Book V · Proposition 15

V.15

Parts have the same ratio as the same multiples of them taken in corresponding order.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• parts have the same ratio as their equimultiplesby 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.15",
    THEOREM,
    sample=_two_magnitudes_and_a_multiple,
)
def prop_V_15(a, b, times: int) -> Out:
    hypothesis("the magnitudes are positive", sign(a) > 0 and sign(b) > 0, guard=True)
    claim("parts have the same ratio as their equimultiples", "Def.5",
          separating_witness(a, b, times * a, times * b) is None)
    return Out()