;74 



NA TURE 



[August i8, 1904 



mental powers that the question may well be raised, 

 whether there is any longer a reason for maintaining in 

 our minds a distinction between mathematical and experi- 

 mental physics, or at all events whether these should be 

 looked upon as separate provinces which may conveniently 

 be assigned to different sets of labourers. It may be held 

 that the highest physical research will demand in the 

 future the possession of both kinds of faculty. We must 

 be careful, however, how we erect barriers which would 

 exclude a Lagrange on the one side or a Faraday on the 

 other. There are many mansions in the palace of physical 

 science, and work for various types of mind. A zealous, 

 or over-zealous, mathematician might indeed make out 

 something of a case if he were to contend that, after all, 

 the greatest work of such men as Stokes, Kirchhoff, and 

 Maxwell was mathematical rather than experimental in its 

 complexion. An argument which asks us to leave out of 

 account such things as the investigation of Fluorescence, 

 the discovery of Spectrum Analysis, and the measurement 

 of the Viscosity of Gases, may seem audacious ; but a survey 

 of the collected works of these writers will show how 

 much, of the very highest quality and import, would 

 remain. However this may be, the essential point, which 

 cannot, I think, be contested, is this, that if these men 

 had been condemned and restricted to a mere book 

 knowledge of the subjects which they have treated with 

 such marvellous analytical ability, the very soul of their 

 work would have been taken away. I have ventured to 

 dwell upon this point because, although I am myself 

 disposed to plead for the continued recognition of mathe- 

 matical physics as a fairly separate field, I feel strongly 

 that the traditional kind of education given to our pro- 

 fessed mathematical students does not tend to its most 

 effectual cultivation. This education is apt to be one- 

 sided, and too much divorced from the study of tangible 

 things. Even the student whose tastes lie mainly in the 

 direction of pure mathematics would profit, I think, by a 

 wider scientific training. A long list of instances might 

 be given to show that the most fruitful ideas in pure 

 mathematics have been suggested by the study of physical 

 problems. In the words of Fourier, who did so much to 

 fulfil his own saying, " L'l^tude approfondie de la nature 

 est la source la plus f^conde des d^'couvertes math^-matiques. 

 Non seulement cette (5tude, en offrant aux recherches un but 

 d^termin^, a I'avantage d'exclure les questions vagues et 

 les calculs sans issue ; elle est encore un moyen assur^ de 

 former 1 'analyse elle-meme, et d'en d^couvrir les ^liSments 

 qu'il nous importe le plus de connaitre, et que cette science 

 doit toujours conserver : ces ^Mments fondamentaux sont 

 ceux qui se reproduisent dans tous les effets naturels." 



Another characteristic of the applied mathematics of the 

 past century is that it was, on the whole, the age of 

 linear equations. The analytical armoury fashioned bv 

 Lagrange, Poisson, Fourier, and others, though subject, 

 of course, to continual improvement and development, has 

 served the turn of a long line of successors. The pre- 

 dorninance of linear equations, in most of the phvsical 

 subjects referred to, rests on the fact that the changes are 

 treated as infinitely small. The electric theorv of light 

 forms at present an exception; but even here' the linear 

 character of the fundamental electrical relations is itself 

 rernarkable, and possibly significant. The theory of small 

 oscillations, in particular, runs as a thread through a great 

 part of the literature of the period in question. It has 

 suggested many important analytical results, and it still 

 gives the best and simplest intuitive foundation for a whole 

 class of theorems which are otherwise hard to compre- 

 hend in their various relations, such as Fourier's theorem, 

 Laplace's expansion, Bessel's functions, and the like. 

 Moreover, the interest of the subject, whether mathematical 

 or physical, is not yet exhausted ; many important problems 

 in Optics and Acoustics, for example, still await solution. 

 The general theory has in comparatively recent times 

 received an unexpected extension (to the case of " latent 

 motions ") at the hands of Lord Kelvin ; and Lord Rayleigh, 

 by his continual additions to it, shows that, in his view, it 

 is still incomplete. 



When the restriction to infinitely small motions is 

 abandoned, the problems become of course much more 

 arduous. The whole theory, for instance, of the normal 

 modes of vibration which is so important in Acoustics, and 



NO. 1816, VOL. 70] 



even in Music, disappears. The researches hitherto made 

 in this direction have, moreover, encountered difficulties of 

 a less patent character. It is conceivable that the modern 

 analytical methods which have been developed in Astronomy 

 may have an application to these questions. It would 

 appear that there is an opening here for the mathematician ; 

 at all events, the numerical or graphical solution of any 

 one of the various problems that could be suggested would 

 be of the highest interest. One problem of the kind is 

 already classical — the theory of steep water-waves discussed 

 by Stokes ; but even here the point of view has perhaps 

 been rather artificially restricted. The question proposed 

 by him, the determination of the possible form of waves of 

 permanent type, like the problem of periodic orbits in 

 Astronomy, is very interesting mathematically, and forms 

 a natural starting-point for investigation ; but it does not 

 exhaust what is most important for us to know in the 

 matter. Observation may suggest the existence of such 

 waves as a fact ; but no reason has been given, so far as I 

 know, why free water-waves should tend to assume a form 

 consistent with permanence, or be influenced in their 

 progress by considerations of geometrical simplicity. 



I have tried to indicate the kind of continuity of subject- 

 matter, method, and spirit which runs through the work 

 of the whole school of mathematical physicists of which 

 Stokes may be taken as the representative. It is no less 

 interesting, I think, to examine the points of contrast with 

 more recent tendencies. These relate not so much to subject- 

 matter and method as to the general mental attitude towards 

 the problems of Nature. Mathematical and physical science 

 have become markedly introspective. The investigators of 

 the classical school, as it may perhaps be styled, were 

 animated by a simple and vigorous faith ; they sought as 

 a matter of course for a mechanical explanation of pheno- 

 mena, and had no misgivings as to the trustiness of the 

 analytical weapons which they wielded. But now the 

 physicist and the mathematician alike are in trouble about 

 their souls. We have discussions on the principles of 

 mechanics, on the foundations of geometry, on the logic of 

 the most rudimentary arithmetical processes, as well as of 

 the more artificial operations of the Calculus. These dis- 

 cussions are legitimate and inevitable, and have led to some 

 results which are now widely accepted. Although they 

 were carried on to a great extent independently, the ques- 

 tions involved will, I think, be found to be ultimately very 

 closely connected. Their common nexus is, perhaps, to be 

 traced in the physiological ideas of which Helmholtz was 

 the most conspicuous exponent. 'I"o many minds such dis- 

 cussions are repellent, in that they seem to venture on the 

 uncertain ground of philosophy. But, as a matter of fact, 

 the current views on these subjects have been arrived at by 

 men who have gone to work in their own way, often in 

 entire ignorance of what philosophers have thought on such 

 subjects. It may be maintained, indeed, that the mathe- 

 matician or the physicist, as such, has no special 

 concern with philosophy, any more than the engineer or 

 the geographer. Nor, although this is a matter for their 

 own judgment, would it appear that philosophers have very 

 much to gain by a special study of the methods of mathe- 

 matical or physical reasoning, since the problems with which 

 they are chieflv concerned are presented to them in a much 

 less artificial form in the circumstances of ordinary life. 

 As regards the present topic I would put the matter in this 

 way, that between Mathematics and Physics on the one 

 hand and Philosophy on the other there lies an undefined 

 borderland, and that the mathematician has been engaged 

 in setting things in order, as he is entitled to do, on his own 

 side of the boundary. 



Adopting this point of view, it would be of interest to 

 trace in detail the relationships of the three currents of 

 speculation which have been referred to. .'Vt one time, 

 indeed, I was tempted to take this as the subject of my 

 Address ; but, although I still think the enterprise a possible 

 one, I have been forced to recognise that it demands a better 

 equipment than I can pretend to. I can only venture to 

 put before you some of my tangled thoughts on the matter, 

 trusting that some future occupant of this Chair may be 

 induced to take up the question and treat it in a more 

 illuminating manner. 



If we look back for a moment to the views currently 

 entertained not so very long ago by mathematicians and 



