>H 



NA TURE 



\Fcb. 3, il 



whence a graphical determination of the values of w is 

 easily inferred (pp. 50, 51, footnote). 



Another interesting paper due to Euler is " De altitu- 

 dine columnarum sub proprio pondere corruentium " 

 (177S), investigating the height at which a mast or tree 

 will begin to bend under its own weight. To this paper he 

 might well have prefixed the old German proverb, quoted 

 by Goethe in " Wahrheit und Dichtung" : — "Es istdafiir 

 gesorgt, dass die Baiime nicht in dem Himmel wachsen." 

 We know now that the functions of Bessel are required 

 for the complete analytical solution of this question, 

 though the Theoreina tnaxiine meinorahilc enunciated by 

 Euler, " Maxima altitude, qua columnas cylindricae ex 

 eadem materia confectae, proprium pondus etiamnunc 

 sustinere valent, tenet rationem subtriplicatam ampli- 

 tudinis," is interesting as one of the first applications of 

 the principle of mechanical similitude, showing why the 

 proportions of the giant of the forest are stunted 

 compared with those of the young tree, and also why it 

 is hopeless to attempt the problem of human flight while 

 .§•1332. 



Lagrange considered the same subject in " Sur la figure 

 des colonnes " (1770), examining and disproving the dictum 

 of Vitruvius that the rcufleincnf of a column was necessary 

 for strength : the dictum can hardly be called an architec- 

 tural fallacy, as the rsnflement corrects the tendency, due 

 to irradiation, of a perfectly cylindrical column to appear 

 attenuated in the middle ; for a similar reason it is neces- 

 sary to slightly blunt the neighbourhood of the point of a 

 Gothic spire to avoid the appearance of concavity. 



Coulomb, a well-known name to electricians, is men- 

 tioned by Saint- Venant as giving about this time (17S0), 

 in " Remarques sur la rupture des corps," the true position 

 of the neutral line of a beam, although it is asserted by Dr- 

 Todhunter that the ancient erroneous idea prevailed into 

 the present century. 



In Chapter II. the work of Young, Gregory, Eytelwein, 

 Plana, Dupin, Belli, Binet, Biot, Rennie, Barlow, Tredgold, 

 Fourier, Nobili, Bordoni, Hodgkinson, and others is 

 analysed. Of these the English writers, who generally 

 were experimentalists as well as theorists, are severely 

 handled by Dr. Todhunter for their heresies on the neutral 

 axis. Considering that the neutral axis is a mathematical 

 fiction, depending on an ignoration of the shearing stress, 

 and the consequent warping of the normal sections of a 

 beam, this treatment of Dr. Todhunter is too severe, 

 compared with the leniency with which he views the 

 metaphysical speculations of the pure theorists. These 

 experimentalists were trusted in their advice on important 

 constructions, and took care their formula erred on the 

 right side of strength. 



To Navier(i82i) we are first indebted for the genera] 

 mathematical equations of the equilibrium and vibrations 

 of an elastic solid, to be satisfied in the interior and at 

 the surface, and henceforth the researches of mathe- 

 maticians take a bolder flight from the treatment of the 

 simple beam of former investigators. 



Mile. Sophie Germain's " Recherches sur la thcorie des 

 surfaces elastiques" (i 821) appears to afford Dr. Todhunter 

 gratification in showing that sex can make itself apparent 

 even in mathematics. However, it is dangerous to argue 

 from this instance, as hardly any mathematician has yet 

 written on elastic surfaces without falling into error in the 



boundary conditions, and the subject is even now not yet 

 certainly settled. 



The vibration of elastic surfaces is important in its 

 bearing on acoustics and music, and received about this 

 time experimental and theoretical treatment from Chladni, 

 Strehlke, Pagani, and Savart. 



Chapters IV. and V. give an account of the treatment 

 of the subject by the celebrated mathematicians Poisson 

 and Cauchy, who practically exhausted the soluble pro- 

 blems, if we except the torsion questions considered by 

 Saint- Venant. Poisson's results are generally expressed 

 by means of definite integrals, most of which we see now 

 can be classified as Bessel's functions. Both Poisson and 

 Cauchy appear to have considered the subject of elasticity 

 principally in its bearing on the new theory of physical 

 optics, then receiving such important experimental and 

 theoretical treatment at the hands of Fresnel. 



Henceforth the theory receives development at the 

 hands of so many writers that it is possible only to 

 specify the honoured names of Gerstner, Green, 

 McCullagh, Poncelet, and Maxwell as having contri- 

 buted important advance to the subject. 



Lame's " Theory of Elasticity," carefully analysed in 

 Chapter VII., still remains a standard text-book, in con- 

 junction with the treatises of F. Neumann and Clebsch. 



The volume concludes with an account of Saint- 

 Venant's researches before 1850, the subsequent work 

 to be recorded in the second volume. Saint-Venant is 

 the name most honoured by practical elasticians and 

 engineers, inasmuch as he has developed his theories 

 from the definite practical problems presented by the 

 large and daring constructions in iron and steel which 

 mark the middle of this century. 



In the appendix Mr. Pearson has carefully analysed 

 the conflicting notations of different writers, and proposed 

 a very convenient terminology and notation, which would 

 save great trouble if universally adopted. He has also 

 given an account of experiments carried out by Prof. 

 Kennedy in his mechanical laboratory, which have an 

 important bearing on the limitations of the truth of 

 Hooke's law, or, in the language of elasticity, the constancy 

 of the ratio of stress to corresponding strain. 



The present volume is an indispensable hand-book of 

 reference for the mathematician and the engineer, and in 

 the editing and printing must be considered a very fitting 

 tribute to the wonderful industry and application of its 

 projector, the late Dr. Todhunter. 



A. G. Greenhill 



THE ENCYCLOP.-EDIA BRITANNICA 



The Encydopadia Britannica. Vol. XX. Pru — Ros. Vol. 



XXI. Rot— Sia. (Edinburgh: A. and C. Black, 1886.) 



THE leading scientific articles in these two volumes are 

 mainly biological. In Vol. XX. Prof A. Newton con- 

 tributes the articles on the various important groups of 

 birds; and in those on the Quail, Screamer, Secretary Bird, 

 Seriema or Cariama, it is truly surprising to find so 

 many facts condensed into so small a compass. Mr. C. 

 T. Newton's article on Pterodactyles gives us the new- 

 est information on this strange group of fossil reptiles. 

 In the article on Reproduction only the broadest as- 

 pects of the phenomena attending it are glanced at, 

 Mr. P. Geddes treating of the Animal, and Mr. S. H. 



