The Quadrature of the Circle and Hippocrates' Lunes - More on Area

If plane geometry is the study of plane figures, then a natural and fundamental problem in plane geometry is: given a figure, determine its area. That is, how much of the plane does it enclose? And while solutions to this problem were already well known for many types of the simplest plane figures, it is the systematic way in which the problem was being addressed by them theoretically that was new. Let us follow some of this development.

First of all, what is meant by area? The simple answer for us modern readers is that area is a measure of the size of a plane figure. But as soon as one attempts to measure, one needs an appropriate scale to measure against. For the same figure might measure 5 square feet on one scale and 0.4645 square meters on another. Which of these measures is the correct one? Well, both are, of course. This example just illustrates that we measure area, like length and volume, relative to some unit. The value of the measure is dependent on the unit used.

Modern geometry is characterized by its highly arithmetized and algebraic form: in treatments of geometry given today, geometric objects are often defined and described through equations or formulas. Another reflection of this perspective more pertinent to our present discussion, however, is the modern need to quantify the chief properties of geometric objects through the measurement of their dimensions. Length, area, angle, volume, and other characteristics of geometric objects are for us numerical measures, relative to some convenient scale (feet, square meters, degrees, cubic centimeters, etc.). This way of doing geometry is quite different from that found in the work of the Greeks.

For them, the area of a figure was not a number computed through some measurement; indeed, area was not conceived of numerically at all. Rather, it was a property attached to the object, the one having to do with its size. For the Greek geometer, the area of a triangle was not a number but a certain extent of two-dimensional space. In fact, in most cases, the area of the object and the object itself were not distinguished each from the other. Instead, to find the area of a figure meant for them to determine the dimensions of a square, the primordial plane figure, having the same two-dimesional content of space as the given figure. As we will see, to the Greek geometer, figures which had the same area were in fact said to be ëqual," illustrating that the figure itself was identified as one and the same with its area. This process was called quadrature (from the Latin word for square, quadratus, literally meaning four-sided), manifesting the importance of reference to the square as the most fundamental of plane figures. Notice also that this form of quadrature of a figure requires no measure, no computation, no number; it is a purely geometric procedure.

We will consider a sequence of propositions from Books I and II of Euclid's Elements, consisting of results that were already well-known by the middle of the fifth century. They systematically solve the quadrature problem for any polygonal figure.