NATURE 



31, 



THURSDAY, FEBRUARY 3, 1887 



A HISTORY OF THE THEORY OF ELASTICITY 

 A History of the Theory of Elasticity and of the Strength 

 of Mnterials,from Galilei to the Present Time. By the 

 late Isaac Todhunter, D.Sc, F.R.S. Edited and com- 

 pleted for the Syndics of the University Press by Karl 

 Pearson, M. A., Professor of Applied Mathematics, Uni- 

 versity College, London. Vol. I. Galilei to Saint-Venant, 

 1 639- 1 850. (Cambridge : at the University Press, 1886.) 

 'T'HIS work was projected by the late Dr. Todhunter 

 on the same lines as his well-known Histories of the 

 '■ Theory of Probabilities," of the " Figure of the Earth,'' 

 and of the " Calculus of Variations," and will doubtless 

 equal them in usefulness to the mathematical student. 



The first object of a writer in the preparation of such 

 a work would be to draw up as complete a bibliography 

 as possible of all books and papers relating to the sub- 

 ject, arranged in chronological order. Afterwards, in read- 

 ing these memoirs, he would make copious notes, extracts, 

 and criticisms ; and then, on reaching the end of this 

 self-imposed task, he would find his materials for a 

 book like the present ready to place in the printer's 

 hands. Incidentally, enough material and ideas would 

 accumulate to form an independent treatise on the subject. 

 Such a task was undertaken by Dr. Todhunter on the 

 " History and Theory of Elasticity," from the standpoint 

 of the mathematician, but he did not live, unfortunately, 

 to complete it. 



Prof. Karl Pearson e.xplains in the preface the circum- 

 stances in which he undertook to edit and complete the 

 work, and, from his own account, the labour thus 

 devolved on him would have been sufficient to enable 

 him to complete the " History " ab initio. 



The present volume, like the previous " Histories," 

 carries the subject and commentaries only to the year 

 1850, although Dr. Todhunter had analysed the chief 

 mathematical memoirs from 1850 to 1870. The prepara- 

 tion of the second volume, to carry the history from 1850 

 up to date, is a task from which Prof. Pearson appears to 

 recoil, with some justification ; but it is to be hoped that he 

 will enlist in his service some of the junior elasticians 

 mentioned in his preface, and, by the application of the 

 modern principle of the subdivision of labour, carry this 

 in\aluable work to its proper conclusion. 



,-\t the outset Prof Pearson gives the palm to Galileo 

 Galilei (1638) as the founder of the subject of elasticity 

 and the strength of materials, while Dr.'Todhunter asserts 

 in § iS that "the first work of genuine mathematical 

 value on our subject is due to James Bernoulli . . . 1695." 

 Galileo treated only the question of the breaking moment 

 of a beam, or rather what we should call the bending 

 moment, exactly as is done now in calculating the stresses 

 in a structure, before proceeding to determine the conse- 

 quent strains and deformations. 



At this point the law enunciated by Hooke (1678) must 

 intervene, which goes by his name, " Ut tensio, sic vis,'' 

 originally published by him, in the fashion of those times, 

 as an anagram, ceiiinossstt u u. Stated in the mod ern 

 form, this law asserts that 



tension pressure stress , , r 1 .■ v 



= - = .- = modulus of elasticity, 



extension compression strain 



Vol. XXXV. — No. 901 



and is the law universally employed to connect mathe- 

 matically the corresponding stresses and strains in an 

 elastic substance, as pointed out by Saint-Venant [8]. 



When the stresses and strains are large enough for 

 variations on Hooke's law to become observable,- a fresh 

 set of phenomena depending on the ductility and vis- 

 cosity of the substance came into play, and the previous 

 mathematical investigations ro longer hold. Much of 

 the confusion pointed out by Dr. Todhunter in the treat- 

 ment of the subject by experimentalists is due to the fact 

 that in experiments it has been usual to test the strength 

 of structures to the breaking-point, and hence the use of 

 the term breaking instead of bending moment. The 

 modern experiments of Wohler show that this point, 

 at which ductility manifests itself, is much sooner 

 reached than was formerly supposed ; consequently, 

 modern engineering practice is much less bold than 

 formerly in large iron structures like bridges. For 

 this reason, the diagrams of the frontispiece, though 

 physically extremely interesting, cannot be considered 

 to bear on the mathematical theory. 



Returning again to the treatment of the subject by the 

 mathematicians, we find a picturesque diagram given by 

 Galileo (p. 2) of a beam built into an old wall and sup- 

 porting a weight, the cross-grained character of the wood 

 of the beam being carefully shown ; so that it is not 

 surprising that Galileo does not attempt any molecular 

 theory to account for the flexure of the beam. This 

 theory, supplied by Hooke's law, was applied by Mariotte, 

 Leibnitz, De Lahire, and Varignon ; but they neglect the 

 compression of the fibres, and So place the neutral plane 

 in the lower face of Galileo's beam. The true position 

 of the neutral plane was assigned by James Bernoulli in 

 1695, who, in his investigation of the simplest case of the 

 bent beam, was led to the consideration of the curve 

 called the " elastica." This " elastica " curve speedily 

 attracted the attention of the great Euler (1744), and 

 must be considered to have directed his attention to the 

 elliptic integrals. Probably the extraordinary divination 

 which led Euler to the formula connecting the sum of two 

 elliptic integrals, thus giving the fundamental theorem of 

 the addition equation of elliptic functions, was due to 

 mechanical considerations concerning the " elastica " 

 curve ; a good illustration of the general principle that 

 the pure mathematician will find the best materials for 

 his work in the problems presented to him by natural 

 and physical questions. The result obtained by Euler 

 for the thrust at which a straight column begins to bend, 

 when the corresponding " elastica " differs from a straight 

 line very slightly in a curve of sines, is of the utmost 

 importance to the architect and engineer ; and, as Prof 

 Kennedy can testify, is employed with the greatest con- 

 fidence in the design of the highest columns and pillars. 



It is interesting to find the complete treatment of the 

 problem of lateral vibrations of elastic bars is also due 

 to Euler, though the analytical difficulties of the period 

 equations seem to have puzzled him. If we employ the 

 modern notation of the hyperbolic functions, we shall find 

 his period equations all reduced to the form — 



cos «> cosh a> = ± I, 

 or, tanh w = ± sin m ; 



and this again is equivalent to 



tanh 4 w = ± tan 4 m, or ip cot 4 w, 



