20 SCIENCE ABSOLUTE OF SPACE. 



The same is easily extended to the case of 

 the incommensurability of x and y. 

 But if q=y-x, manifestly Q=Y'X. 

 It is also manifest that in J, for any x, we 

 have X=l, but in S is X>1, and for any AB [ 

 and ABE there is such a CDF ||| AB, that CDF 

 -AB, whence AMBN^AMEP, though the 

 first be any multiple of the second; which in- 

 deed is singular, but evidently does not prove 

 the absurdity of S. 



25. In any rectilineal triangle, the cir- 

 cles with radii equal to its sides are as the 

 sines of the opposite angles. 



^ For take ZABC=rt.Z, 

 and AMlBAC, and BN and 

 CP II AM; we shall have CAB 

 1 AMBN, and so (since CB_|_ 

 BA), CBlAMBN, conse- 

 quently CPBNl AMBN. 



Suppose the F of ray CP 

 FIG. 17. cuts the straights BN, AM 



respectively in D and E, and the bands CPBN, 

 CPAM, BNAM along the L form lines CD, 

 CE, DE. Then (20) ZCDE^the angle of 

 NDC, NDE, and so = rt.Z; and by like reason- 

 ing Z CED = Z CAB. But (by 21) in the L line 

 A CDE (supposing always here the radius =1), 

 EC:D=l:sin DEC =1: sin CAB. 



