# Proposition: Prop. 10.079: Construction of Apotome is Unique

### (Proposition 79 from Book 10 of Euclid's “Elements”)

(Only) one rational straight line, which is commensurable in square only with the whole, can be attached to an apotome. * Let $AB$ be an apotome, with $BC$ (so) attached to it. * $AC$ and $CB$ are thus rational (straight lines which are) commensurable in square only [Prop. 10.73]. * I say that another rational (straight line), which is commensurable in square only with the whole, cannot be attached to $AB$. ### Modern Formulation

In other words,

$\alpha - \sqrt{\beta} = \gamma - \sqrt{\delta}$ has only one solution: i.e., $\gamma=\alpha\quad\text{ and }\quad\delta=\beta,$ where $$\alpha,\beta,\gamma,\delta$$ denote positive rational numbers. Likewise, $\sqrt{\alpha} - \sqrt{\beta} =\sqrt{\gamma}-\sqrt{\delta}$ has only one solution: i.e., $\gamma=\alpha\quad\text{ and }\quad\delta=\beta.$

### Notes

This proposition corresponds to [Prop. 10.42], with minus signs instead of plus signs.

Proofs: 1

Proofs: 1 2
Propositions: 3

Github: non-Github:
@Fitzpatrick