Book XI · Proposition 37

XI.37

If four straight lines be proportional, the parallelepipedal solids on them which are similar and similarly described will also be proportional; and, if the parallelepipedal solids on them which are similar and similarly described be proportional, the straight lines will themselves also be proportional.Heath, 1908

Every step, checked

• the three arms are not in one planegiven
• the four straight lines are proportionalgiven
• the four solids are similar and similarly describedby XI.Def.9
• as the first solid is to the second, so is the third to the fourthby XI.37
• and a solid on any other line breaks the proportion, so solids proportional give lines proportionalby XI.37

What it needs, and what needs it

Needs: XI.33

Rests on: VI.Def.1, XI.Def.11, XI.Def.9

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("XI.37", THEOREM, sample=samples3.corner_and_arms)
def prop_XI_37(o: Point3, a: Point3, b: Point3, c: Point3) -> Out:
    """Four proportional straight lines carry similar parallelepipeds that are
    proportional, and conversely."""
    hypothesis("the three arms are not in one plane", not coplanar(o, a, b, c))
    ratio, apart = Fraction(3, 2), Fraction(5, 4)
    lines = (Fraction(1), ratio, apart, apart * ratio)
    hypothesis("the four straight lines are proportional",
               lines[0] * lines[3] == lines[1] * lines[2])

    directions = tuple(unit(vector_between(o, point)) for point in (a, b, c))

    def described(reach):
        return parallelepiped(o, *[_arm_at(o, step, reach) for step in directions])

    solids = [described(reach) for reach in lines]
    _built(solids[0])
    _built(solids[3])
    contents = [content(solid) for solid in solids]

    because(prop_XI_33, o, a, b, c)

    claim("the four solids are similar and similarly described", "XI.Def.9",
          all(_solid_angle_of(o, tuple(solid.vertices[i] for i in (1, 3, 4)))
              == _solid_angle_of(o, tuple(solids[0].vertices[i] for i in (1, 3, 4)))
              for solid in solids[1:]))
    claim("as the first solid is to the second, so is the third to the fourth",
          "XI.37", contents[0] * contents[3] == contents[1] * contents[2])
    # The converse is checked where it can fail: a fourth line that is not the
    # fourth proportional, and the solid on it, which the proportion must then
    # refuse. Each content is the cube on its line times one and the same
    # figure, so a proportion between the solids is one between the cubes.
    astray = lines[3] * Fraction(6, 5)
    claim("and a solid on any other line breaks the proportion, so solids "
          "proportional give lines proportional", "XI.37",
          lines[0] * astray != lines[1] * lines[2]
          and contents[0] * content(described(astray)) != contents[1] * contents[2])
    return Out(solids=tuple(solids), lines=lines)