SCIENCE ABSOLUTE OP SPACE. 19 



and D will fall on the path of the point C, and 

 L' and the line CD are the same. Such an L' is 

 designated by I/lliL. 



23. If the L form line CDF ||| ABE ( 22), 

 and AB-BE, and the rays AM, BN, EP are 

 axes; manifestly CD=DF; and if any three 

 points A, B, E are of line AB, and AB=n.CD, 

 we shall also have AE^n.CF; and so (mani- 

 festly even for AB, AE, DC incommensurable), 

 AB:CD=AE'CF, and AB'CD is independent 

 of AB, and completely determined by AC. 



This ratio AB:CD is designated by the cap- 

 ital letter (as X) corresponding to the small let- 

 ter (as x) by which we represent the sect AC. 



_y 



24. Whatever be x and>v (23), Y=X X . 



For, one of the quantities x, y is a multiple 

 of the the other (e. g. y of x), or it is not. 



If^=n.^, take*=AC = CG=GH=&c., until 

 we get AH=jF. 



Moreover, take CD ||| GK ||| HL. 



We have ((23) X=AB:CD = CD:GK=GK: 

 HL; and so AB = f AB 



y HL~ [ CD 

 or Y=X n =X x . 



If x } y are multiples of /, suppose 

 and y^ni; (by the preceding) X=I m , Y=P, 

 consequently n _y 



Y-X m =X x 



