28 



NA TURE 



{Nov. 8, 1888 



sent an open field for the writers of really able didactic 

 treatises. For those who have reached the Elysian fields 

 of the calculus, no better book can be recommended 

 than Prof. Minchin's admirable work ; while for those 

 who have not yet reached that stage, and perhaps never 

 intend to, the " Elementary Statics " by Mr. John 

 Greaves, which is written much on the same lines, can 

 be unreservedly recommended. The science of statics, 

 like everything else, has been obliged to mo7>e with 

 the times, and Mr. Greaves, following the modern 

 views, prefers to consider it as merely a particular 

 case of the science of dynamics, and to base it upon 

 the laws of motion. Thus, instead of the familiar 

 proof of the parallelogram of forces on the principle 

 of transmissibility of force, he deduces it solely by 

 the aid of the parallelogram of velocities, from which, 

 together with the third law of motion, the conditions of 

 equilibrium are obtained more readily, and, in the 

 author's opinion, more clearly, than usual. This ex- 

 pectation can only be tested by actual experience. Mean- 

 while we would recommend that, in a reprint of the book, 

 the more salient propositions and results should be ren- 

 dered more emphatic and conspicuous by being placed 

 either in italics or large type. In their present form and 

 posijtion the plums are too disguised to be readily picked 

 out. The work is characterized by thoroughness, and by 

 a large number of worked-out examples illustrated by 

 excellent figures, the material lines being very properly 

 distinguished from the geometrical and force lines by 

 thicker type. The free use made by the author of the 

 purely geometrical method for solving some statical prob- 

 lems is elegant, but occasionally leads to a neglect of the 

 statical or primary limitations under which they are 

 stated. An example of this occurs on p. 86, where it is 

 required to find the greatest inclination to the horizon at 

 which a uniform rod can rest partly within and partly 

 without a fixed smooth hemispherical bowl. The con- 

 dition assumed for the maximum inclination leaves no 

 fart of the rod outside the bowl, which clearly violates 

 the latter part of the question. 



Machines are deferred to the last chapter in the book, 

 presumably because some of the principles, such as 

 virtual work, are dealt with in preceding chapters ; but 

 we think they might be advantageously introduced, at all 

 events in a preliminary way, much earlier, since their 

 consideration not only enlivens the otherwise dry dis- 

 cussion of abstract principles, but gives concrete expres- 

 sion to their reception. We are glad to see that the 

 merits of this excellent little book are recognized by the 

 authorities of the Mason Science College, who recommend 

 it for one of their courses. 



At the threshold of the higher mathematics we find two 

 books on " Differential Calculus," which, though rivals, 

 will, we trust, often be found in company, since each 

 possesses certain merits and characteristics which dis- 

 tinguish it from the other. One is the well-known and 

 excellent treatise by Prof. Benjamin Williamson, F.R.S., 

 which has now reached a sixth edition. In this edition, 

 besides careful revision, a short discussion is added on 

 the elementary properties of solid and spherical har- 

 monics, which are so frequently employed in the higher 

 developments of electrical and optical theories. As 

 a former edition of the book has been fully noticed 



in Nature, we need only indorse the opinion then put 

 forward, that it is one of the best treatises on the subject 

 in our language. The other work, by Mr. J. Edwards, 

 formerly Fellow of Sidney-Sussex College, Cambridge, 

 is very different in style, and more elementary, in so far 

 as it is, according to the author's design, "unencumbered 

 by such parts of the subject as are not usually read in 

 Colleges and schools." Compared with Prof. William- 

 son's treatise, it is distinctly more geometrical in method, 

 and in this and some other points, such as large type, 

 beautifully-drawn figures, an unusually full and systematic 

 account of curve-tracing and the properties of curves, 

 which, contrary to the usual custom, precede maxima 

 and minima, it is more suited to the wants of the average 

 student as a preliminary course of reading. Some of the 

 geometrical illustrations, such as those of the compressed 

 form of Taylor's theorem, <^(a- -|- h) = <^[x) -j- h<\)'{x -f 6h)^ 

 dz , . dz 



and V)z 



, dx ■\- — dy, are very elegant, and help to 



keep alive the real meaning of differential symbols, which 

 a too exclusive attention to algebraic analysis tends to 

 annihilate. Symmetry and brevity have both been evi- 

 dently studied, and a good example of this may be seen 

 on p. 271, where the radius of curvature . for an implicit 

 function of x and y is deduced. If this be compared 

 with the analogous method on p. 290 of Williamson's 

 book, the difference in the style will be manifest. Re- 

 garding the two books together, we should advise a 

 student to begin with Edwards, and then proceed with 

 WiUiamson. Nothing in the former work need be 

 omitted at a first reading, after which he may plunge 

 fearlessly into the more complete and analytical treatise 

 of the Dublin Professor. 



Two books remain to be noticed, which lie somewhat 

 outside the ordinary run of didactic works. One is an 

 Algebra by Profs, OHver, Wait, and Jones, of Cornell 

 University, U.S. This, though originally intended as a 

 text-book for their own students, seems, in the course of 

 construction, to have developed into a work which, while 

 it might be found really useful as a book of reference to 

 teachers and the rare youth who cultivates mathematics, 

 is quite unsuited to the ordinary student. 



In some respects it appears to be an effort to regard 

 algebra from the modern point of view as the science of 

 finite operations, and to present it in the form of " a 

 stepping-stone to the higher analysis," and there is much 

 that is commendable from this point of view in the 

 exposition of incommensurables, limits, imaginaries, 

 derivatives, and graphic representation of equations, as 

 also in the introduction of some fresh symbols, such as 

 the Gaussian sign of congruence, =. and -^ •^ for 

 smaller than and greater than in the sense of size only. 

 The use of the signs + - as left-hand indices to indicate 

 absolutely positive and negative quantities is also an 

 improvement, and renders it easier to deal with negative 

 and directional quantities. For English didactic pur- 

 poses, however, this book will be chiefly useful as one of 

 reference for the teacher. 



In conclusion, we must not omit to draw attention to a 

 very handy little manual of " Practical Solid Geometry," 

 by Major Gordon Ross, of the Royal Military Academy, 

 Woolwich, which is particularly adapted to military 

 students. The method of orthographic projection by 



