Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/162

138 PROPOSITION 1. THEOREM.

If any number of magnitudes be equimultiples of as many, each of each; whatever multiple any one of them is of its part, the same multiple shall all the first magnitudes be of all the other.

Let any number of magnitudes AB, CD be equimultiples of as many others E, F, each of each: whatever multiple AB is of E, the same multiple shall AB and CD together, be of E and F together.

For, because AB is the same multiple of E, that CD is of F, as many magnitudes as there are in AB equal to E, so many are there in CD equal to F. Divide AB into the magnitudes AG, GB each equal to E; and CD into the magnitudes CH, HD, each equal to F. Therefore the number of the magnitudes CH, HD, will be equal to the number of the magnitudes AG, GB.

And, because AG is equal to E, and CH equal to F, therefore AG and CH together are equal to E and F together; and because GB is equal to E, and HD equal to F, therefore GB and HD together are equal to E and F together. [Axiom 2. Therefore as many magnitudes as there are in AB equal to E, so many are there in AB and CD together equal to E and F together. Therefore whatever multiple AB is of E, the same multiple is AB and CD together, of E and F together.

Wherefore, if any number of magnitudes &c.

PROPOSITION 2. THEOREM.

If the first be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; the first together with the fifth shall be the same multiple of the second that the third together with the sixth is of the fourth.