Book VIII · Proposition 18

VIII.18

Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side.Heath, 1908

'Similar plane numbers' are rectangles of the same shape, and they behave exactly as similar rectangles do in Book VI.

Every step, checked

• the sides are genuine numbersgiven
• the sides are proportional, so the planes are similargiven
• one mean proportional falls between themby VIII.18
• and plane is to plane in the duplicate ratio of the sidesby VIII.11

What it needs, and what needs it

Needs: VIII.11

Used by: VIII.26

Depth: 3 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.18",
    THEOREM,
    sample=similar_planes,
    note="'Similar plane numbers' are rectangles of the same shape, and they "
    "behave exactly as similar rectangles do in Book VI.",
)
def prop_VIII_18(a: int, b: int, c: int, d: int) -> Out:
    """The planes are AB and CD; the sides are given, not multiplied 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
    mean = a * d
    because(prop_VIII_11, a, b)

    claim("one mean proportional falls between them", "VIII.18",
          first * second == mean * mean)
    claim("and plane is to plane in the duplicate ratio of the sides", "VIII.11",
          first * (c * c) == second * (a * a))
    return Out(mean=mean)