Proposition: Prop. 10.051: Construction of Fourth Binomial Straight Line

(Proposition 51 from Book 10 of Euclid's “Elements”)

To find a fourth binomial (straight line).

fig051e

Modern Formulation

If the rational straight line has unit length then the length of a fourth binomial straight line is \[\alpha+\frac \alpha{\sqrt{1+\beta}},\]

where \(\alpha,\beta\) denote positive rational numbers.

Notes

This, and the fourth apotome, whose length according to [Prop. 10.88] is \[\alpha-\frac \alpha{\sqrt{1+\beta}}\] are the roots of the quadratic function \[x^2- 2\,\alpha\,x+\frac{\alpha^2\,\beta}{1+\beta}=0,\]

where \(\alpha,\beta\) denote positive rational numbers.

Proofs: 1

Propositions: 1


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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