Book VII · Proposition 18

VII.18

If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.Heath, 1908

Every step, checked

• all three are numbersgiven
• the products have the same ratio as the multipliersby VII.18

What it needs, and what needs it

Needs: nothing earlier.

Used by: VIII.4 VIII.5 VIII.8

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.18",
    THEOREM,
    sample=three_numbers,
)
def prop_VII_18(a: int, b: int, c: int) -> Out:
    hypothesis("all three are numbers", a > 1 and b > 1 and c > 1, guard=True)
    claim("the products have the same ratio as the multipliers", "VII.18",
          _taken(_taken(a, c), b) == _taken(_taken(b, c), a))
    return Out()