Divide now the Equation propofed by x-r

and it will become

*AxBx Ex3...

··· +Gx—H+ —

I K L

L M

}

In which fubftituting a, b, c, &c. fucceffively for x, we obtain

a-Aa-Bar-2-Car-3...

Ab+Bb-2-Cbr-3

I

3

K L M

Ъ

C~ Ac+B2C-3.

"} = 0

C3

But, by the Notation here used, and explained

And the Sum of these =aG(+a+¿G(+b+cGlte +&c.= (by this Notation) AXG' (by the

Compare this laft Conclufion with that which followed from dividing the propofed Equation by x*, and fubftituting for x the Roots a, b, c &c. and you will have

From these two Theorems Sir Ifaac Newton's Rule manifeftly follows.

But, to illuftrate the Reasoning here used by fome Examples: fuppofe r=3, then we are to take C for H, because three Terms only precede C in the Equation x-Ax+Bx——— Cx-3+ &c. =0; and we are to prove that

+&c. +&c. -&c.

That this may appear, obferve that a+b+ +c3+d3+&c. =a2+b2+c*+a2+ &c. x xa+b+c+d+ &c. —aaxb+c+d+&c. —¿a× xa+c+d+&c.—c2Xa+b+d+&c.—ď3× xa+b+c+ &c. —&c. = (because AB'= = =axab+ac+ad+ &c.+bxab+bc+bd+ &c. + +cXac+bc+de+&c.+dxad+bd+cd+&c. +&c.) =a2+b2+c2+d2+ &c. ×A—A′B'= (by the Lemma) =a+b2+c2+d2+ &c. ×A—AB+ +3C.

In like Manner, a++b+c++d+ +&c. = =a3+b3+c3+d3+ &c. xa+b+c+d+ &c. -a2+b2+c3+d2+&c.Xab+ac+ad+bc+bd+cd

+&c. +a2xbc+bd+cd+&c. +b2xac+ad+cd+

+ &c. +c2xab+ad+bd+&c. +d2xab+ac+bc+ &c. + &c. =a3+b3+c3+d3+ &c. ×A—

3

—a2+b2+c2+d2+ &c. ×B+A'C' = =a3+b3+c3+d3+ &c. ×A—a2+b2+c2+d′′‡ &c ×B+a+b+c+d+ &c. ×C—4D.

End of the Second Pari.

A

Of the Application of Algebra and Geometry to each other.

Of the Relation between the Equations of Curve Lines and the Fi gure of those Curves, in general.

N the two firft Parts we confidered Al

$1.IN

gebra as independent of Geometry; and demonstrated its Operations from its own Principles. It remains that we now explain the Ufe of Algebra in the Refolution of Geometri

cal