512 DR. A. N. WHITEHEAD ON 



The next three axioms, viz. (VI.-VIlI.)HpR, are the axioms establishing the 

 relation of interpoints to points. Intpnt Rp R has been denned in * 1 "4 1 . 



*22'31. VIHpR = intpntHpR Df 



*22'32. VII Hp R is the statement that, if u is an interpoint, there exists a point 

 containing u. In symbols, 



VII Hj> R . = : (t, u):we intpnt . a . fop) . p e pnt . cp Df 



*22-33. VIII Hp R is the statement that, if p be a point, and u and v be two 

 distinct interpoints contained in p, then u and v possess no common member. In 

 symbols, 



VIII Hp R . = : (p, u,v,t):pe pnt K1 .u,ve intpnt R( n cls'p . u ^ v . => . uav = A Df 



The next set of three axioms, viz. (IX.-XI.)HpR, supplies the missing hypotheses 

 requisite to make homaloty a "geometrical property," as defined in *10. 



*22'41. IX HpR is the statement that, if t is an instant of time, j/Hp/* K( is true. 



In symbols, 



IX Hp R . = : tt T . 3( . v Hp ft Df (cf. *10'3) 



*22'42. XHpR is the statement that, if t is an instant of time, irHp/i Ht is Zrwe. 



In symbols, 



. = :*eT.=> . irH^/t,,, Df (c/ *10-4) 



*22'43. XI HpR is the statement that, if t is an instant of time, p Hp/* Ki is true. 



In symbols, 



XIHpR. = :eT.D f ./oHp/iH, Df (cf. *10'5) 



*22'51. XII HpR is the statement that, ifp and q are distinct planes, and there 

 exists a point, not a cogredient point, ivhich is a member of the punctual associates of 

 both planes, then p and q possess a common member. In symbols, 



= : p, 



Df 



The next axiom, XIII HpR, is the " Euclidean" axiom. 



*22'61. XIII HpR is the statement that the cogredient points are points. In 

 symbols, 



XIII Hp R . = . oo R , c pnt Ht Df 



The next three axioms, namely (XIV.-XVI.) Hp R, establish the theory of the 

 order of points as determined by the point-ordering relation (cf. *20'5l). Incidentally 

 some existence theorems can be deduced from them, which would else have to be 

 provided for elsewhere. 



*2271. XIV HpR is the statement that, if A and B are two distinct non- 



