"The geometricians of these schools... were especially interested in three problems: namely (i), the duplication of the cube... (ii) the trisection of an angle; and (iii) the squaring of a circle... Now the first two... (considered analytically) require the solution of a quadratic equation; and, since a construction by means of circles (whose equations are of the form x^2 + y^2 + ax + by + c = 0 and straight lines (whose equations are of the form \alpha x + \beta y + \gamma = 0) cannot be equivalent to the solution of a cubic equation, the problems are insoluble if in our constructions we restrict ourselves to the use of circles and straight lines, that is, to Euclidean geometry. If the use of s be permitted, both of these questions can be solved in many ways. The third problem is equivalent to finding a rectangle whose sides are equal respectively to the radius and to the semiperimeter of the circle. These lines have long been known to be incommensurable, but it is only recently that it has been shown by Lindemann that their ratio cannot be the root of a rational algebraical equation. Hence the problem also is insoluble by Euclidean geometry. The Athenians and Cyzicians were thus destined to fail in all three problems, but the attempts to solve them led to the discovery of many new theorems and processes."