Trisection of the Angle: From Ancient Greece to 1900 Ruth Helgerud Math 646 Texas A&M University April 4, 2011 One of three “classical problems of antiquity” (Boyer, p. 64), the trisection of a general angle using only a straightedge and compass has its roots in ancient Greece in the fifth century B.C., during the time of Plato (429-348 B.C.). For more than 2200 years, many great minds wrestled with the problem, until a proof that it was impossible was finally achieved in 1837. These struggles with difficult problems such as these (and the accompanying modifications of the rules!) formed the basis for most of Greek mathematics, as well as a great deal of the mathematics which followed thereafter. (Boyer, pp. 64-65) By the time these classic problems were gaining notoriety, there were already so many theorems in geometry it was clear there was a need to systematize and organize all of the material into a coherent, logical whole. Geometry was transitioning from a practical science to a more general, abstract view of relationships in an “ideal existence.” (Hartshorne, p. 10) Plato described the changing perspectives of geometers of the time in The Republic: “Although they make use of the visible forms and reason about them, they are not thinking of these, but of the ideals which they resemble; not the figures which they draw, but of the absolute square and the absolute diameter, and so on....” (Hartshorne, p. 9) One of the rules established as part of this transition was that geometric constructions must be done solely with a straightedge and compass, a requirement Plato himself is considered to have originated. (Burton, p. 116) Specifically, a straightedge can be used to draw a line through 1 two given points, and a compass can be used to draw a circle with a given radius around a given point. No other uses were allowed. For example, neither tool could be used to copy distances or lengths. The straightedge could not be marked, and the compass must be thought of as collapsing as soon as either of the two points were removed from the paper. Furthermore, every point, line, or circle had to be able to be produced from what was already given or previously produced, using the straightedge and compass as described, in a finite number of steps. Apparently, in spite of the move toward generalization and abstraction, it is not clear that the Greeks believed something which they could not construct even existed. (Hartshorne, p. 167) They had, after all, physical evidence that the things they could prove in writing worked in the real world. Euclidean geometry seemed to offer truth. (Dudley, p. 2) This worldview no doubt provided some of the impetus for pursuing the solution to trisecting a general angle. The problem was, no one could do it! Great scholars put great effort into solving the problem, and they came up with all kinds of creative ways of solving it without following the rules of only using a straightedge and compass. It is very possible that at some point the ancient Greeks knew, or at least strongly suspected, that the construction under those restrictions was not possible in general, they just did not have the tools available to prove it. (Burton p. 116, Dudley p. 2, Dunham p. 245) The number of “illegal” constructions they came up with supports Now, the Greeks could perform many different constructions within the parameters of the straightedge and compass restriction, some of them fairly complicated, such as a regular pentagon. These skills, along with the facts that bisecting a given angle and trisecting a given line segment with compass and straightedge are fairly simple procedures make it a reasonable to assume that trisecting an angle would be also possible. (Dunham, p.240) In fact, there are many 2 angles that can be trisected, which probably further encouraged the pursuit of a general method. Countless professional and amateur mathematicians obviously thought such a method might exist, since so many devoted so much time to it over the centuries. Although they were wrong in their assumption, it is a good thing they persisted, since their work led eventually not only to the proof that the problem as given was unsolvable, but heavily influenced the development of the theory of equations and algebra as we know it today. Unfortunately, quite a few individuals even today still believe it possible and spend countless hours coming up with “solutions” to this insoluble problem... (Dudley). Part of the problem with such “Trisectors,” as Mr. Dudley calls them, is that, as mentioned before, there are plenty of angles which can be trisected. The simplest one is a 90° angle. It is elementary to construct an equilateral triangle and then bisect one of its angles to get a 30° angle, as shown in Figure 1. A 60° D 30° 30° 30° C B Figure 1 1 Since 30° = 3 ⋅ 90° , clearly, some angles can be constructed with only a compass and straightedge. The Greeks knew this, but they wanted to find a method to trisect any given angle. 3 Hippocrates (460-380 B.C.) was one of the earlier to try his hand at the trisection of the angle problem, along with the other two classical problems of squaring the circle and doubling the cube. Through his work he helped advance the latter two problems and made some important contributions to the development of mathematics, including using letters to denote points and lines in geometric figures. However, he was not able to make any progress on trisecting the angle. (Burton) The earliest known trisection was done by Hippias of Elis (ca 460-399 B.C.) using his quadratrix, the first curve other than the line and the circle seen in mathematics. (Dudley p. 6, Cajori p. 55, Boyer pp.68-69) To construct the quadratrix, start with a square and inscribe a quarter circle. Then move the top (horizontal) side of the square to meet the bottom (horizontal) side of the square at a uniform rate, and rotate the left (vertical) side/radius of the circle clockwise to meet the bottom side of the square at a uniform rate so that they both end up in the same position at the same time. The locus of the point of intersection of the horizontal segment Figure 2 from http://www.geom.uiuc.edu/docs/forum/angtri/ and the radius as they move is the quadratrix curve. Because of the way the curve is created, the 4 distance from the top is proportional to the rotation angle. This provides a simple way to trisect a given angle, as shown in Figure 2. Simply position the given angle on the quadratrix with its initial side coinciding with the bottom horizontal side of the square. Then find the point of intersection of the curve with the terminal side of the angle, and trisect the vertical distance to that point. Draw a horizontal through the point corresponding to one third the original vertical segment height, and the point where it intersects the quadratrix determines the terminal side of the desired angle. This curve will actually divide an angle into any number of equal sections. As ingenious as this curve was, it did not solve the problem, since it cannot be constructed using only a straightedge and compass. Archimedes (287-217 B.C.), who was an engineer, a physicist, and a mathematician, and perhaps “one of the greatest minds humanity has produced” (Dudley) came up with an elegant, uncomplicated solution, bending the rules only slightly by marking the straightedge in two places. One version of his method is illustrated in Figure 3. To construct this figure from a B 2! D r 2! r ! E C r O r A Figure 3 given angle ∠AOB , we start by extending side OA in each direction and constructing a semicircle with center O and radius OA, intersecting the extended side at C . Mark the straightedge with two marks the same distance apart as O and A . Then find point E on the extended side by lining up the straightedge with the point B on the semicircle in such a way that 5 the two marks fall on the semicircle at point D and on the extended side at E . Then DE = OA = OB = OD . Now we can use properties of circles and triangles to show that 1 α = m∠BEA = m∠DEO = 3 m∠AOB as follows. First note that ODB and DOE are isosceles triangles, so we have m∠DEO = m∠DOE = α and m∠OBD = m∠ODB . The exterior angle theorem tells us that m∠DEO + m∠DOE = m∠ODB , so m∠ODB = m∠OBD = 2α . It also tells us that m∠AOB = m∠DEO + m∠OBD, so m∠AOB = α + 2α = 3α . Archimedes’ spiral also provides a way to trisect any angle, but it cannot itself be constructed using only straightedge and compass. Figure 4 illustrates the spiral with trisection. The key is Figure 4 from http://www.uwgb.edu/dutchs/pseudosc/ trisect.htm that the ratio of the distance between the vertex and any point on the spiral (R) to the measure of the angle formed is constant. This proportionality means all we need to do to trisect an angle is to trisect the segment from the vertex to the spiral. Then we rotate the resulting segment 6 (extended from the vertex) until it meets the spiral, and the angle formed by this new segment with the horizontal will be one third the original angle. Like Archimedes, Nicomedes (ca 280-210 B.C.) used a marked straightedge in one of his trisection methods, illustrated in Figure 5.