XI] 



CONCERNING GNOMONS 



761 



ratio of 1 : a/2, it is obvious that, on doubling it, we obtain a similar 

 figure; for 1 : V2 : : a/2 : 2; and each half of the figure, accordingly, 

 is now a gnomon to the other. Were we to make our paper of such 

 a shape (say, roughly, 10 in. x 7 in.), we might fold and fold it, 

 and the shape of folio, quarto and octavo pages would be all the 

 same. For another elegant example, let us start with a rectangle 

 (A) whose sides are in the proportion of the "divine" or "golden 

 section*" that is to say as 1 : J {Vb — 1), or, approximately, as 

 1:0-618.... The gnomon to this rectangle is the square (B) 

 erected on its longer side, and so on successively (Fig. 356). 



V2 



1 

 Fig. 355. 







Fig. 356. 



In any triangle, as Hero of Alexandria tells us, one part is always 

 a gnomon to the other part. For instance, in the triangle ABC 

 (Fig. 357), let us draw BD, so as to make the angle CBD equal to 

 the angle A. Then the part BCD is a triangle similar to the whole 

 triangle ABC, and ABD is a gnomon to BCD. A very elegant case 

 is when the original triangle ABC is an isosceles triangle having 

 one angle of 36°, and the other two angles, therefore, each equal 

 to 72° (Fig. 358). Then, by bisecting one of the angles of the base, 

 we subdivide the large isosceles triangle into two isosceles triangles, 

 of which one is similar to the whole figure and the other is its 

 gnomon f. There is good reason to believe that this triangle was 

 especially studied by the Pythagoreans; for it lies at the root of 



* Euclid, Ti, 11. 



t This is the so-called Dreifachgleichschenkelige Dreieck; cf. Naber, op. infra 

 cit. The ratio 1 : 0-618 is again not hard to find in this construction. 



