NOTE II. To find the cube root of a fraction, apply the

same principles as in the square root, thus :

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,

12

is written 12: 4, or 7; and 3, the quotient of 12 divided by 4, is the index of the ratio.

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.

The geometrical ratio, 12: 4, or

18

12

4'

is equal to the ratio

18:6, or ; and these two ratios form the proportion,

6

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

12-7+7+49-47-4.

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 = 9+7; 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 12-7=9-4; 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 cand d. They will evidently form the arithmetical proportion,

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

a-b+d=c-d+d;

and adding b to each,

a-b+d+b=c-d+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 - b = c -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..

RULE.

If the unknown term be one of the extremes, add together the mean terms, and subtract the other extreme; if it be a mean term, add the extremes and subtract the given

mean.

Let there be the arithmetical proportion 19-7 23-,

=

and we shall have for the fourth term,

7+23-19 = 30 — 19 = 11.

In the proportion 13 – 5 = . - 12, the third term is found by adding the extremes, and subtracting the known mean terms; thus,

13 +12-5 = 25 – 5 = 20, the third term, and we have the complete proportion,

13-520-12.

When the consequent of the first ratio is the same as the antecedent of the second, the proportion is called a continued arithmetical proportion. 18-15 = 15 – 12 is a continued arithmetical proportion; the number 15 being the consequent of the first ratio, and the antecedent of the second.

To such a proportion another ratio, or any number of ratios may be added, all having this same property, and this series of ratios is called an arithmetical series, or progression: thus, 18-15= 15-12- 12-9-9-6=6-3=3-0, is an arithmetical progression.

It is usual in arithmetical progression, to omit one of the mean terms of each proportion, and also the sign of equality. The progression above would be written,

18-15-12-9-6-3-0.

Arithmetical progressions are usually written without the sign between the terms. When the numbers increase from left to right, the series is called an increasing arithmetical progression; when they increase in a contrary order, a decreasing arithmetical progression.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19,

is an increasing progression;

21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1,

is a decreasing progression.

The numbers that enter into an arithmetical progression, are called the terms of the progression. The first and last terms are the extremes, and the number by which each term is greater or less than that which preceded it, is called the common difference.

We proceed now to give certain properties of arithmetical progressions, without demonstrations.*

(61.) 1st. In every arithmetical progression, the sum of the extremes is equal to the sum of any two terms equally distant from them; or, equal to double the middle term, when there is an odd number of terms.

Thus, in the progression, 2, 4, 6, 8, 10, 12, 14, we have, 2+14=4+ 12 = 6 + 10 = 8+8, or 2 × 8.

(62.) 2d. In an increasing arithmetical progression, the last term is equal to the first term, plus the common difference multiplied by the number of terms less 1; and in a decreasing arithmetical progression, the last term is equal to the first, minus the common difference multiplied by the number of terms less 1.

Thus, in the progression 3, 7, 11, &c., the 12th term is 3 + (4 × 11) = 3 + 44 = 47.

In the progression 82, 79, 76, &c., the 25th term is 82-(3×24) = 82—72 = 10.

EXAMPLES.

1. The first term of an increasing arithmetical progression is 5, the common difference 6, what is the 10th term ?

Ans. 59.

2. The 10th term of an arithmetical progression is 34, the common difference 3, required the first term?

whence, 34

In a question like this, we may consider 34 the first term of a decreasing progression, and find the 10th term, which would correspond to the 1st term in an increasing progression ; (3×9) = 34-27 = 7, the ans. 3. The 1st term of a progression is 2, the common difference, what is the 27th term ? Ans. 103. 4. The 12th term of a progression is 3, the common difference, what is the 1st term? Ans..