Book VIII · Proposition 23

VIII.23

If four numbers be in continued proportion, and the first be cube, the fourth will also be cube.Heath, 1908

Every step, checked

• the sides are genuine numbersgiven
• the four are in continued proportiongiven
• the first is cubegiven
• the fourth is cube alsoby VIII.23

What it needs, and what needs it

Needs: nothing earlier.

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.23",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 8), rng.randint(2, 8)),
)
def prop_VIII_23(a: int, b: int) -> Out:
    hypothesis("the sides are genuine numbers", a > 1 and b > 1, guard=True)
    terms = [a ** 3, a * a * b, a * b * b, b ** 3]
    hypothesis("the four are in continued proportion", in_continued_proportion(terms))
    hypothesis("the first is cube", is_cube(terms[0]))
    claim("the fourth is cube also", "VIII.23", is_cube(terms[3]))
    return Out()