694 SCIENCE PROGRESS 



It seems unnecessary to say more of a production which cannot be re- 

 garded as a serious contribution to the solution of the problems involved in the 

 presentation of geometry. One quaint feature deserves notice — the author has 

 a pleasant habit of quoting from the great writers. In this way many an ugly 

 blank in his pages is filled by encouraging aphorisms from Bacon, Emerson, 

 Aristotle, Dante, and Chaucer. 



C. 



Models to Illustrate the Foundations of Mathematics. By C. Elliott. 

 [Pp. viii + 116] (Edinburgh : Lindsay & Co. Price 2s. 6d. net.) 



The title of this book hardly expresses its contents adequately. The book deals 

 with the logical bases of mathematics, and " is intended to assist in bringing some 

 modern views on the Foundations of Mathematics within the scope of school 

 work." The models themselves are mentioned only accidentally, the body of the 

 text consisting of an exposition of mathematics from the standpoint of the ideas 

 of correspondence, classification, multiplexes, etc. Although it is not stated how 

 far the author has been successful in familiarising children with the subject-matter 

 of the book, the writer possesses such a firm grasp of his material and presents it 

 with such clearness that we are quite willing to believe that pupils have enjoyed 

 and benefited from such instruction. Evidence of success would not, however, 

 convince us of the wisdom of such a course of instruction ; a good teacher rises 

 superior to the system which he adopts. It is, however, unwise to dogmatise as to 

 the standpoint from which mathematics may in the future be expounded. Newton 

 and Euler would doubtless be amazed at much they would find in schools to-day, 

 and many a mathematician of the present time would rub his eyes very hard 

 indeed if introduced into a class in which the writer of this book was teaching the 

 foundations of his subject to a class of children. The one thing certain is that 

 the minds of children in the seventeenth century were very much the same as they 

 are in the twentieth, and Boole's dictum holds good for the twentieth as it will for 

 the thirtieth that "a premature converse with abstraction is fatal to a virile growth 

 of the intellect." It is the support of such an authority as Boole on such a subject 

 as this that leads me to expect that the development of mathematical teaching will 

 not follow the lines so lucidly laid down in this tract. 



C. 



Lecons de Mathematiques Generales. Par L. Zoretti, avec une Preface de 

 P. Appell. [In-8 (23-14) de xvi + 753 pages, avec 205 figures. Cartonne.] 

 (Paris : Gauthier-Villars, 1914. 20 fr.) 



It is always profitable to read the better class of French mathematical text- 

 books, because the spirit in which they are written is so different from our own ; 

 and such books are particularly interesting at present when, to judge from 

 M. Zoretti's preface, French mathematical teachers and writers are faced with the 

 same perplexities and doubts as ours. The author modestly says that his book 

 ne veut etre qu'un essai ; we should describe it as an attempt to cover the range 

 of mathematics which should be included in a liberal modern education. The 

 book differs in many respects from those brilliant iraites d'analyse with which we 

 are familiar, though it covers much of the ground usually occupied by such 

 treatises : it is wider in extent, as it includes the analytical portions of mechanics, 

 but it is more restricted, inasmuch as there is no attempt made to discuss number, 

 limits, or continuity. In such a book the choice, as well as the arrangement of 



