THE FOURTH DIMENSION. 



79 



damental propositions, truths on which all other truths are, directly 

 or indirectly, founded, but which themselves are assumed without 

 demonstration as self-evident. 



But the spirit of mathematical research grew in time more and 

 more critical, and finally asked, whether these axioms might not pos- 

 sibly admit of demonstration. Especially was a rigid proof sought 

 for the eleventh* axiom of Euclid, which treats of parallels. 



After centuries of fruitless attempts to prove Euclid's eleventh 

 axiom, Gauss, and with him Bolyai and LobacheVski, Riemann, 

 and Helmholtz, finally stated the decisive reasons why any attempt 

 to prove the axiom of the parallels must necessarily be futile. These 

 reasons consist of the fact that though this axiom holds good enough 

 in the world-space such as we do and can conceive it, yet three- 

 dimensioned spaces are ideally conceivable though not capable of 

 mental representation, where the axiom does not hold good. The 

 axiom was thus shown to be a mere fact of observation, and from that 

 time on there could no longer be any thought of a deductive demon- 

 stration of it. In view of the intimate connection, which both in an 

 historical and epistemological point of view exists between the ex- 

 tension of the concept of space and the critical examination of the 

 axioms of Euclid, we must enter at somewhat greater length into 

 the discussion of the last mentioned propositions. 



Of the axioms which Euclid lays at the foundation of his 

 geometry, only the following three are really geometrical axioms : 



Eighth axiom: Magnitudes which coincide with one another 

 are equal to one another. 



Eleventh axiom: If a straight line meet two straight lines so as 

 to make the two interior angles on the same side of it taken to- 

 gether less than two right angles, these straight lines, being con- 

 tinually produced, shall at length meet on that side on which are 

 the angles which are less than two right angles. 



Twelfth axiom: Two straight lines cannot inclose a [finite] 

 space. 



The numerous proofs which in the course of time were adduced 



*Also called the twelfth axiom, also the fifth postulate. Tr. 



