July 7, 1916] 



SCIENCE 



27 



with in a first course in the calculus. But few 

 of the problems have been worked out fully 

 and the devising of geometric figures has been 

 left to the student under the guidance of the 

 test he is using or of his instructor, but nu- 

 merous cautionary and directive explanations 

 are given clearly and concisely, usually at the 

 beginnings of the various problem lists. The 

 answers to typical exercises of each list are 

 given, but a large percentage of the problems 

 are unanswered. Such a collection of exer- 

 cises ought to make it practicable to teach the 

 elements of the calculus by means of lectures 

 or by means of thin books confined mainly to 

 a presentation of theory. 



There are two special reasons why the ap- 

 pearance of Professor Carmichael's beautiful 

 book should be noted in this journal. One is 

 that the subject treated has made a most ex- 

 traordinary appeal in all scientific times and 

 places. With the exception of geometry, 

 astronomy and logic, hardly any other tech- 

 nically scientific subject has better served to 

 tie together so many centuries, for interest 

 in it probably antedates the school of Pytha- 

 goras. The second reason is that a certain 

 long-outstanding problem of Diophantine 

 analysis has recently come to very popular 

 fame by virtue of the extraordinary prize of 

 $25,000 provided by the German mathemati- 

 cian Wollfskehl for its solution. The problem 

 is to prove the so-called Last Theorem of 

 Fermat (1601-1665) stated by him without 

 proof on the margin of a page of his copy of a 

 fragment of the " Arithmetica " of the Greek 

 mathematician Diophantos. The theorem is : 

 If n is an integer greater than 2 there do not 

 exist integers x, y, z, all different from zero, 

 such that x n -\- y n = z n . The prize, the offer 

 of which does not expire till September 13, 

 2007, will be awarded to one who proves that 

 the theorem is not universally true (if it is 

 not) and who at the same time determines all 

 values of n for which it is true. Long before 

 the prize was announced the problem engaged 

 the efforts of great mathematicians and thus 

 led to important developments in the theory of 

 numbers. Since the announcement thousands 

 of the mathematically innocent have assailed 



the problem. If these innocents could have 

 had access to such a book as Professor Car- 

 michael's where the nature of the problem is 

 explained and the present state of knowledge 

 regarding it is sketched, they might have been 

 deterred from wasting their time and that of 

 others. 



The rendering of such a service was not, 

 however, the author's aim. There is scarcely 

 another branch of mathematics in which the 

 results achieved in course of the centuries are 

 so special, fragmentary and isolated. Pro- 

 fessor Carmichael's aim was two-fold, namely, 

 to produce for beginners an introduction to 

 Diophantine analysis and to bring its frag- 

 mentary and scattered discoveries into organic 

 unity. And he has succeeded admirably. The 

 style is excellent. The content and scope of 

 the book are fairly well indicated by the 

 titles and lengths of its six chapters : Introduc- 

 tion, rational triangles, the method of infinite 

 descent (22 pages) ; problems involving a mul- 

 tiplicative domain (30 pages) ; equations of 

 third degree (20 pages) ; equation of fourth 

 degree (10 pages) ; higher equations, the Fer- 

 mat problem (17 pages) ; the method of func- 

 tional equations (9 pages). The theory thus 

 actually presented and the judiciously selected 

 exercises make the work available for private 

 reading as well as for a short university course 

 in the subject. 



Professor Miller's " Historical Introduction 

 to Mathematical Literature " grew out of a 

 course of lectures designed to supplement reg- 

 ular instruction. It thus employs a more or 

 less expansive style and seeks to be " synoptic 

 and inspirational " for such as may not lay 

 claim to much mathematical discipline. It is 

 guided by a highly commendable aim : namely, 

 to conduct the reader to commanding points of 

 view so that he may judge for himself whether 

 the fields he is thus enabled to glimpse invite 

 him to further exploration. The aim is pur- 

 sued with a notable optimism despite the 

 nation-wide depreciatory utterances of such 

 educational leaders and agitators as Commis- 

 sioner Snedden and Abraham Plexner. Pro- 

 fessor Miller believes that " shameless igno- 

 rance " of mathematics " does not represent a 



