THE EVOLUTION
OF
MATHEMATICAL PHYSICS
Being the Rouse Ball Lecture for 1924,
BY
HORACE LAMB, Sc.D., F.R.S.,
HONORARY FELLOW OF TRINITY COLLEGE
CAMBRIDGE
AT THE UNIVERSITY PRESS
1924
PRINTED IN GREAT BRITAIN
THE EVOLUTION OF MATHEMATICAL PHYSICS
THE founder of this Lecture has chosen as one of his special interests the history of Mathematics, both through the ages and as reflected in the studies of the University. Within a short compass he has given an account of the development of the subject which contrasts with the elaborate treatises of previous writers by its concentration on essentials, and also by the glimpses which it affords of the personalities of the mathematicians whose achievements he records, with their limitations and their failures, as well as their ambitions and successes.
The study of the successive steps in the evolution of any subject is an attractive pursuit, and many years ago, speaking not far from this place, I was led to hazard some speculations as to the ideas which prompted and guided the very first steps in the development of Greek Mathematics. I even ventured to say, not altogether in the spirit of paradox, that if any one scientific invention could claim pre-eminence above all others, I should be inclined to suggest a monument to the unknown inventor of the mathematical point, as the first step in that long process of abstraction and idealization which has culminated in the science (and not merely mathematical science) of to-day. I remember that the eminent engineer who sat near remarked to me afterwards that if the scale of subscriptions was to be appropriate to the dimensions of the object to be commemorated he would gladly head the list. An even more eminent astronomer told me that the whole address was an elaborate scientific joke. Such friendly satire did not disturb my opinion; but speculations on the psychology of the primitive mathematicians, attractive as I think they are, are necessarily precarious, and I am not tempted to venture on this field again. The task which I would attempt to-day is to trace the leading steps in the development of that great tradition of Mathematical Physics, as distinguished from Dynamics and Astronomy, which began in the early years of the last century, and has dominated physical speculation until quite recent times, when new discoveries and new ideas have emerged, calling for newer methods, without, however, rendering the old ones obsolete. The ground has of course been traversed before, but not I think quite from the present point of view. I am not concerned with physical theories as such but rather with the mathematical dress which they have assumed from time to time. My object is to shew how it comes about that we have inherited a mathematical scheme which in its final form embraces subjects physically so different as Heat-Conduction, Hydrodynamics, Elasticity, Magnetism, Electricity, and Light, and can be made to include any one of these by assigning proper names to the symbols. The scheme admits of course of being set forth in a purely abstract form without any physical reference at all, and this has in fact been done; but its chief value is for the physical analogies which it facilitates, and in which it originated. The development has been continuous, although the wide scope of the final result could not have been foreseen.
The time I have indicated as a starting point was peculiarly favourable. The great calculator Euler had ranged over the whole field of Mathematics, and had given to many parts of it almost the final form which we find in our text-books. Lagrange, Laplace, and Legendre had developed the Newtonian Astronomy, and made important contributions to general Dynamics, as well as incidentally to Analysis. So that when attention began to be directed to physical subjects the available mathematical resources were far in advance of what had been within reach at any earlier period.
Isolated questions of course had been treated previously; for instance the flexure of bars had been discussed by Bernoulli and Euler. More important from the present point of view is the foundation of Hydrodynamics by Euler, who formulated the fundamental differential equations, and proceeded to integrate them on the supposition that a velocity-potential exists. He was careful to note, however, that there are cases, such as that of uniform rotation about an axis, where this condition is not fulfilled. The theory of plane waves of sound was also known, and I need hardly recall the subject of vibrating strings with its reactions on Analysis, and the long controversies which resulted. But the starting point of Mathematical Physics, in the now general sense of the term, is to be fixed I think about the time when the storms of the French revolution had subsided and were succeeded by the comparative tranquillity of the early Empire. If a more definite date is required, we may perhaps fix on the year 1807, which was marked by the publication of Poisson’s first memoir on Sound. This deals with spherical waves, with waves in an atmosphere of variable density and, most astonishing of all, with waves of finite amplitude. He finds that the boundaries of such a wave advance with the ordinary velocity of sound, but omits to examine the progressive change of type. This was only done long afterwards by Stokes. It may I think be said of Poisson that, with all his extraordinary power in dealing with a problem when once it had been reduced to an analytical form, and the great achievements which stand to his credit, he was less concerned with the physical interpretation of his results.