78 On the PRINCIPLES of the 



R O p p 

 the fecond to N. Therefore — . -^. is lefs than- 



R-2Q^ R— Q ^ 



R R— Q Q T>.r 1 .^R O t- 



-—.-—->-. — — -==^. N. But Iince the ratio of — . -i^ to E, 



Q. Q. gRiiO. Q. p R-2Q^ 



being compounded of the firfl term to N, and of B to E, is the 

 fame with the ratio of C to D, E is a given magnimde, (2. Data), 

 and B — E a given magnitude, (4. Data). Wherefore the given 



. , R Q B— E . , ^ , R R— Q ^ O ,T 

 magnitude, q^- — rI^- -£", is lefs than q^- -^- ^f^- N, 



which (Cor. 2. Prop, i.) is lefs than any given magnitude,, 

 which is abfurd. 



In like manner is it demonflrated, that the ratio of the fe^ 

 cond expreffion to N, is nearer to the ratio of its firfl term to> 

 N than any given ratio. Q^E. D. 



SCHOLIUM. 



If the fame reafoning be applied to the expreffion, 

 ^-^^ Q- - V^"^ ^= . which is half 



the difference of the two geometrical exprefllons that have 

 refpe(flively to B ratios having to the ratios of A+N to B, and 

 A — N to B, the ratio of R to Q^ we get the ratio of the firfl 

 term to twice the fecond, the fame with that of A to 



R— Q^ R-2Q_ N" . - , ^ , - 



—7; — — ;;— • -— , and the ratio of the fecond to four times the 

 <^ 3Q„ A' 



third. 



