Book V · Proposition 24

V.24

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

Every step, checked

• the magnitudes are positivegiven
• a : b = c : dgiven
• the fifth stands to the second as the sixth to the fourthby Def.5
• so the first and fifth together stand to the second as the third and sixth together to the fourthby 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.24",
    THEOREM,
    sample=lambda rng: _proportional_magnitudes(rng) + (_magnitude(rng),),
)
def prop_V_24(a, b, c, d, e) -> Out:
    """A : B = C : D, and E : B = F : D; the antecedents add."""
    hypothesis("the magnitudes are positive",
               all(sign(x) > 0 for x in (a, b, c, d, e)), guard=True)
    hypothesis("a : b = c : d", a * d == b * c)
    f = e * d / b  # so that e : b = f : d
    claim("the fifth stands to the second as the sixth to the fourth", "Def.5",
          e * d == b * f)
    claim("so the first and fifth together stand to the second as the third and "
          "sixth together to the fourth", "Def.5",
          separating_witness(a + e, b, c + f, d) is None)
    return Out(sums=(a + e, c + f))