760 THE EQUIANGULAR SPIRAL [ch. 



specifically defines a gnomon (as indeed Aristotle had implicitly 

 defined it), as any figure which, being added to any figure what- 

 soever, 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 clbrjTLKot apiOixol, which can be 

 translated into fwm, 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 way of Pythagoras, and the secret 



Fig. 354. Gnomonic figures. 



magick ot numbers." For instance, the triangular numbers, 1, 3, 

 6, 10 etc., have the natural numbers for their "differences"; and 

 so the natural numbers may be called their gnomons, because they 

 keep the triangular numbers still triangular. In hke manner the 

 square numbers have the successive odd numbers for their gnomons, 

 as follows: 



0+ 1- P 



P + 3 = 22 



22 + 5 = 32 



32 + 7 = 42 etc. 



And this gnomonic relation we may illustrate graphically {axr]fJiaTo- 

 ypa<f>€tv) by the dots whose addition keeps the annexed figures 

 perfect squares*: 



• • • • • • 



• • • • • 



There are other gnomonic figures more curious still. For example, 

 if we make a rectangle (Fig. 355) such that the two sides are in the 



* Cf. Treutlein, Ztschr. /. Math. u. Phya. {Hist. litt. Abth.), xxviii, p. 209, 1883. 



