Proposition: Prop. 10.037: First Bimedial is Irrational

(Proposition 37 from Book 10 of Euclid's “Elements”)

If two medial (straight lines), commensurable in square only, which contain a rational (area), are added together then the whole (straight line) is irrational - let it be called a first bimedial1 (straight line).


Modern Formulation

Thus, a first bimedial straight line has a length expressible as \[\delta^{1/4}+\delta^{3/4},\]

for some positive rational number \(\delta\).


The first bimedial and the corresponding first apotome of a medial, whose length is expressible as

\[\delta^{1/4} - \delta^{3/4},\]

(see Prop. 10.74), are the positive roots of the quartic \[x^4-2\,\delta\sqrt{\delta} x^2+ \delta\left(1-\delta\right)^2 = 0.\]

Proofs: 1

Corollaries: 1
Proofs: 2 3 4 5 6 7 8 9
Propositions: 10 11 12 13 14 15

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  1. Literally, "first from two medials" (translator's note).