Infinite . Series * 
95 -: 
D 
t(t — 7t)(t — ^)(r— O-) (r — yj 
c — g° • t + &c. =ra. 
(§ — f) (<>- t/) <r(^-w)(«r — f )(cr — *>)(* — ») 
. — — 7- r\e.e — 7 r • e — t 
v (« — w ■) (y — e) ( V — <r) ( 0 — r) J 
Their refolu tions were firft given in the Meditatlones Ana* 
lytic# 9 publiflhed in the year 177 4, and require the extra&ion 
of a quadratic, cubic, and in general of an equation of ( \n ) 
dimenfions ; which rules will often give a nearer approximate 
than the preceding, when the roots are not nearly equal. 
3. Thefe rules may he applied to find approximations to the 
roots of algebraical equations : for example, let the algebraical 
equation be x n — px n '~ l +: gx n ~ z - &c. = o, fubfHtute in it for x 
two quantities and a + e much nearer to one root than to any 
other, and there refult a n —pa "- 1 +, qa 71 " 7 - - &c — A, and (a 4- gfi 
^p G ? m h e ') n ~ 1 + (<* + e ) n ~~ z “ = B ; then, by the rule of 
falfe, B - A : e :: A 
« , k — I . n — 2 0 
a — pa +qa — -&C. 
(m 
n~~lpa z ~\~n — 2qa n 3 — &c.)-[-&c. 
~h% 
whence a — b a near approximate value to the root fought. If the 
quantities, in which are involved e,e 2 ,e 3 ,&.c. on account of e being 
very fmall, be rejefled, then will the approximate fought b^. 
- a —p a _ _ + <l a , . ; which will nearly.be the fame 
7 ia 1 —n — \pa Zj r7i-*-lqa n 3 — &c. 
as found, where a near approximate is given, from the method 
given by Viet a, Harriot, Oughtred, Newton, De 
Lagny, Halley, Sec. 
4* From this expreffion it follows, that if (a) be a root of 
the equation •na n ~~ 1 -\pa n ~ ml +,&c. ~ o, of which the roots 
are limits between the roots of the given equation, the ap- 
proximation found will be infinite. 
5. I11 finding thefe approximations, when there are two or 
piore quantities contained in the given equation dependent on 
each, i 
