Infinite Series « 
In the MifcelL Anaiyt . and Meditationes the fame principle is 
applied in different rules for finding approximates to the 
greateft and other roots of the given equation ; and alfo limits 
of the ratios of the approximate values of the roots found by 
thefe rules to the roots themfelves are given. 
It is obferved in the Meditationes , that from thefe rules in ge- 
neral to find the greateft root, it is often neceflary that the greateft 
poffible root be greater than the fum of the quantities con- 
tained in the poffible and im poffible part of any impoffible root 
of the given equation: for example, lia^bs/ — r be an im- 
poffible root of the given equation, then it is neceffary that the 
greateft poffible root be greater than a -f b. 
It may further be obferved, that in equations of high dimen- 
fions (unlefs purpofedly made) it is probable, the number of 
impoffible will greatly exceed the number of poffible roots; and 
confequently thefe rules moft commonly fail. 
12 . M. Bernoulli affumed a fraction whofe numerator is a 
rational function of the unknown quantity, and denominator 
the quantity, which conftitutes the equation ; and reduced the 
fradlion into a feries, whofe terms proceed according to the 
dimenfions of the unknown quantity; and thence found an 
approximate value of the greateft or leaft root of the given 
equation or its reciprocal, by dividing the co-efficient of any 
term of the feries refulting by the co-efficient of the preceding 
or fubfequent term : for example, let the equation be 
x n - fx n ~ l q- - rv”~"3 j- sx n ~~* . . ± P v V Q v 2 =±: R v ^ S = o ; aft 
fume the fra&ion 
nx 
ipx 
+ n — 2 qx n ^ — n — o ) rx n 4 -f&c. 
x n -p x n ~ 1 + + &c. 
nx . 1 q- [<% q- (3 q* y q- $ 4* &c .) x z q* q - @ q- y q- ci 2 q- £ q* &c.) 
•K-3 + (V + /3 3 + y + V + 6 3 +&C.) X-* + (* 4 + &*+?* + $' + 6* + 
O 2 &C.) 
