Lecture 6. Three Famous Geometric Construction Problems

Home , Doubling the cube, Ferdinand von Lindemann, Menaechmus, Quadratrix, Squaring the circle

The first Athenian school: the Sophist School After the final defeat of the Persians at Mycale in 479 B.C., Athens became a major city and commercial center in a league of Greek cities. Athens became increasingly wealthy through a rise in trading. At the same time, more scholars, including mathematicians from the Ionian school, Pythagoreans, and other schools, flocked to Athens. The Sophist school was the first Athenian school which had learned teachers in many areas: grammar, rhetoric, dialectics, eloquence, morals, geometry, astronomy, and philosophy. As Pythagoreans did before, one of their major goals was also to 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 construction problems.

37 • 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 a cube of given edge. • Trisecting an angle: to trisect any angle.

There was a rule attached these problems: They must be performed with a straightedge and 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 are their physical analogues. As a result, constructions with these tools were preferable. Very importantly this was insisted by Plato (see Lecture 7).

The origin of the problem of squaring the circle The first Greek to be associated with this problem was Anaxagoras1, who worked on it while in prison. Anaxagoras created real 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 suggested by 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 to trial. It’s not certain what the result of the trial was (records are not preserved), but we do 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. Many people 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 living in 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 the cosmological concept of Nous (mind), the ordering force. As mentioned in Chapter 3, he was in the Ionian School.

38 oracle, and the oracle responded that to stop the plague, they must double the size of their altar. The Athenians dutifully doubled each side of the altar, but the plague increased. Then the Delians realized that doubling the sides would not double the volume. They turned to Plato to get advice, who told them that the God of the oracle had not so answered because he wanted or needed a double altar, and he was not pleased with the Greeks for their indiﬀerence 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 the efforts of Menaechmus2(he not only used a straightedge and a compass, but also some other tools). 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 Since any angle can be bisected, it was natural to consider a problem of trisection. One of the most 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, is not itself constructible with straightedge and compass3. His curve is called the quadratrix and 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 his apparent discovery of conic sections and his solution to the then-long-standing problem of doubling the cube using the parabola and hyperbola. 3This was the ﬁrst such curve discovered in the world at the time.

39 Figure 6.3 Trisect an angle φ

Let AB rotate clockwise about A at a constant speed to the position AD. At the same time let BC move downward parallel to itself at a uniform speed to AD. Suppose AB reaches AE as BC reaches B0C0. Denote by E the intersection of AE and B0C0. Then F is a 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 B0C0 passing FH is also t. Then π φ AB FH the angle speed = 2 = , and the speed = = . T t T t where φ = ∠F AD so that φ FH π = . 2 AB 0 Similarly, let φ = ∠NAD, we obtain φ0 F 0H π = . 2 AB This implies φ FH = . φ0 F 0H

4 −1 πy The curve is indeed given by the equation x = y tan 2a , where x = AH, y = FH and a = AB. Morris Kline Mathematical Thought from Ancient to Modern Times, volume 1, New York Oxford, Oxford University Press, 1972, p.39.

40 0 FH 1 Now if φ is a given angle, we can take the point F such that F 0H = 3 . Then we take the line F 0C00 parallel to AD so that this line intersects the quadratrix BF LG to get a point L. As above, we have 1 φ0 = φ. 3 We have trisected the angle. As we pointed out, however, the trouble is that the quadratrix BF LG cannot be constructed with straightedge and compass only.

Hippocrats of Chois and his result on the problem of squaring the circle For the problem of squaring the circle, the ﬁrst person to come close to a real solution was Hippocrates, who proved that certain lunes (like a crescent moon, made from two circular arcs) 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 physician of the same name, Hippocrates has been cited as the greatest mathematician of the ﬁfth century B.C.

He was born on the isle of Chios, where he originally was a merchant. After some misadventures, 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 and philosophy. During this time, he came under the inﬂuence of a mathematical school based on the principles of Pythagoras (580-500 B.C.). In the end, he stayed in Athens from about 450 to 430 B.C. There he grew into a leading mathematician.

41 Figure 6.5 Hippocrates’ discovery

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

the area of the big disk = 2 the area of the small disk and thus 1/4 of the area of the big disk = 1. 1/2 of the area of the small disk By subtracting the area of the common piece of both disks in the numerator and in the denominator, one gets the area of the triangle ABC = 1. the area of the shaded lune Namely, the area of the shaded lune part equals to the area of the triangle ∆ABC. Excited about this, Hippocrates hoped, by further modiﬁcation, that it would lead to a solution of the 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 using compass-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 all approached the problem in their own ways in the centuries to follow. Even Leonardo da Vinci attempted to square the circle, using mechanical methods instead of mathematical ones.

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

42 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 problem of 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 transcendental number, rather than an algebraic irrational number; that is, π is not the root of any polynomial with rational coeﬃcients. It had been known for some decades before then that if π were transcendental then the construction would be impossible, but that π is transcendental was not proven until 1882.

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

s : x : y : 2s with x2 = sy, y2 = 2sx. Then x3 = 2s3 and hence

1 x = 2 3 s.

1 The problem was reduced to: how to construct a segment of length 2 3 s of a given segment of length s by a straightedge and a compass? Hippocrates’ work had a major inﬂuence on attempts to duplicate the cube, all eﬀorts after this being directed towards the mean proportionals problem. The proof for the impossibility of doubling the cube and trisecting an angle was given by Pierre Wantzel(1814-1848) in 1837. The 23-year-old French mathematician 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.

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