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. q.e.d.
PROPOSITION 7. THEOREM.
Equal magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes.
