XI] CONCERNING GNOMONS 511 



each half of the figure, accordingly, is now a gnomon to the other. 

 Another elegant example is when we start with a rectangle (A) 

 whose sides are in the proportion of 1 : ^(V5 — 1), or, approxi- 

 mately, 1 : 0-618. The gnomon to this figure is a square (B) erected 

 on its longer side, and so on successively (Fig. 252). 



In any triangle, as Aristotle tells us, one part is always a 

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

 (Fig. 253), let us draw CD, so as to make the angle BCD equal to 

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

 whole triangle ABC, and ADC 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, there- 

 fore, each equal to 72° (Fig. 254). Then, by bisecting one of the 



Fig. 254, 



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*. There is good reason to believe 

 that this triangle was especially studied by the Pythagoreans ; 

 for it lies at the root of many interesting geometrical constructions, 

 such as the regular pentagon, and the mystical "pentalpha," and 

 a whole range of other curious figures beloved of the ancient 

 mathematicians f. 



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

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



f 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. 



