Book IX · Proposition 10

IX.10

If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be not square, neither will any other be square except the third from the unit and all those which leave out one. And, if the number after the unit be not cube, neither will any other be cube except the fourth from the unit and all those which leave out two.Heath, 1908

Every step, checked

• the progression is genuinegiven
• the number after the unit is not squaregiven
• then none is square except the third from the unit and the alternate onesby IX.8

What it needs, and what needs it

Needs: IX.8

Rests on: Def.VII.20

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(
    "IX.10",
    THEOREM,
    sample=_from_a_unit,
)
def prop_IX_10(ratio: int, count: int) -> Out:
    """The converse of IX.9, stated negatively."""
    hypothesis("the progression is genuine", ratio > 1 and count >= 5, guard=True)
    hypothesis("the number after the unit is not square", not is_square(ratio))
    terms = [ratio ** k for k in range(count)]
    because(prop_IX_8, ratio, max(count, 7))

    claim("then none is square except the third from the unit and the alternate ones",
          "IX.8",
          all(is_square(terms[k]) == (k % 2 == 0) for k in range(1, count)))
    return Out()