namely,
becomes small as n increases, we see that the difference between the ratio of two incommensurable quantities and that of two commensurable numbers m and n can be made as small as we please. Hence, ultimately, the ratio of incommensurable quantities may be regarded as the limit of the ratio of commensurable quantities.
5. The two terms of a ratio are called the antecedent and the consequent. These correspond to the numerator and the denominator of a fraction. Hence we have the following definition:—“A ratio is the fraction got by making the antecedent the numerator and the consequent the denominator.”
6. The reciprocal of a ratio is the ratio obtained by interchanging the antecedent and the consequent. Thus, 4 : 3 is the reciprocal of the ratio 3 : 4. Hence we have the following theorem:—“The product of a ratio and its reciprocal is unity.”
7. If we multiply any two numbers, as 5 and 7, by any number such as 4, the products 20, 28 are called equimultiples of 5 and 7. In like manner, 10 and 15 are equimultiples of 2 and 3, and 18 and 30 of 3 and 5, &c.
v. The first of four magnitudes has to the second the same ratio which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth, if, according as the multiple of the first is greater than, equal to, or less than the multiple of the second, the multiple of the third is greater than, equal to, or less than the multiple of the fourth.
vi. Magnitudes which have the same ratio are called proportionals. When four magnitudes are proportionals, it is usually expressed by saying, “The first is to the second as the third is to the fourth.”
viii. Analogy or proportion is the similitude of ratios.