If two rational (straight lines which are) commensurable in square only are added together then the whole (straight line) is irrational - let it be called a binomial (straight line).^{1} * For let the two rational (straight lines), $AB$ and $BC$, (which are) commensurable in square only, be laid down together. * I say that the whole (straight line), $AC$, is irrational.
Thus, a binomial straight line has a length expressible as
\[1 +\sqrt{\delta}\]
or, more generally,
\[\rho\left(1+\sqrt{\delta}\right),\]
where $\rho$ and $\delta$ are positive rational numbers.
The binomial and the corresponding apotome, whose length is expressible as
\[1 -\sqrt{\delta},\]
(see Prop 10.73), are the positive roots of the quartic \[x^4-2\left(1+\delta\right)x^2 + \left(1-\delta\right)^2 = 0.\]
Proofs: 1
Corollaries: 1
Proofs: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Propositions: 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
Literally, "from two names" (translator's note). ↩