independent of Space Measurement. 383 



indefinite number of other instants of such a scale will be 

 unequivocally co-determined, provided of course that certain 

 (very inoffensive) axioms are assumed at the outset. 



Every logical (mathematical) theory is bound to start with 

 some undefined terms and with a set of axioms or, better, 

 assumptions. These, being enunciated about the " undefined 

 terms,"" make these latter defined, in part at least. 



2. Now, besides such concepts as "instant of time," 

 distinct instants, "earlier and later" (at a spot at least), 

 and " between " two instants, which we will take for granted, 

 let our principal undefined term be " uniform motion/'' (The 

 objectors will notice that this is also the undefined term of 

 ordinary chronometry and enters as such, whether named or 

 un-named, into every treatise on Kinematics.) To be more 

 explicit, we shall call certain, undefined, motions of a particle 

 along a straight path, uniform motions. About this class of 

 undefined motions we will enunciate our assumptions. 



With regard to space itself it will be assumed that it is the 

 ordinary three-dimensional projective space (independent of 

 any idea of measurement). As a matter of fact, however, we 

 shall require only two of its dimensions, that is to say, a pro- 

 jective plane in the common sense of the word. 



Thus our space-time, necessary to prove the determinateness 

 of the proposed time-scale, will be, to use a now favourite 

 name, a three-dimensional world. And, having proved this, 

 the (actual or mental) experiment leading to the required 

 time-scale will be all performed on a single, one-dimensional, 

 space line. The passage of a particle through a point of the 

 plane at a time instant will be called an event or sometimes 

 a world-point ; and the succession of such events, a history 

 or, more fashionably, a world-line of a particle *. 



3. Let a particle p be capable of moving along a fixed 

 straight drawn in our plane. Let A, B be two points on the 

 straight, and a, b two instants of time, say, b later than a. 

 (Points will be denoted by capitals, and time-instants by 

 small letters.) Let it be required that p should be at 

 or pass through A and B at the instants a and b respectively. 

 Then, in absence of other requirements, there will still be an 



* The latter name is borrowed from Minkowski's relativists 

 vocabulary. But it will be kept in mind that throughout only one 

 and always the same reference system will be contemplated, so that 

 our investigation will have nothing- to do with Einstein's theory of 

 Relativity, whether " classical " (1905) or " new " (1913). 



2 D2 



