# Proposition: Prop. 10.040: Side of Rational plus Medial Area is Irrational

### (Proposition 40 from Book 10 of Euclid's “Elements”)

If two straight lines (which are) incommensurable in square, making the sum of the squares on them medial, and the (rectangle contained) by them rational, are added together then the whole straight line is irrational - let it be called the square root of a rational plus a medial (area) . * For let the two straight lines, $AB$ and $BC$, incommensurable in square, (and) fulfilling the prescribed (conditions), be laid down together [Prop. 10.34]. * I say that $AC$ is irrational. ### Modern Formulation

Thus, the square root of a rational plus a medial (area) has a length expressible as $\sqrt{\frac{\sqrt{1+\rho^2}+\rho}{2\,(1+\rho^2)}} +\sqrt{\frac{\sqrt{1+\rho^2}-\rho}{2\,(1+\rho^2)}},$

for some positive rational number $$\rho$$.

### Notes

This and the corresponding irrational with a minus sign1 whose length is expressible as

$\sqrt{\frac{\sqrt{1+\rho^2}+\rho}{2\,(1+\rho^2)}} -\sqrt{\frac{\sqrt{1+\rho^2}-\rho}{2\,(1+\rho^2)}},$

(see Prop. 10.77), are the positive roots of the quartic $x^4-\frac 2{\sqrt{1+\rho^2}}x^2+ \frac{\rho^2}{(1+\rho^2)^2} = 0.$

Proofs: 1

Proofs: 1 2 3 4
Propositions: 5 6 7

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