MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 491 



comparison. It does not belong to the purely geometrical side of the concept, but is 

 a necessary part of the " physical" ideas. IIHpE, (of Concept IV.), though it does 

 not occur explicitly in the above comparison, is required to give the geometry 

 " existence." Thus the geometry of Concept IV. requires thirteen axioms. 



For the purpose of the transition to projective geometry (cf. VEBLEN, loc. cit.), it is 

 now unnecessary to conceive a new class of " projective points." The points already 

 on hand are exactly the entities required. All that is necessary is to define the class 

 of those linear objective reals (cf. XIV Hp R), coplanar with any given linear 

 objective real and not intersecting it, as the point at infinity on that objective real. 

 Then with these new points at infinity, and the old points, the complete set of 

 " projective points" is obtained. 



The Extraneous Relation. For the purpose of the definition of motion, one 

 extraneous tetradic relation is required, exactly as in Concept III. Also the same 

 hypotheses must hold respecting it. The three mutually rectangular and intersecting 

 punctual lines, thus indicated at each instant, are to be taken as the " kinetic axes," 

 and all motion measured by reference to them. A given set of kinetic axes does not, 

 in general, correspond to the same three linear objective reals at different instants of 

 time. 



Matter. It is necessary to assume that the points in this concept disintegrate, and 

 do not, in general, persist from instant to instant. For otherwise the only continuous 

 motion possible would be representable by linear transformations of coordinates ; and 

 it seems unlikely that sense-perceptions could be explained by such a restricted type 

 of motions. We have therefore to consider what, in this concept, can represent the 

 permanence of matter. A " corpuscle," as we may call it, may be conceived to be a 

 volume with some special property in respect to the linear objective reals " passing 

 through " it. This is the procedure adopted in Concept V. ; and the methods of 

 overcoming the obvious difficulties which suggest themselves will be considered in 

 detail there. It is sufficient here to notice that, in this Concept IV., the special 

 property of the volume must relate merely to the motion of the objective reals. For 

 the only alternative is to make the property consist of the permanence of the points 

 within the volume. But then the difficulty of permanent coll ineat ions, mentioned 

 above, recurs. To find a special property of motion, we require a kinematical science 

 for linear objective reals in this concept analogous to the kinematical parts of 

 hydrodynamics. In the absence at the present time of such a science, we proceed to 

 other alternatives. 



Concept IVA. Conceive a class of particles, each particle being associated at each 

 instant with some point, but not necessarily each point with some particle. Then 

 the particles represent the " matter" which " occupies" space. Laws of motion must 

 then be stated (i) for the particles and (ii) for the linear objective reals. Also the 

 motion of the particles may be conceived to be influenced by that of the linear 

 objective reals, and vice versa. The endeavour to state such laws appears to reduce 



3 R 2 



