the identity or finitude 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 simply called proportionals. 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 that 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 habitude or relation of two ratios when compared together; as ratio is of two quantities. See Algebra, A-Proportion, Arithmetic, and Geometry.

Arithmetical and Geometrical Proportion. See Progression.

Harmonical or Musical Proportion, is a kind of numerical 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 \( \frac{2}{6} = \frac{1}{3} \). 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 \( \frac{24}{9} = \frac{16}{8} = \frac{12}{3} \). 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 their 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 proportional 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 divide the product by the number remaining after the middle or second is subtracted from double the first; the quotient is a third harmonical proportion; thus supposing the numbers 9, 12, 16, a fourth will be found by the rule to be 24.

5. If there be four numbers disposed in order, whereof one extreme and the two middle terms are in arithmetical proportion; and the same middle terms with the other extreme are in harmonical proportion, the four are in geometrical proportion; as here 2 : 3 : 4 : 6, which are geometrical; whereof 2, 3, 4, are arithmetical, and 3, 4, 6, harmonical.

6. If betwixt any two numbers you put an arithmetical mean, and also an harmonical one, the four will be in geometrical proportion; thus betwixt 2 and 6 an arithmetical mean is 4, and an harmonical one 3; and the four 2 : 3 : 4 : 6, are geometrical.

We have this notable difference between the three kinds of proportion, arithmetical, harmonical, and geometrical; that from any given number we can raise a continued arithmetical series increasing in infinitum, but not... PROPORTION not decreasing: the harmonical is decreasable in infinitum, but not increasable; the geometrical is both.

or Rule of Three. See Arithmetic, nos. 13, 14, 15.

Reciprocal PROPORTION. See Reciprocal.

PROPORTION is also used for the relation between unequal things of the same kind, whereby their several parts correspond to each other with an equal augmentation or diminution.

Thus, in reducing a figure into little, or in enlarging it, care is taken to observe an equal diminution or enlargement, through all its parts; so that if one line, e.g., be contracted by one-third of its length, all the rest shall be contracted in the same proportion.

in architecture, denotes the just magnitude of the members of each part of a building, and the relation of the several parts to the whole; e.g., of the dimensions of a column, &c., with regard to the ordonnance of a whole building.

One of the greatest differences among architects, M. Perrault observes, is in the proportions of the heights of entablatures with respect to the thickness of the columns, to which they are always to be accommodated.

In effect, there is scarcely any work, either of the ancients or moderns, wherein this proportion is not different; some entablatures are even near twice as high as others;—yet it is certain this proportion ought of all others to be most regulated; none being of greater importance, as there is none wherein a defect is sooner spied, nor any wherein it is more shocking.

Compass of PROPORTION, a name by which the French, and after them some English authors call the SEC-

TOR.