158 SCIENCE PROGRESS 



MATHEMATICS 



A Course in Mathematical Analysis. By Edouard Goursat, Professor of 

 Mathematics in the University of Paris. Translated by Earle Raymond 

 Hedrick, Professor of Mathematics in the University of Missouri. Vol. I. 

 [Pp. xviii + 548.] Vol. II, Parti. Functions of a Complex Variable. 

 Translated by Earle Raymond Hedrick and Otto Dunkel, Instructor 

 in Mathematics, the University of Missouri. [Pp. x+260.] (Boston, 

 New York, Chicago, and London : Ginn & Co. Price respectively 16s-. 

 and us. 6d.) 

 This is an excellent translation of one of the best of the French treatises on 

 analysis. The original French edition was published in 1902, and the first 

 volume of this translation was apparently published in 1904, although the title- 

 page bears no date. The first part of the second volume is translated from the 

 first half of the second volume of the second edition of Goursat's work, and was 

 published in 191 6. The first volume was not radically altered in the second 

 French edition, so that the present English translation of that volume may be 

 used conveniently as a companion to that of the second volume. A second 

 part of the second volume will presumably contain the theory of differential 

 equations. 



The contents of the first volume concern derivatives and differentials ; implicit 

 functions, functional determinants, and change of variable ; Taylor's series and 

 maxima and minima ; definite integrals ; indefinite integrals ; double integrals ; 

 multiple integrals and integration of total differentials ; infinite series ; power 

 series and trigonometric series ; plane curves ; skew curves ; and surfaces. 



The first part of the second volume treats successively the general principles 

 of the theory of analytic functions ; power series with complex terms and elementary 

 transcendental functions ; conformal representation ; the general theory of analytic 

 functions according to Cauchy's method of complex integration, including a study 

 of the periods of definite integrals ; one-valued analytic functions treated accord- 

 ing to the method of Weierstrass and Mittag-Leffier, and Weierstrass's elliptic 

 functions ; analytic continuation ; and analytic functions of several variables. 



The treatise differs from many of the French ones in laying great stress on 

 what is usually referred to as the " intuitional " as opposed to the logical de- 

 velopment of analysis. Thus the first volume contains a discussion of differentials 

 as well as derivatives (pp. iii, 19), and a particularly welcome feature is the 

 introduction of integrals by a short and almost historical account of the methods 

 of quadrature used before the integral calculus was invented (pp. 134-40). The 

 excellent collection of examples, which brings the book very close to those to 

 which we have been accustomed in this country, should also be noticed. The 

 translators have added several useful notes to both volumes. With regard to 

 the second of the two volumes noticed here, attention should be drawn to that 

 very characteristic part of Goursat's original work on the proof of Cauchy's 

 theorem on complex integration (pp. 66-70). There is one point which it seems 

 that we may criticise. On p. 139 we read that : " Long before Weierstrass's 

 work, Cauchy had deduced from the theory of residues a method by which a 

 function, analytic except for poles, may ... be decomposed into a sum of an 

 infinite number of rational terms." The object of the theorems of Weierstrass 

 and Mittag-Leffier was actually to construct certain analytic functions ; the 

 object of Cauchy's process was to develop a function which was assumed to exist, 

 in a certain form. 



Philip E. B. Jourdain. 



