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

Proof: By Euclid

(related to Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line)

• Let $AB$ be an apotome.

• I say that $AB$ is not the same as a binomial.

• For, if possible, let it be (the same).

• And let a rational (straight line) $DC$ be laid down.
• And let the rectangle $CE$, equal to the (square) on $AB$, have been applied to $CD$, producing $DE$ as breadth.
• Therefore, since $AB$ is an apotome, $DE$ is a first apotome [Prop. 10.97].
• Let $EF$ be an attachment to it.
• Thus, $DF$ and $FE$ are rational (straight lines which are) commensurable in square only, and the square on $DF$ is greater than (the square on) $FE$ by the (square) on (some straight line) commensurable (in length) with ($DF$), and $DF$ is commensurable in length with the (previously) laid down rational (straight line) $DC$ [Def. 10.10] .
• Again, since $AB$ is a binomial, $DE$ is thus a first binomial [Prop. 10.60].
• Let ($DE$) have been divided into its (component) terms at $G$, and let $DG$ be the greater term.
• Thus, $DG$ and $GE$ are rational (straight lines which are) commensurable in square only, and the square on $DG$ is greater than (the square on) $GE$ by the (square) on (some straight line) commensurable (in length) with ($DG$), and the greater (term) $DG$ is commensurable in length with the (previously) laid down rational (straight line) $DC$ [Def. 10.5] .
• Thus, $DF$ is also commensurable in length with $DG$ [Prop. 10.12].
• The remainder $GF$ is thus commensurable in length with $DF$ [Prop. 10.15].
• Therefore, since $DF$ is commensurable with $GF$, and $DF$ is rational, $GF$ is thus also rational.
• Therefore, since $DF$ is commensurable in length with $GF$, $DF$ (is) incommensurable in length with $EF$.
• Thus, $FG$ is also incommensurable in length with $EF$ [Prop. 10.13].
• $GF$ and $FE$ [are] thus rational (straight lines which are) commensurable in square only.
• Thus, $EG$ is an apotome [Prop. 10.73].
• But, (it is) also rational.
• The very thing is impossible.
• Thus, an apotome is not the same as a binomial.
• (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"