Book VIII · Proposition 4

VIII.4

Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the given ratios.Heath, 1908

Every step, checked

• the ratios are genuinegiven
• the first pair keeps its ratioby VII.18
• the second pair keeps its ratioby VII.18
• and the middle is the least that both consequents measureby VII.34

What it needs, and what needs it

Needs: VII.18 VII.34

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.4",
    CONSTRUCTION,
    sample=lambda rng: (rng.randint(2, 5), rng.randint(2, 5), rng.randint(2, 5),
                        rng.randint(2, 5)),
)
def prop_VIII_4(a: int, b: int, c: int, d: int) -> Out:
    """Given the ratios a:b and c:d, find the least continued proportion in them."""
    hypothesis("the ratios are genuine", all(n > 1 for n in (a, b, c, d)), guard=True)
    first, second = least_terms(a, b)
    third, fourth = least_terms(c, d)
    # The middle term must be measured by both consequents, so take the least
    # number they both measure and scale each ratio up to meet it.
    middle = lcm(second, third)
    terms = [first * (middle // second), middle, fourth * (middle // third)]

    because(prop_VII_18, a, b, c)
    because(prop_VII_34, a, b)

    claim("the first pair keeps its ratio", "VII.18", terms[0] * b == terms[1] * a)
    claim("the second pair keeps its ratio", "VII.18", terms[1] * d == terms[2] * c)
    claim("and the middle is the least that both consequents measure", "VII.34",
          measures(second, middle) and measures(third, middle)
          and not any(measures(second, m) and measures(third, m)
                      for m in range(1, middle)))
    return Out(terms=terms)