PART I
HELLENIC FOUNDATIONS
Miletus, in present-day Turkey, is the birthplace of the great Greek mathematician and philosopher Thales. The ancient theater in this photograph was built during the fourth century BCE and expanded during the Roman period.
ONE
GOD IS NUMBER
Through the work of the Greeks, the early mathematics of the Babylonians and Egyptians changed its character and became an abstract discipline rather than a field mostly dedicated to the solution of practical problems arising in astronomy or in everyday life. In creating pure mathematics—mathematics divorced from any necessary applications and constituting sheer knowledge—the Greeks had achieved an intellectual advance of great power and beauty.
THALES OF MILETUS
Mathematics as we know it today, with theorems and proofs, began with the great Greek mathematician Thales of Miletus (ca. 624–548 BCE). Miletus was among the first free city-states within the larger Greek empire, which spanned much of the eastern Mediterranean from Anatolia to the south of Italy and Egypt, including the islands in between. Lying on the coast of Anatolia, Miletus was one of the oldest and most prosperous Greek settlements of the time.
Thales is often called the first philosopher. He is also known for his famous saying “Know thyself,” which was even engraved on the stone entrance to the cave of the Oracle of Delphi, a sacred site where the Greeks sought counsel from their gods. Additionally, Thales was one of the Seven Sages of Greece, though according to the historian Plutarch, he surpassed the others. In his book on Solon, another of the Seven Sages, Plutarch says this about Thales: “He was apparently the only one of these whose wisdom stepped, in speculation, beyond the limits of practical utility: the rest acquired the reputation of wisdom in politics.”1
It is easy to forget just how ancient the Egyptian and Mesopotamian civilizations are, even as compared to Greece, which is to us an ancient civilization. By the time of Thales, who lived during the seventh century BCE, Egypt and Mesopotamia were already two millennia old. In the way a modern tourist may travel to Rome or Athens to view the magnificent ruins of the Roman Forum and Colosseum, or the Parthenon on the Acropolis, the Romans and Greeks who were contemporary with these monuments traveled to Egypt to view the pyramids and to absorb Egyptian culture. For example, the obelisks we see in Rome attest to how much the Romans loved Egyptian artifacts—so much so that they decided to bring some of them home. The obelisk at the Place de la Concorde in Paris attests to the greed of a far more recent visitor to Egypt: the French emperor Napoleon, who arrived on a voyage of conquest in 1798 (with two mathematicians, as we will later see) and plundered the land.
Like other young Greeks interested in philosophy and culture, Thales headed for Egypt, and when he arrived there, “he spent his time with the priests,” as Plutarch tells us.2 The priests taught him about Egyptian religion and philosophy, but he was also given the opportunity to practice some ingenious mathematics and subsequently to propose the first known theorem in history while visiting the pyramids.
Thales stood in the desert plain west of the Nile at Giza and looked up at the imposing Great Pyramid of Cheops, named after the Egyptian pharaoh Khufu (known as Cheops in Greece), for whose burial it was constructed. The colossal tomb was completed around 2560 BCE, so when Thales visited this huge edifice—still the most massive monument on earth and, over much of history, the tallest—the pyramid was two thousand years old. The Great Pyramid was one of the Seven Wonders of the Ancient World and is the only one still standing today.
Thales was awed by this pyramid, the largest and tallest—and, as we now know, also the oldest—of the three pyramids of Giza. Like any curious onlooker, he asked his Egyptian guides how high the pyramid was, but no one knew. When he asked the priests, they hadn’t any idea, either. So Thales decided to measure the height of the pyramid without having to climb it—something that seemed impossible at the time. How could such a measurement be made from ground level?
The great pyramid is square-based, and each of its four sides lines up perfectly with the cardinal directions: north, south, east, and west. Hieronymus, a student of the Greek philosopher Aristotle, who lived two centuries after Thales, described what happened next. As quoted by the third-century-CE historian Diogenes Laertius: “Hieronymus says that he [Thales] even succeeded in measuring the pyramids by observation of the length of their shadow at the moment when our shadows are equal to our own height.”3 Thales knew how tall he was, so he waited for the moment in which his shadow was exactly the same length as his height, and he measured the length of the shadow cast by the Great Pyramid at that moment. The logic seems straightforward, but there are obstacles to take into account. Unlike a pole, or for that matter an obelisk of the kind Thales undoubtedly saw many times, a pyramid has bulk. Because it has a wide, square base, when the sun is high in the sky, the pyramid leaves no shadow along the ground but instead casts a shadow along its own slopes. However, being perfectly aligned with cardinal north, its shadow will exceed the extent of its base when the sun is low enough over the southern horizon. And when the sun is at the zenith—the highest point of its path through the sky—it will be perpendicular to the side of the base that faces north. But when will these conditions allow the shadow of an object to equal its height? This will happen when the sun’s rays are at 45 degrees, twice a year—on November 21 and January 20 in Giza.
But did a great mathematician such as Thales have to wait for one of these two days? Not likely. A more probable scenario is described by the first-century historian Plutarch. In a fictionalized conversation between a Greek scholar named Niloxenus and Thales, the former says, referencing the Pharaoh Amasis II, who ruled Egypt at the time: “Among other feats of yours, [the pharaoh] was particularly pleased with your measurement of the pyramid, when, without trouble or the assistance of any instrument, you merely set up a stick at the extremity of the shadow cast by the pyramid and, having thus made two triangles by the impact of the sun’s rays, you showed that the pyramid has to the stick the same ratio which the shadow has to the shadow.”4 This calculation obviates the need to have the shadow equal the height of an object—i.e., the sun doesn’t have to make an angle of 45 degrees—but the shadow must be longer than the half-length of the base. Half the length of the base must be added to the length of the pyramid’s shadow, and a multiplicative ratio factor must be used. For example, if a yardstick’s shadow is two yards long, then the length of the shadow of the pyramid (plus half the length of the base) must be halved to determine the height of the pyramid. When Thales implemented his method, he found a height of 280 Egyptian cubits, equivalent to 480.6 feet (though erosion has likely reduced the height of the pyramid slightly since that time). If Thales had used the former method on one of the two dates in which the angle of the sun’s rays is 45 degrees, however, then for a pyramid with a base length of 756 feet, half of which is 378 feet, the length of the shadow beyond the base would have been 102.6 feet.
This diagram illustrates how Thales may have computed the height of the Great Pyramid using a stick of height a, which casts a shadow of length b. After computing the ratio A/B, he could then measure the length of the shadow plus half the length of the base of the pyramid (collectively, c). In order to find the height of the pyramid, d, all he has to do is multiply c by the ratio A/B.
This illustration of Thales’ intercept theorem contains a generalization of his pyramid-height calculation method, d/c = a/b. We have DE/BC = AE/AC = AD/AB.
But it is the ability to do the calculation on any day in which there is a shadow longer than half the base length by using a common proportionality factor that brought Thales to his beautiful theorem—the first theorem in history. This theorem, motivated by a real-world problem, is an abstract mathematical statement. In the figure at the bottom of page, two parallel lines intersected by two arbitrary lines cut segments according to the same proportion, illustrating Thales’ theorem.
Thales’ other theorems incl
ude the statement that the base angles of an isosceles triangle are equal and that a circle is bisected by a diameter. Until then, length, breadth, and volume were considered the key elements of geometry. Thales, however, was more concerned with beautiful geometrical theorems than the study of numbers, and he was the first mathematician to consider angles as important in the study of geometry. He provided a key link between triangles and circles by showing that every triangle corresponds to a circumscribing circle that touches all three points of the triangle. Thus he demonstrated that only one circle passes through any three points that are not all on the same straight line, and that the diameter of such a circle corresponds to the circumscribed triangle’s hypotenuse. Additionally, he showed that an angle formed by the extension of two segments from the two endpoints of a diameter to any point on a semicircle is a right angle.
In addition to being the first “pure” mathematician, in the sense that he proposed and proved abstract theorems, Thales was also the first Greek astronomer. One day Thales was so engrossed in observing the stars that he moved forward a few steps without looking and fell into a well. “A clever and pretty maidservant from Thrace” who passed by and helped him out of the well chastised him for “being so eager to know what goes on in the heavens that he could not see what was straight in front of him, nay, at his very feet!”5
Thales’ theorem shows that, if A, B, and C are points on a circle, where segment AC is its diameter, then AC also represents the hypotenuse of a right triangle.
As an astronomer, Thales was so competent that he could predict solar eclipses. In fact, he is credited with predicting the total solar eclipse that took place in his part of Greece on May 28, 585 BCE. Greek mathematics historian Sir Thomas Heath explained that Thales’ prediction was probably based on the fact that the Babylonians, who had been recording solar eclipses for centuries, knew that eclipses recur after a period of 223 new moons. Presumably, there had been a record of an eclipse in that area that had taken place 223 moons, or about eighteen years, earlier. This piece of information was probably transmitted to Thales through his intellectual connections in Egypt. He is also known to have studied the equinoxes and the solstices.6
PYTHAGORAS OF SAMOS
The next great Greek mathematician is the renowned Pythagoras of Samos (ca. 580–500 BCE). As a young man, he was coached by the aging Thales, and he would continue the Greek quest to turn the mathematics of the Egyptians, Babylonians, and early Indians from a practical computational discipline into a beautiful, abstract philosophy. It was Pythagoras who gave us the ubiquitous Pythagorean theorem, which allows us to determine the length of a right triangle’s hypotenuse. Today GPS and maps use this theorem—as well as our very early understanding of numbers and of geometry—to compute distances between two locations.
Pythagoras was born on the Greek island of Samos, a stone’s throw from the Anatolian Plateau of Asia Minor, which at that time was also part of greater Greece. The island is home to the Temple of Hera, one of the Seven Wonders of the Ancient World (although, unlike the almost-intact Great Pyramid, this temple has only one marble column still standing). Today the main town on the island is called Pythagoreion in honor of the island’s native son.
Pythagoras began his life as a precocious intellectual adventurer, curious about nature, life, philosophy, religion, and mathematics. As a young man, he traveled extensively. In Egypt he met with priests in temples to learn about their religion, their knowledge of the world, and their mathematics. In Mesopotamia he visited astronomers to learn how they observed celestial bodies, and he studied their mathematical and scientific methods. Did he learn about the theorem he is now credited with developing, or did he simply absorb related concepts in Mesopotamia? This we do not know. Because mathematics had roots in India as well, and because some Pythagorean ideas appear to be related to Indian mathematical principles, some historians have surmised that Pythagoras may have traveled as far as India. We have no confirmation of this conjecture, however.
The Greek philosopher and mathematician Pythagoras is depicted in this undated illustration.
Neither do we know how the great Thales met the young Pythagoras. We do know that the two men knew each other and that Thales recognized Pythagoras’s budding intellect and encouraged him to expand his horizons. According to the third-century philosopher Iamblichus, who wrote a biography of Pythagoras, “Thales, admiring his remarkable ability, communicated to him all that he knew, but, pleading his own age and failing strength, advised him for his better instruction to go and study with the Egyptian priests.”7
Pythagoras wanted to see much more than Egypt, so he first traveled east to Phoenicia, visiting Byblos, Tyre, and Sidon, where he met with priests and learned about Phoenician rites and customs. He is also reputed to have met with the descendants of the mysterious Mochus, a natural philosopher and prophet credited by some historians as proposing an atomic theory before Democritus. There Pythagoras is said to have been initiated into a strange regimen “to which he submitted, not out of religious enthusiasm, as you might think, but much more through love and desire for philosophic inquiry, and in order to secure that he should not overlook any fragment of knowledge.”8
Pythagoras suspected that the rites and rituals he was observing and learning in Phoenicia had Egyptian roots, and he proceeded to Egypt to find their origin, just as Thales had encouraged him to do.
There, he studied with the priests and prophets and instructed himself on every possible topic … and so he spent 22 years in the shrines throughout Egypt, pursuing astronomy and geometry and, of set purpose and not by fits and starts or casually, entering into all the rites of divine worship, until he was taken captive by Cambyses’ force and carried off to Babylon, where again he consorted with the Magi, a willing pupil of willing masters. By them he was fully instructed in their solemn rites and religious worship, and in their midst he attained to the highest eminence in arithmetic, music, and the other branches of learning. After twelve years more thus spent he returned to Samos, being then about 56 years old.9
When he returned to his native island, Pythagoras was steeped in exotic ideas that he had absorbed during his travels. He developed a religious belief that the soul never dies but rather transmigrates to other living things. Hence, if a person kills another living thing—even a small insect—he could be killing a being with the soul of a deceased friend. This idea, which bears a strong resemblance to the Indian notion of reincarnation, led Pythagoras to a strictly vegetarian lifestyle. He also developed an aversion to eating beans—perhaps another fetish acquired as a result of his travels.
Pythagoras began to think about how he could combine the science of numbers and measurement that he absorbed in Egypt and Mesopotamia with the theorems of his Greek predecessor, Thales. Numbers fascinated him, so much so that eventually he and his followers would come to believe that “God is number.” Further, Pythagoras transformed mathematics into the abstract philosophical discipline we see in pure mathematics today.
Pythagoras’s notion that numbers held powers led to a kind of number mysticism, and he became a sort of guru. A growing group of disciples who adhered to his strict lifestyle principles and devoted their time to studying the abstract concepts of the new discipline of mathematics gathered around him. At some point a fearful island leader who worried that the group might someday vie for political power and unseat him applied political pressure on the Pythagoreans, and they were forced to leave Samos. Pythagoras and his followers moved to a place called Crotona, in the center of the bottom of the Italian “boot,” which was also part of Magna Graecea (greater Greece). Isolated from the surrounding population, members of the secret society dedicated themselves to their religion—number mysticism—and the study of mathematics.
The Pythagoreans considered mathematics a moral beacon that helped them lead a righteous life. In addition to the word philosophy (love of wisdom), the word mathematics, which comes from the Greek phrase meaning “that which is learned,” is believe
d to have been coined by Pythagoras.10 He used both terms to describe the intellectual activity in which he and his followers were engaged. Pythagoras continued the work of Thales in pure mathematics and is seen to have transformed the discipline into “a liberal form of education, examining its principles from the beginning and probing the theorems in an immaterial and intellectual manner.”11 The “educational” aspect of mathematics was pursued in lectures that Pythagoras delivered to the members of his sect. These talks consisted of theorems, results, and discoveries about numbers and their meaning. As a form of public service to the outside community that surrounded the sect’s compound—and, perhaps, to avoid being chased away, as had happened at Samos—Pythagoras gave public lectures to the entire community living in the area. The talks within the sect were strictly confidential, however. Most of the discoveries Pythagoras and his followers made about numbers were kept secret, with only select facts released to the outside world.
Mathematics historian Carl Boyer states, “The Pythagoreans played an important role—possibly the crucial role—in the history of mathematics.”12 What achievements by Pythagoras and his sect merit such an assessment? According to mathematics scholar Sir Thomas Heath, it was probably Pythagoras who discovered that the sum of successive natural numbers (i.e., 1, 2, 3, 4, 5 …) beginning with 1 makes a triangular number—that is, a number that can be drawn as a triangle. For example, 1 + 2 = 3; 1 + 2 + 3 = 6; 1 + 2 + 3 + 4 = 10; 1 + 2 + 3 + 4 + 5 = 15, and so on. Written algebraically, Tn = 1 + 2 + 3 + 4 +…+ n = (½)n(n + 1), where Tn is a triangular number and n is the number of units on a side.
A Strange Wilderness Page 2
