Book X · Proposition 111

X.111

The apotome is not the same with the binomial straight line.Heath, 1908

The thirteen are genuinely thirteen: an apotome is never a binomial, so the two halves of the book do not overlap.

Every step, checked

• the scale is genuinegiven
• the apotome is positivegiven
• one is a binomial and the other an apotomeby X.36
• so an apotome is not the same as a binomialby X.111
• and the thirteen names are distinct from one anotherby X.111

What it needs, and what needs it

Needs: X.36

Rests on: X.48-53

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("X.111", THEOREM, sample=lambda rng: (Fraction(rng.randint(3, 9)),),
             note="The thirteen are genuinely thirteen: an apotome is never a "
             "binomial, so the two halves of the book do not overlap.")
def prop_X_111(scale) -> Out:
    hypothesis("the scale is genuine", sign(scale) > 0, guard=True)
    binomial = scale + sqrt(2)
    apotome = scale - sqrt(2)
    hypothesis("the apotome is positive", sign(apotome) > 0)
    because(prop_X_36, scale, sqrt(2)) if not commensurable(scale, sqrt(2)) else None

    claim("one is a binomial and the other an apotome", "X.36",
          classify(binomial).family == "binomial"
          and classify(apotome).family == "apotome")
    claim("so an apotome is not the same as a binomial", "X.111",
          classify(apotome).family != classify(binomial).family)
    claim("and the thirteen names are distinct from one another", "X.111",
          len(set(SPECIES)) == len(SPECIES) == 13)
    return Out()