Trisection of the Angle:From Ancient Greece to 1900

Ruth HelgerudMath 646

Texas A&M UniversityApril 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

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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

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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.

Since 30° = 13 ⋅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.

30°30°

30°

60° D

A

BC

Figure 1

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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

and the radius as they move is the quadratrix curve. Because of the way the curve is created, the

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Figure 2from http://www.geom.uiuc.edu/docs/forum/angtri/

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

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

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r

2!

2!

!r r

r

E C O A

B

D

Figure 3

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

α = m∠BEA = m∠DEO = 13m∠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 thatm∠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

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

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Figure 4from http://www.uwgb.edu/dutchs/pseudosc/

trisect.htm

(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

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 right triangle, 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 curves that can be used to easily trisect an angle, such as the

cardioid, used by Etienne Pascal (1588-1640), and the trisectrix of Maclaurin, used by Colin

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

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a2

a

G

F

EO A

B DC

Figure 5

Apollonius (250-175 B.C.) solved the trisection problem using conic sections, specifically a

hyperbola (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

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irreducible (unfactorable) cubic equations of the type we will see are at the heart of the proof that

a general angle trisection 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 polygons

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 polynomials

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

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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 cubic equation 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 cube root, 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

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with the trigonometric identity cos3θ = 4 cos3θ − 3cosθ , where θ = 20° and therefore

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

α = cos20° , substituting and simplifying, we now have 12 = 4α3 − 3α , or 8α 3 − 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 field of abstract algebra and

what came to be known as Galois theory. 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.

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

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