Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/189

Rh For, because AB is to BE as CD is to DF; [Hypothesis. therefore, by division, AE is to EB as CF is to FD; [V, 17. and, by inversion, EB is to AE as FD is to CF. [V. B. Therefore, by composition, AB is to AE as CD is to CF. [V. 18.

Wherefore, if four magnitudes &c.

PROPOSITION 20. THEOREM.

If there me three magnitudes, and other three, which have the same ratio, taken two and two, then, if the first be greater than the third, the fourth shall he greater than the sixth; and if equal, equal; and if less, less.

Let A, B, C be three magnitudes, and D, E, F other three, which have the same ratio taken two and two; that is, let A be to B as D is to E, and let B be to C as E is to F: if A be greater than C, D shall be greater than F; and if equal, equal; and if less, less.

First, let A be greater than C: D shall be greater than F. For, because A is greater than C, and B is any other magnitude, therefore A has to B a greater ratio than C has to B. [V. 8. But A is to B as D is to E; [Hypothesis. therefore D has to E a greater ratio than C has to B. [V. 13. And because B is to C as E is to F, [Hyp. therefore, by inversion, C is to B as F is to E. [V. B. And it was shewn that D has to E a greater ratio than C has to B; therefore D has to E a greater ratio than F has to E; [V. 13, Cor. therefore D is greater than F, [V. 10.