The circle is one of the enigma's of mathematics. It is defined as the set of points in a given plane at a given distance from a center point. From a practical position, a compass is an excellent tool for describing such a circle. It is one of the simplest concepts, a cornerstone in the edifice of mathematics. Yet, it eludes mathematical exactness. It is a constant reminder that nothing is exact, even in mathematics. It is not difficult to see why so many wise men pondered the problem in hopes of imposing order upon a reluctant nature.
The earliest evidence of practical attempts to solve the problem of squaring the circle comes from the Egyptian Ahmes Papyrus(circa 1550 BC), where the area of a circle is approximated via the formula (64/81) times the diameter^2 [or 3.16 r^2]. The Egyptian's, though, are considered practical mathematicians and probably never confronted the problem formally, and legendary credit for first formal attempt is given to Anaxagorus of Clazomenae while he was in prison for a time. The problem gained importance as Greek mathematics evolved, and it eventually became such a prominent issue that it even earned ridicule from Aristophanes in his Birds, as the astronomer Meton tries to aid in the division of land(beginning in line 997)
The problem of squaring the circle defeated conventional techniques of compass and straight edge, but it remained until recent times to be proven impossible. The Greeks, realizing the difficulty, were forced to turn to more complicated structures like the quadratrix of Hippias to solve the problem.
Antiphon the Sophist, Socrates' contemporary, proposed one useful method for obtaining the area in question. Beginning with a regular polygon, bisect the sides and produce a new polygon with twice the number of sides. The same procedure can the be performed on this new polygon, again, and if the process is continued indefinitely, a point will eventually be reached(he claimed) where the sides of the polygon will be indistinguishable from the circle. Then, since the area of any polygon can easily be found( by multiplying the number of sides by the area of the isosceles triangles containing those sides, the area of the circle can then be deduced. Unfortunately, this method was completely rejected by Aristotle and Eudemus because it was independent of the accepted mathematical methods of the time. In it, however, were the seeds for the methods of exhaustion and integral calculus, each of which would serve as powerful tools in the development of mathematics beyond the constraints of absolutes.
Bryson, a pupil of either Socrates or Euclid of Megara, extended this idea to "squeeze" out the area by producing both circumscribed and inscribed polygons. Since the area of the circle was between the areas of the two polygons, one now had a method for reducing the region of possibility for the area of a circle.
A more accepted method, but one of less practical use, involved the quadratrix of Hippias, which he also used in the division of an angle into equal parts of any number. The proof can then be found in Pappus, who gives credit to Deinostratus(c. 350 BC) and Nicomedes(c. 180 BC), that the intersection of this curve and the bottom of the square can be used to determine the circle's area. While the complexity of this solution serves as a vivid reminder of the level of Greek advancement, it is of little developmental importance because the solution merely replaces one indefinite value(pi) with another.
Perhaps the most elegant construction of a circle's area, though, would be that of Archimedes. It states that the area of the circle is equal to the area of a right triangle whose perpendicular height is equal to the circle's radius and the base length is equal to the circle's circumference. This can easily be demonstrated algebraically to hold true, but note that this is not a proof of equality.
The base length can be found by rolling the circle down a line for exactly one full revolution, but unfortunately, this is not a very mathematical technique. Archimedes, however, found a much more clever and beautiful way to do it, with the use of a spiral.