684 SCIENCE PROGRESS 



A First Course in Higher Algebra. By Helen A. Merrill, Ph.D., Professor of 

 Mathematics in Wellesley College, and Clara E. Smith, Ph.D., Associate 

 Professor of Mathematics in Wellesley College. [Pp. xvi + 247.] (New 

 York: The Macmillan Company; London : Macmillan & Co., Ltd., 1917.) 



The novelty in this book is the fact that it is " an outgrowth of the conviction of 



the authors that Higher Algebra, to be worthy of the name, must employ advanced 



methods, and that the method which chiefly marks advanced work in analysis is 



that of limits." Thus, besides the usual chapters on the algebra of schools, we 



have a chapter on the elementary theory of integration as well as a chapter 



on the differentiation of algebraic functions. The authors state that the proofs 



are made "as rigorous as seems advisable for immature students," and that it 



is their hope that there is "nothing to be unlearned in later work" (p. vii). 



But surely we are all supposed to know now that logically the theory of 



irrational numbers precedes the theory of convergence, and that the theory of 



irrationals is used implicitly in the theorem of comparison of series (p. 103). Also 



I think that the student would have to unlearn the ideas of Kronecker (pp. I, 181) 



on the indefinability of the integers. The views of the authors that zero, negative 



numbers, and fractions are "symbols" (pp. 2, 3, 6) seem to be not wholly unrelated 



with Kronecker's views ; but complex numbers, constants, and variables are 



"quantities" (pp. 50, 169). The contrast between "it has been proved" and "it 



is true" on p. 154 does not seem to be quite wise, and there are analogous 



difficulties in the note on p. 115 and on p. 126. The introduction of determinants 



in the third chapter seems to be good, and so do certain of the remarks on the 



calculation of logarithms (pp 1 50- 52) and the beginnings of the theory of functions 



of a complex variable (pp. 182-4). At least these things are interesting, and that 



seems to be the main thing in education. There are good and interesting 



examples on pp. 87-90. 



Philip E. B. Jourdain. 



Finite Collineation Groups : with an Introduction to the Theory of Groups 

 of Operators and Substitution Groups. By H. F. BLICHFELDT, Professor 

 of Mathematics in Leland Stanford Junior University. [Pp. xii 4- I94-] 

 (Chicago : University of Chicago Press ; London : Cambridge University 

 Press. Price $1.50 or 6s. net.) 

 The University of Chicago Science Series owes its origin to a feeling that there 

 should be a medium of publication occupying a position between the technical 

 journals with their short articles and the elaborate treatises which attempt to 

 cover several or all aspects of a wide field. Thus, since the theory of finite 

 collineation groups (or linear groups) is at present to be found mainly in scattered 

 articles, in addition to a few books on group theory, Prof. Blichfeldt has given in 

 the present volume an outline of the different principles contained in these 

 publications, and has at the same time made an effort to depend upon a minimum 

 of abstract group theory. " In this and in many other respects the present 

 volume differs from Part II. of" the Theory and Applications of Finite Groups, 

 written by Profs. Miller, Blichfeldt, and Dickson, and published in 1916; "in 

 particular, the present volume contains more of the theory of linear groups" (p. vii). 

 The theory of linear groups of finite order may be said to have been originated by 

 Klein in 1876, and subsequently Klein extended the Galois theory of algebraic 

 equations by the intioduction of linear groups. At about the same time the 

 solutions of Schwartz, Fuchs, and Jordan of a problem connected with linear 

 differential equations were published : these solutions hinge upon the discovery of 



