Book V · Proposition 11

V.11

Ratios which are the same with the same ratio are also the same with one another.Heath, 1908

Transitivity of proportion -- which Eudoxus has to prove, because his ratios are not numbers.

Every step, checked

• a : b = c : dgiven
• c : d = e : fgiven
• therefore a : b = e : fby Def.5
• and no equimultiples separate themby Def.5

What it needs, and what needs it

Needs: nothing earlier.

Used by: V.12

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.11",
    THEOREM,
    sample=_six_proportional,
    note="Transitivity of proportion -- which Eudoxus has to prove, because his "
    "ratios are not numbers.",
)
def prop_V_11(a, b, c, d, e, f) -> Out:
    hypothesis("a : b = c : d", a * d == b * c)
    hypothesis("c : d = e : f", c * f == d * e)
    claim("therefore a : b = e : f", "Def.5", a * f == b * e)
    claim("and no equimultiples separate them", "Def.5",
          separating_witness(a, b, e, f) is None)
    return Out()