Book VIII · Proposition 27

VIII.27

Similar solid numbers have to one another the ratio which a cube number has to a cube number.Heath, 1908

Every step, checked

• the sides are genuine numbersgiven
• similar solid numbers have the ratio of a cube to a cubeby VIII.19

What it needs, and what needs it

Needs: VIII.19

Depth: 2 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.27",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 6), rng.randint(2, 6), rng.randint(2, 6),
                        rng.randint(2, 4)),
)
def prop_VIII_27(a: int, b: int, c: int, scale: int) -> Out:
    hypothesis("the sides are genuine numbers", all(n > 1 for n in (a, b, c, scale)), guard=True)
    first = a * b * c
    second = (a * scale) * (b * scale) * (c * scale)
    because(prop_VIII_19, a, b, c, scale)

    claim("similar solid numbers have the ratio of a cube to a cube", "VIII.19",
          first * scale ** 3 == second and is_cube(scale ** 3))
    return Out(ratio=(1, scale ** 3))