SCIENCE ABSOLUTE OF SPACE. 23 



The path called CD will be denoted by CD 

 illAB. 



28. If BN II ^ AM, and C in ray AM, and 

 AC=x: we shall have ( 23) 



X=sin u : sin v. 

 For if CD and AE are _L BN, 

 *<\F / f^ and BF_L AM; we shall have (as 

 in 27) 



OBF : ODC^sin u : sin v. 

 ButevidentlyBF=AE: therefore 



OEA : OCD=sin u : sin v. 

 But in the F form surfaces of 

 AM and CM (cutting AMBN in AB and CG) 

 (by 21) 



OEA : ODC=AB : CG=X. 

 Therefore also 



M 



FIG. 20. 



29. 



X=sin u 

 If ZBAM=rt. 



sin v. 



and sect ABjy, and 

 BNIIAM, we shall 

 have in S 



Ycotan ^ u. 

 For, if sect AB- 

 sect AC, and CPU 

 AM (and so BNll ^ 

 CP), and ZPCD = 

 ZQCD; there is given (19) DS_j_ray CD, so 

 that DS II CP, and so ( 1) DT II CQ. Moreover, 

 if BE 1 ray DS, then ( 7) DS II BN, and so ( 6) 



FiG. 21. 



