[ 4 l 1 ] 
for their fucceffive Denominators, is One-fixth Part of 
the Number, which expreffes the Ratio of the Square 
or the Periphery of a Circle to the Square of its Dia- 
meter ; which is deduced by Mr. Euler, Comment, 
l etrophol. Tom. 7. in a different manner ; and other 
Theorems of this kind may be demonftrated from 
the fame or like Principles. 
The Series that is deduced by the ufual Methods 
for computing the Area or fluent, converge in fome 
Caies at lo flow a Rate, as to be of little or no Ufc 
without fome farther Artifice. For Example : The 
Sum of the firft Thoufand Terms of Lord Brounkers 
Series for the Logarithm of 2, is deficient in the 
nfth Decimal. In order therefore to render the Ac- 
count of the inverfe Method more complete, the 
Author fhews how this may be remedied, in many 
Cafes, by Theorems derived from the Method of 
Fluxions itfelf, which likewife ferve for approxi- 
mating readily to the Values of Progreffions, and for 
refolving Problems that are commonly referred to 
other Methods. Thofc Theorems had been defcribed 
in the Firft Book, Art. 352, but the Demonftra- 
tion and Examples were referred to this Place, as 
requiring a good deal of Computation. The Bale 
being fuppofed equal to Unit, and its Fluxion alfo 
equal to Unit, let half the Sum of the extreme Or- 
dinates be reprefented by a , the Difference of the 
hrft Fluxions of thefe Ordinates by b , the Difference 
ot their Third, Fifth, Seventh and higher alternate 
Fluxions by c, d> e, &c. then the Area fhall be equal 
3024.0^" 1209600 which is the firft 
Them cm for finding the Area. The reft remaining, 
let 
