If two straight lines (which are) incommensurable in square, making the sum of the squares on them rational, and the (rectangle contained) by them medial, are added together then the whole straight line is irrational - let it be called a major (straight line). * For let the two straight lines, $AB$ and $BC$, incommensurable in square, and fulfilling the prescribed (conditions), be laid down together [Prop. 10.33]. * I say that $AC$ is irrational.
Thus, a major straight line has a length expressible as \[\sqrt{\left(1+\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12} + \sqrt{\left(1-\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12}\]
for some positive rational number \(\rho\).
The major and the corresponding minor, (see Prop. 10.76), whose length is expressible as
\[\sqrt{\left(1+\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12} - \sqrt{\left(1-\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12}\]
are the positive roots of the quartic \[x^4-2\,x^2+ \rho^2/(1+\rho^2) = 0.\]
Proofs: 1
Corollaries: 1
Proofs: 2 3 4 5 6 7 8
Propositions: 9 10 11 12 13 14