SCIENCE ABSOLUTE OP SPACE. 25 



But (by 27) sin v : sin v' =cos u : cos u ; 



,, sin u _ sin ' 

 consequently- .Qy= 



COS U COS U 



or OjF : Oy' =tan u' : tan u=tan w : tan w' . 



Moreover, take CN and C'N' II AB, and CD, 

 C'D' L-form lines j_ straight AB; we shall 

 have also (21) 



Qy : Qy' =r : r' , and so 



r : r =tan w : tan w' . 



Now let p beginning from A increase to in- 

 finity; then w=z, and w '=z ' , whence also 

 r : r tan z : tan z ' . 



Designate by i the constant 



r : tan z (independent of r) ; 



whilst y=0, 



r i tan z , ^ -. 



-=- =1, and so 



y y 



-2^i. From 29, tan z=y 2 (Y-Y- 1 ); 

 tan z 



therefore -^_^=i^ 



or (24) ^_L.i. 



~\ i~ 



But we know the limit of this expression 

 (where y=0) is 



/I 



-. Therefore 



nat. log I 



