Book X · Proposition 19

X.19

The rectangle contained by rational straight lines commensurable in length is rational.Heath, 1908

Every step, checked

• both lines are rationalgiven
• they are commensurable in lengthgiven
• the rectangle they contain is rationalby X.19

What it needs, and what needs it

Needs: nothing earlier.

Used by: X.27

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.19",
    THEOREM,
    sample=_rational_pair_commensurable,
)
def prop_X_19(a, b) -> Out:
    hypothesis("both lines are rational", is_rational_in_square(a) and is_rational_in_square(b))
    hypothesis("they are commensurable in length", commensurable(a, b))
    claim("the rectangle they contain is rational", "X.19", is_rational_area(a * b))
    return Out(rectangle=a * b)