ABSTRACT
Our previous discussion of the three geometric problems of antiquity — trisecting the angle, duplicating the cube, and squaring the circle — left one key fact unproved. To complete the proof of the impossibility of squaring the circle by a ruler-and-compass construction, crowning three thousand years of mathematical effort, we must prove that π is transcendental. The proof we give is analytic, which should not really be surprising since π is best defined analytically. The techniques involve symmetric polynomials, integration, differentiation, and some manipulation of inequalities, together with a healthy lack of respect for apparently complicated expressions.
