Having found the means to effect the quadrature of any polygonal figure (as we saw in Elements, II.14), the inability to square the circle by similarly direct means stood as an inviting challenge to geometers of the fifth and fourth centuries. So when Hippocrates' lune was found to be equal to a triangle, a tantalizing hope was raised that some similar analysis might effect the quadrature of the full circle. Thus Hippocrates' result stands as one of the more important advances in geometry of that time. But the problem of the quadrature of the circle remained unresolved as no way was found to replicate Hippocrates' success in application to the full circle. The best that could be done was to achieve approximate quadratures.
Antiphon, a contemporary of Hippocrates who taught in Athens, attempted to resolve the problem by inscribing a polygon within the circle and effecting the quadrature of the polygon by finding lower bound approximations for the area of the circle. He performs a similar process to find an upper bound approximation for the area of the circle by circumscribing a polygon about its circumference. Then, by successively doubling the number of sides of these polygons, one inside and the other outside the circle, the error of approximation can be reduced. Effectively, the area of the circle is ëxhausted" by taking more and more sides for these approximating polygons. Because the sources we have are quite fragmentary, we cannot be quite sure whether Antiphon believed that this procedure would actually result in finding the exact quadrature of the circle, or whether he knew that the best this would accomplish is the determination of a pair of values which are close but never equal to the true area.
Figure 11: "Exhausting" the area of the circle with inscribed and circumscribed polygons.
Another contemporary, Bryson of Heraclea, is said to have asserted that the area of the circle was at once greater than the area of any and all possible polygons that could be inscribed within it and smaller than the area of any and all possible polygons that could be circumscribed about it. This principle would come to fruition in the work of Eudoxus of Cnidos, who working at the start of the fourth century BCE would give the first proof of Elements, XII.2, the theorem that circles are to each other as the squares on their diameters. This method of exhaustion would be one that later geometers would return to again and again over many centuries to apply to quadratures of a variety of curved shapes. Archimedes (in the third century BCE) used it often, to demonstrate results he had discovered regarding the quadrature of regions like the parabolic segment44 and the cubature45 of the sphere. Much later, when Greek geometry was studied once again in the Europe of the sixteenth and seventeenth centuries CE, a resurgence of interest in these methods took place. Geometers like Gregoire a Saint-Vincent (1584-1667) and his student Alfonso Antonio de Sarasa (1618-1667), both Jesuit scientists, applied the method to show that the area between a hyperbola and its asymptote behaves like a logarithm.
As for the original problem of the quadrature of the circle, final resolution would not come until the nineteenth century, when it was shown using the tools of modern algebra (not geometry!) that although the Greek construction tools of straightedge and compass are capable of producing an infinite number of line segments, these segments all have lengths that belong to a restricted subset of real numbers, and p is not in this set, making the quadrature, in its strictest form, impossible.
The work of the fifth century geometers set a course for the pursuit of quadratures that led to the generation of much new and important mathematics. As geometers came across new types of curves, they considered new sorts of plane regions and solids and asked questions about their quadrature and cubature. This work came to a high point in the seventeenth and eighteenth centuries CE with the development of integral calculus. The fifth century area problem was the first mathematical "program," an enterprise representing the efforts of many individuals over a long period of time, all contributing to the understanding of a single type of problem through the development of new and more powerful mathematical methods. Even early on, as we have seen from direct reference to some of the texts written at the time, it involved surprising discoveries, patient systematization, and the realization that to adequately resolve the problem would require not just cleverness, but willingness to change the way in which the problem was being considered. These features have characterized progress in mathematical programs throughout history.
44Like a circular segment, a parabolic segment is the closed region bounded on one side by an arc of a parabola and on the other by a line that cuts the parabola in two points.
45Naturally, cubature is for three-dimensional solids what quadrature is to two-dimensional regions. The cubature of a solid is obtained by constructing a line segment equal to the side of a cube which is equal in volume to the given solid.