MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 517 



and B is any other point, then A n B possesses one and only one member. In 

 symbols, 



|- .-. (II.-XIV.) Hp R . D : A e pnt R( - OO R( . B e pnt R( .A^B.o.AnBel 



Proof. (i) If B is non-cogredient, cf. *14'21 and *20'51 and *2271. (ii) If B is 

 cogredient, then (cf. *14'14) let b be a member of B. Then (cf. *26'24) there is on b 

 a non-cogredient point D. Hence by (i) A n D possesses a single objective real, d 

 say. Hence (cf. *14'12) b and d are coplanar and cm, w '(i'6 u i'd) is a plane whose 

 punctual associate possesses both A and B. Also, since a point is three-dimensional, 

 B possesses another objective real, c say, not coplanar with b and d. Hence by 

 similar reasoning cm R( '(i'6 u i'c) is a plane, not identical with cm R( '(i'6 u i'd), whose 

 punctual associate also possesses A and B. Hence (cf. *27'13) these two planes have 

 one objective real in common, and hence (cf. *16'33) this objective real is a member 

 of A n B, and hence (cf. *14'21) A n B possesses one and only one member. 



*27'52. Proposition. Assuming (II.-XIV.) HpR, if A be a non-cogredient point 

 and B be any other point, then ass R( ' (A n B) is a punctual line with a non-cogredient 

 point, and conversely, if m be a punctual line with a non-cogredient point, then there 

 exist two points A and B, such that A is not cogredient and m is identical with 

 ass R( '(A n B). In symbols, 



(gA, B) . A e pnt R( - OO R( . B e pnt K( . A j B . m = ass R( '(A n B) 

 Proof. Cf. *27'22-31-42-51. 



*28. On Figures. 



*28'01. Proposition. Assuming (II.- VI., IX.-XI.)HpR, if t be an instant of 

 time, there exists at least one punctual plane, not the plane OO R( . In symbols, 



[- . '. (II.-VL, IX -XL) HpR. 3 :*eT.D.a! (pple w - t' 



Proof. Cf. *12'42 and *22'41. 



*28'11. Proposition. Assuming (II.-XIII.) HpR, if p be any punctual plane, not 

 the plane oo R( . it possesses at least three non-cogredient points, which are not 

 collinear. In symbols, 



| . '. (II.-XIII.) Hp R . D : p e pple R( . i' 



(gw) . u f. (3 n cls'p). u n O R = A . -~ coll Ht !w 



Proof. Cf. *14'2l and *26'24 and *27'21'31. 



Note.Cf. *16'42 and the note on it. 



*28'12. Proposition. Assuming (II.-XIII.)HpR, if p be any punctual plane, not 



