^NfECEDENT^L CALCULUS.' 73 



Scholium. In like manner is it £hewn, that, if AC be a 

 third part of AB, CD of AC, DE of CD, EF of DE, and fo on, 

 the ratio of each term to all the fucceeding ones taken together, 

 be their number ever fo great, exceeds the ratio of two to one ; 

 and, in general, if the ratios AB to AC, AC to CD, CD to DE, 

 DE to EF, &c. be refpedlively the fame with that of A to N, 

 that the ratio of each term to all the fucceeding ones, be their 

 number ever fo great, exceeds the ratio of A— N to N. This is 

 alfo evident from the well known method of finding the aggre- 

 gates of geometrical progrefTions ; and if the ratio of AC to- 

 CD be greater than that of AB to AC, the ratio of CD to DE 

 greater than that of AC to CD, and fo on, the ratio of any 

 term to all the fucceeding ones, be their number ever fo great, 

 exceeds the ratio of A — N to N, more, than the ratio it has to tlie 

 fame number of fucceeding terms, exceeds it, when each term 

 has to the immediately fucceeding one the ratio of A to N. 



I NOW proceed to prove, that each of the general geometi'icaL 

 exprefSons in p. 3. Antecedental Calculus, viz. 



R , R— ^, R R— Q R-zQ ,, R R— Q R-2Q R-?0 



and 



R ,R— Q^,^ R R— Q„ R-:Q^,. R R— Q R-iO R-^g 

 -— ■ A ^ .N — -— -. ->-A ->-N H . ^. ->rA — 52irN3_ + _&.c 



B- 



Q. 



r__^r-q: 



has to N a ratio nearer to the ratio of -=:-- — ~- to B than 



B 



R-.Q^ 



any 



given or afTigned ratio, or than by any given or affigned mag- 

 nitude, when A+N' and A -N have either to A or B ratios 

 nearer to that of eqxiality than any given, or afCgned ratio, or 

 Vol. IV. K - than. 



