7. What is the square root of 0,000729? Ans. 0,027. 8. What is the square root of 3?

Ans. 1,732050 &c.

9. What is the square root of 343396 ? 10. What is the square root of 300304? 11. What is the square root of 220900 ? 12. What is the square root of 10?

Ans. 586.

Ans. 548.

Ans. 470.

Ans. 3,162277 &c.
Ans. 43,4.

13. What is the square root of 1883,56 ?

NOTE III.-The square root of a vulgar fraction is found, by extracting the roots of the numerator and denominator. But, if either the numerator or denominator should not be a perfect square, the fraction should be reduced to a decimal, and the approximate root extracted.

(57.) Divide the given number into periods of three figures each, beginning at the right; and if there be decimals, point off the decimals in a contrary direction from the decimal point; observing, that if the last period of decimals does not contain three figures, the deficiency is to be supplied by annexing ciphers.

Find the greatest cube in the left hand period, and place its root at the right, after the manner of a quotient in division. Subtract its cube from the aforementioned period, and to the remainder bring down the next period for a dividend.

Multiply the square of the root already found by 300,

for a divisor. Seek how often the divisor is contained in the dividend, and place the result in the root.

Complete the divisor by adding to it thirty times the product of the root before found by this last figure, and also the square of the last figure; then multiply the divisor, so increased, by the same figure, and subtract the product from the dividend.

To the remainder bring down the next period, and proceed as before; and continue the operation in the same manner, through all the periods of the given number.

NOTE I.-If the last mentioned product should be greater than the dividend, the last figure in the root must be diminished.

(563)2 = 316969 × 300 = 95090700 380633104 3d dividend. 563 × 4 × 30+ 16 =

=

67576

NOTE II. To find the cube root of a fraction, apply the same principles as in the square root, thus:

-

0,875 = 0,9564, &c.

3

RATIOS AND PROPORTIONS.

(58.) Any relation which subsists between two numbers is called a ratio.

An arithmetical ratio is the difference of two numbers, and is expressed by writing the less number after the greater, with the sign of subtraction between them. Thus, the arithmetical ratio of 7 to 3, is written 7-3; that of 9 to 4, is written 9- -4.

A geometrical ratio is the quotient of one number divided by another, or the number of times one number is contained in another. The particular value of this quotient is called the index of the ratio. The geometrical ratio of 12 to 4, is written 12: 4, or; and 3, the quotient of 12 divided by 4, is the index of the ratio.

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Ratio can only exist between numbers expressing quantities of the same kind; as it would be absurd to enquire, how

much more than a rod is two pounds? or how many times five yards of cloth is contained in forty dollars?

Two ratios may be equal to each other, and such ratios, when written together, with the sign of equality between them, constitute a proportion. Thus, the arithmetical ratio 9 — 4, is equal to the ratio 12-7, and from these two ratios, the proportion 9-4 = 12-7, may be formed.

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The geometrical ratio, 12: 4, or is equal to the ratio

and these two ratios form the proportion,

18: 6, or

The first term of a ratio is called the antecedent, and the second term the consequent. The first and last terms of a proportion are called the extreme terms, the second and third the mean terms; or simply, the extremes and means.

ARITHMETICAL PROPORTIONS.

(59.) An arithmetical ratio being the difference of two numbers, and an arithmetical proportion expressing an equality between two ratios; it follows, that an arithmetical proportion shows that the difference of two given numbers is equal to the difference of two others. Thus, 12 — 7 = 9 —— 4,. shows that the difference of 12 and 7 is equal to the difference of 9 and 4. Another propriety of arithmetical proportions, and one on which most of the calculations relating to them are based, is the following:-

In every arithmetical proportion, the sum of the means is equal to the sum of the extremes.

To prove this, take the proportion 12-79-4. Since equals added to equals make equal sums, the quantities on each side of the sign of equality will still be equal, if 7 be added to each; and we shall have 12-7+7=9-4+7: adding also 4 to each; we shall have

127+7+49-4+7-4. But since a quantity both added to, and subtracted from, another, can neither increase nor diminish it, we may expunge 7 from the first, and 4 from the second, and there remains 12+4 97; that is, the sum of the extremes equal to the sum of the means. We have now proved this property

to belong to the particular proportion 12-79-4; but to show that it is general and independent of the particular numbers used, let a, b, c, and d, represent any numbers whatever which have this property, namely, that the difference between the numbers represented by a and b, is equal to the difference of those represented by c and d. They will evidently form the arithmetical proportion,

α b = C d.

If we now add d to each of these equal quantities, we shall have

ab+d=c-d+d;

and adding b to each,

ab+d+b=cd+d+b;

and as b is both added to, and subtracted from, the first, and d added to, and subtracted from, the second, b may be omitted in the first, and d in the second, and we have

a + d = c + b ;

or the sum of the extremes, equal to the sum of the means.

The letters of the alphabet are here introduced to accustom the pupil to their use, preparatory to his entering upon the study of Algebra. He will easily perceive, that they are here subjected to the same arithmetical operations as the numbers in the preceding proportion, and will see how, by means of signs, we can apply the rules of arithmetic to letters as the representatives of numbers without giving to the letters any particular numerical value. Thus, in the proportion a bc-d, any numbers whatever which have such a relation to each other, that the second shall be as much less than the first, as the fourth is less than the third, may be put in the place of the letters; and all that we have proved of the letters, will be true of the numbers.

Suppose a 21, b = 12, c = 19, and d = 10; then

=

21-12-19-10;

and since we had a + d = c +b, we shall have

21+ 10 = 12+ 19.

Again, let a = 11, b = 6, c = 18 and d = 13, then

(60.) From the property which has just been demonstrated, it follows, that if three terms of an arithmetical proportion be given, the fourth can always be found by the following rule.