I have also enlarged upon and illustrated the meaning of the symbols in the general formula wherever they occur in the work.

In the chapter on Quadratic Equations several examples are given to explain the meanings of the two values of the unknown quantity, and a general rule given in page 168 to obtain the interpretation of the second value.

In the proof of the Binomial Theorem for positive integers, the appeal to the principle of induction has not been made, because it seems to me to be unnecessary. I have also discarded Euler's proof for fractional indices, because I have found very few beginners in Algebra who could make anything of it.

Very few examples of multiplication and division with fractional or negative indices are given in the early part; but the examples in page 116, &c., may be used for exercise in them.

R. F.

ALGEBRA.

CHAPTER I.

DEFINITIONS.

1. In addition to the figures used in Arithmetic, the letters of the Alphabet, a, b, c, x, y, z, &c., are used in Algebra to denote numbers and quantities. Thus we may speak of a line a feet in length, of a stone weighing x lbs.; also of the product of two quantities a and b, &c.

As examples of the difference in Arithmetical and Algebraical notation, take the following:

(1.) If 2 lbs. of sugar cost 14d., what will 7 lbs. cost?

By the Rules of Proportion we know that the price of 7 lbs. will be

(2.) If a lbs. of sugar cost p pence, what will b lbs. cost?

Here a represents the given number of lbs.

b represents the number of lbs. whose price is required.

p represents the given number of pence.

And question (2) is of the same nature as (1); therefore the price of b lbs. will be

bx p

a

(1) is an Arithmetical question.

(2) is an Algebraical question.

2.

pence.

(read plus, more) signifies that the quantity to

which it is prefixed is to be added, when possible.

(read minus, less) signifies that the quantity to which

it is prefixed is to be subtracted, when possible.

Thus, 5+3 means that 3 is to be added to 5;

So,

and 5-3 means that 3 is to be subtracted from 5.

a+b means that b is to be added to a; a-b means that b is to be subtracted from a.

When no sign is prefixed to the letter at the beginning of an expression, + is understood.

Thus, a+b means +a+b; or, a and b are both to be added.

B

Quantities with the plus sign before them are called positive: those with the minus sign are called negative.

3. The sign x (read into) is the sign of Multiplication. Thus, 4x3 means 4 multiplied into 3.

axbxxxy means the product of the four quantities a,
b, x, and y.

Sometimes a dot (.) is used instead of the sign ×.
Thus, a.b.x.y means the same as axbxxxy.

When letters are written in a row without any sign between them, the sign × is understood.

Thus, a bxy means the same as a xbxxxy.

4. The sign (read by) is the sign of Division. Thus, 12÷4 means that 12 is to be divided by 4;

Also

a

b

and a÷b mean that a is to be divided by b.

5. The sign (read equal to) is the sign of equality.

=

Thus, 5x=3a+4 means that five times x are equal to three times a with four added.

6. The sign

means greater than ;

The sign <means less than.

Thus, a>2b means that a is greater than twice b.

7. The sign... means therefore;

and ... means because.

8. The sign () is called a bracket. It is used to enclose several separate quantities in one expression.

Thus, (a+2b-3c) is a bracketed expression.

If any quantity or sign be written outside the bracket, each of the quantities within the bracket is equally affected by it.

9. When a quantity consists of two or more quantities multiplied together, each of the separate quantities is called a factor, and the whole quantity is called the product of the separate factors.

Thus, 3abc is the product of the four factors, 3, a, b, c.

10. When all the factors of a product are equal, the product is said to be a power of the factor. The order of the power is the same as the number of the factors.

Thus, 5×5 or 25 is the second power of 5.

5×5×5 or 125 is the third power of 5.

Also, axaxaxa or a.a.a.a is the fourth power of a.

11. Instead of writing all the factors of a power as in the last article, the power is represented by writing the factor once, and denoting, by a small figure at the top, the number of times the factor is repeated.

Thus, a.a.a.a is represented by a1.

The small figure that denotes the power to which the quantity is raised is called the index of the power, or the exponent of the power.

Thus 3 is the index in the last expression.

12. When a quantity is the product of a number of equal factors, each of those factors is called a root of that quantity. The order of the root is the same as the number of factors. Thus, 6 is the second or square root of 36.

And 2 is the fifth root of 32.

And a is the fourth root of a1.

When any root of a quantity is required, it is expressed. by writing the sign√ on the left hand of the quantity, and the figure denoting the root required on the left of the sign. Thus we have

36 = 6.

= 2.

These roots may also be written as follows :—

36 may be written 363 or √/36.

13. When the different parts of an expression are separated by the signs + and -, each part that is contained between two contiguous signs is called a term.

Thus, 3a-4b+4c-10x2yz contains 4 terms.

14. A quantity of one term is called a Monomial.

A quantity of two terms is called a Binomial.

A quantity of three terms is called a Trinomial.

A quantity of more than three terms is called a Multinomial.

15. The number denoting how many times a quantity is taken is called its coefficient.

If no coefficient be written, 1 is always understood.

16. Like quantities are such as differ only in their numerical coefficients.

Thus, xy, 4xy, -10xy are like quantities.
But xy, x2y, abc are unlike quantities.

CHAPTER II.

NUMERICAL VALUES, ADDITION, ETC.

17. The expression a+b represents in general the sum of the two quantities a and b, and if we suppose a = 4, b = 3: Then, a+b=4+3. Or, a+b=7.

... 7 is the numerical value of a+b, when a and b have the values 4 and 3 respectively.

If a and b have different values from what we have supposed above, the numerical value of the expressions will be found by writing those values in the place of a and b respectively.

Thus, if a=5 b=1

Then a+b=6.

18. In order then to find the numerical value of any