Sum of Antecedent and Consequent of Proportion

Theorem

As Euclid defined it:

If four magnitudes be proportional, the greatest and least are greater than the remaining two.

(The Elements: Book V: Proposition $25$)

That is, if $a : b = c : d$ and $a$ is the greatest and $d$ is the least, then:

$a + d > b + c$

Proof

Let the four magnitudes $AB, CD, E, F$ be proportional, so that $AB : CD = E : F$.

Let $AB$ be the greatest and $F$ the least.

We need to show that $AB + F > CD + E$.

Let $AG = E, CH = F$.

We have that $AB : CD = E : F$, $AG = E, F = CH$.

So $AB : CD = AG : CH$.

So from Proportional Magnitudes have Proportional Remainders $GB : HD = AB : CD$.

But $AB > CD$ and so $GB > HD$.

Since $AG = E$ and $CH = F$, it follows that $AG + F = CH + E$.

We have that $GB > HD$.

So add $AG + F$ to $GB$ and $CH + E$ to $HD$.

It follows that $AB + F > CD + E$.

$\blacksquare$

Historical Note

This is Proposition 25 of Book V of Euclid's The Elements.