Infinite Series,. j x j 
of curves, which depend on feme of the firft terjais of the 
feries ; but will very feldom be ufed for finding the roots of an 
equation; the rule of falfe, or method given by Vieta, will 
ever be fubftituted in its ftead. 
13. The values of x may be required between which the above- 
mentioned feries will converge, as 
the infinite feries anfwers no purpofe when it diverges* Firft, 
if an afeending be required, write fory in the given algebraical 
equation an infinite quantity, and find the roots of v in the 
equation thence refulting P~o ; which fory write in the fame 
equation, and find the roots of x in the refulting equation 
which contain irrational quantities, viz. if one root be x = aj 
then let it contain (x — where m is not a whole number; 
find the roots of the equations refulting from making every 
irrational fun&ion of (*) contained in the given equation = o, 
there being no irrational function of y contained in it ; then, 
if x be greater than the leaft root not =0 of the above-men- 
tioned equations, the feries will not converge ; but if it be a 
feries defeending according to the dimenfions of x 9 and a; be 
lefs than the greateft root of the above-mentioned equations, 
the feries will not converge. 
In interpolating feriefes to inveftigate the fluent contained 
between two values a and bo£ the fluxion 
it is preferable to make the abfeiffie begin from every one of 
the above-mentioned roots contained between a and b . 
Moft commonly thefe feriefes will not converge unlefs x 
be lefs, &c. than other quantities not inveftigated by this rule. 
14. Sir Isaac Newton gave an elegant example of this rule 
in the reverfion of the feries, y = ax-{-bx z -\-cx 1i -\-lkc. from which 
the inveftigation of the law of the feries has never been 
..attempted, In the year 1757 I fent the firft edition of my 
Vol , LXXVL Q Meditationes 
