Proposition: Prop. 10.036: Binomial is Irrational

Euclid's Formulation

If two rational (straight lines which are) commensurable in square only are added together then the whole (straight line) is irrational - let it be called a binomial (straight line).1 * For let the two rational (straight lines), $AB$ and $BC$, (which are) commensurable in square only, be laid down together. * I say that the whole (straight line), $AC$, is irrational.

Modern Formulation

Thus, a binomial straight line has a length expressible as

$1 +\sqrt{\delta}$

or, more generally,

$\rho\left(1+\sqrt{\delta}\right),$

where $\rho$ and $\delta$ are positive rational numbers.

Notes

The binomial and the corresponding apotome, whose length is expressible as

$1 -\sqrt{\delta},$

(see Prop 10.73), are the positive roots of the quartic $x^4-2\left(1+\delta\right)x^2 + \left(1-\delta\right)^2 = 0.$

Proofs: 1

Corollaries: 1
Proofs: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Propositions: 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

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References

Adapted from (subject to copyright, with kind permission)

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

Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016

Footnotes

1. Literally, "from two names" (translator's note).