Book IX · Proposition 16

IX.16

If two numbers be prime to one another, the second will not be to any other number as the first is to the second.Heath, 1908

Every step, checked

• the numbers are prime to one anothergiven
• the first does not measure the secondgiven
• the second is to no other number as the first is to the secondby VII.20

What it needs, and what needs it

Needs: VII.20

Used by: IX.17

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(
    "IX.16",
    THEOREM,
    sample=coprime_pair,
)
def prop_IX_16(a: int, b: int) -> Out:
    hypothesis("the numbers are prime to one another", coprime(a, b))
    hypothesis("the first does not measure the second", not measures(a, b))
    because(prop_VII_20, a, b, a, b)

    claim("the second is to no other number as the first is to the second", "VII.20",
          not any(a * c == b * b for c in range(1, b * b + 1)))
    return Out()