Jan. 17, 1889] 



NATURE 



267 



their work, and of the amount contributed by each of 

 them to the advancement of the science. 



The remaining portion — about half— of the book is 

 divided into six chapters (numbered XIV. to XIX. inclu- 

 sive), in which the history of modern mathematics is briefly 

 considered. These arc so full of great discoveries and 

 illustrious names that they must be read to be appreciated. 

 We can only, in the limited space at our disposal, quote 

 their titles and add some remarks. 



Chapter XIV. "Features of Modern Mathematics." 

 In this chapter, which is a sort of summary of the other 

 five, we read that " five distinct stages in the history of this 

 period can be discerned." Turning to the table of con- 

 tents, we find the five stages thus described : (i) " inven- 

 tion of analytical geometry and the method of indivisibles," 

 (2) " invention of the calculus," (3) " development of mech- 

 anics," (4) "application of mathematics to physics," (5) 

 " recent development of pure mathematics.' The mere 

 remark that each of these might be made the title of a 

 bulky volume, will show at once the enormous extent and 

 importance of modern mathematics. 



Chapter XV. " History of Mathematics from Des- 

 cartes to Huygens." The principal names in this chapter 

 are Descartes, Cavalieri, Pascal, Wallis, Fermat, Barrow, 

 and Huygens. In many of their writings may be found 

 the germs of those ideas which have since been developed 

 in the infinitesimal calculus. Especially would we men- 

 tion Cavalieri's method of indivisibles, of which our in- 

 tegral calculus is the modified descendant, and Barrow's 

 method of drawing tangents to curves, substantially the 

 same as that given at the beginning of any modern differ- 

 ential calculus. Full explanations of both methods may 

 be found in the present chapter. 



The history of modern mathematics dates from the 

 publication of the " Geometric ' of Descartes, and we 

 wish to call attention to a bibliographical point connected 

 with it. M. Marie (" Histoire," &c., t. iv. p. 20) speaks 

 of " quatre trait^s separ^s : ' Le Discours de la Methode,' 

 ' La Dioptrique,' ' Les Met^ores,' et ' La Geometric,' " all 

 of them published in 1637 ; Mr. Ball (p. 241) says that 

 " Descartes's researches in geometry are given in the 

 third section of the ' Discours.' " We cannot positively 

 say which is correct, but our impression is that we have 

 seen a copy of the separately-published " Geometric." 

 The point is of small importance, but it should be cleared 

 up in subsequent editions. 



Chapter XVI. "The Life and Works of Newton." 

 There are two sections— one devoted to the life, the other 

 to an analysis of the works, of our Enghsh Archimedes ; 

 his three capital discoveries — fluxions, the decomposition 

 of light, and universal gravitation — will occur to most of 

 our readers. Most of the well-known facts relating to 

 Newton's private and public life are mentioned in this 

 chapter, together with some others that have only recently 

 come to light. 



Chapter X\TI. "Leibnitz and the Mathematicians of 

 the First Half of the Eighteenth Century." The following 

 sentence occurs in the opening paragraph : — 



" Modern analysis is, however, derived directly from 

 the works of Leibnitz and the elder Bcrnouillis ; and 

 it is immaterial to us whether the fundamental ideas 

 of it were obtained by them from Newton, or discovered 

 independently." 



It forms a fitting sequel to the tale told in the preceding 

 chapter of the celebrated controversy between Newton 

 and Leibnitz. 



The present chapter is in three sections : (i) " Leibnitz 

 and the Bei nouillis," (2) " The Development of Analysis 

 on the Continent," (3) "The English Mathematicians of 

 the Eighteenth Century." The two greatest French 

 names in the chapter are those of Clairaut and D'Alem- 

 bert ; the two greatest English ones, those of Taylor and 

 Maclaurin. Matthew Stewart succeeded Maclaurin as 

 Professor at Edinburgh, and was " almost the only other 

 British writer of any marked eminence in pure mathe- 

 matics during the eighteenth century." After recounting 

 his chief works, our author proceeds to say : — 



" These prove him to have been a mathematician of 

 great natural power, but, unfortunately, he followed the 

 fashion set by Newton and Maclaurin, and confined 

 himself to geometrical methods." 



This sentence gives the history, in epitome, of the decline 

 and fall of British mathematics in the last century. 



Chapter XVIII. "Lagrange, Laplace, and their Con- 

 temporaries." There are four sections : (i) "The Deve- 

 lopment of Analysis and Mechanics," (2) " The Creation 

 of Modern Geometry," (3) " The Development of Mathe- 

 matical Physics," (4) " The Introduction of Analysis into 

 England" The greatest foreign name in this chapter 

 (we single it out from a number of others) is that 

 of Euler ; the greatest English one is possibly that of 

 Thomas Simpson, who seems to be rather harshly treated 

 by being alli)tted only three lines in a footnote, whens 

 others of less ability are noticed in the text. 



In Section 4 we read : " The introduction of the nota- 

 tion of the differential calculus into England was due to- 

 three undergraduates at Cambridge — Babbage, Peacock,, 

 and Herschel — to whom a word or two may be devoted."" 



Doubtless the success of the movement was largely due 

 to their efforts, but the initiative was taken by Wood- 

 house in 1803 (see J. W. L. Glaisher on the "Tripos," 

 Proc. Lond. Math. Soc, vol. xviii. p. 18). The name of 

 Woodhouse is surely as deserving of mention as the other 

 three. 



Chapter XIX. "Recent Times." The author begins 

 with a long list of names well known in the mathematical 

 world. This list, however, " is not and does not pretend 

 to be exhaustive." He then classifies the writers he has. 

 enumerated " acccrding to the subjects in connection with 

 which they are best known, arranging the latter in the 

 following order: elliptic and Abelian functions, theory oi' 

 numbers, higher algebra, modern geometry, analytical 

 geometry, analysis, astronomy, and physics." 



The section on the theory of numbers is, in our opinion,^ 

 the best. It contains biographies of Gauss and the late 

 Prof. Smith (about four pages being allotted to each), 

 and mentions the researches of Cauchy, Liouville, 

 Eisenstein, Kummer, Kronecker, Hermite, Dedekind, 

 and Tchebycheff. 



W^e may now say with old Martial — 



" Ohe jam satis est, ohe libelle : 

 Jam pervenimus usque ad umbilices." 



But we have yet to record the impression left by the 

 perusal of the entire work. The most desirable thing in 

 a book of reference is that the reader should be enabled " 



