Book VII · Proposition 11

VII.11

If, as whole is to whole, so is a number subtracted to a number subtracted, the remainder will also be to the remainder as whole to whole.Heath, 1908

Every step, checked

• the four are proportionalgiven
• the parts are less than the wholesgiven
• the remainder is to the remainder as whole to wholeby VII.11

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.11",
    THEOREM,
    sample=_proportional_with_smaller_parts,
)
def prop_VII_11(a: int, b: int, c: int, d: int) -> Out:
    hypothesis("the four are proportional", a * d == b * c)
    hypothesis("the parts are less than the wholes", c < a and d < b)
    claim("the remainder is to the remainder as whole to whole", "VII.11",
          (a - c) * b == (b - d) * a)
    return Out(remainders=(a - c, b - d))