Book V · Proposition 13

V.13

If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourth a greater ratio than a fifth has to a sixth, the first will also have to the second a greater ratio than the fifth to the sixth.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• a : b = c : dgiven
• c : d is greater than e : fgiven
• therefore a : b is greater than e : fby Def.7
• and equimultiples can be produced that show itby Def.7

What it needs, and what needs it

Needs: nothing earlier.

Used by: V.20 V.21

Rests on: Def.7

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.13",
    THEOREM,
    sample=_proportion_and_a_smaller_ratio,
)
def prop_V_13(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 = c : d", a * d == b * c)
    hypothesis("c : d is greater than e : f", ratio_cmp(c, d, e, f) > 0)
    claim("therefore a : b is greater than e : f", "Def.7",
          ratio_cmp(a, b, e, f) > 0)
    claim("and equimultiples can be produced that show it", "Def.7",
          separating_witness(a, b, e, f) is not None)
    return Out()