Infinite Series* 103, 
feme as given by Mr. Simpson and others.- When two or 
snore values of the quantity x are nearly = a 7 then in a refult- 
ing equation or equations, quantities of two or more dimens- 
ions of the approximate ,x A are to be included. 
PART III. 
1. THE firfl: algebraifts divided quantities, and extrafted 
their roots no further than the quantities themfelves : they 
did not perceive the utility of proceeding any further, other- 
wife the operation would have been the fame continued. Mr« 
Gregory St. Vincent, and Mr. Mercator divided, and 
Sir Isaac Newton divided and extracted the roots of quan- 
tities (in which only one unknown quantity fix) is contained) 
by the operations then ufed by arithmeticians, into feries 
afcending or defeending, according to the dimenfions of x in 
infinitum . They clearly faw the utility of it in finding the 
fluents of fluxions, as Dr. Wallis and others feme little 
in 
time before had found the fluent of the- fluxion ax n x; or, which' 
m 
is the fame,, the area of a curve whole ordinate is ax 11 and ab~ 
feiffa is 
2. M. Leibnitz afked from Mr. Newton the cafes in 
which the above-mentioned feriefes would converge; for it 
would be altogether ufelefs when it diverges, and of little ufe 
when it converges flowly. 
