Book X · Proposition 32

X.32

To find two medial straight lines commensurable in square only, containing a medial rectangle, and such that the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater.Heath, 1908

Every step, checked

• the radicand is not a squaregiven
• both are medialby X.21
• and the excess is commensurable with the greaterby X.29

What it needs, and what needs it

Needs: X.21 X.29

Rests on: X.Def.3, X.Def.4

Depth: 5 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.32",
    CONSTRUCTION,
    sample=lambda rng: (rng.choice([2, 3, 5, 7]),),
)
def prop_X_32(radicand: int) -> Out:
    """Medials in square only, medial rectangle, excess commensurable."""
    hypothesis("the radicand is not a square", not _is_square_int(radicand))
    quarter = sqrt(sqrt(radicand))
    a, b = 5 * quarter, 3 * quarter
    excess = sqrt(a * a - b * b)

    because(prop_X_29, 2, 1)

    because(prop_X_21, Fraction(1), sqrt(radicand))

    claim("both are medial", "X.21", is_medial(a) and is_medial(b))
    claim("and the excess is commensurable with the greater", "X.29",
          commensurable(excess, a))
    return Out(lines=(a, b), excess=excess)