THE MATHEMATICAL SCIENCES 153 



for instance, the summation of the series of even whole 

 numbers is equivalent to the summation of all whole 

 numbers, since between the terms of these two sum- 

 mations we can establish a univocal and reciprocal 

 correspondence. It is easy to verify this by writing 

 the two series as follows : 



1234 .... 11 .... 

 2468 .... N .... 



To every whole number a corresponding even number 

 can be found ad infinitum. 



Thirdly, the primary propositions must be in 

 sufficient number, without any being superfluous. 

 The Elements, in spite of their endeavour to be 

 complete, sometimes leave much to be desired in this 

 respect. Often they omit to justify by an axiom facts 

 regarded as evident, even when they are not derived 

 from the principles primarily postulated ; for example, 

 the following statement : if A, B, C be three points 

 belonging to the same straight line and if B be between 

 A and C, it will also be between C and A. 1 



Finally, it is essential that the primary propositions 

 considered necessary for the building up of geometry 

 should form a logically indissoluble whole, that is com- 

 posed in such a way that not one part can be suppressed 

 or altered without involving the ruin of the whole edifice. 

 If the suppression or change of one of the primary 

 propositions should lead to consequences which, 

 without being logically absurd, were simply different 

 from what they were before, the necessary conclusion 

 would be that various types of geometry are equally 

 possible, that is to say equally true from a logical point 

 of view. 



This problem did not present itself to Euclid ; but 

 he has intuitively understood its importance, by 

 claiming as a postulate that from a point taken outside 



1 5 Boutroux, Les mathematiques, p. 73. 



