n6 SCIENCE IN GRECO-ROMAN ANTIQUITY 



certain property of numbers or a certain class of figures 

 rather than at any other ? The standard by which 

 they made their choice of figures was the construction. 

 This construction, as P. Boutroux points out, has 

 nothing in common with the concrete measurements of 

 surveyors. "It is a rational operation by which the 

 theoretical existence of the figures on which the reason- 

 ing is based can be stated and proved. To attain this 

 object, the most simple means evidently consist in 

 constructing the figure, or rather in defining a 

 theoretical process which would permit the construc- 

 tion to be made if it were possible to draw perfectly." * 

 It is quite possible, however, to conceive of a figure 

 being constructed or drawn by means of straight lines 

 and circles, or even by considering the path traced by 

 a point which moves on a plane or in space according to 

 a given law (cycloid, spiral, etc.). Here a choice need 

 not necessarily be made. The Greeks, after some 

 hesitation, would only admit as legitimate construc- 

 tions those which could be made by means of the 

 straight line and the circle, or, in concrete terms, by 

 means of the rule and compass. The objects of plane 

 geometry are thus clearly defined. In dealing with 

 spatial geometry, however, a difficulty at once arises. 

 Solid bodies cannot be represented by a plane draw- 

 ing without using descriptive geometry. The Greek 

 geometers did not think of having recourse to this 

 expedient, and did not at first know how to get over 

 this difficulty, for which Plato reproaches them very 

 severely {Laws, 528 B). They ended by admitting 

 a priori the legitimacy of constructions, which corre- 

 spond spatially to plane constructions made with rule 

 and compass ; the construction of a plane, a straight line 

 or a circle in space, and also of round bodies such as the 

 cylinder, cone, sphere, generated respectively by the 

 revolution of a rectangle, triangle, and circle round a 



1 4 Boutroux, Ideal, p. 38. 



