Book V · Proposition 12

V.12

If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.Heath, 1908

As one antecedent is to one consequent, so is the sum to the sum -- the step that lets Book V add proportions together.

Every step, checked

• the magnitudes are positivegiven
• a : b = c : dgiven
• and c : d = e : fgiven
• so all three pairs stand in one ratioby V.11
• so one antecedent is to one consequent as all are to allby Def.5

What it needs, and what needs it

Needs: V.11

Used by: VI.20

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.12",
    THEOREM,
    sample=_three_pairs_in_one_ratio,
    note="As one antecedent is to one consequent, so is the sum to the sum -- the "
    "step that lets Book V add proportions together.",
)
def prop_V_12(a, b, c, d, e, f) -> Out:
    hypothesis("the magnitudes are positive",
               all(sign(x) > 0 for x in (a, b, c, d, e, f)), guard=True)
    hypothesis("a : b = c : d", a * d == b * c)
    hypothesis("and c : d = e : f", c * f == d * e)
    because(prop_V_11, a, b, c, d, e, f)

    claim("so all three pairs stand in one ratio", "V.11",
          a * d == b * c and a * f == b * e)
    claim("so one antecedent is to one consequent as all are to all", "Def.5",
          separating_witness(a, b, a + c + e, b + d + f) is None)
    return Out(total=(a + c + e, b + d + f))