To find the numerical value of an algebraic quantity

To reduce the similar terms of an algebraic expression to

a single term

Addition of algebraic quantities

Subtraction of algebraic quantities

Of subtractive quantities

Multiplication of algebraic quantities-Of monomials

Multiplication of polynomials

Division of algebraic quantities-Of monomials

To divide a polynomial by a monomial

To divide one polynomial by another

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Application of algebra to investigating the properties of
numbers

Of the division of algebraic quantities into factors

FRACTIONS.-To reduce fractions to the lowest terms

To find the greatest common divisor of two polynomials

To reduce a mixed quantity to an improper fraction

To reduce improper fractions to a whole or mixed quantities

Multiplication of fractions

Solution of problems producing simple equations

Solution of simple equations which involve more than one

unknown quantity

General rule for extracting the roots of all powers

Demonstration of the rules for extracting the root of num-

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Multiplication of quantities having fractional exponents

Division of quantities having fractional exponents.

Formation of powers

Extraction of roots

Of polynomials having radical terms

Equations involving radical quantities

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Reduction of exponental equations by logarithms

Properties of the common system of logarithms formed

upon the base 10

Application of logarithms to algebraic formulas

Proportion by logarithms

INTRODUCTION TO ALGEBRA.

CHAPTER I.

PRELIMINARY REMARKS AND DEFINITIONS.

(1.) Quantity, or magnitude, is any thing whatever that will admit of increase or diminution; for example, a sum of money, the length of a line, and a given weight, or bulk, are quantities, since they can be increased or diminished. Mathematics is the science that investigates the means of measuring quantity, and hence is called the science of quantity.

A quantity cannot be great or small, much or little in itself; it is only by comparing it with another quantity of the same kind that it becomes so. Thus, a mile is a large quantity compared with an inch, but small compared with the diameter of the earth; and this again is a small quantity compared with the distance of the earth from the sun.

(2.) It is evident, therefore, that in order to measure a quantity, we must compare it with some known quantity of the same kind, assumed as the unit or measure of quantity. For example, if the quantity to be measured is a sum of money, and we assume one dollar as the unit, the number of times one dollar is contained in that quantity, is the magnitude of the quantity. Thus 300 dollars is conceived to be composed of 300 parts, each of which is equal in value to one dollar.

If the quantity be the length of a line, and we assume one foot as the unit of measure, the number of times a line one foot in length can be applied to this line, is its magnitude. If it can be applied 10 times, we say the given line is ten feet in length.

Quantities of every kind whatever can therefore be expressed by numbers, the number always expressing how many times the assumed unit of quantity is contained in the given quantity.

(3.) Arithmetic and Algebra both treat of the relations of quantity as expressed by numbers.