Feb. 23, 1882] 
NATURE 
Madagascar, a Common Coot (/wdica atra), British, a Blaubok 
(Cephalophus pygmaeus) from South Africa, three Pluto Monkeys 
(Cercopithecus pluto) from West Africa, purchased; an Axis 
Deer (Cervus axis 6), born in the Gardens. 
THE INFLUENCE OF MATHEMATICS ON 
THE PROGRESS OF PHYSICS + 
N discussing the value of a given study, a lecturer is by 
common consent allowed—sometimes even in private duty 
bound—to exaggerate the importance of his subject, and to pre- 
sent it to his audience enlarged, as it were, through the magni- 
fying power of a projecting lens, so that the details with which 
he has necessarily to deal may be brought into more prominent 
view. In an introductory lecture such as it is my duty to give 
to-day, the speaker need the less feel any scruples in following 
the usual custom, as differeat subjects are treated of in successive 
years, and the hearer may, after the lapse of a short cycle, strike 
a pretty fair balance between the various branches which have 
successively been brought before him. But although I might 
have felt tempted to-day to insist on the advantages of Applied 
Mathematics as a separate subject not only worthy of study, but 
second to none in interest and importance, and though I feel no 
doubt you would have accorded to me the indulgence which 
everybody requires who endeavours to lay an abnorial stress on 
the merits of a single branch of human knowledge, I prefer to 
found the claims of the subject which I have the honour to repre- 
sent in this college, not so much on its intrinsic value as on the 
influence it has had on the progress of other sciences. For no 
subject can stand by itself, and the utility of each must be 
measured by the part it takes in the play of the acting and re- 
aetng forces which weave together all sciences into a common 
web. 
The growing importance of mathematics as an aid to the study 
of all sciences is daily becoming more apparent, and it may 
indeed be questioned whether at the present time we can speak of 
physics as apart from applied mathematics. iemann’s opinion 
that a sczence of physics only exists since the invention of differ- 
ential equations is intelligible ; but however close the connection 
between physics and mathematics may be or may become, their 
growth in the earlier stages has been altogether independent. 
Galileo may be said to have been the founder of mathematical 
physics, and amongst his successors have been many who showed 
a greater inclination towards pure mathematics than towards 
physics proper. On the other hand, we can trace back the 
ancestry of our experimental physicist and that of our modern 
popular books on science to the Middle Ages, where we reach J. 
Baptista Porta and his books on natural magic. Eyen eighty 
years ago the fullest account of the state of experimental 
science was to be found in ‘‘ Wiegler’s Natiirliche Magie,” a 
book of twenty volumes, in which scientific experiments and 
conjurers’ tricks are alternately described, But since the 
beginning of this century the importance of the mathematical 
treatment of purely physical subjects has steadily grown, and 
fifty years ago the two sciences were already sufficiently united 
to induce the founders of the British Association to join them 
together into one section. From that time until the present year, 
when the mass of work necessitated a temporary separation, the 
experimentalist and the pure mathematician could be seen at the 
annual meetings listening, or at least appearing to listen, to each 
other’s investigations, and the influence which men of science on 
these occasions had on each other may be taken to represent 
roughly the mutual influence of the two sciences themselves ; it 
was substantial, though in great part unconscious. I could not 
attempt to-day to give you a complete historical survey of the 
effect which the contact—one might ofien say the collision—of 
the two sciences had on the progress of each; even that part of 
the subject which I have chosen for special consideration is too 
vast to be successfully confined within the limits of a single 
lecture, and an incomplete sketch is all I can offer. 
The influence of mathematical investigations on physical 
theories is not restricted to any single stage, but makes itself 
apparent throughout the whole course of their evolution. Before 
a theory is even started, the mathematician is often necessary to 
prepare its way. He has to classify complicated facts in a sys- 
tematic manner, and working backwards from the phenomena 
* A lecture introductory to the Session 1881-82 of Owens College, 
Manchester, by Arthur Schuster, Ph.D., F.R.S., Professor of Applied 
Mathematics. 
1 
57 
presented by nature, he endeavours to find out which of them 
are necessary consequences of others, and which of them require 
independent hypotheses for their explanation. It is in this way 
that the works of Poisson, Green, Gauss, and of all those who 
have followed in their footsteps, may be said to have laid the 
foundation of the theory of magnetism and electricity, although 
we do not yet as possess any physical notions as to the causes of 
these phenomena. The true power of mathematics, however, 
comes into play only when the physical inventor has done his 
work, and has formed distinct materialistic conceptions which 
allow themselves to be expressed by mathematical symbols. 1t 
is then that the consequences of the theory are to be worked out 
and tested by experiment. In order to be convinced of the truth 
of any hypothesis, the scientific world wants quantitative experi- 
ments. Numbers form the connecting link between theory and 
verification, and they always imply mathematical formule, how- 
ever simple these may be. Often two rival theories are on their 
trial and the mathematician is supposed to find out where their 
conclusions differ and where crucial experiments are most likely 
to decide definitely between them. It is remarkable, however, 
how much more often physical or even metaphysical considera- 
tions have decided between two theories than arguments derived 
from mathematical reasoning. So-called crucial experiments, as 
a rule, come either too early or too late. Sir Humphry Davy’s 
experiment was absolutely conclusive against the corpuscular 
theory of heat, but scientific ideas were not ripe yet for the dis- 
covery, and his experiment had no marked effect on the progress 
of science. The crucial experiment here did not involve any 
mathematical deductions ; it is otherwise with that which might 
have decided between the two theories of light. According to 
the corpuscular theory, light travels more quickly in water than 
in air; according to the undulatory theory, the passage through 
water is the slower, and this distinction is founded on the neces- 
sity to account mathematically for the Jaws of refraction, But 
when Foucault actually made the experiment, and gave a death- 
blow to the corpuscular theory, that theory was already dead. 
There was then only one scientific man of note left who still 
viewed the undulatory theory with suspicion, and his suspicions 
were not allayed by the crucial experiment. But if mathematical 
deductions have not decided as often as they might have done 
between two rival theories, they have constantly strengthened 
and confirmed our belief in physical hypotheses by inventing 
new cases which might test the theory, and which might, if 
experiment snpported the mathematical deduction, establish it 
on a yet firmer basis. 
The most important of all the functions of mathematical 
physics, hewever, and perhaps the only one through which 
mathematics has had an unmitigated beneficial influence on the 
progress of physics is derived from its power to work out to 
their la-t consequences the assumptions and hypotheses of the 
experimentalist. All our theories are necessarily incomplete, 
for they must be general in order to avoid insurmountable diffi- 
culties. It is for the mathematician to find out how far experi- 
mental confirmation can be pushed, and where a new hypothesis 
is necessary. Facts apparently unconnected are found to have 
their origin in a common source, and often only a mathematician 
can trace their connection, It is here that the pure experi- 
mentalist most often fails, A new experiment gives results to 
him unexpected, and he is tempted to invent a new theory to 
account for a fact which may only be a remote consequence of a 
long-established truth. Many examples might be given to show 
how mathematics often finds a connection unsuspected by the 
pure experimentalist, but one may be sufficient. A ray of light 
passing through heavy glass placed in a magnetic field, in the 
direction of the lines of force, is doubly refracted as it comes 
out. To none but a mathematician is it clear that this is only a 
direct consequence of Faraday’s discovery that the magnet turns 
the plane of polarisation of the ray on its passage through the 
glass. Happily this fact was first worked out theoretically ; 
had it been otherwise, we should have heard much of the power 
of the magnet to produce double refraction. 
In addition to the many services actually rendered by mathe- 
matical treatmont, the mere attempt to put physical theories into 
a form fit for such a treatment has of*en been invaluable in clearing 
the theory of all unnecessary appendages and presenting it in the 
simple purity which may bring its hidden failings to light, or 
may suggest valuable generalisations. Instead of dealing, how- 
ever, ina general manner with the various ways in which mathe- 
matics have been useful in the prosecution of physical investiga- 
tions, it will be better to give a short account of the growth of 
