Book V · Proposition 8

V.8

Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.Heath, 1908

The constructive core of Book V: when two ratios differ, a pair of equimultiples can be produced that proves it.

Every step, checked

• the ratios a : b and c : d are unequalgiven
• equimultiples can be found which separate the two ratiosby Def.5
• with m = 1 and n = 1 the multiples fall on opposite sidesby Def.5

What it needs, and what needs it

Needs: nothing earlier.

Used by: V.14 V.20 V.21

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.8",
    THEOREM,
    sample=_unequal_ratios,
    note="The constructive core of Book V: when two ratios differ, a pair of "
    "equimultiples can be produced that proves it.",
)
def prop_V_8(a, b, c, d) -> Out:
    hypothesis("the ratios a : b and c : d are unequal", a * d != b * c)
    witness = separating_witness(a, b, c, d)
    claim("equimultiples can be found which separate the two ratios", "Def.5",
          witness is not None)
    m, n = witness
    claim(f"with m = {m} and n = {n} the multiples fall on opposite sides", "Def.5",
          sign(m * a - n * b) != sign(m * c - n * d))
    return Out(witness=witness)