Again, if the square on the whole is greater than (the square on) the attached (straight line) by the (square) on (some straight line) incommensurable [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 fourth apotome.
The fourth apotome is a straight line whose length is \[\alpha-\frac \alpha{\sqrt{1+\beta}},\]
where \(\alpha,\beta\) denote positive rational numbers.
Proofs: 1 2 3 4 5 6 7
Propositions: 8 9 10 11