3. Divide 06314 by 007241 | 4. To divide 7438 by 12.9476.

Numbers. Logs.

Divid. 7438 -1.871456

Divisor 12.9476 1.112189

0-940506 Quot. 057447 -2.759267

Here 1 carried from the decimals to the -3, makes it become-2, which taken from the other 2, leaves 0 remaining.

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

Note. The Rule-of-Three, or Rule of Proportion, is performed by adding the logarithms of the 2d and 3d terms, and subtracting that of the first term from their sum. Instances will occur in Plain Trigonometry.

INVOLUTION BY LOGARITHMS.

RULE.

TAKE out the logarithm of the given number from the table. Multiply the logarithm thus found, by the index of the power proposed. Find the number answering to the product, and it will be the power required.

Note. In multiplying a logarithm with a negative index, by an affirmative number, the product will be negative. But what is to be carried from the decimal part of the logarithm, will always be affirmative. And therefore their difference will be the index of the product, and is always to be made of the same kind with the greater.

TAKE the log. of the given number out of the table. Divide the log. thus found by the index of the root. Then the number answering to the quotient will be the root.

Note. When the index of the logarithm, to be divided is negative, and does not exactly contain the divisor, without some remainder, increase the index by such a number as will make it exactly divisible by the index, carrying the units borrowed, as so many tens, to the left-hand place of the decimal, and then divide as in whole numbers.

10) 0-301030 Power 1.045 365) 0.019116 0-030103 Root 1.000121 0.000052

162

ALGEBRA.

DEFINITIONS AND NOTATION.

1. ALGEBRA is the science of investigation by means of symbols. It is sometimes also called Analysis; and is a general kind of arithmetic, or universal way of computa

tion.

2. In this science, quantities of all kinds are represented by the letters of the alphabet. And the operations to be performed with them, as addition or subtraction, &c. are denoted by certain simple characters, instead of being expressed by words at length.

3. In algebraical inquiries, some quantities are known or given, viz. those whose values are known and others unknown, or are to be found out, viz. those whose values are not known. The former of these are represented by the leading letters of the alphabet, a, b, c, d, &c. ; and the latter, or unknown quantities, by the final letters, z, y, x, u, &c. 4. The characters used to denote the operations, are chiefly the following:

+ signifies addition, and is named plus.

signifies subtraction, and is named minus.

X or signifies multiplication, and is named into. signifies division, and is named by.

signifies the square root; the cube root; the

4th root, &c.; and the nth root.

::: signifies proportion.

= signifies equality, and is named equal to.

And so on for other operations.

Thus ab denotes that the number represented by b is to be added to that represented by a.

α b denotes that the number represented by b is to be subtracted from that represented by a.

a b denotes the difference of a and b, when it is not known which is the greater.

DEFINITIONS AND NOTATION.

ab, or a X b, or a. b, expresses the product, by multipli cation of the numbers represented by a and b.

a

a÷b, or, denotes, that the number represented by a b

is to be divided by that which is expressed by b.

a:b::c:d, signifies that a is in the same proportion to b, as c is to d.

a-b+c is an equation, expressing that x is equal to the difference of a and b, added to the quantity c.

✔a, or a3, denotes the square root of a;

a, or a3, the cube root of a; and 2/a2 or a3 the cube root of the square of a;

1

m

also n/a, or a is the nth root of a; and "a" or a

is the

n

power.

of the mth root of a, or it is a to the

m

nth power a3 denotes the square of a; a3 the cube of a ; a' the fourth power of a; and a" the nth power of a.

a+b× c, or (a + b) c, denotes the product of the compound quantity a + b multiplied by the simple quantity c. Using the bar, or the parenthesis () as a vinculum, to connect several simple quantities into one compound.

a+b

a+b÷a− b, or abexpressed like a fraction, means

the quotient of a + b divided by a-b.

✔ab+cd, or (ab+cd), is the square root

pound quantity ab+cd. denotes the product of c pound quantity ab + cd.

of the comAnd cab + cd, or c And c√ab + cả, or c (ab+cd), into the square root of the com

a+b-c3, or (a + b −c)3 denotes the cube, or third power, of the compound quantity a+b-c.

3a denotes that the quantity a is to be taken 3 times, and 4(a + b) is 4 times a + b. And these numbers, 3 or 4, showing how often the quantities are to be taken, or multiplied, are called Co-efficients.

Also denotes that x is multiplied by; thus Xx or z.

5. Like quantities, are those which consist of the same letters, and powers. As a and 3a; or 2ab and 4ab; or 3a2bc and -5a2bc.

6. Unlike Quantities, are those which consist of different letters, or different powers. As a and b ; or 2a and a2; or 3ab2 and 3abc.

7. Simple Quantities are those which consist of one term only. As 3a, or 5ab, or 6abc.