154 SCIENCE IN GRECO-ROMAN ANTIQUITY 



a straight line only one parallel can be drawn to it. 

 Seeing the hypothetical character which he gives to 

 this proposition, Euclid has had regard to the exigencies 

 of the modern theory of axioms, but if, as he believed, 

 only one geometry is possible, his hypothesis would 

 appear strange and superfluous, for one would neces- 

 sarily be able to affirm the singleness of the parallel 

 and deduce from it the definitions already postulated 

 of the straight line, the plane and angles. It would 

 seem that he must speak of a theorem of parallels and 

 not of a postulate if logically there only exists but one 

 geometry. 



The successors of Euclid were of this opinion, and 

 not without reason, and this is why they endeavoured 

 to demonstrate the proposition which Euclid had 

 enunciated as a hypothesis, but all their attempts in 

 this direction were in vain. 



In the nineteenth century they surrendered to 

 evidence. It is possible to abandon the postulate of the 

 parallels, whilst keeping the other primary propositions. 

 Geometries can then be constructed which have other 

 properties than that of Euclid and which for this reason 

 are called non-Euclidean (Lobatschewsky, Riemann). 

 These geometries, the truth of which is guaranteed by 

 logic, deal with mathematical facts (lines, surfaces, 

 angles) which are real and in no wise fanciful, although 

 we cannot picture them by intuitive perception. The 

 field of geometry is therefore vaster than Euclid sup- 

 posed, but although he did not entirely construct the 

 modern theory of axioms, to him belongs the merit of 

 having established it upon a permanent basis. 



The primary notions having once been elucidated, 

 it is possible by logical deduction to link to them a series 

 of propositions entirely derived from one another. 

 These propositions are classified and distinguished 

 according to their nature. There is first the theorem 



