So, again, if the square on the greater term is larger than (the square on) [the lesser] by the (square) on (some straight line) incommensurable in length with (the greater) then, if the greater term is commensurable in length with the rational (straight line previously) laid out, let (the whole straight line) be called a fourth binomial (straight line).
The fourth binomial 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 8
Propositions: 9 10 11