Book VIII · Proposition 22

VIII.22

If three numbers be in continued proportion, and the first be square, the third will also be square.Heath, 1908

Every step, checked

• the sides are genuine numbersgiven
• the three are in continued proportiongiven
• the first is squaregiven
• the third is square alsoby VIII.22

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.22",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 12), rng.randint(2, 12)),
)
def prop_VIII_22(a: int, b: int) -> Out:
    hypothesis("the sides are genuine numbers", a > 1 and b > 1, guard=True)
    terms = [a * a, a * b, b * b]
    hypothesis("the three are in continued proportion", in_continued_proportion(terms))
    hypothesis("the first is square", is_square(terms[0]))
    claim("the third is square also", "VIII.22", is_square(terms[2]))
    return Out()