Book VIII · Proposition 6

VIII.6

If there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will any other measure any other.Heath, 1908

Every step, checked

• the ratio is a genuine onegiven
• a genuine progression is asked forgiven
• the first does not measure the secondgiven
• then no one of them measures any otherby VIII.6

What it needs, and what needs it

Needs: nothing earlier.

Used by: VIII.7 IX.17

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.6",
    THEOREM,
    sample=progression,
)
def prop_VIII_6(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)
    hypothesis("the first does not measure the second", not measures(terms[0], terms[1]))
    claim("then no one of them measures any other", "VIII.6",
          not any(measures(terms[i], terms[j])
                  for i in range(count) for j in range(count) if i != j))
    return Out()