Book X · Proposition 76

X.76

If from a straight line there be subtracted a straight line which is incommensurable in square with the whole and which with the whole makes the squares on them added together rational, but the rectangle contained by them medial, the remainder is irrational; and let it be called minor.Heath, 1908

Every step, checked

• the greater term is the greatergiven
• the remainder is irrationalby X.76
• and it is minorby X.76

What it needs, and what needs it

Needs: nothing earlier.

Used by: XIII.11 XIII.16

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(ref, THEOREM, sample=_kind_sample(kind))
def _difference(a, b, _k=kind) -> Out:
    hypothesis("the greater term is the greater", sign(a - b) > 0)
    remainder = a - b
    claim("the remainder is irrational", ref,
          not isinstance(remainder, Fraction))
    claim(f"and it is {SUBTRACTED_NAMES[_k]}", ref,
          classify(remainder).name == SUBTRACTED_NAMES[_k])
    return Out(remainder=remainder)