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.
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