152 SCIENCE IN GRECO-ROMAN ANTIQUITY 



granted as possible without requiring proofs. The 

 axioms or common notions (xoival evvoiai) are truths 

 which cannot be demonstrated but are self-evident (for 

 example, the whole is greater than the part). It 

 appears, however, that Euclid only admitted two kinds 

 of primary propositions, definitions and postulates, 

 and that he classified as one or the other propositions 

 which were afterwards called axioms. This question is 

 of secondary importance ; it is of greater interest to 

 examine whether the primary propositions of the 

 Elements are *in agreement with the conditions laid 

 down by Euclid himself, and whether, on the other 

 hand, they satisfy the exigencies of the modern use of 

 axioms. With regard to the first point, it must be 

 noted that the form of the definitions often leaves 

 something to be desired. Such is the definition of the 

 straight line, the empirical origin of which is purposely 

 concealed, thus rendering it obscure. 1 Further, certain 

 definitions, such as that of the diameter, contain useless 

 elements. If the diameter be defined as passing 

 through the centre, it is superfluous to add that it 

 divides the circle into two equal parts. 



As to the relation of the Elements to the modern 

 theory of axioms, the following statements may be 

 made : 



Firstly, the primary propositions must be compatible, 

 that is to say, not contradictory to each other, other- 

 wise the consequences deduced from their combinations 

 would necessarily be contradictory. The Elements 

 fulfil this condition without proving it theoretically. 



Secondly, the enunciation of a primary proposition 

 must be rigorously complete. When we say that the 

 whole is greater than the part, we must add, which 

 Euclid has not done, that such an enunciation only 

 concerns finite magnitudes and numbers. We know, 

 in fact, that in infinity the part is equal to the whole ; 



1 See page 119. 



