ant:ecedentjl calculus. 



71 



Secondly, Let the ratio of F to D be lefs than that of A to 

 B. Then (id. Euc. 5.) the magnitude, which B A D F E 

 has to D the fame i-atio with that of A to B, 

 is greater than F. If E therefore be that 

 magnitude, the ratio of F to D, compounded 

 with the ratio of B to A, is the fame with the 

 ratio of F to D, compounded with the ratio 

 of D to E, (Propofitions B. and F. 5. Euc. 

 Sim.). Wherefore the ratio produced by 

 compounding the ratio of F to D with that 

 of B to A, is the fame with the ratio of F to 

 E. But fince F is lefs than E, the ratio of 

 F to E is lefs than that of E to E, (10. Euc. 5.), or a I'atio of 

 equahty, Q^E. D. 



LEMMA m. 



If any ratio be compounded with a ratio of equality, it is 

 not altered thereby. 



For the ratio of C to D, compounded with the ratio of A to 

 A, is the fame with the ratio of C to D, compounded with the 

 ratio of D to D^ (Prop. F. 5. Sim. Euc), which, by th,e defini- 

 tion of compound ratio, is that of C to D. Q. E. D. 



These three Lemmata are alfo evident from Formula i. 

 Theorem i. Univerful Comparifon. 



Cor. From this and Lemma i. with the definition of com- 

 pound ratio, it is evident, that if with any ratio there be com- 

 pounded a greater one, there arifes a ratio greater than it ; and 

 that, if with any ratio diere be compounded a lefs one, there 

 arifes a ratio lefs than it. 



I L EM- 



