SCIENCE ABSOLUTE OF SPACE. 19 



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

 I/ and the line CD are the same. Such an I/ is 

 designated by I/lllL. 



23. If the L form line CDF III 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 y; (23), Y=X X . 



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

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



If^=n.#, take ;r=AC=CG=GH=&c., until 

 we get AH^jy. 



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



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

 HL; and so AB_ f ABJ n 



y HIT I CD j 



or Y=X n =X x . 



If x, y are multiples of /, suppose xmi } 

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

 consequently n _y 



