io 6 Dr . Waring on 
lefs than ±(a — b ), and in the fecond cafe greater than 
(a -f b) ; or, more general, it will converge in the firft 
cafe when x n is always infinitely lefs than the reciprocal 
(a + b 
when 
of the quantity 
, where n is infinite % 
(* 2 + £ 2 )” 
and in the latter cafe it will converge when x n is infinitely 
greater than P. 
It may not be improper to obferve, that the fame values of 
the root are to be applied in the equations, which are applied 
in the feries. 
3. Sir Isaac Newton, in the binomial theorem, reduced 
the power or root of a binomial into a feries proceeding ac- 
cording to the dimenfions of the terms contained in the bino- 
mial. M. de Moivre reduced the power or root of a multi- 
nomial into a like feries ; but in all cafes, except the moft 
fimple, we muft ftill recur to the common divifion, extraction 
of roots, &c. 
4. Melfi Euler, Maclaurin, and other mathematicians,, 
finding that the feriefes before-mentioned often converged ilowly, 
or, if the truth may be confeffed, commonly not at all, to 
deduce the afea of a curve contained between two values a and 
b of the abfcifs, or fluent of a fluxion between two values a 
and b of the variable quantity x, interpolated the feries or area 
between a and b ; that is, found the area or fluent contained be- 
tween the abfci ffre a and rf-f a, then between the abfciffie a + & 
and a + 2*, and then between the abfciffae a -f 2 oc and a -f 3^ and 
fo on, till they came to the area between b — « and b . M. 
Euler obferved, that when the ordinate became o or infinite, 
the feries expreffing the area converges flowly ; and therefore,,. 
2 Ini 
