REVIEWS 517 



sketches of mathematical progress are hardly full enough to make them valuable 

 for a learner, while such remarks as those on the honours and personal appearance 

 of Poincare (p. 272) can hardly be thought relevant or inspiring for a history which 

 should presumably be a history of a branch of thought. 



When dealing with the definitions that have been proposed for mathematics 

 (pp. 78-81), there is the usual failure to distinguish between attempts, such as that 

 of Russell, to define the logical nature of the subject-matter of mathematics and 

 attempts to describe the processes of thought used in mathematics. The mis- 

 understanding of Russell's definition, which was meant to describe humorously a 

 profound truth about implication and variables, is rather amusing : it may, I think, 

 be traced back to another American, Prof. J. W. Young. There are other state- 

 ments which seem to be open to criticism. On p. 92 we read : " In our own day 

 the theory of aggregates tends more and more to cease as a distinct subject " ; on 

 p. 96 John Bernoulli is said to have been the first to use the word function "with 

 its precise actual meaning " ; on p. 186 the fundamental theorem of algebra is said 

 to have been first proved by d'Alembert, " which [proof] was, however, not entirely 

 correct." D'Alembert, as a matter of fact, entirely missed the point of an existence- 

 theorem. On p. 207 the work of Boole on the early theory of invariants might 

 have been referred to. 



The reference to Hamilton's work on the groups of the regular polyhedra 

 (pp. 173-4) is very useful, and so is the list of references on the history of mathe- 

 matics in the nineteenth century (pp. 7-8). The remark about economy of thought 

 in mathematics (bottom of p. vii) is profound and true. Finally, surely there must 

 be some mistake in the Preface (p. ix) — written in October 191 5 — that "with our 

 gradual evolution from the state of barbarism the history of war and bloodshed is 

 being slowly replaced by that of political and intellectual movements." 



Philip E. B. Jourdain. 



(1) Guida alio Studio della Storia delle Matematiche. By Gino Loria, 



Professor at the University of Genoa. [Pp. xvi + 228.] (Milano : Ulrjco 

 Hoepli, 1916. Price 3 lire.) 



(2) Biblioteca Scientifico-Politecnica Internazionale : Bibliografia delle piu 



importanti opere italiane e straniere sulle scienze esatte, dell'ingegnere, le 

 arti belle e le arte utili, con speciale riguardo alia lore applicazione pratica 

 industriale, pubblicate dal 1905 al 191 5, disposte in ordine alfabetico delle 

 materie (con indice alfabetico degli autori). Proemio del Senatore Giuseppe 

 Colombo. By Ulrico Hoepli. [Pp. xxiv + 468.] (Milano : Ulrico 

 Hoepli, 1916. Price 3 lire.) 



(1) This seems to me quite one of the most useful for its purpose of the admirably 

 produced " Manuali Hoepli." Prof. Loria's idea, which was first published in 1908, 

 that it would be a good thing to compile a* manual for the use of those who wish 

 to make a study of the history of mathematics is here actualised in a satisfactory 

 manner. The first Part contains : (1) Generalities and references to works on 

 the historical method, the early history of philosophy and the arts, and literary 

 history ; (2) a critical list of the principal works on the history of mathematics ; 



(3) a summary of the contributions made in periodical literature. The second 

 Part contains details about manuscripts, biographies, bibliographies, and collected 

 works. A separate index of names for each Part is provided. 



We may shortly indicate some omissions. In Part I. the histories of Burnet, 

 Fink and Gow, and Brunschvicg's Etapes de la philosophic tnathe'matique might 



