(related to Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line)
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)
Proofs: 1
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. ↩