of lines by projection from four dimensions 135 



respectively with U, V, W, being the intersections of these 

 planes respectively with the lines a', b' , c' . Thus/^ =/is a common 

 transversal of the lines a' , b' , c' , d' , e' ; as was to be shown. 



§ 2. Now take four arbitrary lines a, b, c, d in four dimensions, 

 of which no two intersect. Two of these lines, determined by four 

 points, two on each, determine a threefold space, defined by the 

 four points, and this meets a third line in the four dimensional 

 space in a point. From this point, in the threefold space, can be 

 drawn an unique transversal to the two lines spoken of. Thus three 

 lines in four dimensions, of which no two intersect, have an unique 

 transversal. Let then a' be the transversal of b, c, d, and similarly 

 6', c', d' the transversals respectively of c, a, d; a, b, d and a, b, c. 

 Denote the points (6', c), (6, c'), (c', a), (c, a'), (a', 6), (a, b') 

 respectively by A, B, C, A', B', C' and the points (a, d'), (6, d'), 

 (c, d'), {a', d), (6', d), (c', d) respectively by P, Q, R, P', Q\ R'. 



In general use the word plane for the planar twofold space 

 which is determined by three points, and the word space, or 

 threefold for the planar threefold space determined by four 

 points; as above remarked two lines determine a space, each 

 line being determined by two points; reciprocally two spaces, in 

 the most general case, intersect in a plane, there being a duahty 

 of properties in four dimensions wherein a space is reciprocal to 

 a point and a plane to a line. The points A, A', being respectively 

 on the lines C'Q', BR', are in the space {a, d), and evidently are 

 in the space (6, c); the points P, P', being on the lines QR, B'C 

 respectively, are in the space (6, c), and are evidently in the space 

 {a, d). Thus the four points A, A', P, P' lie in a plane, which we 



