May 6, 1922] 



NATURE 



575 



ential calculus. The treatment is almost without 

 blemish and is so simple and clear that the beginner 

 should have no serious difficulty.' Chaps. 1-4 intro- 

 duce only algebraic functions ; chaps. 5-8 are con- 

 cerned with trigonometric and exponential functions 

 and the corresponding inverse functions. The treat- 

 ment of infinitesimals and differentials in chap. 5 is 

 specially to be commended. The author has one or 

 two hobby-horses. One that he ought not to have 

 ridden here is the denial of the existence of " in- 

 finity." He says (p. 27) : " We should not read * Z 

 approaches infinity/ . . . but ' Z becomes infinite ' ; 

 , . . the statement sometimes made that ' Z becomes 

 greater than any assignable quantity ' is absurd. 

 There is no quantity greater than any assignable 

 quantity." This last remark contains a certain mis- 

 understanding, and, in any case, such subtleties are 

 not suited to beginners. Among minor points it is 

 curious to note that there is no definition of a limit in 

 the book. Some proof, or at least a reference, should 

 be given for the proposition quoted on p. 113 : "A 

 convex curved line is less than a convex broken line 

 which envelops it and has the same extremities." 



(2) It is a pity that authors who do not " make 

 reference to difficulties which seldom arise in the 

 minds of elementary students " generally manage 

 to make their subject so uninteresting. We do not 

 ask for proofs. Mr. Sydney Jones effectually sup- 

 presses such an unreasonable desire by putting the 

 word " proofs " in inverted commas in his preface. 

 But could we not have a little colour .? Let us take 

 MacLaurin's theorem as a sample (p. 158). Mr. 

 Jones says : " Assuming that a function /(x) can 

 be expanded in positive integral ascending powers 

 of X ... . 



/(x) = Ao-i-AiX-HA2x7i.2+ . . . , 

 to determine the coefficients A(„ Aj, Ag, . . ." He 

 then differentiates the series and determines the co- 

 efficients, as if this were a most ordinary and most 

 dull proceeding. His pupils no doubt wonder vaguely, 

 learn their lesson by rote, and pass on. If he would 

 only pause to tell them what a wonderful theorem 

 this is, or point out how great are the assumptions 

 he is making, it would be well worth the space. Judg- 

 ing the book from the author's own point of view, 

 there is little to find fault with in it. But he should 

 not call a differential coefficient a " differential " (p. 

 18, etc.). 



(3) This misuse of the word " differential " is a 

 bad habit that appears to be gaining ground. The 

 authors of " A First Course in the Calculus " are also 

 addicted to it (preface and p. 215). Their text-book 

 is mainly manipulative ; it contains the usual treat- 

 ment of the infinitesimal calculus, and concludes with 



NO. 2740, VOL. 109] 



a chapter on differential equations. The proof, 

 depending on the area of a circular sector, for the 

 limit of sin OjO (p. 181), is open to the objection that 

 students are generally taught to use the limit in 

 question for finding this area. It is not necessary to 

 use a formula for an area at all (see, for example, Levett 

 and Davison's " Plane Trigonometry," p. 82). Many 

 mathematicians would be pained by the author's 

 statement on p. 343 : " If we proceed indefinitely, 

 taking only a fractional part of a given object, it is 

 perfectly plain that the fractional portion will soon be 

 very small indeed." There is no doubt that an in- 

 telligent person can convince himself that the limit of 

 x" is zero, if x is less than unity, but a more exacting 

 logician demands a proof of the proposition. The 

 authors hope that " the student will have nothing 

 to unlearn if he afterwards . . . proceeds to a rigor- 

 ous course of modem analysis." But it would be 

 a pity if he learnt to regard analysis as the proving 

 of the " perfectly plain." 



-' (4) Mr. Gheury de Bray calls his book a " little 

 brother " of '* Calculus Made Easy." We do not 

 know whether the late Prof. S. P. Thompson would 

 have been pleased with this facetious httle relative. 

 The only portion of the work that we can unreservedly 

 recommend is a long preliminary quotation from 

 Henri Fabre (pp. 1-12). There follow part i on 

 indices, binomial series, etc., and part 2 on the ex- 

 ponential series, the equiangular spiral, the hyper- 

 bola (because its area is a logarithmic function), the 

 catenary, the parabola (because it resembles a catenary), 

 the probabihty curve, and " exponential analysis." 

 The method consists in " talking round " the subject ; 

 something may be said for it, but it requires skilful 

 handling, and in this author's hands it is often long- 

 winded and obscure. The unwary should be warned 

 that the method, which is stated on p. 55, is not 

 " mathematical induction," but a kind of sampling ; 

 the statement on p. 147 that the centroid of a catenary- 

 arc is its middle point is, of course, incorrect. The 

 last chapter is interesting, but too difficult for any one 

 who would care to read the rest of the book. In sum 

 the author had an excellent idea, which he has not 

 quite managed to reaUse. 



(5) " Mathematics for Technical Students " is 

 designed for the first two years' work following on 

 an elementary school course. The treatment is apt 

 to be rather too formal in places — Mr. Forrest teaches 

 algebra in the old style like a game of patience with 

 jc's and y's for playing cards, and only hints that 

 algebra has something to do with the workaday 

 world after his pupil has learnt to play the game. 

 The treatment is, of course, still quite defensible, but 

 it is now generally thought better to reverse this order 



