Book X · Proposition 21

X.21

The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called medial.Heath, 1908

Every step, checked

• both lines are rational in squaregiven
• they are commensurable in square onlygiven
• the rectangle they contain is irrationalby X.Def.4
• the side of the equal square is medialby X.21
• and the classifier names it soby X.21

What it needs, and what needs it

Needs: nothing earlier.

Used by: X.27 X.28 X.32 XIII.11

Rests on: X.Def.4

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.21",
    THEOREM,
    sample=_rational_pair_in_square_only,
)
def prop_X_21(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))
    rectangle = a * b
    side = sqrt(rectangle)
    claim("the rectangle they contain is irrational", "X.Def.4",
          not isinstance(rectangle, Fraction))
    claim("the side of the equal square is medial", "X.21", is_medial(side))
    claim("and the classifier names it so", "X.21", classify(side).name == "medial")
    return Out(medial=side)