42J 



NATURE 



[December i, 192 i 



that the whole time considered has been divided 

 into three successive stages. The first of these 

 ends when the classical Greek methods had prac- 

 tically lost their prestige and fertility ; the second 

 when (at the end of the eighteenth century) the 

 new methods introduced by Descartes, Newton, 

 Leibniz, and Euler seemed to have reached in like 

 manner the limit of their powers and to be incap- 

 able of suggesting really new and important fields 

 of research. 



Now that so much good work has been done on 

 Greek mathematics we can realise with fair accu- 

 racy what ideas the Greek mathematician had of 

 his subject. It included the arithmetic of whole 

 numbers as well as geometry, and for the Pytha- 

 goreans at any rate the former branch was the 

 most fundamental. By the time of Euclid the 

 Greeks had a perfect theory of proportion, and a 

 method of exhaustion capable of solving problems 

 for which we now use integral calculus. In 

 ApoUonius's "Conies" we have numerous pro- 

 positions which are so easily translated into a 

 modern algebraical form that one is almost 

 tempted to think that the Greeks anticipated 

 Descartes' analytical method. This, however, is 

 not the case ; and in the same way it is a mistake 

 to fancy that they had a theory of irrational 

 numbers. The discovery of incommensurable 

 quantities led to a theory of proportion applicable 

 to them, but the ratio of two similar quantities 

 was never regarded as a number even if one of the 

 quantities was an exact multiple of the other. 

 Computation and measurement as such did not 

 form part of mathematics ; limitations of construc- 

 tion, such as those of Euclid's " Elements," seem 

 due to a mixture of aesthetic and philosophical 

 considerations. Archimedes' tract " On Method " 

 shows that the greatest Greek mathematicians did 

 not disdain the use of mechanical considerations 

 as an aid to mathematical research ; but they never 

 were contented with a mechanical proof, and 

 always tried to replace it by one strictly mathe- 

 matical in their sense of the term. 



Coming now to the second period. Prof. 

 Boutroux considers that the great innovation of 

 the seventeenth century does not consist in the 

 use of new methods such as the infinitesimal cal- 

 culus, but rather in the development of the notion 

 of " function," especially in the form of an infinite 

 series. Few things are more interesting than the 

 history of the mathematical term " function." 

 For a long period all the functions actually con- 

 sidered were those expressible by power-series. 

 Even these were discussed in what would now 

 be called a scandalously superficial way. In 

 spite of the fact that as early as the time of the 

 NO. 2718, VOL. 108] 



Bernoullis the problem of vibrating strings led to 

 a prolonged discussion as to the possible or proper 

 representation of so-called " arbitrary " functions, 

 the subject was left " wropped up in mystery " : 

 it was often tacitly assumed that a power-series 

 representation, at least for some intervals, was 

 always possible if only it could be found. 

 Fourier's introduction of trigonometrical series 

 and his bold application of them to physical prob- 

 lems gave a sort of electrical shock to the mathe- 

 matical world, and the use of complex quantities 

 was still regarded by the orthodox as a sort of 

 juggling trick which led to correct results in a 

 quite inscrutable way. 



The beginning of the modern period may be 

 conveniently marked by Abel's memoir on the 

 binomial series ; in this we have for the first time 

 a correct and rigorous treatment of an analytical 

 function. After this, in the hands of Riemann, 

 Dirichlet and others the subject made great 

 strides, and the vast subject of the classification 

 of functions has been attacked with a gratifying 

 measure of success. 



In tracing the currents of mathematical thought 

 the author has frequently to deal with controver- 

 sial matters on which he cannot pretend to give 

 more than a plausible opinion. Perhaps the most 

 important of these depends on the definition of 

 mathematics. In our own opinion it is best for 

 the present not to attempt any formal definition 

 of mathematics, but to be content with the inter- 

 national schedule on which the Royal Society 

 Subject-Index is based. So long as it is agreed 

 to include such things as the theory of groups oni 

 one hand, and abstract dynamics on the other,| 

 a formal definition is practically impossible. But 

 there is one point on which a difference of opinion^ 

 is both significant and important. Is mathe-^ 

 matics a science in the true sense, or is it an| 

 organised method of a special type reducible 

 almost entirely to a branch of symbolic logic ^ 

 We are very glad to see that Prof. BoutrouJ 

 emphatically adopts the former view. To himg 

 as to us, mathematics proceeds from a body oi| 

 indefinable data which it discusses by means o% 

 special methods peculiarly its own, while its 

 general method is simply that of all scientific] 

 research. The school of which Mr. Bertranc 

 Russell may be taken as a representative has per- 

 formed a great service by its critical discussion of 

 what we may call mathematical first principles;! 

 but few of its members, we fancy, w^ould claim 

 that it has contributed to mathematical theory. 

 At any rate the great body of progressive mathe- 

 maticians have refused to admit any such claim; 

 and we may venture to suggest one reason for 



