Book VIII · Proposition 13

VIII.13

If there be as many numbers as we please in continued proportion, and each by multiplying itself make some number, the products will be proportional; and, if the original numbers by multiplying the products make certain numbers, the latter will also be proportional.Heath, 1908

Every step, checked

• the ratio is a genuine onegiven
• a genuine progression is asked forgiven
• the squares of the terms are proportionalby VIII.13
• and so are the cubesby VIII.13

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.13",
    THEOREM,
    sample=progression,
)
def prop_VIII_13(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)
    squares = [term * term for term in terms]
    cubes = [term ** 3 for term in terms]
    claim("the squares of the terms are proportional", "VIII.13",
          in_continued_proportion(squares))
    claim("and so are the cubes", "VIII.13", in_continued_proportion(cubes))
    return Out(squares=squares, cubes=cubes)