While the ancient Greeks get a lot of the credit for establishing geometry as we know it today, the Egyptian empire asked and answered many of the same mathematical questions several millennia earlier. They were devised not just to construct the Great Pyramids and temples, but to construct them in very specific ways; the Golden Ratio was used widely, for instance. So was the squared circle, which is shorthand for "**find a square with an area exactly equal to the area of a given circle**." The object was to calculate the exact area of a circle by creating a square with the same area, and then measuring its side with a ruler.

The problem of squaring the circle (a.k.a. finding the **quadrature of the circle**) is an impossible one to solve, for reasons a high school student can easily explain today. The area of a circle is πr^{2}, the area of a square is s^{2}, and setting them equal means that s = r√π. The fact that π is a transcendental number means that it's impossible to draw or reduce it to a root of a rational number, to say nothing of the square root of π. (Non-transcendental irrational numbers do not suffer this problem, and indeed almost any integer square root can be drawn using to the Pythagorean Theorem.)

If you're willing to settle for an approximation of π, however, the problem is easier to conquer. The Egyptians recorded on the Rhind papyrus that one can compute eight-ninths of the diameter of the circle (16*r/9) and use this as the side of the square; we can compute its exact area as (256/81)*r^{2}, which means the computed value for "π" is about 3.1605, missing the mark by only about 0.6%. Not good enough for a mathematician, but close enough for a pharoah.

The Greeks took on the problem with more enthusiasm. Anaxagoras worked on the problem around 450 B.C. during a prison sentence, using the straightedge and compass approach. After he was freed, the problem became one of considerable interest among other Greek thinkers, and the expression "to square the circle" became a cliche meaning "to attempt the impossible." Plenty of false mathematical proofs were attempted and recorded in the years to come.

Bryson, a student of Socrates, took the obvious approach of using inscribed and circumscribed polygons to "squeeze" the area of the circle. Imagine drawing a square inside a circle such that all four of its corners touch the circle, and a second square around the circle such that each side is tangent to the circle. The area of the circle would be somewhere between the areas of these two squares. By using pentagons, hexagons and so on, the circle's area is constrained to smaller and smaller ranges. Bryson tried, but failed, to prove that it was possible to compute the circle's area exactly by taking this approach far enough.

The first person to come anywhere close to a real solution was Hippocrates, who proved that certain lunes (like a crescent moon, made from two circular arcs) could be squared. Specifically, he used the following special case:

___
__A_______ B
--- /| --- |\
/ | | E. \| |
| \| |/
| .-------| D
| C -___- |
\ /
---_____---

*(Gawd, I hate doing math with ASCII art.)* We have a circle C and a square ABDC, where the points A and D lie on the circle C. Now find point E halfway between A and D (but not on circle C) and circumscribe circle E around the square. We now have a lune built from arcs ABD on circle E and AD on circle C.

Now look at arc AB on circle E. Since angle AEB and angle ACD are both right angles, the sections AB from circle E and AED from circle C are similar. Applying the Pythagorean Theorem to the triangle formed from these angles shows that AE/AC = 1/√2, so AE^{2}/AC^{2} = 1/2 and the ratio of the area of segment AB to segment AED is also 1/2. Since segment BD is congruent to segment AB, the areas of segments AB and BD add up to equal the area of segment AED. Therefore, since the lune can be created by removing segment AED from the triangle ABD and adding the segments AB and AD, **the area of the lune equals the area of the triangle ABD.** And since any triangle can be squared (a separate problem, already solved by the Greeks), so can the lune.

In the above proof, the arc ABD is an exact semicircle; Hippocrates also proved that lunes where the outer arc was greater or less than a semicircle could also be squared (although this did not comprise the set of all possible lunes). His goal was to show that certain shapes bounded by circular arcs could be squared, even if circles themselves could not. If he ever hoped to produce a method of squaring the circle by combining these methods, however, he was not successful.

Archimedes took an interesting approach to the problem using spirals, a type of curve he studied at length. His definition of a spiral was: "If a straight line drawn in a plane revolves uniformly any number of times about a fixed extremity until it returns to its original position, and if, at the same time as the line revolves, a point moves uniformly along the straight line beginning at the fixed extremity, the point will describe a spiral in the plane." In other words, spin a drawing compass around and have the pencil move away from the fixed point at a constant speed.

___
/ \
_____|__./_____.______
\ |C / P
\|____/
| /
| /
| /
T |/

Say you draw exactly one complete turn of an Archimedean spiral, starting and ending on the line CP. Drop a line perpendicular to CP through C, and draw a tangent line to the spiral at point P. This tangent will cross the perpendicular line at some point, call it T. Archimedes proved that the segment CT is exactly equal to the circumference of a circle C with radius CP.

He also proved geometrically that **this can be used to square the circle**, but we can do it more neatly with algebra: if the circumference of circle C is 2πr = CT, then the area of the triangle CPT is r(2πr)/2 = πr^{2}, and by squaring the triangle we've squared the circle. If it weren't for the fact that the Archimedean spiral is impossible to draw *precisely* with an ordinary compass, the problem would now be solved.

While the Greeks seemed to understand that squaring the circle was unsolvable using compass-and-straightedge techniques, they never proved it was so, and so the problem continued to be attacked. Mathematicians in India, China, Arabia and medieval Europe all approached the problem in their own ways in the centuries to follow. Even Leonardo da Vinci attempted to square the circle, using mechanical methods instead of mathematical ones.

In 1761 Johann Heinrich Lambert proved that π was irrational, but not necessarily transcendental, inspiring literally hundreds of amateur mathematicians to try and publish false proofs of how to square the circle. It wasn't until 1882 that Ferdinand von Lindemann proved that **π is transcendental**, which should have been the final nail in the coffin. It can be shown that straightedge-and-compass constructions can only produce lengths which are equal to the sum (addition), difference (subtraction), product (multiplication), quotient (division), or square root of other lengths; a transcendental number by definition can never be computed using these operations. However, other geometric approaches continued to flow, and some of them were remarkably close in the same way the ancient Egyptians were. Most notable were the straightedge-and-compass contructions by Srinivasa Ramanujan in 1913 and 1914, which used approximations of π accurate to six and eight decimal places, respectively.

Today, of course, it's easier and more accurate to just use a pocket calculator and a good ruler or CAD program. But it's hard to say which is more remarkable: that the problem continues to invite solutions to this day, or that we've been able to come as close as we have.

*
Sources:*

http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Squaring_the_circle.html

http://www.geom.umn.edu/docs/forum/square_circle/