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 or other of these two heads: 1. How much A exceeds B, or B exceeds A; which is found by taking A from B, or B from A, and is called arithmetical reason or ratio. 2. How many times, or parts of a time, A contains B; or B contains A; which is called geometrical reason or ratio, or, as Euclid defines it, the mutual habitude, or respect, of two magnitudes of the same kind, according to quantity, that is, how often the one contains or is contained in the other; and this is found by dividing A by B, or B by A. But observe, 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. 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 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 observe, that the quantities thus compared must be of the same kind; that is, such as by multiplication may be made the one to exceed 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 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 equal to a superficies, however small, so these can never be compared together, and consequently have no ratio or respect to one another, according to quantity; that is, how often the one contains or is contained in the other.