As already stated, the usual plan of the textbooks is in part the method followed by Archimedes. It is possible to start with any regular polygon of which the side can conveniently be found in terms of the radius. In particular we might begin with an inscribed square instead of a regular hexagon. In this case we should have _Length of Side_ _Perimeter_ _s__{4} = 1.414... = 1.41 5.66 _s__{8} = [sqrt](2 - [sqrt](4 - 1.414^2)) = 0.72 5.76 and so on. It is a little easier to start with the hexagon, however, for we are already nearer the circle, and the side and perimeter are both commensurable with the radius. It is not, of course, intended that pupils should make the long numerical calculations. They may be required to compute _s__{12} and possibly _s__{24}, but aside from this they are expected merely to know the process. If it were possible to find the value of [pi] exactly, we could find the circumference exactly in terms of the radius, since c = 2[pi]_r_. If we could find the circumference exactly, we could find the area exactly, since _a_ = [pi]_r_^2. If we could find the area exactly in this form, [pi] times a square, we should have a rectangle, and it is easy to construct a square equivalent to any rectangle. Therefore, if we could find the value of [pi] exactly, we could construct a square with area equivalent to the area of the circle; in other words, we could "square the circle." We could also, as already stated, construct a straight line equivalent to the circumference; in other words, we could "rectify the circumference." These two problems have attracted the attention of the world for over two thousand years, but on account of their interrelation they are usually spoken of as a single problem, "to square the circle." Since we can construct [sqrt]_a_ by means of the straightedge and compasses, it would be possible for us to square the circle if we could express [pi] by a finite number of square roots. Conversely, every geometric construction reduces to the intersection of two straight lines, of a straight line and a circle, or of two circles, and is therefore equivalent to a rational operation or to the extracting of a square root. Hence a geometric construction cannot be effected by the straightedge and compasses unless it is equivalent to a series of rational operations or to the extracting of a finite number of square roots. It was proved by a German professor, Lindemann, in 1882, that [pi] cannot be expressed as an algebraic number, that is, as the root of an equation with rational coefficients, and hence it cannot be found by the above operations, and, furthermore, that the solution of this famous problem is impossible by elementary geometry.[86]