Book IX · Proposition 9

IX.9

If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be square, all the rest will also be square. And, if the number after the unit be cube, all the rest will also be cube.Heath, 1908

Every step, checked

• the progression is genuinegiven
• if the number after the unit is square, all the rest are squareby IX.8
• and if it is cube, all the rest are cubeby 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.9",
    THEOREM,
    sample=_from_a_unit,
)
def prop_IX_9(ratio: int, count: int) -> Out:
    hypothesis("the progression is genuine", ratio > 1 and count >= 4, guard=True)
    squares = [(ratio * ratio) ** k for k in range(count)]
    because(prop_IX_8, ratio, max(count, 7))

    claim("if the number after the unit is square, all the rest are square", "IX.8",
          is_square(squares[1]) and all(is_square(term) for term in squares[1:]))
    cubes = [(ratio ** 3) ** k for k in range(count)]
    claim("and if it is cube, all the rest are cube", "IX.8",
          is_cube(cubes[1]) and all(is_cube(term) for term in cubes[1:]))
    return Out()