# Proposition: Prop. 10.080: Construction of First Apotome of Medial is Unique

### (Proposition 80 from Book 10 of Euclid's “Elements”)

Only one medial straight line, which is commensurable in square only with the whole, and contains a rational (area) with the whole, can be attached to a first apotome of a medial (straight line). * For let $AB$ be a first apotome of a medial (straight line), and let $BC$ be (so) attached to $AB$. * Thus, $AC$ and $CB$ are medial (straight lines which are) commensurable in square only, containing a rational (area) - (namely, that contained) by $AC$ and $CB$ [Prop. 10.74]. * I say that a(nother) medial (straight line), which is commensurable in square only with the whole, and contains a rational (area) with the whole, cannot be attached to $AB$.

### Modern Formulation

In other words, the first apotome of a medial straight line is unique, i.e. $\alpha^{1/4} - \alpha^{3/4} = \beta^{1/4} - \beta^{3/4}$ has only one solution: i.e., $\beta=\alpha,$ where $$\alpha,\beta$$ denote positive rational numbers.

### Notes

This proposition corresponds to [Prop. 10.43], with minus signs instead of plus signs.

Proofs: 1

Propositions: 1

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