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APPENDIX OF ALGEBRA.

LGEBRA (or the great Art) is a Method of managing arithmetical and geometrical Computations by letters, by means whereof any Question may be clearly solved, and curions Theorems deduced for solving all Questions wherein Numbers or lines are concerned, which would be in vain to attempt either by Arithmetic or Geometry. I have here subjoined the principles of Algebra, with sundry examples to exercise the young Algebraist. Having made every thing short and plain, that the Rules may not be burthensome to Youth.

The Characters or Signs which are used in the follow- ing Appendix are in the Preceding Book, except the fol lowing; x2 signify that x is squared x3 the Cube of x, and x any Power of x, &c.

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any Square Root. ✔✅ he Cube Root, and

✔any Root at Pleasure.

Quantities that are not known are represented by x, y, z, and v, those that are known by a, b, c, d, &c.

ADDITION may be comprehended in one Case, provided the following Directions be well understood.

1. When Quantities are of the same Kind, whether they have co-efficients or not, add them together, and' their Sum will be the Sum required; but if unlike, subtract the co-efficients (if any) and set down the Difference with the Sign of the greater,+or-as the Question may require.

Example 1.
x+y+z

Example 2.
3x-4y+2z

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N. B. Where no Sign is, the Sign+is understood.

Example 3. x2-4x+5y2

6x2+9xy+7y2

-2x-3xy+8y2

Sum 5x+2xy+6y2

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Subtraction may be performed by changing all the terms that are to be fubtracted,-in+,or+into-,and then adding them together, and the Sum will be the Difference.

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MULTIPLICATION is performed by one general Rule; obferving that like Signs produce+,and unlike in the Product. N. B. If the quantities have co-efficients multiply them, if not, join the Letters like the Letters of a Word, obferv ing to place the fign+or-(as above) before them,

Exam. Multiply x

by x

Product x'.

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x6

x+y+z

x+y+z

x2+xy+xz

xy+y2+yż

xz+yz+z2

x2+2xy + y2+2x2+2yz+z2`

x3+y3+z3

x3+y3+z3

x6+x3у3+x3z}

xy3+y+y3z3

x3z3+y1z3+26

x+2x3y3+y+2x3z3+2y3z3+z6

N. B. The addition of Exponents is the fame as Mültiplication of Quantities.

Multiply x+y
by x+y

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This laft fhews that the Rectangle or Product of the Sum and Difference of any two Quantities is equal to the Difference of their Squares, which is very useful.

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DIVI

DIVISION may be performed by one general Rule, being nothing but the Proof of Multiplication; and therefore if the Quantities in the Divifor have like Signs to thofe in the Dividend, the Quotient will be Affirmative; but if unlike, it will be Negative.

Exam. 1. x)xy+x(y+1

Exam. z. xy)xy2 (y Exam. 3. x2)x(x3
Exam, 4. x+v)x2+2xv+v2(x+v3

x2+xv

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Involution of Quantities is nothing but multiplying them continually together as the Power they are to be raised toThus xxxx3 ×x=x3 &c.

Sir Ifaac Newton has given the following rule for raifing any Binominal to any Power, which is this.

Rule

Rule.

If the Index of the firft Letter of any Term be multiplied into its own Co-efficient, and the Product be divided by the Number of Terms to that Place; the Quotient will be the Co-efficient of the next fucceeding Term forward.

Required to raife x+v to the 6th Power regularly up. √x+v/2 = x2+2xv+v2 the Square. 'x+v/3x3+3x2v+3xv2+v3 the Cube. x+v4=x4+4x3v+6x2v2+4xv3+v4 4th power.

2

√x+5=x5+5x4v+10x3v2+10x2v3+5xv4-tv5=5th

Power.

√x+v\°=x6+6x5v+15x4v2+20x3v3+10x2v+4xv5

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v66th Power.
you please.

2 = x2 — 2 xv + 5x2

2

+

And fo for as many Powers as

v3=x3-3x2v+5xv2 —v3

4x44x3v+6x2v2-4xv3+v4

√x—vs=x5—5x4v+10x3y2-10x2v3+5xv4_v5

√x—v 6—x6—6x5v+15x4v2—20x3 v3+15x2v4—6

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Note, All even Terms end with + and thofe that are odd with

Evolution, or the Extraction of Roots being the reverse of Involution, or raifing of Powers, is performed by converfe Operations (viz.) by the Divifion of Indices, as Involution was by their Multiplication-Thus the fquare Root of xo, x=x3 and x5=x=2x51 and univerfally, xnxn= xn--Required the Square Root of x-a2 it will be

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If the Cube Root of any Quantity were to be extracted, put the Radical Sign over the Quantity, with the Index 3 above the Radical, thus: the Cube Root of ax. will be 3ax of x=x4 and xn=x", &c.

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Required the Cube Root of x3-3x2y+3xy2—y3.. •/3x3-3x2y+3x2y—y3=x—xy.

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See

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N. B. As Surd quantities are not easily managed without the help of a Mafter, I thought it beft to omit them.

Of Equations.

An Equation is when two equal quantities, differently xpreffed, are compared together, by means of the fign= placed between them.

Thus, 9-4-5 is an equation, expreffing the Equality of the quantities 9-4 and 5.

Alfo xnm is an equation shewing that x is equal to the Sun of the Quantities n and m.

Equations are the means whereby we come at fuch conclufions as to anfwer the conditions of any problem that may be propofed; and this is called reduction of equations. Reduction of Equations,

Has fundry Rules, according as the Problem is proposed; therefore when a Problem is propofed, having but one unknown Quantity, it is called a fingle Equation, though before the Quantity can be cleared we muft examine it whether it requires Addition, Subtraction, Multiplication, or Divifion, &c. And then having cleared the unknown Quantity, and brought it to one fide of the Equation, and all the known Quantities to the other, the Problem will be done by some of the above Rules.

Here follow a few Examples to illuftrate the Rule. Note. Any Quantity may be transposed to either fide of the Equation, by changing its Sign. Thus x+6=14..x-14— 6-8- -And if x-9=3..x=12. Let 5 x—9=4x+8. Then it is plain 5x muft be=4x+-17 and by taking 4x out of both fides of the Equation we have x=17,

Reduction where two or more unknown Quantities are concerned; find the value of any of them, and then com pare them together, remembering to get as many Values of each unknown Quantity as you have Equations, other. wife it will be in vain to attempt the Solution of any Pro

blem

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