REVIEWS 159 



Differential and Integral Calculus. By Clyde E. Love, Ph.D., Assistant 

 Professor of Mathematics in the University of Michigan. [Pp. xviii + 344.] 

 (New York : The Macmillan Co. ; London : Macmillan and Co., Ltd., 191 6. 

 Price gs. net.) 



This text-book possesses some features of great value. Thus, the applications of 

 the calculus to mechanics are dealt with very fairly thoroughly ; there is some 

 insistence on the importance of checking the results of exercises, either directly or 

 by solving in more than one way ; and a section on line integrals is included which 

 is of interest and importance to students of the integral calculus. The chapters 

 of the book deal with functions, limits, and continuity ; the derivative ; differ- 

 entiation of algebraic functions ; geometric applications ; differentiation of 

 transcendental functions ; the differential ; curvature ; applications of the de- 

 rivative in mechanics ; curve tracing in Cartesian co-ordinates ; curve tracing 

 in polar co-ordinates ; the indefinite integral ; standard formulas of integration ; 

 integration of rational fractions ; the definite integral ; the definite integral 

 as the limit of a sum ; integral tables ; improper integrals ; centroids and 

 moments of inertia ; law of the mean and evaluation of limits ; infinite series and 

 Taylor's theorem ; functions of several variables ; envelopes and evolutes ; 

 multiple integrals; fluid pressure; differential equations of the first order; 

 differential equations of higher order ; and applications of differential equations in 

 mechanics. 



It will be seen that an integral is defined as the inverse of a differential before 

 the integral is^regarded as the limit of a sum, and a definite integral is (p. 143) 

 defined first of all as the change of value in the integral between the limits of 

 integration. It is stated that " the text is intended to contain a precise statement 

 of the fundamental principle involved. . . . Wherever possible, except in the 

 purely formal parts of the course, the summarising of the theory into rules or 

 formulas which can be applied blindly has been avoided " (p. v). It does not seem 

 that the frequent use of " It can be shown " (cf. pp. 56, 238, 296) and " It is 

 evident" (p. 121 ; cf. pp. 148-9), quite without any further explanation, is good 

 policy in teaching higher mathematics. Further, even if there is no harm in 

 assuming fundamental points without attempting to prove them, it seems a mistake 

 not to call attention to the fact that the points are unproved (cf. pp. 7, 212). 



Philip E. B. Jourdain. 



Functions of a Complex Variable. By Thomas M. Macrobert, M.A., B.Sc, 

 Lecturer in Mathematics in the University of Glasgow. [Pp. xiv + 298.] 

 (London : Macmillan & Co., Ltd., 1917. Price 12.?. net.) 



Of this somewhat comprehensive treatise the author says in the preface : " I have 

 abstained from the use of strictly arithmetical methods, and have, while endeavour- 

 ing to make the proofs sufficiently rigorous, based them chiefly on geometrical 

 propositions." Thus the method is chiefly that of Cauchy with some of the 

 additions made by Riemann and Weierstrass. Cauchy's fundamental theorem on 

 complex integration is proved by the help of Green's theorem, and a slight 

 modification of Goursat's proof is also given, which is credited (p. 54) to Knopp in 

 his Funktionentheorie. The essential point in this proof is, however, not brought 

 out, because the tacit supposition is made that the derivative is continuous. The 

 exposition (p. 57) of the theory of "residues" seems to have some advantages 

 from the teacher's point of view, and the expositions of the theorems of Mittag- 

 Leffler (p. 105) and Weierstrass (p. 108) avoid the usual mistake of considering 



