September ii, 1890] 



NA TURE 



465 



pretend that there are not many branches of mathematics which 

 partake of both these characters, nor do I suppose that the 

 description I have given of a mathematical theory is at all 

 peculiar to pure mathematics. Much of it is common to all 

 scientific research in a fruitful field, though, possibly, we may 

 not find elsewhere such profusion of ideas or perfection of form. 



I have been tempted to speak at such length on the objects 

 and aims of the mathematician by the feeling that they are not 

 infrequently misunderstood by the workers in the less abstract 

 sciences. I do not think that mathematical formulae or pro- 

 cesses, merely as such, are much more interesting to the pure 

 than to the applied mathematician. The one studies number, 

 quantity, and position, the other deals with matter and motion ; 

 and in both cases the investigations are carried on by means of 

 the same symbolic language. 



The order in which the subjects which form an ordinary 

 mathematical course are presented to the student is regulated by 

 the fact that portions of the elements of the pure sciences are 

 required for the explanation and development of any exact 

 science ; for example, a knowledge of the elements of trigono- 

 metry, analytical geometry, and differential and integral calculus, 

 must necessarily precede any adequate treatment of mechanics, 

 light, or electricity. The majority of students, after mastering 

 a sufficient amount of pure mathematics to enable them to pass 

 on to the physical subjects, continue to devote their attention to 

 the latter, and never know more of the nature of the pure 

 sciences than they can derive from the processes and methods 

 which they learned at the very outset of their mathematical 

 studies. This is necessarily the case with many of the wranglers, 

 as the first part of the Mathematical Tripos includes no true 

 mathematical theory. Most of the mathematical text-books 

 in use at Cambridge are so admirably adapted to the purposes 

 for which they are intended that it seems ungracious to make an 

 adverse criticism of a general kind. But I cannot help feeling 

 regret that their writers have had so much in view the imme- 

 diate application of the principles of the pure subjects to the 

 treatment of physical problems. In the case of the differential 

 and integral calculus, for example, there seems an increasing 

 tendency to introduce into the book-work and examples propo- 

 sitions which really belong to the physical subjects. This is an 

 important tribute to the growth and influence of physical mathe- 

 matics in this country, and a zealous physicist might even con- 

 sider it satisfactory that the student should not be required to 

 encumber himself with knowledge which was not directly ap- 

 plicable to the theory of matter. But from the mathematician's 

 point of view it is unfortunate, for, while shortening by very 

 little the path of the student, it cannot fail to give an incom- 

 plete, if not erroneous, idea of the relations of the pure to the 

 applied sciences. How can he help feeling that the former are 

 merely ancillary to the latter when he finds that the mathe- 

 matical problems which arise naturally in physical investigations 

 have been already dealt with out of their place in the treatises 

 which should have been devoted solely to the sciences of quantity 

 and position? 



Perhaps few persons who have not had the matter forced upon 

 their attention fully realize how fragmentary and unsatisfactory 

 is the treatment of even those fundamental subjects in pure 

 mathematics which form the groundwork of any course of mathe- 

 matical study. Algebra is necessarily the first subject set before 

 the student ; it has therefore to be adapted to the beginner, who 

 at that time is only learning the first elements of the language 

 of analysis. It is customary to regard trigonometry as primarily 

 concerned with the solution of triangles ; the geometrical defi- 

 nitions of the sine and cosine are therefore adopted, and after 

 the application of the formulse to practical measurement and 

 calculation a new departure is made with De Moivre's theorem. 

 ■J he elementary portions of the theory of equations, and the 

 differential and integral calculus and differential equations, are 

 valuable collections of miscellaneous principles, processes, and 

 theorems, useful either as results or as instruments of research, 

 but possessing no great interest of their own. Analytical 

 geometry fares the best, for it includes one small subject — 

 curves of the second order — which is treated scientifically and 

 with thoroughness. It is true, however, that the course of 

 reading just mentioned includes one theory which, though itself 

 an imperfect one, receives a tolerably complete development — 

 I mean the theory of singly periodic functions ; but it is 

 dispersed in such small fragments among the various subjects 

 that it does not naturally present itself to the mind as a whole. 

 If we could commence this theory by considering analytically 



NO. 1089, VOL. 42] 



the forms and necessary properties of functions of one period 

 (thus obtaining their definitions as series and products), and 

 could then proceed to a detailed discussion of the functions so 

 defined — including their derivatives, the integrals involving 

 them, the representation of functions by their means in series 

 (Fourier's theorem), &c. — we should obtain a connected 

 system of results relating to a definite branch of knowledge 

 which would give a good idea of the orderly development of 

 a mathematical theory ; but the fact that the student at the 

 time of his introduction to sines and cosines is supposed to be 

 ignorant of all but the most elementary algebra, jjlaces great 

 difficulties in the way of any such systematic treatment of the 

 subject. 



Passing now to the consideration of pure mathematics itself, 

 that is to say, of the abstract sciences, which can only be con- 

 quered and explored by mathematical methods, it is difficult 

 not to feel somewhat appalled by the enormous development 

 they have received in the last fifty years. The mass of investi- 

 gation, as measured by pages in Transactions and Journals, 

 which is annually added to the literature of the subject, is so 

 great that it is fast becoming bewildering from its mere magni- 

 tude, and the extraordinary extent to which many special lines 

 of study have been carried. To those who believe, if any such 

 there are, that mathematics exists for the sake of its applications 

 to the concrete sciences, it must indeed seem that it has long 

 since run wild, and expanded it-elf into a thousand useless 

 extravagances. Even the mathematician must sometimes ask 

 himself the question — not infrequently put to him by his friends 

 — " To what is it all tending ? What will be the result of it all ? 

 Will there be any end ? " The last question is readily answered. 

 There certainly can be no end ; so wide and so various are the 

 subjects of investigation, so interesting and fascinating the 

 results, so wonderful the fields of research laid open at each suc- 

 ceeding advance — no matter in what direction — that we may be 

 sure that, while the love of learning and knowledge continue to 

 exist in the human mind, there can be no relaxation of our 

 efforts to penetrate still further into the mysterious worlds 

 of abstract truth which lie so temptingly spread before us. 

 The more that is accomplished, the more we see remaining to 

 be done. Every real advance, every great discovery, suggests 

 new fields of inquiry, displays new paths and highways, gives 

 us new glimpses of distant scenery. This wonderful suggestive- 

 ness is itself one of the marks of a true theory, one of the signs 

 by which we know that we are investigating the actual, existing 

 truths of Nature, and that our symbols and formulae are ex- 

 pressing facts quite independent of themselves, though decipher- 

 able only by their means. As for the other questions, it is very 

 difficult to render intelligible, even to a mathematician, the 

 kind of knowledge acquired by mathematical research in a new 

 field until he has made himself acquainted with its processes 

 and notation, and we cannot hope to find in the remote regions 

 of an abstract science many results so simple and striking as to 

 appeal forcibly to the imagination of those who are unfamiliar 

 with its conceptions and ideas. It would seem, therefore, that 

 the question, "To what is it all tending?" could never be 

 answered in general terms. I do not think any mathematician 

 could see his way to a reply, or even give definite meaning to 

 the question. He might feel daring enough to predict the 

 probable drift of his own subject, but he could scarcely get a 

 broad-enough view to enable him to indulge his fancy with re- 

 spect to more than a very minute portion of the field already 

 open to mathematical investigation. To the outsider I am 

 afraid that the subject will continue to present much the same 

 appearance as it does now ; it will always seem to be stretching 

 out into limitless symbolic wastes, without producing any results 

 at all commensurate with its expansion. 



Instead of attempting to consider the general question of 

 what may be expected to result from the progress of mathe- 

 matical science, we may restrict ourselves to asking whether the 

 great extension of the bounds of the subject which is taking 

 place in our time will materially add to its powers as a weapon 

 of research in the concrete subjects. This is a question of the 

 highest interest, and one that cannot fail to have occupied the 

 thoughts of every mathematician at some time or another in the 

 course of his work. For my own part, I do not think that the 

 bearing of the modern developments of mathematics upon the 

 physical sciences is likely to be very direct or immediate. It 

 would indeed be rash to assert that there is any branch of mathe- 

 matics so abstract or so recondite that it might not at any 

 moment find an application in some concrete subject ; still it 



