Book X · Proposition 6

X.6

If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• the numbers are genuinegiven
• the two stand in the ratio of the given numbersgiven
• so they are commensurableby X.6

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.6",
    THEOREM,
    sample=_in_the_ratio_of_two_numbers,
)
def prop_X_6(a, b, numerator: int, denominator: int) -> Out:
    """The converse of X.5."""
    hypothesis("the magnitudes are positive", sign(a) > 0 and sign(b) > 0, guard=True)
    hypothesis("the numbers are genuine", numerator > 0 and denominator > 0, guard=True)
    # Both magnitudes are given. Built as multiples of one base, the ratio below
    # came out as commutativity and held whatever the numbers were, so the
    # hypothesis of the proposition was never tested against anything.
    hypothesis("the two stand in the ratio of the given numbers",
               a * denominator == b * numerator)
    claim("so they are commensurable", "X.6", commensurable(a, b))
    return Out(measure=a / numerator)