762 THE EQUIANGULAR SPIRAL [ch. 



many interesting geometrical constructions, such as the regular 

 pentagon, and its mystical "pentalpha," and a whole range of other 

 curious figures beloved of the ancient mathematicians * : culminating 



A 



C B C B 



Fig. 357. Fig. 358. 



in the regular, or pentagonal, dodecahedron, which symbolised the 

 universe itself, and with which Euclidean geometry ends. 



If we take any one of these figures, for instance the isosceles 

 triangle which we have just described, and add to it (or subtract 



Fig. 359. 



from it) in succession a series of gnomons, so converting it into larger 

 and larger (or smaller and smaller) triangles all similar to the first, 

 we find that the apices (or other corresponding points) of all these 



* See, on the mathematical history of the gnomon, Heath's Euclid, i, passim, 

 1908; Zeuthen, Theoreme de Pythagore, Geneve, 1904; also a curious and 

 interesting book, Das Theorem des Pythagoras, by Dr H. A. Naber, Haarlem, 1908. 



