Book V · Proposition 4

V.4

If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.Heath, 1908

Ratio survives being scaled on both sides -- what makes a ratio a thing about the pair rather than about the magnitudes.

Every step, checked

• a : b = c : dgiven
• the magnitudes are positivegiven
• the equimultiples stand in the same ratioby Def.5
• and no equimultiples separate themby Def.5

What it needs, and what needs it

Needs: nothing earlier.

Rests on: Def.5

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(
    "V.4",
    THEOREM,
    sample=lambda rng: _proportional_magnitudes(rng) + (rng.randint(2, 6), rng.randint(2, 6)),
    note="Ratio survives being scaled on both sides -- what makes a ratio a thing "
    "about the pair rather than about the magnitudes.",
)
def prop_V_4(a, b, c, d, of_first: int, of_second: int) -> Out:
    hypothesis("a : b = c : d", a * d == b * c)
    hypothesis("the magnitudes are positive",
               all(sign(x) > 0 for x in (a, b, c, d)), guard=True)
    first, third = of_first * a, of_first * c
    second, fourth = of_second * b, of_second * d
    claim("the equimultiples stand in the same ratio", "Def.5",
          first * fourth == second * third)
    claim("and no equimultiples separate them", "Def.5",
          separating_witness(first, second, third, fourth) is None)
    return Out()