Book V · Proposition 14

V.14

If a first magnitude have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• a : b = c : dgiven
• as the first stands to the third, so the second stands to the fourthby V.8

What it needs, and what needs it

Needs: V.8

Rests on: Def.5

Depth: 1 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.14",
    THEOREM,
    sample=_proportional_magnitudes,
)
def prop_V_14(a, b, c, d) -> Out:
    hypothesis("the magnitudes are positive",
               all(sign(x) > 0 for x in (a, b, c, d)), guard=True)
    hypothesis("a : b = c : d", a * d == b * c)
    # V.8 wants two ratios that differ, and these are equal by hypothesis. The
    # pair Euclid actually sets against each other is a : b and c : b, which
    # differ exactly when A and C do.
    if a != c:
        because(prop_V_8, a, b, c, b)

    claim("as the first stands to the third, so the second stands to the fourth",
          "V.8", sign(a - c) == sign(b - d))
    return Out()