510 



THE LOGARITHMIC SPIRAL 



[CH. 



Pythagoras, and the secret magick of numbers." Thus for 

 example, the odd numbers are "gnomonic numbers," because 



+ 1 = 1-, 

 12 + 3 = 22, 



22 + 5-32, 



32 + 7 = 42 etc., 



which relation we may illustrate graphically {(T)(^r)ijLaToypa<f>€iv) 

 by the successive numbers of dots which keep the annexed figure 

 a perfect square * : as follows : \ 



There are other gnomonic figures more curious still. For 

 instance, if we make a rectangle (Fig. 251) such that the two 



Fig. 251. 



Fig. 252. 



sides are in the ratio of 1 : V2, it is obvious that, on doubhng it, 

 we obtain a precisely similar figure ; for 1 : V2 : : V2 : 2 ; and 



* Cf. Treutlein, Z. f. Math. u. Phys. (Hist. litt. Abth.), xxvm, p. 209, 1883. 



