# Proposition: Prop. 10.038: Second Bimedial is Irrational

### (Proposition 38 from Book 10 of Euclid's “Elements”)

If two medial (straight lines), commensurable in square only, which contain a medial (area) , are added together then the whole (straight line) is irrational - let it be called a second bimedial (straight line).1 * For let the two medial (straight lines), $AB$ and $BC$, commensurable in square only, (and) containing a medial (area) , be laid down together [Prop. 10.28]. * I say that $AC$ is irrational.

### Modern Formulation

The second bimedial is a straight line whose length is expressible as $\alpha^{1/4}+\frac{\sqrt{\beta}}{\alpha^{1/4}},$

where $\alpha$ and $\beta$ are positive rational numbers.

### Notes

The second bimedial and the second apotome of a medial (see Prop. 10.75), whose length is expressible as

$\alpha^{1/4}-\frac{\sqrt{\beta}}{\alpha^{1/4}},$

are the positive roots of the quartic $x^4-2\,\left(\frac{\left(\alpha+\beta\right)}{\sqrt{\alpha}}\right)\,x^2+ \left(\frac{\left(\alpha-\beta\right)^2}{\alpha}\right) = 0.$

Proofs: 1

Corollaries: 1
Proofs: 2 3 4 5 6 7
Propositions: 8 9 10 11 12

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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, "second from two medials" (translator's note).