REVIEWS 535 



cussion of fundamental assumptions has not been neglected, and the subject is 

 developed systematically, the one omission being the absence of any mention of 

 the existence-theorem for parallel lines, with the fruitful discussion of the 

 axioms upon which Euclid's definition is based. Some gain would have resulted 

 from following the traditional order in Chapter II., thus bringing the discussion of 

 trigonometry into closer relation with the simpler geometrical theorems and 

 postponing the discussion of the absolute. Though the arrangement of this part 

 of the book may be challenged, it could be defended. But no justification can be 

 imagined for the section which includes pp. 94-100. In these few pages the 

 author summarises pure projective geometry. The summary is introduced, it is 

 true, with an apology that "unfortunately most English text-books start by 

 assuming metrical geometry," but there are fortunately in the English language 

 excellent text-books which give a thorough treatment of the subject, and to two 

 of these reference is made by the author. It is a sad thing to find in such an 

 excellent book a belief expressed that a summary of six or seven pages can provide 

 any reader with a working knowledge of projective geometry. Surely if a writer 

 expounds the subject of which he treats coherently and consistently, he may 

 expect his readers to provide the necessary preliminary information. 



The book begins with a historical section which, coming from the pen of the 

 historiographer of the subject, is, as we should expect, admirable. Mathematics 

 contains in its history many romances, but the historians of few subjects have a 

 more fascinating chapter to write than that which tells of the discovery of 

 metageometry. Dr. Sommerville tells here the story of the two generations 

 of the Bolyai who worked in this field, he recalls the solemn adjuration of the 

 father who, foiled in his attempts to prove the parallel-postulate, took refuge in 

 poetry and bade his son avoid the loathsome subject : then he gives the 

 triumphant letter of the son in which he announces to his father the great 

 discovery which he has made. " I cannot say more now, except that out of 

 nothing I have created a new and another world." This new world is described 

 in Chapter II. In Chapter III., the world of elliptic geometry, even more 

 bewildering in its structure than Bolyai's, is explained — the geometry in which the 

 straight lines are of finite length. In the following chapter elliptic geometry is 

 developed and explained by analytical methods. This chapter is of great interest, 

 and the reader will probably leave it only with regret that it was not a little 

 longer, and that the author had not included some account of hyperbolic geometry. 

 Perhaps this subject was passed over because it receives in Prof. Liebmann's 

 Nicht-euklidische Geometrie such full treatment. In Chapter V. the representa- 

 tion of non-Euclidian geometry in Euclidian space is discussed, and leads up to 

 a chapter on space-curvature and the general philosophy of the subject. Here the 

 author has received expert advice from high philosophical authorities, but at the 

 end he shows true mathematical instinct in giving a quotation from La Science et 

 PHypothese, for the mathematician even when deaf to the charms of the most 

 cunning philosopher always listens to the greatest master of his subject in this 

 generation. After this the reader will be conscious of a certain bathos when his 

 attention is called to radical axes, homotethic centres, and the like. Interesting 

 as such elaborations of the theory are (and every reader should study the sections 

 on "marginal images " and "the conic ''), it may be suggested that some of these 

 discussions would have been better placed in small-print appendices. The 

 arrangement of the subject-matter is the weak side of the book. Students who 

 read Prof. Liebmann's book will feel the value of a carefully devised sequence in 

 the subject. There is no object in instituting a comparison between the two 



