ANTECEDENTAL CALCULUS, 75 



lition, that any given ratio C to D is greater than it, is ab- 

 furd. 



And, fince the ratio of the fecond term to thrice the third, is 



that of A to -^N, it is proved exadlly in the fame manner, 



that this ratio is greater than any given ratio. And precifely 

 in the fame way is it demonftrated, that the ratio of the third 

 term to four times the fourth, is greater than any given ratio ; 

 and fo on. 



CoR. I. If R — Q^be equal to Q^ the ratio compounded of " 

 C to D, and R — Qjo Q^is the fame with that of C to D, (Lem- 

 ma 3.) ; and if R — Qj3e greater or lefs than Q>_ the ratio com- 

 pounded of C to D and R — Qj:o Q>_ is accordingly greater or 

 lefs than that of C to D, (Cor. to Lemma 3.). 



R-Q^ R-2Q, 



Cor. 2. The magmtudes, -^— ^N, _.-^.__^N, 



^c. -Q--N, —Q— N, -^J^N, &.C. are lefs than any given or 

 afTigned magnitude. 



Cor. 3. The ratio of each term to all the fucceeding ones, be 

 their number ever fo great, is greater than any given ratio, 

 (Scholium to Lemma 4.). 



CoR. 4. The magnitudes A t^^^N, A+^II^N, &c. have 



refpeaively to A ratios nearer to that of equality than any given 

 ratio, or than by any given magnitude. 



CoR. 5. The magnitude which has to B a ratio, having to the 

 ratio of A to B the ratio of R to Q^ has to twice the firft term, 

 m each of thefe general geometrical expreffions, a ratio greater 

 than any given ratio. 



K2 PRO=^ 



