70 



On the PRINCIPLES of the 



rar'o of A to B compounded with the ratio of B to A, or a ra- 

 tio of equalitj. Q^E. D. 



LEMMA II. 



If with the inverfe of any ratio there be compounded a ratio 

 greater than it, the compofition produces a ratio of greater ine- 

 quality, or a ratio of which the antecedent is greater than the 

 confequent ; and if with the inverfe of any ratio, there be com- 

 pounded a ratio lefs than it, the compofition produces a ratio of 

 lefs inequality, or a ratio of which the antecedent is lefs than 

 the confequent. 



First, Let the ratio of C to D be greater than that of A to 

 B. Then ( I o. Euc. 5.) the magnitude, which BADGE 

 has to D the ratio of A to B, is lefs than C. 

 If E therefore be that magnitude, the ratio of 

 C to D, compounded with the ratio of B to 

 A, is the fame with the ratio of C to D, com- 

 pounded with the ratio of D to E, (Propoli- 

 tions B. and F. 5. Euc. Sim.). Wherefore, the 

 ratio produced by compounding the ratio of 

 C to D with that of B to A, is the fame with 

 the ratio of C to E. But lince C is greater 

 than E, the ratio of C to E is greater than 

 that of E to E, (10. Euc» 5.), or a ratio of 

 equality. Q;_E. D. 



Secondly, 



