Book V · Proposition 10

V.10

Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• A and B are unequalgiven
• that which has the greater ratio to the same is the greaterby Def.7
• and that to which the same has the greater ratio is the lessby Def.7

What it needs, and what needs it

Needs: nothing earlier.

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.10",
    THEOREM,
    sample=_three_magnitudes,
)
def prop_V_10(a, b, c) -> Out:
    """A and B both have a ratio to C."""
    hypothesis("the magnitudes are positive",
               all(sign(x) > 0 for x in (a, b, c)), guard=True)
    hypothesis("A and B are unequal", a != b)
    claim("that which has the greater ratio to the same is the greater", "Def.7",
          (ratio_cmp(a, c, b, c) > 0) == (sign(a - b) > 0))
    claim("and that to which the same has the greater ratio is the less", "Def.7",
          (ratio_cmp(c, a, c, b) > 0) == (sign(a - b) < 0))
    return Out()