122 SCIENCE IN GRECO-ROMAN ANTIQUITY 



a great difference between the shape of a right-angled 

 triangle and that of a scalene triangle ; nevertheless, 

 if the bases and heights of these triangles be equal, 

 their areas will be expressed by exactly the same 

 number. A regular hexagon and an equilateral triangle 

 appear to us very different, but the hexagon can be 

 decomposed into six equilateral triangles. 



But it is not only motionless figures which can be 

 measured, the movements of the stars are likewise 

 subject to the law of number. And furthermore, 

 musical sounds are heterogeneous as to quality with 

 respect to each other, for a number of low notes cannot 

 produce a high note and inversely ; but there exist 

 numerical relations between the quality of sounds and 

 the dimension of the objects producing them. Thus 

 number is at the basis of everything. To the Pytha- 

 goreans it was not an abstract symbol, but a concrete 

 reality, 1 occupying a determinate place in space, hav- 

 ing clearly defined qualities and affinities, both moral 

 and physical, something like the chemical atom. 

 Under these conditions numbers are identified with 

 space, they not only resemble it, but they create it. 

 Thus, by a suitable analysis it is possible to find groups 

 of numbers which correspond to certain spatial forms. 

 According to the Pythagoreans the best analysis is 

 that obtained by means of the gnomon or set-square. 

 As defined by Hero of Alexandria (iv Definitiones, p. 44, 

 13) the gnomon is that which, being added to a number 

 or figure, gives a whole similar to that to which it has 

 been added. 2 This being so, let us suppose a set of 

 gnomons (or set-squares) which fit into one another. 

 If the first encloses one point, the second three points, 

 etc., then it will be seen that the sum of the uneven 

 numbers forms squares (Fig. 10). If the gnomons 

 enclose even numbers, the result is no longer squares, 



1 7 Brunschvicg, E tapes, p. 34. 

 * 20 Milhaud, Phi. geo., p. 88. 



