II] THE PRINCIPLE OF SIMILITUDE 27 



fruit on slender branches, and that melons and pumpkins must lie 

 upon the ground. And yet again, that in quadrupeds a large head 

 must be supported on a neck which is either excessively thick and 

 strong like a bull's, or very short like an elephant's*. 



But it was Gahleo who, wellnigh three hundred years ago, had 

 first laid down this general principle of simiUtude; and he did so 

 with the utmost possible clearness, and with a great wealth of illustra- 

 tion drawn from structures living and deadf. He said that if we 

 tried building ships, palaces or temples of enormous size, yards, 

 beams and bolts would cease to hold together; nor can Nature 

 grow a tree nor construct an animal beyond a certain size, while 

 retaining the proportions and employing the materials which suffice 

 in the case of a smaller structure J. The thing will fall to pieces of 

 its own weight unless we either change its relative proportions, which 

 will at length cause it to become clumsy, monstrous and inefficient, 

 or else we must find new material, harder and stronger than was 

 used before. Both processes are famihar to us in Nature and in 

 art, and practical apphcations, undreamed of by Gahleo, meet us at 

 every turn in this modern age of cement and steel §. 



Again, as Galileo was also careful to explain, besides the questions 

 of pure stress and strain, of the strength of muscles to hft an 

 increasing weight or of bones to resist its crushing stress, we have 

 the important question of bending ynornents. This enters, more or 

 less, into our whole range of problems ; it aifects the whole form of 

 the skeleton, and sets a limit to the height of a tall tree||. 



* Cf. W. Walton, On the debility of large animals and trees, Quart. Journ. 

 of Math. IX, pp. 179-184, 1868; also L. J. Henderson, On volume in Biology, 

 Proc. Amer. Acad. Sci. ii, pp. 654-658, 1916; etc. 



t Discorsi e Dimostrazioni matematiche, intorno a due nuove scienze attenenti 

 alia Mecanica ed ai Muovimenti Locali: appresso gli Elzevirii, 1638; Opere, 

 ed. Favaro, viir, p. 169 seq. Transl. by Henry Crew and A. de Salvio, 1914, p. 130. 



X So Werner remarked that Michael Angelo and Bramanti could not have built 

 of gypsum at Paris on the scale they built of travertin at Rome. 



§ The Chrysler and Empire State Buildings, the latter 1048 ft. high to the foot 

 of its 200 ft. "mooring mast," are the last word, at present, in this brobdingnagian 

 architecture. 



II It was Euler and Lagrange who first shewed (about 1776-1778) that a column 

 of a certain height would merely be compressed, but one of a greater height would 

 be bent by its own weight. See Euler, De altitudine columnarum etc.. Acta Acad. 

 Sci. Imp. Petropol. 1778, pp. 163-193; G, Greenhill, Determination of the greatest 

 height to which a tree of given proportions can grow, Cambr. Phil. Soc. Proc. rv, 

 p. 65, 1881, and Chree, ibid, vu, 1892. 



