148 SCIENCE PROGRESS 



done between 1894 and 1917 by Borel, Hadamard, Mittag-Leffler, and others, 

 but the nearest approach to this is a reference— it is hardly more— on pp. 69-70 

 to some papers by Mittag-Leffler and Poincare\ Even in this third edition 

 Prof. Forsyth is unpardonably obscure about the values of a function "at" an 

 essential singularity (pp. 66,115, JI 7), and no account is given of Picard's proof 

 by modular functions of his very important theorem on "exceptional values." 

 But sometimes, as on p. in, a decided attempt is made, even in the first part, 

 to bring references to investigations up-to-date, while the added note already 

 mentioned, on the applications of conformal representation, is quite modern. 



As is so often the case with mathematical textbooks, so it is here : the book 

 vastly improves when the first few chapters are left behind, and would be still 

 further improved if they were left out. Prof. Forsyth's useful book does not now, 

 it seems, make many attempts to be an encyclopaedia or to be quite accurate ; 

 perhaps this is why it is a good textbook. 



Philip E. B. Jourdain. 



Infinitesimal Calculus. By F. S. CAREY, M.A., Professor in the University of 

 Liverpool [Section I. : Pp. xiv+ 144 + v, with diagrams, 1917 ; Section II. 

 Pp. x+ 145-352 + iv, with diagrams, 1918.] (London and New York: 

 Longmans, Green &' Co. Price 6^. net and 10s. 6d. net respectively.) 



This book is one of " Longmans' Modern Mathematical Series," and the first 

 section "deals with those parts of the Infinitesimal Calculus which have been 

 recently introduced into the syllabus of some examinations for higher school 

 certificates, while the two sections taken together correspond fairly closely to 

 the curriculum of students reading for the first part of an honours course in 

 mathematics or for the ordinary degree in arts, science or engineering " (p. v). 

 The first section passes through " those domains of number and function with 

 which the student is probably already acquainted, while the functions which are 

 likely to be unfamiliar to him have been reserved for the second section " (p. v). 

 In symbolism there are some good innovations in this book : there is a convenient 

 notation for open and closed ranges of real variables by square and round 

 brackets (p. 14), and arrows with a single (upper or lower) barb are used to 

 express convergence of the general term of a sequence down or up to a limit 

 (p. 24). This notation is a decided advance on the fully-barbed arrow now so 

 generally used to replace signs often involving the notion of " equality to infinity." 

 The most striking difference of this book from others is that "no attempt has been 

 made in the first section to deal with the definite integral, nor has the usual 

 notation for the indefinite integral been introduced until a comparatively advanced 

 stage" (p. vi), because of the impossibility of justifying the use of the usual symbol 

 for an indefinite integral as a representation of inverse differentiation (see p. no, 

 cf. p. 122) until the nature of a definite integral has been explained. 



"The book is not written for any particular group of students ; it is designed 

 for those who wish to use the Infinitesimal Calculus as an instrument in the 

 attainment of further knowledge" (p. vi) ; and yet more attention is paid to logic 

 than in the usual textbook : in fact, if we describe, as it seems that we may, the 

 object of any textbook to be to give, for teaching purposes, a judicious com- 

 promise between history and logic, this book must be accused of being logical. 

 Though it is on "infinitesimal" calculus, infinitesimals are banished (cf. p. 38). 

 This is a good book, and it is pleasing to see (p. 245) a treatment of integrals 

 giving mean values — an important notion in the theory of functions. 



Philip E. B. Jourdain. 



