I have referred mainly to the development of general principles and methods, but that is, of course, not the whole of the story. A complete history would have to treat in some detail the special problems which suggested themselves from time to time. The impulse to general theory indeed often came about in this way. For instance, the problem of the two electrified spheres gave the impulse to Electrostatics, whilst Chladni’s figures of nodal lines led up by degrees to the theory of Elasticity. It is, moreover, in the special applications that the skill of the analyst is particularly evoked, with results often of great interest and value even from the purely mathematical point of view. We need not go back to the theory of Attractions, or of the Figure of the Earth, which evoked Spherical Harmonics. The Conduction of Heat led incidentally to Bessel Functions, and above all to the theorems specially associated with the name of Fourier, whilst Poisson’s problem of the two electrified spheres is a signal instance of the treatment of a functional equation. To Kelvin we owe the method of electric inversion, including the astonishing solution of the problem of the electrified spherical bowl, which had engaged the attention of Green, and the symmetrical treatment of Spherical Harmonics. To Maxwell are due the singularly beautiful solution of the problem of current sheets, a new interpretation of Spherical Harmonics, and other interesting results and points of view scattered through his treatise. As an example of a more systematic application of mathematical technique we may refer again to Cauchy’s wave-problem, where the integrals afterwards attributed to Fresnel first make their appearance.
I have tried in this rapid sketch to do justice especially to the pioneers in the period; the merits and achievements of their more recent successors are fresh in our memories. It was I think fortunate that the first essays in the development of mathematical physics were by men whose accomplishments ranged over the whole of mathematics, and who thus had abundant analytical resources at their disposal. It may be claimed indeed that they provided almost the entire analytical equipment for their successors down to a comparatively recent time. You may search for instance the volumes of Lord Kelvin’s papers and find hardly an appeal to any result of Pure Mathematics later than Cauchy, with the very important exception of what he had discovered himself. The most important province of later analysis which has found a direct application to physical questions is the Theory of Functions, and this again, so far as is necessary for the purpose, dates back to Cauchy, whom I should be disposed to place, after Fourier, as highest among the pioneers of mathematical physics.
I should like to be able to tell more about these men, about their characters, the vicissitudes of their lives and how these reacted on their work, their ambitions, their friendships, and even their quarrels and jealousies. Much that would be interesting is not to be found in official obituary notices. Sometimes an indication of these more human qualities has survived, such as the charming account of Ampère’s early career, of the tragedy of his father’s death in the Revolution, and of his idyllic love-story, and even the foible attributed to him in his later years, of carrying off in all innocence the wrong umbrella, even when there was no right one!
Some points of contrast with present conditions may be noted. The scientific work was largely academical, not so much that the men held as a rule official posts, or were trained in strict schools, but that they were under the influence of scientific Academies, which jealously guarded admission, and narrowly scrutinized the memoirs submitted to them. Consequently there was a tendency towards what I have called the ‘grand style,’ with great attention to form and presentation. One result is that their memoirs can often even now be referred to with interest, the absence of novelty in the subject matter being compensated by the literary charm.
But the great and I think the enviable point of difference is that there was little specialization, and no idea at all of a divorce between Pure and Applied Mathematics. The names I have so often had to quote testify how fruitful the alliance has been. And with all recognition of modern difficulties, I would quote the words of Fourier, but in a somewhat more catholic sense than he had in mind: “L’étude approfondie de la nature est la source la plus féconde des découvertes mathématiques.”
The absence of English names from the first part of the record has often been remarked upon and deplored. The whole story and its lessons are given in Mr Rouse Ball’s well-known History of Mathematics at Cambridge. We may point with pride however to the later achievements of our countrymen, most of them more or less connected with this University. Some features, specially characteristic, which we may claim as of English origin have been already indicated, the search for definite geometrical images of physical relations, and especially the cultivation of graphical methods. I may in particular mention the instructive diagrams which are appended to Maxwell’s treatise, and which have been so great an assistance to the imagination of his readers, and so valuable as an example to later writers.
The period we have been surveying had I think a fairly definite beginning, and an almost equally definite close. From the mathematical point of view its most striking achievement is the wide-embracing scheme of relations, which can be applied to so many diverse subjects, with hardly more than a change in the names of the various concepts. In their purely abstract form, in the rarefied atmosphere of Vector Fields, Triple Tensors, and so on, these relations might almost be developed in an hour, though they could hardly be understood or appreciated without reference to their physical aspects, to which they owe all their value. That such generality should have been attained is an instance of the constant endeavour of Mathematics to reduce to simple laws the infinite variety of nature. With a wider view than was possible to Fourier, we may echo his Newtonian quotation: Quod tam paucis tam multa praestet geometria gloriatur.
CAMBRIDGE: PRINTED BY W. LEWIS AT THE UNIVERSITY PRESS