Feb. 23, 1882] 
Young and Fresnel became more numerous and confident, and 
the scientific societies duly acknowledged the services rendered 
by both. Young was elected one of the eight f reign members 
of the French Academy, and Fresnel received the Rumford 
Medal of the Royal Society, which, however, only reached him 
on his deathbed. 
The undulatory theory now entered on a stage in which it 
could be taken up by the mathematician pure and simple. Its 
foundations had to be rendered more secure, and its consequences 
had to be worked out to a greater extent than even Fresnel had 
don>. 
The scruples which hindered raost of the French mathemati- 
cians from accepting Fresnel’s views were shared by Poissou, 
who deduced from his equations a resultap arently paradoxical. 
According to Fresnel’s formulz, the ceutre of the shadow of a 
small circular dise formed by a Juminous point should be as 
bright as if the disc were absent. But, however curious this 
result mizht be, it had been obseived just 100 years before 
Fresnel’s time, and as that experiment had been completely for- 
gotten, Poisson’s theoretical conclusion had agvin to be subjected 
to the test of experiment, when it was found to be completely 
in accordance with fact. 
But the most remarkable disc»very made solely by calculation 
was the so-called conical refraction, theo etically deduced from 
Fresnel’s wave surface by Sir Wm. Hamilton, ‘That gr-at 
mathe natician had found that a point, when looked at through a 
crystalline plate cut in a certain direction, should appear not as 
a point, but as aring, and the fact was verified experimentally 
by Prof. Lloyd. ‘This discovery has always been considered 
one of the greatest triumphs of mathematical physics, and justly 
ranks on equal terms with the discovery of the planet Neptune 
by Adams and Leverrier. It is necessary to remark, however, 
that strange and unexpected conclusions, especia!ly when they 
have been arrived at after complicated mathematical transforma- 
tions, tempt us sometimes to exaggerate the additional support 
which their verification gives to the theory by means of which 
those conclusions have been reached. It is extremely unlikely 
that any theory should acc unt for all the facts explained by 
Fresnel, and not al.o for all those discovered by his successors. 
As a matter of fact, Fresnel’s wave surface is not the only one 
which has been suggested, but as they all contain the singular 
points at which the conical refracti»n is produced, this pheno- 
menon is no proof that Fresuel’s equations are strictly correct, 
It often happens in mathematical explanations of physical phe- 
nomena that the equations originally deduced contain a series of 
constants which are then determined to fit the experiments. This 
-process, which is perfectly legitimate, does however often prove 
only that the theory is successful in giving us a useful formula of 
interpolation, and need not be conclusive in favour of the ideas 
which have led to the formula. In a considerable number of 
cases, such as the reflection of light from metals, and even the 
theory of double refraction, we have different formula which all 
give, as far as we can test them, a sufficiently correct account of 
the fact-, and none of them therefore prove anything in favour 
of the views which the different authors of the equations have 
put forward, 
Before leaving our consideration of the services rendered by 
mathematics to the undulatory theory, we must not forzet to 
notice the mathematical investigations by means of which its 
foundations have been placed on a safe dynamical basis. The 
investigations of Cauchy, those of Green, which followed, but 
especially those of Stokes, have secured for this theory such a 
firm support tiat even Laplace might have accepted it without 
further scruples. As a matter of history these investigations 
have done litle towards the final victory of the theory. They 
came too late to affect the course of events, but they have in- 
creased the confidence of mathematicians in phy-ical theories, 
and have prepared the way for further investigations. 
As I have already remarked, it is one of the great objects of 
mathematical physics to investizate how far we can safely push 
certain assumptions and where a new hypothesis must be brought 
into play. And, indeed, when we ave carried our calculations 
as far as we can, when we have experimented and measured as 
much as we can, we find that the undulatory theory as it stands 
at present, though following up to a certain point with mar- 
vellous accuracy the true course of nature, shares the common 
fate of all theories, and leaves a’ vast quantity of facts unex- 
plained and waiting fur more complete investigations. Nor is 
this to be wondered at; cur assumptions as regards material 
media may in many cases give correct results and no doubt 
NATURE 
eel, 
answer very well as a first approximation, but we arrive at a 
point where such a material medium can no longer be considered 
homogeneous, and here our conclusions must break down; but 
it is to mathematics that we must look for the next great step. 
The progress of the science of optics during this century has 
shown us how much mathematical calculation can help to esta- 
blish a great and important fact such as the existence of that all- 
pervading mediu, the vibrations of which constitute light, and 
I may review more quickly the recent progress of other branches 
of science. 
In the science of heat we do not require mathematical calcula- 
tions to show the superiority of the mechanical over the corpus- 
cular theory. Sir Humphy Davy’s experiment shows conclusively 
that heat cannot be a substance, and Joule’s experiments served 
further to illustrate the great advantages of the mechanical 
theory. ‘The mathematical treatment of thermic problems was 
not required to establish a theory, but was suggested by practical 
considerations. The important question, how much work we 
caa get out of a steam-engine first attracted mathematicians, and 
out of this question the present science of thermodynamics may 
be said to have arisen.! Carnot, who gave the initial impulse to 
these mathematical investigations, assumed in his papers that 
heat was indestructible, though he seemed personally inclined to 
prefer the mechanical theory, which denied that indestructibility. 
Carnot’s investigations were only gradually appreciated, and it 
was only when Clausius and Thomson corrected his theory so as 
to bring it into accordance with modern ideas, that general atten- 
tion was directed to the subject. It was found that so many 
important consequences of physical interest (as the lowering of 
the freezing-point of water by pressure) followed out of Carnot’s 
corrected reasoning that the mechanical theory now rapidly made 
its way, and thou_h, as already mentioned, the proof of its truth 
rests on a perfectly simple experiment, mathematics must be 
considered to have had an important share in the final establish- 
ment of that theory. 
It seems impossible to speak of the services rendered by 
mathematics to the progress of our knowledge of heat without 
mentioning the great law of the dissipation of energy. No two 
sciences seem further apart than mathematics and metaphysics, 
yet mathematical proposi ions have often furnished material for 
m-taphysical speculations on the workings of nature. Thus the 
many dynamical propos.tions involving minimum or maximum 
properties, such as the principle of least action, have been taken 
to show that nature always works with the least expenditure of 
force, and thus the important law of dissipation of energy, 
which as erts that the worl must have a slow and gradual end, 
cculd not fail to be used in the discussion of its sudden and 
abrupt beginning. ‘These metaphysical speculations react again 
on the progress of physics, but it seems doubtful how far this 
indirect influence of mathematics has been beneficial; at any 
rate mathematicians cannot be held responsible for such an ex- 
tension of their power. 
An offshoot of the mechanical theory of heat is the molecular 
theory of gases. The idea on which that theory is bared is not 
new, but it remained a speculation merely until, chiefly through 
the labours of Joule, the mechanical theory of heat was experi- 
mentally established, and its laws investigated. Vhere is perhaps 
no branch of science in which mathematics has had such unex- 
pected results in forming and confirming our faith in purely 
physical conceptions. That matter is made up of atoms and 
molecules is an hypothesis which simplifies many physical and 
chemical problems. It may, on chemical grounds especially, be 
considered a highly probable hypothesis, but we ce uld hardly 
have obtained the confirmation amounting to proof which the 
idea has received of late years, without the mathematical treat- 
ment which it has received at the hands of Clerk Maxwell and 
those who have followed in his footsteps. One of the most 
astonishing results obtained by Maxwell is the one subsequently 
verified by experiment, that so long as Boyle’s law is true, the 
coefficient of viscosity, as well as that of the thermal conductivity 
in a gas, is independent of the pressure. ‘This fact alone, 
which could never have been obtained without the aid of mathe- 
matics, is a sufficiently strong foundation on which we may rest 
our belief in molecules. It would be extremely interesting to 
follow out the more recent developments of the mechanical 
theory of gases, and to show how both mathematic; and the 
absence of mathematics have advanced its prozre-s, but if it i 
* Foucault’s investigations, though of enormous mathematical importan 
cannot be sail to have had a direct influence on the progress of phys.cs. 
