44 PHILOSOPHICAL TRANSACTIONS. [aNNO 1791. 



before tano-ht; but the continuation of them in infinitum would have been use- 



rt 



less, as the arese of curves, whose ordinates are ax'" (where x denotes the absciss, 

 and a, n, and ?« invariable quantities) had not been discovered many years before 

 the time of Mercator's publication, and consequently it would have been of little 

 use to transform an ordinate or fluxion, whose area or fluent is unknown, into 

 another form, of which the area, &c. is equally unknown. 



Sir Isaac Newton extended the rule for raising a binomial, to any affirmative 

 power, to negative powers, the extraction of roots and fractional indexes, by ap 

 plying the law of the series for affirmative powers to them, and continuing it in 

 infinitum. M. de Moivre extracted the root, &c. ot a multinomial by a series of 

 a similar nature ; but these methods will only apply in the most simple cases, 

 when not more than one root is to be extracted. In every complicate case, viz. 

 the extraction of roots of quantities which involve the roots of compound quan- 

 tities, of irrational quantities, recourse must be had to the old methods of multi- 

 plication, division, and extraction of roots. If a root of a complicate irrational 

 quantity be required by a series proceeding according to the dimensions of ar; first 

 reduce all the irrational quantities contained under the root by multiplication, 

 division, and extraction of roots into serieses proceeding according to the dimen- 

 sions of X, so that the terms of the least dimensions be constituted first, if an 

 ascending series be required, and so on ; and the contrary, if a descending; then 

 add the several sums together, and extract the root of the resulting sum by a 

 series which proceeds according to the dimensions of x, and it will be the root 

 required. This is then illustrated by an example. 



The principal use of reducing quantities into series, proceeding according to 

 the dimensions of the variable quantity, is, as before mentioned, for finding the 

 area of a curve from its ordinate ; or, which corresponds, the integral from its 

 nascent or evanescent increment ; but the serieses deduced should converge, 

 otherwise from them cannot be found the area or integral. In the Meditationes 

 Analyticse a method was first published of finding when these series will converge 

 and when not. Hence most commonly the series for the area contained between 

 •2 ordinates, or integral between 2 diffisrent increments, deduced by the com- 

 mon method, will diverge ; on which account, in the same book, is given a 

 method by interpolation of finding the area or integral contained between any 2 

 different values of a: by converging series, if the area, &c. is finite. 



To find whether a given value (-f- a) is less than the least affirmative or nega- 

 tive root (x) of a given algebraical equation a -\- bx -\- cx'^ -\- dx^ -}- &c. = O, 

 if all its roots are possible; transform the equation into another, whose root z is 



the reciprocal of the root x z= - of the given equation, and for z in the resulting 



equation write respectively v -\- a and v — a ; and if from the former substitu- 



