last of which signifies as much as a multiplied four times into itself; and so of all the rest.

In algebra, given or known quantities are usually denoted by the leading letters of the alphabet; and unknown quantities, or quantities sought, are denoted usually by the final letters.

These quantities are connected by certain signs or symbols, which serve to shew their mutual relation; while at the same time, they simplify the science, and reduce its operations into smaller compass.

Thus, the sign + plus, or more, denotes Addi

tion.

notes the biquidratic root. of 16, or 2.

denotes Equality.

Quantities which have the sign + prefixed to them are called positive, or affirmative, quantities; and such as are preceded by the sign are termed negative quantities. When no sign

is prefixed to a qnantity, + is always understood. Quantities which have the same sign, either + or, are said to have like signs. A quantity consisting of one term, is called a simple quantity; but if it consist of two or more connected terms, it is called compound.

A number prefixed to a letter is called a numerical coefficient, and denotes how often that quantity is to be taken. Thus, 3 a signifies that a is to be taken three times; when no number is prefixed, the coefficient is supposed to be unity.

The quotient arising from the division of one quantity by another, is expressed by placing the dividend above a line, and the divisor below it. Thus denotes the quotient arising from the division of 12 by 3, and denotes the quotient

b

α

arising from the division of b by a.

Quantities that have no radical sign (√) or index annexed to them, are called rational tities.

quan

The mark signifies that the quantities between which it stands, are proportional: thus, a b c d, denotes that a is in the same proportion to b as c is to d.

QUESTIONS.

What is algebra ? By what signs or characters does algebra represent quantities of every kind? Whence is the term algebra derived? What is numeral algebra? What is literal algebra? How are known and unknown quantities usually denoted in algebra? What are the signs or marks by which algebra connects quantities? What are positive, and what are negative quantities? What is a coefficient? How is the quotient, arising from the division of one quan

tity by another, expressed? What are rational quantities? How are the square, and the cube roots of quanties expressed? How is a biquadratic root expressed ? What is a simple quantity? What is a compound quantity? When no sign is prefixed to a quantity, what is understood?

CHAP. XVI.

ALGEBRA - continued.

THE fundamental operations in algebra are performed by the four first rules of arithmetic. Addition, in algebra, is the connecting quantities together by their proper signs, and uniting into term or sum such as are similar. In this rule there are three cases :

1. When the quantities are like, and the signs like also.

2. When the quantities are like, but their signs unlike.

3. When the quantities are unlike.

In the first case, the coefficients are added. together, and the sum set down; to which sum is prefixed the common sign, and the common letter, or letters, subjoined; as,

a

bn

3 z

In the second case, subtract the lesser coefficient from the greater, and to the remainder prefix the sign of the greater; then subjoin the common quantity, or letters.

Examples.

In the third case, collect all the like quantities as in the preceding cases, and set down those that are unlike, one after another, with their proper signs.

Example.

Rule for subtracting quantities.

Change the sign of the quantity to be subtracted into the contrary sign; and, when so changed, add it to the quantity from which it was to be subtracted.

General rule.

MULTIPLICATION.

If the quantities to be multiplied have like signs, the sign of the product is; but if they have unlike signs, the sign of the product is.

The examples of multiplication may be referred to two cases. 1. When both the quantities are simple. 2. When one, or both of them,

is compound.

To Multiply Simple Quantities.

Rule. Find the sign of the product of the preceding rule, and annex to it the product of the numeral coefficients; then set down all the letters one after another, as in one word.

Rule. Multiply every term of the multiplicand by all the terms of the multiplier, one after another, according to the preceding case; and collect their products into one sum, which will be the product required.