sign in sign up
<<
Home , Angle trisection, Extended side, Quadratrix

Trisection of the : 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 and has its roots in ancient Greece in the fifth century B.C., during the time of (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 , 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 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 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 through

1 two given points, and a compass can be used to draw a with a given 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. 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 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 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 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 and . 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 of Elis (ca 460-399 B.C.) using his , the first 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 and 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 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. (Burton, p. 123) In the figure, BE ⊥ OA , the point C

B C D

a G 2 a

F A O E Figure 5 is chosen so that FC = 2 OB , and G is the midpoint of FC . Then ∠AOC ≅ ∠BCO and

∠CBE ≅ ∠BEO by alternate interior angles, and BG = FG = CG = OB (since G is the midpoint of the hypotenuse of a , it is equidistant from the endpoints of its sides).

So GBC and BOG are isosceles and therefore ∠CBG ≅ ∠BCO and ∠BOG ≅ ∠BGO . By the external angle theorem, m∠BGO = 2m∠BCO . Therefore m∠BOC = 2m∠AOC , which verifies that m∠AOB = 3m∠AOC . Nicomedes also discovered a curve called the conchoid

(shaped like a normal curve) which can be used to trisect any given angle, an example of how in the process of studying one problem scholars often discovered something entirely new to study

(Bunt, p. 106) There are other that can be used to easily trisect an angle, such as the cardioid, used by Etienne Pascal (1588-1640), and the of Maclaurin, used by Colin

Maclaurin (1698-1746). (Dudley, p.10)

7 Apollonius (250-175 B.C.) solved the trisection problem using conic sections, specifically a (Katz, p. 125), as did Pappus (early fourth century). Pappus’ Collection also contains a “high level” discussion of the trisection problem (Bunt, p. 202)

The ancient Greeks were able to construct many complicated figures, including many angles, by starting with the basic tools of the straightedge and compass, and adding, subtracting and bisecting fundamental angles (such as 60° 90° and 108° angles). They were focused on the geometric constructions, but “it is a remarkable tribute to the wisdom of the Greek geometers that two of the problems they highlighted, the duplication of the cube and the trisection of the angle, provide precisely the tools needed for the solution of cubic and quartic equations.” (Hartshorne, p. 271) Those working on the trisection of the angle problem eventually moved beyond physical constructions, incorporating first trigonometric relationships in their explorations, then related algebraic equations. The evolution of the theory of equations is thus closely tied to the work done on this problem.

In the sixteenth century, Gerolamo Cardano (1501-1576) published a general formula for the solution to the cubic in his Ars Magna, sive de regulis algebraicis (The Great Art, or the Rules of

Algebra) in 1545, giving geometric justifications for his methods. François Viète (1540-1603) and René Descartes (1596-1650) were then instrumental in introducing and perfecting the symbolic notation necessary to really manipulate variables and coefficients in equations, an advance without which the search for solutions for higher order equations would have been practically impossible. Especially of note to our discussion is the fact that Viéte noticed the link between the equations he was working with and trigonometry, in particular when working with

8 irreducible (unfactorable) cubic equations of the type we will see are at the heart of the proof that a general cannot be constructed using straightedge and compass. (Boyer p. 310,

Stillwell [16] p. 56) He really “developed the earliest consciously articulated theory of equations.” (Hartshorne, pp. 266-267) Another related development was Decartes’ discovery that the straightedge and compass constructions in Euclidean geometry correspond to quadratic equations. (Hartshorne, pp. 120-123)

In addition to the three “classic problems of antiquity”, the construction of regular with prime numbers of sides higher than 5 (i.e. 7, 11, 13, 17, 19, and so on) had not been accomplished, but in 1796, Carl Friedrich Gauss (1777-1855) came up with a way to construct a regular 17-gon using only a straightedge and compass, the first such construction to be accomplished since antiquity. It was this success which made him finally decide to go into mathematics. (Katz p. 723, Stillwell [15] p. 16) More importantly, it was a significant event in the evolution of modern algebra, since it revealed “hidden, abstract properties of just as much as manipulative skill.” (Stillwell [15], p. 16)

Gauss also claimed to have proven that any regular n-gon is constructible when n-1 is a power of 2, although he never published it. Constructing regular polygons is directly related to constructing angles, and thus connected to our problem of trisection. In fact, the trisection problem generalized (dividing a given angle into n parts) is precisely the problem of constructing an n-gon. (Stillwell [15], p. 10)

Interestingly enough, the impossibility of the angle trisection, the duplication of the cube, and the Gauss’ conjecture about regular n-gons were first proven by an obscure mathematician,

Pierre Laurent Wantzel (1814-1848) in his 1837 paper, “Research on the Means of Knowing If a

9 Problem of Geometry Can Be Solved with Compass and Straightedge,” merely seven pages long, published in Liouville’s Journal de Math (vol. 2, 1837, pp. 366-372). (Dunham, Dudley, Katz,

Stillwell [15]) Finally, more than 2200 years after their introduction, these classical problems had definitive answers.

Wanzel’s proof of the impossibility of the trisection involved, as has already been stated, the fact that the construction can be represented by a which has no “constructible” roots. The essence is that since straightedge and compass allow one to represent addition, subtraction, multiplication, division, and square roots using lines and circles (and their intersections). In order for a number to be constructible, it must be able to be represented by some combination of these operations on positive integers. Since the trisection problem boils down to an irreducible cubic equation (a fact already known by Viéte, remember), the only way to find a solution is to take a , which cannot be arrived at through any combination of additions, subtractions, multiplications, divisions, and square roots on positive integers. By the time Wanzel put together his proof, the conclusion had already been reached that there was actually no general method for trisecting an arbitrary angle. Therefore, the proof shows that assuming the trisection can be done leads to a contradiction (a number which cannot be arrived at through only combinations of additions, subtractions, multiplications, divisions, and square roots on positive integers). (Dudley p. xv, Quine)

Now we will look at an algebraic argument based on the trigonometric identities and the cubic equation used as early as Viéte. Just as with Wanzel’s proof, we show that the trisection is not possible in one case, thereby proving it cannot be done in general. We use the case where the given angle is 60°. If we could trisect it, we would be able to construct a 20° angle. We start

10 with the trigonometric identity cos 3θ = 4 cos3 θ − 3cosθ , where θ = 20° and therefore

3 60° = 3θ . Substituting, we get the equation cos60° = 4(cos20°) − 3cos20° . Letting

1 3 3 α = cos20° , substituting and simplifying, we now have 2 = 4α − 3α , or 8α − 6α − 1 = 0 .

Now let x = 2α , and we can substitute again to get the equation x3 − 3x − 1 = 0 , the aforementioned irreducible cubic equation. We know it is irreducible because the only possible rational roots are ±1 (from the coefficients of the first and last terms), neither of which satisfy the equation. At this point we could use Cardano’s formula for the solutions to a cubic equation, which would give us complex solutions for x, and thus complex solutions for α = cos20° , which are clearly not constructible as described in the previous paragraph. Therefore, a 60° angle cannot be trisected using a straightedge and compass alone, which means the general trisection is not possible.

The 19th century saw a big turning point in algebra, through the work of those already mentioned, Gauss and Wantzel, but also through the work on solvability of equations of Niels

Henrik Abel (1802-1829) and his contemporary Évariste Galois (1811-1832). The results led to the ability to show that the general trisection of the angle is not only impossible in Euclidean geometry, but also in an even more generalized sense using the new of and what came to be known as . Thus, a geometry problem in ancient Greece which seemed at first to be straightforward not only turned out to be impossible, but contributed significantly to the development of modern algebra.

11 References

(1) Anglin, W.S., Mathematics: A Concise History and Philosophy, Springer-Verlag, New

York, 1994, pp. 75-80

(2) Artemiadis, Nicolaos K., , American Mathematical Society, 2004,

pp.59-66

(3) Boyer, Carl B., A History of Mathematics, 2nd Ed., John Wiley & Sons, Inc, 1991

(4) Bunt, Lucas N.H., Jones, Phillip S., and Bedient, Jack D., The Historical Roots of

Elementary Mathematics, Prentice-Hall, Inc, New Jersey, 1976

(5) Burton, David M., A History of Mathematics: An Introduction, 4th Ed., McGraw-Hill,

1999

(6) Cajori, Florian, A History of Elementary Mathematics, The Macmillan Company, London,

1896

(7) Dedron, P. and Itard, J., Mathematics & Mathematicians, Vol. 1, The Open University

Press, 1973

(8) Dudley, Underwood, The Trisectors, The Mathematical Association of America, 1994

(9) Dunham, William, The Mathematical Universe, John Wiley & Sons, Inc., New York, 1994,

pp. 237-247

(10) Eves, Howard, Great Moments in Mathematics (Before 1650), The Mathematical

Association of America, 1980, pp.92-94

(11) Eves, Howard, Great Moments in Mathematics (After 1650), The Mathematical

Association of America, 1981, p.226

(12) Hartshorne, Robin, Geometry: and Beyond, Springer-Verlag, 2000

(13) Katz, Victor J., A History of Mathematics, Addison-Wesley, Boston, 2009

12 (14) Quine, W.V., “Elementary Proof That Some Angles Cannot Be Trisected by Ruler and

Compass”, Mathematics Magazine, Vol. 63, No. 2 (Apr., 1990), pp. 95-105

(15) Stillwell, John, Elements of Algebra, Springer-Verlag, 1994

(16) Stillwell, John, Mathematics and Its History, Springer-Verlag, New York, 1989

(17) B.L. van der Waerden, A , Springer-Verlag, 1985

13