Corollary: Cor. 10.111: Thirteen Irrational Straight Lines of Different Order

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

(Corollary to Proposition 111 from Book 10 of Euclid's “Elements”)

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.

Modern Formulation

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Proofs: 1

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Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"


  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.