Book IX · Proposition 21

IX.21

If as many even numbers as we please be added together, the whole is even.Heath, 1908

Book IX ends with a run of parity theorems, which read as trivial now and were not: Euclid has no notation for 'even' but 'divisible into two equal parts'.

Every step, checked

• all the numbers are evengiven
• the sum of them is evenby Def.VII.6

What it needs, and what needs it

Needs: nothing earlier.

Used by: IX.22 IX.24 IX.28

Rests on: Def.VII.6

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(
    "IX.21",
    THEOREM,
    sample=_even_numbers,
    note="Book IX ends with a run of parity theorems, which read as trivial now "
    "and were not: Euclid has no notation for 'even' but 'divisible into two "
    "equal parts'.",
)
def prop_IX_21(given: list) -> Out:
    hypothesis("all the numbers are even", all(measures(2, n) for n in given) and given)
    claim("the sum of them is even", "Def.VII.6", measures(2, sum(given)))
    return Out(total=sum(given))