If four magnitudes are proportional, then the sum of the greatest and the least is greater than the sum of the remaining two.

Let the four magnitudes *AB, CD, E,* and *F* be proportional so that *AB* is to *CD* as *E* is to *F,* and let *AB* be the greatest of them and *F* the least.

I say that the sum of *AB* and *F* is greater than the sum of *CD* and *E.*

Make *AG* equal to *E,* and *CH* equal to *F.*

Since *AB* is to *CD* as *E* is to *F,* and *E* equals *AG,* and *F* equals *CH,* therefore *AB* is to *CD* as *AG* is to *CH.*

And since the whole *AB* is to the whole *CD* as the part *AG* subtracted is to the part *CH* subtracted, therefore the remainder *GB* is also to the remainder *HD* as the whole *AB* is to the whole *CD.*

But *AB* is greater than *CD,* therefore *GB* is also greater than *HD.*

And, since *AG* equals *E,* and *CH* equals *F,* therefore the sum of *AG* and *F* equals the sum of *CH* and *E.*

And if, *GB* and *HD* being unequal, and *GB* greater, the sum of *AG* and *F* is added to *GB,* and the sum of *CH* and *E* is added to *HD,* it follows that the sum of *AB* and *F* is greater than the sum of *CD* and *E.*

Therefore, *if four magnitudes are proportional, then the sum of the greatest and the least is greater than the sum of the remaining two.*

Q.E.D.

This proposition says that if *w* : *x* = *y* : *z* and *w* is the greatest of the four magnitudes while *z* is the least, then *w* + *z* > *x* + *y.* All four magnitudes must be of the same kind.

The argument goes as follows. Since *w* : *x* = *y* : *z*, therefore (*w* – *y*) : (*x* – *z*) = *w* : *x*, by V.19. But since *w* > *x*, therefore *w* – *y* > *x* – *z*. Thus *w* + *z* > *x* + *y*.

This proposition is not used in the rest of the *Elements* but is an end in itself.

- The arithmetic mean of two magnitudes is less than their geometric mean.

This proposition relies on treating V.Def.4 as an axiom of comparability. Infinitesimal counterexample: when *y* is infinitesimal with respect to *x,* consider the proportion (*x* + 5*y*) : (*x* + 2*y*) = (*x* + 4*y*) : *x.*