REVIEWS 



MATHEMATICS 



Elementary Mathematical Analysis. By John Wesley Young, Professor of 

 Mathematics, Dartmouth College, and Frank Millett Morgan, Assistant 

 Professor of Mathematics, Dartmouth College. [Pp. xiv + 548.] (New 

 York: The Macmillan Company ; London : Macmillan & Co., Ltd., 191 7. 

 Price us. net.; 



The purpose of this book is to present a course suitable for students in the first 

 year at American colleges, universities, and technical schools. " It presupposes 

 on the part of the student only the usual minimum entrance requirements in 

 elementary algebra and plane geometry " (p. v). The most interesting feature is 

 that more emphasis is " placed on insight and understanding of fundamental 

 conceptions and modes of thought, and less on algebraic technique and facility of 

 manipulation" (p. v). A good example of this will be found on p. 259; after 

 some instructions are given about the working of the slide-rule, we read : " These 

 rules are not to be memorised. They will be used almost instinctively by one who 

 has made the reason for each rule thoroughly clear to himself and who is in 

 practice." " The concept of functionality and the mathematical processes 

 developed for the study of functions are precisely the things in mathematics that 

 have most effectively contributed to human progress in more modern times ; and 

 the thinking stimulated by this concept and these processes is fundamentally 

 similar to the thought which we are continually applying to our daily problems " 

 (p. viii). Thus : " This course in mathematics is primarily concerned with the 

 study of certain of the simpler kinds of functions and their applications " (p. 2). 

 The conception of function is introduced at the very beginning and treated with a 

 great wealth of illustration ; then certain elementary functions are considered, 

 such as simple algebraic, trigonometric, and logarithmic functions ; then there is a 

 chapter on numerical computation. The third Part is on applications to geometry 

 and constitutes a fairly complete elementary course of plane analytical geometry. 

 The fourth Part is on miscellaneous algebraic methods, and includes much of 

 what we in Great Britain understand by " higher algebra " and the " theory of 

 equations." The fifth Part is on functions of two variables and solid analytic 

 geometry. The book ends with several useful tables. 



A feature of the method of exposition is the continual interruption of the text 

 by the word "Why?" (see, for example, p. 61), and this is not unlikely to 

 stimulate inquiry. The successive treatment of " continuous " functions from 

 a very intuitional basis (pp. 19, 102, 451) seems quite good, and the treatment of 

 logarithms (pp. 220-35) seems excellent with the exception of the mention of the 

 number e on p. 224. It would possibly have been an improvement to have 

 introduced on p. 34 a simple proof that the measure of the diagonal of a unit 

 square is incommensurable. This is an excellent working text-book of the same 

 nature as Mr. Hardy's book, but far more elementary. 



Philip E. B. Jourdain. 



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