Book V · Proposition 3

V.3

If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also ex aequali the magnitudes taken will be equimultiples respectively, the one of the second and the other of the fourth.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• the equimultiples taken are multiples of the original multiplesby Def.2
• so ex aequali they are equimultiples of the second and the fourthby V.1

What it needs, and what needs it

Needs: V.1

Rests on: C.N.2, Def.2

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(
    "V.3",
    THEOREM,
    sample=_two_magnitudes_and_two_multiples,
)
def prop_V_3(b, d, times: int, again: int) -> Out:
    """A is `times` B as C is of D; then `again`-fold A and C are `times*again`
    fold B and D."""
    hypothesis("the magnitudes are positive", sign(b) > 0 and sign(d) > 0, guard=True)
    a, c = times * b, times * d
    because(prop_V_1, b, d, times)

    claim("the equimultiples taken are multiples of the original multiples", "Def.2",
          again * a == again * (times * b) and again * c == again * (times * d))
    claim("so ex aequali they are equimultiples of the second and the fourth", "V.1",
          again * a == (again * times) * b and again * c == (again * times) * d)
    return Out(taken=(again * a, again * c))