TRISECTION OF AN ANGLE (from tri sect, from Lat. //Ts, three + sect as, p. of secure, to cut). One of the three famous prob lems of antiquity, the others being the duplica tion of the cube (q.v.) and the squaring of the' circle. (See CIRCLE; QUADRATITRE. ) This prob lem, like the quadrature of the circle, is almost as old as geometry itself, but first received thorough investigation at the hands of the Soph ists (Re. 400). Hippias of this school invented the quadratrix (see QuADRATNRE), by which any angle may be trisected. In the figure BD is a quadrant of a circle, BG is an arc of the quadratrix, and the construction involves the relation = -2. Hence by dividing BA into segments having any given ratio, the quad rant or any arc BD can be divided into arcs the angle to be trisected, C any point on OB, CM 1 OA and CP 11 OA, it is easily seen that if P can be found so that PN=2C0 then PO is a trisection line of angle AOB. But the trammel of the conchoid with its directrix rest ing on CM, parameter its pale at 0, will describe a curve cutting CP in the required point. Many other methods have been devised for the solution of this problem. Viete (1591) showed its relation to the solution of the cubic equation. Gauss ( IS01) showed its relation to cyclotomic equations. Other mathe
maticians of the nineteenth century have de dared its solution impossible by means of the straight edge and compasses (that is, by the postulates of Euclidean geometry). But the real reason for the failure of this method to solve the problem and also its associates was set forth by Klein.
The argument may be briefly stated as follows: ( I) According to the formula of De Moivre (q.v.) the roots of the equation isin 271" , 47r are x,=cts 3 , — -- (2) These roots are represented geometrically by the vertices of an equilateral triangle inscribed in the unit circle with its centre at the origin. The figure shows that to the root x, corresponds the argument Hence the equa tion = cos0 isin 0 is the analytic expres sion of the problem of the trisection of the angle. No root of the equation can be expressed as a rational function of cos and sin0. That is, the equation is irreducible and can be solved by the aid of a finite number of square roots only for special values of 0. Hence the trisection of an arbitrary angle cannot be ef fected with straight edge and compasses. Con sult: Klein, Famous Problems of Elementary Geometry (1806, Eng. ed., Boston, 1897).