516 SCIENCE PROGRESS 



A Course in Mathematical Analysis. Differential Equations: being Part II. 

 of Volume II, By Edouard Goursat, Professor of Mathematics in the 

 University of Paris. Translated by Earle Raymond Hedrick, Professor 

 of Mathematics in the University of Missouri, and Otto Dunkel, Assistant 

 Professor of Mathematics in Washington University. [Pp. viii + 300.] 

 (Boston, New York, Chicago and London : Ginn & Co., 19 17. Price 

 1 1 j. 6d. net.) 



This volume consists of a translation of the second half of the second volume of 

 the Cours d' 'Analyse Mathimatique of Goursat, and the former parts of the English 

 translation have already been reviewed in SCIENCE PROGRESS (July 1917, p. 158). 

 The book under review will form a very useful text-book on the subject of differ- 

 ential equations from a fairly modern point of view. In the first chapter some 

 simple types of equations of the first order, and, more briefly, of higher order, are 

 considered, whose integration can be effected by quadratures ; and existence- 

 theorems are only considered in the next chapter. " If this order of procedure 

 seems subject to criticism from the point of view of pure logic, we may at least 

 observe that it conforms to the historical development of the subject " (p. 6). In 

 the chapter on existence-theorems, Cauchy's method of the calculus of limits is 

 first described, then the method of successive approximations, and lastly the 

 Cauchy-Lipschitz method ; there are then short and useful sections on first 

 integrals and multipliers, and on infinitesimal transformations. The third chapter 

 is on linear differential equations ; the fourth chapter is on non-linear equations ; 

 and the fifth and last chapter is on partial differential equations of the first order. 



Philip E. B. Jourdain. 



The Continuum and other Types of Serial Order. With an Introduction to 

 Cantor's Transfinite Numbers. By Edward V. Huntington, Associate 

 Professor of Mathematics in Harvard University. Second Edition. [Pp. 

 viii + 82.] (Cambridge, Mass., U.S.A. : Harvard University Press, 1917.) 



This exceedingly useful introduction to a most interesting part of modern 

 mathematics originally appeared in 1905 in the Annals of Mathematics, and the 

 present book is much added to and the references are brought up to date. The 

 seven chapters are on classes in general ; simply ordered classes or series ; discrete 

 series, and especially the type of the natural numbers arranged in order of 

 magnitude ; dense series, and especially the type of the rational numbers ; con- 

 tinuous series, and especially the type of the real numbers ; continuous series of 

 more than one dimension, with a note on multiply-ordered classes ; and well- 

 ordered series, with an introduction to Cantor's transfinite numbers. The most 

 notable addition to this edition is the account on pp. 77-9 of Hartog's theorem on 

 the well-ordering of sets, of which some account has been given in a former 

 number of Science Progress (July 1917, p. 7). It may be remarked that in this 

 edition, as well as in the preceding one, there is no mention that the " multiplica- 

 tive axiom ); is necessary to prove the theorem mentioned at the top of p. 66, nor 

 that the theorem that the square of every Aleph number leaves the number 

 unaltered is necessary for the proof that there is a series of Alephs. The fact also 

 that the multiplicative axiom also enters into the proof that there is such a series 

 is not mentioned. Indeed, much that comes out in the recent work of Whitehead 

 and Russell is left out of account, although the work of Whitehead and Russell is 

 referred to. 



Philip E. B. Jourdain. 



