Book VIII · Proposition 17

VIII.17

If a cube number do not measure a cube number, neither will the side measure the side; and, if the side do not measure the side, neither will the cube measure the cube.Heath, 1908

Every step, checked

• the sides are genuine numbersgiven
• if the cube does not measure the cube, neither does the sideby VIII.15

What it needs, and what needs it

Needs: VIII.15

Depth: 1 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(
    "VIII.17",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 10), rng.randint(2, 10)),
)
def prop_VIII_17(a: int, b: int) -> Out:
    hypothesis("the sides are genuine numbers", a > 1 and b > 1, guard=True)
    because(prop_VIII_15, a, b)

    claim("if the cube does not measure the cube, neither does the side", "VIII.15",
          (not measures(a ** 3, b ** 3)) == (not measures(a, b)))
    return Out()