NATURE 



[November 22, 191 



information of those who have actually to handle 

 the objects illustrated, they at least adorn the 

 book. Another great feature is the number of 

 numerical illustrations of the formula?. Even if 

 the student does not properly understand the 

 formulae, or the physical principles on which they 

 depend, the g^uidance afforded by the numerical 

 illustrations will probably enable him to deal with 

 concrete cases. The book, in short, seems in- 

 tended for the man for whom facts are a necessity, 

 but reasons a luxury. If the form and contents 

 of the book were dictated, as one would naturally 

 suppose, by the wants -of U.S. military cadets, the 

 most natural inference is that when the cadet 

 commences the study of ordnance he does not 

 possess that knowledge either of mathematical 

 analysis or of the mathematical theory of elas- 

 ticity desirable for a critical study of the problems 

 presented by wire-wrapped guns and recoil springs. 



The author begins his treatment of wire- 

 wrapped guns by quoting from Lissak's 

 " Ordnance and Gunnery " formulse for the 

 strains and stresses in a hollow circular cylinder. 

 The first formula, as ill-luck will have it,' suffers 

 from a printer's error, Rf, for R„2. The differ- 

 ences between stresses and strains and the rela- 

 tions between them are not made altogether clear, 

 the expressions for the strains being multiplied 

 by E, Young's modulus, and there being no 

 explicit reference to Poisson's ratio, which is 

 tacitlv assumed to be 1/3. This, no doubt, 

 simplifies the mathematics, and a further simpli- 

 fication is effected by accepting a common value 

 of E for the forged steel of the tube, the steel 

 wire of the w^inding, and the cast steel of the 

 jacket. These materials are supposed to differ 

 only in their "elastic limits." These assumptions 

 may be necessary to bring the problem within the 

 powers of the average cadet, but there are, it is 

 to be hoped, superior cadets who would benefit 

 by having the limitations of the formulae pointed 

 out. It is to be feared that the reader will find 

 the way of reaching the formulas relating to the 

 elastic strains and stresses produced by wire- 

 wrapping rather a feat of jugglery. He is also 

 not unlikely to miss the fact that the inferences 

 as to elastic limits are generally based on a 

 greatest strain theory. 



The student who will derive benefit from the 

 treatment of elementary elastic problems given 

 on pp. 106-20 has not reached the stage of know- 

 ledge desirable when tackling wire-A\Tapped guns. 

 There is, moreover, no clear statement of prin- 

 ciples. Formula? are quoted from various sources, 

 apparently simply that thev may be available for 

 reference in connection with the numerical illus- 

 trations. No warning seems to be given as to 

 the risks in applyino- to short and irregularly 

 shaped beams formulae based on the Euler- 

 Bernoulli treatment of bendine. 



The treatment of helical springs in the last 

 chapter, though very arbitrary, seems fairlv satis- 

 factory so far as concerns snrinQ-s in which the 

 section of the orip-inal bar is circular: but the 

 extension to cases in which the section is rectan- 

 gular invites criticism. The formulae obtained 

 NO. 2508, VOL. 100] 



for the circular section involve a quantity I^, what 

 is called the "polar moment of inertia " (other- 

 wise 7rd*/32, where d is the diameter). The same 

 formulae are applied to springs coiled from bars 

 of rectangular section, h x h, with the following 

 explanation : " As first shown by Saint-Venant 

 ... a plane section whose axes are unequal 

 becomes a warped surface when subjected to great 

 torsional strain. . . . Reuleaux states that the 

 polar moment of inertia of a rectangle when sub- 

 jected to great torsional strain is 



I,= (/z&)3-f{3(/l2+b2)}, 



and that the distance from the centre of gravity 

 to the point of the section most distant from it 

 is r=}ih{h^-\-\fi)~-i.'' The author then inserts these 

 expressions for I,, and r in the formulae deduced for 

 the circular section. The student will naturally 

 infer that the " warping " appears only when the 

 torsional couple is large, and his ideas as to the 

 geometry of a rectangle must receive something 

 of a shock. The author does not seem well 

 advised in using the same letter E to denote the 

 rigidity and Young's modulus. 



A work which contains so much information 

 about U.S. ordnance, even if not the absolutely 

 latest patterns, and the methods employed by 

 U.S. ordnance experts will naturally appeal to an- 

 unusually wide circle at present. 



I ISAAC BARROW. 



I The Geometrical Lectures of Jsaac Barroiv. 



i Translated, with Notes and Proofs, by J. M. 



j Child. Pp. xiv + 218. (Chicago and London: 

 Open Court Publishing Co., 1916.) Price 

 45. 6d. net. 



• jV/f R. CHILD begins by laying down the 

 ^^ ^ startling thesis that "Isaac Barrow was 



I the first inventor of the Infinitesimal Calculus: 

 Newton got the main idea of it from Barrow by 

 personal communication ; and Leibniz also was 



I in some measure indebted to Barrow's work." 



I To interpret this according to the writer's inten- 



I tion we must use the term " calculus " to mean 

 a set of analytical rules applied to analytical ex- 

 pressions ; with this restriction, Mr. Child has 

 made out a case that is convincing enough in 

 this sense, that if Barrow had been given any 

 function likely to be constructed in his time, he 

 would have been able to differentiate it by apply- 

 ing a few standard rules. 



It is extremely interesting to read Barrow's 

 lectures, because they were written at a time 

 when the power of the new analysis was be- 

 coming apparent, whereas every mathematician 

 of note had been thoroughly grounded in the 

 classical geometry of the Greeks. Barrow makes 

 considerable use of algebraic symbols- — otherwise 

 we could only say that he generalised the methods 

 of Fermat and others ; even the fact that he 

 practically gives rules for differentiating a sum, 

 product, quotient, etc., would not make him the 

 inventor of the calculus. At the same time 

 Barrow's treatment is, in the main, geometrical, 

 and we feel that he would like to make it com- 

 pletely so, if he could. 



