Book VIII · Proposition 10

VIII.10

If numbers fall between each of two numbers and an unit in continued proportion, however many numbers fall between each of them and an unit in continued proportion, so many also will fall between the numbers themselves in continued proportion.Heath, 1908

Every step, checked

• the ratio is a genuine onegiven
• a genuine progression is asked forgiven
• progressions run from a unit up to each numbergiven
• then as many fall between the two numbers themselvesby VIII.10

What it needs, and what needs it

Needs: nothing earlier.

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(
    "VIII.10",
    THEOREM,
    sample=progression,
)
def prop_VIII_10(p: int, q: int, count: int) -> Out:
    """The converse of VIII.9."""
    hypothesis("the ratio is a genuine one", p > 1 and q > 1, guard=True)
    hypothesis("a genuine progression is asked for", count >= 3, guard=True)
    from_unit_first = [q ** k for k in range(count)]
    from_unit_last = [p ** k for k in range(count)]
    hypothesis("progressions run from a unit up to each number",
               in_continued_proportion(from_unit_first)
               and in_continued_proportion(from_unit_last))
    between = continued_proportion(1, (p, q), count)
    claim("then as many fall between the two numbers themselves", "VIII.10",
          in_continued_proportion(between) and len(between) == count)
    return Out(between=between)