Book V · Proposition 23

V.23

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, they will also be in the same ratio ex aequali.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• a : b = e : fgiven
• b : c = d : egiven
• ex aequali in perturbed proportion, a : c = d : fby 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.23",
    THEOREM,
    sample=_two_triples_perturbed,
)
def prop_V_23(a, b, c, d, e, f) -> Out:
    hypothesis("the magnitudes are positive",
               all(sign(x) > 0 for x in (a, b, c, d, e, f)), guard=True)
    hypothesis("a : b = e : f", a * f == b * e)
    hypothesis("b : c = d : e", b * e == c * d)
    claim("ex aequali in perturbed proportion, a : c = d : f", "Def.5",
          separating_witness(a, c, d, f) is None)
    return Out()