Book VII · Proposition 33

VII.33

Given as many numbers as we please, to find the least of those which have the same ratio with them.Heath, 1908

Every step, checked

• both are numbersgiven
• the pair found has the same ratioby VII.33
• it is in least terms, being prime to one anotherby VII.22
• and no smaller pair has that ratioby VII.21

What it needs, and what needs it

Needs: VII.21 VII.22

Used by: X.5

Depth: 2 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(
    "VII.33",
    CONSTRUCTION,
    sample=lambda rng: (rng.randint(2, 200), rng.randint(2, 200)),
)
def prop_VII_33(a: int, b: int) -> Out:
    hypothesis("both are numbers", a > 1 and b > 1, guard=True)
    least = least_terms(a, b)
    because(prop_VII_21, a, b) if coprime(a, b) else None
    because(prop_VII_22, a, b)

    claim("the pair found has the same ratio", "VII.33", least[0] * b == least[1] * a)
    claim("it is in least terms, being prime to one another", "VII.22",
          coprime(*least))
    claim("and no smaller pair has that ratio", "VII.21",
          not any(x * b == y * a
                  for x in range(1, least[0]) for y in range(1, least[1])))
    return Out(least=least)