August 17, 1905] 



NA TURE 



2>77 



saw the awakening from what now might be regarded 

 as the dark ages. Nor is the field of possible application 

 nearing exhaustion ; on the contrary, it seems to be in- 

 creasing by reason of new discoveries in pure science that 

 yet will find some beneficent outcome in practice. Invisible 

 rays and wireless telegraphy may be cited as instances 

 that are occupying present activities, not to speak of 

 radium, the unfolding of the future of which is watched 

 by eager minds. 



One gap, indeed, in this subject strikes me. There are 

 great histories of mathematics and great histories of 

 astronomy ; I can find no history of physics on the grand 

 scale. Some serviceable manuals there are, as well as 

 monographs on particular topics : what seems to me to 

 be lacking is some comprehensive and comparative survey 

 of the whole range. The history of any of the natural 

 sciences, like the history of human activity, is not merely 

 an encyclopaedic record of past facts ; it reveals both the 

 spirit and the wealth which the past has bequeathed to 

 the present, and which, in due course, the present will 

 influence before transmission to the future. Perhaps all 

 our physicists are too busy to spare the labour needed for 

 the production of a comprehensive history ; yet I cannot 

 help thinking that such a contribution to the subject would 

 be of great value, not to physicists alone. 



But, as you hear me thus referring to astronomy and to 

 physics, some of you may think of the old Roman proverb 

 which bade the cobbler not to look above his last ; so I 

 take the opportunity of referring very briefly to my own 

 subject. One of the features of the century has been 

 the continued development of mathematics. As a means 

 of calculation the subject %vas developed as widely during 

 the earlier portion of the century as during the preceding 

 century ; it soon began to show signs of emergence as an 

 independent science, and the later part of the century has 

 witnessed the emancipation of pure mathematics. It was 

 pointed out, in connection with the growth of theoretical 

 astronomy, that mathematics developed in the direction 

 of its application to that subject. When the wonderful 

 school of French physicists, composed of Monge, Carnot, 

 Fourier, Poisson, Poinsot, Ampere, and Fresnel (to mention 

 only some names), together with Gauss, Kirchhoff, and 

 von Helmholtz in Germany, and Ivory, Green, Stokes, 

 Maxwell, and others in England, applied their mathematics 

 to various branches of physics, for the most part its 

 development was that of an ancillary subject. The result 

 is the superb body of knowledge that may be summarised 

 under the title of "mathematical physics"; but the final 

 interest is the interest of physics, though the construction 

 has been the service of mathematics. Moreover, this 

 tendency was deliberate, and was avowed in no uncertain 

 tone. Thus Fourier could praise the utility of mathe- 

 matics by declaring that " there was no language more 

 universal or simpler, more free from errors or obscurity, 

 more worthy of expressing the unchanging relations of 

 natural entities " ; in a burst of enthusiasm he declares 

 that, from the point of view he had indicated, " mathe- 

 matical analysis is as wide as Nature herself," and " it 

 increases and grows incessantly stronger amid all the 

 changes and errors of the human mind." Mathematicians 

 might almost blush with conscious pleasure at such a 

 laudation of their subject from such a quarter, though 

 it errs both by excess and defect; but the exultation of 

 spirit need not last long. The same authority, when 

 otTicially expounding to the French Academy the work of 

 Jacobi and of Abel upon elliptic functions, expressed his 

 chilling opinion (it had nothing to do with the case) that 

 " the questions of natural philosophy, which have the 

 mathematical study of all important phenomena for their 

 aim, are also a worthy and principal subject for the 

 meditations of geometers. It is to be desired that those 

 persons who are best fitted to improve the science of 

 calculation should direct their labours to these important 

 applications." Abel was soon to pass beyond the range 

 of admonition ; but Jacobi, in a private letter to Legendre, 

 protested that the scope of the science was not to be 

 limited to the explanation of natural phenomena. I have 

 not quoted these extracts by way of even hint of reproach 

 against the author of such a wonderful creation as 

 Fourier's analytical theory of heat; his estimate could 

 have been justified on a merely historical review of the 



NO. 1868, VOL. 72] 



circumstances of his own time and of past times ; and I 

 am not sure that his estimate has not its exponents at 

 the present day. But all history shows that new dis- 

 coveries and new methods can spread to issues wider than 

 those of their origins, and that it is almost a duty of 

 human intelligence to recognise this possibility in the 

 domain of progressive studies. The fact is that mathe- 

 matical physics and pure mathematics have given much 

 to each other in the past and will give much to each other 

 in the future ; in doing so, they will take harmonised 

 action in furthering the progress of knowledge. But 

 neither science must pretend to absorb the activity of the 

 other. It is almost an irony of circumstance that a 

 theorem, initiated by Fourier in the treatise just men- 

 tioned, has given rise to a vast amount of discussion and 

 attention, which, while of supreme value in the develop- 

 ment of one branch of pure mathematics, have hitherto 

 offered little, if anything, by way of added explanation of 

 natural phenomena. 



The century that has gone has witnessed a wonderful 

 development of pure mathematics. The bead-roll of names 

 in that science — Gauss; Abel, Jacobi; Cauchy, Riemann, 

 Weierstrass, Hermite ; Cayley, Sylvester ; Lobatchewsky, 

 Lie — will on only the merest recollection of the work with 

 which their names are associated show that an age has 

 been reached where the development of human thought is 

 deemed as worthv a scientific occupation of the human 

 mind as the most profound study of the phenomena of 

 the material universe. 



The last feature of the century that will be mentioned 

 has been the increase in the number of subjects, appar- 

 ently dissimilar froin one another, which are now being 

 made to use mathematics to some extent. Perhaps the 

 most surprising is the application of mathematics to the 

 domain of pure thought ; this was effected by George 

 Boole in his 'treatise "Laws of Thought," published in 

 18^4 : and though the developments have passed consider- 

 ably beyond Boole's researches, his work is one of those 

 classics that mark a new departure. Political economy, 

 on the initiative of Cournot and Jevons, has begun to 

 employ symbols and to develop the graphical methods ; 

 but, there, the present use seems to be one of suggestive 

 record and expression, rather than of positive construction. 

 Chemistry, in a modern spirit, is stretching out into 

 mathematical theories ; Willard Gibbs, in his memoir on 

 the equilibrium of chemical systems, has led the way ; 

 and. though his way is a path which chemists find strewn 

 with the thorns of analysis, his work has rendered, in- 

 cidentally, a real service in coordinating experimental 

 results belonging to physics and to chemistry. A new and 

 generalised theory of statistics is being constructed ; and 

 a school has grown up which is applying them to biological 

 phenomena. Its activity, however, has not yet inet with 

 the sympathetic goodwill of all the pure biologists ; and 

 those who remember the quality of the discussion that 

 took place last year at Cambridge between the biometricians 

 and some of the biologists will agree that, if the new 

 school should languish, it will not be for want of the tonic 

 of criticism. 



If I have dealt with the past history of some of the 

 sciences with which our Section is concerned, and have 

 chosen particular epochs in that history with the aim of 

 concentrating your attention upon them, you will hardly 

 expect me to plunge into the future. Being neither a 

 prophet nor the son of a prophet, not being possessed of 

 the knowledge which enabled Halley to don the prophet's 

 mantle with confidence, I shall venture upon no prophecy 

 even so cautious as Bacon's — " As for the mixed mathe- 

 matics I may only make this prediction, that there cannot 

 fail to be more kinds of them as Nature grows further 

 disclosed " — a declaration that is sage enough, though a 

 trifle lacking in precision. Prophecy, unless based upon 

 :onfident knowledge, has passed out of vogue, except 

 perhaps in controversial politics ; even in that domain, it 

 is helpless to secure its own fulfilment. Let ine rather 

 exercise the privilege of one who is not entirely unfamiliar 

 with the practice of geometry, and let me draw the pro- 

 verbial line before indulgence in prophetic estimates. The 

 names that have flitted through my remarks, the dis- 

 coveries and the places associated with those names, 

 definitelv indicate that, notwithstanding all appearance of 



