The Three Unsolved Problems of Ancient Greece
The Three Unsolved Problems of Ancient Greece
The geometry of ancient Greece, as characterized by Euclid's famous book, the Elements, has formed the basis of much of modern mathematical thought. For example, the Greek insistence on strict methods of proof has survived to this day. The methods and theorems found in the Elements were taught to schoolchildren almost unchanged until the twentieth century. Even today, school geometry is essentially the same geometry as that composed by Euclid (c. 325-c. 265 b.c.) well over two millennia ago.
It became the practice in traditional Greek mathematics to accept geometrical constructions only if they could be performed with an unmarked straightedge and a compass. This custom is derived from the first three postulates of Euclid's Elements. A postulate is a statement that is accepted as true without proof. In the Elements, Euclid gives five postulates that are the starting points for the propositions or theorems given in the body of the book. The first three of these postulates address the construction of a straight line and a circle:
- A straight line can be drawn between any two points.
- A finite straight line can be extended indefinitely.
- A circle can be drawn with any center point and any line segment as a radius.
Although none of these postulates (or any others) refer directly to a straightedge or a compass, this tradition, usually attributed to Plato (427-347 b.c.), became an integral part of Greek geometry. The Greeks referred to constructing geometric figures using only a straightedge and a compass as the plane method.
Although the bulk of Greek geometry was constructed using plane methods, three problems defied solution by these methods for centuries. The ancient problems of squaring the circle (or quadrature of a circle), trisecting an angle, and doubling a cube (or duplicating a cube) have produced countless attempted solutions. These attempts have come from the great mathematicians in history, as well as from numerous amateurs and cranks. Each attempted solution had one thing in common; they all failed.
Although the Greeks were unable to prove the impossibility of solving the three problems by plane methods alone, they were certainly aware of the difficulties in solving each problem. Instead of despairing over their futile attempts to solve the problems using only straightedge and compass, the Greek mathematicians attacked the problems using other, less traditional means. In fact, we now know that none of the three problems can be solved using only a straightedge and a compass. In these failed attempts, however, we find important advances made in many areas of mathematics.
Squaring the Circle
To "square" a given geometric figure (such as a triangle or a circle) means to construct a square whose area is equal to the area of the given figure. Greek geometers succeeded in squaring figures bounded by straight lines, such as rectangles and triangles. Finding the exact areas of such figures was a relatively easy task. The next logical step, according to the Greek author Proclus (411-485), was to square regions bounded by nonlinear curves, the simplest such figure being the circle. The problem called "squaring the circle," often referred to as "quadrature of a circle," is to construct a square equal in area to a given circle; a simple-sounding explanation for a problem that has intrigued mathematicians for several millennia. Although the ancient Babylonians, Indians, and Chinese attempted the problem of squaring the circle, it was the work of the classical Greek mathematicians that made the problem famous for centuries.
The problem of squaring the circle takes on different meanings depending on how one approaches the solution. Beginning with the Greeks, many geometric methods were devised that allowed the construction of a square whose area was equal to a given circle. However, none of these methods accomplished the task at hand using the plane methods requiring only a straightedge and a compass. Of all the methods through the centuries that have been found to square the circle, none has involved the exclusive use of the straightedge and the compass. Rather, all required more sophisticated geometric methods such as the use of conic sections or complicated mechanical devices.
Therefore, when we say that the problem of squaring the circle is unsolved, what we mean is that it has never been solved using plane methods. In fact, it was not until late in the nineteenth century that it was proven that the problem of squaring the circle could not be solved by plane methods. In spite of the history of futility behind the problem, the work done in squaring the circle by means other than plane methods has proven to be fertile soil for the growth of mathematics. Additionally, as so often happens in science and mathematics, even the unsuccessful attempts at squaring the circle using only a straightedge and a compass have proven valuable to the development of mathematics.
We owe the credit for giving the problem of squaring the circle an important place in mathematics to the Greeks. The Greek mathematician Anaxagoras (499-428 b.c.) was among the first to attempt to solve the problem (while in prison, no less), but his work on squaring the circle has not survived to modern times. The first recorded progress made comes from two Greek mathematicians named Antiphon and Bryson. Antiphon (480-411 b.c.) approximated the area of a circle by inscribing first a square inside of the circle, then an octagon, then a 16-sided polygon, and so forth. Inscribing a figure inside a circle means to draw the figure, such as the square, so that the vertices just touch the inside of the circle. As the number of sides of the inscribed polygon doubled, the area of that polygon became closer to the area of the circle. Obviously, no matter how many sides the inscribed polygon was composed of, the area would always be smaller than that of the circle. Bryson (fl. 450 b.c.) improved Antiphon's approximation by circumscribing (drawing the figures around the outside of) the circle with polygons, thus guaranteeing the correct answer to be between the area of the inscribed polygon and the circumscribed polygon.
Hippocrates of Chios (c. 470-c. 410 b.c.) made what seemed to be important progress on the problem when he was able to construct a square equal in area to a region called a lune. Since a lune is a region bounded by the arcs of two circles, Hippocrates (not the same man as the physician for whom the Hippocratic Oath is named) seemed headed in the right direction. In fact, Hippocrates was able to solve the problem of squaring three kinds of lunes in his lifetime. Alas, his work did not lead to a successful solution to the problem of squaring a circle. In fact, the eighteenth-century Swiss mathematician Leonhard Euler (1707-1783) was the next to successfully square a new kind of lune, actually squaring two such figures that had eluded mathematicians since the time of Hippocrates. This, it turns out, marked the end of "lune-squaring," as it was eventually proven that only five such figures (the three found by Hippocrates plus the two by Euler) were squarable.
It is important to remember that the "unsolved" label attached to the problem of squaring the circle comes from the countless attempts throughout history to solve the problem using only a straightedge and a compass. However, many methods involving other geometric techniques have been used throughout history to successfully solve the problem of squaring the circle. For example, Greek mathematicians such as Dinostratus (c. 390-c. 320 b.c.) and Nicomedes (c. 280-c. 210 b.c.) used a curve called the quadratix to square the circle. The quadratix could not be constructed, however, using only a straightedge and compass.
Archimedes (287-212 b.c.), considered the greatest mathematician of ancient times, made several advances on the problem of squaring the circle. In his treatise On the Measurement of a Circle, Archimedes stated and proved a theorem that equated the area of a circle to that of a right triangle. It would seem that this had solved our problem, since it is a relatively simple matter to construct a square whose area is equal to any triangle. This is not the case, however. Archimedes had not solved the problem of squaring the circle, because his method did not actually allow the construction of a triangle equal in area to a circle using only straightedge and compass. Although to us this might seem like an insignificant technicality, it was very important to the Greeks. The figures must actually be constructable for the problem to be considered solved. Archimedes also used a curve that he invented, now called the spiral of Archimedes, to square the circle. Unfortunately, the spiral of Archimedes, like the quadratix, could not be constructed using only a straightedge and compass. Therefore, the problem of squaring the circle remained unsolved in the tradition of Greek geometry.
Squaring the circle became a popular problem wherever it was introduced in the world. There is evidence that the problem was attempted in India, China, and in the Arabic empires during the medieval period. European mathematicians of the Renaissance, including Leonardo da Vinci (1452-1519), were also interested in the problem of squaring the circle. Famous mathematicians such as Carl Friedrich Gauss (1777-1855), Gottfried Leibniz (1646-1716), and Isaac Newton (1642-1727), all of whom searched for better and more accurate methods for approximating the value of π, were in essence working on the problem of squaring the circle. The reason that the calculation of π is so closely tied to squaring the circle is that the problem essentially reduces to constructing a square whose side is the square root of π times the radius of the circle. Therefore, to square the circle requires that a line of length π be constructed using only straightedge and compass.
The number π is defined as the ratio of the circumference of a circle to its diameter. It was also understood by ancient mathematicians to be intimately related to the area of a circle, thus its importance to the problem of squaring a circle. Ancient cultures, long before classical Greek times, were concerned with the value of π. The famous Rhind Papyrus, an Egyptian relic (c.1650 b.c.), essentially gives an estimate of π at 3.16. Babylonian clay tablets, from roughly the same era, give an estimate equivalent to 3⅛ Archimedes found an excellent approximation for π (about 3 1/7) by inscribing and circumscribing polygons with 96 sides. Over two millennia later, the Indian mathematician Srinavasa Ramanujan (1887-1920) found several remarkably accurate approximations to π and his techniques continue to interest research mathematicians to the present day. Using modern computers, mathematicians have calculated π to millions and even billions of decimal places. (Remember, π is an irrational number, which means the decimal never ends or repeats.)
In modern times, the huge number of supposed solutions to the problem of squaring the circle, submitted to various scientific societies throughout Europe from amateurs and cranks, prompted the Paris Academy of Sciences and the Royal Society of London to cease considering the "solutions" sent to them by fame-seeking amateur mathematicians. But this was not before many respected would-be mathematicians tried their hand at solving the problem. For instance, in the seventeenth century the British political philosopher Thomas Hobbes (1588-1679) claimed to have solved the problem and to have revolutionized geometry in the process. Only after a protracted battle of words with the mathematician John Wallis (1616-1703) were all the fallacies in Hobbes's argument exposed and Hobbes himself became a castaway from the mainstream mathematical community. At one point, the quest to square the circle became such an obsession that the British mathematician Augustus De Morgan (1806-1871), coined the term "morbus cyclometricus," or circle-squaring disease.
Eventually, in the nineteenth century, the German mathematician Ferdinand von Lindemann (1852-1939) proved that π was a transcendental number (a transcendental number is one that cannot be the root of an algebraic equation with integer coefficients). This in turn confirmed that no straightedge and compass construction of a solution to the circle-squaring problem was possible. However, unconvinced amateurs continued to seek a solution.
The numerous attempts to square the circle, both with plane methods and using other geometric constructions, have led to many advances in mathematics. One of the most important modern advances that have grown at least partially from circle-squaring work is the calculus technique called integration. In fact, the technique of approximating the area of a circle by inscribing and circumscribing polygons around a circle, first used by the ancient Greeks, is a forerunner of integration.
Trisecting an Angle
The construction of regular polygons (polygons that have equal sides and equal angles) and the construction of regular solids (solids whose faces are equal regular polygons) was a traditional problem in Greek geometry. Some regular polygons, such as equilateral triangles and squares, and some regular solids, such as a cube, were relatively easy to construct. In fact, the Greeks were able to construct any regular polygon with an even number of sides, as well as some with an odd number of sides (such as a triangle and a pentagon), using only plane methods, in other words a straightedge and a compass. They could even construct complicated solids like regular hexagons (6 sides), octagons (8 sides), decagons (10 sides), dodecagons (12 sides), and pentadecagons (15 sides). In 1796 Carl Friedrich Gauss was able to construct a regular polygon with 17 sides using only a straightedge and a compass. But in order to construct a regular polygon with an arbitrary number of sides, it was required that an arbitrary angle be divided an arbitrary number of times. For instance, to construct a regular polygon with 9 sides requires that a 60° angle be trisected. The Greeks knew that any angle could be bisected using only a straightedge and a compass. Cutting an angle into equal thirds, or trisection, was another matter altogether. This was required to construct other regular polygons. Hence, trisection of an angle became an important problem in Greek geometry.
The Greeks found that certain angles could be trisected rather easily. The problem of trisecting a right angle is a relatively simple process. There are other angles that can be trisected with relative ease. In fact, Hippocrates of Chios, who we have already seen was instrumental in finding solutions to the problem of squaring the circle, found a relatively simple method to trisect any given angle. Unfortunately (at least for traditional Euclidean geometry), Hippocrates's method did not use only a straightedge and compass in its construction. Others succeeded in solving the problem, but never by using the plane methods that required only straightedge and compass.
The methods that the Greek mathematicians did find to trisect an angle involved curves such as conic sections or more complicated curves requiring mechanical devices to construct. The curve called the quadratix, which we have seen was used to square the circle, was also used to trisect an angle. Curves such as Nicomedes's (c. 280-c. 210 b.c.) conchoid and the spiral of Archimedes were also used to trisect an angle. The Greeks found several methods for trisecting an angle using curves called conic sections. A conic section is a curve obtained by the intersection of a cone and a plane. Examples of conic sections are circles, ellipses, parabolas, and hyperbolas. The method of solving the trisection problem by the use of conic sections lived on for many centuries. Even the great French mathematician and philosopher René Descartes (1596-1650) found a way to trisect an angle using a circle and a parabola. None of these curves, however, could be constructed using the restrictions required by traditional Greek geometry.
The problem of trisecting an arbitrary angle using only a straightedge and a compass generated interest for centuries just as the problem of squaring the circle had. If bisecting an arbitrary angle was so easy, and trisecting certain angles was also relatively simple, surely, thought some, the problem of trisecting an arbitrary angle could be solved. Mathematicians found that very close approximations to trisecting an angle could be made by continued bisections. In fact, if this process were repeated an infinite number of times, an exact trisection could be made. In addition to the straightedge and compass restrictions, however, the Greeks required that a process be accomplished in a finite number of steps to be valid. Continued bisection, then, did not represent an acceptable solution to the trisection problem.
Like the problem of squaring the circle, it had become evident to most trained mathematicians by the eighteenth century that a solution to the problem of trisection probably did not exist. François Viète (1540-1603), the mathematician credited with introducing systematic notation into algebra, warned in his lectures about the many flawed proofs from enthusiastic amateurs. In fact, in 1775 the Paris Academy of Sciences discontinued examination of angle trisection methods submitted by the public, much like they had done with solutions to the circle-squaring problem. It was not until 1837 that Pierre Wantzel (1814-1848) completed a proof that the problem was impossible using only a straightedge and a compass. Wantzel essentially showed that trisecting an angle could be reduced to solving a cubic equation. Since most cubic equations could not be solved with straightedge and compass, neither could the trisection problem. This put a stop to attempts by serious mathematicians to solve the problem, but unconvinced amateurs continue to seek fame by looking for methods to trisect an angle.
Doubling a Cube
For a cube with a given volume, can one construct another cube whose volume is double the original using only a straightedge and compass? This is the third problem of ancient Greek geometry. Like the problems of squaring the circle and trisecting an angle, the origin of the problem of doubling a cube (also referred to as duplicating a cube) is not certain. Two stories have come down from the Greeks concerning the roots of this problem. The first is that the oracle at Delos commanded that the altar in the temple (which was a cube) be doubled in order to save the Delians from a plague. After failing to solve the problem, the men from Delos questioned Plato as to how this might be done. Plato's response was that the command was actually a reproach from the gods for neglecting the study of geometry. Needless to say, the plague at Delos continued. The problem of doubling a cube is often referred to as the Delian problem after the citizens of Delos who suffered for their ignorance.
A second version of the origin of the cube-doubling problem relates that King Minos commanded that a tomb be erected for his son, Glaucus. After its completion, however, Minos was dissatisfied with its size, as its sides were only 100 feet (30.5 m) in length. He commanded that the cubic tomb be made twice as large by doubling each side of the tomb. Since the volume of a cube is the length multiplied by the width multiplied by the height, the volume of the original cube was
By doubling the length of each side, the new cube would have a volume given by
So if each side were doubled, the resulting volume would not be double the original, but eight times the original volume. Like the men of Delos, King Minos's subjects could not solve the problem of doubling the cube. Although both of these stories may contain as much myth as fact, the problem of doubling a cube using only a straightedge and a compass became important in Greek geometry.
Eratosthenes (276-194 b.c.) is credited with one of the first solutions to the problem of doubling a cube, using a mechanical instrument of his invention to construct the required cube. Interestingly, Plato is also credited with a mechanical solution to the problem, although he is said to have abhorred the use of mechanical devices in geometry. Archytas of Tarentum (c. 428-c. 350 b.c.) produced a remarkable construction of the problem, relying on the intersection of several three dimensional objects, including a cylinder, a cone, and a surface known as a tore. Eudoxus (408-355 b.c.), the inventor of an important mathematical technique called the method of exhaustion, is also said to have constructed a solution to the problem. In addition, Nicomedes (c. 280-c. 210 b.c.) used the same curve (conchoid) to solve the problem of doubling the cube as he had used to solve the problem of trisecting an angle.
Just as with the other two problems of Greek geometry, the problem of doubling a cube was solved using conic sections. Menaechmus (c. 380-320 b.c.) was able to find two solutions using the intersection of conic sections. In fact, it is said that Menaechmus discovered conic sections while attempting to solve the problem. Many other famous Greek mathematicians, including Apollonius, Heron, Philon, Diocles, Sporus, and Pappus, constructed their own solutions to the problem of doubling the cube. None, however, succeeded in solving the problem using the straightedge and compass alone.
Hippocrates of Chios, who was an important figure in the history of the problems of squaring the circle and trisecting an angle, also worked on the problem of doubling a cube. Hippocrates found that the problem of doubling a cube could be solved if the related problem of finding two mean proportionals between one line and its double was solved. This means, in modern notation, to find two values, x and y, such that a/x = x/y = y/2a. This leads to the equation x3 = 2a3, which means that a cube with side x has twice the volume of a cube with side a. Although Greek mathematicians were able to find ways to perform this construction, none met the requirements of the exclusive use of straightedge and compass.
Attempts to solve the problem of doubling the cube have led to many other important discoveries in mathematics, just as the same has happened with the other two problems of Greek geometry. Conic sections, discovered by Menaechmus while he was trying to solve the problem of doubling the cube, have been an extremely important part of mathematics throughout history. The Persian mathematician (and poet) Omar Khayyam (1048-1131) used the intersection of conic sections to solve cubic equations, a problem closely related to duplicating the cube.
There is a very good reason that neither the Greeks nor anyone else were ever able to find a solution to the problem of doubling the cube using plane methods; such a solution does not exist. In the sixteenth century, François Viète was able to show a relation between the solutions of cubic equations and the problems of duplication of a cube and trisection of an angle. René Descartes later showed that any cubic equation could be solved using a parabola and a circle, but not with a line and a circle (the beginning point of straightedge and compass constructions). The final nail in the cube-doublers coffin came from Pierre Wantzel in 1837. Wantzel proved that a geometric construction to double a cube using only straightedge and compass could not exist, just as he had proved that a similar construction for trisecting an angle was impossible.
The history of the three unsolved problems of Greek geometry is interesting in itself, but it is made even more interesting by the impact that these three problems have had on mathematics through the centuries. These problems, and related problems, were addressed by Arabic mathematicians of the medieval period such as Omar Khayyam, by great European mathematicians of the Renaissance and early modern period such as Leonardo da Vinci and René Descartes, and by modern mathematicians like Ramanujan. Usually when these problems were addressed, new advances in mathematics resulted. The great German mathematician Carl Friedrich Gauss was at least partially motivated by these problems in his work on the solutions of algebraic equations. Gauss asserted a connection between a certain type of equation, called a cyclotomic equation, and the construction of regular polygons. And this sort of construction was tied closely to our three Greek problems. In fact, Wantzel's proof of the impossibility of trisecting an angle or doubling a cube using plane methods was the culmination of the work started by Gauss.
The problems of squaring the circle, trisecting an angle, and doubling a cube are three of the most famous mathematical problems in history. They challenged the minds of the greatest mathematicians of ancient Greece and have interested mathematicians well into modern times. Their influence on mathematics through the ages makes these problems an important part of the history of man's search for answers to the questions of science.
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