XVII] THE COMPARISON OF RELATED FORMS 1053 



sheep (extracted from the above table) are seen to form a more or 

 less regular and even curve. This simple graphic result implies 

 the existence of a comparatively simple equatiqn between y 

 and y'. 



An elementary application of the principle of coordinates to the 

 study of proportion, as we have here used it to illustrate the varying 

 proportions of a bone, was in common use in the sixteenth and 

 seventeenth centuries by artists in their study of the human form. 

 The method is probably much more ancient, and may even be 

 classical*; it is fully described and put in practice by Albert Diirer 

 in his Geometry, and especially in his Treatise on Proportio7f\ . In 

 this latter work, the manner in which the human figure, features, 

 and facial expression are all transformed and modified by slight 

 variations in the relative magnitude of the parts is admirably and 

 copiously illustrated (Fig. 508). 



:^Z3t 



Fig. 508. (After Albert Durer.) 



In a tapir's foot there is a striking difference, and yet at the same 

 time there is an obvious underlying resemblance, between the middle 

 toe and either of its unsymmetrical lateral neighbours. Let us 

 take the median terminal phalanx and inscribe its outline in a net 

 of rectangular equidistant coordinates (Fig. 509, a). Let us then 

 make a similar network about axes which are no longer at right 

 angles, but inclined to one another at an angle of about 50° (h). 



* Cf. Vitruvius, ni, 1. 



t Les quatres livres d' Albert Diirer de la proportion des parties et pourtraicts des 

 corps humains, Arnheim, 1613, folio (and earlier editions) . Cf. also Lavater, 

 Essays on Physiognomy, iii, p. 271, 1799; also H. Meige, La geometrie des visages 

 d'apres Albert Diirer, La Nature, Dec. 1927. On Diirer as mathematician, cf. 

 Cantor, ii, p. 459; S. Giinther, Die geometrische Ndherungsconstructione Albrecht 

 DUrers, Ansbach, 1866; H. Staigmuller, Diirer als Mathematiker, Stuttgart, 1891. 



