Professor Baker, On a proof of the theorem of a double six, etc. 133 



On a 'proof of the theorem, of a double six of lines by projection 

 from four dimensions. By Professor H. F. Bakek. 



[Read 9 February 1920.] 



The theorem in question is that if five lines in three dimensions, 

 of which no two intersect, say a, b, c, d, e, have a common trans- 

 versal, say / ', and we take the five transversals other than / ' of 

 every four of these five given lines, the five new lines so obtained 

 have also a common transversal. Namely if a' be the transversal, 

 beside/', of b, c, d, e, and b' be the transversal, beside/', of 

 a, c, d, e, and so on, so that we have the scheme 



a b c d e 



a' b' c' d' e' f 



in which every line intersects those not occurring in the same row 

 or column with itself, but not the others, in general, then there is 

 a transversal/ of a' , b', c' , d' , e' . 



We see that the theorem is that if we take eight lines a, b, c, d 

 and a', b', c' , d' , so related that a' meets b, c, d, while b' meets 

 a, c, d, and c' meets a, b, d and d' meets a, b, c, and if e',/' be 

 the two transversals of a, b, c, d and e, f be the two transversals 

 of a', b', c', d', then the meeting of one of the two former, / ', 

 with one of the two latter, e, involves the meeting of the other, e', 

 of the two former, with the remaining one, /, of the two latter. 

 But the original relation of the eight lines a, b, c, d, a', b', c', d' 

 has a certain artificiality; the object of the present note is to show 

 that there is a simple figure in four dimensions, possessing perfect 

 naturalness, being determinate when four arbitrary lines of that 

 space are given, from which the figure in three dimensions may 

 be derived by projection ; and that the condition for this derivation 

 is precisely the intersection of the two transversals e and/'. The 

 naturalness of this figure lies in the fact that three lines in four 

 dimensions have just one transversal. 



§ 1. In order to show this, it is necessary to enter into some 

 detail in regard to the elements of the geometry of four dimensions; 

 this appears worth while for its own sake; and in order not to 

 over-emphasize the importance of the theorem in three dimensions 

 which is here made the excuse for this, we first give an elementary 

 proof of this theorem, employing only three dimensions (Proc. 

 Roy. Soc. A, Lxxxiv, 1911, 597), 



