MATHEMATICAL CONCEPTS OF THE MATEKIAL WOULD. 483 



points ; and the descriptive point, from which in the current theory the projective 

 point is ultimately derived, is here abolished. The "Theory of Interpoints" [cf. 

 Part III. (ii)] and the "Theory of Dimensions" [cf. Part IV. (i)] represent two 

 distinct methods of overcoming the following initial and obvious difficulty of these 

 "linear" concepts: A point is to be defined as the class of objective reals 

 " concurrent" at a point. But this definition is circular. How can this circularity 

 be removed ? The Theory of Interpoints and the Theory of Dimensions give two 

 separate answers to this question. The points in the linear concepts, being only 

 classes of objective reals, are capable of disintegration. In fact, when motion is 

 considered, it will be found that the points of one instant are, in general, different 

 from the points of another instant, not in the sense of Concept III. that they are the 

 same entities with different relations, but in the sense that they are different entities. 

 More difficulty will probably be felt in conceiving anything analogous to a line as a 

 simple unity. Here it is to be observed that a linear objective real does not replace 

 a line of points of ordinary geometry. On the contrary, the class of those points 

 (here called a, punctual line), which have a given linear objective real as a common 

 member, is this ordinary geometrical line. A punctual line has parts and segments 

 in the ordinary way. The idea of a single unity underlying a straight line is not 

 wholly alien to ordinary language. The idea of a direction, as it could also be used 

 in non-Euclidian geometries where each line will have its own peculiar direction, may 

 be conceived as being that of a line taken as a unit. But it is unnecessary to 

 elaborate these considerations, as they have no relation to the logic of the subject. 



In the dualistic Concept IVA. the particles form another class of objective reals in 

 addition to the linear objective reals. Each particle is associated at each instant 

 with some one point, that is, with some class of linear objective reals. Thus the two 

 points, respectively associated at any instant with two particles, have in common one 

 linear objective real. Thus, when mutually determined motions are considered, these 

 linear objective reals assume the aspect of lines of force. In the monistic Concept V. 

 the analogy of objective reals to lines of force arises in a similar way. In this case 

 particles, in the sense used above, do not exist. Corpuscles, to use another term, are 

 defined entities, analogous to the corpuscles of Concept III. ; any general consideration 

 of them is best deferred till the definitions can be understood. 



In Concepts IV. and V. the conception of an ether is (in a sense) rendered 

 unnecessary, or (in another sense) is largely modified. The collection of linear 

 objective reals (i.e., in Concepts IVs. and V., of all objective reals) now forms the 

 entity (the ether) which " lies between " the corpuscles of gross matter. These 

 corpuscles must be conceived as volumes with some peculiarity either of motion or of 

 structure. Of course it might be found useful, for the explanation of physical 

 phenomena, to assume that corpuscles of some sort are generally distributed between 

 bodies of gross matter, thus forming an ether in a secondary sense. The ancient 

 controversy concerning action at a distance becomes irrelevant in these concepts. In 



3 Q 2 



