This has the same fault as some of the others—it is not a definition, but an inference. It occurs when the means in a proportion are equal, so that, in fact, there are four terms. As an illustration, let us take the numbers 4, 6, 9. Here the ratio of 4 : 6 is , and the ratio of 6 : 9 is , so that 4, 6, 9 are continued proportionals; but, in reality, there are four terms, for the full proportion is 4 : 6 :: 6 : 9. x. When three magnitudes are continual proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second. xi. When four magnitudes are continual proportionals, the first is said to have to the fourth the triplicate ratio of that which it has to the second. xii. When there is any number of magnitudes of the same kind greater than two, the first is said to have to the last the ratio compounded of the ratios of the first to the second, of the second to the third, of the third to the fourth, &c. We have placed these definitions in a group; but their order is inverted, and, as we shall see, Def. xii. is a theorem, and x. and xi. are only inferences from it. 1. If we have two ratios, such as 5 : 7 and 3 : 4, and if we convert each ratio into a fraction, and multiply these fractions together, we get a result which is called the ratio compounded of the two ratios; viz. in this example it is , or 15 : 28. It is evident we get the same result if we multiply the two antecedents together for a new antecedent, and the two consequents for a new consequent. Hence we have the following definition:—“The ratio compounded of any number of ratios it the ratio of the product of all the antecedents to the product of all the consequents.” 2. To prove the theorem contained in Def. xii. Let the magnitudes be a, b, c, d. Then the ratio of 1st : 2nd = , 2nd : 3rd = , 3rd : 4th = . Hence the ratio compounded of the ratio of 1st : 2nd, of 2nd : 3rd, of 3rd : 4th 3. If three magnitudes be proportional, the ratio of the 1st : 3rd is equal to the square of the ratio of the 1st : 2nd. For the ratio of the 1st : 3rd is compounded of the ratio of the 1st : 2nd, and of the ratio of the 2nd : 3rd; and since these ratios are equal, the ratio compounded of them will be equal to the square of one of them. Or thus: Let the proportionals be a, b, c, that is, let a : b :: b : c; hence we have And multiplying each by , we get or a : c :: a2 : b2—that is, 1st : 3rd :: square of 1st : square of 2nd. Now, the ratio of 1st : 3rd is, by Def. x., the duplicate ratio of 1st : 2nd. Hence the duplicate ratio of two magnitudes means the square of their ratio, or, what is the same thing, the ratio of their squares (see Book VI. xx.).