I 3*3 3 
makes it flop at a finite Number of Terms, in Series 
that either converge or diverge. Whence in di- 
verging Series it muft neceflarily be found and ad- 
mitted, orotherwife the Conclusion will not be true 5 
but in converging Series, where it can feldom be 
known, it may fafely be omitted, becaufe it conti* 
nually diminifhes with the Terms of the Series, and 
finally becomes lefs than any aflignable Quantity. 
The Nature of infinite Series being thus difplay'd, 
he applies them to the Refolution of all kinds of Al- 
gebraical Equations. He explains in a very general 
Manner, the Author's famous Artifice, for finding the 
Forms of the Series for the Roots, and their initial 
Approximations, by means of a Parallelogram and 
Ruler, and (hews its Application in all Cafes. Then 
he invents many ways of Analyfis, by which the 
Roots are further profecuted, and may be produced 
to any Degree of Accuracy required. Alfo many other 
Speculations are added, to compleat the Do&rine of 
Series > particularly a very general and ufeful Theo- 
rem, for the Solution of all affe&ed Equations in 
Numbers. 
From the Refolution of Equations and the Doc- 
trine of Infinite Series, which finifhes the firft Part 
of this Work, Sir Ifaac Newton proceeds to lay 
down the Principles of his Method of Fluxions, which 
is the chief Defign of the prefent Treatife. This Me- 
thod he founds upon the abftrad or rational Mecha- 
nicks, by fuppofing ail Mathematical Quantities to be 
generated, as it were, by local Motion, and there- 
fore to have relative Velocities of Increafe orDecreafe, 
which Velocities he calls Fluxions . And the Quan- 
tities fo generated by a continual Flux he calls Flu- 
T t 2 ents 
