Pappus on the trisection of an angle
When ancient geometers desired to divide a given rectilinear angle into three equal parts, they were baffled for the following reason. There are, we say, three types of problem in geometry, the so-called "plane," "solid," and "linear" problems. Those that can be solved with straight line and circle are properly called "plane" problems, for the lines by which such problems are solved have their origin in a plane. Those problems that are solved by the use of one or more sections of the cone are called "solid" problems. For it is necessary in the construction to use the surfaces of solid figures, that is to say, of cones. There remains the third type, the so-called "linear" problem. For the construction in these cases curves other than those already mentioned are required, curves having a more varied and forced origin and arising from more irregular surfaces and from complex motions. Of this character are the curves discovered in the so-called "surface loci" and numerous others even more involved discovered by Demetrius of Alexandria in his Treatise on Curves [unknown except for this reference] and by Philo of Tyana [unknown except for this reference] from the interweaving of plectoids and of other surfaces of every kind. These curves have many wonderful properties. More recent writers have indeed considered some of them worthy of more extended treatment, and one of the curves is called "the paradoxical curve" by Menelaus. Other curves of the same type are spirals, quadratrices, cochloids, and cissoids.
Now it is considered a serious type of error for geometers to seek a solution to a plane problem by conics or linear curves and, in general, to seek a solution by a curve of the wrong type. Examples of this are to be found in the problem of the parabola in the fifth book of Apollonius's Conics, and in the use of a solid verging with respect to a circle in Archimedes' work on the spiral. For in the latter case it is possible without the use of anything solid to prove Archimedes' theorem, viz., that the circumference of the circle traced at the first turn is equal to the straight line drawn at right angles to the initial line and meeting the tangent to the spiral.
In view of the existence of these different classes of problem, geometers of the past who sought by planes to solve the aforesaid problem of the trisection of an angle, which is by its nature a solid problem, were unable to succeed. For they were as yet unfamiliar with the conic sections and were baffled for that reason. But later with the help of the conics they trisected the angle using the following 'vergings' for the solution.
JOC/EFR August 2006
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