THE RELATION 
BETWEEN 
THE ATOMIC WEIGHTS. 
: ■ A 7;’ V 
OF 
THE CHEMICAL ELEMENTS. 
BY 
JOSIAH P. COOKE, Jr., 
ERVING PROFESSOR OF CHEMISTRY IN HARVARD UNIVERSITY. 
PROM THE MEMOIRS OF THE AMERICAN ACADEMY, NEW SERIES, VOL. V. 
CAMBRIDGE: 
METCALF AND COMPANY, 
PRINTERS TO THE UNIVERSITY. 
1 8 54. THE RELATION 
BETWEEN 
THE ATOMIC WEIGHTS 
OF 
THE CHEMICAL ELEMENTS. 
BY 
JOSIAH P. COOKE, Jr., 
ERVING PROFESSOR OF CHEMISTRY IN HARVARD UNIVERSITY. 
PROM THE MEMOIRS OF THE AMERICAN ACADEMY, NEW SERIES, VOL. V. 
CAMBRIDGE: 
METCALF AND COMPANY, 
PRINTERS TO THE UNIVERSITY. 
1 8 54. XI. 
The Numerical Relation between the Atomic Weights, with some Thoughts on the Classifi- 
cation of the Chemical Elements. 
By JOSIAH P. COOKE, Jr., A.M., 
ERVING PROFESSOR OF CHEMISTRY IN HARVARD UNIVERSITY. 
(Communicated February 28, 1854.) 
Numerical relations between the atomic weights of the chemical elements have been 
very frequently noticed by chemists. One of the fullest expositions of these relations 
was that given by M. Dumas of Paris, before the British Association for the Advance- 
ment of Science, at the meeting of 1851. This distinguished chemist at that time 
pointed out the fact, that many of the elements might be grouped in triads, in which the 
atomic weight of one was the arithmetical mean of those of the other two. Thus the 
atomic weight of Bromine is the mean between those of Chlorine and Iodine; that of 
Selenium is the mean between those of Sulphur and Tellurium, and that of Sodium, the 
mean between those of Lithium and Potassium. M. Dumas also spoke of the remark- 
able analogies between the properties of the members of these triads, comparing them 
with similar analogies observed in Organic Chemistry, and drew, as is well known, 
from these facts arguments to support the hypothesis of the compound nature of many 
of the now received elements. Similar views to those of Dumas have been advanced 
by other chemists. 
The doctrine of triads is, however, as I hope to be able to show in the present me- 
moir, a partial view of this subject, since these triads are only parts of series similar 
in all respects to the series of homologues of Organic Chemistry, in which the differ- 
ences between the atomic weights of the members is a multiple of some whole number. 
All the elements may be classified into six series, in each of which this number is 
1 2 
THE NUMERICAL RELATION 
different, and may be said to characterize its series. In the first it is nine, in the 
second eight, in the third six, in the fourth five, in the fifth four, and in the last three. 
The discovery of this simple numerical relation, which includes all others that have 
ever been noticed, was the result of a classification of the chemical elements made for 
the purpose of exhibiting their analogies in the lecture-room. A short notice of this 
classification will, therefore, make a natural introduction to the subject. 
Every teacher of Chemistry must have felt the want of some system of classification 
like those which so greatly facilitate the acquisition of the natural-history sciences. 
In most elementary text-books on Chemistry, the elements are grouped together with 
little regard to their analogies. Oxygen, Hydrogen, and Nitrogen are usually placed 
first, and therefore together, although there are hardly to be found three elements more 
dissimilar; again, Phosphorus and Sulphur, which are not chemically allied, are fre- 
quently placed consecutively, while Arsenic, Antimony, and Bismuth, in spite of their 
close analogies with Phosphorus, are described in a different part of the book. This 
confusion, which arises in part from retaining the artificial classification* of the elements 
into metals and metalloids, is a source of great difficulty to the learner, since it obliges 
him to retain in his memory a large number of apparently disconnected facts. In 
order to meet this difficulty, a classification of the elements into six groups, differing 
but slightly from that given in the table accompanying this memoir, was made. The 
object of the classification was simply to facilitate the acquisition of Chemistry, by 
bringing together such elements as were allied in their chemical relations considered 
collectively. As the classification has been in use for some time in the courses of lec- 
tures on Chemistry given in Harvard University, I have had an opportunity for ob- 
serving its value in teaching, and cannot but feel that the object for which it was made 
has been in a great measure attained. The series which is headed The Six Series 
will illustrate the advantage gained from the classification in a course of lectures, the 
elements which compose it being among those especially dwelt upon in lectures to 
medical students, and, generally, very widely separated in a text-book on the science. 
As Chemistry is usually taught, the properties of the members of this series, Nitro- 
gen, Phosphorus, Arsenic, and Antimony, as well as the composition and properties 
of their compounds, make up a large body of isolated facts, which, though without 
any assistance for his memory, the student is expected to retain. Certainly it cannot 
be wondered at, that he finds this a difficult task. The difficulty can, however, be in a 
great measure removed, if, after he has been taught that Nitrogen forms two important 
acids with Oxygen, N03 and N05, that it unites with Sulphur and Chlorine to form 
NS3 and NC13, and also with three equivalents of Hydrogen to form NH3, he is also 3 
BETWEEN THE ATOMIC WEIGHTS. 
told, that, if in these symbols of the Nitrogen compounds he replaces N by P, As, or 
Sb, he will obtain symbols of similar compounds of Phosphorus, Arsenic, and Anti- 
mony ; for he thus learns, once for all, the mode of combination of all four elements, 
so that when he comes to study the properties, in turn, of Phosphorus, Arsenic, and 
Antimony, he has not to learn with each an entirely new set of facts, but finds the 
same repeated with only a few variations. Moreover, these very variations he will 
learn to predict, if he is shown that the elements are arranged in the series according 
to the strength of their electro-negative properties, or, in other words, that their affin- 
ities for Oxygen, Chlorine, Sulphur, etc. increase, while those for Hydrogen decrease, 
as we descend. He will then readily see why it is that, though Nitrogen forms N03 
and N05, it forms only NC13 and NS3, and that this reason is correct he will be pleased 
to find confirmed when he learns that Phosphorus, which is more electro-positive 
than Nitrogen, and has, therefore, a stronger affinity both for Chlorine and Sulphur, 
forms not only PC13 and PS3, but also PC15 and PS5. Again, he will not be surprised, 
after seeing the affinity of the elements for Hydrogen growing constantly weaker as 
he descends in the series, to learn that a compound of Bismuth and Hydrogen is not 
certainly known. Should he inquire why, though NH3 has basic properties, PH3, 
AsH3, and SbII3 have not, he can be shown that the loss of basic properties in passing 
from NH3 to PH3 corresponds to a decrease in the strength of the affinity between the 
elements, and that if in PH3, SbH3, or AsH3, atoms of Methyle, Ethyle, or other organic 
radicals analogous to Hydrogen, are substituted for the Hydrogen atoms, and more 
stable compounds thus obtained, strong bases are the result. The other series would 
afford similar illustrations, and, from my own experience, I am confident that no teach- 
er who will once use the classification of the elements here proposed, or one similar 
to it, will ever think of attempting to teach Chemistry without its aid. 
Classifications of the elements, more or less complete, have been given by many au_ 
thors ; but the fact that no one has been generally received, is sufficient to prove that 
they are all liable to objections, and would, indeed, also seem to show that a strictly 
scientific classification is hardly possible in the present state of the science. The 
difficulty with most of the classifications is, undoubtedly, that they are too one-sided, 
based upon one set of properties to the exclusion of others, and often on seeming, 
rather than real resemblances. This is the difficulty with the old classification into 
metals and metalloids, which separated Phosphorus and Arsenic, Sulphur and 
Selenium, because Arsenic and Selenium have a metallic lustre, while Phosphorus 
and Sulphur have not, though there could hardly be found another point of difference. 
For a zoologist to separate the ostrich from the class of birds because it cannot fly 4 
THE NUMERICAL RELATION 
would not be more absurd, than it is for a chemist to separate two essentially allied 
elements, because one has a metallic lustre and the other has not. Yet it is surpris- 
ing to see how persistently this classification is retained in every elementary work on 
the science; and if it is sometimes so far modified as to transfer elements analogous 
to Selenium and Arsenic to the class of metalloids, this is only acknowledging the 
worthlessness of the principle, without being willing to abandon it. If there were 
any fundamental property common to all the elements, the law of whose variation was 
known, this might serve as the basis of a correct classification. Chemistry, however, 
does not as yet present us with such a property, and we must, therefore, here, as in 
other sciences, base our classification on general analogies. The most fundamental of 
all chemical properties is, undoubtedly, crystalline form, but a classification of the 
elements based solely on the principles of isomorphism is defective in the same way 
as it is in mineralogy. It brings together, undoubtedly, allied elements, but it also 
groups with them those which resemble each other only in their crystalline form. 
The mode of combining seems to be also a fundamental property; but, like crystalline 
form, it would bring together in some instances elements differing very widely in their 
chemical properties. A classification of the elements which shall exhibit their nat- 
ural affinities, must obviously pay regard to both of these properties. It must at the 
same time seek to group together isomorplious elements, and those which form anal- 
ogous compounds. Moreover, in such a classification, other less fundamental prop- 
erties must not be disregarded. There are many properties both physical and chem- 
ical, which, although they cannot be exactly measured, and are oftentimes difficult 
to define, (such properties as those by which a chemist recognizes a familiar substance, 
or a mineralogist a familiar mineral,) and which on account of their indefinite char- 
acter cannot be used as a basis of classification, may, nevertheless, render important 
aid in tracing out analogies. Judging from such properties as these, chemists are 
generally agreed in grouping together Carbon, Boron, and Silicon, although they can- 
not be proved to be isomorphous, and are not generally thought to form similar 
compounds. 
It is, however, much easier to point out what a classification should be, than to 
make one which shall fulfil the required conditions. Indeed, as has been already 
said, past experience would seem to show that a perfect scientific classification of the 
elements is hardly possible in the present state of Chemistry. At best, the task is 
attended with great difficulties, and it cannot be expected that these should be sur- 
mounted at once. The classification which is offered in this memoir will, undoubtedly, 
be found to contain many defects. If, however, it is but one step in advance of those BETWEEN THE ATOMIC WEIGHTS. 
5 
which have preceded it, it will be of value to the science. It was originally made, 
as has already been said, simply for the purpose of teaching, and never would have 
been published had it not led to the discovery of the numerical relation between the 
atomic weights. 
On turning to the table which accompanies this memoir, it will be seen that the 
elements have been grouped into six series. These correspond entirely to the series 
of homologues of Organic Chemistry. In the group of volatile acids homologues of 
Formic Acid, for example, we have a series of compounds yielding similar derivatives, 
and producing similar reactions, and many of whose properties, such as boiling and 
melting points, specific gravity, etc., vary as we descend in the series according to a 
determinate law. From Formic Acid, a highly limpid, volatile, and corrosive fluid, the 
acids become less and less volatile, less and less fluid, less and less corrosive; first 
oily, then fat-like, and finally hard, brittle solids, like wax. As is well known, the 
composition of these acids varies in the same way, and the variation follows a regular 
law, so that by means of a general symbol we can express the composition of the 
class. This symbol for the volatile acids may be written (C2H)03, HO -f- n (C2H2). 
This description of the well-known series of the volatile acids, applies, word for 
word, nominibus mutantis, to each of the six series of chemical elements. The ele- 
ments of any one series form similar compounds and produce similar reactions; more- 
over, they resemble each other in another respect in which the members of the organic 
series do not. Their crystalline forms are the same, or, in other words, they are iso- 
morphous. Although this may be true of the volatile acids, yet it cannot be proved 
in the present state of our knowledge. Still further, many of their properties vary 
in a regular manner as we descend in the series. In one case, at least, the law of the 
variation is known, and can be expressed algebraically, though in most instances it 
cannot be determined. Finally, as one general symbol will express the composition 
of a whole organic series, so a simple algebraic formula will express the atomic weight, 
or, if you may be pleased so to term it, the constitution of a series of elements. 
These points may be illustrated with any of the series in the table; with the first, 
for example, which consists of Oxygen, Fluorine, Cyanogen, Chlorine, Bromine, and 
Iodine. All these elements form similar compounds, as will be seen by inspecting 
the symbols of their compounds given at the right hand of the list of names, where 
the similar or homologous compounds are arranged in upright columns. Moreover, 
they are all isomorphous, as may be seen by referring to the left hand side of the list, 
where the similar compounds in each upright series are isomorphous, the numbers at 
the heads of the columns indicating the systems of crystallization, as is described in 6 
THE NUMERICAL RELATION 
the explanation accompanying the table. That the properties of these elements vary 
as we descend, can be easily shown. Oxygen is a permanent gas, as is also Fluorine. 
Cyanogen is a gas, but may be condensed to a liquid. Chlorine, a gas also, can be 
condensed more easily than Cyanogen. Bromine is a fluid at the ordinary tempera- 
ture ; and, finally, Iodine is a solid. Moreover, starting from Cyanogen, the solubility 
of these elements in water decreases as we descend in the series; and, again, the 
specific gravity of their vapors follows the inverse order of progression, gradually 
increasing from Oxygen down. The atomic weights vary in the same order, and 
admit of a general expression, which is 8 -f- n 9, or, in other words, the differences 
between the atomic weights of these elements are always a multiple of nine. This 
general formula may be said to represent the constitution of these elements, in the 
same way that the symbol (C2H) 03, HO -f- n (C2H2) represents the composition of the 
volatile acids before mentioned. In the place of (C2H) 03, HO we have 8 = 0 = the 
weight of one atom of Oxygen, and in the place of C2H2 we have nine. What it is 
that weighs nine (for it must be remembered that those numbers are weights) we can- 
not at present say, but it is not impossible that this will be hereafter discovered. In 
order to bring the general symbol of the volatile acids into exact comparison with 
that of the Nine Series, we must reduce the symbols to weights, when the two formula? 
become 
46 -f- n 14, where 46 = (C2H) 03, HO and 14 = C2H2; 
and 8 -f- n 9, where 8 = 0 and 9 = oc. 
The numbers 46 and 14 are known to represent the weights of aggregations of atoms. 
The number 8 represents the weight of one Oxygen atom, but we cannot as yet say 
what the 9 represents. After this comparison, it does not seem bold theorizing to 
suppose that the atoms of the members of this series are formed of an atom of Oxy- 
gen as a nucleus, to which have been added one or more groups of atoms, the weight 
of which equals nine, or perhaps one or more single atoms each weighing nine, 
to which the corresponding element has not yet been discovered. As it will be con- 
venient to have names to denote the two terms of the formulee which represent the 
constitution of the different series, we will call the first term, in accordance with this 
theory, the nucleus, and the number in the second term multiplied by n the common 
difference of the series. 
From what has been said, it will be seen that the idea of the classification is that 
of the organic series. It is in this that the classification differs from those which 
have preceded it. Other authors, in grouping together the elements according to the 
principles of isomorphism, have obtained groups very similar to those here presented. BETWEEN THE ATOMIC WEIGHTS. 
7 
Indeed, this could not be otherwise, since, as has been already said, the members of 
each series are isomorphous, while, as a general rule, to which, however, there are 
many exceptions, no isomorphism can be established between members of different 
series. These groups, however, have been merely groups of isomorphous elements, 
and not series of homologues like those in which the elements are here classed. 
These general remarks will suffice to indicate the principles upon which the classi- 
fication has been made, and the character of the numerical relation between the 
atomic weights which has been established. The details of the classification can be 
best studied by referring to the table, so that it will be only necessary to speak of 
those points which are of special interest, or which may require explanation, or in 
regard to which there may be doubt. The series I have named from their common 
differences. The first I have called the Nine Series, the second the Eight Series, &c. 
Let us examine the doubtful points in each, commencing with the first. 
The last five members of the Eight Series are connected by so many analogies, 
that they have been invariably grouped together in the elementary books. There can 
be no doubt, therefore, in regard to the propriety of placing them in the same series, 
on the ground of general analogies. Fluorine, it is true, presents some striking points 
of difference from the rest. Fluoride of Calcium is almost insoluble in water, while 
the Chloride, Bromide, and Iodide of Calcium are all very soluble. We must, how- 
ever, remember that we have to do with series, and must not therefore expect to find 
close resemblances except between adjacent members. If, then, we consider that Oxy- 
gen is one of the series, and that Fluorine stands but one step removed from Oxygen, 
while it is two steps removed from Chlorine, the discrepancy in a measure vanishes, 
for Lime CaO is but slightly soluble in water. Nevertheless, the difficulty does not 
entirely disappear, for CaFl is much less soluble than CaO, although it should be 
more soluble judging from the law of the series and the fact that CaCl is so much 
more soluble than CaFl. 
The solubility of a series of homologous elements or compounds in water, may be 
regarded as a function of one or more variables. In the case of elements there may 
be but one variable, but it is easy to see that in the case of compounds there must 
be several. One of these variables is probably the same which determines the com- 
mon difference of the series to which the elements or compounds belong; (it will be 
hereafter shown that the atomic weights of the homologous compounds are related in 
the same way as those of the elements;) the other variables are perhaps the atomic 
forces which determine the hardness, density, &c. of the solid. We may, therefore, 
with justice, compare the relative solubilities of a series of homologues to a curve 8 
THE NUMERICAL RELATION 
which should be the same function of the same variables, and what mathematics 
teaches we ought reasonably to expect in the case of this curve, we ought to expect 
also in the variations of solubility of these substances. Now every mathematician 
is familiar with the remarkably rapid changes which a curve undergoes that is a 
function of several variables, and we cannot be surprised that similarly rapid changes 
should be observed in the solubility of homologous substances in passing from one to 
the next in the series. In the curve which corresponds to the relative solubility of CaO, 
CaFl, CaCy, CaCl, CaBr, and Cal, it would seem that at CaFl there is a singular point 
where the curve, after rising for some distance above the axis, bends down again 
towards it. Several of the other series of compounds of these elements present sim- 
ilar anomalies; for example, KO, KF1, KCy, KC1, KBr, and KI. Here the solubility 
diminishes until we come to KC1, which is less soluble than KCy; then it increases 
to the last. Here, of course, the singular point is at KC1. With the corresponding 
compounds of Sodium, the solubility diminishes to NaFl, which is the least soluble of 
the series, and then increases constantly to the end. 
These facts at least seem to show that apparent variations from the law of series 
in properties, which evidently are unknown functions of several variables, should not 
be allowed to outweigh strong analogies, and certainly the analogies between Fluorine 
and the other haloids are very marked. Fluorine itself possesses properties such as 
we should expect to find in a member of the series above Chlorine. The strong and 
active affinities of Fluorine might be indeed predicted, after seeing the rapid increase 
both in the strength and activity of the affinities in passing from Iodine to Chlorine. 
In passing from Bromine to Chlorine, we pass from a liquid to a gas, permanent under 
any natural conditions; and we should expect, therefore, in rising still higher in the 
series, to find in Fluorine a gas less easily reduced to a liquid than Chlorine. Now 
although, on account of its remarkably active affinities, this fact cannot be demon- 
strated on the gas itself; it can, nevertheless, be inferred with perfect certainty from 
its compounds. Finally, the isomorphism of Fluorine and the other haloids may be 
urged as indicating close analogy. From these considerations, I cannot but think that 
those chemists who have questioned the propriety of classing Fluorine with the other 
haloids will, on reviewing the facts, and regarding the haloids in the light of a series, 
and not simply as a group of elements possessing certain general properties, be led 
to change their opinion. 
Cyanogen, though a compound radical, has been classed with the other haloids, not 
only from its atomic weight, but also from its other analogies. Its properties are in 
most cases those which we should expect from an element occupying its position in BETWEEN THE ATOMIC WEIGHTS. 
9 
the series; but in others it presents remarkable variations, owing probably to the fact 
that it contains a radical which is easily decomposed. As well known, it is perfectly 
isomorphous with Chlorine. 
The propriety of classing Oxygen in this series seems to be placed beyond doubt by 
the discovery of Ozone, which, though it does not seem to possess such energy as we 
should expect in an element higher in the series than Fluorine, may, nevertheless, be 
found to fulfil all anticipations should it ever be obtained in a perfectly unmixed con- 
dition. The isomorphism of Oxygen with Chlorine, and therefore with the other ha- 
loids, seems sufficiently established by the determination both of Proust and Misterlich 
of the tetrahedral form of Cu2Cl. It must, however, be admitted that Oxygen presents 
as strong analogies with Sulphur as it does with Chlorine; and since, not only from its 
analogies, but also from its atomic weight, it appears to be the nucleus in all the first 
three series, I have placed it at the head of each. It may be mentioned here, that in 
all cases the fact that the atomic weight of an element is included in the general formula 
of a series, is an argument for classifying it in that series, if the relation between the 
atomic weights pointed out in this memoir is admitted to be a law of nature; but as I 
wish to show that the relation is not that of a mere accidental group of numbers, but 
is connected with the most fundamental properties of the elements, and has, therefore, 
the claims of a law, I have endeavored to establish the correctness of the classification 
which conforms to the law, and, indeed, suggested the law on other grounds. 
The atomic weights of the numbers of the Nine Series, as determined by experiment, 
present greater deviations from the numerical law already explained, than are to be 
found in any of the others. The weights which would exactly conform to the general 
formula 8 -j- n 9 are given in the column of the table headed Theoretical, while in the 
next column at the right are given the weights of experiment. These for the most 
part (in this as well as in the other series) have been taken from the table of Atomic 
Weights given in the last volume of Liebig and Kopp’s Jahresbericlit (for 1852), 
which was supposed to give the most accurate and latest results. In the few cases in 
which the numbers have not been taken from this table, the initial letter of the name 
of the observer has been annexed. It will be seen, on comparing the two columns 
that the greatest deviation from the law is in the case of Fluorine, if we consider the 
care which was taken both by Berzelius and Louyet in the determination of the atomic 
weight of this element. It may, however, he remarked, that, as the processes used by 
both experimenters were essentially identical, any hidden constant source of error 
would produce the same effect on both results; so that the atomic weight of Fluorine 
cannot be regarded as yet as absolutely fixed. Nevertheless, it is not possible that so 10 
THE NUMERICAL RELATION 
great a difference between the true and observed weights as two units could have es- 
caped detection in the numberless analyses which have been made, by the most expe- 
rienced chemists, of the Fluorine compounds. It must, therefore, be admitted, and not 
only in the case of Fluorine, but also in other instances, that there are deviations 
from the law; but these deviations are not greater than those from similar numerical 
laws in astronomy and other sciences, and indeed, judging from the analogy of these 
sciences, ought to be expected. 
Those who are not familiar with the amounts of probable error in the determination 
of the different atomic weights wrould judge, on comparing together the columns of the- 
oretical and observed values, that the deviations from the law were much greater than 
they are in reality. It should, therefore, be stated, that, in by far the larger number of 
instances, the deviations are within the limit of possible errors in the determinations, 
leaving only a few exceptional cases to be accounted for. It must be remembered that, 
other things being equal, the amount of probable error is the greater the greater the 
atomic weight, so that a difference of 1.9 in the case of Iodine is not a greater actual 
deviation from the law than only 0.5 in the case of Chlorine. Indeed, it is very possi- 
ble that on more accurate determinations the atomic weight of Iodine will be found to 
correspond to the law, which cannot be expected of that of Chlorine. It is well known 
that many of the larger atomic weights, especially those of the rarer elements, cannot 
be regarded as fixed within several units. 
I have calculated, as well as the data I have would permit, the amount of probable 
error in the determinations of many of the atomic weights, and by comparing together 
the results from different processes, and by different experimenters, I have endeavored 
to detect the existence of constant errors, which seem to be the great errors in all these 
determinations, those accidental errors which are made in the repetitions of the same 
process by equally careful experimenters being comparatively insignificant. The re- 
sults of this investigation will be published in a subsequent memoir. It is sufficient 
for the present purpose to state, that, while they show that, in the greater number of 
cases, the apparent variations from the law are within the limit of probable error, 
there are yet several instances, where, after allowing for all possible errors of observa- 
tion, there is a residual difference. I do not therefore look alone to more accurate ob- 
servations for a confirmation of the law, but, regarding the variations as ascertained 
facts, hope that future discovery will reveal the cause. Whether the variations will be 
found to be a secondary result of the very cause which has determined the distribution 
of the atomic weights according to a numerical law, as the perturbations in astronomy 
are a necessary consequence of the very law they seemed at first to invalidate, or BETWEEN THE ATOMIC WEIGHTS. 
11 
whether they are due to independent causes, can of course, for the present, be only a 
matter of speculation. There are, however, facts which seem to indicate that the vari- 
ations are not matters of chance, but correspond to variations in the properties of the 
elements. 
From the beautiful discovery of Professor Schonbein we have learnt that Oxygen has 
two allotropic modifications, and that besides its ordinary condition it is capable of as- 
suming another highly active state when its properties resemble those of Chlorine. 
Cyanogen is only known in a quiescent state. The other haloids, Fluorine, Chlorine, 
Bromine, and Iodine are only known in a highly active state. Now it will be seen on 
examining the table, that the atomic weights of the highly active elements, as determined 
by experiment, exceed slightly the theoretical numbers, and that where the affinities are 
the most intense, in Fluorine, the deviation is the greatest. A similar fact may be 
observed in the atomic weights of the members of the Six Series. Arsenic has been 
proved to be capable of existing in two allotropic modifications. In its ordinary state, 
it has a crystalline form belonging to the Rhombic System. In the other condition, in 
which it may be obtained by sublimation at a low temperature, it crystallizes in regular 
octahedrons. The other members of this series are probably isodimorphs with Arsenic. 
The ordinary condition of Phosphorus is its monometric modification, while the rhombic 
state seems to be the normal condition of Arsenic, Antimony, and Bismuth. Now the 
atomic weights of the last three are either equal to, or slightly exceed, the theoretical 
number, while that of the first fall short, perhaps even by a unit. Other facts, which 
also tend to show that the deviations are not matters of chance, may be found in the 
affiliations of the series. There are some elements which seem to be most remarkably 
double-faced, having certain properties which connect them closely with one series, and 
at the same time others which unite them nearly as closely to another. In such cases 
we find that the atomic weight either falls naturally into both series, or, not corre- 
sponding exactly with the theoretical number of the series to which the element prop- 
erly belongs, it inclines towards that of the other, and sometimes equals it. Such is 
the case with Chromium, Manganese, and Gold, as will be seen by referring to the affil- 
iations at the bottom of the Nine Series. These various facts force upon me the con- 
viction, that this relation between the atomic weights is not a matter of chance, but 
that it was a part of the grand plan of the Framer of the universe, and that in the very 
deviations from the law, there will, hereafter, be found fresh evidence of the wisdom 
and forethought of its Divine Author. 
The general formulae for the Eight Series are 8 n 8 and 4 -f n 8. The two nuclei 
correspond to two different sets of elements, or sub-series, one consisting of Oxygen, 12 
THE NUMERICAL RELATION 
Sulphur, Selenium, and Tellurium, the other of Molybdenum, Vanadium, Tungsten, 
and Tantalum. The atomic weights of the first are all equal to 8 + w 8 ; those of the 
second to 4 -f-n 8. The sub-series exhibit marked analogies, as well as certain differ- 
ences. They resemble each other chiefly in that the members of both form analogous 
acids with Oxygen, while they differ in that, though the members of the first sub-series 
form compounds with Hydrogen, those of the second do not. The isomorphism of the 
members of each sub-series among themselves, with the exception of Vanadium, is com- 
plete; but there seems to be no proof of any isomorphism between the sub-series. 
Johnston attempted to establish the isomorphism of Chromic and Molybdic Acids from 
the red variety of Molybdate of Lead from Rezbanya, which he supposed to be a Chro- 
mate ; but the fact has been disproved by G. Rose, who has shown that the supposed 
Chromate is a Molybdate mixed with a small amount only of Chromate. There seems, 
nevertheless, to be some reason for believing that Chromic Acid may replace Molybdic 
Acid to a certain extent. If this is proved, it establishes another link of connection 
between the members of the two sub-series, since Chromic Acid is isomorphous with 
Sulphuric Acid. For the present, however, we must regard them as sub-series, related, 
but distinct, the second being in a measure supplementary to the first. They are dis- 
tinguished in the table by printing the names of the second sub-series a little to the 
right of those of the first, and the fact that their atomic weights are intermediate to 
those of the first, I have indicated to the eye by giving to the names also an interme- 
diate position. 
The analogies between Oxygen and Sulphur are so numerous, that, were we to place 
Oxygen in but one series, we should place it in this. HO and HS, H02 and HS2, resemble 
each other very closely, as do also the Oxygen salts the corresponding Sulphur salts. 
Moreover, there can be no doubt in regard to the isomorphism of the two elements, 
since it has been established upon the authority both of Mitscherlich and Becquerel, 
and from two different compounds. The only doubtful case in the series was that of 
Vanadium, which in some of its properties resembles Arsenic more closely than it does 
Molybdenum. The reasons for giving it the place which it occupies were the facts that 
its acids correspond to those of Molybdenum, and that it forms remarkably highly 
colored oxides which are repeated also in Molybdenum. It is true that the properties 
of the element itself are not those we should expect from the position which it occupies 
in our table; yet, if it were placed in the Six Series, it would fall between Phosphorus 
and Arsenic, which on the whole it resembles less than it does Molybdenum, for al- 
though it is combustible, yet neither it nor its oxides are volatile. I consider it, there- 
fore, as a member of the Eight Series, but affiliating very closely with the Six. Its BETWEEN THE ATOMIC WEIGHTS. 
13 
atomic weight favors this hypothesis. Vanadate of Lead has been considered isomor- 
phous with the Phosphate; but as this isomorphism does not rest on any measurement 
of angles, and as, moreover, the received symbols of the two minerals, Vandinite and 
Pyromorphite, on whose crystalline forms the isomorphism was determined, show a 
very different constitution, I have not given much weight to this fact* The observed 
atomic weights of the members of this series are almost precisely the same as the the- 
oretical members, and, with the exception, perhaps, of Molybdenum, there appears 
to be no instance in which the difference is greater than the amount of possible error. 
The members of the Six Group form a well-characterized family, so that, with the ex- 
ception of Oxygen, there can be no doubt in regard to the justice of classifying them 
together, and any discrepancies will disappear on considering the group in the light of 
a series. They form acids containing three and five atoms of Oxygen which are com- 
pletely homologous, and make two series parallel to that of the elements. They form 
also a remarkable series of compounds with three atoms of Hydrogen. The idea which 
has been advanced by some authors, that NH3 is the Nitride of Hydrogen, while PH3 is 
the Hyduret of Phosphorus, or, in other words, that Hydrogen is electro-positive with 
reference to Nitrogen and electro-negative with reference to Phosphorus and those 
lower in the series, does not seem to me correct, since the remarkable bases which may 
be formed from PH3, AsH3, SbH3, and BiH3, by replacing the Hydrogen atoms by or- 
ganic radicals, seem to indicate that they have the same type as NH3, and are therefore 
homologues of it. 
The isomorphism of the four lower members of the series is perfect. It has been 
shown in the table, both by the crystalline forms of the elements themselves, as well as 
by those of their coumpounds. In the other series, wherever it was possible, the same 
double proof has been given. The doubt expressed by G. Pose in regard to the dimor- 
phism of Arsenic, as I hope to be able to show in a paper soon to be published, has 
been removed. In one state Arsenic crystallizes in perfect octahedrons of the regular 
system, and is therefore isomorphous, not only with Antimony and Bismuth, but also, 
in its allotropic state, with Phosphorus. Isomorphism, as is well known, is not abso- 
lute, except in forms of the regular system. The rhombic angles of the crystals of 
Arsenic, Antimony, and Bismuth are respectively 85° 41', 87° 35', 87° 40', and there- 
fore conform to the general rule. It will be observed that the angle varies constantly in 
the same way as we descend in the series. Now, although these few instances do not 
afford sufficient ground for any general conclusion, yet they show that similar varia- 
* See G. Rose’s Mineral System. 14 
THE NUMERICAL RELATION 
tions are possible in the other systems, and therefore that we cannot be expected to 
establish isomorphism in any case except between nearly consecutive members. 
The atomic weights of the members of this series, with the exception of Phosphorus, 
do not present any important deviations from the theoretical numbers, taking into ac- 
count always, of course, the amount of possible error. The deviation in the case of 
Phosphorus has already been noticed. Oxygen, it must be admitted, is not connected 
with the series from any similarity of properties, though the Phosphides, Arsenides, and 
Antimonides present certain analogies with the Oxides. As has already been said, Ox- 
ygen was placed at the head of this, as well as of the last two series, because its atomic 
weight seemed to be the nucleus of all three. 
The Five Series is the shortest of all, consisting of only three members, Carbon, Boron, 
and Silicon. Of these, the last two are as closely allied as are any two members of the 
other series, Silicon having precisely the properties we should expect in a homologue 
of Boron, which was lower in the series; and the same is also true of their compounds. 
The analogies, however, between these two elements and Carbon are by no means so 
close, for not only Carbon cannot be proved to be isomorphous with them, but it 
does not form similar compounds. Carbonic Acid, it is true, presents some points of re- 
semblance to Boracic and Silicic Acids; like them it unites in a large variety of propor- 
tions with bases, its alkaline salts give a basic reaction, &c.; but according to the gen- 
erally received opinion, its symbol is C02, while those of Boron and Silicon are B03 and 
SiOa. In its uncombined state, however, Carbon resembles Boron and Silicon, not only 
in its outward properties, but also in its action before the blowpipe. Two of the allo- 
tropic states of Carbon, Graphite and Charcoal, are probably repeated in Boron, and 
are known to be in Silicon. The principle of exclusion would also seem to place Car- 
bon in this series, for it certainly presents no analogies with the members of any other. 
The correspondence of the atomic weights of the members of this series to the law is 
remarkably close. 
The Four Series is by far the largest of all, including the greater number of what are 
generally known as the heavy metals. The members of the series resemble each other 
in the following respects. First, they are isomorphous ; for although each member can- 
not be directly proved to be isomorphous with every other, yet isomorphism can be 
established between consecutive members, which, as has before been said, is all that can 
be expected. Second, the members of this series all form, by uniting with Oxygen, 
either Protoxides or Sesquioxides, or both, which, as a general rule, are strong bases. 
Third, these Oxides are either insoluble, or nearly insoluble, in water. And finally, the 
elements of the series have all those physical properties which are known as metallic 
properties. BETWEEN THE ATOMIC WEIGHTS. 
15 
This series may be naturally divided into two sub-series. The first contains those 
elements whose protoxide bases are their characteristic compounds, and which do not 
form acids with Oxygen. The second contains those elements whose characteristic 
compounds are their sesquibases. They generally unite with two or more equivalents 
of Oxygen, and form acids. These sub-series are distinguished in the table in the same 
way as those of the Six Series. Corresponding to these sub-series we have two sets of 
atomic weights, each having the same common difference, but differing in their starting- 
point or nucleus. The first set is expressed by the formula 4 -f- n 4, the second by 
2 ~h?i 4. 
The sub-series affiliate with each other in a most remarkable manner. Manganese, 
for example, not only forms a strong protoxide base, but also unites with a larger 
amount of Oxygen, forming both a sesquibase and acids. Its atomic weight places it 
in the first group, and it has therefore been classed there, although by its properties it 
is equally allied to the second. Cobalt and Nickel certainly resemble much more 
closely the members of the first than of the second sub-series, although their atomic 
weights place them in the second. With this exception, the subdivision of the series 
which the atomic weights require does not differ from that suggested by the properties 
of the elements. The members of this series may of course be still further subdivided 
into groups according to their special properties, as they are in all works on Chemistry. 
They are placed together here because the atomic weights form but one numerical 
series. 
The isomorphism of the members of this series will be found well established with 
the limitations before given. In order to establish the isomorphism of Cobalt and 
Nickel with Iron, the isomorphism of one atom of Arsenic with two atoms of Sulphur 
has been assumed. This is generally admitted; but if it is not, no one can doubt in 
regard to the isomorphism of these three metals, as they constantly replace each other. 
Glucinum, Zirconium, Lanthanum, Cerium, and Thorium cannot be shown to be iso- 
morphous with the other metals by any of their compounds, but their oxides are known 
to replace the analogous oxides of the other metals. So also is Ruthenium known to 
replace Rhodium. There have been doubts expressed in regard to the existence of a 
monometric form of Zinc; but as we have established its isomorphism with the other 
members of the series, not only by its own crystalline form, but also by those of its 
compounds, the fact is of no importance to the present question. 
The atomic weights of the members of this series, as determined by observation, very 
nearly correspond with the theoretical numbers, which is the more remarkable, as the 
limit of error in the determination of the atomic weights of the greater number, espe- 
cially of the rarer metals, is quite wide. 16 
THE NUMERICAL RELATION 
The Three (and last) Series is composed of Hydrogen and the metals of the alkalies. 
The analogies between Lithium, Sodium, and Potassium are very close, as is well 
known, and there can be no doubt in regard to the propriety of classing them together. 
It may be said, however, in regard to Hydrogen, that it resembles as closely some of the 
metals of the Pour Series as it does those of the alkalies. Though this cannot be denied, 
yet the fact that the atomic weight of Hydrogen is the nucleus of the series, and the 
great solubility of the alkalies in water, may be urged as reasons for placing it at the 
head of the Three Series. 
The isomorphism of Lithium, Sodium, and Potassium is fully established; but I can 
find no data which prove Hydrogen isomorphous either with them or with the metals 
of the other group. 
The unit of the atomic weights which has been used thus far throughout the table, 
is the double atom of Hydrogen ; but the nucleus of the Three Series is the weight of 
the single atom, so that the unit in this series is one half of the unit of the weights in 
all the other series. This fact must be kept in mind in comparing the atomic weights 
of this with those of the other series. All the weights might have been made uniform 
by doubling them throughout; but as this would not have changed the relation, and 
would have been departing from the general custom, it was thought best to confine the 
doubling to the Three Series, into which alone Hydrogen enters. The general symbol 
of this series is 1 -j- n 3, where of course the unit is one half of that of the symbols at 
the head of the other series. The observed atomic weights will be found to correspond 
very closely with the theoretical numbers; indeed, the two coincide, except in the case 
of Potassium, where the difference is 0.6. This, however, it must be remembered, is 
0.6 of the single Hydrogen atom. Compared with the double atom, as the weight of 
Potassium is generally given, the difference amounts to but 0.3. 
One of the most remarkable points of the classification which has been now ex- 
plained, is the affiliation of the series. We find in Chemistry, as in other sciences, that 
Nature seems to abhor abrupt transitions, and shades off her bounding lines. Many 
of the elements, while they manifestly belong to one series, have properties which ally 
them to another. Several examples of this have already been noticed. In such cases, 
we find invariably that there is a similar affiliation of the atomic weight. Of all the 
elements Chromium and Manganese are the most protean. Two atoms of these ele- 
ments unite with seven atoms of Oxygen and form acids analogous to Perchloric Acid, 
and, as has already been shown, the weight of two atoms of either element falls into 
the Nine Series. Moreover, one atom of Chromium or of Manganese unites with 
three atoms of Oxygen to form Chromic or Manganic Acid. Chromic Acid is a BETWEEN THE ATOMIC WEIGHTS. 
17 
strong oxidizing agent, and resembles closely Nitrous Acid, and the atomic weight of 
Chromium falls into the Six Series just below that of Nitrogen. Manganic Acid, on 
the other hand, resembles Sulphuric Acid, with which it is isomorphous, and the atomic 
weight of Manganese would place it in the Eight Series. In like manner Osmium in 
many of its properties resembles Platinum and the other metals with which it is asso- 
ciated in nature; but, unlike them, it forms a very remarkable volatile acid, whose 
insupportable and suffocating odor as well as composition reminds one of the acids of 
the Nine Series, and its atomic weight seems to justify the apparent analogy. Gold 
likewise, though the noblest of metals, yet in some of its chemical relations resembles 
much more closely the members of the Nine than of the Four Series, and here again 
its accommodating atomic weight seems to account for its double-sided character. 
Several other examples of similar affiliations are given in the Table, but do not need 
explanation. 
In the description just concluded of the classification of the chemical elements, 
which is offered in this memoir, I have not entered into details, for to have done so 
would have been to write a treatise on Chemistry. I have confined myself almost 
exclusively to general points, and only referred to those particulars which I thought 
might present doubts. I hope that I have been able to show, first, that the chemical 
elements may be classified in a few series similar to the series of homologues of 
Organic Chemistry; second, that in those series the properties of the elements follow 
a law of progression; and finally, that the atomic weights vary according to a similar 
law, which may be expressed by a simple algebraic formula. As already intimated, I 
have endeavored to prove the correctness of the classification on general grounds, in 
order that it might appear that the simple numerical relation which has been discovered 
between the atomic weights is not a matter of chance, but is connected with the most 
fundamental properties of the elements. I might leave the subject at this point, but 
the existence of the law which I wish to establish will be proved more conclusively if 
it can be shown, not simply that the general properties of the members of each series 
vary in a regular manner, but also if in one or more cases the exact law of the varia- 
tion can be pointed out. 
There are but few properties of the elements which are subjects of measurement, 
and which therefore can be compared numerically. Such are the specific gravity in 
the three states of aggregation, the boiling and melting points, the capacity of heat, 
and a few others. It is easy to see that there are but few of these properties the law 
of whose variation in the series we could reasonably expect to discover in the present 
state of science. Most of them evidently depend upon molecular forces with which 18 
THE NUMERICAL RELATION 
we are entirely unacquainted. Such in solids is undoubtedly the case with so simple and 
fundamental a property as specific gravity, and most, if not all, of the other properties 
of solids belong to the same category. It cannot therefore he expected that we should 
point out the laws by which these properties vary, although the remarkable investiga- 
tions of Dana, Filhol, Kopp, Schroder, and others, on the relations between the density 
of substances and their atomic weights, and those of Kengott on the relation of hard- 
ness to atomic volume, give grounds for expecting that even they will before long be 
discovered. In liquids and gases, however, most of these molecular forces which pro- 
duce the apparent irregularities in solids have less influence, as we should naturally 
expect, probably because the atoms are removed out of the sphere of their action. 
We may therefore hope, on comparing together the properties of the liquid or gaseous 
states of the elements in any series, to discover some numerical relation between them. 
Unfortunately, however, we have not sufficient data for making such a comparison 
except in the case of one property, the specific gravity. The boiling point, which 
would be a very valuable property for the purpose, is known only in a few instances. 
That the specific gravity of the elements in their gaseous state varies in each series 
according to a numerical law, follows necessarily from what is already known. It is a 
well-known fact, that the specific gravities of the gaseous states of the elements divided 
by their atomic weights give quotients which are either equal, or which stand in a 
very simple relation to each other. For any series, as far as w'e have data, this quo- 
tient is the same for all the elements with only a few exceptions. That is = p. 
But we have found that At. W. may be expressed in general by a + n b, and substituting 
this for At. W. in the above equation, it becomes or Sp. Gr. —p a-{-npb; so 
that p a -j- np b is a general expression for the specific gravity of all the elements of 
any series, in the same way that a -f- n b is for the atomic weight. The value of p will 
differ according as the specific gravities used are referred to Hydrogen or Air. Below 
will be found tables which give the calculated and observed specific gravities of the 
elements of the Nine and Six Series referred to Hydrogen, which has been taken as 
the unit instead of Air, as we thus in great measure avoid fractions. In the Nine 
Series p= 1, so that the numbers representing the specific gravities are the same as those 
representing the atomic weights. In the Six Series it equals two, so that the numbers 
representing the specific gravities are in this series twice as large as those representing 
the atomic weights. When the specific gravity has not been observed, the calculated 
number only is given. The observed numbers are taken from the “ Table of Specific 
Gravity of Gases and Vapors,” in Graham’s Elements of Chemistry, which is a very 
complete collection of all known data. For the other series, we have only occasional 
data, so that no complete tables of their specific gravities are possible. BETWEEN THE ATOMIC WEIGHTS. 
19 
THE NINE SERIES. 
SP. Gr. 
At. W. ~ L 
Sp. Gr. = 8 —j— 9. 
THE SIX SERIES. 
. 
Sp. Gr. 
AtW. ~ 
Sp. Gr. = 16 -f n 12. 
SPECIFIC GRAVITIES. 
SPECIFIC GRAVITIES. 
Names. 
Names. 
Theoret. 
Observed. 
Theoret. 
Observed. 
Oxygen 
8 
16 
Oxygen 
16 
16 
Fluorine 
17 
Nitrogen 
28 
14 
Cyanogen 
26 
26 
Phosphorus 
64 
64 
Chlorine 
35 
35.5 
Arsenic 
148 
150 
Bromine 
80 
78 
Antimony 
256 
Iodine 
125 
126 
Bismuth 
412 
It is evident, then, that at least one property of the elements varies in the series 
according to an ascertained numerical law. But, it may be said, this proves nothing, 
for these specific gravities are connected so closely with the atomic weights that what 
is true of the one must he to the same extent true of the other. It must be remem- 
bered, however, that the specific gravities are a distinct set of observed facts, and that 
the probability of a law is in exact proportion to the number of facts which accord 
with it. Moreover, the closeness of the connection is unimportant. Whether the 
value of p be expressed by a single digit, or by a complicated algebraic formula, is 
evidently a matter of indifference so far as the confirmation of the law is concerned. 
I regret exceedingly that there are not sufficient data in the case of any of the other 
properties of the elements in the state of gas to allow comparison, as I feel confident 
that the law which governs their variation in the series might easily be discovered; 
but I look forward to the time when in the general formula pa -\-npb the value of p 
shall be known, not only for the properties of the elements in their gaseous state, but 
for every property capable of numerical expression. 
In this memoir I have confined myself entirely to the elements, but it is evident 20 
THE NUMERICAL RELATION 
that the classification here offered, and the numerical law here explained, may be ex- 
tended to all compounds. The elements of any one series, by combining, give rise to 
perfectly parallel series of homologous binaries, some of which are given in the table. 
The binaries of those series which have the greatest common difference are generally 
acids; and of those which have the smallest, they are generally bases. These acids and 
bases unite together and form series of homologous salts. As in Organic Chemistry, 
many of the series are very incomplete; but they are much more generally perfect 
than in that newer department of the science, and almost every day fills up some gap. 
It will be seen, then, that not merely a plan has been given for classifying the ele- 
ments, but one which will also embrace all inorganic compounds, and affiliate with the 
similar classification which has already been established in Organic Chemistry. We 
have not attempted to develop such a classification, since to do it would require a vol- 
ume ; nor is it necessary, as any one can develop it for himself. 
That the atomic weights of the series of homologous compounds follow the same 
numerical law as those of the elements is easily shown. Take as an example the series 
of salts homologous with ICO, N05, which may be expressed in general by KO, R05, 
where It is any member of the Six Series after Oxygen, and whose atomic weight, 
therefore, equals 8 + The atomic weight of KO, ROg must be necessarily 
39.5 -f- 48 -j- (8 -f- n 6), or 95.5 +w6. As this symbol differs from that of the Six 
Series only in the nucleus, the atomic weights of the salts which are represented by it 
must progress by the same differences as those of the corresponding elements. 
The properties of these series of homologous compounds will also be found to vary 
in a regular manner, and the law of the progression of the specific gravities in the 
gaseous state can be easily expressed algebraically, since in each series the quotient of 
the specific gravity divided by the atomic weight is a constant quantity. As an illus- 
tration, we may take the series of binaries homologues of water given in the Nine 
Series of our table. It follows from what has been said, that the atomic weights of 
these compounds equals 9 fi- n 9. With each |^r9 = therefore Sp. Gr. = 4.5 + n 4.5. 
We give below a table of the observed or calculated specific gravities, not only of 
these compounds, but also of those homologues of NH3 whose specific gravity has 
been observed. BETWEEN THE ATOMIC WEIGHTS. 
21 
HOMOLOGUES OF WATER. 
Sp. Gr. , 
At. W. ~ 2< 
Sp. Gr. = 4.5 -f- n 4.5. 
HOMOLOGUES OF AMMONIA GAS. 
Sp. Gr. , 
At. W. — 2' 
Sp. Gr. = 5.5 -f- n 3. 
SPECIFIC GRAVITIES. 
SPECIFIC 
GRAVITIES. 
Symbols. 
Symbols. 
Theoret. 
Observed. 
Theoret. 
Observed. 
HO 
4.5 
9 
NH3 
8.5 
8.5 
HF1 
9 
ph3 
17.25 
17.5 
HC1 
13.5 
13.5 
AsH3 
39 
38.5 
HBr 
40.5 
39.5 
HI 
63 
63.5 
As the series of compounds give a greater scope for investigating the relations of 
properties than is presented by those of the elements, we may expect that these rela- 
tions will be first discovered in the former, and to my conceptions Chemistry will then 
have become a perfect science, when all substances have been classed in series of hom- 
ologues, and when we can make a table which shall contain, not only every known 
substance, but also every possible one, and when by means of a few general formulse 
we shall be able to express all the properties of matter, so that when the series of a 
substance and its place in its series are given, we shall be able to calculate, nay, pre- 
dict, its properties with absolute certainty; and when our chemical treatises shall have 
been reduced to tables of homologues, and our laws comprised in a few algebraic 
formulee, then the dreams of the ancient alchemist will be realized, for the problem of 
the transmutation of the elements will have been theoretically, if not practically, 
solved. 22 
NUMERICAL RELATIONS BETWEEN THE ATOMIC WEIGHTS. 
EXPLANATION OF THE TABLE. 
The formula at the head of each series is a general expression for the atomic weights 
of that series. The names of the series are derived from the “ Common Differences,” 
which are the numbers multiplied by n in the general formulae. In the columns 
headed “ Theoretical ” are given the atomic weights calculated from these formulae and 
the values of n given in the last columns at the right of each division of the table. 
In the columns headed “ Observed ” will be found the observed values of the same 
atomic weights. These have been taken from the table of atomic weights given in the 
last volume of Liebig and Kopp’s Jahresbericht (for 1852), with the exception of those 
against which are placed the initials of the observers. The last were taken from 
Weber’s Atomgewichts Tabellen. In some cases the atomic weight is taken at twice its 
received values, but it is then underlined. The compounds in any one column at the 
right of the names of the elements are homologous. In the same way, those in any 
one at the left are isomorphous. The numbers at the head of these last columns 
indicate crystalline systems as follows: 1. Monometric; 2. Dimetric; 3. Trimetric; 
4. Monoclinic; 5. Triclinic; 6. Rhombic. The data from which the table was com- 
piled were drawn from numerous sources, but especially from the following works: 
Gmelin’s Handbook of Chemistry, Graham’s Elements of Chemistry, Phillips’s Miner- 
alogy by Brooke and Miller, and Gustav Rose’s Krystallo-chemische Mineralsystem. 
References have been given only in a few cases, to avoid crowding the tables. For 
authorities in other cases, the author would refer to the above-mentioned works.