Sum of Antecedent and Consequent of Proportion
Theorem
In the words of Euclid:
- If four magnitudes be proportional, the greatest and least are greater than the remaining two.
(The Elements: Book $\text{V}$: 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 proof is Proposition $25$ of Book $\text{V}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{V}$. Propositions