384 Dr. L. Silberstein on a Time-Scale 



infinite variety of possible histories or motions of the particle 

 between these two events. In other words, there is an infinity 

 of world-lines joining the world-points a = A,a and ft = B,b. 

 Let, now, our first assumption be : 



Among all the possible motions leading from A, a to B,b 

 there is one and only one uniform motion. 



We may say, equivalently : between two world-points 

 a and ft there is one and only one uniform or right world-line, 

 to have a short name. 



As a consequence of this assumption we have at once : 



Two particles moving uniformly (along the same or distinct 

 straights), and not permanently united, do not meet more than 

 once. (Or, equivalently, two right world-lines do not cross 

 more than once.) For if they met twice they would, by the 

 first assumption, never part company. 



It does not follow, however, that they will meet, or have 

 met, even once only. 



It is scarcely necessary to say that the above assumption 

 is simply a translation of the foremost axiom of projective 

 geometry, the terms u point " and " straight " being replaced 

 by " event" or " world-point " and by u uniform motion " or 

 " right world-line." Similarly let us assume the correlates 

 of all the remaining axioms of projective geometry, well- 

 known by the name of axioms of order and connexion. Thus, 

 calling shortly a, ft, etc. the eA r ents A, a; B, b, etc. : 



IE 7 is an event of the world-line segment aft, then any 

 fourth event 8 belonging to this segment belongs 

 either to ay or to yft, but never to both segments. 



If y, an event distinct from ft, belongs to aft and if the 

 event ft belongs to yB, then 7, and therefore also ft, 

 belong to the segment aS. (This gives the extension 

 of the right world-line beyond the terminal event B, b, 

 and similarly A, a.) 



If a, ft, 7 do not belong to the same right world-line, 

 and if h belongs to fty and e to uS, then there is 

 always an event f belonging to a/3, such that* e 

 belongs to yf. 



Each of these assumptions can be enunciated in terms 

 of places and time-instants relating to uniformly moving 

 particles. This is left to the reader, who will find the 

 use of representative drawings helpful, but by no means 

 necessary for the purpose of logical deductions. 



If these projective axioms were valid only for particles 

 moving uniformly all along the same straight path, we should 



