18 SCIENCE ABSOLUTE OF SPACE. 



their intersection (by 7 and 11); and so also 

 ray AP and ray BD intersect. 



From this it is evident that Euclid's Axiom 

 XI and all things which are claimed in geome- 

 try and plane trigonometry hold good abso- 

 lutely in F, L lines being substituted in place 

 of straights: therefore the trigonometric 

 functions are taken here in the same sense as 

 in .1; and the circle of which the L form ra- 

 dius = r in F, is = 2*r; and likewise area of 

 Or (in F) = -r* (by - understanding ^Ol in F, 

 or the known 3.1415926. . .) 



22. If ray AB were the L of ray AM, and 

 C on ray AM; and the ZCAB (formed by the 

 straight ray AM and the L form 

 line ray AB), carried first along 

 the ray AB, then along the ray 

 BA, always forward to infinity: 

 the path CD of C will be the 

 line L of CM. 



For let D be any point in line 

 CD (called later L'), let DN be II CM, and B 

 the point of L falling on the straight DN. We 

 shall have BN=^ AM, and sect AC=sect BD, and 

 so DN ^ CM, consequently D in I/ . But if D in 

 I/ and DN II CM, and B the point of L on the 

 straight DN; we shall have AM^BN and CM 

 whence manifestly sect BD=sect AC, 



