Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/171

Rh PROPOSITION D. THEOREM.

If the first he to the second as the third is to the fourth, and if the first he a multijyle, or a part, of the second, the third shall he the same multiple, or the same part, of the fourth.

Let A be to B as C is to D. And first, let A be a multiple of B: C shall be the same multiple of D.

Take E equal to A; and whatever multiple A or E is of B, make F the same multiple of D.

Then, because A is to B as C is to D, [Hypothesis. and of B the second and D the fourth have been taken equimultiples E and F; [Construction. therefore A is to E as C is to F. [V. 4, Corollary.

But A is equal to E; [Construction. therefore C is equal to F. [V. A.

And F is the same multiple of D that A is of B; [Construction. therefore C is the same multiple of D that A is of B.

Next, let A be a part of B: C shall be the same part of D. For, because A is to B as C is to D; [Hypothesis. therefore, inversely, B is to A as D is to C. [V. B. But A is a part of B; [Hypothesis. that is, B is a multiple of A; therefore, by the preceding case, D is the same multiple of C; that is, C is the same part of D that A is of B.

Wherefore, if the first &c.

PROPOSITION 7. THEOREM.

Equal magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes.