Book X · Proposition 34

X.34

To find two straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational.Heath, 1908

The pair behind *the side of a rational plus a medial area*.

Every step, checked

• the radicand is not a squaregiven
• the two are incommensurable in squareby X.34
• the sum of the squares on them is medialby X.34
• and the rectangle they contain is rationalby X.34

What it needs, and what needs it

Needs: nothing earlier.

Depth: 0 steps of argument above the first principles. Parallel postulate: not needed.

What it takes on trust

Nothing. It draws no intersections and reads nothing off the picture.

The proposition as code

@proposition(
    "X.34",
    CONSTRUCTION,
    sample=lambda rng: (rng.choice([2, 3, 5, 7]),),
    note="The pair behind *the side of a rational plus a medial area*.",
)
def prop_X_34(radicand: int) -> Out:
    hypothesis("the radicand is not a square", not _is_square_int(radicand))
    squares = sqrt(radicand)
    rectangle = Fraction(1, 2)
    a, b = _pair_from(squares, rectangle)

    claim("the two are incommensurable in square", "X.34",
          not commensurable_in_square(a, b))
    claim("the sum of the squares on them is medial", "X.34",
          is_medial_area(a * a + b * b))
    claim("and the rectangle they contain is rational", "X.34",
          is_rational_area(a * b))
    return Out(lines=(a, b))