independent of Space Measurement. 391 



the notion of a cyclical time (although logically not incon- 

 sistent) seems repugnant, but also because it would amount 

 to assuming that every two right world-lines meet, whereas 

 we cannot say, for instance, that two fixed particles ever 

 meet, without straining very much our common language 

 and ideas. 



But the remaining two possibilities deserve some careful 

 attention. For the sake of shortness we will henceforth say 

 simply " particle " instead of " particle moving uniformly 

 along the ^-line.' ; 



Parabolic. Let p be one such particle, and q another, 

 never meeting the former, say always lagging behind p. 

 Then the parabolic axiom, which is also tacitly being 

 assumed in ordinary kinematics, amounts to this : 



P. If q be sped up *, no matter how little, it will overtake p 

 at some future instant ; and if q be slackened, p will 

 have overtaken it at some past instant. In other \\ ords, 

 through a world-point ex. (not on p) there is one and 

 only one world-line q not intersecting p. 



This axiom is simply the kinematical or world-analogue of 

 the much debated parallel-postulate of Euclid. It has hitherto 

 been tacitly assumed by all physicists. But, as in the case of 

 Euclid's postulate, the axiom P is by no means a consequence 

 of our previous axioms. On the contrary, it is in no logical 

 relation with them ; it is a further possible assumption entirely 

 independent of them. 



Without any real loss to generality we may consider the 

 particle p as fixed, say at a point A. Then the parabolic axiom 

 will run, somewhat more drastically, thus : 



The only particle q which, being at a certain instant at B 

 (distinct from A), never met and never will meet p, is that 

 which does not move at all, i. e. which is always at B. Every 

 particle which moves, uniformly, towards A (or away from A), 

 no matter how slowly, will reach A in future (or was at A in 

 its past). 



In this form it sounds almost as a truism. Yet it is but one 

 more axiom or assumption added to the projective ones. 



Hyperbolic. The parabolic axiom is but the limiting case 

 of the following, more general one, which it will again be 

 enough to state for the case of one particle, />, being kept 

 fixed, at a point A, thus : 



* That is to say, if q be replaced at a certain instant by a particle q' 

 which after that is ahead of q. In a similar sense we speak of 

 " slackening." 



