359. An accurate and familiar acquaintance with the doctrine of ratios, is necessary to a ready understanding of the principles of proportion, one of the most important of all the branches of the mathematics. In considering ratios, we compare two quantities, for the purpose of finding either their difference, or the quotient of the one divided by the other. But in proportion, the comparison is between two ratios. And this comparison is limited to such ratios as are equal. We do not inquire how much one ratio is greater or less than another, but whether they are the same. Thus the numbers 12, 6, 8, 4, are said to be proportional, because the ratio of 12:6 is the same as that of 8:4.

360. Proportion, then, is an equality of ratios. It is either arithmetical or geometrical. Arithmetical proportion is an equality of arithmetical ratios, and geometrical proportion is an equality of geometrical ratios. Thus the numbers 6, 4, 10, 8, are in arithmetical proportion, because the difference between 6 and 4 is the same as the difference between 10 and 8. And the numbers 6, 2, 12, 4, are in geoinetrical proportion, because the quotient of 6 divided by 2, is the same as the quotient of 12 divided by 4.

361. Care must be taken not to confound proportion with ratio. This caution is the more necessary, as in common discourse, the two terms are used indiscriminately, or rather, proportion is used for both. The expenses of one man are said to bear a greater proportion to his income, than those of another. But according to the definition which has just been given, one proportion is neither greater nor less than another. For equality does not admit of degrees. One ratio may be greater or less than another. The ratio of 12:2 is greater than that of 6:2, and less than that of 20:2. But these differences are not applicable to proportion, when the term is used in its technical sense. The loose signification which is so frequently attached to this word, may be proper enough in familiar language: for it is sanctioned by a general usage. But for scientific purposes, the distinction between proportion and ratio should be clearly drawn, and cautiously observed.

362. The equality between two ratios, as has been stated, is called proportion. The word is sometimes applied also to the series of terms among which this equality of ratios exists. Thus the two couplets 15:5 and 6:2 are, when taken together, called a proportion.

363. Proportion are expressed by the common sign of equality.
            8 •• 6 = 4 •• 2,
            a •• b = c •• d
are arithmetical proportions.

            12 : 6 = 8 : 4,
            a : b = d : h
are geometrical proportions.
The latter is read, 'the ratio of a to b equals the ratio of d to h', or more concisely, 'a is to b, as d to h'.

364. The first and last terms are called the extremes, and the other two theO means. Homologous terms are either the two antecedents or the two consequents. Analogous terms are the antecedent and consequent of the same couplet.

365. As the ratios are equal, it is manifestly immaterial which of the two couplets is placed first.

If a : b = c : d, then c : d = a : b. For if $\frac{a}{b } =\frac{c}{d } $ then $\frac{c}{d } =\frac{a}{b } $.

366. The number of terms uraat be, at least, four. For the equality is between the ratios of two couplets; and each couplet must have an antecedent and a consequent. There may be a proportion, however, among three quantities. For one of the quantities may be repeated, so as to form two terms. In this case the quantity repeated is called the middle term, or a mean proportional between the two other quantities, especially if the proportion is geometrical.

Thus the numbers 8, 4, 2, are proportional. That is, 8:4 = 4:2. Here 4 is both the consequent in the first couplet, and the antecedent in the last. It is therefore a mean proportional between 8 and 2.

The last term is called a third proportional to the two other quantities. Thus 2 is a third proportional to 8 and 4.

367. Inverse or reciprocal proportion is an equality between a direct ratio, and a reciprocal ratio.

Thus 4:2 = ⅓:⅙; that is, 4 is to 2, reciprocally, as 3 to 6. Sometimes also, the order of the terms in one of the couplets, is inverted, without writing them in the form of a fraction.—(Art. 346.)
Thus 4:2 = 3:6 inversely. In this case, the first term is to the second, as the fourth to the third; that is, the first divided by the second, is equal to the fourth divided by the third.

368. When there is a series of quantities, such that the ratios of the first to the second, of the second to the third, of the third to the fourth, etc. are all equal; the quantities are said to be in continued proportion. The consequent of each preceding ratio is, then, the antecedent of the following one. Continued proportion is also called progression, as will be seen in a following section.
Thus the numbers 10, 8, 6, 4, 2, are in continued arithmetical proportion. For 10 - 8 = 8 - 6 = 6 - 4 = 4 - 2.
The numbers 64, 32, 16, 8, 4, are in continued geometrical proportion. For 64:32 = 32:16 = 16:8 = 8:4.
If a, b, c, d, h, etc. are in continued geometrical proportion; then a:b = b:c = c:d = d:h, etc.
One case of continued proportion is that of three proportional quantities. (Art. 366.)

369. As an arithmetical proportion is, generally, nothing more than a very simple equation, it is scarcely necessary to give the subject a separate consideration.
      The proportion a..b = c..d
      Is the same as the equation a - b = c - d.

It will be proper, however, to observe that, if four quantities are in arithmetical proportion, the sum of the extremes is equal to the sum of the means.
      Thus if a..b = h..m, then a + m = b + h
      For by supposition, a - b = h - m
      And transposing -b and -m, a + m = b + h
So in the proportion, 12..10 = 11..9, we have 12 + 9 = 10 + 11.

Again if three quantities are in arithmetical proportion, the sum of the extremes is equal to double the mean.
      If a..b = b..c, then, a - b = b - c
      And transposing -b and -c, a + c = 2b.

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