Book VIII · Proposition 5

VIII.5

Plane numbers have to one another the ratio compounded of the ratios of their sides.Heath, 1908

A 'plane number' is a product of two sides. Their ratio is the ratio of the sides compounded -- Book VI's VI.23 in arithmetic.

Every step, checked

• the sides are genuine numbersgiven
• the plane numbers have the ratio compounded of the ratios of the sidesby VII.17
• which is to say, the product of the two ratiosby VII.18

What it needs, and what needs it

Needs: VII.17 VII.18

Depth: 1 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.5",
    THEOREM,
    sample=lambda rng: tuple(rng.randint(2, 12) for _ in range(4)),
    note="A 'plane number' is a product of two sides. Their ratio is the ratio "
    "of the sides compounded -- Book VI's VI.23 in arithmetic.",
)
def prop_VIII_5(a: int, b: int, c: int, d: int) -> Out:
    """The plane numbers are a*b and c*d."""
    hypothesis("the sides are genuine numbers", all(n > 1 for n in (a, b, c, d)), guard=True)
    first, second = a * b, c * d
    because(prop_VII_17, a, b, c)
    because(prop_VII_18, a, b, c)

    claim("the plane numbers have the ratio compounded of the ratios of the sides",
          "VII.17", first * (c * d) == second * (a * b))
    claim("which is to say, the product of the two ratios", "VII.18",
          first * d * c == second * a * b)
    return Out(planes=(first, second))