484 DR A. N. WHITEHEAD ON 



one sense there is something, not mere space, between two distant -corpuscles, namely, 

 the objective reals possessed in common ; in another sense there is a direct action 

 between two distant corpuscles not depending on intervening corpuscles. In fact, the 

 premises common to both bands of disputants are swept away. 



The Essential Relation. In both of the Concepts IV. and V. the essential relation 

 (R) is a pentadic relation, and has for its field both the class of instants of time and 

 that of linear objective reals, that is, in Concept V. the field is the complete class of 

 ultimate existents. The proposition ~R-(abcdt) can be read as the statement that the 

 objective real a intersects the objective reals b, c, d in the order bed at the instant t. 

 This conception of " the intersection in order of three linear objective reals by a 

 fourth at an instant of time " must be taken as a fundamental relation between the 

 five entities. But the properties of the relation are not to be limited by the 

 suggestion of the technical name " intersection." The axioms will be so assumed 

 that ~R'-(abcdt) implies that a, b, c, and d are distinct. Also, when points are defined, 

 it will be found that the axioms secure that a intersects b, c, and d in distinct points. 

 Furthermore, in general, b, c, and d are not co-punctual ; so that the case when a is 

 a transversal of the pencil b, c, d of co-punctual lines is only a particular case of the 

 satisfaction of T\,'(abcdt). 



Definitions. The notation of the general symbolism provides us with the symbol 

 E/(;;;;-) for the class of linear objective reals, and with R ; ( ;) for the class of 

 instants. But these symbols are long. Accordingly will be defined to stand for 

 the class of linear objective reals, and T for the class of instants. Thus, in symbols, 



= B/(;;;;-) Df 



T = R ; (----;) Df 



When "particles" (in Concept IV.) are not being directly considered, the term 

 " objective real " will be used instead of " linear objective real," or " member of 0." 



(ii) THE THEORY OF INTERPOINTS.* 



*1. The theory of intersection-points (shortened into interpoints) is required in both 

 of the Concepts IV. and V. Accordingly, it is convenient to investigate it before the 

 special consideration of either concept. In Concept IV. the interpoints are the points, 

 and there are no other points. In Concept V. the interpoints are, in general, only 

 portions of points, and a point may contain no interpoint or many interpoints. Thus 



* From this point a continuous argument commences, and the sections and included propositions are 

 numbered by a combined integral and decimal system, the whole number for the section and the decimal 

 part for the proposition, also the symbol (*) is placed before an integral number marking a section. All 

 the easier proofs of propositions are omitted, those proofs remaining being retained either as specimens, or 

 as containing some point of difficulty. The omitted proofs are often replaced by references to the 

 preceding propositions used in them, as a guide to their reconstruction. Note that "cf. *2'31 -41 '5 " is 

 used as a shortened form of "r/ *2'31 and *2'41 and *2'5." 



