XI] 



CONCERNING GNOMONS 



509 



on two of its sides, as in c. All this is closely associated with the 

 fact, which we have already noted, that the Nautilus shell is but 

 a cone rolled up; in other words, the cone is but a particular 

 variety, or "limiting case," of the spiral shell. 



This property, which we so easily recognise in the cone, would 

 seem to have engaged the particular attention of the most ancient 

 mathematicians even from the days of Pythagoras, and so, with 

 little doubt, from the more ancient days of that Egyptian school 

 whence he derived the foundations of his learning* ; and its bearing 

 on our biological problem of the shell, though apparently indirect, 

 is yet so close that it deserves our further consideration. 



If, as in Fig. 249, we add to two sides of a square a symmetrical 

 L-shaped portion, similar in shape to what we call a " carpenter's 

 square," the resulting figure is still a square; and the portion 



Fig. 249. 



Fig. 250. 



which we have added is called, by Aristotle {Phys. iii, 4), a 

 "gnomon." Euclid extends the term to include the case of any 

 parallelogram f, whether rectangular or not (Fig. 250); and Hero 

 of Alexandria specifically defines a "gnomon" (as indeed Aristotle 

 impHcitly defines it), as any figure which, being added to any 

 figure whatsoever, leaves the resultant figure similar to the 

 original. Included in this important definition is the case of 

 numbers, considered geometrically; that is to say, the elSrjTiKol 

 dptO/uLol, which can be translated into form, by means of rows of 

 dots or other signs (cf. Arist. Metaph. 1092 b 12), or in the 

 pattern of a tiled floor: all according to "the mystical w^ay of 



* I am well aware that the debt of Greek science to Egypt and the East is 

 vigorously denied by many scholars, some of whom go so far as to believe that the 

 Egyptians never had any science, save only some "rough rules of thumb for measur- 

 iiig fields and pyramids "-(Burnet's Greek Philosojihy, 1914, p. 5). 



t Euclid (n, def. 2). 



