226 SCIENCE IN GRECO-ROMAN ANTIQUITY 



more complex and more justified. There are gravita- 

 tional and electromagnetic properties which confer on 

 space, in every region, its geometrical qualities (curva- 

 ture, possible kinds of triangles, etc.). 



This being so, there cannot be a universal system 

 of reference, granted once for all, to which the study 

 of a group of localized phenomena, in any part what- 

 ever of the universe, can be related. The system of 

 reference must in every case be intrinsic to this group 

 of phenomena, which are then studied by methods 

 necessitating the use of tensorial calculus and absolute 

 differential calculus. As G. Juvet points out, " the 

 characteristic feature of these methods arises from 

 the fact that they enable a geometrical entity to be 

 studied from a purely intrinsic point of view. The 

 Greeks never studied their geometry in any other way, 

 when they were searching for the properties of a figure, 

 they always examined the figure itself, considered by 

 itself and taken independently of any system of 

 reference." x It is evident that in Greek geometry, 

 as in the algorithm of relativity, the relations of a 

 figure are sufficient in themselves, and although they 

 may be studied by means of a method and by universal 

 formulae, it is not necessary, as in Cartesian geometry, 

 to relate them to an exterior system of co-ordinates. 



We know, besides, that the universe of the physics 

 of relativity, while lending itself to questions of 

 infinity, remains finite in its dimensions by virtue of 

 its curvature. Now, as we have seen, the hypothesis 

 of finiteness is characteristic of Greek astronomy. As 

 we have pointed out, Empedocles expressed an idea 

 concerning the universe, considered as finite, which 

 recalls that of phantom stars ; he declared in fact 

 that the sun has no real existence, that it is formed by 

 a simple concentration of luminous rays which are 



1 G. Juvet, Introduction au calcul tensor iel, A. Blanchard, 

 Paris, 1922. 



