Book VIII · Proposition 24

VIII.24

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

Every step, checked

• the numbers are genuinegiven
• the two have the ratio of a square to a squaregiven
• the first is squaregiven
• the second is square alsoby VIII.24

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.24",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 10), rng.randint(2, 10), rng.randint(2, 6)),
)
def prop_VIII_24(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 * a * scale * scale, b * b * scale * scale
    hypothesis("the two have the ratio of a square to a square",
               first * (b * b) == second * (a * a))
    hypothesis("the first is square", is_square(first))
    claim("the second is square also", "VIII.24", is_square(second))
    return Out()