NATURE 



M5 



THURSDAY, JUNE i6, 1892. 



MECHANICS. 



A Treatise on Analytical Statics. With numerous Ex- 

 amples. Vol. I. By Edward John Routh, Sc.D., LL.D., 

 F.R.S., Hon. Fellow of Peterhouse, Cambridge ; Fellow 

 of the Senate of the University of London. (London: 

 Macmillan and Co., 1891.) 



77/1? Elementary Part of a Treatise on the Dynamics of a 

 System of Rigid Bodies. Being Part L of a Treatise 

 on the Whole Subject. With numerous Examples. 

 By the Same. (London : Macmillan and Co., 1891.) 



WITH these two volumes the mathematical student 

 is completely equipped for the course of Analytical 

 Mechanics, as required for Part L of the Cambridge 

 Mathematical Tripos. 



A second volume is promised of the ^* Analytical 

 Statics," to cover the parts in Attraction, Astatics, and 

 the Bending of Beams ; and this, in conjunction with 

 Part n. of the " Dynamics," will complete his library for 

 the second part of the Mathematical Tripos, according 

 to present regulations. 



The great feature of these works is the very complete 

 collections of examples which the author has brought 

 together with great labour, and enriched with many of his 

 own invention, fit to rank among the theorems of the 

 science, rather than as mere problems. 



The author is of the opinion that in order to learn 

 Mechanics it is essential to the student to work many 

 examples, taken as far as possible from questions that 

 have actually arisen. 



In this opinion he agrees with Fourier, who says : — 



" L'^tude approfondie de la nature est la source la plus 

 f^conde des d^couvertes math^matiques. Non seulement 

 cette etude, en offrant aux recherches un but ddtermind, 

 a I'avantage d'exclure les questions vagues et les calculs 

 sans issue ; elle est encore un moyen assurd de former 

 I'Analyse elle-meme," &c. 



This is an opinion, however, that has always divided 

 mathematicians into rival camps, and we find Jacobi 

 remonstrating with these words of Fourier by retali- 

 ating : — 



" II est vrai que M. Fourier avait I'opinion que le but 

 principal des mathematiques etait I'utilitd publique et 

 I'explication des phdnom^nes naturels ; mais un philo- 

 sophe comme lui aurait du savoir que le but unique de la 

 science c'est I'honneur de I'esprit humain ; et que sous 

 ce titre, une question de nombres vaut autant qu'une 

 question du systeme du monde." 



The developments of mathematics are now so great 

 that specialization is a necessity, so that these rival 

 theories need not come into collision ; and the pure 

 mathematician may allow the writer on Mechanics to 

 treat of what the name of the subject implies without 

 being compelled to regard his own Geometry as mere 

 Land-Surveying, according to the strict meaning of the 

 word. 



There is a tendency in operation among certain mathe- 

 maticians, as illustrated by Poincard's remarks on Max- 

 well's writings, to degrade mathematical argument to 

 mere Calcul, by reducing the experimental facts on which 

 NO. 1 181, VOL. 46] 



the theory is based to the barest minimum, and that not 

 always clearly established (we venture to instance the 

 Newtonian Law of Universal Gravitation). A vast array 

 of Analysis is in consequence balanced upon a very small 

 amount of axiomatic experiment, which in many cases 

 the smallest divergence of experimental fact is sufficient 

 to upset. 



We had hoped at the outset that Duchayla's proof of the 

 Parallelogram of Forces had disappeared, never to re- 

 appear again, but it unfortunately pops up on p. 16. 



Considering that Static Jeals with the Equilibrium of 

 Bodies would make a great simplification if the word 

 Resultant was abolished, unless when required to mean a 

 single force reversed of a system of equilibrating forces. 



In this way a much simpler proof of the Parallelogram 

 of Forces can be constructed, as indicated by Prof. Max- 

 well in the Mathematical Tripos ; and one figure will now 

 serve for all the possible cases arising in the equilibrium of 

 three parallel forces (p. 47). 



Again, when the system is in equilibrium, there is no 

 need to introduce the restriction that the bodies are rigid 

 (p. 12) ; the conditions are precisely the same for elastic 

 bodies ; but the system having come to rest, the parts 

 are of invariable form. Every structure (the Forth Bridge, 

 for instance) is composed of elastic parts, but the theorems 

 of elementary Statics are still applicable in the investi- 

 gation of the principal stresses. 



Again, by considering balancing couples, the refined 

 theorems concerning the equivalence of couples in the 

 same or parallel planes, and the composition of couples 

 in different planes, are rendered much more convincing. 



In accordance with its title of " Analytical Statics," the 

 theorems concerning the composition and equilibrium of 

 forces in space are treated with reference to co-ordinate 

 axes ; but Sir Robert Ball's purely geometrical concep- 

 tions of the Wrench, Screw, and Cylindroid are introduced, 

 and discussed from a fundamental standpoint. 



A chapter on the determination of Centre of Gravity 

 appears in all treatises on Analytical Statics, just as works 

 on Rigid Dynamics begin with a long and tedious chapter 

 on Moments of Inertia : these subjects should form part 

 of the ordinary treatises on Integral Calculus, and so 

 relieve treatises on Mechanics from at least the principal 

 elements of such calculations. 



In the application of the Barycentric Calculus to 

 geometry, the author has made a very interesting collec- 

 tion of problems, well calculated to illustrate the power 

 of this method. 



The principal theorems of Statics involve profound geo- 

 metrical argument, and consequently prove difficult to 

 the majority of students, whose proclivities are usually 

 analytical ; but in the applications to Catenaries the ana- 

 lytical interest comes again to the front. Considering 

 that the hyperbolic functions can now be obtained tabu- 

 lated numerically — for instance, in a table by Mr. T. H. 

 Blakesley, published by the Physical Society — it is curious 

 that the author does not employ them in the discussion 

 of the ordinary Catenary, where their use introduces great 

 elegance and simplicity into the analysis. The figure ot 

 the Catenary on p. 316 might with advantage be re- 

 drawn, so as to exhibit accurately the principal properties 

 of this curve. 



Again, in Example 6, p. 352, where the problem ot 



H 



