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to that particular science. This classification is the most natural 
from a physical point of view, and it is generally the first in order 
of time. But when the student has become acquainted with 
several different sciences, he finds that the mathematical pro- 
cesses and trains of reasoning in one science resemble those 
in another so much that his knowledge of the one science may 
be made a most useful help in the study of the other. When he 
examines into the reason of this, he finds that in the two sciences 
he has been dealing with systems of quantities, in which the 
mathematical forms of the relations of the quantities are the same 
in both systems, though the physical nature of the quantities may 
be utterly different. He is thus led to recognise a classification 
of quantities on a new principle, according to which the physical 
nature of the quantity is subordinated to its mathematical form. 
This is the point of view which is characteristic of the mathe- 
matician ; but it stands second to the physical aspect in order of 
time, because the human mind, in order to conceive of different 
kinds of quantities, must have them presented to it by nature. 
T do not here refer to the fact that all quantities, as such, are 
subject to the rules of arithmetic and algebra, and are therefore 
capable of being submitted to those dry calculations which re- 
present, to so many minds, their only idea of mathematics. The 
human mind is seldom satisfied, and is certainly never exercising 
its highest functions, when it is doing the work of a calculating 
machine. What the man of science, whether he be a mathe- 
matician or a physical inquirer, aims at is, to acquire and de- 
velop clear ideas of the things he deals with. For this purpose 
he is willing to enter on long calculations, and to be for a season 
a calculating machine, if he can only at last make his ideas 
clearer. But if he finds that clear ideas are not to be obtained 
by means of processes, the steps of which he is sure to forget 
before he has reached the conclusion, it is much better that he 
should turn to another method, and try to understand the subject 
by means of well-chosen illustrations derived from subjects with 
which he is more familiar. We all know how much mere 
popular the illusirative method of exposition is found, than that 
in which bare proces:es of reasoning and calculation form the 
principal subject of discourse. Now a truly scientific illustra- 
tion is a method to enable the mind to grasp some conception or 
law in one branch of science, by placing before it a conception or 
a law in a different branch of science, and directing the mind to 
lay hold of that mathematical form which is common to the cor- 
responding ideas in the two sciences, leaving out of account for 
the present the difference between the physical nature of the real 
phenomena. ‘The correctness of such an illustration depends on 
whether the two systems of ideas which are compared together 
are really analogous in form, or whether, in other words, the 
corresponding physical quantities really belong to the same 
mathematical class. When this condition is fulfilled, the illus- 
tration is not only convenient for teaching science in a pleasant 
and easy manner, but the recognition of the mathematical 
analogy between the two systems of ideas leads to a knowledge 
of both, more profound than could be obtained by studying each 
system separately. 
There are men who, when any relation or law, however com- 
plex, is put before them in a symbolical form, can grasp its full 
meaning as a relation among abstract quantities. Such men 
sometimes treat with indifference the further statement that 
quantities actually exist in nature which fulfil this relation. The 
mental image of the concrete reality seems rather to disturb than 
to assist their contemplations. But the great majority of man- 
kind are utterly unable, without long ‘training, to retain in their 
minds the unembodied symbols of the pure mathematician ; so 
that if science is ever to become popular and yet remain scien- 
tific, it must be by a profound study and a copious application 
of those principles of truly scientific illustration which, as we 
have seen, depend on the mathematical classification of quanti- 
ties. There are, as I have said, some minds which can go on 
contemplating with satisfaction pure quantities presented to 
the eye by symbols, and to the mind in a form which none but 
mathematicians can conceive. There are others who feel more 
enjoyment in following geometrical forms, which they draw on 
paper, or build up in the empty space before them. Others, 
again, are not content unless they can project their whole physi- 
cal energies into the scene which they conjure up. They learn 
at what a rate the planets rush through space, and they ex- 
perience a delightful feeling of exhilaration. They calculate the 
forces with which the heavenly bodies pull at one another, and 
they feel their own muscles straining with the effort. 
To such men impetus, energy, mass, are not mere abstract ex- 
pressions of the results of scientific inquiry. They are words 
of power which stir their souls like the memories of childhood. 
Tor the sake of persons of these different types, scientific truths 
should be presented in different forms, and should be regarded 
as equally scientific, whether it appears in the robust form and 
the vivid colouring of a physical illustration, or in the tenuity 
and paleness of a symbolical expression. Time would fail me 
if I were to attempt to illustrate by examples the scientific value 
of the classification of quantities. I shall only mention the 
name of that important class of magnitudes having direction in 
space which Hamilton has called Vectors, and which form the 
subject-matter of the Calculus of Quaternions—a branch of 
mathematics which, when it shall have been thoroughly under- 
stood by men of the illustrative type, and clothed by them with 
physical imagery, will become, perhaps under some new name, 
a most powerful method of communicating truly scientific know- 
ledge to persons apparently devoid of the calculating spirit. The 
mutual action and reaction between the different departments of 
human thought is so interesting to the student of scientific 
progress, that, at the risk of still further encroaching on the 
valuable time of the Section, I shall saya few words on a branch 
of science which not very long ago would have been considered 
rather a branch of metaphysics: I mean the atomic theory, or, 
as it is now called, the molecular theory of the constitution of 
bodies. Not many years ago, if we had been asked in what 
regions of physical science the advance of discovery was least 
apparent, we should have pointed to the hopelessly distant fixed 
stars on the one hand, and to the inscrutable delicacy of the 
texture of material bodies on the other. Indeed, if we are to 
regard Comte as in any degree representing the scientific opinion of 
his time, the research into what takes place beyond our own solar 
system seemed then to be exceedingly unpromising, if not alto- 
gether illusory. The opinion that the bodies which we see and 
handle, which we can set in motion or leave at rest, which we 
can break in pieces and destroy, are composed of smaller bodies 
which we cannot see or handle, which are always in motion, 
and which can neither be stopped nor broken in pieces, nor in 
any way destroyed or deprived of the least of their properties, 
was known by the name of the Atomic Theory. It was asso- 
ciated with the names of Democritus and Lucretius, and was 
commonly supposed to admit the existence only of atoms and 
void, to the exclusion of any other basis of things from the uni- 
verse. 
In many physical reasonings and mathematical calculations we 
are accustomed to argue as if such substances as air, water, or 
metal, which appear to our senses unifo.m and continuous, were 
strictly and mathematically uniform and continuous. We know 
that we can divide a pint of water into many millions of portions, 
each of which is as fully endowed with all the properties of 
water a3 the whole pint was, and it seems only natural to con- 
clude that we might go on subdividing the water for ever, just as 
we can never come to a limit in subdividing the space in which it 
is contained. We have heard how Faraday divided a grain of 
gold into an inconceivable number of separate particles, and we 
may see Dr. Tyndall produce from a mere suspicion of nitrite 
of butyle an immense cloud, the minute visible portion of which 
is still cloud. and therefore must contain many molecules of nitrite 
of butyle. But evidence from different and independent sources 
is now crowding in upon us which compels us to admit that if 
we could push the process of subdivision still further we should 
come to a limit, because each portion would then contain only 
one molecule, an individual body, one and indivisible, unalter- 
able by any power in nature. Even in our ordinary experiments 
on very finely divided matter we find that the substance is begin- 
ning to lose the properties which it exhibits when in a large 
mass, and that effects depending on the individual action of 
molecules are beginning to become prominent. The study of 
these phenomena is at present the path which leads to the de- 
velopment of molecular science. That superficial ten-ion of 
liquids which is called capillary attraction is one of these phe- 
nomena. Anotherimportant class of phenomena are those which 
are due to that motion of agitation by which the molecules of a 
liquid or gas are continually working their way from one place 
to another, and continually changing their course, like people 
hustled in a crowd. On this depends the rate of diffusion of 
gases and liquids through each other, to the study of which, as 
one of the keys of molecular science, that unwearied inquirer into 
nature’s secrets, the late Prof. Graham, devoted such arduous 
labour. 
The rate of electrolytic conduction is, according to Wiede- 
manu’s theory, influenced by the same cause ; and the conduction 
of heat in fluids depends probably on the same kind of action, 
