Book VIII · Proposition 19

VIII.19

Between two similar solid numbers there fall two mean proportional numbers; and the solid number has to the similar solid number the ratio triplicate of that which the corresponding side has to the corresponding side.Heath, 1908

Every step, checked

• the sides are genuine numbersgiven
• two mean proportionals fall between themby VIII.19
• and solid is to solid in the triplicate ratio of the sidesby VIII.12

What it needs, and what needs it

Needs: VIII.12

Used by: VIII.27

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.19",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 6), rng.randint(2, 6), rng.randint(2, 6),
                        rng.randint(2, 4)),
)
def prop_VIII_19(a: int, b: int, c: int, scale: int) -> Out:
    """Similar solid numbers: three sides each, in proportion."""
    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)
    means = [first * scale, first * scale * scale]
    because(prop_VIII_12, a, b)

    claim("two mean proportionals fall between them", "VIII.19",
          in_continued_proportion([first] + means + [second]))
    claim("and solid is to solid in the triplicate ratio of the sides", "VIII.12",
          second == first * scale ** 3)
    return Out(means=means)