Book VIII · Proposition 14

VIII.14

If a square measure a square, the side will also measure the side; and, if the side measure the side, the square will also measure the square.Heath, 1908

Every step, checked

• the sides are genuine numbersgiven
• if the square measures the square, the side measures the sideby VIII.14

What it needs, and what needs it

Needs: nothing earlier.

Used by: VIII.16

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.14",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 15), rng.randint(2, 15)),
)
def prop_VIII_14(a: int, b: int) -> Out:
    hypothesis("the sides are genuine numbers", a > 1 and b > 1, guard=True)
    claim("if the square measures the square, the side measures the side", "VIII.14",
          measures(a * a, b * b) == measures(a, b))
    return Out()