500 SCIENCE PROGRESS 



are given in essentials after Stolz, to whose Allgemeine Arithmetik Prof. Hill's 

 work is greatly indebted. Then some propositions on their positive integral 

 multiples are given ; and the "ratio" of two such multiples of the same magnitude 

 is (p. 12) defined as an integer or fraction. Chapter IV. : By a proof probably due 

 to Pythagoras himself, it is shown that there are magnitudes of the same kind (the 

 side and diagonal of a square) which are not multiples of the same magnitude. 

 In such cases, can the magnitudes have a ratio to one another? (p. 19): "if so 

 there must be numbers which are not rational "(p. 21). Prof. Hill is evidently 

 more interested in the process of discovery of mathematical objects than in their 

 logical nature. Logically, Prof. Hill's method is like first defining "foreigners " as 

 Frenchmen and then asking if there are any non-French foreigners. But we 

 all know how easily one can make fun of the " extension of the idea of number," 

 which is described much as usual in Chapter V., and what important heuristic 

 methods badly expressed have been used in this "extension." We can all, with a 

 little sympathy, appreciate the value of the vague question : " Does anything exist 

 which is not a rational number, which is nevertheless entitled to be ranked as a 

 number?" (p. 24). And most of us will agree that " an argument which does not 

 follow the course of discovery is frequently very difficult to follow" (p. viii). 



Chapter VI. : Construction of a new theory of the ratio of commensurable or 

 incommensurable magnitudes which satisfies the conditions: If ArB, then 

 (A : C)r{B : C), where the relation r is either >, <, or =. " Two ratios are said 

 to be equal when no rational number lies between them " (p. 33) ; and the test for 

 equal ratios is that, if {A : B) r (pjq) then (C : D) r (p/q), where r has the above 

 meaning and p and q are unrestrictedly variable integers (p. 34). " Stolz's 

 theorem" (p. yi) is that the condition obtained when = is put for r is superfluous ; 

 and applications of this theorem in the method of exhaustions are dealt with in 

 Chapter X. Chapters VII. -IX. give properties of equal ratios; Chapter XI. 

 contains further remarks on irrational numbers, and includes a statement of the 

 Cantor-Dedekind axiom ; and Chapter XII. is a commentary on the Fifth Book 

 of Euclid. 



On p. x it is said that Dedekind acknowledged that he drew his inspiration 

 especially from Euclid's fifth definition (of the equality of ratios) of his Fifth Book. 

 This does not seem to be the case. In the passage referred to by Prof. Hill 

 (Beman's translation of Dedekind's Essays, Open Court Co., p. 40), Dedekind 

 merely remarks that the conviction that an irrational number is defined by the 

 specification of all rationals that are less and all that are greater than it was, put 

 in another way of course, at the bottom of Euclid's definition, and was the source 

 of Bertrand's and others' considerations and of his own theory. 



This is a book which should stimulate an intelligent student to research. 



Philip E. B. Jourdain. 



The General Theory of Dirichlet's Series. By G. H. Hardy, M.A., F.R.S., 

 Fellow and Lecturer of Trinity College, and Cayley Lecturer in Mathe- 

 matics in the University of Cambridge, and Marcel Riesz, Dr. Phil. 

 (Budapest), Docent in Mathematics in the University of Stockholm. 

 [Pp. vi + 78. No. 18 of the "Cambridge Tracts in Mathematics and 

 Mathematical Physics."] (Cambridge: University Press, 191 5. Price 

 3-r. 6d. net.) 



It was primarily with a view to applications in the theory of numbers that 

 Dirichlet first introduced into analysis the series called after him. The series are 

 infinite sums in which each term consists of a coefficient multiplied by e raised to 



