Book X · Proposition 36

X.36

If two rational straight lines commensurable in square only be added together, the whole is irrational; and let it be called binomial.Heath, 1908

Every step, checked

• both lines are rational in squaregiven
• they are commensurable in square onlygiven
• the whole is irrationalby X.36
• Book X calls it a binomialby X.36
• and assigns it one of the six speciesby X.48-53

What it needs, and what needs it

Needs: nothing earlier.

Used by: X.48 X.49 X.50 X.51 X.52 X.53 X.111

Rests on: X.48-53

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.36",
    THEOREM,
    sample=_rational_pair_in_square_only,
)
def prop_X_36(a, b) -> Out:
    hypothesis("both lines are rational in square", is_rational_in_square(a) and is_rational_in_square(b))
    hypothesis("they are commensurable in square only", not commensurable(a, b))
    whole = a + b
    claim("the whole is irrational", "X.36", not isinstance(whole, Fraction))
    named = classify(whole)
    claim("Book X calls it a binomial", "X.36", named.family == "binomial")
    claim("and assigns it one of the six species", "X.48-53",
          named.species in ("first", "second", "third", "fourth", "fifth", "sixth"))
    return Out(binomial=whole, species=named.species)