QUADRATURE OF THE CIRCLE. The problem involved in the quadrature of the circle requires the determination of the length of a straight line such that the square con structed thereon shall have an area equal to that of a given circle. It can be shown in a variety of ways that, if r is the radius of the circle, the area will be equal to wr* where w can only be obtained approximately in terms of a finite number of fractions. On the other hand, it has become a matter of general information that the quadrature of the circle is impossible, and this is true only when the ancient Greek problem is understood, which involves a serious limitation; that is, the quadrature must be effected by means of a geometrical construction in which the mathematician is limited to the use of but two instruments, the straight edge and a pair of compasses. The problem is not solved, there fore, if any other instrument or any equivalent analytic method is employed. For 4,000 years innumerable attempts have been made to discover this construction, all destined to fail, as it was demonstrated by Lindemann in 1882 to be impossible. In reaching this conclusion we are confronted by the fundamental question: What geometrical constructions are, and what are not, possible, when restricted to the use of these instruments? In analysis, operations cor respond to constructions. The operation, a X b, involves taking b units a times, which, with an assumed unit of length, is equivalent to laying off a line having a X b units of length, which is accomplished by using the method of propor tions. In a similar manner, the rational opera tions of addition, subtraction, multiplication and division find a geometrical solution involving the straight edge alone.
Irrational operations are divided into alge braic and transcendental. Any operation that involves the extraction of a square root only Presents the simplest case of an algebraic irrationality, and any construction involving the determination of the points of intersection of two circles, or a circle and straight line, leads to an equation of the second, of the fourth or of some higher degree, whose solution involves the extraction of square roots and rational opera tions only. Conversely, the necessary and suffi
cient condition that unknown quantities can be constructed with the straight edge and com passes is that the unknown quantities can be ex pressed explicitly in terms of the known by an analytic expression involving only a finite num ber of rational operations and square roots. In other words a Euclidean geometrical solution is impossible when no corresponding algebraic equation exists. When a number like V 2 is the root of an algebraic equation with integral co efficients, for example, and still can not be expressed exactly in terms of a finite series of numbers it is an algebraic irrational number. When the number, like e, the natural base in the theory of logarithms, or 7r, the ratio of the circumference to the diameter, is not the root of any algebraic equation, with in tegral coefficients, it is a transcendental num ber. Lindemann provided that Tr is a transcen dental number and, hence, since it is not the root of any algebraic equation, it cannot be con structed to an assumed unit by the extraction of square roots, that is, by using the straight edge and compasses.
possiblity of a geometrical solution of a problem in general depends upon a theorem in the theory of numbers to the effect that the degree of the irreducible equation satisfied by an expression composed of square roots only is always a power of 2; whence, if an irreducible equation u not of degree 2 w,it cannot be solved by square roots.
Next to the squaring of the circle, the most famous problems of antiquity are the Delium problem of the duplication of the cube and the trisection of an arbitrary angle. Granting the preceding general theorem, these, are easily shown to be impossible when restricted to the straight edge and compass.