the identity or similitude of two ratios. Hence quantities that have the same ratio between them are said to be proportional; e.g., if A be to B as C to D, or 8 be to 4 as 30 to 15; A, B, C, D, and 8, 4, 30, and 15, are said to be in proportion, or are simply called proportional. Proportion is frequently confounded with ratio; yet have the two in reality very different ideas, which ought by all means to be distinguished. Ratio is properly the relation or habi- PRO

tude of two things, which determines the quantity of one from the quantity of another, without the intervention of any third: thus we say the ratio of 5 and 10 is 2, the ratio of 12 and 24 is 2. Proposition is the sameness or likeness of two such relations; thus the relations between 5 and 10 and 12 and 24 being the same, or equal, the four terms are said to be in proportion.

Hence ratio exists between two numbers, but proportion requires at least three. Proportion, in fine, is the habitude or relation of two ratios when compared together; as ratio is of two quantities. See ALGEBRA, ARITHMETIC, and GEOMETRY.

Arithmetical and Geometrical Proportion. See PROGRESSION.

Inordinate Proportion, is where the order of the terms compared is disturbed or irregular. As, for example, in two ranks of numbers, three in each rank, viz. in one rank, 2, 3, 9; and in the other, 8, 24, 36, which are proportional, the former to the latter, but in a different order, viz. 2 : 3 :: 24 : 36, and 3 : 9 :: 8 : 24, then casting out the mean terms in each rank it is concluded that 2 : 9 :: 8 : 36, that is, the first is to the third in the first rank, as the first is to the third in the second rank.

Harmonical or Musical Proportion, is a kind of numeral proportion formed thus: of three numbers, if the first be to the third as the difference of the first and second to the difference of the second and third; the three numbers are in harmonical proportion.

Thus 2, 3, 6, are harmonical, because 2 : 6 :: 1 : 3. So also four numbers are harmonical, when the first is to the fourth as the difference of the first and second to the difference of the third and fourth.

Thus 24, 16, 12, 9, are harmonical, because 24 : 9 :: 8 : 3. By continuing the proportional terms in the first case, there arises an harmonical progression or series.

1. If three or four numbers in harmonical proportion be multiplied or divided by the same number; the products or quotients will also be in harmonical proportion: thus, if 6, 8, 12, which are harmonical, be divided by 2, the quotients 3, 4, 6, are also harmonical; and reciprocally the products by 2, viz. 6, 8, 12.

2. To find an harmonical mean between two numbers given; divide double the product of the two numbers by their sum, the quotient is the mean required; thus suppose 3 and 6 the extremes, the product of these is 18, which doubled gives 36; this divided by 9 (the sum of 3 and 6) gives the quotient 4. Whence 3, 4, 6, are harmonical.

3. To find a third harmonical proportion to two numbers given.

Call one of them the first term, and the other the second; multiply them together, and divide the product by the number remaining after the second is subtracted from double the first; the quotient is a third harmonical proportional: thus, suppose the given terms 3, 4, their product 12 divided by 2 (the remainder after 4 is taken from 6, the double of the first), the quotient is 6, the harmonical third sought.

4. To find a fourth harmonical proportion to three terms given: multiply the first into the third, and di-

Vol. XVII. Part II.