Commensurability of Squares on Proportional Straight Lines/Lemma
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Lemma to Commensurability of Squares on Proportional Straight Lines
In the words of Euclid:
- Given two unequal straight lines, to find by what square the square on the greater is greater than the square on the less.
(The Elements: Book $\text{X}$: Proposition $14$ : Lemma)
Proof
Let $AB$ and $C$ be two unequal straight lines.
Let $AB > C$.
Let the semicircle $ADB$ be described with $AB$ as the diameter.
Using Fitting Chord Into Circle, let $AD$ be fitted into $ADB$ equal to $C$.
Let $DB$ be joined.
From Relative Sizes of Angles in Segments, $\angle ADB$ is a right angle.
From Pythagoras's Theorem:
- $AB^2 = AD^2 + DB^2$
and so $AB^2$ is greater than $AD^2$, that is, $C^2$, by $DB^2$.
Conversely, given two straight lines $A$ and $B$ the same technique can be used to find the straight line the square of whose length equals the sum of the squares on $A$ and $B$.
$\blacksquare$
Historical Note
This proof is Proposition $14$ of Book $\text{X}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions