Book VII · Proposition 13

VII.13

If four numbers be proportional, they will also be proportional alternately.Heath, 1908

Every step, checked

• the four are proportionalgiven
• they are proportional alternately as wellby VII.13

What it needs, and what needs it

Needs: nothing earlier.

Used by: VII.14

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.13",
    THEOREM,
    sample=_proportional_numbers,
)
def prop_VII_13(a: int, b: int, c: int, d: int) -> Out:
    hypothesis("the four are proportional", a * d == b * c)
    claim("they are proportional alternately as well", "VII.13", a * d == c * b)
    return Out()