Example 2.-Multiply a-b by a-b, or find the square of a-b.

Operation...a-b

a-b

a2-ab

-ab+b2

a2-2ab+b2=(a - b)2

105. Hence the square of any binomial, which has one of its parts negative, is the same which ever of the parts be negative; for the square of each of the parts is always affirmative, and twice the product of the two parts is negative; so that the square of a-b is the very same as the square of b-a.

106. Hence we shall always know, by bare inspection only, that the difference of two squares is the product of the sum and difference of the roots: hence x2-y2= (x+y) (x− y), and, reciprocally, that the product of the sum and difference of any two quantities is equal to the difference of their squares. Thus (a+x) (a-x)= a2 — x2.

ALGEBRAIC DIVISION AND FRACTIONS.

107. DIVISION is the converse of Multiplication; therefore, if the signs be alike in the divisor and dividend, the quotient will be affirmative; but if unlike, the quotient will be negative. The general rule is to place the dividend in the form of a numerator, and the divisor in that of a denominator; expunge like quantities from both, and divide the co-efficients by the greatest common

108. Powers of the same root are divided by subtracting their exponents.

109. A FRACTION is multiplied by any quantity by joining it to the numerator; thus, the product of r and is represented by, which is the product of rm divided by n, or the number of times that the product rm contains n.

n

Since the operation of division is opposite to that of multiplication, if a fraction be multiplied by a quantity equal to its denominator, both the denominator and the multiplier may be taken away from the result; thus, if be multiplied by b, the result is, which is equivalent to the quantity a only.

ს

110. The Terms of a fraction are its numerator and denominator.

If the terms of a fraction be equally multiplied, that is, multiplied by the same quantity, the value of that fraction will be the same as before; thus

a

ma

=mb

= and, if the terms of a fraction be equally divided by the same quantity, the result will be equal to the original quantity.

ALGEBRAIC EQUATIONS.

111. The resolution of an equation is the mode of finding the value of the unknown quantity, in terms of those which are given.

The principles of this resolution depend on the following AXIOMS, which are similar to those already given at the beginning of the Geometry, in

page 15.

1. If to each side of an equation the same quantity be added, the sums will be an equation still.

2. If from each side of an equation the same quantity be subtracted, the remainders will still be an equation.

3. If each side of an equation be multiplied by the same quantity, the

ducts will be an equation.

pro

4. If each side of an equation be divided by the same quantity, the quotients will still be an equation.

OF PROPORTION.

112. DEFINITION.-Four quantities are proportionals when the first contains some part of the second, as often as the third contains the like part of the fourth.

THEOREM 39.

113. If four quantities, a, b, c, d, are proportionals, the product of the two extremes will be equal to the product of the two means.

Let the first, a, contain the nth part of the second b, m times; then, by the definition, the third, c, will contain the nth part of d also m times; now the nth part of b is, and the nth part of d is; therefore =m; and=m; wherefore=

a

Multiply each side of this equation by bd, and abd bed; therefore, dividing the terms of the fraction on the first side by b, and the terms of the fraction on the second side by d, which are common, ad=bc.

THEOREM 40.

114. If any equation consist of the product of two quantities on each side, and if the four factors be placed in a row, so that the two factors on either side may occupy the middle place of the four, and the other two each one of the extreme places; the four factors, thus taken in order, will be proportionals.

=

Let ad bc: Now here are four different ways of taking out the quantities; but in which ever of these ways they are taken, we shall always have ad=bc, by multiplying the two extreme terms together, and the two middle terms together.

Let, therefore, a, b, c, d, be one of the four; then, since ad = bc, divide both sides by bd and ad=b; that is, =; now let h be the common measure of a

bc

a

с

and b, supposing h contained in a, m times, and in b, n times; then === ; therefore =m; but is the nth part of b; therefore the nth part of b is con

a

Again, let c=mi, and we shall have; wherefore, by this equation,

C mi m
ni

d=ni; therefore ==; and, consequently, m: wherefore, also, e con

tains the nth part of d, m times.

c

115. COROLLARY.-Hence, if two fractions are equal, the numerator of the one will be to its denominator, as the numerator of the other is to its denominator. To indicate that four quantities, a, b, c, d, are proportionals, the sign is placed between the first two and between the last two; and the sign :: is placed between the two terms that stand in the middle; thus, ab :: c: d, is read, as a is to b, so is e to d,

116. Four proportionals are termed a proportion.

Of four proportional quantities, the last term is called the fourth proportional to the other three.

The first and third terms of a proportion are called the antecedents, and the second and fourth terms the consequents.

SCHOLIUM.

117. Since, from ad be, we may choose four different ways of taking out the first term, in order to make a proportion; and since there are two ways of making choice of the second term after the first, there are eight ways in which the positions of the terms will be different: these are--

118. So that, in every two sets of proportionals, where the same quantity stands first, the terms of the one set are placed alternately to those of the other; and we may also observe that, four of these sets of proportionals have their terms inverted with regard to the other four, and are therefore the same when read in contrary order.

THEOREM 41.

119. If the corresponding terms of any number of proportions are multiplied together, the products, taken in the same order, will be proportionals.

Thus, as a b c d

e: ƒ :: g h

ik :: l: m

then, as aei bfk :: cgl dhm

For ad= bc

eh=fg

im-kl

therefore, adehim befgkl.

Consequently, aei : bfk :: cgl: dhm; therefore the proposition is manifest.

120. COROLLARY.-Hence, if the proportions are the same as the first, we shall have a3 : b3 :: c3 : d3.

121. The proportion which is formed by the multiplication of the corresponding terms of two or more proportions is said to be compounded of these proportions.

When two of the proportions to be compounded are the same, the proportion compounded of the two is said to be the duplicate of either: thus, a2 : b2 : : c2 : d2 is in duplicate proportion to that of a: b::c: d.

When three proportionals, which are the same, are compounded, the compound is said to be triplicate to either of the simple ones: thus, a3: b3 :: c3 : d3, is the triplicate proportion of a: b:: c: d.

And so on to the succeeding orders.