And if neither of (the whole or the attached straight line) is commensurable in length with the (previously) laid down rational (straight line), and the square on the whole is greater than (the square on) the attached (straight line) by the (square) on (some straight line) commensurable (in length) with (the whole), then let the (apotome) be called a third apotome.
The third apotome is a straight line whose length is \[\sqrt{\alpha}\,\left(1-\sqrt{1-\beta^{\,2}}\right),\]
where \(\alpha,\beta\) denote positive rational numbers.
Proofs: 1 2 3 4 5 6
Propositions: 7 8 9 10