Second Bimedial is Irrational/Mistake

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Source Work

1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.)

In the words of Euclid:

If two medial straight lines commensurable in square only and containing a medial rectangle be added together, the whole is irrational; and let it be called a second bimedial straight line.

(The Elements: Book $\text{X}$: Proposition $38$)

The mistake can be seen in the edition as published by Dover: ISBN 0-486-60089-0.


For let a rational straight line $DE$ be set out, and let the parallelogram $DF$ equal to the square on $AC$ be applied to $DE$, producing $DG$ as breadth. [I.44]

The reference is incorrect.

Proposition $44$ of Book $\text{I} $: Construction of Parallelogram on Given Line equal to Triangle in Given Angle

is not the result needed here.

The correct reference should be:

Proposition $45$ of Book $\text{I} $: Construction of Parallelogram in Given Angle equal to Given Polygon.