 
64-3 
75-9 
74-5 
79-8 
82.6 
85-5 
73.0 
that when the discrimination was equally good the confidence 
was less with lights than with weights. 
Sec. 50. The Constant Error.-All the observers showed a 
tendency to underestimate the second light, though the tend- 
ency was much more marked with some than with others. The 
amount of this error for each observer and for each degree of 
difference is shown in Table XXXII.1 
TABLE XXXIII. 
Constant Error in Distinguishing Lights-Method of Right and Wrong 
Cases-9 Observers-4000 Experiments. 
ioo : 
IIO. 
120. 
130. 
140. 
AV. 
R 
6.2 
13-2 
22.2 
22.7 
16.1 
K 
0.9 
0.9 
3-7 
0.0 
1.4 
G 
18.1 
12.9 
8.8 
17-4 
14.3 
L 
7-7 
5-8 
0.0 
0.0 
3.4 
S 
19.2 
16.4 
i9-4 
9.9 
16.2 
N 
2.9 
0.9 
5-9 
0.2 
2.5 
D. I 
9.2 
8-7 
14.2 
16.7 
12.2 
D. II. . . . 
6.7 
6.2 
8.1 
2-5 
5.9 
F 
8.4 
9.4 
5-7 
0.0 
5.9 
C 
4.1 
4-5 
2.6 
6.9 
4.5 
Av 
8.3 
7.9 
9.1 
7.6 
8.2 
V 
4.3 
4.2 
5.7 
7.2 
5.2 
: 
- 
' 
, 7~ 
1 The constant error was obtained here as in the experiments on lifted weights. 146 
It will be observed that the error varies considerably with 
different observers, and, indeed, with the same observer as the 
difference is greater or less. The averages do not increase with 
the difference, as is the case in the experiments with lifted 
weights. It is interesting to note that in no case was the first 
light underestimated, and in this respect the error is more con- 
stant than it is in the experiments with weights. 
As we have seen, when two weights are successively lifted, 
there is a sufficiently marked tendency to overestimate the 
second. When two lights are successively looked at, there is a 
still more marked tendency to underestimate the second. The 
cause of this underestimation is not clear. Fatigue would not 
seem to account for it in an entirely satisfactory manner, for the 
second light in each pair was the one which seemed fainter, and 
the successive pairs of lights did not grow proportionally fainter 
as the series was continued. To test this more fully we made 
two series of ten experiments each upon C. and two upon F. 
The pendulum was allowed to swing freely eleven times, while 
the observer compared the eleven resulting sensations, and de- 
scribed them to the recorder. The intensity of the objective 
stimulus was kept the same. The results of these experiments 
were as follows: 
i. 
2. 
3- 
4- 
5- 
6. 
7- 
8. 
9- 
10. 
C. I, 
f 
e 
f 
f 
f 
e 
f 
e 
f 
f 
C. II, 
e 
f 
f 
e 
f 
e 
f 
e 
f 
f 
F. I, 
f 
f 
e 
e 
e 
e 
e 
f 
f 
e 
F. II, 
f 
e 
f 
e 
f 
f 
e 
e 
e 
f 
Here f indicates that the light it represents appeared fainter 
than the one preceding, e that the light it stands for appeared 
the same as the one preceding. The letters in italics indicate 
that the difference was distinct. The writers have little con- 
fidence in the judgments not marked by letters in italics. The 
judgments had to be made rapidly, and the differences were felt 147 
to be slight. They are, however, confident that they were never 
inclined to regard a light as brighter than the one before it. 
Sec. 51. Memory for Weights and Lights.-In making experi- 
ments on the perception of small differences, the time elapsing 
between the sensations to be compared should not be neglected. 
In the experiments hitherto described the interval was always 
one second. It is possible that two seconds or longer might be 
a more favorable interval for comparison, but were the interval 
further lengthened the first sensation might be expected to fade 
from memory, while the rate of forgetting would be measured in 
terms of the error of observation. The study of memory for 
sensation is thus an attractive subject for psychological re- 
search, one quantity being studied as the function of another. 
Indeed, the earliest workers1 in the field of psychophysics made 
a few experiments on the subject, but they did not continue 
them long enough to obtain definite results. Wolfe2 carried out 
extended experiments on memory for the pitch of tones. He 
found about two seconds the most favorable interval; after this 
the percentage of right cases decreased, at first rapidly and then 
more slowly, until with an interval of sixty seconds the probable 
error is about five times as great as with one second. Lehmann3 
made a few experiments with shades of gray produced by re- 
volving wheels, and obtained results agreeing with Wolfe's. 
On the other hand, Paneth4 recently found that intervals of 
time could be reproduced nearly as well after longer (up to five 
minutes) as after shorter intervals; and Exner, who communi- 
cates these results, says that another experimenter obtained 
similar results with lights. The important monograph on 
1 Weber, Hagelmayer and Fechner. 
2 H. K. Wolfe, Philos. Stud., III. 1886. Wolfe finds that the ratio of right to wrong 
cases is inversely proportional to the logarithm of the time. But accuracy of discrimi- 
nation is not directly measured by the percentage of right cases. 
3 Alfred Lehmann, Philos. Stud., V. 1888. 
4 J. Paneth, Centr. f. Physiol., IV. 1890. 148 
Memory by Ebbinghaus,1 in which more complex impressions 
were used, is well known. 
Our own experiments on memory were the first we made. 
They were postponed in order that we might study the method, 
and were not afterward completed. We used seven intervals 
varying from one second to about one minute, the experiments 
otherwise being conducted in a manner similar to those already 
described. The time of stimulation was one second. The normal 
weight was about 100 gm., and the difference usually 8 gm. 
The light was that described, the difference being one-fifth 
of the whole amount. The weights and lights were usually 
varied within narrow limits in order that the observer might 
not learn them, and thus be helped in his decision. On the 
weights 100 experiments were. made with each interval, and on 
the lights 180, except with the 31 and 61 second intervals, with 
which only 80 were made. Ten different observers took part in 
the experiments. Owing to the large number of experiments 
required by the method of right and wrong cases, the probable 
errors given in the table are only approximate. The variable and 
constant errors have not been distinguished. In taking the aver- 
age the probable errors with the lights have been halved in order 
to reduce them approximately to the standard of the weights. 
TABLE XXXIV. 
Memory for Weights and Lights-Probable Errors for Intervals from i to 
6i Secs.-io Observers and 1760 Experiments-V. to XII. 1888- 
140 Experiments-XII. 1891. 
INTERVAL IN SEC. 
i. 
3- 
5- 
9- 
15- 
31- 
61. 
AV. 
Weights ioo gm. P. E. 
Lights ioo P. E. 
9.8 
8 
5-8 
11 4 
8.1 
10.2 
7.6 
7-7 
15-2 
9-3 
12.4 
[1i-8] 
8-3 
[10.2] 
9.6 
9.8 
Average 
8.9 
8.6 
9.1 
7.6 
12.2 
12.1 
9.2 
9.7 
1 Herm. Ebbinghaus, Ueber das Gedachtniss, Leipzig, 1885. 149 
The accompanying curve shows the relation between the 
probable error and the interval between the stimuli. 
Fig. 70. 
The experiments were not sufficient to determine exactly the 
probable errors with the different intervals, but these evidently 
do not become much larger as the interval is taken longer. The 
error of observation seems to be nearly the same so long as the 
interval is not over 9 sec., after which it is increased by about 
one-third. The smaller probable error with 61 sec. is most 
likely accidental. We had supposed that the increase in the 
error of observation with the larger interval would be more 
decided; the comparison becomes much more difficult to the 
observer, especially for the three longest intervals. The 
memory-image seems to last up to 9 sec., after which the ob- 
server does not so much compare the sensations as decide on 
the approximate intensity of each sensation separately, and 
compare the decisions. For example, the first sensation seems 
weaker than was expected, and this decision is remembered 
during the interval rather than the intensity itself. 150 
Sec. 52. Psychophysical Methods.-The method of the just 
noticeable difference-in which an observer finds a difference 
which he can just perceive-is not satisfactory. If the observer 
simply choose a difference which he thinks he can always or 
usually perceive, the result is without objective criterion. In- 
deed our experiments show that those who think they can per- 
ceive the smallest difference are apt to be the worst observers. 
If the percentage of mistakes made by the observer be re- 
corded, this method becomes a case of the following. But 
" the just noticeable difference " is not a convenient differ- 
ence to use in the method of right and wrong cases. If the 
percentage of right cases be very large, a single chance 
variation greatly affects the average. If there be no mistake 
we have indeed found a difference which can be perceived, but 
not the difference which can just be perceived, nor any other 
quantity which can be used as a measure of discrimination. If 
the just noticeable difference be interpreted by the observer as 
a difference apparently equal to some other difference, the 
method is reduced to that of estimated amount of difference. 
The method of right and wrong cases-in which two stimuli 
nearly alike are presented to an observer, and he is required to 
say which seems the greater-is the most accurate of the 
methods. It requires a considerable number of experiments- 
at least 100-and the number must be the greater, the less 
practised the observer. The method is consequently not well 
suited for provisional, anthropometric or clinical purposes. 
The percentages of right cases obtained do not directly 
measure the fineness of discrimination. The probable error, 
that is the difference with which the observer should be right 
75 per cent, of the time, is the most convenient measure of 
CONCLUSION. 151 
discrimination. The probability integral may be used to cal- 
culate the probable error, when the amount of difference is 
known, and the percentage of right cases is greater or less than 
75. It is better not to allow the observer to give doubtful as 
his decision, but the confidence felt by him in its correctness 
may be recorded with advantage. The observer is more apt to 
be right than wrong even when he feels little or no confidence 
in his decision. Some observers are not confident unless they 
are, in fact, right, while others are often confident when they are 
wrong. By recording the degree of confidence and using com- 
paratively large differences the method of the just noticeable 
difference may be combined with that of right and wrong cases. 
The probable error and the constant error may be determined 
separately. The latter is more irregular than the former. It is 
worth noting that this most accurate method of observation has 
not been used in the physical sciences. Psychology has 
hitherto been much indebted to physics for its methods, but the 
obligation will soon be mutual. 
The method of average error-in which an observer makes one 
stimulus as nearly as possible like another-is in many cases the 
most convenient of the methods. It is closely related to the 
preceding, as the probable error can be found either from the 
average error or from the percentage of right cases. The 
probable error of the just noticeable difference or of an esti- 
mated amount of difference may also be determined, and the 
several methods thus combined. The error obtained by the 
method of average error is complex, being partly an error of 
adjustment and partly an error of perception. These errors 
may be separately determined by requiring the observer to 
judge the stimuli by the method of right and wrong cases after 
they have been adjusted. The average error may be analyzed 
into a constant and a variable error. The distribution of the 152 
errors tends to follow the probability curve. This method can 
be used to special advantage when only a few experiments are 
made, as a result is reached more quickly than by the method 
of right and wrong cases. 
The method of estimated amount of differences-in which an 
observer judges the definite quantitative relations of stimuli, as 
in making one difference equal to or double another-gives vari- 
able results. The observer probably does not estimate quanti- 
tative relations in sensation, but quantitative differences in the 
stimuli learned by association. It is, consequently, an open 
question whether the differences in sensation are qualitative or 
quantitative. 
Great care should be taken in psychological experiments to 
keep all the conditions constant except the variable to be inves- 
tigated. The observer should not know the results of preceding 
experiments nor the objective relations of the stimuli. Experi- 
ments should not be rejected because they make the averages 
less accordant. The results of experiment depend on accom- 
modation to the conditions of experiment as well as on differ- 
ences in senses or faculties, and these factors should be sepa- 
rately studied. 
Sec. 53. The Error of Observation and the Magnitude of the 
Stimulus.-Weber's law, according to which the least noticeable 
difference is proportional to the magnitude of the stimulus, does 
not hold for movement, as the least noticeable difference (or 
the error of observation) increases more slowly than the stim- 
ulus. Fechner's law, according to which the sensation increases 
as the logarithm of the stimulus, does not hold, as it rests on 
Weber's law and on assumptions which are probably incorrect. 
As there is no logarithmic relation between mental and physical 
processes, the psychophysical, physiological and psychological 
theories put forward to account for it are superfluous. 153 
When amounts of difference in movements are estimated, the 
stimuli tend to be judged in their objective relations, and not as 
the logarithm of these. The results obtained by the method of 
right and wrong cases, and, by the method of average error, 
determine the error of observation. This is a physical quantity. 
Its correlation with other physical quantities (for example, 
the magnitude of the stimulus) depends on psychophysical 
conditions, and offers an important subject for psychological 
research. A mental quantity is not, however, directly measured. 
The error of observation usually increases as the stimulus is 
taken greater, but more slowly than in direct proportion to the 
magnitude. If the errors made in observing two stimuli of the 
same sort be combined, they will not be twice as large as the 
average error, but will equal the average error multiplied by the 
square root of two. This results both from theory and from 
our experiments; consequently, if two magnitudes, say two 
seconds, be observed continuously, the combined error in ob- 
serving the two seconds would tend to equal the error in ob- 
serving one second multiplied by square root of two, and 
generally the error in observing a magnitude, extensive or 
intensive, would increase as the square root of the magnitude. 
The summation of errors in this manner seems to account per- 
fectly for the usual increase of the error of observation ("just 
noticeable difference ") with larger magnitudes. The error 
would increase as the square root of the magnitude if each frac- 
tion of the magnitude, physically equal, were, in fact, subject to 
the same error of observation. In actual perception this would 
seldom or never be the case, but most of our experiments give 
an error of observation more nearly proportional to the square 
root of the magnitude than directly proportional to the magni- 
tude. We, therefore, substitute for Weber's law the following : 
The error of observation tends to increase as the sqtiare root of 154 
the magnitude, the increase being subject to variations whose 
amount and cause must be determined for each special case) 
Sec. 54. The Extent of Movement.-Experiments on the 
extent of movement were made by moving the finger along a 
scale fastened to the edge of the table. The four distances 
chosen were 100, 300, 500 and 700 mm. Time was kept by a 
seconds pendulum, one second being allowed for each movement, 
and one second for the interval between the two movements 
to be compared. Experiments were made by the four psycho- 
physical methods on one observer, and by the method of average 
error upon two others. 
The attempts to mark off a distance just greater and one just 
less than 500 mm. resulted in, respectively, 539.4 and 477.2 mm. 
The distance marked off in separate experiments was highly 
variable. Even for groups of 100 experiments the average just 
noticeable difference varied, for the attempt at just greater, be- 
tween 60.1 and 21.5 mm., and for the attempt at just less, be- 
tween 37.7 and 4.8 mm. In striking contrast with these figures 
was the slight degree of variation in the variable error and its 
variation. For instance, where in two groups of 100 experi- 
ments each, the just noticeable difference was 60.1 and 21.5 mm., 
the corresponding variable error was 9.8 and 8.9. The highly 
variable character of the just noticeable difference makes it 
of small value in psychophysical experiment. 
By the method of estimated amount of difference three kinds 
of experiments were made. It was attempted to halve 500 mm., 
to double 300 mm., and to find the mean between 300 and 700 
mm. The results of these experiments were all contrary to 
Fechner's law, the attempt to halve 500 mm. resulting in a dis- 
tance of 305.2, the attempt to double 300 giving one of 560.1, 
1 One of the writers (G. S. F.) does not give unqualified assent to the subject-matter of 
this section. See the note to Sec. 8. 155 
and that to find the mean between 300 and 700 giving one of 
512.4. In these experiments the variable error was, in relation to 
the whole extent of the movement made, greater than in the 
experiments by the method of just noticeable difference. 
The experiments by the method of average error consisted in 
attempts to measure off on the scale 100, 300, 500 and 700 mm. 
The variable errors for these magnitudes (with three observers) 
were 10.0, 18.7, 21.1 and 17.2. The best of the observers was 
about three times as accurate as the worst. The variable error 
does not accord with Weber's law, but increases much more 
slowly than the stimulus. For all the observers it was actually 
smaller for 700 than for 500 mm., this probably because the 
distance was nearly the limit which could be reached, and the 
observer was helped by the strain. 
In the experiments by the method of right and wrong cases 
the stimuli used were 500 and 510 mm. When the second 
stimulus was the greater, 75 per cent, of the judgments were 
right, and when the second was the less, 71.8 per cent. The 
probable error (the difference which could be distinguished 75 
per cent, of the time) is 11 mm. The first movement was slightly 
underrated, the constant error being .7 mm. The degree of confi- 
dence expressed by the observer was a fair index of the objective 
correctness of his judgment. 
Sec. 55. The Force of Movement.-A dynamometer may be 
used to advantage in studying the discrimination of the force of 
movements, but the clinical dynamometers are too inaccurate 
for scientific experiment. In making experiments on movement 
the observer can himself give the first or normal movement as 
well as the second or judgment movement, and the two move- 
ments will thus be made and perceived under like conditions. 
"The just noticeable difference" in the force of movement 
varied greatly, not being proportional to the error of observation, 156 
but more accordant results are obtained if the probable error be 
found by taking into account the number of mistakes made by 
the observer. The average error of the just noticeable differ- 
ence may also be used as a measure of discrimination. The just 
noticeable differences (for two observers) for about 2, 4, 8 and 
16 kg. were, respectively, about X, |, | and of the stimulus. 
The variable errors of observation were, respectively, .12, .20, .37 
and .41 kg., and the probable errors obtained by taking the 
percentage of the errors into account were, respectively, .14, 
.26, .40 and .45 kg. 
Experiments by the method of average error gave (for five 
observers) variable errors .19, .29, .43 and .46 kg.; for the 
magnitudes 2, 4, 8 and 16 kg., respectively. The worst of 
the five observers had an error about one-third larger than the 
best. Some observers are relatively better with the weaker, 
some with the stronger movements. There were considerable 
constant errors varying with different observers. The second 
pull was too great with the smallest magnitude, and usually too 
small with the largest. Neither the just noticeable difference 
nor the error of observation is a proportional part of the stim- 
ulus. Weber's law, consequently, does not hold for the force of 
movement. The error of observation is nearly proportional to 
the square root of the magnitude. 
The error, when two movements are made as nearly alike as 
possible, is partly an error of perception and partly an error of 
adjustment, and these two factors may be separated. The error 
of perception was, on the average, about twice as great as the 
error of adjustment, but the error of adjustment was relatively 
the smallest for the best observers. The errors in making two 
movements as nearly alike as possible tend to be distributed as 
required by the probability curve. The combined error obtained 
by adding algebraically the errors in pairs is nearly equal to the 
average error multiplied by the square root of two. 157 
Experiments by the method of estimated amount of differ- 
erence showed that the force of movements tends to be esti- 
mated in their objective relations, and not as the logarithm of 
these. The results are variable and subject to large constant 
errors. 
Sec. 56. The Time of Movement.-Apparatus can be con- 
structed which will measure accurately and conveniently the 
time either of a slow movement or of a quick blow'. When 
movements are discriminated it is an advantage to let the ob- 
server adjust the time of the first as well as of the second 
movement. 
The average errors in judging the time of movements (50 
cm. in extent) with the arm, lasting about |, | and 1 sec., 
were (as the average of four observers) .014, .028, .052 and 
.100 sec., respectively. The error of the worst of the observers 
was about twice that with the best. 
With | and 1 sec., when the time of the two movements 
seemed equal, the second was the slower; when two blows in 
succession are made as quickly as possible, the second is the 
quicker and seems the quicker. | sec. seemed less than half of 
1 sec., and more than double | sec. 
The results obtained by analyzing the error into an error of 
perception and an error of adjustment, and from the distribu- 
tion of errors and summation of errors, were nearly the same 
as with the force of movement. 
The time of the quickest possible blow (50 cm. in extent) 
varied (with four observers) from .085 to .181 sec. While 
the rate of movement varies considerably with different ob- 
servers, its average variation under like conditions is small- 
for a good observer .005 sec. The time was about the same 
for the right and left hand, and the rate was nearly uniform. 
The rate of movement should be used in the study of diseases 
of the nervous system. 158 
Within the limits investigated the extent of movements can be 
judged better than the force, and the force better than the time. 
Sec. 57. Lifted Weights.-The probable error in discrimi- 
nating lifted weights weighing about 100 gm. varied (for nine 
observers) from 5 to 8.2 gm., the average being 6.2 gm. This is 
the difference which could be correctly distinguished three- 
fourths of the time. The difference which could be correctly 
given 99 times out of 100 would be about 21 gm. The probable 
error is nearly the same, whether calculated from a large differ- 
ence and large percentage of right cases, or from a small differ- 
ence and smaller percentage of right cases. The confidence felt 
by different observers in the correctness of their judgment 
varies greatly, and is not proportional to their fineness of dis- 
crimination. The constant error can be calculated. In these 
experiments it varied from .5 to 6.8 grammes. The second of 
the two weights seemed relatively the heavier to nearly all the 
observers. In judging the accuracy of discrimination of an ob- 
server, both variable and constant errors should be considered. 
The probable error is not greatly altered when the manner of 
lifting the weights is altered. It becomes larger when the 
weights are lifted with different hands, or up or down only, or 
are pressed with the fingers. It is scarcely altered when one 
weight is lifted four times as high or four times as fast as the 
other. 
Sec. 58. Lights.-In our experiments on lights, apparatus 
was devised to give the observer two sensations of light in suc- 
cession, each lasting one second and one second apart. The 
conditions were thus similar to those in the experiments with 
lifted weights. The lights compared were as 100 to no, 120, 
130 and 140. The probable error (given in hundredths of the 
intensity of the stimulus) varied, for nine observers, from 9.9 to 
18.7, with an average of 13.9. Reckoning upon this basis, a 159 
difference, to be correctly given 99 times out of 100, would 
have to be about 48, or nearly half the stimulus. This large 
figure may be due partly to the fact that the illuminated area on 
the retina was small and the lights comparatively faint, but it 
was probably chiefly due to the sensations being successive. 
We consider it an advantage to have the sensations succes- 
sive, as the conditions can thus be kept constant, and sight 
can be compared with the other senses, muscular sense, hear- 
ing, etc. Different observers differed much in their degree 
of confidence in the correctness, of their judgment, and their 
degree of confidence was no indication of the relative fineness 
of their power of discrimination. . For the same observer, how- 
ever, the degree of confidence corresponded fairly well to the 
degree of objective accuracy. All the observers showed a 
tendency to underestimate the second light, the constant error 
varying from 1.4 to 16.2. Under the conditions employed the 
muscular sense is about again as accurate as the sense of sight. 
Memory for sensations may be studied by increasing the in- 
terval between the two stimuli to be compared, the probable 
error of an observer measuring his rate of forgetting. Observers 
remembered lifted weights and lights so well up to 9 sec. that 
their error of observation was not increased. When the time 
was from 15 to 61 sec., the error was increased by about one- 
third. This is contrary to the common view, according to which 
we are supposed to forget most rapidly at first. PHILOSOPHICAL SERIES. 
No. I. On Sameness and Identity. A Contribution to 
the Foundations of a Theory of Knowledge. 
By George Stuart Fullerton. 
No. II. On the Perception of Small Differences, 
with Special Reference to the Extent, Force and 
Time of Movement. By the Editors. 
IN PREPARATION. 
No. III. Descartes' " Meditations," with Latin, French 
and English Texts, and Philosophical Analysis. 
, By George Stuart Fullerton and William Ro- 
maine Newbold.