MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 481 



and similarly for the other two axioms, and (iii) that one additional axiom (the axiom 

 of persistence) must be introduced, namely, 



XIIIHpR. = :eT.D ( .R ; (;;r)cR ; (;;;0 Df 



This axiom of persistence is unnecessary for the geometrical reasoning, but is an 

 integral part of the " physical" side of the concept. Also the hypothesis eT, which 

 is introduced in the three axioms (I, VII, IX) Hp R, is unnecessary in the other 

 axioms, since it is implied by the hypotheses already existing. The same explanation 

 holds of the absence of the hypothesis, t e T, from many axioms and propositions of 

 subsequent concepts. 



Thus at each instant the objective reals may be considered as the points of the 

 classical concept, and the whole of Euclidean geometry holds concerning them. But 

 at another instant the points will not have preserved the same geometrical relations 

 as held between them at the previous instant. Thus, in the comparison of the states 

 of the objective reals at different instants, the objective reals assume the character of 

 particles. 



Tke Extraneous Relation. A single extraneous relation is necessary to obviate the 

 difficulty of comparing straight lines and planes at one instant with similar entities at 

 another instant. In what sense can a point at one instant be said to have the same 

 position as a point at another instant ? This definition can be effected by introducing 

 into the concept a single tetradic extraneous relation S, so that, when S'(uvwt) holds, 

 t is an instant of time, and u, v, w are intersecting straight lines mutually at right 

 angles. Also corresponding to any instant t in the fourth term, there is one and 

 only one line for each of the other terms respectively. This last condition, expressed 

 in symbols, is 



<T.D ( .S ; (;--)el S ; ( ;-t)fl . &(--;t) el 



The straight lines indicated at each instant by this relation are to be taken as the 

 " kinetic axes."* Velocity and acceleration can now be defined, and a general 

 continuity of motion (in some sense) must be included among the axioms. 



This concept has the advantage over Concepts I. and II. that it has reduced the 

 class of extraneous relations to one member only, in the place of the innumerable and 

 perhaps infinite number of extraneous relations in the other two concepts. The 

 concept pledges itself to explain the physical world by the aid of motion only. It 

 was indeed a dictum with some eminent physicists of the nineteenth century that 

 " motion is of the essence of matter." But this concept takes them rather sharply at 

 their word. There is absolutely nothing to distinguish one part of the objective reals 

 from another part except differences of motion. The " corpuscle" will be a volume in 

 which some peculiarity of the motion of the objective reals exists and persists. Two 



* Cf. W. H. MACAULAY, ' Bulletin of the Amer. Math. Soc.,' 1897. 

 VOL. CCV. A. 3 Q 



