386 Dr. L. Silberstein on a Time-Scale 



This graphical construction shows ns at once how to 

 arrange what we mny call the four-p articles experiment. 

 In fact, it is enough for this purpose to remember the 

 meaning of the allegorical figure 1. 



Thus, the three instants 0, 1, T being fixed arbitrarily, let 

 the two particles p u p 2 start together at the instant £ = from 

 any point of the X-straight. At the instant £ = 1 let p x 

 and p 2 send off two uniformly running messengers^ and p 4 , 

 such as will catch or meet p 2 and jt? 3 , respectively, at the same 

 instant T. Then, these two messengers p Zi p± will meet with 

 one another at a perfectly definite instant (no matter, where 

 p s meets p 2 and where p± meets pi), and this will be the 

 instant t~2 of our time-scale. 



A glance at the upper part of fig. 1 will suffice to see how 

 to construct the instants £ = 3, 4, and so on. Thus, let p± 

 send off, at t = 2, a fifth particle p$ so that it should meet p 2 

 at the instant T. Then p 5 will meet p± at a perfectly definite 

 instant £ = 3. Similarly a sixth particle p G parting from p 2 

 at £ = 3 will meet p± at £ = 4, and so on. The proof that the 

 instant 1 will precede 2, and that this will be earlier than 

 t = 3, and so on, may bo left to the care of the reader. The 

 construction of fractional-^ instants and of negative ones 

 (preceding t=^0) will be obtnined on similar lines. At this 

 stage we may introduce the generalized archimedean postulate, 

 i.e. assume that every instant of time between and T (and 

 distinct from T) can be exceeded by a finite repetition of the 

 process described nbove. 



The index to be given to the instant T itself, as approached 



