Book X · Proposition 51

X.51

To find the fourth binomial straight line.Heath, 1908

Every step, checked

• both terms are rational in squaregiven
• they are commensurable in square onlygiven
• the sum is a binomialby X.36
• and it is the fourth binomialby X.51

What it needs, and what needs it

Needs: X.36

Used by: X.57

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(ref, CONSTRUCTION,
             sample=lambda rng, _i=index: _binomial_of_species(_i))
def _species(a, b, _i=index) -> Out:
    hypothesis("both terms 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
    named = classify(whole)
    wanted = BINOMIAL_SPECIES[_i - 1]
    because(prop_X_36, a, b)
    claim("the sum is a binomial", "X.36", named.family == "binomial")
    claim(f"and it is the {wanted} binomial", ref, named.species == wanted)
    return Out(binomial=whole, species=named.species)