Book VIII · Proposition 8

VIII.8

If between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between them in continued proportion, so many will also fall in continued proportion between the numbers which have the same ratio with the original numbers.Heath, 1908

Every step, checked

• the ratio is a genuine onegiven
• a genuine progression is asked forgiven
• as many fall between the scaled pair as between the originalby VIII.8
• and the outer ratio is unchangedby VII.18

What it needs, and what needs it

Needs: VII.18

Used by: VIII.11

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(
    "VIII.8",
    THEOREM,
    sample=progression,
)
def prop_VIII_8(p: int, q: int, count: int) -> Out:
    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)
    terms = continued_proportion(1, (p, q), count)
    scaled = [term * 3 for term in terms]
    because(prop_VII_18, p, q, count)

    claim("as many fall between the scaled pair as between the original", "VIII.8",
          len(scaled) == len(terms) and in_continued_proportion(scaled))
    claim("and the outer ratio is unchanged", "VII.18",
          terms[0] * scaled[-1] == terms[-1] * scaled[0])
    return Out(between=scaled[1:-1])