482 DR- A. N. WHITEHEAD ON 



different developments, viz., Concept III A. and Concept Hie., are now possible, 

 according as the persistence is taken to be of one or other of two possible types. 



Concept IIlA. Here the persistence is that of the same objective reals in the same 

 special type of motion. KELVIN'S vortex ring theory of matter can be adapted to 

 such a concept. 



Concept IIIn. Here the persistence is that of the type of motion in some volume, 

 but not necessarily of the identity of the objective reals in the volume. The 

 continuity of motion of a corpuscle as a whole becomes then the definition of the 

 identity of a corpuscle at one instant with a corpuscle at another instant. 



PART III. (i) GENERAL EXPLANATIONS OF LINEAR CONCEPTS. 



These concepts depart widely from the classical concept. The objective reals (at 

 least those which, with the instants of time, form the field of the essential relation) 

 have properties which we associate with straight lines, considered throughout their 

 whole extent as single indivisible entities. These objective reals, which in Concept V. 

 are all the objective reals, will be called linear objective reals. Perhaps, however, a 

 closer specification of the linear objective reals of these concepts is to say that they are 

 the lines of force of the modern physicist, here taken to be ultimate unanalysable 

 entities which compose the material universe, and that geometry is the study of a 

 certain limited set of their properties. But this mode of realizing the nature of the 

 linear objective reals has also its pitfalls, for a line of force suggests ends, while these 

 linear objective reals have no properties analogous to the properties of the ends of 

 lines of force. The whole of a straight line, viewed as a point-locus, will be found to 

 be associated with a linear objective real. The " linear " concepts here considered are 

 all Leibnizian. 



Concept IVA. is dualistic, and requires among the objective reals a class of 

 "particles" in addition to the linear objective reals. Concept IVB. is the monistic 

 variant of Concept IVA., obtained exactly as Concept II. is derived from Concept I. 

 Both of the Concepts IVA. and IVfi. labour under the same defect as Concepts I. 

 and II. in requiring an indefinitely large class of extraneous relations. Concept V. is 

 monistic, and is by far the most interesting of the set of linear concepts. It requires 

 only one extraneous relation to perform a similar office to that of the extraneous 

 relation in Concept III. 



Points are now defined complex entities, being certain classes of linear objective 

 reals. Geometers are already used to the idea of the point as complex. In 

 projective geometry, as derived from descriptive geometry, the projective point is 

 nothing but a class of straight lines.* This idea will now be extended to all 



* Cf. PASCH, loc. cit., and SCHUR, " Ueber die Einfiihrung der sogennanten idealen Elemente in die 

 projective Geometrie," 'Math. Annal.,' vol. XXXIX., and BONOLA, "Sulla Introduzione degli Enti 

 improprii in Geometria projettiva," ' Giorn. di Mat.,' vol. XXXVIII. 



