Hence it is manifeft that m and n being equal to affirma

tive Numbers,

ax, when evolved as above, is (by what was before

If any Binomial or Refidual ax be rais'd to any Power whofe Index is 2 (n reprefenting any Number whatsoever) the nth Power of that Binomial or Refidual will be a"

aq (:

= a × 1 = q); and a a q" is an → n ar

Or, putting A

2—3 arq4, &c.

4

1ft Term, B = 2d, C=3d, D=4th,

&c ; then it will be a±a q["=a” + n A ×±q+2 —

Bx+q+

n -- 2

3

Cx±922—3Dx±q, &c.

4

I

If it were required to raife a Trinomial, Quadrinomial, &c. or an Infinitomial to any given Power, it may be done by the foregoing Corollary: As for inftance, fuppofe the Infinitomial az-|- bz2 + c z3 +-dz+ &c. were to be raised to the Power whofe Index is 22 ; then

2

a z + b z 2 + c z 3-dz+--&c." is an " + na z

See Pages

65,66, and 67.

R. Raphson's Method in raifing particular Theorems for extracting the Roots of all Equations whatfoever is as follows; viz. He fuppofes the Root in any Equation, which he calls a to be a Binomial of g a known, and x an unknown Quantity, and then raises the Binomial g -- to all the Powers of a in the propos'd Equation, and with and by these Values of a and its Powers he reduces the propos'd Equation to another exclufive of a. Then he rejects, or

leaves out all the Members of the new Equation wherein there is any Power of x, exceeding the Root; and with the other Members he makes a fuppos'd Equation, and thereby finds the fuppos'd Value of x, or the Theorem required.

Now if this Method be applied to an univerfal Equation, an univerfal Theorem will thereby be raised which will include all his first Set of Theorems, and indeed all others of that Kind for extracting the Roots of all Equations whatfoever: Let us therefore fuppofe the propos'd Equation to be represented by the univerfal one

In which Nis the given abfolute Number.
the Exponent of the highest
Power of athe Root fought. S

NB. n must be not

2.

I, p, q, r, &c. = to the refpective Coefficients of the
Powers of a in the propos'd Equation.

Let us alfo fuppofe gxa, which g is the known Part of the Root fought a, and ought to be taken as near the fought Root as may be, whether it be greater or lefs than the faid Root, and x is the unknown Part of the Root fought, whofe Value may be negative or affirmative, according as that of g is taken greater or less than the Truth: Then the foregoing Univerfal Equation will become

Now, by rejecting all the Members in the last Equation wherein any Power of x, exceeding the Root, is contain'd, we have

Which Theorem exhibits all poffible particular ones for extracting Roots, according to the first fort of Mr. Raphfon's, agreeing exactly with them, as will be found on Trial;

always

always remembring that the Signs in the Dividend must be contrary to thofe in the Equation, and in the Divifor the fame refpectively. And likewife it will be proper to take notice, that if any Term be wanting in the Equation, the fame must be omitted in the Theorem. Thus,

univerfal Canon or Theorem for extracting the Roots of all pure Powers whatsoever.

Now the Method of finding the affirmative Roots of any Equation whatsoever by the above univerfal Theorem is this;

First, Deduce from the univerfal Theorem the particular one for extracting the Root (or Roots) of the propos'd Equation; then fuppofe g = fome Number as near to the required Value of a as you can guefs, and, by the particular Theorem, find the fuppos'd Value of x, which add to the faid fuppos'd Value of g, and call that Sum g, that is your 2dg; with which 24 g (by the faid particular Theorem) find another x, and call the Sum of the last found x and of g the 2d your 3d g; and fo proceed 'till fome g be found as near a, the true Root required, as fhall be judged fuffi

cient.

That this Method of proceeding will give the true Value of a proxime in the foregoing univerfal fimple Equation I demonftrate, thus

The fuppos'd Value of g must be either equal to the af firmative Value of a (for it is manifeft it hath but one Value that is affirmative, which is what we feek), or less than it, or greater.

Cafe 1. If it be equal to it (that is, if g=a), then the Theorem we are about demonftrating, to wit x=

N-g"

ng"

I

that is to fay xo, as it must,

Cafe