But to bring this to Converging Series, it is first neceffary to prove, that every Power raifed from a Binomial (without regarding the Co-efficients) confifts, or is compofed of two Ranks or Series of Powers, one in

creafing from n or I to n and the other de

n

creafing from m to m

or I; and each Member in

one is multiplied into its correfponding Member, in the other respectively, as may appear thus.

That the Co-efficient of the fecond Term in any Power raised from a Binomial is always n, the Exponent of the highest Power.

COROLLARY

2.

That the Root or Side of n, the unknown Quantity, is always multiplied into the Second Term of the known.

Now

Now from the Latter, it is evident we are (in this Cafe, but to make use of the two Members of the Power of fuch Binomial, and by the first we may exprefs the Co-efficient of the fecond Term by n, the Exponent of the Power: Therefore the former Equation will now ftand thus.

Now to find the Value of n, or the unknown Quantity: It is plain, that thofe Members into which it is multiplied will be the Divifor with the fame Signs, as being to be tranfpofed to the other Side of the Equation. Therefore first we get the following

Which Theorem exhibits all poffible particular ones, for extracting of Roots according to the first Sort of Mr. Ralphfon's, agreeing exactly with them, as will be found on Trial, always remembring that the Signs in the Dividend must be contrary to those in the Equation, and in the Divifor the fame respectively.

But m+n=x. Therefore Secondly,

THEOREM,

Which gives all those of the fecond Sort univerfally.

But in this Cafe the Signs, both in the Dividend and Divifor, will be the fame, as in the given Equation refpectively; as likewise it may be proper to take Notice, That if any Term be wanting in the Equation, the fame mnft be omitted in either Theorem respectively.

Now from either of these two Generals, to deduce any particular Theorem for finding the Root of any given

Equation, we need only confider, that m = I. or

alfo that n-n" =0, or n -n = 0; and any Quan

tity multiplied into Nothing is

Cafe happens (which always will,

o, and when either

except where the last

Term is wanting, the Theorem is determined.

THEREFOR

After the fame Manner for any Equation whatsoever. Thus having the particular Theorem, the Application in either Cafe is as follows.

Let m be any Number taken at Pleasure as before. T = Theorem, in which m must be of its laft Value found.

Then the Procefs will be of the

First General Theorem.

m the Ift.

m the 2d.

m the 3d. m the 4th.

Second General Theorem.

Tm the 2d. Then [T=m the 2d. Then
T=m the 3d. Then | Tm the 3d. Then
Tm the 4th. Then | Tm the 4th. Then
Tm the 5th, &c. Tm the 5th. &c.

Some of which true Values of m will terminate in the true Root fought, if it have one: But if it be a Surd, then the Value of m will proceed into an Infinite Series, but may be profecuted nearer the Truth than any affignable, which Series, each Operation, will proceed in Number of Places, in a Geometrical Progreffion, whofe first

Term

Term is 1, and Ratio=2, viz. First 1, then 2, then 4, then 8, then 16, then 32, then 64. &c. Places.

It is likewife obfervable, that the firft General Theorem converges by finding out a Number to be added to, or fubtracted from the last Value of m (as it fhall be adfected with or ) untill m bex fought. So the laft converges by m itfelf, whofe Value, at each Operation, fhall grow nearer and nearer, untill it be = fought.

We may also take Notice, that tho' m be affumed never fo far from the Root, yet it will converge to it by renewing the Operation.

But the Work may be much fhortened, in Cafe we point the given Equation (if it will admit of it) both in the abfolute Number and Co-efficient, according to their refpective Degrees of Adfection; and take first 1. then 2, then 4, &c. of thofe Points (from the firft) each Operation: For it is evident, the Co-efficients increase their Powers, as the highest known Term decreases; therefore the abfolute Number is of the fame Power, with the higheft unknown Quantity.

One Inftance may be fufficient to explain it. Suppose this Cubic Equation to be pointed, viz.

·×3+25x2+836x = 53297.

or x3+px2+qx = N.

Then it would be x2+25x2+836x=53297.

For the abfolute Number is a Cube

Co-efficients q a Square
pa Lateral

and are pointed

accordingly.

And the like Method for any other Equation, where it

will admit of it.

Now to apply this we are to take

The firft Operation x3+2x2+8x=53.

Second Operation +25+836x=53297, and confequently the Value of the Co-efficients, as well as the abfolute Numbers alters, fo long as there are Punctations.

But