January 20, 1905.] 



SCIENCE. 



101 



bors were elected. The association was 

 well represented at the dinner of the 

 naturalists and affiliated societies on the 

 evening of the twenty-eighth, and the next 

 night, following the president's address, 

 an enjoyable smoker was held in conjunc- 

 tion with the Psychological Association. 



H. N. Gardiner. 



SCIEyriFIC BOOKS. 

 An Iniroduction io the Modern Theory of 

 Equations. Florian Cajori. New York, 

 The Macmillan Company. 1904. Pp. ix + 

 239. $1.75 net. 



The present work falls into two nearly equal 

 parts. The first 103 pages treat the following 

 topics : Elementary properties and transforma- 

 tions of equations ; location and appi'oximation 

 of the roots of numerical equations; solution 

 of cubic, biquadratic, binomial and reciprocal 

 equations; the linear and Tschirnhausian 

 transformations. The remaining 120 pages 

 are devoted to substitution groups and Galois's 

 theory of the solution of algebraic equations. 



The work has much that may be praised; 

 in particular, its very moderate size, its choice 

 of topics, copious references for further study, 

 and a large number of illustrative examples 

 and problems. 



We mention now a few points which we 

 believe might be improved in a later edition. 



The definition of algebraic and transcen- 

 dental functions in § 1 is not quite satisfac- 

 tory. The author really defines explicit al- 

 gebraic functions, and the reader might easily 

 infer that all other functions were trans- 

 cendental. 



Would it not be well to give a mathematical 

 definition of continuity of a function in § 25? 

 The reader would then have a clearer idea of 

 the import of the theorem of this section. 



In § 26 the author assumes that a continu- 

 ous function which has opposite signs in an 

 interval must vanish in this interval. This 

 requires demonstration unless an appeal to 

 our intuition is allowed. If so, the demon- 

 stration the author gives, that every equation 

 has at least one root, might well be replaced 

 by a simpler one which rests on the property 



that a continuous function attains its ex- 

 tremes. 



In § 65 the author makes use of continued 

 fractions to prove the relation mb — na = 

 + 1 where m, n are relative prime. It seems 

 preferable, because more elementary, to prove 

 this by means of the algorithm of the greatest 

 common divisor. 



In § TO the assumption is made that nu- 

 merator and denominator of a symmetric 

 rational function are also symmetric. The 

 definition of incommensurable in § 53 might 

 be improved; we would also suggest the repre- 

 sentation of complex numbers by points and 

 not by vectors, as in § 22. 



Let us turn now to the second half of the 

 book which deals with Galois's theory. As 

 the author tells us, he follows the exposition 

 given by Weber. We must, however, in justice 

 to Weber, note that the latter's treatment is 

 not only more general, but is also free from a 

 lack of precision of statement which mars the 

 work under review and which is at times quite 

 provoking. 



The author restricts himself to equations 

 whose coefficients are either constants or in- 

 dependent variables ; why, we are unable to 

 see. Certainly not because a greater sim- 

 plicity is gained. 



But this restriction once made, the reader 

 should have clearly in view whether the coef- 

 ficients of the equation dealt with in a given 

 case are constant or variable. For results true 

 when they are variable may be false when 

 these coefiicients are supposed constant. We 

 regret to say the author is extremely careless 

 in this important particular. Thus in chapter 

 XL we are informed in a footnote that the 

 coefiicients in this chapter are variables. In 

 chapter XIII. we are left entirely in doubt; 

 yet the theorems of Exs. 1, 2, § 119, which 

 are used in a later chapter, may be incorrect 

 if the coefficients are not independent vari- 

 ables. 



This lack of explicitness is manifest in 

 other parts of the book, e. g., in the chapter 

 on cyclic equations. The casual reader might 

 well believe that the results established here 

 are true for all cyclic equations. This, how- 

 ever, is not the intention of the author, for in 



