78 On the PRINCIPLES of the 



A- 



the fecond to N. Therefore — -. -=i— . is lefs thanr 



R R_0 O -D r V .^R Q ^ 



Q:-~Qr— ^''- ^""^ '^"''^ '''' '■'''° °^ Qi— R=k'° • 



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



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



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

 R— Q^ R-2Q^ 



. . R "qT S— e . , ^ , R R— Q^ ^~q7~ ,. 

 magnitude, q]^'— rI^-— IT"' ^^ ^^^^ ^^^^ Q1'~Q;^ ^C 



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

 which is abfurd. 



In Hke manner is it demonftrated, that the ratio of the fe- 

 cond expreflion to N, is nearer to the ratio of its firft term tOv 

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



SCHOLIUM, 



If the fame reafoning be applied to the exprefTion, 

 ^--^ '^^^^ 2. , which is h.lf 



the difference of the two geometrical exprefhons that have 

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



third, 



