Book VII · Proposition 12

VII.12

If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents.Heath, 1908

Every step, checked

• the four are proportionalgiven
• as one antecedent is to one consequent, so are all to allby VII.12

What it needs, and what needs it

Needs: nothing earlier.

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(
    "VII.12",
    THEOREM,
    sample=_proportional_numbers,
)
def prop_VII_12(a: int, b: int, c: int, d: int) -> Out:
    hypothesis("the four are proportional", a * d == b * c)
    claim("as one antecedent is to one consequent, so are all to all", "VII.12",
          a * (b + d) == b * (a + c))
    return Out(total=(a + c, b + d))