independent of Space Measurement. 385 



have a two- dimensional manifold of events or world-points 

 (X, t) which would have all the properties o£ a projective 

 plane. Many interesting theorems would hold tor such a 

 world-plane. Yet the Desargues theorem about perspective 

 triangles (built up of right w r orld-lines) which is the most 

 important tiling for our purposes, would not necessarily 

 hold *. But if, as was announced, the said axioms are 

 assumed to hold for every straight of (at least) one space- 

 plane (X, Y) as the possible path of uniformly moving 

 particles, then w r e shall have a three-dimensional world 

 (X, Y, t) endowed with all the properties of projective 

 space. And the world-plane (X, t) being now "part" of 

 this projective three-dimensional manifold, the theorem 

 of Desargues will hold in that plane. 



In other words, the world-plane X,t will be a Desarguesian 

 plane. Such being the case, any three collinear world-points 

 a, {3, 7 (i. e. belonging to the history of a uniformly moving- 

 particle) will determine a unique fourth point, their fourth 

 harmonic conjugate to a, say. Likewise, any three con- 

 current right world-lines in the plane X, t will co-determine 

 a unique fourth line. 



4. Now, this enables us to build up at once the desired 

 " projective " scale of time-instants. All that is required 

 for this purpose is a number of uniformly moving particles 

 which we will call pi, P2, Pb, p*-, etc. We shall use these 

 symbols also for the histories of the particles or their 

 representative straights in the world-plane (X, t). 



In fact, let all these particles move along the same 

 X-straight. Let £ = 0, 1, T be three conventionally fixed 

 instants of time, say 1 later than 0, and T later than 1. Let 

 us represent the X-straight at various instants by straight 

 lines drawn in the X, £-plane through an arbitrarily fixed 

 centre which we will call fl x . Thus the X-straight at 

 the said three instants will be represented by the three 

 lines marked £ = 0, 1, T (fig. 1). Now, through any point 

 of the first line draw two straights p x , p 2 crossing the second 

 line in a, /3, and the third line in 7, $ respectively. Draw p 5} 

 the join of a, 6, and p 4 , the join of /3 and 7. These will cross 

 in e, and fl. f e will be a definite line through fl x , independent, 

 that is, of the choice of the two lines p ij p 2 . The line D, x € 

 will thus represent the X-seraight at a perfectly definite 

 instant. Let us give to this instant the label t = 2. 



* Since this, as is well-known, would require either three-dimensionality 

 or the cono-ruence axioms. 



