Thus, aXb shows that the quantity represented by a is to be multiplied by that represented by b; and is read a into b.

The multiplication of simple quantities is also frequently denoted by a point, or by joining the letters together in the form of a word.

Thus, aXb, a. b, and ab, all signify the product of a and b also, 3Xa, or 3a, is the product of 3 and : is read 3 times a.

a; and

by, the sign of division; signifying that the former of the two quantities between which it is placed is to be divided by the latter.

Thus, ab shows that the quantity represented by a is to be divided by that represented by b; and is read a by b, or a divided by b.

Division is also frequently denoted by placing one of the two quantities over the other, in the form of a fraction.

b

Thus, b÷a and both signify the quotient of b di

vided by a; and

by a+c.

a

a+c

signifies that ab is to be divided

equal to, the sign of equality; signifying that the quantities between which it is placed are equal to each other.

Thus, x=a+b shows that the quantity denoted by x is equal to the sum of the quantities a and b ; and is read x equal to a plus b.

= identical to, or the sign of equivalence; signifying that the expressions between which it is placed are of the same value, for all values of the letters of which they are composed.

Thus, (x+a) x (x− a) = x2—a2, whatever numeral values may be given to the quantities represented by x and a.

= greater than, the sign of majority; signifying that the former of the two quantities between which it is placed is greater than the latter.

Thus, a=b shows that the quantity represented by a is greater than that represented by b; and is read a greater than b.

= less than, the sign of minority; signifying that the former of the two quantities between which it is placed is less than the latter."

Thus, a=b shows that the quantity represented by a is less than that represented by b; and is read a less than b.

: as, or to, and :: so is, the signs of an equality of ratios; signifying that the quantities between which they are placed are proportional.

Thus, a : b :: c; d denotes that a has the same ratio to b that c has to d, or that a, b, c, d, are proportionals; and is read, as a is to b so is c to d, or a is to b as c is to d.

the radical sign, signifying that the quantity before which it is placed is to have some root of it extracted.

Thus, a is the square root of a; a is the cube root of a; and a is the fourth root of a; &c.

The roots of quantities, are also represented by figures placed at the right hand corner of them, in the form of a fraction.

Thus, a is the square root of a ; as is the cube root

1

of a; and a is the nth root of a, or a root denoted by any number n.

In like manner, a2 is the square of a; a3 is the cube of a; and am is the mth power of a, or any power denoted by the number m.

is the sign of infinity, signifying that the quantity

standing before it is of an unlimited value, or greater than any quantity that can be assigned.

The coefficient of a quantity is the number or letter which is prefixed to it.

2

3

Thus, in the quantities 36, -6, 3 and are the coefficients of b; and a is the coefficient of x in the quantity ax.

A quantity without any coefficient prefixed to it is supposed to have 1 or unity; and when a quantity has no sign before it, is always understood.

Thus, a is the same as +a, or + la; and a is the

A term is any part or member of a compound quantity, which is separated from the rest by the signs +

or -.

Thus, a and b are the terms of a+b; and 3a, 26, and+5cd, are the terms of 3a 26+5cd.

In like manner, the terms of a product, fraction, or proportion, are the several parts or quantities of which they are composed.

Thus, a and b are the terms of ab, or of;

b, c, d, are the terms of the proportion a

and a,

b::c: d.

A factor is one of the terms, or multipliers which form the product of two or more quantities.

Thus, a and b are the factors of ab; also, 2, a, and. 62, are the factors of 2ab2; and a-x and b-x are the factors of the product (a−x) × (b−x).

A composite number, or quantity, is that which is produced by the multiplication of two or more terms or fac

tors.

Thus, 6 is a composite number, formed of the factors 2 and 3, or 2×3; and 3abc is a composite quantity, the factors of which are 3, a, b, c.

Like quantities, are those which consist of the same letters or combinations of letters; as a and 3a, or 5ab and 7ab, or 2a2b and 9a2b.

Unlike quantities, are those which consist of different letters, or combinations of letters; as a and b, or 3a and a2, or 5ab2 and 7a2b.

Given quantities, are such as have known values, and are generally represented by some of the first letters of the alphabet; as a, b, c, d, &c.

Unknown quantities, are such as have no fixed values, and are usually represented by some of the final letters of the alphabet; as x, y, z.

Simple quantities, are those which consist of one term only; as 3a, 5ab, -8a2b, &c.

Compound quantities, are those which consist of seve ral terms; as 2a+b, or 3a-2c, or a+26 - 3c, &c.

Positive, or affirmative quantities, are those which are to be added; as a, or +a, or +3ab, &c.

Negative quantities, are those which are to be subtracted; as —a, or — 3ab, or — 7ab2, &c.

Like signs, are such as are all positive, or all negative; as+and+, or and-.

Unlike signs, are when some are positive and others negative; as + and or and +.

A monomial, is a quantity consisting of one term only : as a, 2b, 3a2b, &c.

A binomial, is a quantity consisting of two terms; as a+b, or a-b; the latter of which is, also, sometimes called a residual quantity.

A trionomial, is a quantity consisting of three terms, as a+26-3c; a quadrinomial of four, as a-2b+3c d; and a polynomial, or multinomial, is that which has many terms.

The power of a quantity, is its square, cube, biquadrate, &c.; called also its second, third, fourth power, &c.; as a2, a3, a1, &c.

The index, or exponent of a quantity, is the number which denotes its power or root.

Thus, -1 is the index of a1, 2 is the index of a2, and

of alorα.

When a quantity appears without any index, or exponent, it is always understood to have unity, or 1.

Thus, a is the same as a1, and 2x is the same as 2x1; the 1, in such cases, being usually omitted.

A rational quantity, is that which can be expressed in finite terms, or without any radical sign, or fractional index; as a, ora, or 5a &c.

2

An irrational quantity, or surd, is that which has no cact root, or which can only be expressed by means of the radical sign, or a fractional index; as 2 or 2, a2 or a3, &c.

α

A square or cube number, &c. is that which has an exact square or cube root, &c.

Thus, 4 anda2 are square numbers; and 64 and a3 are cube numbers, &c.

A measure of any quantity, is that by which it can be divided without leaving a remainder.

Thus, 3 is a measure of 6, 7a is a measure of 35a, and 9ab of 27 a2b2.

Commensurable quantities, are such as can be each divided by the same quantity, without leaving a remainder.

Thus, 6 and 8, 2√2 and 3 √2, 5a26 and 7ab2, are commensurable quantities; the common divisors being 2,√2, and ab.

Incommensurable quantities, are such as have no common measure, or divisor, except unity.

Thus, 15 and 16, 2 and 3, and a + b and a2 + b2, are incommensurable quantities.

A multiple of any quantity, is that which is some exact number of times that quantity.

Thus, 12 is a multiple of 4, 15a is a multiple of 3a, and 20a2b2 of 5ab.