The apotome and the irrational (straight lines) after it are neither the same as a medial (straight line) nor (the same) as one another. (In particular,) there are, in order, 13 (different)1 irrational (straight lines) in all: * Medial, * Binomial, * First bimedial, * Second bimedial, * Major, * Square root of a rational plus a medial (area) , * Square root of (the sum of) two medial (areas), * Apotome, * First apotome of a medial, * Second apotome of a medial, * Minor, * That which with a rational (area) produces a medial whole, * That which with a medial (area) produces a medial whole.
(not yet contributed)
Euclid's expression "(not) being the same in order" means that the resulting irrational number is "(not) of the same kind" as that irrational number, with which it is commensurable. ↩