120 SCIENCE IN GRECO-ROMAN ANTIQUITY 



of sides, and it is this identity which logically 

 guarantees the properties of the figure generated. 

 The instruments which form the second group are, on 

 the contrary, composed of one or more parts which 

 change their respective positions whilst the figure is 

 described. Consequently these parts do not occupy 

 the same position at the beginning and the end of the 

 operation. In tracing a quadratrix, for example, the 

 radius of the circle moves angularly whilst the straight 

 line which cuts it moves so as to remain constantly 

 parallel to itself (p. 55). How can the point of inter- 

 section resulting from the combination of these two 

 movements be logically defined ? This intersection 

 involves the indefinite divisibility of the radius and 

 the straight line, and thus runs counter to the objections 

 raised by Zeno of Elea. It would seem that it was a 

 reason of this kind that consciously or unconsciously 

 impelled the Greek geometers to admit only figures 

 constructed by rule and compass, and the solids of 

 revolution generated by these figures. 



2. ARITHMETIC AND ALGEBRA 



The Greek scientists took little interest in concrete 

 applications of science, and they early distinguished 

 between theoretical arithmetic and the art of calculat- 

 ing numerically concrete magnitudes. According to 

 Plato's saying, we must reason about numbers as 

 abstractions and not about numbers which are visible 

 and tangible (Rep. 252 D). Hence " when we speak 

 of Greek arithmetic, we understand the theory of the 

 properties of numbers and exclude all that concerns 

 calculation, namely, that which, since Plato at least, 

 has been called logistic." 1 A scholium on the Char- 

 mides, translated by P. Tannery, 2 endeavours to define 



1 25 Tannery, Science Hellene, p. 369. 



2 26 Tannery, Geo. grecque, pp. 48 and 49. 



