DIVISION BY LOGARITHMS.

RULE.

From the logarithm of the dividend subtract the logarithm of the divisor, and the number answering to the remainder will be the quotient required.

NOTE. If 1 be to be carried to the index of the subtrahend, apply it according to the sign of the index; then change the sign of the index to, if it be +, or to +, if it be; and proceed according to the second note under the last rule.

Here 1, carried from the decimals to the -3, makes it become -2, which, taken from the other -2, leaves O remaining.

4. To divide '7438 by 12'9476.

Here the 1, taken from the -1, makes it become —2, to be set down.

INVOLUTION BY LOGARITHMS.

RULE

Multiply the logarithm of the given number by the index of the power, and the number answering to the product will be the power required.

NOTE. A negative index, multiplied by an affirmative number, gives a negative product; and as the number, carried from the decimal part, is affirmative, their difference with the sign of the greater is, in that case, the index of the product.

Here 4 times the negative index being -8, and 3 to be carried, the difference -5 is the index of the product.

4. To raise 1'0045 to the 365th root.

Divide the logarithm of the given number by the index of the power, and the number answering to the quotient will be the root required.

NOTE. When the index of the logarithm is negative, and cannot be divided by the divisor without a remainder, increase the index by a number, that will render it exactly divisible, and carry the units borrowed, as so many tens, to the first decimal place; and divide the rest as usual.

Here the divisor 2 is contained exactly once in the negative index-2, and therefore the index of the quotient is -1.

6. To find the third root of '00048.

Here the divisor 3 not being exactly contained in 4, 4 is augmented by 2, to make up 6, in which the divisor is contained just 2 times; then the 2, thus borrowed, being carried to the decimal figure 6, makes 26, which, divided by 3, gives 8, &c. For-4--6+2.

ALGEBRA.

DEFINITIONS AND NOTATION.

1. ALGEBRA

LGEBRA is the art of computing by symbols. It is sometimes also called ANALYSIS; and is a general kind of arithmetic, or universal way of computation.

2. In Algebra, the given, or known quantities are usually denoted by the first letters of the alphabet, as a, b, c, d, &c. and the unknown, or required quantities, by the last letters, as x, y, z.

NOTE. The signs, or characters, explained at the beginning of Arithmetic, have the same signification in Algebra. 3. Those quantities, before which the sign + is placed, are called positive, or affirmative; and those, before which the sign is placed, negative.

And it is to be observed, that the sign of a negative quantity is never omitted, nor the sign of an affirmative one, except it be a single quantity, or the first in a series of quantities, then the sign + is frequently omitted: thus a signifies the same as +a, and the series a+b-c+d the same as +a+b -c+d; so that, if any single quantity, or if the first term in any number of terms, have not a sign before it, then it is always understood to be affirmative.

4. Like signs are either all positive, or all negative; but signs are unlike, when some are positive and others negative. 5. Single, or simple quantities consist of one term only, as a, b, x.

In multiplying simple quantities, we frequently omit the sign x, and join the letters; thus, ab signifies the same axb; and abc, the same as axbxc. And these products,

as