Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/159

Rh When four magnitudes are proportionals it is usually expressed by saying, the first is to the second as the third is to the fourth.

7. When of the equimultiples of four magnitudes, taken as in the fifth definition the multiple of the first is greater than the multiple of the second, but the multiple of the third is not greater than the multiple of the fourth, then the first is said to have to the second a greater ratio than the third has to the fourth ; and the third is said to have to the fourth a less ratio than the first has to the second.

8. Analogy, or proportion, is the similitude of ratios.

9. Proportion consists in three terms at least.

10. When three magnitudes are proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second.

[The second magnitude is said to be a mean proportional between the first and the third.]

11. When four magnitudes are continued proportionals, the first is said to have to the fourth, the triplicate ratio of that which it has to the second, and so on, quadruplicate, &c. increasing the denomination still by unity, in any number of proportionals.

Definition of compound ratio. When there are any number of magnitudes of the same kind, the first is said to have to the last of them, the ratio which is compounded of the ratio which the first has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and so on unto the last magnitude.

For example, A, B, C, D be four magnitudes of the same kind, the first A is said to have to the last D, the ratio compounded of the ratio of A to B, and of the ratio of B to C, and of the ratio of C to D ; or, the ratio of A to B is said to be compounded of the ratios of A to B, B to C, and C to D.

And if A has to B the same ratio that E has to F; and B to C the same ratio that G has to H ; and C to D the same ratio that K has to L ; then, by this definition, A is said to have to D the ratio compounded of ratios which are the same with the ratios of E to F, G to H, and K to L.