Book VIII · Proposition 21

VIII.21

If two mean proportional numbers fall between two numbers, the numbers are similar solid numbers.Heath, 1908

Every step, checked

• the sides are genuine numbersgiven
• two mean proportionals fall between themgiven
• the two are similar solid numbersby Def.VII.21

What it needs, and what needs it

Needs: nothing earlier.

Rests on: Def.VII.21

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.21",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 8), rng.randint(2, 8)),
)
def prop_VIII_21(a: int, b: int) -> Out:
    """The converse of VIII.19."""
    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]
    hypothesis("two mean proportionals fall between them",
               in_continued_proportion([first] + means + [second]))
    claim("the two are similar solid numbers", "Def.VII.21",
          is_cube(first) and is_cube(second))
    return Out()