independent of Space Measurement. 387 



through 1, 2, 3, etc., will of course be t = co. The same 

 instant T, when approached through t= — 1, etc., will ob- 

 viously have also the index t= — oo. Thus, T itself will be a 

 singular point of the projective time-scale. But, exactly as 

 is the case with the familiar staudtian space-scale, this formal 

 singularity will give rise to no real confusion. 



5. Having thus set up a projective time-scale, let us now 

 construct a similar space-scale, i.e. a staudtian scale of points 

 (places) on the X-straight. This can be done either in the 

 well-known way, by means of von Staudt's construction in a 

 space-plane passing through the X-line, or — which in the 

 present connexion seems more elegant — by an experiment 

 performed upon the X-line, without invoking any other 

 dimension of space proper. 



In fact, replace in the reasoning guided by fig. 1 the pencil 

 of world-lines of constant date by world-lines representing 

 particles fixed at points x = 0, 1, S, all these lines passing 

 through a conventional world-point £l t as centre. Then, if 

 Pi-> V^ e ^c. De a g aln uniformly moving particles, the required 

 experiment will easily be seen to be as follows : 



pi and p 2 start simultaneously from the same place x = ; 

 when pi arrives at the place x = l, it sends p d to meet p 2 at S, 

 and when p 2 arrives at x = l, it sends p 4 to meet p x at S. Then 

 ;>3 and j9 4 ivill meet with one another at a perfectly definite place 

 (no matter when p z meets p 2 , and when p 4 meets p x ), and this 

 will be the point a: = 2. Similarly, a fifth particle p~, will give 

 ,.r = 3, and so on. 



In short, everything will be as in Section 4, the words 

 "instant" and "simultaneously" (same instant) being sub- 

 stituted for " place " (point) and " same place," and vice 

 versa. 



Having thus obtained a scale of instants of time and of 

 points along the straight in question, we can now set up a 

 system of projective " coordinates " #, t defining an event in 

 the world a?, t, in much the same way as in projective geometry 

 (the supplementary axioms of continuity being properly intro- 

 duced). Thus, graphically, let Q x , fit (fig. 2) represent two 

 fixed events or world-points, the above "centres," and let ot, 

 a third fixed event, be the "origin " of our coordinates. Let 

 the lines trOt, t**£Ix or axes be provided with their scales as 

 explained above. Then, if a be any event in the contemplated 

 two-world, draw the joins I2 r a, tit*. The former will find on 

 the first axis the date t, and the latter, on the second axis, the 

 place x of the event a.. Any right world-lines through i\ or 



