438 



SCIENCE 



[N. S. Vol. XXVI. No. 666 



College Algehra. By Charles H. Ashton 

 and Walter E. Marsh. New York, Charles 

 Scribner's Sons. 1907. Pp. ix + 279. 

 For more than a century after the inven- 

 tions of analytical geometry and the cal- 

 culus, mathematicians and physicists may be 

 said to have fairly rioted in applications of 

 these instruments to geometric, mechanical 

 and physical problems without concerning 

 themselves about the nicer questions of funda- 

 mental principles, cogency and precision. The 

 efforts of Euler, Lacroix and others to sys- 

 tematize results served to reveal in a surpri- 

 sing way the need of improving foundations. 

 Constructive work was not arrested by that 

 disclosure. On the contrary, new doctrines 

 continued to spring up and old ones to expand 

 and flourish. But a new spirit began to mani- 

 fest itself. Mathematics became increasingly 

 critical as its towering edifices more and more 

 challenged attention to their foundations. 

 Already manifest in the work of Gauss and 

 Lagrange, the new tendency, under the power- 

 ful impulse and leadership of Cauchy, rapidly 

 developed into a powerful movement. It was 

 the foundations of the calculus that were first 

 overhauled, and, while its instrumental eificacy 

 was greatly improved, the calculus was ad- 

 vanced from the level of a tool to the rank and 

 dignity of a science. Accordingly every 

 genuine university to-day oilers two courses 

 in the calculus : an elementary course designed 

 to equip the student with the calculus viewed 

 as an instrument for making rough investiga- 

 tions, and an advanced course designed to ac- 

 quaint him with the intimate structure of the 

 subtlest of the sciences and to qualify him to 

 use the calculus in the finest and exactest 

 thinking. The work of Messrs. Veblen and 

 Lennes deals with the calculus in the latter 

 conception of it. Their work has but a single 

 English rival, viz., " The Theory of Functions 

 of Eeal Variables," by Professor James Pier- 

 pont, which appeared in 1906. Prior to the 

 appearance of the latter work, an American 

 or English student of the modern critical 

 calculus had to depend upon such foreign 

 works as Jordan's " Cours d' Analyse " and 

 Stolz's " Allgemeine Arithmetik." The aim 



of such critical work being precision and 

 logical perfection, it tends at first to be prolix 

 and only at last succeeds in becoming concise. 

 The most conspicuous among the merits of 

 the work by Messrs. Veblen and Lennes is 

 the union of conciseness with rigor. This 

 union was effected by means of two principles 

 of economy. One of these is the happy defini- 

 tion of the all-important notion of the limit 

 of a function in terms of the notion of " value 

 approached." The other consists in the sys- 

 tematic employment throughout of a recently 

 established theorem in the modern doctrine of 

 assemblages (ensembles, manifolds, sets), 

 namely, the Borel theorem, so called after its 

 discoverer. The value of the book might be 

 improved by the introduction of more numer- 

 ous illustrative examples. 



The books by Professor Chandler and Pi'o- 

 fessor Osborne, as designed for the beginner, 

 have numerous English rivals. Professor 

 Chandler in this third edition of his book has 

 made some changes to bring the treatment and 

 content into fuller accord with modern de- 

 mands both of rigor and of utility. The basis 

 is laid in the doctrine of limits. The dif- 

 ferential notation is introduced at an early 

 stage, and, everywhere throughout, the reader 

 finds the abstract notions and processes 

 illuminated by simple applications to concrete 

 problems, chiefly of geometry. The closing 

 chapters afford an excellent introduction to 

 differential equations and mechanical integra- 

 tion. Taken all in all, it is one of the more 

 substantial books for the student of engineer- 

 ing, for whom it is primarily designed. It 

 is not one of those emasculated, merely " prac- 

 tician," works that some teachers and students 

 of engineering seem to crave. 



By introducing a chapter on series, by re- 

 arranging the order of topics, by the earlier 

 geometric illustration of the notion of deriva- 

 tive, and by the incorporation of physical and 

 mechanical applications, Professor Osborne 

 has amply justifled the revision of his well- 

 known book, though his decision to give several 

 (not obviously equivalent) deflnitions of the 

 differential instead of one can hardly fail to 

 annoy the instructor and confuse the pupil. 



