REVIEWS 



MATHEMATICS 



A First Course in the Calculus. By William P. Milne, M.A., D.Sc, and G. J, 

 B. Westcott, M.A. [Pp. xv + xxxix.] (London : G. Bell & Sons, 1920. 

 Price 55. net.) 



The first part of Dr. Milne and Mr. Westcott's book was reviewed by the late 

 Mr. P. E. B. Jourdain in Science Progress, vol. xiv, p. 152. It dealt only 

 with differentiations and integrations of powers of x, and applications. This 

 second volume considers trigonometrical and logarithmic functions, the 

 differentiation of products and quotients, and of inverse trigonometrical 

 functions, and then goes on to the differentiation and integration of logarithms, 

 expansions and calculations, and differential equations. The historical intro- 

 duction gives short biographies of Taylor and Maclaurin. Packed in between 

 the main propositions there is also useful mention of many theorems in dy- 

 namics and physics, and geometrical and graphical methods are very largely 

 employed. There are many examples for the student to work out. Like 

 most modern writings, the work is loose rather than rigid ; but we question 

 whether on the whole the student gains any advantage from this looseness — 

 whether, indeed, the older dissertations such as those of Bartholomew Price 

 and Isaac Todhunter are not really capable of giving him the same knowledge 

 with less effort, or at least with more satisfaction. The older, rigid, and logical 

 expositions supposed the student to be a rational person, whereas the modern 

 method insists upon treating' him more as a child. The former reached the 

 summit by a steep ascent, perhaps ; but the latter keeps him wandering 

 about along a gentle slope where he is very apt to lose himself without ever 

 reaching the summit at all. On the other hand, this book often does reach 

 particular points more quickly, or even more easily, than the rigid system 

 used to do — as in the cases of expansions and of integrating factors, for 

 example ; and integration is put very simply, and, as it ought to be, almost 

 pari passu with differentiation. The greatest gain is obtained in the examples 

 and graphs. We should have liked to see Boole's elementary exposition of 

 Taylor's Theorem by means of finite differences, and also some note of his 

 symbolic methods, given in this book. 



A History o£ the Conceptions of Limits and Fluxions in Great Britain from 

 Newton to Woodhouse. By Florian Cajori, Ph.D. [Pp. viii + 299, 

 with portraits of Berkeley and Maclaurin.] (Chicago and London : 

 The Open Court Publishing Co., 1919. Price 8s. 6i. net.) 



The way in which the Americans are fast out-pacing British scientific workers 

 in certain lines is demonstrated by the issue of this much-needed little book. 

 Although we have many teachers of mathematics, few of them trouble to 

 write on such a fundamental matter as the history — much less to write in 

 the detail adopted by Prof. Cajori. The book deals, however, only with 

 part of the discovery of the Calculus, namely from Newton to Woodhouse. 

 It gives numerous and full excerpts from various writings with translations 

 where required, and deals fully with discussions such as the polemics between 



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