04 Dr. Waring < 9 ?i 
how much nearer the approximate aflumed is to one value 
of the quantity fought poffible or impoffible than to any other, 
and not to the quantity itfelf : hence, when two or more ( n ) 
values of the quantity fought are nearly equal, it is neceflary 
to recur to more difficult rules, viz. to three or more trials ; 
as, for example, let two roots be nearly equal, and write 
a, a + 7 t 9 and a + ^ for the unknown quantity in the given 
equation made =:o, and let the quantities refulting be A, B, 
and C, then will more near approximations to the tw T o roots 
nearly equal of the given equation be a + the two roots(^) of the 
quadratic 
r,K ~ ( A 
7 T+ P 
Bx 
+ 
p 7C[7T p) ' p(p — v)J ' IV p ' "tv. {vr—p) 
^ x th A “ o z for write a, 7r, and o, refpedlively for x in 
f • (p — ■ n/ 
the equation,, and there will refult the quantities A, B, and C. 
More generally, fubftitute for x in the given equation the 
quantities a , a + 7 r, a + p 9 a -p <r, a -p r, a + u, See. where n r, , <r, r, 
&c. are very fmall quantities ; and let the quantities refulting 
be A, B, C, D, B, F, &c. ; then will more near approximations 
to the (ft) roots of the given equation b e.a + a, a+y 9 
& c. where u, j@, y , See. &c. are the n roots (e) of the 
given equation — ' 
■ x + e(« -a:). («-»).(« T 
-<A- 
V TT^cr 
7 ^ 7 + ( 
D 
A' 
B 
D 
£-) 
e-v)/ 
x e 
€ — 7T 
B 
r(v—e) (w— o‘)(’ r - T j&C. 
!> + (- + 
K / \ Kg 
c 
(t 
=r.) 
er(<7 — w)(o— f ) 
c 
x £ 
£ — 7 T 
w — 
A 
par 
■§)(«-”cr 
4 
p(p — 9r)(p — cr 
B 
w(?r- 
$)(«:— tr)(< 
E 
t) 
f ( e -r)( ? -c7)(g~r)&C. ' cr(<r — *)(o--e}(ir-T>&C. t(t ^(t- 
+ (j * ^ ^ (^wgcrrv — — <r) pr— r) (t— v) 
:) 
?((r-o-)&c 
c 
(e~ T ) 
