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
or, more generally,
where $\rho$ and $\delta$ are positive rational numbers.
The binomial and the corresponding apotome, whose length is expressible as
(see Prop 10.73), are the positive roots of the quartic \[x^4-2\left(1+\delta\right)x^2 + \left(1-\delta\right)^2 = 0.\]
Literally, "from two names" (translator's note). ↩