( 177 ) 
by Mr. Newton as the firft Rule upon which he founds his 
Quadrature of Curves. Dr. Wallis demonftrated this Pro- 
portion by Steps in many particular Propofitions, and then 
collected all the Prcpofitions into One by a Table of the Cafes. 
Mr. Newton reduced all the Cales into One, by a Dignity 
with an indefinite Index, and at the End of his Compendium 
demonftrated it at once by his method of Moments, he being 
the firft who introduced indefinite Indices of Dignities into 
the Operations of Analjfis. 
By the 108 th Propofition of the Paid Arithmeticalnfinitoriim , 
and by feveral other Propofitions which follow it; if the Or- 
dinate be compofed of two or more Ordinates taken with 
their Signes -j- and — , the Area fhall be compos’d of two or 
more Areas taken with their Signes + and — refpe&ively. 
And this is afTumed by Mr. Newton as the fecond Rule upon 
which he founds his Method of Quadratures. 
And the third Rule is to reduce Fractions and Radicals, 
and the affeded Roots of Equations into converging Series, 
when the Quadrature does not otherwife fucceed ; and by the 
firft and fecond Rules to fquare the Figures, vvhofe Ordinates 
are the Tingle Terms of the Series Mr Newton y in his Let- 
ter to Mr. Oldenburgh dated June 13. 1676. and communicated 
to Mr. Leibnitz y taught how to reduce any Dignity of any 
Binominal into a converging Series, and how by that Series to 
fquare the Curve, whofe Ordinate is that Dignity. And be- 
ing defired by Mr. Leibnitz to explain the Original of this 
Theoreme, he replied in his Letter dated Offob. zq. 1676, 
that a little before the Plague ( which raged in London in the 
Year 16 65) upon reading the Arithmetica Infinitorum of 
Dr. Wallis , and confidering how to interpole the Series x, 
X — j x\ X ^4?^) X — | X[ + y X S , _ ~ x\ &C. 
he found the Area of a Circle to be* — 
1 \ 
\x* 
x‘ 
— — &c% And by purfuing the Method of Interpolati- 
°g 3 
on 
