The Pythagorean Theorem
The Story of Its Power and Beauty
By ALFRED S. POSAMENTIER
Copyright © 2010 Alfred S. Posamentier
All right reserved.
Chapter One Pythagoras and His Famous Theorem
As we embark on our exploration of the Pythagorean Theorem, we are faced with some questions. Chief among them is why is the relationship that historically bears that name-the Pythagorean Theorem-so important? There are many reasons: perhaps because it is easy to remember; perhaps because it can be easily visualized; perhaps because it has fascinating applications in many fields of mathematics; or perhaps because it is the basis for much of mathematics that has been studied over the past millennia. We shall explore these aspects in the chapters that follow. But, perhaps it is best to begin at its roots, with the mathematician whom we credit as being the first to prove this theorem, and examine the man himself, his life, and his society.
The first biography of Pythagoras was written about eight hundred years after his death by Iamblichus, one of many Pythagoras enthusiasts, who tried to glorify him. And, although Pythagoras has been mentioned numerous other times throughout history, by well-known writers such as Plato, Aristotle, Eudoxus, Herodotus, Empedocles, and others, we still do not have reliable information about him. Some of his contemporary followers actually believed that he was a demigod, a son of Apollo-a conviction they supported by noting that his mother was said to be a very beautiful woman. Some reported that he even worked wonders.
But even though he was called the greatest mathematician and philosopher of antiquity by some, he was not without critics who tried to revile him. They say that he was merely the founder and chief of a sect-the Pythagoreans-and that the many scientific results that came from them were written by the members of the sect and dedicated to its leader, and thus were not the work of Pythagoras himself. The critics considered him a collector of facts without any deeper understanding of the related concepts, and therefore felt he did not really contribute to a deep understanding of mathematics. Similar criticism also was aimed at such luminaries as Plato, Aristotle, and Euclid. We must remain mindful of these uncertainties when we consider the "facts" about Pythagoras's life and work.
Pythagoras was born circa 575 BCE on the island Samos (located off the west coast of Asia Minor). His initial and perhaps most influential teacher was Pherecydes, who was primarily a theologist who taught him religion and mysticism along with mathematics. As a young man he traveled to Phoenicia, Egypt, and Mesopotamia, where he advanced his knowledge of mathematics and also pursued a variety of other interests, such as philosophy, religion, and mysticism. Some biographers believe that, while in his late teens, Pythagoras traveled first to Miletus, a coastal town in Asia Minor near Samos, where he continued his studies in mathematics under the tutelage of the famous philosopher and mathematician Thales of Miletus. It is very likely that he also attended lectures from another Miletic philosopher, Anaximander, who further inspired Pythagoras in geometry. When he returned to Samos, the tyrant Polycrates, who ruled Samos from 538 to 522 BCE, had come to power. It is not clear if Pythagoras disagreed with Polycrates' leadership. But soon after returning home, he moved to Croton (today, Crotone in southern Italy), about 530 BCE, a region that had had a considerable Greek population since the eighth century. There he founded a community-or society-whose main interests were religion, mathematics, astronomy, and music (acoustics). The Pythagoreans' conviction that all aspects of nature and the universe could be explained and expressed by means of the natural numbers and the ratios of numbers suffered a setback when they learned that the emblem of their community, the pentagram, contradicted their core numerical principles.
The Pythagoreans tried to explain the nature of the world and the universe with the help of numbers. In particular, they studied vibrating strings and found that two strings sound harmonious if their lengths can be expressed as the ratio of two small natural numbers such as 1:2, 2:3, 3:4, 4:5, and so on. They came to believe that the entire universe is ordered by such simple relations of natural numbers. This ties in with their study of the three most popular means: the arithmetic mean, the geometric mean, and the harmonic mean, which relate to each other. We will visit these means in chapter 5.
One of the core beliefs of the Pythagoreans is that there is a strong connection between religion and mathematics. They believed that the sun, the moon, the planets, and the stars were of a divine nature and therefore they could move only along circular paths. Furthermore, they believed that the movements of these bodies caused sounds of different frequencies because of their different velocities, which in turn depended on their radii. These sounds were said to generate a harmonic scale, which they called the "harmony of the spheres." Yet they believed that man cannot actually hear this sound, as it surrounds humans constantly from birth. Even the great scientist Johannes Kepler (1571-1630) was sometimes characterized as a late Pythagorean since he believed that the diameters of the orbits of the planets could be explained by inscribed and circumscribed Platonic solids (see figure 1-1), an idea he published in his work De Harmonice Mundi (About the Harmony of the World).
By investigating the courses of celestial bodies the Pythagoreans sought to purify their souls and to prepare them for their final passage into the heavens. Before that final stage-so they believed-their souls would transmigrate, not only from human to human but also into animals. Therefore, among their other rules, those of modesty, discipline, and secrecy, they were said to have enforced a ban on sacrificing animals and eating meat, for they thought that the soul of a deceased person might be in the animal. To further their ability to focus on their beliefs, they also refrained from eating beans, as that produced flatulence and interfered with intellectual thinking. Some biographers contend, however, that the Pythagoreans enforced their animal-sacrificing ban only on some animals-namely, those that they believed had a soul. In particular, one anecdote reports that the Pythagoreans sacrificed twenty oxen whenever they came up with and proved a mathematical concept.
In contrast to the Pythagoreans, the philosophers Anaxagoras and Democritus believed that planets and stars were only glowing stones. Anaxagoras was sentenced to death because of his so-called godlessness in espousing this belief; but his sentence was commuted to banishment after the intervention of the revered statesman Pericles.
Part of the reason Pythagoras gained such a large following was because he was an eloquent speaker; in fact, four of his speeches, given to the public in Croton, are still remembered today. The Pythagoreans also gained political influence in that region, even over the non-Greek population. But sometimes-as is frequent in politics-they faced resistance and animosity. Later (in approximately 510 BCE), when the Pythagoreans got involved in various political disputes, they were expelled from Croton. The displaced Pythagoreans tried to move to other towns, such as Locri, Caulonia, and Tarentum, but the people in these towns did not allow them to settle. Finally, in Metapontum, they found their new home. Here Pythagoras died of old age around 495 BCE.
As there was no appropriately charismatic leader to succeed Pythagoras, the Pythagoreans split up into several small groups and tried to continue their tradition, while continuing to exert political influence in various towns in southern Italy. They were rather conservative and well connected to established influential families, which got them into conflict with their common counterparts. As soon as their opponents gained the upper hand, bloody persecutions of the Pythagoreans began. Given the political situation, many of them emigrated to Greece. This was-more or less-the end of the Pythagoreans in southern Italy. Very few individuals tried to continue the tradition and to advance the Pythagorean ideals. Two groups that persisted were the Acusmatics and the Mathematics. The former believed in acusma (i.e., what they had heard Pythagoras say) and did not give any further explanations. Their only justification was "He said it." This gave Pythagoras a level of importance, or popularity, in his day, which to another extent still exists today. In contrast to the Acusmatics, the Mathematics tried to develop his ideas further and provide precise proofs for them.
One of the very few Pythagoreans who remained in Italy was Archytas of Tarentum (ca. 428-350 BCE). He was not only a mathematician and philosopher but also a very successful engineer, statesman, and military leader. He befriended Plato in about 388 BCE, giving rise to the belief that Plato learned the Pythagorean philosophy from Archytas, and that is why he discussed it in his works. Aristotle, who was first a student in Plato's academy but soon became a teacher there, wrote rather critically about the Pythagoreans. While Plato may have adopted many ideas from the Pythagoreans, such as the divine nature of planets and stars, in other cases he disagreed with them. Plato mentioned Pythagoras only once in his books, but not as a mathematician, despite his being in close contact with all of the mathematicians of his time and holding them in high regard. It is probable that Plato did not consider Pythagoras a proper mathematician. Similarly, Aristotle also mentioned the Pythagoreans, but said almost nothing about Pythagoras.
In the fourth century BCE the Greeks distinguished between "Pythagoreans" and "Pythagorists." The latter were extremists of the Pythagorean philosophy and consequently often the target of sarcasm because of their unusual ascetic lifestyle. Still, among the Pythagoreans there were some members who were able to command respect from outsiders.
After the fourth century BCE, the Pythagorean philosophy disappeared from sight until the first century CE when Pythagoras came into vogue in Rome. This "Neo-Pythagoreanism" remained alive in subsequent centuries. In the second century CE Nicomachus of Gerasa wrote a book about the Pythagorean number theory whose Latin translation by Boethius (ca. 500 CE) was widely distributed. Today, Pythagorean ideas permeate our thinking in a variety of fields, as we will see.
The Pythagorean Theorem
Let us now focus on the geometric relationship that made Pythagoras famous in today's world and that, of course, bears his name. We would do well to consider his prominent role (or that of his society) in the development of this amazing relationship.
Although the relationship was already known before Pythagoras (as you will see in the ensuing pages), it is appropriate that the theorem should be named for him, since Pythagoras (or one of the Pythagoreans) was the first to give a proof of the theorem-at least as far as we know. Historians suppose that he used the squares as shown in figures 1-2 and 1-3-perhaps inspired by the pattern of floor tiles. We will briefly demonstrate the proof here.
To show that [a.sup.2] + [b.sup.2] = [c.sup.2], you need only subtract the four right triangles, with sides a, b, and c from each of the two larger squares, so that in figure 1-2 you end up with [a.sup.2] + [b.sup.2], and in figure 1-3 you end up with [c.sup.2]. Therefore, since the two original squares were the same size and we subtracted equal quantities from each, we can conclude that [a.sup.2] + [b.sup.2] = [c.sup.2], which is shown in figure 1-4 with the two figures of the same area.
To prove a theorem is one thing, but to come up with the idea establishing this geometric relationship is quite another. It is likely that Pythagoras learned about this relationship on his study trip to Egypt and Mesopotamia, where this concept was known and used in construction for special cases.
During his travels to Egypt, Pythagoras probably witnessed the measuring method of the so-called Harpedonapts (rope stretchers). They used ropes tied with 12 equidistant knots to create a triangle with one side of length 3 units, one of 4, and a third side of 5, knowing that this enabled them to "construct" a right angle. (See figure 1-5.)
They applied this knowledge to survey the banks of the river Nile after the annual floods in order to rebuild rectangular fields for the farmers. They also employed this method in laying the foundation stones of temples. To the best of our knowledge, the Egyptians did not know of the generalized relationship given to us by the Pythagorean Theorem. They seem to have only known about the special case of the triangle with side lengths 3, 4, and 5, which produced a right triangle. This was arrived at by experience and not by some sort of formal proof.
In Mesopotamia, mathematicians were even able to produce further triples of numbers, fulfilling the Pythagorean condition of [a.sup.2] + [b.sup.2] = [c.sup.2], as we can see on a Babylonian clay tablet from ca. 1800 BCE, known as the Plimpton 322. (See figure 1-6.) The tablet was part of a collection of about a half million of such tablets found in the mid-nineteenth century as a result of Mesopotamian digs, of which about three hundred were identified as having mathematical significance. The tablet is written in Old Babylonian (or cuneiform) script and uses the sexagesimal system (base 60). It shows us the high level of mathematical knowledge that existed well before the Greeks.
The two shaded columns of figure 1-7 translate the Babylonian numerals to our base-10 system and give a strong indication of their knowledge of the Pythagorean triples. These two columns list the leg and hypotenuse of several Pythagorean triples.
Here, we notice that the three left-hand numbers in each row satisfy the Pythagorean Theorem, [a.sup.2] + [b.sup.2] = [c.sup.2], and are called Pythagorean triples.
Pythagorean triples have also been discovered in northern Europe in megalithic rings, where they are displayed as triples of numbers that are, in large measure, accurate Pythagorean triples. However, in Babylonia we not only find Pythagorean triples but we also find problems, which can only be solved with a proper knowledge of the Pythagorean Theorem.
The Babylonians derived the Pole against the wall problem. If a pole of length 0;30 units slips down 0;6 units along the 0;30-unit wall, how far is the base of the pole from the base of the wall? (See figure 1-8.)
This calculation is seen geometrically in figure 1-9.
The calculation in our words and our number symbols would translate to the following:
Square 0.5, you will get 0.25. Subtract 0.1 from 0.5, you will get 0.4. Square 0.4, you will get 0.16. Subtract 0.16 from 0.25, you will get 0.09. 0.09 is the area of a square, so its side is 0.3.
The stick has slipped by 0.3, which, when converted to the sexagesimal system, is 0;18.
On another clay tablet, YBC 7289 (Yale University Babylonian Collection), we can see that the Babylonians already applied the Pythagorean Theorem to calculate a rather accurate approximation of [square root of] 2. In figure 1-10, there are three pictures of this tablet; the first shows the original tablet, the second displays it with accentuated marking, and the third shows the values of the lengths of its sides. On this tablet there is a square, whose sides have length 1, with a value written along the diagonal. That is, there is an isosceles right triangle-half the square-where, by using the Pythagorean Theorem, we can determine that in that isosceles right triangle with legs of length 1, the hypotenuse is length [square root of] 2 , since [1.sup.2] + [1.sup.2] = 2 = [c.sup.2], so that c = [square root of] 2.
Excerpted from The Pythagorean Theorem by ALFRED S. POSAMENTIER Copyright © 2010 by Alfred S. Posamentier. Excerpted by permission.
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