# Proposition: Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely

### (Proposition 46 from Book 10 of Euclid's “Elements”)

The square root of a rational plus a medial (area) can be divided (into its component terms) at one point only. * Let $AB$ be the square root of a rational plus a medial (area) which has been divided at $C$, so that $AC$ and $CB$ are incommensurable in square, making the sum of the (squares) on $AC$ and $CB$ medial, and twice the (rectangle contained) by $AC$ and $CB$ rational [Prop. 10.40]. * I say that $AB$ cannot be (so) divided at another point.

### Modern Formulation

In other words, $\sqrt{\frac{\sqrt{1+\alpha^2}+\alpha}{2\,(1+\alpha^2)}} + \sqrt{\frac{\sqrt{1+\alpha^2}-\alpha}{2\,(1+\alpha^2)}}=\sqrt{\frac{\sqrt{1+\beta^2}+\beta}{2\,(1+\beta^2)}} + \sqrt{\frac{\sqrt{1+\beta^2}-\beta}{2\,(1+\beta^2)}}$ has only one solution: i.e., $\beta=\alpha,$

where $$\alpha,\beta$$ denote positive rational numbers.

### Notes

This proposition corresponds to [Prop. 10.83], with plus signs instead of minus signs.

Proofs: 1

Propositions: 1

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