Book V · Proposition 25

V.25

If four magnitudes be proportional, the greatest and the least are greater than the remaining two.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• a : b = c : dgiven
• A is the greatest and D the leastgiven
• the greatest and the least together exceed the other twoby V.19

What it needs, and what needs it

Needs: V.19

Rests on: Def.5

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(
    "V.25",
    THEOREM,
    sample=_four_proportional_ordered,
)
def prop_V_25(a, b, c, d) -> Out:
    hypothesis("the magnitudes are positive",
               all(sign(x) > 0 for x in (a, b, c, d)), guard=True)
    hypothesis("a : b = c : d", a * d == b * c)
    hypothesis("A is the greatest and D the least",
               sign(a - b) > 0 and sign(a - c) > 0 and sign(b - d) > 0 and sign(c - d) > 0)
    because(prop_V_19, a, b, c, d)

    claim("the greatest and the least together exceed the other two", "V.19",
          sign((a + d) - (b + c)) > 0)
    return Out()