Book VIII · Proposition 26

VIII.26

Similar plane numbers have to one another the ratio which a square number has to a square number.Heath, 1908

Every step, checked

• the sides are genuine numbersgiven
• the sides are proportional, so the planes are similargiven
• similar plane numbers have the ratio of a square to a squareby VIII.18

What it needs, and what needs it

Needs: VIII.18

Depth: 4 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.26",
    THEOREM,
    sample=similar_planes,
)
def prop_VIII_26(a: int, b: int, c: int, d: int) -> Out:
    hypothesis("the sides are genuine numbers",
               a > 1 and b > 1 and c > 1 and d > 1, guard=True)
    hypothesis("the sides are proportional, so the planes are similar",
               a * d == b * c)
    first, second = a * b, c * d
    because(prop_VIII_18, a, b, c, d)

    claim("similar plane numbers have the ratio of a square to a square", "VIII.18",
          first * (c * c) == second * (a * a)
          and is_square(a * a) and is_square(c * c))
    return Out(ratio=(a * a, c * c))