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 D is to E, [Hypothesis.
and C is to B as F is to E, [Hyp. V. B.
therefore D is to E as F is to E; [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 she^vn in the first case, C is to B as F is to E;
and, in the same manner, B is to A as E is to D;
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.
PROPOSITION 21. THEOREM.
If there be three magnitudes, and other three, which the same ratio, taken two and two, but in a cross order, then if the first he 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, but in a cross order; that is, let A be to B as E is to F, and let B to C as D is to E: 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.
