January 30, 1914] 



SCIENCE 



175 



simple expedient of purifying the water 

 supply. 



Chahles E. Woodruff 



SCIENTIFIC BOOKS 

 Lectures on the Differential Geometry of 



Curves and Surfaces. By A. R. Forsyth. 



University Press, Cambridge, 1912. Large 



octavo. Pp. xxiii + 585. Price, $8. 



Professor Forsyth's skill and versatility in 

 the writing of mathematical treatises, already 

 proved by his well-known works on differential 

 equations and the theory of functions, is 

 again illustrated by this new volume, his first 

 in the field of geometry. The lectures were 

 delivered, in substantially their present form, 

 during the author's tenure of the Sadlerian 

 professorship at Cambridge. They make very 

 interesting reading. The style is graceful, 

 and the technical discussions are illuminated 

 by many passages on the history and develop- 

 ment of the special topics considered. 



Naturally no attempt is made to cover the 

 whole field of differential geometry. Not even 

 the classic four-volume treatise of Darboux 

 pretends to include all the applications of the 

 methods of the infinitesimal calculus to the do- 

 main of geometry. In particular, the author 

 omits all extensions to hyperspace and non- 

 Euclidean geometries. His main aim is to 

 " expound those elements with which eager 

 and enterprising students should become ac- 

 quainted," and to provide such students, who, 

 later, may devote themselves to original work 

 " with some of the instruments of research." 



The author restricts himself to curves and 

 surfaces in ordinary Euclidean space and uses 

 the direct methods introduced by Gauss. " I 

 have made no attempt to give what could only 

 have been a rather faint reproduction of Dar- 

 boux's treatment, which centers round the 

 tri-rectangular trihedron at any point of a 

 curve or surface or system. My hope is that 

 students may experience an added stimulus 

 when they find that different methods combine 

 in the development of growing knowledge." 

 It must be admitted that, by showing the 

 power of the more natural methods (combined, 

 of course, with typical Cambridge skill in 



analytical manipulation) in the solution of 

 extremely difficult problems, the author's pro- 

 cedure is amply justified. 



As regards logical rigor the work is about 

 on a level with the texts of Bianchi, Scheffers, 

 and Eisenhart. No attempt is made to lay 

 precise function-theoretic foundations for the 

 geometric structure which is erected. In par- 

 ticular the concepts of analytic curve and 

 surface, employed throughout the work, 

 are never formulated precisely. Professor 

 Study's vigorous criticism of the new edition 

 of Bianchi in this aspect applies in fact to 

 all standard treatises on differential geometry. 

 It must be confessed that the claims of rigor 

 are not emphasized in geometry to nearly the 

 same extent as in analysis. In this respect 

 geometry in fact occupies a position between 

 analysis and physics, and to that extent be- 

 longs to applied rather than to pure mathe- 

 matics. Study has himself outlined a proper 

 basis for the treatment of ajialytic curves,^ 

 but this has not yet been digested into a form 

 suitable for an introductory text, and the cor- 

 responding discussion of surfaces is still to be 

 undertaken. No doubt, in the future — how 

 near one can not say — Study's high and 

 beautiful ideal will become realized. Mean- 

 while, most geometers, at least when they 

 write on differential geometry, follow the 

 older and what they considered the most ex- 

 pedient approach. Perhaps a distinction 

 should be made, even in the domain of gradu- 

 ate mathematics, between pedagogic books and 

 logical books. The evolution toward a rigor- 

 ous treatment (never perfect, but at least up 

 to the highest standard of a given period of 

 mathematics) is obviously inevitable. 



As regards the introduction of imaginary 

 configurations in geometry the author follows 

 the traditional half-hearted policy of consider- 

 ing them only when it is convenient, or at 

 least traditional, to do so. Thus, in connec- 

 tion with a real surface, it is analytically 

 expedient to introduce certain curves, of 

 course imaginary, whose length (between any 

 two points) is zero. [These the author desig- 



1 In two memoirs published in the Trans. Araer, 

 Math. Sac, 1909, 1910. 



