Book XI · Proposition 32

XI.32

Parallelepipedal solids which are of the same height are to one another as their bases.Heath, 1908

Every step, checked

• the three arms are not in one planegiven
• the two solids are of the same heightby XI.14
• as the base is to the base, so is the solid to the solidby XI.32

What it needs, and what needs it

Needs: XI.14 XI.31

Used by: XI.34

Rests on: C.N.1, C.N.2, C.N.3, C.N.4, C.N.5, Def.10, Def.15, Def.4, Post.5, XI.Def.10, XI.Def.3, XI.Def.8

Depth: 15 steps of argument above the first principles. Parallel postulate: needed.

What it takes on trust

Nothing. It draws no intersections and reads nothing off the picture.

The proposition as code

@proposition("XI.32", THEOREM, sample=samples3.corner_and_arms)
def prop_XI_32(o: Point3, a: Point3, b: Point3, c: Point3) -> Out:
    """Parallelepipeds of the same height are to one another as their bases."""
    hypothesis("the three arms are not in one plane", not coplanar(o, a, b, c))
    first = _built(parallelepiped(o, a, b, c, "the first solid"))
    wider = posit3(_along(o, a, Fraction(5, 2)), "P")
    second = _built(parallelepiped(o, wider, b, c, "the second solid"))

    because(prop_XI_31, o, a, b, c)

    here, there = _base_area(o, a, b), _base_area(o, wider, b)
    claim("the two solids are of the same height", "XI.14",
          first.face_plane(0) == second.face_plane(0)
          and first.face_plane(1) == second.face_plane(1))
    claim("as the base is to the base, so is the solid to the solid", "XI.32",
          here * content(second) == there * content(first))
    return Out(solids=(first, second), bases=(here, there))