(related to Lemma: Lem. 10.13: Finding Pythagorean Magnitudes)

- Let $AB$ and $C$ be the two given unequal straight lines, and let $AB$ be the greater of them.
- So it is required to find by (the square on) which (straight line) the square on $AB$ (is) greater than (the square on) $C$.

- Let the semicircle $ADB$ have been described on $AB$.
- And let $AD$, equal to $C$, have been inserted into it [Prop. 4.1].
- And let $DB$ have been joined.
- So (it is) clear that the angle $ADB$ is a right angle [Prop. 3.31], and that the square on $AB$ (is) greater than (the square on) $AD$ - that is to say, (the square on) $C$ - by (the square on) $DB$ [Prop. 1.47].
- And, similarly, the square root of (the sum of the squares on) two given straight lines is also found likeso.
- Let $AD$ and $DB$ be the two given straight lines.
- And let it be necessary to find the square root of (the sum of the squares on) them.
- For let them have been laid down such as to encompass a right angle - (namely), that (angle encompassed) by $AD$ and $DB$.
- And let $AB$ have been joined.
- (It is) again clear that $AB$ is the square root of (the sum of the squares on) $AD$ and $DB$ [Prop. 1.47].
- (Which is) the very thing it was required to show.∎

**Fitzpatrick, Richard**: Euclid's "Elements of Geometry"