# Proposition: Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely

### (Proposition 44 from Book 10 of Euclid's “Elements”)

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

### Modern Formulation

In other words,

$\alpha^{1/4}+\frac{\beta^{1/2}}{\alpha^{1/4}} = \gamma^{1/4}+\frac{\delta^{1/2}}{\gamma^{1/4}}$ has only one solution: i.e., $\gamma=\alpha\quad\text{ and }\quad \delta=\beta,$ where $$\alpha,\beta,\gamma,\delta$$ denote positive rational numbers.

### Notes

This proposition corresponds to [Prop. 10.81], with plus signs instead of minus signs.

Proofs: 1

Propositions: 1

Thank you to the contributors under CC BY-SA 4.0!

Github:

non-Github:
@Fitzpatrick

### 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