which, taken two and two in a cross order, have the same ratio,

and so on, whatever be the number of such magnitudes.

PROP. XXIV. THEO.

If the first has to the second the same ratio which the third has to the fourth, and the fifth to the second the same which the sixth has to the fourth, the first and fifth together shall have to the second the same ratio which the third and sixth together have to the fourth.

and, because these magnitudes are proportionals, they are proportionals when taken jointly,

Cor. 1. If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second as the excess of the third and sixth to the fourth. The demonstration of this is the same with that of the proposition, if division be used instead of composition.

Cor. 2. The proposition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to a fourth magnitude; as is manifest.

Arithmetical Illustration

Let the number 6 (the first) have to 4 (the second) the same ratio which 3 (the third) has to 2 (the fourth), and 12 (the fifth) to 4 (the second) the same ratio which 6 (the sixth) has to 2 (the fourth).

PROP. XXV. THEO.

If four magnitudes of the same kind are proportionals, the greatest and least of them together are greater than the other two together.

of the same kind, be proportionals, that is to say,

and

be the greatest of the four, and, consequently,

by proposition A and the 14th of this book,

is the least;