Book VII · Proposition 38

VII.38

If a number have any part whatever, it will be measured by a number called by the same name as the part.Heath, 1908

Every step, checked

• the second has a part named after the firstgiven
• then it is measured by the number of that nameby Def.VII.3
• and the quotient is the part itselfby VII.37

What it needs, and what needs it

Needs: VII.37

Used by: VII.39

Rests on: Def.VII.3

Depth: 1 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.38",
    THEOREM,
    sample=_a_part_of,
)
def prop_VII_38(part: int, whole: int) -> Out:
    """The converse of VII.37."""
    hypothesis("the second has a part named after the first",
               measures(part, whole) and part > 1)
    because(prop_VII_37, part, whole) if measures(part, whole) and part > 1 else None

    claim("then it is measured by the number of that name", "Def.VII.3",
          measures(part, whole))
    claim("and the quotient is the part itself", "VII.37",
          whole // (whole // part) == part)
    return Out()