Book V · Proposition 21

V.21

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then, if ex aequali the first magnitude is greater than the third, the fourth will also be greater than the sixth; if equal, equal; and if less, less.Heath, 1908

'Perturbed' means the second triple is matched to the first crosswise, which is how ratios get composed in the wrong order and still behave.

Every step, checked

• the magnitudes are positivegiven
• a : b = e : fgiven
• b : c = d : egiven
• as the first stands to the third, so the fourth stands to the sixthby V.8, V.13

What it needs, and what needs it

Needs: V.8 V.13

Rests on: Def.5, Def.7

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.21",
    THEOREM,
    sample=_two_triples_perturbed,
    note="'Perturbed' means the second triple is matched to the first crosswise, "
    "which is how ratios get composed in the wrong order and still behave.",
)
def prop_V_21(a, b, c, d, e, f) -> Out:
    hypothesis("the magnitudes are positive",
               all(sign(x) > 0 for x in (a, b, c, d, e, f)), guard=True)
    hypothesis("a : b = e : f", a * f == b * e)
    hypothesis("b : c = d : e", b * e == c * d)
    # The ratios compared are a : b against c : b, which differ exactly when A
    # and C do. V.13 speaks of a ratio greater than another, and no figure
    # meeting these hypotheses has one, so it stays cited.
    if a != c:
        because(prop_V_8, a, b, c, b)

    because(prop_V_13, a, b, e, f, c, d) if a * f == b * e and ratio_cmp(e, f, c, d) > 0 else None

    claim("as the first stands to the third, so the fourth stands to the sixth",
          ["V.8", "V.13"], sign(a - c) == sign(d - f))
    return Out()