534 SCIENCE PROGRESS 



has worked at improving this book. English mathematicians are deeply indebted 

 to their American colleagues for many excellent translations of German books, 

 and this last contribution is by no means the least under which we are placed. 

 Unfortunately, in England an ignorant dogmatism about the uselessness of 

 translations has suppressed many attempts to render into English some of the 

 best German books ; this policy has placed difficulties in the way of young 

 students, who are deterred often from reading German books by the technical 

 terms for which they cannot find the English equivalent terms, even if such 

 exist. 1 hope that Professor Rasor will add to the debt which we all owe him 

 by continuing the series of translations and giving us at least another volume 

 of Dr. Burkhardt's Vorlcsungen. It is needless to describe the merits of the 

 original work, which are well known. Professor Rasor has added to the book 

 collections of examples which will be welcome to the teacher ; they are well 

 chosen and apposite ; many of them are original and skilfully devised. 



C. 



The Elements of Non-Euclidean Geometry. By D. M. J. Sommerville, M A. 

 D.Sc. [Pp. xvi + 274.] (London : G. Bell & Sons, Ltd., 1914. Price $s.) 



The book which Dr. Sommerville has written upon non-Euclidian geometry is an 

 extension of lectures delivered in 1913 at the summer colloquium arranged by 

 the Edinburgh Mathematical Society. These courses, if estimated by their first- 

 fruits here presented, must be of great value. The book itself is one of a series 

 of mathematical books for schools and colleges published by Messrs. George 

 Bell & Sons. The author in a preface seems to justify its presence in such a 

 series by expressing a hope that the book may prove serviceable to scholarship 

 candidates. This expectation belies either the book or its hypothetical bene- 

 ficiaries. It should be regarded rather as addressed to the class of readers to 

 whom the lectures were originally delivered : it is to mathematicians and to 

 teachers of mathematics that the subject of metageometry appeals. The school- 

 boy has other tasks, and it is an error to place before an immature judgment 

 subjects which demand philosophical insight. The book really concerns the 

 school-boy in quite another way ; for the study of this subject by the masters who 

 lay down his course of ordinary geometry cannot fail to have direct beneficial 

 effect. It is unfortunate that we have had to wait so long for such a book as 

 Dr. Sommerville here offers ; the book might have been written fifteen years ago, 

 and in that case would have exerted a powerful influence in the movement which 

 has ejected the elements of Euclid from its firm position as the foundation of our 

 geometrical instruction, and has yet done so little to rebuild where it has pulled 

 down. Geometrical reformers have too often lacked the respect for the genius of 

 Euclid which they might have acquired from a study of non-Euclidian geometry. 

 It is hard for those of us who were taught " Euclid " in our school-days, with the 

 clear, logical sequence of its propositions, to realise the lack which is now felt by 

 those who have been trained upon modern substitutes. Many of this later school 

 will, on reading Dr. Sommerville's book, be sensible of the loss which they 

 have sustained in studying geometry in books whose claims are based solely 

 upon either utility or elegance. 



The purist may with some justice find fault with Dr. Sommerville's present- 

 ment of the earlier portions of the subject ; this weakness is almost inherent in 

 a book which is an elaboration of lectures. The author himself is conscious of 

 this joint in his armour, and refers to another book those students who wish to 

 study the development of the subject from a set of axioms. However, the dis- 



