March 31, 1921] 



NATURE 



135 



to which no meaning has been givers The follow- 

 ing definition is free from this objection. 



" Def. I, A straight line is a line such that any 

 part of it, however placed, may be made to lie 

 upon any other part of it." 



Alas ! there are twenty-five words in this defini- 

 tion, and " no meaning has been given " to at least 

 twenty-one of them. It is here that trouble lies 

 for every beginner, and here lies also one reason 

 why the modern teacher has taken to experimental 

 .geometry. 



In a geometry which is essentially logical and 

 metrical we should like to find the mechanism of 

 measurement treated more fully. Dr. Davison 

 L does define "greater" and "less" in terms of 

 I "between" for segments of straight lines; but 

 this is scarcely satisfactory for angles (p. 4), 

 although thereafter the idea of magnitude is sup- 

 posed to be sufficiently known. Again, in the 

 subject of proportion incommensurables are given 

 only a little paragraph in small type (p. 186). It is 

 to be regretted that the author has not followed 

 Prof. M. J. M. Hill in giving adequate treatment 

 to this important notion. 



(2) There is now a great host of books on mathe- 

 matics for engineers, and most of them are bad. 

 Here is a good book. Mr. Dunkley's programme 

 is modest ; it covers the ground as far as the solu- 

 tion of triangles. The main text is clear and 

 sound, and it is incorporated with well-chosen ex- 

 amples of mechanisms which are fully worked out 

 and illustrated with excellent diagrams. Each 

 theme that is entered upon is followed through 

 to the finish in a way that will give mental satis- 

 faction to the student. Mr. Dunkley describes 

 himself as a machine tool designer, and is appar- 

 ently not a teacher. 



(3) These two unhappy volumes are in con- 

 trast with the last. The author sets himself 

 the task of covering the w^hole range of pure 



I mathematics from the beginnings of algebra to 

 ' volume integration, but it is difficult to see why the 

 book is "for engineers," as there is scarcely a 

 reference to engineering in the whole of it. The 

 ■quality of the mathematics may be judged by a 

 single citation (p. 97) : — 



"... for a series to be convergent the follow- 

 ing conditions must be satisfied. . . . 



" (3) The ratio of each term to the preceding 

 must always be less than i." 



(4) This book is considerably better than the 

 last. Much less ground is covered and there is 

 a large collection of relevant exercises, which is 

 the chief merit of the book. The text is not good. 

 There is the tendency, familiar in such books, to 



NO. 2683, VOL. 107] 



introduce advanced subjects too early, e.g. in- 

 finite series on pp. 28-32 before simple equations. 

 The treatment of the calculus is pedagogically 

 unsound. Thus a differential coefficient is ex- 

 plained graphically as the slope of a graph 

 (p. 150), but the authors say nothing about tan- 

 gents until the foot of p. 154. When a tangent is 

 at length introduced, it is explained as "the tan- 

 gent to a circle which most closely coincides with 

 the small portion of the graph on either side of the 

 point." The introduction of differentials (p. 158 

 seq.) without any explanation is to be depre- 

 cated. It becomes unpardonably loose later on : 

 " dy = ^{^x^) = x^. Hence the integral of x^.dx is 

 i^^"(p. 181). 



(5) " Applied mathematics " does not here mean 

 mechanics. The book is written " to meet the 

 needs of students who want some elementary 

 mathematical training that they can use in every- 

 day affairs." The chief, need of American stu- 

 dents appears to consist in having every problem 

 turned inside out, and may be judged by the fol- 

 lowing typical exercise (p. 74) :• — 



"Sold cotton on 5% commission, invested 

 the proceeds in sugar at 2% commission. 

 My whole commission was $210. What was the 

 price of the sugar and cotton? " 



We do not need to work out such problems this 

 side of the pond : we know the price of our 

 sugar before we start; we find it easier. 



There is, however, one thing in Prof. Webber's 

 book that we do need, and do not often find in 

 English books, and that is half a dozen pages on 

 statistics. 



(6) We like this book. The author calls it "a 

 supplementary text-book," and makes no effort at 

 completeness of exposition, but selects a number 

 of subjects which are not often satisfactorily dealt 

 with in ordinary text-books, and he dwells upon 

 them at length. His style has the naivete and 

 freshness of first discovery, and there is nothing 

 of that supercilious haste that one is accustomed 

 to expect in a book designed for engineers. 

 Although the author does not aim at completeness, 

 he achieves nevertheless a certain continuity and 

 unity. Among subjects that receive a chapter each 

 are : — Dimensional arithmetic, calculus notation, 

 the laws of equilibrium, and velocity diagrams. 

 The last chapter contains an illuminating discussion 

 of four or five examples of applied mechanics. 



(7) Mr. Landon's book combines a text on more 

 or less academic lines with well-selected examples 

 from engineering as it is taught in colleges. The 

 examples are clearly stated and neatly worked out, 

 but somehow the author does not seem to dwell 



