It is now generally admitted that the duplication problem, as well as the trisection and quadrature problems, cannot be solved by means of the ruler and compass only.
Bryson of Heraclea, a contemporary of Antiphon, advanced the problem of the quadrature considerably by circumscribing polygons at the same time that he inscribed polygons. He erred, however, in assuming that the area of a circle was the arithmetical mean between circumscribed and inscribed polygons.
Archimedes studied also the ellipse and accomplished its quadrature, but to the hyperbola he seems to have paid less attention. It is believed that he wrote a book on conic sections.Florian Cajori