jected, taking care to point out the exceptions to which these transformations are liable.

Two inequalities are said to subsist in the same sense, when the greater quantity is in the first member in both, or in the second member in both; and in a contrary sense, when the greater quantity is in the first member of one and in the second member of the other.

68 and 79, Thus, 25 20 and 18> 10, or are inequalities which subsist in the same sense; and the in equalities

15 13 and 12 < 14,

subsist in a contrary sense.

1. If we add the same quantity to both members of an inequality, or subtract the same quantity from both members, the resulting inequality will subsist in the same sense.

Thus, take 8>6; by adding 5, we still have

+56 +5;

When the two members of an inequality are both negative, that one is the least, algebraically considered, which contains the greatest number of units.

Thus, 25-20; and if 30 be added to both members, we have 5 < 10. This must be understood entirely in an alge braic sense, and arises from the convention before established, to consider all quantities preceded by the minus sign, as subtractive. The principle first enunciated serves to transpose certain terms from one member of the inequality to the other. Take. for ex ample, the inequality

a2 + b2 >3b2-2a2;

2. If two inequalities subsist in the same sense, and we add them member to member, the resulting inequality will also subsist in the

same sense,

Thus, if we add

there results

ab and c>d, member to member,

a+c>b+d.

But this is not always the case, when we subtract, member from member, two inequalities established in the same sense.

Let there be two inequalities 4 <7 and 2 <3, we have 4-2 or 2<7 3 or 4.

But if we have the inequalities 910 and 68, by subtracting, we have

9-6 or 3 > 10 8 or 2.

We should then avoid this transformation as much as possible, or if we employ it, determine in which sense the resulting inequality subsists.

3. If the two members of an inequality be multiplied by a positive quantity, the resulting inequality will exist in the same

we deduce, by multiplying by bad,

3a(a2 — b2) > 2d (c2 — d2),

and the same principle is true for division. But,

When the two members of an inequality are multiplied or divided by a negative quantity, the resulting inequality will subsist in a contrary sense.

Take, for example, 8>7; multiplying by

-24-21.

- 3, we have

Therefore, when the two members of an inequality are multiplied or divided by a quantity, it is necessary to ascertain whether the multiplier or divisor is negative; for, in that case, the inequality will exist in a contrary sense.

4. It is not permitted to change the signs of the two members of an inequality, unless we establish the resulting inequality in a contrary sense; for, this transformation is evidently the same as multiplying the two members by - 1.

--

5. Both members of an inequality between positive quantities can be squared, and the inequality will exist in the same sense. Thus, from 5>3, we deduce, 25 > 9; from a + b>c, we find

(a + b)2 > c2.

6. When the signs of both members of the inequality are not known, we cannot tell before the operation is performed, in which sense the resulting inequality will exist.

For example, 2<3 gives (-2)2 or 49.

But, 35 gives, on the contrary, (3)2 or 9 (-5)2 or 25.

We must, then, before squaring, ascertain the signs of the two members.

Let us apply these principles to the solution of the following examples. By the solution of an inequality is meant the oper ation of finding an inequality, one member of which is the unknown quantity, and the other a known expression.

CHAPTER V.

EXTRACTION OF THE SQUARE

ROOT OF

NUMBERS.--FORMATION OF THE

SQUARE AND EXTRACTION OF THE SQUARE ROOT OF ALGEBRAIC QUANTITIES. TRANSFORMATION OF RADICALS OF THE SECOND DEGREE.

93. THE square or second power of a number, is the product which arises from multiplying that number by itself once: for example, 49 is the square of 7, and 144 is the square of 12. The square root of a number, is that number which multiplied by itself once will produce the given number. Thus, 7 is the square root of 49, and 12 the square root of 144: for, 7 x7 and 12 x 12 = 144.

49,

The square of a number, either entire or fractional, is easily found, being always obtained by multiplying the number by itself once. The extraction of the square root is, however, attended with some difficulty, and requires particular explanation. The first ten numbers are,

1, 2, 3, 4, 5, 6, 7. 8, 9, 10,

aud their squares,

1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Conversely, the numbers in the first line, are the square roots of the corresponding numbers in the second line.

We see that the square of any number, expressed by one figure, will contain no unit of a higher order than tens.

The numbers in the second line are perfect squares, and, generally, any number which results from multiplying a whole number by itself once, is a perfect square.

If we wish to find the square root of any number less than 100, we look in the second line, above given, and if the number is there written, the corresponding number in the first line

is its square root. If the number falls between any two num bers in the second line, its square root will fall between the corresponding numbers in the first line. Thus, 55 falls between 49 and 64; hence, its square root is greater than 7 and less than 8. Also, 91 falls between 81 and 100; hence, its square root is greater than 9 and less than. 10.

If now, we change the units of the first line, 1, 2, 3, 4, &c., into units of the second order, or tens, by annexing 0 to each,

we shall have,

10, 20, 30,

40, 50, 60, 70, 80,

90,

100,

and their corresponding squares will be,

100, 400, 900, 1600, 2500, 3600, 4900, 6400, 8100, 10000 : Hence, the square of any number of tens will contain no unit of less denomination than hundreds.

94. We may regard every number as composed of the sum of its tens and units.

Now, if we represent any number by N, and denote the tens by a, and the units by b, we shall have,

N=a+b;

whence, by squaring both members,

N2 a2+2ab + b2:

=

Hence, the square of a number is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units.

For example,

78708, hence,

(78)2 = (70)2 + 2 × 70 × 8 + (8)2 = 4900 + 1120 + 64 = 6084.

60 84

95. Let us now find the square root of 6084. Since this number is expressed by more than two figures, its root will be expressed by more than one. But since it is less than 10000, which is the square of 100, the root will contain but two places of figures; that is, units and tens.

Now, the square of the number of tens must be found in the number expressed by the two left-hand figures, which we will separate from the other two, by placing a point over the place