# Definition: Def. 10.11: First Apotome

### (Definition 11 from Book 10 of Euclid's “Elements”)

Given a rational (straight line) and an apotome, if the square on the whole is greater than the (square on a straight line) attached (to the apotome) by the (square) on (some straight line) commensurable in length with (the whole), and the whole is commensurable in length with the (previously) laid down rational (straight line), then let the (apotome) be called a first apotome.

### Modern Formulation

The first apotome is a straight line whose length is $\alpha-\alpha\,\sqrt{1-\beta^{\,2}},$ where $$\alpha,\beta$$ denote positive rational numbers.

Corollaries: 1
Proofs: 2 3 4 5 6 7 8 9 10 11 12 13 14
Propositions: 15 16 17 18 19 20 21 22 23 24

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