REVIEWS 487 



Berkeley, Robins, and Jurin ; and the reader is grateful to the author for the 

 excellent judgment shown by him in selecting the various passages. 



But the book concerns itself chiefly with the philosophical side of fluxions, 

 the questions of infinitesimals, differentials, and Newton's opposite method 

 of fluxions. Really this side does not appear to the reviewer to be so 

 important. There are many ideas in philosophy and science which are almost 

 obvious to anyone, but which it is nearly impossible to expound rigorously 

 in words, chiefly owing to the fact that words have variable and discontinuous 

 meanings. That acute writer Berkeley detected this difficulty at once and 

 set himself to his attack, in the interests of his own cloth, against the philo- 

 sophers. His portrait is given in the book, and the reviewer thinks that 

 he was not quite as candid a person as Huxley and de Morgan and many 

 others seem to think. The mere fact that he professedly set out to " coun- 

 teract atheism " shows not only that he was biased in his opinions before 

 he started, but that he was not aware of what a fatal blemish such a bias 

 may be. His attack upon fluxions appears to have been largely in the 

 nature of a jest ; and he must have enjoyed the evident confusion which 

 he created among the ranks of the mathematicians by his well-directed shafts, 

 especially when these gentlemen began to fight amongst themselves after 

 attacldng him. The fact remains, however, that Newton was right and that 

 the method of fluxions is sound ; whereas, on the other hand, Berkleyism 

 leads merely to solipsism. 



We think that Prof. Cajori should have dealt not only with one part of 

 the subject, but with the whole of it. This would, of course, have meant a 

 much larger book, but then the reader would have had the whole matter in 

 his hand at once. For instance, no mention is made of the recent discovery 

 by Mr. J. M. Child of the connection of Isaac Barrow, Newton's teacher, with 

 the Calculus (The Geometrical Lectures of Isaac Barrow, Open Court Pub- 

 lishing Co., 1916), nor is the Newton-Leibniz controversy dealt with. Really 

 these matters are much more interesting than the philosophical points exam- 

 ined in the present volume. The fundamental discovery was that from a 

 given function a second one may be derived which gives the first function's 

 rate of change, and that, conversely, the first function gives the integration 

 of the second one — just as set out in Newton's famous anagram to Leibniz. 

 Opinions differ as to who originated this momentous advance in human 

 knowledge ; but, pending an exhaustive analysis, the reviewer thinks it most 

 probable that Newton was the originator — that he not only taught his teacher, 

 Barrow, but that he gave the whole idea to Leibniz, who merely embroidered 

 the details while Newton was fully engaged upon applying his methods for 

 the advance of natural knowledge in every direction. Newton built the 

 house, though Barrow may have helped him to dig the foundations, and 

 Leibniz to place the chimney-pots. It is even questionable whether d-ism 

 is really better than dot-age, because, for instance, the latter is applicable 

 in quaternions, whereas the former is not. Might we not also have had some 

 more of Brook Taylor than the five lines given to him on p. 50 — which do 

 not even mention his great theorem. But we are grateful to Prof. Cajori 

 for what he has given us, and hope that he may attack the rest of the history 

 some day. 



R. R. 



An Introduction to Combinatory Analysis. By Major P. A. MacMahon, 

 D.Sc, Sc.D., LL.D., F.R.S. [Pp. viii + 71.] (Cambridge: at the 

 University Press, 1920. Price 7s. 6d. net.) 



This little book is intended to be an introduction to the author's two 

 masterly volumes on Combinatory Analysis published by the Cambridge 

 University Press in 19 15-16, and reviewed in Science Progress, vol. x, 

 p. 601, and vol. xi, p. 695. In writing it the author designs to place the 



