Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/170

146 PROPOSITION C. THEOREM.

If the first he the same multiple of the second or the same part of it, that the third is of the fourth, the first shall he to the second as the third is to the fourth.

First, let A be the same multiple of B that C is of D: A shall be to B as C is to D.

Take of A and C any equimultiples whatever E and F; and of B and D any equimultiples whatever G and H.

Then, because A is the same multiple of B that C is of D; [Hypothesis. and that E is the same multiple of A that F is of C; [Construction therefore E is the same multiple of B that F is of D; [V. 3. that is, E and F are equimultiples of B and D.

But G and H are equimultiples of B and D; [Construction. therefore if E be a greater multiple of B than C is of B, F is a greater multiple of D than H is of D; that is, if E be greater than G, F is greater than H.

In the same manner, if E be equal to G, F may be shewn to be equal to H; and if less, less.

But E and F are any equimultiples whatever of A and C, and G and H are any equimultiples whatever of B and D; [Construction. therefore A is to B as C is to D. [V. Definition 5.

Next, let A be the same part of B that C is of D: A shall be to B as C is to D.

For, since A is the same part of B that C is of D, therefore B is the same multiple of A that D is of C; therefore, by the preceding case, B is to A as D is to C; therefore, inversely, A isto B as C is to D.

Wherefore, if the first &c.