Book X · Proposition 11

X.11

If four magnitudes be proportional, and the first be commensurable with the second, the third will also be commensurable with the fourth; and, if the first be incommensurable with the second, the third will also be incommensurable with the fourth.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• the four are proportionalgiven
• commensurability passes across the proportionby X.11

What it needs, and what needs it

Needs: nothing earlier.

Used by: X.14

Depth: 0 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(
    "X.11",
    THEOREM,
    sample=_proportional_magnitudes,
)
def prop_X_11(a, b, c, d) -> Out:
    hypothesis("the magnitudes are positive", all(sign(x) > 0 for x in (a, b, c, d)), guard=True)
    hypothesis("the four are proportional", a * d == b * c)
    claim("commensurability passes across the proportion", "X.11",
          commensurable(a, b) == commensurable(c, d))
    return Out()