Lecture 6. Three Famous GeometricConstruction Problems

The first Athenian school: the Sophist School After the final defeat of the Persiansat Mycale in 479 B.C., Athens became a major city and commercial center in a league ofGreek cities. Athens became increasingly wealthy through a rise in trading. At the sametime, more scholars, including mathematicians from the Ionian school, Pythagoreans, andother schools, flocked to Athens. The Sophist school was the first Athenian school which hadlearned teachers in many areas: grammar, rhetoric, dialectics, eloquence, morals, geometry,astronomy, and philosophy. As Pythagoreans did before, one of their major goals was alsoto use mathematics to understand the universe.

Figure 6.1 Ancient Athens.

The three geometric problems In that period, many of the mathematical results ob-tained were by-products of efforts to solve the following three famous geometric constructionproblems.

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• Squaring the circle: to construct a square equal in area to a given circle.

• Doubling the cube: to construct the side of a cube whose volume is double that of acube of given edge.

• Trisecting an angle: to trisect any angle.

There was a rule attached these problems: They must be performed with a straightedgeand compass only. Why “straightedge and compass only?” According to the Greek view,the straight line and the circle were the basic figures, and the straightedge and compass aretheir physical analogues. As a result, constructions with these tools were preferable. Veryimportantly this was insisted by Plato (see Lecture 7).

The origin of the problem of squaring the circle The first Greek to be associatedwith this problem was Anaxagoras1, who worked on it while in prison. Anaxagoras createdreal troubles for himself and his friends when he proposed that the sun was a red hot stone.All the planets and stars were made of stone, he said. His belief may have been suggestedby the fall of a huge meteorite near his home when he was young. However, Anaxagoras’belief about the sun made him a prime target for his enemies so that he was brought totrial. It’s not certain what the result of the trial was (records are not preserved), but wedo know that while he was in jail, Anaxagoras made the first attempt to square the circle.This was the first time that such an effort had been made and preserved on record. Manypeople tried, claimed and failed on this problem.

Figure 6.2 Squaring the circle and Anaxagoras

The origin of the problem of doubling the cube According to legend, people livingin Delos, an island in the Mediterranean, were suffering from a plague. They consulted the

1Anaxagoras (c. 500 B.C.-428 B.C.) was a Pre-Socratic Greek philosopher famous for introducing thecosmological concept of Nous (mind), the ordering force. As mentioned in Chapter 3, he was in the IonianSchool.

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oracle, and the oracle responded that to stop the plague, they must double the size of theiraltar. The Athenians dutifully doubled each side of the altar, but the plague increased. Thenthe Delians realized that doubling the sides would not double the volume. They turned toPlato to get advice, who told them that the God of the oracle had not so answered becausehe wanted or needed a double altar, and he was not pleased with the Greeks for theirindifference to mathematics and their lack of respect for geometry.

This proved to be a most difficult problem indeed. It was solved in 350 B.C. due to theefforts of Menaechmus2(he not only used a straightedge and a compass, but also some othertools). By the way, the plague was finished several decades before Menaechmus’ solution.It is due to this legend that the problem is often known as the “Delian problem.”

Hippias of Elis, the quadratrix curve and the problem of trisecting angle Sinceany angle can be bisected, it was natural to consider a problem of trisection. One of themost famous attempts to this problem is due to Hippias of Elis.

Hippias, a leading Sophist, was born about 460 B.C. and was a contemporary of Socrates.In his attempts to trisect an angle, Hippias invented a new curve, which, unfortunately, isnot itself constructible with straightedge and compass3. His curve is called the quadratrixand is generated as follows.

2Menaechmus (380 - 320 BC) was an ancient Greek mathematician and geometer born in Alopeconnesus(in Turkey today), who was known for his friendship with the renowned philosopher Plato and for hisapparent discovery of conic sections and his solution to the then-long-standing problem of doubling the cubeusing the parabola and hyperbola.

3This was the first such curve discovered in the world at the time.

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Figure 6.3 Trisect an angle φ

Let AB rotate clockwise about A at a constant speed to the position AD. At the sametime let BC move downward parallel to itself at a uniform speed to AD. Suppose ABreaches AE as BC reaches B′C ′. Denote by E the intersection of AE and B′C ′. Then F isa typical point on the quadratrix BFG where G is the final point on the quadratrix.4

Suppose it needs time T to rotate AB to AD and it needs time t to rotate AE to AD.Since all movement are in constant speed, the time to move B′C ′ passing FH is also t. Then

the angle speed =π2

T=φ

t, and the speed =

AB

T=FH

t.

where φ = ∠FAD so thatφπ2

=FH

AB.

Similarly, let φ′ = ∠NAD, we obtain

φ′

π2

=F ′H

AB.

This impliesφ

φ′ =FH

F ′H.

4The curve is indeed given by the equation x = y tan−1 πy2a , where x = AH, y = FH and a = AB.

Morris Kline Mathematical Thought from Ancient to Modern Times, volume 1, New York Oxford, OxfordUniversity Press, 1972, p.39.

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Now if φ is a given angle, we can take the point F ′ such that FHF ′H

= 13. Then we take the

line F ′C ′′ parallel to AD so that this line intersects the quadratrix BFLG to get a point L.As above, we have

φ′ =1

3φ.

We have trisected the angle. As we pointed out, however, the trouble is that the quadra-trix BFLG cannot be constructed with straightedge and compass only.

Hippocrats of Chois and his result on the problem of squaring the circle Forthe problem of squaring the circle, the first person to come close to a real solution wasHippocrates, who proved that certain lunes (like a crescent moon, made from two circulararcs) could be squared.

Figure 6.4 Original road from 400 B.C.

Hippocrates of Chios was an ancient Greek mathematician (geometer) and astronomer,who lived 470-410 B.C.

Designated as Hippocrates of Chios to distinguish him from the better-known physicianof the same name, Hippocrates has been cited as the greatest mathematician of the fifthcentury B.C.

He was born on the isle of Chios, where he originally was a merchant. After somemisadventures, he went to Athens to prosecute pirates who had robbed him of all his goods.While waiting for his case to come to court, he attended lectures on mathematics andphilosophy. During this time, he came under the influence of a mathematical school basedon the principles of Pythagoras (580-500 B.C.). In the end, he stayed in Athens from about450 to 430 B.C. There he grew into a leading mathematician.

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Figure 6.5 Hippocrates’ discovery

Here is a proof for Hippocrates’ discovery (see Figure 6.5):

the area of the big disk

the area of the small disk= 2

and thus1/4 of the area of the big disk

1/2 of the area of the small disk= 1.

By subtracting the area of the common piece of both disks in the numerator and in thedenominator, one gets

the area of the triangle ABC

the area of the shaded lune= 1.

Namely, the area of the shaded lune part equals to the area of the triangle ∆ABC. Excitedabout this, Hippocrates hoped, by further modification, that it would lead to a solution ofthe squaring circle problem.

Many Greeks including Archimedes attempted to square the circle, but were not success-ful.

While the Greeks seemed to understand that squaring the circle was unsolvable usingcompass-and-straightedge techniques, they never proved it was so, and so the problem con-tinued to be attacked. Mathematicians in India, China, Arabia and medieval Europe allapproached the problem in their own ways in the centuries to follow. Even Leonardo daVinci attempted to square the circle, using mechanical methods instead of mathematicalones.

Even after more than 1000 years, the problem was still not been solved.

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Figure 6.6 Squaring the circle and Lindemann

Let r be the radius of the above circle and x the side of the above square. The problemof squaring the circle is to ask: given r, what is the x such that

x2 = πr2.

In 1882, the task was proven to be impossible. Lindemann5 proved that π is a tran-scendental number, rather than an algebraic irrational number; that is, π is not the rootof any polynomial with rational coefficients. It had been known for some decades beforethen that if π were transcendental then the construction would be impossible, but that π istranscendental was not proven until 1882.

Hippocrates of Chois and his result on the problem of doubling the cube Anotherachievement of Hippocrates was that he showed that “a cube can be doubled if two meanproportionals can be determined between a number and its double.”

s : x : y : 2s

with x2 = sy, y2 = 2sx. Then x3 = 2s3 and hence

x = 213 s.

The problem was reduced to: how to construct a segment of length 213 s of a given segment

of length s by a straightedge and a compass? Hippocrates’ work had a major influence onattempts to duplicate the cube, all efforts after this being directed towards the mean pro-portionals problem. The proof for the impossibility of doubling the cube and trisecting anangle was given by Pierre Wantzel(1814-1848) in 1837. The 23-year-old French mathemati-cian showed that the two problems of trisecting an angle and of solving a cubic equation

5Carl Louis Ferdinand von Lindemann (1852 - 1939) was a German mathematician.

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are equivalent. Moreover, he showed that only a very few cubic equations can be solvedusing the straightedge-and-compass method. He thus deduced that most angles cannot betrisected. He died at the early age of 34 due to overwork on mathematical theory.

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