Book VIII · Proposition 12

VIII.12

Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of that which the side has to the side.Heath, 1908

Every step, checked

• the sides are genuine numbersgiven
• two mean proportionals fall between the cubesby VIII.12
• and cube is to cube in the triplicate ratio of side to sideby VIII.12

What it needs, and what needs it

Needs: nothing earlier.

Used by: VIII.19

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(
    "VIII.12",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 10), rng.randint(2, 10)),
)
def prop_VIII_12(a: int, b: int) -> Out:
    hypothesis("the sides are genuine numbers", a > 1 and b > 1, guard=True)
    first, second = a ** 3, b ** 3
    means = [a * a * b, a * b * b]
    claim("two mean proportionals fall between the cubes", "VIII.12",
          in_continued_proportion([first] + means + [second]))
    claim("and cube is to cube in the triplicate ratio of side to side", "VIII.12",
          first * b ** 3 == second * a ** 3)
    return Out(means=means)