76 On the PRINCIPLES of the 



PROPOSITION II. 



The ratio of each of thefe two general geometrical expref- 



R Q^ 



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



given or afTigned ratio. 



For, lince the firft term in each has to twice the fecond a ra- 

 tio greater than any given or affigned 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 



^~Q- 7H. Wherefore each of thefe exprefhons has to the firft 



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

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

 Prop. I.). Confequently the ratios which thefe expreffions 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-- — rI^^^ ^- Therefore, &c. Q.E.D. 



otherwise: 



In the firft expreffion, 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 exprefllon itfelf 

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

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

 tio 



