that other Index denominates. And because the Word Power does here fignify both what is in a more particular Definition call'd Power and Root, therefore this Rule comprehends both Involution and Evolution.

RULE IV. If to the Product or Quote of two Numbers any Index is applied, the Expreffion is equal to the Product or Quote of the fame Powers of these two Numbers.

If the Power of one Number is equal to the m Power of another, then is the m Root of the firft equal to the Root of the fecond: Thus if An=Bm, then is AmB". Demonftr. Since AB, then is AB, (Ax. 1.) and B≈Am", (Ax. 1.) =ATM, (Theor XI.)

I

I

n

The Rever fe is alfo true, viz. That if AB, therefore A" Bm; for AmBB,and therefore ABT.

SCHOLIUM.

SCHOLIUM. This Theorem may be made yet more univerfal by taking any Mixt

Index, and making the Theorem thus: If Am=B, then is Ans=Bm; and hence again,

I

n

I

Arm Bus: For the firft, fuppofe Ana, and Br=b, then is Amam and B = AmB": wherefore ambs; and hence, (as is already fhewn) abm. But a=A", and b=

T

B", therefore a3— A^s, and bm=Brm; that is, Ans=Brm; whence again, ATM=Bs. as COROLL. I A"=Bm, and if Am is Rational, fo alfo is B, fince they are equal.

THEOREM XVIII.

IF A" Bm, call m-n=d. Then is Ad a Power of the Order m, and Bd a Power of the

Order ; i. e. Am, and B are both Rational, tho' they are not always equal.

n;

Demonftr. Since A" Bm B+d, (for m=n+d) multiply both by A; and then An+B+dxAd; therefore Ad is a Power of the Order n+d or m, (by Coroll. 2. Theor. II.) Divide both by B and it is A"BB"; therefore Bd is a Power of the Order n, (Coroll. Theor. II.) i. e. A and B are both Rational.

3.

d

COROLL. If d=1, that is, m=n+1, then A has an + 1 Root, and B an " Root:

i. e. A and B are both rational; as they are also equal by the Theorem.

LEMMA.

is in its leaft Terms, or is not fo, then, accordingly, any

If any Fraction Power of this Fraction, as an Converfe, if ""

is or is not in its lowest Terms, accordingly is or is not fo. The Demonftration of this Truth must be referred to another Place, because it depends upon Principles not yet explained: You'll find it demonftrated in Book V. Ch. I. Theor. 13. Coroll. In the mean time we must fuppofe it to be true, for the fake of fome things belonging to this Book, whofe Demonftration depends upon this Lemma, and which could not be fo regularly referred.

THEOREM XIX.

IF any Integer A has not a propofed Root in Integers, it can have no determinate Root of that Order; i. e. it is a Surd Power of that Order. Or thus: If an Integer A is not the Power of a certain Order of an integral Root, it cannot be fo of a fractional. Example. 7 has no Square or Cube Root in Integers, and therefore has no fuch determinate Root in a Fraction.

Demonftr. Let be any Fraction in its lowest Terms, and fuch as is not equal to any

an

Inreger, i. e. let r be greater than 1; then, by the Lemma, is alfo a Fraction in its loweft Terms: and confequently, is not an aliquot Part of A"; nor, confequently, is an

Integer; for in this Cafe would not be in its loweft Terms: Hence again it's clear, that

no

no Fraction (fuch, as is not equal to an Integer) can be any Root to a Whole Number:

not an Integer is equal to A an Integer, which is abfurd. Again, if

is not in its leaft Terms, let

"

T

be its leaft Terms; then, because therefore

an

m

m!

A. But being in its leaft Terms, fo is whence the fame Abfurdity as before.

§. I. Probl. Of INVOLUTION, or Raifing of Powers.

T

HE General Rule for the Practice of Involution is plainly contained in the Definition, and is nothing else but a continued Multiplication of the Root into it felf, whereby, to come at any higher Power, we must make up the Series of all the interior Powers: Thus, the Series of the Powers of 3 is 3:9:27:81:243, &c. this Operation 3x3x3x3x3, &c.

By

But there is a particular Method whereby all Powers above the Cube or 3d Power may be found, without actually finding all the inferior Powers: Which Rule is this:

RULE. Find, by the general Rule, two or more fuch Powers of the given Root as that the Sum of their Indexes be equal to the Index of the Power required; then multiply thefe Powers continually into one another; the Product is the Power fought. Or find any one Power whofe Index is an aliquot Part of the Index of the Power fought, and involve that Power to an Index equal to the Denominator of that aliquot Part.

Example 1. To find the 7th Power of 4, I find the 2d, 3d, and 4th Powers; viz. 16, 64 256; then the 3dx 4th, or 64 X 256=16384 the 7th Power; becaufe 3+4=7. Or inftead of finding the 4th Power, I find the 5th, by multiplying the 2d and 3d; viz. 16X641024; then the 2dx 5th, or 16 x 1024 16384 the 7th Power.

=

Example 2. To find the 12th Power of 3, I find the Square, viz. 3×3 9; the Square of the Square, or 9×981 is the 4th Power; again, the 2d x 4th, or 9x81729 the 6th Power; then 6th X 6th, or 729 x 729531441 the 12th Power.

I need infift no more on this Practice; the Reafon of which is plainly contained in the preceding Theor. VI. But I muft obferve, That it does not in every Cafe give any Advantage either of Eafe or Expeditioufnefs to the Work; yet as it will do fo in many Cafes, and in none can it make the Operation more tedious, it will always be a very convenient Method.

Of the Practice in Univerfal Characters.

As to the literal Practice, or Involution of Numbers reprefented by Letters and Indexes, this is alfo fufficiently explained in the preceding Definitions and Theore ms.

:

But

But obferve, That when a Root is reprefented as a complex Quantity; for Example, If inftead of S we put A+B, its Powers may be reprefented two different Ways; Thus, A+B A+B or A+B; which Method is in fome Cafes fufficient; but in others it's neceffary to have the Operation performed, and the Power expreffed according to the Refult of the Multiplication; fo that the Index be applied only to the fingle Letters: Thus, A+B2= A2+2AB+B. The moft confiderable and important of all thefe Cafes of complex Roots, with their Powers, is that wherein there are only two Members in the Root, as A+B, called hence a Binomial Root; or AB, called a Refidual Root; the Confideration of, whofe Powers, i. e. of their Compofition by the various Powers and Multiples of the different Members of the Root, has been found of very great Ufe in Mathematicks, and in Arithmetick, efpecially for the Bufinefs of the Extraction of Roots; in order to which I fhall here explain it.

Of the Compofition of the Powers of a Binomial and Refidual Root.

In the annex'd Operation you fee the feveral Powers of A+B raised by Multiplying, according to the common Rules, each Member of the Root into each Member of the feveral Powers, which produces the next Powers; in which these Things are remarkable.

Obfervations on the Table of Powers.

I. In the Expreffion of each Power there are as many and no more different Members, (which contain different Powers of the Parts of the

TABLE of Powers raised from the Root A+B.

4th Power. A4+4 A3 B÷6 A2 B2 +4 A B+B

Root A and B) as the Index + 1 expreffes: For tho' each Member of any of the Powers being multiplied by A and B feparately, do make in all twice as many Products as the Terms in the Power multiplied; yet each of the Series of Products by A and B have all their Terms fimilar, except the first Product by A, and the laft by B; for thefe two are A and B .e. the two Series of Products by A and B contain in their several Members the fame Powers of A and B, except the firft Term of the Line of Products by A, which is A", and the laft Term of the Line by B, which is B"; confequently, thefe fi

'milar Products are reducible to a more fimple Expreffion by adding them together, (i. e. addding the Numbers by which they are multiplied, and joining the common or fimilar Letters with their Indexes) thus, A3B+3 A3 B4 A3 B; alfo 3A B2+3 A2 B26 A2 B2; which explains the Reafon of placing the Lines of Products by A and B, as is here done, viz. in order to the Addition of fimilar Products. That this Obfervation will hold true however far the Powers are carried, is eafily feen from the Nature of the Thing; which will yet farther appear from the next Obfervation, in which we have a joint Demonstration of this.

II. Each Power of A+B contains a Series of gradually different Powers of A and of B: Thus, The Index of any Power of A+B being ", the firft Term is fimply A", and the

laft

laft is B; the intermediare Terms containing each a different Power of both A and B, multiplied together; the Index of A decreafing gradually by 1 in each Term from A" to the Term preceding the laft, or B", in which it is fimply A; and the Indexes of B increafing the fame way from the Term next after A", in which it is only B, to the laft, or B"; fo that in all the intermediate Terms there is fome Power of A and of B; and the Sum of their Indexes is equal ton, the Index of the Power propofed of A+B. Therefore, omitting the other Numbers, which are Multipliers in the feveral Terms of any Power of a Binomial, whofe Index is n, these Terms, in as far as they are compofed of the Powers of A and B, may be represented thus;

A+AXB+ An-2x B2+ AnxB+ &c. A2x BAX BB". wherein there are as many Members as n+1, according to the firft Obfervation; and the Index of B, or the Number taken from in the Index of A, is the Number of Terms after A to any Term.

This Obfervation we fee to be true fo far as the Table of Powers is carried; and that it must be true for ever, is eafy to perceive. Or it may be demonftrated, thus: Suppose it's true in any one Cafe or Power of AB, as the n Power, it must be true in the next Cafe, or the n+1 Power: because when each Term of the given Power is multiplied by

0-2

+A*XB+AXB+ A × B3 +, &c, +A2x B+Ax B+ B+

A, the Products must have A once more involved in them than in the Term multiplied: And fince the Indexes of A decrease gradually from An

in the Terms multiplied, confequently they will decrease gradually from Ant in the Series of Products; the Powers of B continuing as they were. Again; The Series of Products made by B muft have B once more involved in each; and confequently increafing gradually from Bto Bn+', leaving the Powers of A as they were: But again, Thefe Products made by B are all of them, (except the laft B+ fimilar to the feveral Products, (after the firft Ant') made by A; because the Indexes of A in the given Power decreafing from the first Term A, which has no Power of B multiplied into it, and thofe of B increafing from the fecond Term An-1X B to the last Term Bn, it's plain that A multiplied into any Term except the first A", and B multiplied into the preceding, muft make fimilar Products; for A multiplied into any Term raifes the Index of the Power of A by 1, which makes it equal to the Index of Á in the preceding Term, without changing that of B; and B multiplied in the preceding Term, raifes the Power of B in it to the Index of B in the following Term, without changing that of A; confequently these Products; are fimilar, which makes the thing obferved manifeftly true in any Cafe, in confequence of its being true in the preceding: But it's true in the Root or ift Power, and as far as we have carried the Powers, therefore it's univerfally true. And this alfo is manifeft, that the Sum of the Indexes of A and B that are in any Term, is always equal to the Index of the Binomial Power, viz. n. Add also this Obfervation, that the Index of A or B is always I less than the Number of Terms from An or B, to that Term.

SCHOLIUM. If any one Member of a Binomial is 1, as A+ 1, then the Powers of 1 being all 1, the Powers of fuch a Root will confift only of the Series of the Powers of A, and i added; thus, A+ A+ A+ &c. + 1. Or thus, 1+A+A2+&c. + AR.

III. The Numbers which in every Power are multiplied into the feveral Terms are called Coefficients (ie. joint Multipliers or Factors) of thefe Terms; and from the Manner of raifing X 2

the