40 SCIENCE ABSOLUTE OP SPACE}. 



intersection of the straight BN with the 

 straight CS (if there is one) having been found, 

 the perpendicular drawn through this inter- 

 section, in PCQ, to the straight CS will mani- 

 festly be the intersection of plane MAB and 

 plane PCQ. 



37. On the straight AM II BN, is found such 

 an A, that AM ^=BN. If (by 

 34) we construct outside 

 of the plane NBM, GT II 

 IC BN, and make BG1GT, 

 GC=GB, and CPllGT; 

 and so place the hemi- 

 plane TGD that it makes 

 with hemi-plane TGB an angle equal to that 

 which hemi-plane PCA makes with hemi-plane 

 PCB; and is sought (by 36) the intersection 

 straight DQ of hemi-plane TGD with hemi- 

 plane NBD; and BA is made -LDQ. 



We shall have indeed, on account of the sim- 

 ilitude of the triangles of Iy lines produced on 

 the F of BN ( 21^, manifestly DB-DA, and 

 AM^BN. 



Hence easily appears (L-lines being given by 

 their extremities alone) we may also find a 

 fourth proportional, or a mean proportional, 

 and execute in this way in F, apart from Ax- 

 iom XI, all the geometric constructions made 



