# Proposition: Prop. 10.081: Construction of Second Apotome of Medial is Unique

### (Proposition 81 from Book 10 of Euclid's “Elements”)

Only one medial straight line, which is commensurable in square only with the whole, and contains a medial (area) with the whole, can be attached to a second apotome of a medial (straight line). * Let $AB$ be a second apotome of a medial (straight line), with $BC$ (so) attached to $AB$. * Thus, $AC$ and $CB$ are medial (straight lines which are) commensurable in square only, containing a medial (area) - (namely, that contained) by $AC$ and $CB$ [Prop. 10.75]. * I say that a(nother) medial straight line, which is commensurable in square only with the whole, and contains a medial (area) with the whole, cannot be attached to $AB$. ### 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. 10.44], with minus signs instead of plus signs.

Proofs: 1

Propositions: 1

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