76 On the PRINCIPLES of the 



PROPOSITION 11. 

 The ratio of each of thefe two general geometrical expref- 



fions to N, is nearer to the ratio of -^- — rI^ to B than any 



given or afligned ratio. 



For, fince the fir ft term in each has to twice the fecond a ra- 

 tio greater than any given or afligned ratio, (Prop, i.), and the 

 fecond has to all the fucceeding terms, be their number ever fo 

 great, a ratio greater than any given ratio, (Cor. 3. Prop, i.) the 

 ratio of the firft term to all the fucceeding ones is a fortiori 

 greater than any given ratio, being greater than that of A to 



5— ^N. Wherefore each of thefe expreflions has to the firft 



term a ratio nearer to that of equality than any given or aflign- 

 ed ratio, or than by any given or afligned magnitude, (Cor. 4. 

 Prop. I.). Confequently the ratios which thefe expreflions have 

 to N, are nearer to the ratio of the firft term in each to N, than 

 any given or afligned ratio. But the ratio of the firft term in 



each to N^ is that of -q-- — jt— rr to B. Therefore, &c. Q.E.D. 



^ B ->■ 



otherwise: 



In the firft expreflion, the firft term, with twice the fecond, 

 is much greater than the whole of it, (Cor. 3. Prop, i.), and 

 confequently has to N a greater ratio than the expreflion itfelf 

 has to N, (8. E. 5.). But this ratio exceeds the ratio of the firft 

 term to N lefs than any given or afligned ratio. For, if the ra- 

 tio 



