MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 519 



Proof. The direct proposition follows from *20'22'61. For the converse, let p 

 and q be a pair of punctual planes. If neither p nor q be oo 8f , then cf. *20'22'61. 

 Consider now p n oo K( , where, p is distinct from OO K( . Take (cf. *28'12) a non- 

 cogredient point A, not in p. Also (cf. proof of *28'3l) there are two distinct points 

 B and C in p n OO K( . Hence (cf. *27'5l) cm R< '{(A n B) u (A n C)} is a plane, and its 

 punctual associate, q say, possesses A and B and is distinct from p. Hence 

 (cf. *27'23 and *28'22) p n q is identical with p n OO K( . But (cf. *28'22) p n q is a 

 punctual line, and hence p n oo u( is a punctual line. 



*28'33. Proposition. Assuming (II.-XIV.) HpR, two distinct points lie in one 

 and only one punctual line. In symbols, 



|- .'. (II. -XIV.) Hp R . D : A, Bepnt K , . A * B . D . m{meliri JM . A, Bern} e 1 



Proof. Firstly, only one punctual line (if any) possesses both A and B (cf. *27'31 

 and *28'22 - 32). Secondly, to prove that a punctual line exists possessing both A 

 and B. If either point is non-cogredient, cf. *27'52. If both points are cogredient, 

 then (if. *28'11'12) two non-cogredient points C and D exist such that the four 

 points A, B, C, D are not punctually coplanar. Hence (cf. *27'5l) the meeting of 

 the punctual associates of cm m ' {(A n C) u (B n C)} and of cm K( ' {(A n D) u (B n D)} is 

 a punctual line possessing A and B. 



*28'41. Proposition. Assuming (II.-XIV.) HpR, three points, which are not 

 collinear, lie in one and only one plane. In symbols, 



h . '. (II.-XIV.) Hp II . D : u e 3 n pnt K( . -*~ coll K( ! u . D . p {p e pple IH . u c p} e 1 



Proof. If the three points are all cogredient, then (cf. *28'22'32) ( R4 is the only 

 punctual plane which possesses them all. If the three points are A, B, C, and 

 A be non-cogredient, then (cf. *27'51 and *28'32) the punctual associate of 

 cm n( '{(A n B) u (A n C)} is a punctual plane, possessing A, B, and C, and is the only 

 one. 



*28'42. Proposition. Assuming (II.-XIV.) HpR, three punctual planes, which do 

 not meet in a punctual line, meet in one point. In symbols, 



h . . (II.-XIV.) Hp R . D : u e 3 n els' pple m . ^ . n ' u e lin K u 1 



Proof. Let p, q, r be the three punctual planes. Assume that p and q are 

 neither the punctual plane oo Bi . If q n r is contained in OO H( , then cf. *20'22'61 and 

 *27'23 and *28'22'32. If q n r is not contained in o> R( , then cf. *27'12 and *28'21. 



*30. Perspective. 



A few propositions on perspective (cf. *20'3t'32'33) are required as a preliminary 

 to the discussion of the point-ordering relation (cf. *20'5l). 



