Definition:Major (Euclidean)

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Let $a, b \in \R_{>0}$ in the forms:

$a = \dfrac \rho {\sqrt 2} \sqrt {1 + \dfrac k {\sqrt {1 + k^2} } }$
$b = \dfrac \rho {\sqrt 2} \sqrt {1 - \dfrac k {\sqrt {1 + k^2} } }$


$\rho$ is a rational number
$k$ is a rational number whose square root is irrational.

Then $a + b$ is a major.

In the words of Euclid:

If two straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial, be added together, the whole straight line is irrational; and let it be called major.

(The Elements: Book $\text{X}$: Proposition $39$)

Also see