Book V · Proposition 22

V.22

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

Ex aequali: ratios compose end to end, which is what makes Book VI's chains of similar triangles work.

Every step, checked

• the magnitudes are positivegiven
• a : b = d : egiven
• b : c = e : fgiven
• ex aequali, 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.22",
    THEOREM,
    sample=_two_triples_in_ratio,
    note="Ex aequali: ratios compose end to end, which is what makes Book VI's "
    "chains of similar triangles work.",
)
def prop_V_22(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 = d : e", a * e == b * d)
    hypothesis("b : c = e : f", b * f == c * e)
    claim("ex aequali, a : c = d : f", "Def.5",
          separating_witness(a, c, d, f) is None)
    return Out()