i6 



alternately affirmative and negative ; and therefore the two feries have 

 tlieir alternate correfpondent terms aflefted by unlike figns ; thefe terms 

 therefore may vanifli by addition. But in order to this, it is alfo ne- 

 celTary that the quantities (both numeral and literal) of thefe terms 

 fliould be the fame. Novi^ the numeral quantity, or coefficient, mult be 

 the fame in correfpondent terms of thefe feries, becaufe each is i 

 divided by the index of the literal quantity e in that term ; it only 

 remains therefore that care be taken to have the quantity e of the 

 fame value in both feries ; and this is done by providing that both 

 its numerator and denominator be the fame : and that its numerator is 

 the fame, follows from />, the quantity inferted between a and b, being 

 an arithmetical mean between them ; for /> - a and b - p are the nu- 

 merators ; and that the denominator is the fame, follows from its being 

 the firfl: of the two ratios that is inverted, for p, the quantity inferted, 

 mud always be the denominator of both fraftions. This appears, when 

 the given ratio is afcending, from what was faid in art. 15 and 16: 

 and if the given ratio had been defcending, as of b to a, ftill it is 

 to be refolved into the ratios oi b to p and of /> to a; and if the 

 firfl of them be inverted, it becomes the ratio of p to b, or afcend- 

 ing, and therefore by art. 13, ^ will be the denominator of the frac- 

 tion: and the other ratio, that of p to a, being ftill defcending, by 

 art. 13, p will be the denominator of the fraftion here alfo ; and thus 

 the fecond terms of the two feries, and the alternate terms from 

 them, being compofed of the fame quantities, both literal and numeral, 

 and having unlike figns, they will entirely vanifh when the feries are 

 added together. 



It may be proper here to obferve, that the two rules, (that for 

 making />, the inferted term, an arithmetic mean between a and b ; and 

 that for inverting the former of the two ratios,) become necelTary to- 

 gether; that is, they are fo connefted together, as that when either is 

 obferved, the other muR be obferved alfo. But we may negleft both 

 thefe rules, and yet arrive at the fame conclufion, by the following 

 rules: divide the difference of the given terms into two parts propor- 

 tional 



