14 SCIENCE ABSOLUTE OF SPACE. 



FGN=st.Z. Therefore also ZDCP+ZCDQ 

 =st.Z. 



II. If R falls without the sect FG; then 

 ZNGR=ZMFR, and let MFGN^NGHL= 

 LHKO, and so on, until FK>FR or begins to 

 be >FR. Then KO II HL II FM (7). 



If K falls on R, then KO falls on RS (1); 

 and so ZRFM+ZFRS=ZKFM+ZFKO=Z 

 KFM+ZFGN=st.Z; but if R falls within the 

 sect HK, then (by I) ZRHL+^KRS=st.Z = 

 ZRFM+ZFRS=ZDCP+ZCDQ. 



14. If BN II AM, and CP II DQ, and ZBAM 

 +ZABN<st.Z; then also ZDCP+ZCDQ< 

 st.Z. 



For if ZDCP+ZCDQ were not <st.Z, and 

 so (by 1) were =st.^, then (by 13; also Z. 

 BAM+ZABN=st.Z (contra hyp.). 



15. Weighing 13 and 14, the System of 

 Geometry resting on the hypothesis of the 

 truth of Euclid's Axiom XI is called j/ and 

 the system founded on the contrary hypoth- 

 esis is S. 



All things which are not expressly said to 

 be in ^ or in S, it is understood are enunci- 

 ated absolutely , that is are asserted true 

 whether I or S is reality. 



