REVIEWS 339 



Plane Trigonometry. By H. Leslie Read, M.A. [Pp. xiii + 290 + xvi.] 

 (London : G. Bell & Sons. Price 3s. 6d.) 



This book is an attempt to present the principles of elementary plane trigonometry 

 in a simple and practical light. It is edited with some care, and examples in large 

 numbers are provided at all stages to suit many tastes. The methods used are 

 those of the orthodox teacher, and need little description. It is, however, a little 

 disconcerting to find that a single-valued function tan x has amongst its values 

 tan 90 = + 00. Again, we find + cos 90 = o. The facts are perhaps explained 

 satisfactorily in the text, but the author's notation surely misrepresents an argument 

 which needs most careful treatment. It is a pity that a subject which after all is 

 comparatively limited and concise should not be expounded in briefer compass. 

 Brevity is the soul of other things besides wit, and if the subject could have been 

 contained within 150 pages, the beginner would have had a less formidable task, 

 and the end of his preliminary labours would have been reached with less 

 expenditure of both time and money. Publishers and authors should remember 

 that the reduction in price of mathematical text-books is an urgent educational 

 need. We hope that schools will never buy bad books because they are cheap ; 

 we should prefer to see good books written concisely and published at a low price. 



C. 



Functions of a Complex Variable. By James Pierpont, LL.D., Professor of 

 Mathematics in Yale University. [Pp. xiv + 583.] (Boston, New York, 

 Chicago, and London; Ginn & Company, 1914. Price 20s. net.) 



The first six chapters of Prof. Pierpont's treatise are devoted to the usual pre- 

 liminaries to the theory of functions, but many of the subjects are treated in a 

 refreshing way ; for example, we may instance the discussion (pp. 22-4) of the 

 irreducible case in cubic equations. The seventh chapter begins with a definition 

 (p. 210) of an analytic function, in which the condition that the first derivative is 

 continuous is included. This is of course necessary if, as is done here (pp. 211-14), 

 Cauchy's fundamental theorem is proved by using Stokes's theorem : the proof 

 of Goursat, which avoids the assumption of the continuity of the derivative, not 

 being used. Prof. Pierpont's account of the theory of functions may be roughly 

 described by saying that little use is made of the work of Riemann, and for the 

 most part the theory is developed by the methods due to Cauchy and his school. 

 However, it was of course necessary to pay attention to some parts of the subject 

 first clearly treated by Weierstrass. Thus we have an account of analytic con- 

 tinuation (p. 224), in which, by the way, a welcome feature is the working out 

 of an application showing that an analytic relation of plane trigonometry is valid 

 when the variable is complex, and the theory of essential singularities (p. 244). 

 Weierstrass's factor-theorem is dealt with in Chapter VIII, and here it must be 

 remarked that the application given on p. 293 is somewhat misleading. Weier- 

 strass's theorem is to show that an integral function with certain assigned zeros 

 exists ; not to give a means of developing a particular function, like the sine, in 

 a product form. Such a development is best effected by a method due to Cauchy 

 or by the method sketched on p. 281. Weierstrass's formula gives a general 

 expression for all the integral functions whose roots are distributed something like 

 those of the sine, and it is only, so to speak, by good luck that a great part of this 

 expression is the same as the well-known development for the sine function. 

 Chapter IX is devoted to Beta and Gamma functions and asymptotic expansions ; 

 Chapter X to Weierstrass's elliptic functions ; Chapter XI to the elliptic integrals 



