PROPOSITION 25.
If four magnitudes be proportional,
the greatest and the least are greater than the remaining two.
Let the four magnitudes
AB,
CD,
E,
F be proportional so that, as
AB is to
CD, so is
E to
F, and let
AB be the greatest of them and
F the least; I say that
AB,
F are greater than
CD,
E.
For let
AG be made equal to
E, and
CH equal to
F.
Since, as
AB is to
CD, so is
E to
F, and
E is equal to
AG, and
F to
CH,
therefore, as AB is to CD, so is AG to CH.
And since, as the whole
AB is to the whole
CD, so is the part
AG subtracted to the part
CH subtracted,
the remainder GB will also be to the remainder HD as the whole AB is to the whole CD. [V. 19]
But
AB is greater than
CD;
therefore GB is also greater than HD.
And, since
AG is equal to
E, and
CH to
F, therefore
AG,
F are equal to
CH,
E.
And if,
GB,
HD being unequal, and
GB greater,
AG,
F be added to
GB and
CH,
E be added to
HD,
it follows that AB, F are greater than CD, E.
Therefore etc. Q. E. D.