Book X · Proposition 8

X.8

If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes will be incommensurable.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• no numbers put them in proportiongiven
• they are incommensurableby X.8

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.8",
    THEOREM,
    sample=_incommensurable_pair,
)
def prop_X_8(a, b) -> Out:
    """The converse of X.7."""
    hypothesis("the magnitudes are positive", sign(a) > 0 and sign(b) > 0, guard=True)
    hypothesis("no numbers put them in proportion",
               not any(a * q == b * p for p in range(1, 25) for q in range(1, 25)))
    claim("they are incommensurable", "X.8", not commensurable(a, b))
    return Out()