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

Proof: By Euclid

(related to Proposition: Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational)

• For let the straight line $BC$, which is incommensurable in square $AB$, and fulfils the (other) prescribed (conditions), have been subtracted from the (straight line) $AB$ [Prop. 10.35].
• I say that the remainder $AC$ is the irrational (straight line) called that which makes with a medial (area) a medial whole.

• For let the rational (straight line) $DI$ be laid down.
• And let $DE$, equal to the (sum of the squares) on $AB$ and $BC$, have been applied to $DI$, producing $DG$ as breadth.
• And let $DH$, equal to twice the (rectangle contained) by $AB$ and $BC$, have been subtracted (from $DE$) [producing $DF$ as breadth].
• Thus, the remainder $FE$ is equal to the (square) on $AC$ [Prop. 2.7].
• Hence, $AC$ is the square root of $FE$.
• And since the sum of the squares on $AB$ and $BC$ is medial, and is equal to $DE$, $DE$ [is] thus medial.
• And it is applied to the rational (straight line) $DI$, producing $DG$ as breadth.
• Thus, $DG$ is rational, and incommensurable in length with $DI$ [Prop. 10.22].
• Again, since twice the (rectangle contained) by $AB$ and $BC$ is medial, and is equal to $DH$, $DH$ is thus medial.
• And it is applied to the rational (straight line) $DI$, producing $DF$ as breadth.
• Thus, $DF$ is also rational, and incommensurable in length with $DI$ [Prop. 10.22].
• And since the (sum of the squares) on $AB$ and $BC$ is incommensurable with twice the (rectangle contained) by $AB$ and $BC$, $DE$ (is) also incommensurable with $DH$.
• And as $DE$ (is) to $DH$, so $DG$ also is to $DF$ [Prop. 6.1].
• Thus, $DG$ (is) incommensurable (in length) with $DF$ [Prop. 10.11].
• And they are both rational.
• Thus, $GD$ and $DF$ are rational (straight lines which are) commensurable in square only.
• Thus, $FG$ is an apotome [Prop. 10.73].
• And $FH$ (is) rational.
• And the [rectangle] contained by a rational (straight line) and an apotome is irrational [Prop. 10.20], and its square root is irrational.
• And $AC$ is the square root of $FE$.
• Thus, $AC$ is irrational.
• Let it be called that which makes with a medial (area) 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"