^NTECEDENl'JL CALCULUSr 73 



Scholium. In like manner is it fhewn, 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 j 

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

 DE to EF, &c. be refpecflively 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 geometrical, 

 expreffions in p. 3. Antecedental Calculus, viz. 



R R— O ,, R R— Q R-2Q ,,. R R— Q R-2a R-?0 

 and 





i.A'^-'^ 



has to N a ratio nearer to the ratio of -^^-— — ^^ to B than any 



given or afligned 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 equality than any given, or afligned ratio, or 

 Vol. IV. K - than. 



