SCIENCE ABSOLUTE OF SPACE. 25 



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



,1 sin u - sin u 

 consequently- .Qv= 



cos u cos u 



or QjK : OjX =tan 2/ ; : tan u=tan w : tan w . 



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

 C D L-form lines _L straight AB; we shall 

 have also (21) 



OjF : OX 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 i whence also 

 r : r =tan z : tan z . 



Designate by i the constant 



r : tan z (independent of r} ; 

 whilst jF=0, 



r i tan z , -, -, 

 -=- =1, and so 



y y 



From 29, tan z=y&amp;gt; (Y-Y- 1 ); 



tan z 

 therefore 



or ( 24) 



But we know the limit of this expression 

 (where y=0) is 



-. Therefore 

 nat. log I 



