July ^, 1878] 



NATURE 



247 



RECENT MATHEMATICAL WORKS 



An Elementary Treatise on the Dynamics tf a System of 

 Rigid Bodies^ with Numerous Examples. By E. J. 

 Routh, M.A., F.R.S., F.R.A.S., F.G.S. Third Edition, 

 Revised and Enlarged. 8vo, pp. 564, xii. (London : 

 Macmillan and Co., 1877.) 



A Treatise on Statics, containing some of the Fitnda- 



■ mental Propositions in Electro-Statics. By G. M. 

 Minchin, M.A. 8vo, pp. 450, xii. (Longmans, 1877.) 



Lectures on the Eleinents of Applied Mechanics, com- 

 prising (i) Stability of Structures j (2) Strength of 

 Materials. By M. W. Crofton, F.R.S. Printed for 

 the Use of the Royal Military Academy. Pp. 107. 

 (C. F. Hodgson and Son, 1877.) 



Handbook of Natural Philosophy — Mechanics. By D. 

 Lardner, D.C.L. New Edition, Edited and consider- 

 ably Enlarged, by B. Loewy, F.R.A.S. Pp. 489, xxii. 

 (Crosby Lockwood and Co., 1877.) 



The Book of Mechanics. Part I. — Statics. By R. Oscar 

 Thorpe, M.A. (Stewart's Local Examination Series, 

 1877.) 



THE main features of Mr. Routh's admirable treatise 

 are well known to students. The first edition, of 

 336 pages, appeared in i860 ; the second, of 492 pages, 

 in 1868 ; the present consists of 564 pages, each page 

 containing from a third to a half as much matter more 

 than the page of the earlier editions. Some idea may thus 

 be formed of the great amount of new matter. Of this 

 increase take another proof: in the second edition the 

 chapter on Small Oscillations took up pp. 273 to 322 ; in 

 this edition the subject occupies pp. 325 to 403 ! The 

 author assigns as a reason for this increase, " I have been 

 led " to make these additions " because there are so many 

 important applications which it did not seem proper to 

 pass over without some notice." 



An interesting feature is the increased number of his- 

 torical notices, though these are confessedly very slight, 

 drawn from Montucla (by a misprint Montuela), Prof. 

 Cayley's Report on Theoretical Dynamics (British Asso- 

 ciation Report, vol. xxvi.), and other sources. Some of 

 these are relegated to an appendix. A great number of 

 original memoirs have been consulted and some of these 

 of very recent date. We do not notice in Articles 282, 

 475, in which a discussion of the problem of Laplace's 

 three particles is given, any reference to the author's 

 paper on the subject in the Proceedings of the London 

 Mathematical Society (vol. vi. No. 81, pp. 86-97), though 

 of course the substance" of this paper is given in the text. 

 We note this, because in both places Mr. Routh cites a 

 reference, by M. Jullien, to a Thfese de M^canique, by 

 M. Gascheau, which he has not succeeded in verifying. 

 Perhaps a notice of this point in Nature may lead to 

 the matter being cleared up for Mr. Routh's satisfaction. 

 We have not ourselves met with this pamphlet by M. 

 Gascheau. We could dwell much longer on this fine 

 work, pointing out the numerous places where new proofs 

 are given and entirely new matter is introduced, but we 

 need only say that it must claim a very high place in our 

 mathematical literature, and go far to remove the re- 

 proach brought against Cambridge text-books by students 

 who have become familiar with the works of continental 

 mathematicians. There is an ample and diversified col- 



lection of problems which are given in the several chap- 

 ters and appended to them. Following a common practice, 

 the author gives a list of articles to which beginners should 

 first turn their attention. 



Prof. Minchin purposely omits the prefix " Elementary," 

 his main object being to give " a tolerably comprehensive 

 view of statics." Very early in his book he introduces 

 the conception of " virtual work," a term he adopts from 

 the best French writers (Collignon, Delaunay, and others) 

 in preference to "virtual velocities." His reason for 

 bringing the subject so soon before the student is "the 

 conception of work is the most prominent in modem 

 physics, and, therefore, at the risk of being charged with 

 prolixity, I have shown in the earliest chapters how all 

 the conditions of equilibrium of a system may be obtained 

 from the principle of virtual work independently of the 

 usual mode of the reduction of forces." Graphic methods 

 are used in the earlier portions ; a good feature, now that 

 the treatises of Culmann, Bauschinger, and Cremona are 

 in the hands of many English students. The subjects 

 treated of are much the same as in other treatises in our 

 hands, and the last chapter (pp. 403 to 450) is devoted to 

 the theory of the potential ; the modes of treatment^ 

 however, are different. 



Prof. Minchin attaches great importance to the solution 

 of problems, and so takes care to solve a great many 

 leading cases, and has done good service to students by 

 these solutions and by the figures which he gives. The 

 following remarks speak for themselves : — " It is charac- 

 teristic of the system of * cramming,' which has been 

 called into existence by modem competitive examinations, 

 that the applications of mathematics, as exhibited in the 

 solution of examples, are greatly neglected. A cause con- 

 tributing to this objectionable system appears to me to 

 exist in our mathematical treatises, many of which are 

 almost wholly filled with unsolved problems and dry 

 ' book- work,' which the student never learns to apply. I 

 have therefore very largely illustrated the principles of 

 the subject by solved examples, and I have attached at 

 least as much importance to examples, all through, as to 

 the abstract principles which they illustrate." 



We cordially commend the book, and hold that it is no 

 unworthy companion of such text-books as those of Dr. 

 Salmon and of Mr. Williamson. 



Prof. Crofton's book is a " Synopsis of a Course of Lec- 

 tures on the Elements of the Theory of Structures and 

 the Strength of Materials, forming the First Part of the 

 Course of Applied Mechanics at present studied by the 

 Gentlemen Cadets of the Royal Military Academy." The 

 book requires to be read with some care, as the author's 

 idea is that it should be supplemented by viva voce 

 instruction and by experimental illustrations. It is 

 thoroughly elementary, however, and avoids all aid from 

 the differential and integral calculus. Great importance 

 has been deservedly attached to the elegant method of 

 diagrams of forces and to Culmann's graphical method. 

 In the first part are considered such matters as frames, 

 roofs, trussed beams, chains, and cords, and the stability 

 of walls. In the second part come under notice resistance 

 to stretching and to compression, theory of beams, 

 moment of resistance in rectangular beams, girders, open 

 girders, partially loaded beams, and other thoroughly 

 practical matters. Prof. Crofton has wisely given a great 



