Book VII · Proposition 37

VII.37

If a number be measured by any number, the number which is measured will have a part called by the same name as the measuring number.Heath, 1908

Every step, checked

• the first measures the secondgiven
• the measured number has a part named after the measureby Def.VII.3

What it needs, and what needs it

Needs: nothing earlier.

Used by: VII.38

Rests on: Def.VII.3

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.37",
    THEOREM,
    sample=_a_part_of,
)
def prop_VII_37(part: int, whole: int) -> Out:
    hypothesis("the first measures the second", measures(part, whole) and part > 1)
    named = whole // part
    claim("the measured number has a part named after the measure", "Def.VII.3",
          measures(named, whole) and whole // named == part)
    return Out(part=named)