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 E is to F; [Hypothesis.
therefore E has to F a greater ratio than C has to B. [V. 13.
And because B is to C as D is to E, [Hypothesis.
therefore, by inversion, C is to B as E is to D. [V. B.
And it was shewn that E has to F a greater ratio than C has to B;
therefore E has to F a greater ratio than E has to D; [V. 13, Cor.
therefore F is less than D; [V. 10.
that is, D is greater than F.
Secondly, let A be equal to C: D shall be equal to F.
For, because A is equal to C, and B is any other magnitude,
therefore A is to B as C is to B. [V. 7.
But A is to B as E is to F; [Hyp.
and C is to B as E is to D; [Hyp. V. B.
therefore E is to F as E is to D; [V. 11.
and therefore D is equal to F. [V. 9.
Lastly, let A be less than C: D shall be less than F.
For C is greater than A;
and, as was shewn in the first case, C is to B as E is to D;
and in the same manner, B is to A as F is to E;
therefore, by the first case, F is greater than D;
that is, D is less than F.
Wherefore, if there be three &c. q.e.d.
