◀ ▲ ▶Branches / Geometry / Elements-euclid / Book-10-incommensurable-magnitudes / Proposition: Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely

Proposition: Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely

(Proposition 43 from Book 10 of Euclid's “Elements”)

A first bimedial (straight line) can be divided (into its component terms) at one point only.

• Let $AB$ be a first bimedial (straight line) which has been divided at $C$, such that $AC$ and $CB$ are medial (straight lines), commensurable in square only, (and) containing a rational (area) [Prop. 10.37].
• I say that $AB$ cannot be (so) divided at another point.

Modern Formulation

In other words, \[\alpha^{1/4} + \alpha^{3/4} = \beta^{1/4} + \beta^{3/4}\] has only one solution: i.e., \[\beta=\alpha,\] where \(\alpha,\beta\) denote positive rational numbers.

Notes

This proposition corresponds to [Prop. 10.80], with plus signs instead of minus signs.

Table of Contents

Proofs: 1

Mentioned in:

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