Book V · Proposition 18

V.18

If magnitudes be proportional separando, they will also be proportional componendo.Heath, 1908

Componendo, the converse of V.17.

Every step, checked

• the magnitudes are positivegiven
• separando: a : b = c : dgiven
• componendo, (a+b) : b = (c+d) : dby 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.18",
    THEOREM,
    sample=_proportional_magnitudes,
    note="Componendo, the converse of V.17.",
)
def prop_V_18(a, b, c, d) -> Out:
    hypothesis("the magnitudes are positive",
               all(sign(x) > 0 for x in (a, b, c, d)), guard=True)
    hypothesis("separando: a : b = c : d", a * d == b * c)
    claim("componendo, (a+b) : b = (c+d) : d", "Def.5",
          separating_witness(a + b, b, c + d, d) is None)
    return Out()