The Quadrature of the Circle and Hippocrates' Lunes - The Area Problem in the Fifth Century BCE

The ancient Babylonians and the Egyptians developed a rich knowledge of basic geometric concepts, like how the areas of simple plane figures - triangles, squares, rectangles, parallelograms, trapezoids, and the like - relate to the lengths of their sides, and how volumes of simple solid bodies - parallelopipeds and pyramids - relate to corresponding appropriate linear dimensions. Indeed, every one of the great human civilizations of the world have geometrized, often quite independently of each other. It seems to be in the nature of civilized human beings to deal in such abstractions.

It is important to note, however, that the geometrical relationships discovered by these early geometers were not expressed algebraically, as we do when, for example we claim that the statement A = [1/2]bh describes the area of a triangle. Algebra, the science of finding unknown numerical values from arithmetical conditions placed on them, especially in its familiar modern symbolic form, did not exist in the ancient world as such. This version of geometry tends to the utilitarian, meant for use by surveyors, builders, and astronomers. The Greeks, however, turned geometry into a theoretical science, a brand new discipline that concerned itself with the properties of a very refined collection of shapes and forms - points, lines, circles, polygons and polyhedra; a science founded on the deductive methods they were systematizing through philosophical dialectics. And for the first time in history, epistemological questions were being studied about mathematical ideas: how do we know that the geometric results we have discovered are true? Can we justify their truth so that others can come to agree to their truth independent of our authority on the matter? Are these ideas structurally interrelated? A tradition of proof through deduction arose to organize these ideas in theoretical, rather than utilitarian, form.

Geometers of the fifth century BCE were more interested in determining what could be asserted, say, about the relationship between the lengths of the pieces into which two intersecting chords in a circle cut each other than in how much land there was in some particular grower's quadrangular pomegranate orchard. Indeed, they understood the power of the logical, deductive method of theory: that once general theorems of geometry were established, they could then be applied in an infinite variety of particular instances to deal with these more prosaic utilitarian issues. And while their objects of study (circles, squares, etc.) were certainly conceived of as representing the underlying forms of physical reality, their theoretical consideration was viewed as applying to highly abstracted versions of this reality.

In any case, by the middle of the fifth century BCE, Greek geometers had worked out a theory for the area of plane figures that successfully handled all rectilinear figures. It is this theory to which we turn our attention.