520 DR. A. N. WHITEHEAD ON 



*30'1. Proposition. Assuming (II.-XIV.)H/)R, if two figures are in perspective, 

 their cardinal numbers are equal and each greater than one. In symbols, 



h .-. (II. -XIV.) HpR . = : upersp Rt v . D . Nc'w = Nc'v . Nc'w > 1 



Proof. The equality of the cardinal numbers follows from the definition ; also if 



both figures were unit classes, then (cf. *28'33) they would be collinear. 



*30'3. Proposition. Assuming (IL-XIV.)HpK, if the figure u is in perspective 

 with the figure v, and also with the figure w, and if u, v, w are respectively collinear, 

 and the punctual lines respectively containing u, v, w possess a common meeting, 

 then either v is in perspective with w, or the joint class of v and w (i.e., v u w) is 

 collinear. In symbols, 



I" : : (II. -XIV.) Hp R . D . '. u persp, i( v . u persp B( v . m, m', m" e lin m . 



u c m . v c m' . w c m" . g; ! (m n mf n m") . D : v persp B , w . V . m' = m" 



Proof. DES ARGUES' well-known propositions respecting triangles in perspective being 

 coaxial, and its converse, can now (cf. *28"11"12"31 1 32'33"41'42) be proved. Then by 

 drawing a figure for the present proposition the conclusion easily follows from some 

 pure geometrical reasoning. 



*31. Tfie Point-Ordering Relation. 



It will be proved in this section that the point-ordering relation (R pn ) has at any 

 instant the same properties as the essential relation of Concept I. (cf. *31'3). It 

 follows that the ordinary Euclidean geometry holds of the figures of Concept V., the 

 points at infinity being the points of the punctual plane oo R( , and the metrical ideas 

 being introduced by appropriate definitions. 



*31-11. Proposition. Assuming (II. -XL, XIII, XIV.) HpR, the class R pll ; (;;;) is 

 identical with the class of non-cogredient points. In symbols, 



I- .'. (II.-XL, XIII, XIV.)H^R . =,. R pn ;(;;;0 = pnt M - oo f 



'St 



Proof.Cf. *3'42 and *20'51 and *22'7l and *26'24. 



*31'12. Proposition. Assuming (II. -XV.) HpR, if a is a punctual line, and A 

 and B are two non-cogredient points on it, then a, without its cogredient point, is 

 identical with the whole class formed by R im ; (;AB) and R pn : (A;B) and R ; (AB;) 

 together with A and B added as members. In symbols, 



I-.-. (II-XV.)HpR.D: aelin K . A, Bea-oo^. A ^ B .3. 



a- QO B( = R pn ; (;AB*) u R pn ; (A;BO u R pn >(AB;) u I'A u i'B 



Proof The identity is to be proved by showing that each class contains the 

 other. For one half of the proof, cf. *2272 and *27'2l. For the other half, cf. 

 *20-5l and*27-21'31. 



