REVIEWS 499 



second volume begins with Chapter XIII., on partial differential equations of 

 families of surfaces. This chapter has been divided into three chapters, and the 

 last two of these chapters deal respectively with rectilinear complexes, rectilinear 

 congruences, and ruled surfaces ; and with triply orthogonal systems of surfaces 

 and normal congruences of curves. These chapters contain much new matter. 

 There is also much new matter added to the subsequent chapters dealing with 

 cubic and quartic surfaces. The Chapter XVII. of the fourth edition was on the 

 general theory of surfaces, and this has been subdivided into two chapters, and 

 much of the chapters has been rewritten or added to. In this second volume Mr. 

 Rogers has had the very active co-operation of Mr. G. R. Webb, Miss Hilda P. 

 Hudson, and Mr. Robert Russell. Mr. Rogers has adopted the convenient plan — 

 rare in books printed in the English language — of providing two indexes, one for 

 subjects and the other for authors. 



Philip E. B. Jourdain. 



The Theory of Proportion. By M. J. M. Hill, M.A., LL.D., Sc.D., F.R.S., 

 Astor Professor of Mathematics in the University of London. [Pp. xx + 

 ioS.] (London : Constable & Co., Ltd., 1914. Price 8.y. 6d. net.) 



ALMOST all teachers agree that Euclid's treatment of proportion in the fifth book 

 of his Elements is so difficult, both in form and matter, that it is quite unsuitable 

 for purposes of teaching. Prof. Hill completely assents to this, but maintains that 

 a treatment of the theory of proportion, which is valid when the magnitudes con- 

 cerned are incommensurable, should be included in the mathematical curriculum. 

 It must be remembered that Prof. Hill has a very large experience of teaching 

 and is also author of an admirable Contents of the Fifth and Sixth Books of 

 Euclid's Eletnents and three important papers on the Fifth Book in the Cambridge 

 Philosophical Transactions. The third of these papers is noticed in this number 

 in " Recent Advances in Science : Mathematics," but the other two were published 

 some years ago. Prof. Hill arrived at the conclusion that, in addition to the 

 difficulties arising out of Euclid's notation and out of the fact that Euclid did not 

 sufficiently define ratio, two reasons could be assigned for the great difficulty of 

 his argument. In the first place, the only two of the many definitions in the Fifth 

 Book which effectively count are the test for deciding when two ratios are equal 

 (fifth definition), and the test for distinguishing between unequal ratios (seventh 

 definition). Further, Euclid takes the unnecessary course of deducing some of 

 the properties of equal ratios from the seventh definition. In this book Prof. Hill 

 only uses the fifth for this purpose. In the second place, Prof. Hill thinks that it 

 is very probable that the two assumptions : If A rB, then (A : C)r(B : C), where 

 the relation r may be either = or >, form the real bed-rock of Euclid's ideas, and 

 that he first of all deduced his fifth and seventh definitions from these two 

 fundamental assumptions (pp. vii-viii). In any case, the appearance of the above 

 definitions at the beginning of Euclid's argument and without explanation presents 

 grave difficulties to the student, which are avoided in this work. This work is a 

 modification of Euclid's method, which requires for its understanding a knowledge 

 of elementary algebra. 



Prof. Hill remarks (pp. x-xi) that Chapters I. -IX. contain an elementary 

 course and Chapters X. and XI. contain an advanced course. However, for those 

 who are more interested in what the book contains than in what parts of it are 

 suitable to be taught to whom, the following summary may be useful. The 

 first three chapters introduce "magnitudes of the same kind" as undefined entities 

 with which examples make us more or less familiar and of which characteristics 



