ANTECEDENT'JL CALCULUS. 69 



has to B a ratio having to the ratio of A to B the fame ratio of 

 R to Q^ is truly expreffed by 



and that half tlie difference of thefe exprefllons is 



Before I proceed farther, however, in the confideration of 

 thefe expreffions, it may not perhaps be improper to premife the 

 few following lemmata, which are almoft too evident to re- 

 quire demonflration. 



LEMMA I. 



If any ratio be compounded with its inverfe, or the inverfe 

 of any ratio the fame with it, the compofition produces a ratio 

 of equality. . 



For of the three magnitudes A, B, A, by the definition of 

 compound ratio, (5. Euc. Sim son's edit.), the ratio of A to B, 

 compounded witli the ratio of B to A, is the ratio of A to A, 

 or a ratio of equahty ; and if the ratio of C to D be equal to, 

 or the fame with the ratio of A to B, its inverfe, D to C, is 

 equal to, or the fame with the ratio of B to A, (Prop. B. ibid.) : 

 Therefore, (Prop. F. Euc. 5. Simson's edit.), the ratio of A to 

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



ratio 

 3 



