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190

To extract the Square Root of a Polynomial...

To extract the Square Root of a Trinomial..

... 194

SECTION XII.

EQUATIONS OF THE SECOND DEGREE.

Definitions of Terms........

.. 196

Method of solving an incomplete Equation ...

Number of the Roots ..

. 198

Method of solving a complete Equation ......

.. 201

Mode of completing the Square..............

• Second Method of completing the Square.......

.......... 207

Problems producing complete Equations of the Second Degree .. 211

Equations of the Second Degree with two Unknown Quantities .. 216

Examples .....

.......... 217

SECTION XIII.

RATIO AND PROPORTION.

Arithmetical Ratio defined-Geometrical Ratio ..

223

Proportion defined ....

Distinction between Ratio and Proportion .......

Property of four Proportional Quantities ........

226

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Property of three Proportional Quantities ....

Equal Ratios compared ........

228

Alternation and Inversion ..........

Composition and Division....

230

Equal Multiples of two Quantities......

Like Powers or Roots of Proportional Quantities .......

232

SECTION XIV.

PROGRESSIONS.

Arithmetical Progression.................

234

To find any Term of an Arithmetical Progression ..

235

Sum of the Terms of an Arithmetical Series .......

237

To find several Arithmetical Means between two Numbers..

Geometrical Progression .....................

To find any Term of a Geometrical Progression.......

244

Sum of the Terms of a Geometrical Progression ......

245

Sua of the Terms of a Decreasing Progression..........

To find a Mean Proportional between two Numbers............ 249

Decreasing Progressions having an infinite Number of Terms .

ELEMENTS OF ALGEBRA.

SECTION I.

PRELIMINARY DEFINITIONS AND FIRST PRINCIPLES.

(Article 1.) ARITHMETIC is the art or science of numbering. It treats of the nature and properties of num. bers, but it is limited to certain methods of calculation which occur in common practice.

(2.) Algebra is a branch of mathematics which enables us to abridge and generalize the reasoning employed in the solution of all questions relating to numbers. It has been called by Newton Universal Arithmetic.

(3.) One advantage which Algebra has over Arithmetic arises from the introduction of symbols, by which the operations to be performed are readily indicated to the eye.

(4.) The following are some of the more common symbols employed in. Algebra.

The sign + (an erect cross) is named plus, and is employed to denote the addition of two or more numbers. Thus, 5+3 signifies that we must add 3 to the number 5, in which case the result is 8. In the same manner, 5+7 is equal to 12; 11+8 is equal to 19, etc.

QUESTIONS.—What is Arithmetic? What is Algebra? What advant age has Algebra over Arithmetic? What does the sign plus denote ?

We also make use of the same sign to connect sev. eral numbers together. Thus 7+5+3 signifies that to the number 7 we must add 5 and also 3, which make 15.

So, also, the sum of 6+5+11+9+2+8 is equal to 41.

(5.) The sign – (a horizontal line) is called minus, and indicates that one quantity is to be subtracted from another. Thus, 7–4 signifies that the number 4 is to be taken from the number 7, which leaves a remainder of 3. In like manner, 11-6 is equal to 5, and 16–10 is equal to 6, etc. .

Sometimes we may have several numbers to subtract from a single one. Thus, 16–5–4 signifies that 5 is to be subtracted from 16, and this remainder is to be further diminished by 4, leaving 7 for the result. In the same manner, 30-8-6-2-5 is equal to 9.

(6.) The sign X (an inclined cross) is employed to denote the multiplication of two or more numbers. Thus, 5x3 signifies that 5 is to be multiplied by 3, making 15.

(7.) The character = (a horizontal line with a point above and another below it) shows that the quantity which precedes it is to be divided by that which fol. lows.

Thus, 24:6 signifies that 24 is to be divided by 6, making four.

Generally, however, the division of two numbers is indicated by writing the dividend above the divisor and drawing a line between them.

Quest.-What does the sign minus indicate? What is the sign of multiplication? How may division be denoted ?

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Thus, instead of 24-6, we usually write

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(8.) The sign = (two horizontal lines), when placed between two quantities, denotes that they are equal to each other.

Thus, 7+6=13 signifies that the sum of 7 and 6 is equal to 13. So, also, $1=100 cents, is read one dollar equals one hundred cents; 3 shillings=36 pence, is read three shillings are equal to thirty-six pence..

(9.) The following examples will afford an exercise upon the preceding symbols.

EXAMPLE 1. 5x8+12-4=6x9· This may be read as follows: The product of 5 by 8, increased by 12 and diminished by 4, is equal to 6 times 9, diminished by one third of 12, and again diminished by 2.

To find the value of each side of this equation, we multiply 5 by 8, which gives 40; adding 12 to this product, we obtain 52, and, subtracting 4, we have 48. Again, the product of 6 by 9 is 54, which, diminished by one third of 12, leaves 50, and, subtracting 2 from this result, we have 48, as before.

Verify the following examples in the same manner : Ex. 2. 7x9–5+14=8x6+20+*

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