Book V · Proposition 7

V.7

Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• A and B are equalgiven
• equal magnitudes have the same ratio to the sameby Def.5
• and the same has the same ratio to equal magnitudesby 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.7",
    THEOREM,
    sample=_equal_pair_and_a_third,
)
def prop_V_7(a, b, c) -> Out:
    """A and B are equal; C is the magnitude they are compared with."""
    hypothesis("the magnitudes are positive",
               all(sign(x) > 0 for x in (a, b, c)), guard=True)
    hypothesis("A and B are equal", a == b)
    claim("equal magnitudes have the same ratio to the same", "Def.5",
          separating_witness(a, c, b, c) is None)
    claim("and the same has the same ratio to equal magnitudes", "Def.5",
          separating_witness(c, a, c, b) is None)
    return Out()