in Arithmetic and Geometry, is that relation of homogeneous things which determines the quantity of one from the quantity of another, without the intervention of a third.

The numbers, lines, or quantities, A and B, being proposed, their relation one to another may be considered under one of these two heads: 1. How much A exceeds B, or B exceeds A? And this is found by taking A from B, or B from A, and is called arithmetic reason or ratio. 2. Or how many times, or parts of a time, A contains B, or B contains A? and this is called geometric reason or ratio; (or, as Euclid defines it, it is the mutual habitude, or respect, of two magnitudes of the same kind, according to quantity; that is, as to how often the one contains, or is contained in, the other;) and is found by dividing A by B, or B by A. And here note, that that quantity which is referred to another quantity is called the antecedent of the ratio; and that to which the other is referred is called the consequent of the ratio; as in the ratio of A to B, A is the antecedent, and B the consequent. Therefore any quantity, as antecedent, divided by any quantity as a consequent, gives the ratio of that antecedent to the consequent.

Thus the ratio of A to B is $\frac{A}{B}$, but the ratio of B to A is $\frac{B}{A}$; and, in numbers, the ratio of 12 to 4 is $\frac{12}{4} = 3$, or triple; but the ratio of 4 to 12 is $\frac{4}{12} = \frac{1}{3}$, or subtriple.

And here note, that the quantities thus compared must be of the same kind; that is, such as by multiplication may be made to exceed one the other, or as these quantities are said to have a ratio between them, which, being multiplied, may be made to exceed one another. Thus a line, how short soever, may be multiplied, that is, produced so long as to exceed any given right-line; and consequently these may be compared together, and the ratio expressed: but as a line can never, by any multiplication whatever, be made to have breadth, that is, to be made equal to a superficies, how small soever; these can therefore never be compared together, and consequently have no ratio or respect to one another, according to quantity; that is, as to how often the one contains, or is contained in, the other. See QUANTITY.