XI] CONCERNING GNOMONS 759 



any cone; for evidently, in Fig. 353, the little inner cone (repre- 

 sented in its triangular section) may become identical with the larger 

 one either by magnification all round (as in a), or by an increment 

 at one end (as in b) ; or for that matter on the rest of its surface, 

 represented by the other two sides, as in c. All this is associated 

 with the fact, which we have already noted, that the Nautilus shell 

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

 particular variety, or "hmiting case," of the spiral shell. 



This singular property of continued similarity, which we see in 

 the cone, and recognise as characteristic of the logarithmic spiral, 

 would seem, under a more general aspect, to have engaged the 

 particular attention of ancient mathematicians even from the days 

 of Pythagoras, and so, with little doubt, from the still 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, however indirect, is close enough to deserve our very careful 

 consideration. 



There are certain things, says Aristotle, which suffer no alteration 

 (save of magnitude) when they growf . Thus if we add to a square 

 an L-shaped portion, shaped Hke a carpenter's square, the resulting 

 figure is still a square ; and the portion which we have so added, with 

 this singular result, is called in Greek a "gnomon." 



EucUd extends the term to include the case of any parallelogram J, 

 whether rectangular or not (Fig.^ 354); and Hero of Alexandria 



* 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 

 measuring fields and pyramids" (Burnet's Greek Philosophy, 1914, p. 5). 



t Categ. 14, 15a, 30: ?(Ttl tipo. av^au 6 fxeva a ouk dWoiourat, olov to Terpdyuvop, 

 •)vu)ixouo^ wepiTedevTo^, rjv^rjrat. fiiv dWoidrepov 5e oudev yeyiurjTai. 



I Euclid (II, def. 2). 



