Mathematics for the liberal arts (2013)

Part I. MATHEMATICS IN HISTORY

Chapter 1. THE ANCIENT ROOTS OF MATHEMATICS

1.3 Early Greek Mathematics: The First Theorists

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stem perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.

BERTRAND RUSSELL (1872–1970)

Modem mathematics is distinguished not only by its techniques and results, but by its logical structure. A mathematician does not merely discover formulas, but proves theorems, starting from well-understood assumptions and definitions. This logical structure is the invention of the Greeks, surely one of the greatest inventions in human history. They also applied this new invention, especially in geometry, to produce some very sophisticated mathematics.

More than only mathematics, much of Western intellectual tradition dates to classical Greece. Mathematics was part of a larger philosophical movement in which the Greeks attempted to understand the world in rational, not mythical or religious, ways. This movement extended to the political and social spheres as well. The idea of democracy is usually dated to 5th century BCE Athens.

To understand the enormity of the Greek accomplishment, and appreciate that this advance was not inevitable, consider that most of recorded history occurred before classical Greek civilization.

Historians have learned a great deal about Greek mathematics. However, unlike the case in Egypt and especially Mesopotamia, none of this knowledge is first-hand. Papyrus did not last long in the moist climate of the Greek world, so what we have is copies of copies of Greek texts.

Greece before 600 BCE

The Middle East and the Mediterranean Sea.¹

Unlike Mesopotamia and Egypt, Greece was not a great agricultural center. The land was too mountainous. Greece did have the sea, however. Very little of the mainland is far from the water, and there are many Greek islands in the Aegean Sea. Hence the Greeks were a seafaring people. By the 11th century BCE (1100–1000), they had spread across the Aegean to Ionia, on the shores of what is now Turkey.

By the middle of the 8th century BCE, the Greeks entered a period of expansion, physically and culturally. This was the time of Homer, author of the Iliad and the Odyssey. In 750 BCE the Greeks established a colony near modern-day Naples. Over the next three centuries, they set up many more colonies around the Mediterranean, in southern Italy, Sicily, Spain, and northern Africa.

The Greeks were borrowers. During this period, they adopted papyrus from Egypt, and adapted the Phoenician alphabet. (Our word “alphabet” comes from the first two Greek letters, alpha and beta.) As we will see, the early mathematicians were also travelers, and learned much from Mesopotamia and Egypt.

The early Greeks did not have great empires. The basic political unit was the city-state, the polis, the origin of our word “politics.” There were many forms, from democracies to monarchies, but they were all distinguished by a respect for law. Another notable feature of Greek public life was debate and argumentation. But public life was not shared by all; slavery was common.

Numeration

The Greeks had a variety of numeral systems. The best known, which appeared in the 6th century BCE and was standard by the 3rd century BCE, was the Ionic system. It had 27 symbols: the 24 letters of the alphabet plus three others. These symbols represented the numbers 1, 2, 3, …, 9, 10, 20, 30, …, 90, 100, 200, 300, …, 900. For example, γ was 3 and μ was 40, so 43 would be written μγ.

There were various ways of writing larger numbers, usually involving adding an extra symbol on top of, or next to, the existing symbols. A similar system handled fractions. The fraction 1/3, for example, would be written (since α = 1).

As you can see, this system was closer to the Egyptian system than to the superior Mesopotamian system. The development of Greek mathematics was not held back by the limitations of this system, since Greek mathematics did not rely heavily on numerical calculations.

Ionia, Miletus

From Thales to the death of Alexander the Great. All dates are BCE.

Classical Greek civilization began in the 6th century BCE, in Ionia, the Greek colony located on the western shores of modem Anatolia in Turkey and some nearby islands. Unlike the Greek mainland, Ionia enjoyed good land for agriculture. Miletus, its greatest city, was an important trading center on the Meander River (which gave us our word “meander”). It was connected to Mesopotamia via overland trading routes. It was also a seaport whose ships traded with Egypt. Thus Miletus had access to the knowledge of these great civilizations.

Some time in the 6th century BCE, Western philosophy was bom, and with it theoretical mathematics. In ancient times, philosophy was not distinct from science. The goal was to understand the world. The explanations that the new philosophers gave were natural, as opposed to supernatural. Religion was not abandoned; rather it was no longer considered adequate to explain the natural world by means of the actions of capricious gods in myths. In mathematics, it was not enough to empirically demonstrate results. One should argue why they were true. This was the beginning of formal deductive reasoning.

No one knows why this major intellectual development started in this place at this time. Certainly, it made a difference that Miletus had access to the major intellectual traditions of the Near East. Perhaps the new philosophy was an attempt to explain the contradictions in these traditions. It has also been suggested that the Greek habit of public debate fostered the notion that all assertions should be justified by careful argument.

Thales of Miletus (c. 625–547 BCE)

Thales was credited with beginning the new philosophy. Very little is known about his life. He was from Miletus, probably bom into an aristocratic family. He was said to be a merchant, and to have traveled to both Egypt and Mesopotamia. He was famous as a statesman, astronomer, and engineer, as well as mathematician and philosopher.

Many stories have been told of Thales, but all of them date from well after his death, and most are no doubt apocryphal. There is a famous story, perhaps the first absent-minded professor story, of how, intent on studying the heavens, he fell into a ditch. On the other hand, there is another tale that when he was criticized for being impractical, he shrewdly cornered the market on olive presses, thereby making a fortune. Nothing he wrote has been preserved, so we have mostly the legends from later times.

It is said of Thales that, on a trip to Egypt, he impressed his hosts by demonstrating a method to determine the height of a pyramid by measuring shadows. Here is how he did it. Let A be the top of the pyramid, C the tip of the pyramid’s shadow, and angle ABC be a right angle. (See Figure 1.5.) Suppose that a staff is held perpendicular to the ground. Let D be its tip, E its base, and F the tip of its shadow. Then the triangles ABC and DEF are similar, which means that the ratios of corresponding sides are equal. In particular, if the height of pyramid and staff are h₁ and h₂, respectively, and the lengths of the shadows are s₁ and s₂, then h1/h₂ = s₁/s₂. Since s₁, and s₂ can easily be measured, h₁ can be computed.

Figure 1.5 Measuring a pyramid’s height by its shadow.

For example, if the staff is 6 feet, its shadow is 8 feet, and the pyramid’s shadow is 640 feet, then h₁/6 = 640/8. Solving this, we get h₁ = 480 feet.

Eudemus of Rhodes wrote a book on early Greek geometers in the 4th century BCE. No copies of it remain, but a short bit of it was included in a book of the commentator Proclus (411–485 CE). Thales is credited with five theorems. They are fairly basic; one of them is that a circle is bisected by its diameter. Such a result would surely be known by anyone who had a practical need for it. Thales, however, was said to have been the first to seek a logical, not merely practical, basis for such theorems.

The most famous student of Thales was Anaximander. As with Thales, we know little of Anaximander. He was credited with introducing into Greece, from Mesopotamia, the gnomon, the center of the sundial, which casts the shadow. Anaximander also contributed to geography, drawing a circular map of the world.

In the last half of the 6th century BCE, due largely to the expansion of the Persian empire, Ionia declined as a cultural center. The center of the new Greek philosophy shifted west to Magna Graecia (“greater Greece” in Latin), the Greek colonies in southern Italy.

The Pythagoreans

In the 6th century BCE, an important group of thinkers emerged in Magna Graecia, centered around Pythagoras.

Pythagoras of Samos (c. 572–497 BCE)

Pythagoras was from the Ionian island of Samos. As with Thales, our knowledge of Pythagoras has been pieced together from reports written long after his death. He was said to have studied in Miletus, perhaps with Anaximander. He also traveled to Egypt, and reportedly spent seven years in Babylon, after which he returned to Samos. He was forced to leave Samos around 530 BCE, and settled in Crotona, a Greek seaport in southern Italy. It was there that he founded his society, known as the Pythagoreans, which also spread to neighboring cities, and was for a time very influential. Around 500 he was forced to move again, to the neighboring town of Metapontum, where he died.

Pythagoras left no writings, and his followers had a habit of attributing all of the group’s discoveries to him. This practice, and the secrecy surrounding his organization, make it difficult to sort out his individual mathematical accomplishments. He certainly was a leading religious, philosophical, and political figure of his time, but perhaps his greatest accomplishment was founding the society that left such an important mark on our intellectual history.

The Pythagoreans were a society of a few hundred aristocrats. It is sometimes called a brotherhood, but there is one story that its original members included dozens of women. It was certainly selective and hierarchical. Members were divided into the akousmatikoi—listeners—who were expected to learn the master’s teachings, and the mathematikoi, who could develop the teachings. The mathematikoi were among the first pure mathematicians. Our words “mathematics” and “mathematician” derive from mathematikoi (which in turn was based on mathesis, learning).

Religiously, the Pythagoreans practiced an asceticism, were probably vegetarian, and eschewed wine. They believed in the transmigration and reincarnation of souls, where the soul is reborn in another body after death.

Politically, the Pythagoreans were anti-democratic. In fact, democratic forces attacked them and burned their buildings in about 450. Their political influence waned thereafter, although the sect continued beyond that time and continued to produce important mathematics.

What distinguished the Pythagoreans from other mystery cults of the time was their philosophy that numbers, by which they meant counting numbers, were the foundation of the universe. An example of this was their discovery of the connection between musical harmonies and simple ratios. If one measures two strings on a lyre tuned an octave apart, their lengths are in the ratio 2:1. The musical interval of a fifth is associated with the ratio 3 : 2, a fourth with 4 : 3, and so on.

The Pythagoreans thought that these musical ratios were reflected in the heavens as well. They theorized that the planets, including the Sun and Moon and a couple of extra ones necessary to produce the number ten, traveled around on invisible spheres. These spheres were spaced according to the harmonic ratios they discovered, and produced sounds, the “music of the spheres.”

The Pythagoreans dealt in numerology as well as number theory. The number 1 is associated with reason, 3 with harmony, odd numbers are masculine, even numbers are feminine. Ten, the sum of the first four numbers, was magical. They had the notion of “perfect” numbers, numbers with the property that the sum of their proper factors equals the number itself. The smallest such number is 6, as 6 = 1 + 2 + 3. This may seem an odd notion to us, endowing numbers with human characteristics, but it inspired some lovely mathematics. Perfect numbers turn out to be associated with a family of prime numbers that is still studied. (See Section 5.7.)

Another area of interest to the Pythagoreans was the study of figurate, or polygonal, numbers, obtained from drawing regular figures with dots. Figure 1.6 shows the first few triangular (bowling pin) numbers. Note that the nth triangular number is the sum of the first n counting numbers. They attached special significance to the tetractys, the figure with 10 = 1 + 2 + 3 + 4 dots. (See the prayer on p. 30.)

Figure 1.6 Triangular numbers.

A type of figurate numbers with which you may be familiar are the square numbers, which are numbers of the form n². As the nth triangular number is the sum of the first n counting numbers, the nth square is the sum of the first nodd numbers. For example, 4² = 1 + 3 + 5 + 7. If you study Figure 1.7, you can see a geometric demonstration of this. The Pythagoreans also studied rectangular and pentagonal numbers. Figurate numbers have continued to fascinate mathematicians into modem times.

Figure 1.7 Square numbers.

Pythagoras is best known for the Pythagorean Theorem, that the area of the square on the hypotenuse of a right triangle is equal to the sum of the areas of the squares on the legs.

With the angles of the triangle labeled A, B, and C (the right angle), and the side lengths labeled a, b and c, as in Figure 1.8, the Pythagorean Theorem says

Figure 1.8 A labeled right triangle.

Figure 1.9 illustrates the Pythagorean Theorem. The area of the darker shaded square is equal to the sum of the areas of the two lighter shaded squares.

Figure 1.9 The Pythagorean Theorem (the area of the darker square equals the sum of the areas of the two lighter squares).

We have remarked that this theorem was known at least a thousand years before Pythagoras. But Pythagoras gets the credit for the theorem because his school was the first to provide a proof that covers all possible right triangles.

We offer the simplest proof of the Pythagorean Theorem that we know of. Let a right triangle be given. As in Figure 1.10, four copies of the right triangle are arranged inside a square whose side length is the sum of the two legs of the triangle. The shaded area is the area outside the four triangles and inside the large square. In the square on the left, the shaded area is the square on the hypotenuse. In the square on the right, the shaded area is the union of the squares on the legs of the triangle. Since the amount of shaded area doesn’t change when we move the four triangles, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the two legs.

Figure 1.10 Proof of the Pythagorean Theorem.

The Pythagoreans also studied Pythagorean triples (as did the Babylonians earlier), which are three positive integers a, b, and c such that a² + b² = c². They discovered an infinite family of such triples, namely,

where n is any positive integer. For example, if n = 2, we get

You can check that 5² + 12² = 13². To see that these are always Pythagorean triples:

and

Thus a² + b² = c².

The Pythagoreans proved that the sum of the angles in a triangle equals 180 degrees. Their proof is based on the equality of alternate angles, e.g., angles α and α′ in Figure 1.11, where the two horizontal lines are parallel. This equality follows from the figure’s symmetry; imagine rotating by 180°, exchanging the parallel lines and leaving the diagonal line unchanged. This rotation also exchanges the angles α and α′.

Now let ABC be a triangle, and draw a line through B parallel to AC, as in Figure 1.12. Note that the angles α′, β, and γ′ together make a straight line, so α′ + β + γ′ = 180°. Using our theorem on alternate angles, α = α′ and γ = γ′. If we substitute these, we obtain α + β + γ = 180°. In other words, the sum of the angles in the triangle ABC is 180°.

One of the most important Pythagorean discoveries was the existence of irrational numbers. A rational number is one that can be expressed as a ratio of integers. For example, 2/3 is rational, as is 5 = 5/1. It is a natural assumption that all numbers are rational, but this turns out not to be the case. For example, , the length of a diagonal from a square of side 1, turns out to be irrational, that is, not expressible as the ratio of two integers. This discovery, reportedly made by the Pythagorean Hippasus of Metapontum, certainly complicated theoretical mathematics. One can only guess at its effect on the Pythagoreans, whose philosophy was so dependent on whole numbers and their ratios. Legend has it that Hippasus made the discovery while at sea, and that the others, appalled by the idea, threw him overboard. More on this important theorem can be found in Section 5.3.

Figure 1.11 Equality of alternate angles.

Figure 1.12 The sum of angles in a triangle is 180°.

The Pythagoreans also had a profound effect on education. They identified four basic areas of education: arithmetic (which essentially meant the theory of numbers), geometry, music, and astronomy. These four were later extolled by Plato and Aristotle, and they loomed large into the Middle Ages, where they were known as the quadrivium.

Archytas of Tarentum (c. 438–347 BCE)

The Pythagorean Archytas was a politician and mathematician in southern Italy one hundred years after the death of Pythagoras. He was a number theorist and a geometer, devising a clever mechanical solution to the Delian problem (see “Three Construction Problems” below). He is also credited with devising the educational system mentioned above, which became the quadrivium.

Archytas had two vastly influential students, Eudoxus and Plato.

Finally, lest we forget that these great mathematicians lived in a very different age, we end with one of their prayers.

Bless us, divine number, thou who generated gods and men! O holy, holy Tetractys, thou that containest the root and source of the eternally flowing creation! For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-bounding, the first-bom, the never-swerving, the never-tiring holy ten, the keyholder of all.

PYTHAGOREAN PRAYER

Elea

Another important center of philosophy in Magna Graecia, not far from Crotona and Metapontum, was the city of Elea.

Parmenides (c. 515–450 BCE)

Parmenides, the founder of the Eleatic school, was bom in Elea, into a wealthy family. We know little of his life. Philosophically, he was influenced by the poet Xenophanes, and was perhaps his pupil. Since Xenophanes was from Ionia, Parmenides was aware of the Ionian philosophers. It seems likely, given their proximity, that the Pythagoreans were also known to him. Plato writes that Parmenides visited Athens in 450, when he was an old man, and there met the young Socrates.

The only known work of Parmenides was a philosophical poem titled On Nature. Only fragments of it remain.

When Parmenides sang this poem (yes, he sang his philosophy), he entreated his listeners to ignore their senses, and instead follow pure reason. In particular, he insisted that movement and change were illusory; reality is unchanging, eternal.

The Eleatic school was important for its insistence on logic in philosophy. One did not merely assert beliefs, but needed to make formal, logically rigorous arguments in support of them. Members of the school were fond of the type of argument called reductio ad absurdum, in which a proposition is proved by demonstrating that its denial leads to a logical absurdity.

In mathematics, a reductio ad absurdum proof is often called a proof by contradiction. Here is a simple example, a proof that the number of integers is infinite. We begin by supposing the opposite, that the number of integers is finite. In this case, there must be a largest integer, say n. But then consider n + 1. It is clearly an integer, and it is larger than n, which gives us a contradiction (logical absurdity). Since this follows logically from assuming that the number of integers is finite, there must be an infinite number of integers.

Zeno (c. 490–425 BCE)

At this point, you will not be surprised to learn that our knowledge of Zeno’s life is sketchy. Most of what we know is from Plato’s book Parmenides. Zeno was bom in Elea, and was a student of Parmenides. His importance rests on a book of paradoxes he wrote, in defense of Parmenides’ philosophy. We do not even have copies of this book, only commentaries on parts of it written after his death.

Zeno’s book was said to contain 40 paradoxes, of which nine have survived, although only as rephrased by other authors. Here are three of the most famous.

The Dichotomy: A runner is running toward a goal. In order to reach this goal, he must first reach the halfway point. He then must go halfway from that point to the goal, and so on. At every point, he must still traverse half the distance to the goal, so can never arrive.

Achilles and the Tortoise: Achilles, the fastest runner in the world, is chasing after a tortoise. In order to catch the tortoise, he must reach the point at which the tortoise started. (We assume that both are running in a straight line.) But when Achilles arrives at that point, the tortoise, slow as he may be, has moved on. So Achilles must then reach the new point at which the tortoise has arrived. This process continues; when Achilles reaches the point at which the tortoise is, the tortoise is no longer there. So Achilles can never catch the tortoise.

The Arrow: Consider a moving arrow at a particular instant of time. At that instant, the arrow occupies a particular place. But the place does not move, therefore the arrow is motionless. Thus motion is impossible.

A paradox, in English, can mean an argument or assertion that defies intuition. However, the sense in which Zeno’s arguments are paradoxes is more specific. He presented arguments which ended in absurd conclusions, but the important point is that it was not clear exactly where the argument went wrong. The issue is not whether the conclusions are correct, but what exactly is wrong with the reasoning. (The word paradox comes from the Greek para, alongside or beyond, and doxa, opinion.)

Since we do not have Zeno’s original words, we must infer his intent in presenting these paradoxes. One of his goals was probably to support his teacher’s assertion that motion was an illusion. Another perhaps was to probe the nature of space and time. Are they infinitely divisible?

Zeno’s paradoxes have inspired philosophers and mathematicians for millennia, because they force us to confront the thorny issues of infinity and continuity. These paradoxes are now considered solved, but the solution took the development of a sophisticated theory of limits, and was not completed until late in the 19th century, more than 2300 years after the paradoxes were first posed.

Democritus (c. 460–370 BCE)

Democritus was a native of Abdera, in Greece. He came from a wealthy family, was said to have traveled to many countries, and talked to many scholars before returning to Abdera.

Although not a resident of Elea, he was an academic grandson of Zeno, being a student of Leucippus who was in turn a student of Zeno. He wrote many works on mathematics, but none survive.

Democritus is most famous for developing (with Leucippus) the atomic theory. The word “atom” comes from the Greek an (not) and temnein (to cut). This is the essence of the atomic theory, that the world consists of atoms that are indivisible. This theory was in part a response to Zeno’s paradoxes.

Archimedes also gave Democritus credit for stating, but not proving, that the volume of a cone is one-third that of its related cylinder, and the volume of a pyramid is one-third that of its related prism. (See Figures 1.13 and 1.14.) The first proofs were given by Eudoxus, about 50 years after Democritus.

Figure 1.13 A cone and its cylinder.

Figure 1.14 A pyramid and its prism.

There is also a tantalizing report by Plutarch (c. 100 CE) that Democritus studied thin sections cut from a cone by planes parallel to its base. This is an idea pursued fruitfully by Archimedes, and is an important part of the integral calculus developed in the 17th century.

Athens

By the middle of the 5th century, and until the late 4th century BCE, Athens was the most important center of Greek mathematics. It was the largest of the Greek city-states; we don’t have good data, but estimates place its peak population at 300,000.

Athens was an important Mediterranean trading center. Its ships carried wine and olive oil, marble and silver. It had to import much of its grain. The economy of Athens was heavily dependent on slavery. Some scholars estimate that one-third of the population were slaves.

The political power of Athens waxed and waned in this period, but throughout it remained an important cultural center. It produced some of the most famous art, architecture, theater, science, and philosophy of antiquity. People still go to Athens to admire the Parthenon, and still stage productions of the plays of Aeschylus, Sophocles, Euripides, and Aristophanes. Its most famous philosophers were Socrates, Plato, and Aristotle.

In mathematics, Athenian scholars were responsible for refining the logical structure, overcoming the problem of irrational numbers, and developing theoretical geometry. They also established important institutions, where professional philosophers and mathematicians could flourish.

Three Construction Problems

Among the most important problems in ancient Greece, and beyond, were these three.

1. Quadrature (or squaring) of the circle: Construct a square whose area is the same as that of a given circle.

2. Duplication of the cube (The Delian Problem): Construct a cube whose volume is twice that of a given cube.

3. Trisection of an angle: Divide a given angle into three equal parts.

Obtaining the quadrature of a figure means constructing a square with the same area. The quadrature of a circle is in a class of problems that concern studying curved figures using simpler, straight-edged ones. Mathematicians had limited success with these problems until the advent of the calculus in the 17th century.

To duplicate a cube of side a, we need another cube of side . For then the volume of the cube of side b is , twice the volume of the cube of side a. So the difficulty is geometrically representing .

What it means to “construct” a figure has been subject to different interpretations, the most famous being the use only of a straightedge and compass. The straightedge allows one to draw a straight line; the compass allows one to draw a circle of given center and radius.

Many mathematicians have worked on these problems, inventing important areas of mathematics in an attempt to solve them. The problems have provided inspiration far beyond ancient Greece. They were finally solved, for straightedge and compass, only in the 19th century, when it was shown that all three are impossible. (We should mention that Pappus, in the 4th century CE, asserted this impossibility, but without proof.)

Hippocrates of Chios (c. 470–410 BCE)

Not to be confused with the physician Hippocrates of Cos (of Hippocratic Oath fame), this Hippocrates was bom on the Ionian island of Chios, not far from Pythagoras’ birthplace of Samos. Early on, Hippocrates was a merchant, but, after setbacks in his business, he made his way to Athens where he became one of the foremost scholars.

Hippocrates was the leading geometer of his time and wrote an influential text on geometry, Elements of Geometry, which has been almost entirely lost. He taught mathematics, being one of the first to make his living that way.

Hippocrates worked on at least the first two of the three construction problems. His advances were typical in the history of difficult mathematical problems, reducing the unsolved problem to another that may be easier to solve. In the case of the quadrature of the circle, he studied a type of intersection of circular arcs called a lune. He showed that if one could always square the lune, then one could square the circle. He further showed how to square a particular type of lune. He was not able to complete this program, however, being unable to square an arbitrary lune.

Similarly, Hippocrates reduced the Delian problem (doubling the cube) to another problem in two dimensions, instead of three. Specifically, he reduced the problem of doubling the cube of side a to that of finding two mean proportionals between a and 2a, that is, numbers x and y such that a : x = x : y = y : 2a, where a : x, for example, is the ratio of a to x.

Although Hippocrates’ text has been lost, historians can make informed guesses about its content based on fragments of his writings included in later works. His work may have been the first to array geometric theorems in a logical sequence, from the simplest to the more advanced. The logic of his surviving proofs is not perfect, but it does demonstrate a sophistication well beyond that of the scholars of a century earlier. One area that had yet to be developed is a system of axioms, or assumptions, upon which to build a geometry.

Plato and His School

Plato (c. 429–347 BCE)

Plato was a son of Athenian aristocrats. When he was young, he became a student of Socrates, the stonemason turned philosopher. After Socrates was executed for irreverence in 399, Plato left Athens, reportedly traveled around Greece, to Egypt, and to Tarentum in Magna Graecia, where he studied with Archytas, the Pythagorean.

In 388 Plato returned to Athens, where he taught, wrote some of the most influential philosophical works ever, and founded his famous Academy. Except for two trips to Syracuse in Sicily, where the Pythagoreans were ensconced, Plato lived the rest of his life in Athens.

Plato was probably not much of a mathematician himself, being more interested in ethics. He had considerable influence on mathematics, however, in two important ways. The first was his philosophy. He believed that the world of the senses was imperfect, that what we experience is but a shadow of what he called “forms” or “ideas.” For example, we can draw a circle but it is merely an imperfect representation of the Circle idea. Many mathematicians, though by no means all, subscribe to a form of Platonism that claims that mathematical objects have a real existence, independent of us. All that mathematicians do is study what is already out there. (Others reject this, believing that mathematics is an invention of humans, perhaps only a game we play.)

Plato’s other major contribution to mathematics, and to learning in general, was his founding of his school, the Academy. The name, which gave us our word “academic,” derived from the name of the site where the school was built. It was founded around 387 BCE.

Above the entrance gate of the Academy was inscribed “Let no man ignorant of geometry enter.” The school’s curriculum was based on the quadrivium—number theory, plane geometry, music, and astronomy—together with solid (3-dimensional) geometry. After the completion of these studies, the best students went on to study dialectics, a method of critical, persistent questioning, which Plato considered the way to arrive at truth.

Plato, very much the aristocrat, viewed the Academy as an institution to educate the ruling class. Its mathematics was therefore theoretical, not practical, with the exception of military applications. Interestingly, the word “school” derives from the Greek skhole, which means “leisure.”

The Academy was more than a school; the best scholars from the Greek world came to Athens to study and teach at the school. The closest modem equivalent is the research university. Throughout most of the 4th century BCE, until the rise of the Museum in Alexandria, the Academy was the home of the cream of Greek scholarship. Even after that, it remained an important center of learning. It was finally closed, in 529 CE, by the Christian emperor Justinian, who viewed it as a pagan institution. It thus lasted more than 900 years. This is roughly the age of the oldest current European university, the University of Bologna, founded in 1088.

Eudoxus’ Theory of Proportions

Eudoxus (c. 408–355 BCE)

Eudoxus was the most illustrious astronomer and mathematician of his time, and is generally considered to be the second best ancient mathematician, after Archimedes. He was bom in the Ionian city of Cnidus. As a young man he traveled to Tarentum, in Sicily, where he studied both mathematics, with Archytas, and medicine. He then spent some months studying with Plato’s circle in Athens. He was apparently too poor to live in Athens proper, so he lived in nearby Piraeus, and walked seven miles daily each way to the Academy. After Athens, he returned for a while to Cnidus.

Later, Eudoxus traveled to Egypt, where he worked primarily on astronomy. After Egypt, he returned to Ionia, specifically Cyzicus, where he founded a school and wrote his greatest astronomical works. He returned for a time to Athens, with some of his students, working there with Plato, Aristotle, and others. Finally, he returned to Cnidus, where he helped write a constitution for their new democracy, founded an observatory, taught, and practiced medicine and astronomy. He died at age 53.

Eudoxus’ greatest mathematical contribution was his theory of proportions. To understand its importance, recall that the Pythagoreans had proved that the square root of 2 is irrational, that is, not representable as the ratio of two integers. The difficulty this presented to mathematicians was in making precise arguments about irrational numbers, and about geometric figures whose magnitudes might be irrational.

Modern mathematicians get around this difficulty by approximating an irrational number by a sequence of rational numbers, then taking a limit. Eudoxus did something similar, but a bit more complicated.

To start with, Eudoxus’ theory did not deal with numbers as such, but with magnitudes, e.g., lengths or areas. The key to dealing with magnitudes was knowing how to compare them. Here is how he did it, as it appears in a definition from Euclid’s Elements.

Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equal multiples whatever are taken of the first and third, and any equal multiples whatever of the second and fourth, the former multiples alike exceed, are alike equal to, or alike fall short of, the latter multiples respectively taken in corresponding order.

Here “multiple” means integer multiple. Let us translate this into more familiar terms. If we represent the ratio a to b as a : b, the question is when we have equality of two ratios, say a : b = c : d. The above definition says that we have equality provided the following three conditions are met, for every choice of positive integers m and n.

1. If ma < nb then mc < nd.

2. If ma = nb then mc = nd.

3. If ma > nb then mc > nd.

It is understandable if this definition leaves you underwhelmed. It is difficult to fully appreciate its power unless you see it used in proofs, an exercise that is beyond this text. Mathematicians, however, were not able to replace this treatment of irrationals with anything of equal precision and usefulness until late in the 19th century.

Eudoxus also made rigorous a method called exhaustion, earlier invented by Antiphon, which is a limiting procedure to compute areas and volumes. He used the method of exhaustion to prove a result stated by Hippocrates, that the ratio of the areas of two circles is proportional to the square of the ratio of their diameters, or in modem terms, A = kd², where A is the area of a circle, d its diameter, and k some fixed constant of proportionality.

The method used successive approximations of a circle by polygons (see Figure 1.15). As the number of sides of the polygons grows, they “exhaust” the area of the circle. Combining this with Eudoxus’ definition of equality of ratios, and some careful arguments, leads to the desired proof.

Figure 1.15 Approximating the area of a circle.

One effect of Eudoxus’ subtle theory was to reinforce the Greek preference for geometry over algebra, a preference that influenced the course of mathematics for the next two thousand years. Greeks did solve some algebraic problems, usually by converting them first to geometric ones.

Finally, we note that Eudoxus introduced the use of spherical geometry in astronomy. His astronomical theory had stars and planets rotating on spheres centered at the Earth. Although this model wasn’t accurate, it was sophisticated for its time and was immortalized by being included (in a modified form) in Aristotle’s supremely influential works.

Logic

Aristotle (384–322 BCE)

Aristotle was from the Ionian colony of Stagira. His father was physician to the kings of neighboring Macedon. When he was seventeen or eighteen, Aristotle came to Athens to study at the Academy. He stayed on as a scholar until Plato’s death in 347 BCE, after which he left Athens.

In 342 Aristotle became the tutor to the young prince Alexander of Macedon, later called Alexander the Great. He stayed as advisor until 335, when Macedonia took control of Athens. In that year, Aristotle returned to Athens and set up his own school, the Lyceum. He remained there until Alexander’s death in 323, when he found it prudent to leave. He died the next year in Chalcis.

Aristotle is one of the most influential philosophers of all time. He wrote on a wide variety of topics and was the preeminent authority on the physical sciences for two thousand years. His importance to mathematics lies in his work on logic. He constructed a formal theory of logic, building on the ideas developed over the preceding 250 years or so of Greek philosophy.

Aristotle believed that the only way to certain knowledge was by the use of logic, deducing new knowledge based on old. One has to start somewhere, however. His starting point was a set of axioms and postulates, which were truths that needed no argument. An axiom, for Aristotle, was a truth that was not particular to any science, for example, “take equals from equals and equals remain.” A postulate was a truth that concerned a particular area; for example, “through every two points a straight line may be drawn” is a geometric postulate. The philosopher should start with the minimum number of axioms and postulates needed, and some definitions, then proceed by logical argument to prove things.

This logical structure is still the basis for mathematics, although modem mathematicians do not distinguish between axioms and postulates, usually calling any initial assumption an axiom. We also tend not to use the word “truth” for our axioms, although axioms must be carefully constructed to be of use.

As an example of this aspect of logic, let us reconsider the earlier proof we gave that the number of integers is infinite.

We begin by supposing the opposite, that the number of integers is finite. In this case, there must be a largest integer, say n. But then consider n + 1. It is clearly an integer, and it is larger than n, which gives us a contradiction. Since this follows logically from assuming that the number of integers is finite, there must be an infinite number of integers.

This argument assumes some things, for example, that every finite set of integers has a largest element, and that for every integer n, there is an integer n + 1. These may seem obvious to you, but they are still logically required. So we may take them as axioms. The argument also assumes that we know what an integer is. Perhaps we might add a definition for integer. (We won’t attempt such a definition here; it turns out to be a delicate matter.)

Thus, the revised argument would start with a definition of “integer” and two axioms: (1), that every finite set of integers has a largest element, and (2), that for every integer n, there is an integer n + 1. We would then proceed with the argument proper.

Perhaps you can think of other axioms or definitions that this argument needs. If so, you have begun to appreciate how difficult and subtle is the task of establishing a logical foundation for mathematics. The Greeks started this process, but as we shall see in later chapters, theirs was not the final word.

At the Lyceum, his school, Aristotle was famed for lecturing while walking the grounds with his students. As a result, teachers and students there were known as Peripatetics, from the Greek peri (around) and patein (to walk). They didn’t spend all of their time walking, however; the Lyceum introduced written examinations into the educational system.

Conic Sections

Menaechmus (c. 380–320 BCE)

Biographical details of Menaechmus’ life are sketchy. He and his brother studied at Eudoxus’ school at Cyzicus, perhaps with Eudoxus himself. He was a friend of Plato, and perhaps a tutor to Alexander the Great. He later headed the school in Cyzicus, where he died.

Among his many contributions, Menaechmus studied, and probably discovered, conic sections. A conic section is a curve obtained by intersecting a double cone with a plane. (See Figure 1.16.) If the plane intersects both parts of the cone, we get a hyperbola. An ellipse, of which a circle is a special case, intersects only one part, and is finite. In between, if the plane is parallel to a side of the cone, we get a parabola, which, unlike the hyperbola, has only one branch and, unlike the ellipse, goes on to infinity.

Menaechmus used conic sections to solve the Delian problem: doubling the cube. Recall that Hippocrates had reduced the problem of doubling the cube of side a to that of finding two mean proportionals between a and 2a, that is, numbers x and y such that a : x = x : y = y : 2a. We would write the equality of the ratios as

These are equivalent to ay = x² and 2ax = y², the equations of two parabolas. So the problem of finding x and y is equivalent to the problem of finding a point (x, y) on both of the parabolas, i.e., finding an intersection of the parabolas. This is what Menaechmus discovered. We don’t know how he constructed his conic sections; they cannot be constructed via straightedge and compass, but the Greeks knew other techniques.

Figure 1.16 Conic sections.¹

Menaechmus’ solution to the Delian problem is an example, of which there are many in mathematics, of a discovery that turned out to be more important than its original inspiration would suggest. The Delian problem has faded into obscurity, but conic sections are very important. For instance, comets travel around the Sun in orbits that are conic sections.

EXERCISES

Below is a partial table of Ionic numerals.

1.21 What numbers would each of these represent?

a) κ

b) λα

c)

d) pα

1.22 Write each of these numbers in the Ionic system.

a) 21

b) 333

c) 220

1.23 Suppose that the shadow of a building measures 60 meters, at the same time that a 2 meter stick casts a 5 meter shadow. How tall is the building?

1.24 If the two legs of a right triangle have lengths 5 and 12, what is the length of the hypotenuse?

1.25 If the two legs of a right triangle both have length 1, what is the length of the hypotenuse?

1.26 If one of the legs of a right triangle has length 1 and the hypotenuse has length 2, what is the length of the other leg?

1.27 If the two legs of a right triangle have lengths 2n and n² – 1, where n is an integer greater than 1, what is the length of the hypotenuse?

1.28 A triangle with side lengths 8,15, and 17 is inscribed in a circle. What is the diameter of the circle?

1.29 Find angles α, β, and γ.

1.30 Another figurate number is the oblong number, which is a number of the form n(n + 1). The first two oblong numbers are 2 = 1 · 2 and 6 = 2 · 3.

a) Find the first ten oblong numbers.

b) Recall that triangular numbers are of the form l + 2 + 3 + · · · + n, and square numbers are of the form 1 + 3 + 5 + · · · + (2n – 1). Find a similar pattern for oblong numbers.

1.31 Use the formula for Pythagorean triples to find two other sets of triples, larger than (5, 12, 13).

1.32 The first perfect number is 6. The next one is between 20 and 30. Find it.

1.33 Zeno’s dichotomy paradox is related to a famous infinite sum. Find the sum 1/2 + 1/4 + 1/8 + · · ·.

1.34 Eudoxus proved that a circle of diameter d has an area given by A = kd², for some constant k. What is k? (Hint: you know the area of a circle in terms of its radius r.)

1.35 How might the atomic theory of Leucippus and Democritus be used to explain Zeno’s paradoxes?

1.36 Let a = 1, x = = 2^1/3, and y = 2^2/3.

a) Verify that x and y are mean proportionals between a and 2a. (Hint: remember that 2^r/2^s = 2^r–s.)

b) Show that this solves the Delian problem for a = 1, i.e., that the cube of side x has twice the volume of the cube of side a.

1.37 Consider the proof of the equality of alternate angles given in the text. What axioms and definitions might be required to make this logically rigorous?

1.38 Look up the word theorem. What is its origin?