38 SCIENCE ABSOLUTE OF SPACE. 



will have ( 27) OCD : OAB = 1 : sin z, pro- 

 M vided that DM II BN. But sin 

 z is not >1; and so AB is 

 not >DC. Therefore a quad- 

 - rant described from the cen- 



A B S 



FIG. so. ter A in BAG, with a radius 



DC, will have a point B or O in common with 

 ray BD. In the first case, manifestly ^=rt.^; 

 but in the second case ( 25) 



(OAO-OCD) : OAB=1 : sin AOB, 

 and so ^=AOB. 



If therefore we take ^=AOB, then DM will 

 be II BN. 



35. If S were reality; we may, as follows, 

 draw a straight _L to one arm of an acute angle, [211 

 which is II to the other. 



Take AMI BC, and 

 suppose AB=BC so 

 small (by 19), that 

 if W e draw BN II AM 



( 34), ABN > the 

 FIG. 31. . . 



given angle. 



Moreover draw CP II AM (34); and take 

 NBG and PCD each equal to the given angle; 

 rays BG and CD will cut; for if ray BG (fall- 

 ing by construction within NBC) cuts ray CP 

 in E; we shall have (since BN^CP), ZEBC< 

 ZECB, and so EC<EB. Take EF=EC, EFR 



