To “square” a figure, for the Greeks, was to find a square, or another rectilinear figure, with the same area. Squaring the circle was the impossible problem par excellence. Archimedes managed to square a curved figure: the segment of a parabola, the region between a parabola and a chord. Every segment bounded by a straight line and a section of a right-angled cone is four thirds of the triangle that has the same base and the same height as the segment. — Archimedes, Quadrature of the Parabola, letter to Dositheus
In the treatise On Spirals, also dedicated to Dositheus, Archimedes studies a new curve, defined by a motion. If a straight line in a plane revolves at a uniform rate about one of its extremities, which remains fixed, and returns to the position from which it started, and if, at the same time, a point moves at a uniform rate along the line, starting from the fixed extremity, the point will describe a spiral in the plane. — Archimedes, On Spirals, definition