MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 505 



*16'42. Proposition. Assuming (\-/>)H/><, if p is a <-plane and is not ^-co- 

 punctual, then p is the ^-concurrence of O^ with p. In symbols, 



h . '. (X - p) Up (f> . D : p c ple^ . > copnt^ ! p . => . p = O^'p 

 Proof. From *8'21 and *16'11 we have 



Let x be any member of the ^-concurrence of O^ with p. Hence (cf. *16'll) 

 p u L'X is a self-(-concurrence. Hence (cf. 16 '31) p u i'x is either ^-copunctual or 

 t^-coplanar. But on the first alternative p is </>-copunctual. Hence p u i'x is <-coplanar. 

 Hence x is a member of p. Hence from (l) the proposition follows. 



Note. In Concept V. the hypothesis of *16'42, that a <-plane is not copunctual, is 

 verified (cf. *28'll), where <j> represents " homaloty," and the axioms of that concept 

 are assumed. 



Summary of the Complete Development of this Subject. By the use of further 

 axioms the whole theory of projective geometry, apart from " order " and apart from 

 FANG'S axiom t respecting the distinction of harmonic conjugates, can be proved for 

 (/-points and the associated geometrical entities. Then FANG'S axiom can be added, 

 and the theory of order and continuity can be introduced, as in PIEPII'S memoir 

 (loc. cit.). In the sequel a somewhat different line of development is adopted, suitable 

 for the special ideas of Concept V. 



(ii) CONCEPT V. 



This concept is linear and monistic. It makes use both of the theory of interpoints 

 and of the theory of dimensions. The points are classes of objective reals, and 

 disintegrate from instant to instant. The corpuscles are capable of various and 

 complicated structures, and are thus well fitted to bear the weight of modern 

 physical ideas. The concept is Leibnizian, and only requires one extraneous relation 

 for the same purposes as that of Concept III. 



The essential relation is the pentadic relation T&-(abcdt), as explained at the 

 commencement of Part III. The four first terms, namely, a, b, c, d, are objective 

 reals and are mutually distinct, the fifth term is an instant of time. 



The relation R,-(abcdt) can be read, a intersects b, c, d in the order bed at the 

 instant t. In this concept copunctual objective reals do not necessarily intersect, 

 though two intersecting objective reals are necessarily copunctual. The relation of 

 intersection is not to be limited in properties by the mere geometrical suggestion of its 

 technical name. 



Since points are defined by the aid of the theory of dimensions, it follows (cf. note 



t Cf. FIERI, loc, tit. 

 VOL. CCV. A. 3 T 



