# Areas of Circles are as Squares on Diameters/Lemma

## Contents

## Theorem

In the words of Euclid:

*I say that, the area $S$ being greater than the circle $EFGH$, as the area $S$ is to the circle $ABCD$, so is the circle $EFGH$ to some area less than the circle $ABCD$.*

(*The Elements*: Book $\text{XII}$: Proposition $2$ : Lemma)

## Proof

Let it be contrived that:

- $S : ABCD = EFGH : T$

for some area $T$.

It is to be demonstrated that $T$ is less than the circle $ABCD$.

From the relationship:

- $S : ABCD = EFGH : T$

it follows from Proposition $16$ of Book $\text{V} $: Proportional Magnitudes are Proportional Alternately that:

- $S : EFGH = ABCD : T$

But:

- $S > EFGH$

Therefore:

- $ABCD > T$

Hence the result.

$\blacksquare$

## Historical Note

This theorem is Proposition $2$ of Book $\text{XII}$ 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 $\text{XII}$. Propositions