◀ ▲ ▶Branches / Geometry / Elements-euclid / Book-10-incommensurable-magnitudes / Proof: By Euclid

Proof: By Euclid

(related to Proposition: Prop. 10.110: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted)

• For, as in the previous figures, let the medial (area) $BD$, incommensurable with the whole, have been subtracted from the medial (area) $BC$.
• I say that the square root of $EC$ is one of two irrational (straight lines) - either a second apotome of a medial (straight line), or that (straight line) which with a medial (area) makes a medial whole.

• For since $BC$ and $BD$ are each medial (areas), and $BC$ (is) incommensurable with $BD$, accordingly, $FH$ and $FK$ will each be rational (straight lines), and incommensurable in length with $FG$ [Prop. 10.22].
• And since $BC$ is incommensurable with $BD$ - that is to say, $GH$ with $GK$ - $HF$ (is) also incommensurable (in length) with $FK$ [Prop. 6.1], [Prop. 10.11].
• Thus, $FH$ and $FK$ are rational (straight lines which are) commensurable in square only.
• $KH$ is thus as apotome [Prop. 10.73], [and $FK$ an attachment (to it).
• So, the square on $FH$ is greater than (the square on) $FK$ either by the (square) on (some straight line) commensurable, or by the (square) on (some straight line) incommensurable (in length) with ($FH$).]
• So, if the square on $FH$ is greater than (the square on) $FK$ by the (square) on (some straight line) commensurable (in length) with ($FH$), and (since) neither of $FH$ and $FK$ is commensurable in length with the (previously) laid down rational (straight line) $FG$, $KH$ is a third apotome [Def. 10.3] .
• And $KL$ (is) rational.
• And the rectangle contained by a rational (straight line) and a third apotome is irrational, and the square root of it is that irrational (straight line) called a second apotome of a medial (straight line) [Prop. 10.93].
• Hence, the square root of $LH$ - that is to say, (of) $EC$ - is a second apotome of a medial (straight line).
• And if the square on $FH$ is greater than (the square on) $FK$ by the (square) on (some straight line) incommensurable [in length] with ($FH$), and (since) neither of $HF$ and $FK$ is commensurable in length with $FG$, $KH$ is a sixth apotome [Def. 10.16] .
• And the square root of the (rectangle contained) by a rational (straight line) and a sixth apotome is that (straight line) which with a medial (area) makes a medial whole [Prop. 10.96].
• Thus, the square root of $LH$ - that is to say, (of) $EC$ - is that (straight line) which with a medial (area) makes a medial whole.
• (Which is) the very thing it was required to show.
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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"