the identity or similitude of two ratios. Hence quantities that have the same ratio between them are said to be proportional. Thus, 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 habitude 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. Proportion 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 habitus or relation of two ratios when compared together; as ratio is of two quantities.

Proportion, Harmonical or Musical. This subject, as usually treated, has little or no practical nor even theoretical utility. Enough regarding the ratios of musical sounds and intervals will be found under the articles Music and Temperament.

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, namely,

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.