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.

Power 2

Root 1.071773

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

Here the divisor 2 is con

tained exactly

gative index

once in the pe

-2, and there

6. To find the /:00048. Numb. Log. Power 00048 3)-4-681241 0782973 -2.893747 Root

Here the divisor 3, not being exactly contained in -4, it is augmented by 2, to make up 6, in which the divisor is contained just 2 times; then the 2,

fore the index of the quotient thus borrowed, being carried to the de

is — 1.

cimal figure 6, makes 26, which divided by 3, gives 8, &c.

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.

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

signifies the square root;

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

signifies proportion.

the cube root; the

= signifies equality, and is named equal to.

And so on for other operations.

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

a 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.

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

a

ab, or denotes, that the number represented by a Ъ

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

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

-

bc 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; Va, or a3, the cube root of a; and a2 or as the cube root of the square of a;

1

m

is the nth root of a; and "/a" or a

m

also n/a, or a is the nth power of the mth root of a, or it is a to the power.

n

m

a2 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
α- -b'

expressed like a fraction, means

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

✔ab+cd, or (ab+cdj3, is the square root of the compound quantity ab+cd. And c√ab+cd, or c (ab+cd), denotes the product of c into the square root of the compound quantity ab + cd.

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 a denotes that x is multiplied by; thus Xx or 2x. 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 8a, or 5ab, or 6abc2.

8. Compound Quantities are those which consist of two or more terms. As a+b, or 2a-3c, or a+2b-3c.

9. And when the compound quantity consists of two terms, it is called a Binomial, as a+b; when of three terms, it is a Trinomial, as a+26-3c; when of four terms, a Quadrinomial, as 2a-3b+c-4d; and so on. Also a Multinomial or Polynomial, consists of many terms.

10. A Residual Quantity, is a binomial having one of the terms negative. As a-2b.

11. Positive or affirmative Quantities, are those which are to be added, or have the sign +. As a or +a, or ab: for when a quantity is found without a sign, it is understood to be positive, or have the sign + prefixed.

12. Negative Quantities, are those which are to be subtracted. Asa, or -2ab, or —3ab2.

13. Like Signs, are either all positive (+), or all nega tive (-).

14. Unlike Signs, are when some are positive (+), and others negative (-).

15. The Co-efficient of any quantity, as shown above, is the number prefixed to it. As 3, in the quantity 3ab.

16. The power of a quantity (a), is its square (a2), or cube (a), or biquadrate (a'), &c.; called also, the 2d power, or 3d power, or 4th power, &c.

17. The Index or Exponent, is the number which denotes the power or root of a quantity. So 2 is the exponent of the square or second power a2; and 3 is the index of the cube or 3d power; and is the index of the square root, a2 ora; and is the index of the cube root, or /a.

18. A Rational Quantity, is that which has no radical sign (✔) or index annexed to it. As a, or 3ab.

19. An Irrational Quantity, or Surd, is that of which the value cannot be accurately expressed in numbers, as the square root of 2, 3, 5. Surds are commonly expressed by means of the radical sign ✔✅: as、/2, or ✅✅a, or √/a2, or ab1. 20. The Reciprocal of any quantity, is that quantity inverted, or unity divided by it. So, the reciprocal of a, or

a 1

is

α b b a'

that of

ī, the reciprocal of is

a

x + y

x +

is

a

21. The letters by which any simple quantity is expressed, may be ranged according to any order at pleasure. So the product of a and b, may be either expressed by ab, or ba;