Book X · Proposition 12

X.12

Magnitudes commensurable with the same magnitude are commensurable with one another also.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• both are commensurable with the samegiven
• each is commensurable with the thirdby X.12
• so magnitudes commensurable with the same are commensurable with one anotherby X.12

What it needs, and what needs it

Needs: nothing earlier.

Used by: X.66 X.67 X.68 X.69 X.70 X.103 X.104 X.105 X.106 X.107

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.12",
    THEOREM,
    sample=lambda rng: _commensurable_pair(rng)
    + (Fraction(rng.randint(1, 5), rng.randint(1, 4)),),
)
def prop_X_12(a, b, scale) -> Out:
    hypothesis("the magnitudes are positive", sign(a) > 0 and sign(b) > 0, guard=True)
    hypothesis("both are commensurable with the same", commensurable(a, b))
    c = b * scale
    claim("each is commensurable with the third", "X.12",
          commensurable(b, c) and commensurable(a, c))
    claim("so magnitudes commensurable with the same are commensurable with one "
          "another", "X.12", commensurable(a, c))
    return Out()