which may be expressed by whole numbers, which the latter have not.

If we conceive unity to be divided into five parts, for example, we express the quotient arising from the division of 9 by 5, or 3, by nine of these parts; then, being contained five times in unity, and nine times in, is the common measure of unity and the fraction, and the relation these quantities have to each other is that of the entire numbers 5 and 9.

Since whole numbers, as well as fractions, have a common measure with unity, we say that these quantities are commensurable with unity, or simply that they are commensurable; and since their relations or ratios, with respect to unity, are expressed by entire numbers, we designate both whole numbers and fractions, by the common name of rational numbers.

On the contrary, the square root of a number, which is not a perfect square, is incommensurable or irrational, because, as it cannot be represented by any fraction, into whatever number of parts we suppose unity to be divided, no one of these parts will be sufficiently small to measure exactly, at the same time, both this root and unity.

In order to denote, in general, that a root is to be extracted, whether it can be exactly obtained or not, we employ the character, which is called a radical sign;

16 is equivalent to 4,

2 is incommensurable or irrational.

100. Although we cannot obtain, either among whole numbers or fractions, the exact expression for 2, yet we may approximate it, to any degree we please, by converting this number into a fraction, the denominator of which is a perfect square. The root of the greatest square contained in the numerator will then be that of the proposed number expressed in parts, the value of which will be denoted by the root of the denominator.

If we convert, for example, the number 2 into twenty-fifths, we have. As the root of 50 is 7, so far as it can be expressed in whole numbers, and the root of 25 exactly 5, we obtain 7, or 1 for the root of 2, to within one fifth.

101. This process, founded upon what was laid down in article 96, that the square of a fraction is expressed by the square of the numerator divided by the square of the denominator, may evidently be applied to any kind of fraction whatever, and more

readily to decimals than to others. It is manifest, indeed, from the nature of multiplication, that the square of a number expressed by tenths will be hundredths, and that the square of a number expressed by hundredths will be ten thousandths, and so on; and consequently, that the number of decimal figures in the square is always double that of the decimal figures in the root. The truth of this remark is further evident from the rule observed in the multiplication of decimal numbers, which requires that a product should contain as many decimal figures, as there are in both the factors. In any assumed case, therefore, the proposed number, considered as the product of its root multiplied by itself, must have twice as many decimal figures as its root.

From what has been said, it is clear, that in order to obtain the square root of 227, for example, to within one hundredth, it is necessary to reduce this number to ten thousandths, that is, to annex to it four cyphers, which gives 2270000 ten thousandths. The root of this may be extracted in the same manner, as that of an equal number of units; but to show that the result is hundredths, we separate the two last figures on the right by a comma. We thus find that the root of 227 is 15,06, accurate to hundredths. The operation may be seen below;

If there are decimals already in the proposed number, they should be made even. To extract, for example, the root of 51,7, we place one cypher after this number, which makes it hundredths; we then extract the root of 51,70. If we proposed to have one decimal more, we should place two additional cyphers after this number, which would give 51,7000; we should then obtain 7,19 for the root.

If it were required to find the square root of the numbers 2 and 3 to seven places of decimals, we should annex fourteen cyphers to these numbers; the result would be

102. When we have found more than half the number of figures, of which we wish the root to consist, we may obtain the rest simply by division. Let us take, for example, 32976; the square root of this number is 181, and the remainder, 215. If

we divide this remainder 215, by 362, double of 181, and extend the quotient to two decimal places, we obtain 0,59, which must be added to 181; the result will be 181,59 for the root of 32976, which is accurate to within one hundredth.

In order to prove that this method is correct, let us designate the proposed number by N, the root of the greatest square contained in this number by a, and that which it is necessary to add to this root to make it the exact root of the proposed number by b; we have then

From this result it is evident, that the first member may be

taken for the value of b, so long as the quantity

b2

is less than 2 a

a unit of the lowest place found in b. But as the square of a number cannot contain more than twice as many figures as the number itself, it follows, that if the number of figures in a exceeds double those in b, the quantity

b2

2 a

will then be a fraction.

In the preceding example, a 181 units, or 18100 hundredths, and consequently contains one figure more than the square of

b2

59 hundredths; the fraction then becomes, in this case,

(59)2

2 a

3481 and is less than a unit of the second part

2 X 18100 36200'

59, or than a hundredth of a unit of the first.

103. This leads to a method of approximating the square root of a number by means of vulgar fractions. It is founded on the circumstance, that a, being the root of the greatest square contained in N, b is necessarily a fraction, and

ler than b, may be neglected.

b2

2 a

being much smal

If it were required, for example, to extract the square root of 2; as the greatest square contained in this number is 1, if we subtract this, we have a remainder, 1. Dividing this remainder by double of the root, we obtain ; taking this quotient for the value of the quantity b, we have, for the first approximation to

the root, 1+, or 3. Raising this root to its square, we find, which, subtracted from 2 or, gives for a remainder. In this case the formula

Substituting

for b, we have for the second approximation

— = 1; taking the square of, we find, a quantity,

This operation may be easily continued to any extent we please. I shall give, in the Supplement to this treatise, other formulas more convenient for extracting roots in general.

tor;

104. In order to approximate the square root of a fraction, the method, which first presents itself, is, to extract, by approximation, the square root of the numerator and that of the denominabut with a little attention it will be seen, that we may avoid one of these operations by making the denominator a perfect square. This is done by multiplying the two terms of the proposed fraction by the denominator. If it were required, for example, to extract the square root of, we might change this fraction into

by multiplying its two terms by the denominator, 7. Taking the root of the greatest square contained in the numerator of this fraction, we have for the root of, accurate to within 4.

If a greater degree of exactness were required, the fraction must be changed by approximation or otherwise into another, the denominator of which is the square of a greater number than 7. We shall have, for example, the root sought within, if we convert into 225ths, since 225 is the square of 15; thus the fraction becomes 75 of one 225th, or, within; the root of

96

falls between

and, but approaches nearer to the second fraction than to the first, because 96 approaches nearer to a hundred than to 81; we have then or for the root of within' By employing decimals in approximating the root of the numerator of the fraction, we obtain 4,583 for the approximate root of the numerator 21, which is to be divided by the root of the new denominator. The quotient thence arising, carried to three places of decimals, becomes 0,655.

105. We are now prepared to resolve all equations involving only the second power of the unknown quantity connected with known quantities.

We have only to collect into one member all the terms containing this power, to free it from the quantities, by which it is multiplied (11); we then obtain the value of the unknown quantity by extracting the square root of each member.

Let there be, for example, the equation

Making the divisors to disappear, we find first

15x2 168 84 14x2.

Transposing to the first member the term 14 x2, and to the second the term 168, we have

It should be carefully observed, that to denote the root of the fraction, the sign is made to descend below the line, which separates the numerator from the denominator. If it were

252

written thus, the expression would designate the quotient

9 29

arising from the square root of the number 252 divided by 29; a result different from 252, which denotes, that the division is to be performed before the root is extracted.

Let there be the literal equation

α x2 + b3 = c x2 + d3 ;