Book VIII · Proposition 25

VIII.25

If two numbers have to one another the ratio which a cube number has to a cube number, and the first be cube, the second will also be cube.Heath, 1908

Every step, checked

• the numbers are genuinegiven
• the two have the ratio of a cube to a cubegiven
• the first is cubegiven
• the second is cube alsoby VIII.25

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.25",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 6), rng.randint(2, 6), rng.randint(2, 4)),
)
def prop_VIII_25(a: int, b: int, scale: int) -> Out:
    hypothesis("the numbers are genuine", a > 1 and b > 1 and scale > 1, guard=True)
    first, second = (a * scale) ** 3, (b * scale) ** 3
    hypothesis("the two have the ratio of a cube to a cube",
               first * b ** 3 == second * a ** 3)
    hypothesis("the first is cube", is_cube(first))
    claim("the second is cube also", "VIII.25", is_cube(second))
    return Out()