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The Pythagorean Theorem
In high school mathematics, students study the Pythagorean Theorem, which relates the lengths of the three sides of a right triangle with a mathematical equation.
The theorem is attributed to a man named Pythagoras and his cult-like group from the sixth century BCE in ancient Greece. Little is known of the details of his life; what we do know of the man and his followers comes from writers, including other ancient philosophers, in subsequent centuries. The Pythagoreans, followers of Pythagoras, have been called a cult, religious order, sect, and a brotherhood, and these are just a few of the names history has given this enigmatic group.
Though many of the cult’s customs seem completely absurd by today’s standards, the group did produce some fundamental work in mathematics and music theory that is still relevant today. Their work has been debated and discussed by such greats as Plato and Aristotle, as well as countless other scholars since ancient times.
Pythagoras was born on the island of Samos off the coast of modern-day Turkey around 570 BCE. It is believed he traveled to Babylonia, Phoenicia, and Egypt as a young man in search of knowledge. About 530 BCE he settled in the Grecian city of Croton, modern day Crotone, in southern Italy. When he was about the age of 40, he began to gather male and female disciples. He organized some 300 followers and imposed strict dietary and lifestyle regimens on his disciples. When a member questioned one of his decisions, they were given the standard reply, “He himself has said it.” Members who continued to question the master and not adhere to the rules were subject to expulsion.
The group developed elaborate rituals and ceremonies, enforced moral self-control, and were mainly vegetarians. They cultivated the arts, mathematics, and music, and gained considerable political influence in Greek cities of southern Italy. The school established by Pythagoras was multifaceted, functioning as a philosophical school, a religious brotherhood, and a political association. As the Pythagoreans grew in number and political power in southern Italy, opposition arose to the group, forcing him to migrate to Metapontum around 495 BCE and his followers to disperse. After this point, nothing else is known about the life of Pythagoras, but theories abound as to the cause and place of his death. None of his original writings have survived, though later writers, his students and other philosophers, have attributed authorship to him.
The Pythagoreans were a cult that demanded secrecy from its members, imposed an ascetic lifestyle, and held many mystical beliefs. Some of the unusual beliefs of the Pythagoreans were: never poke a fire with a knife; never eat or even touch fava beans; do not touch a white cock; when you get out of bed, roll up the bedclothes and smooth out the impression of the body; and never urinate while facing the sun. The Pythagoreans were similar to the other mystery cults that were prevalent in ancient Greece; however, they differed due to their interest in mathematics and astronomy. Pythagoras seemed to have divided his followers into two groups, the “listeners,” who remained in silence while absorbing the teachings of the master, and a second group called the “learners” that was made up of more advanced followers who asked questions and participated in discussions. The learners were more involved in the study of mathematics and the sciences, while the listeners focused more on the religious and mystical aspects of his teachings.
As the cult grew in numbers and influence in the Grecian cities of southern Italy, they began to be persecuted by their fellow Greeks. Tensions arose with the surrounding communities as a form of democratic government was becoming more prevalent. The Pythagoreans became unpopular for their political activity and when the violent democratic revolution occurred in southern Italy about 450 BCE, the Pythagoreans were attacked and their meeting houses destroyed. Many fled to the mainland of Greece, while others settled in Tarentum and the surrounding area. By the middle of the fourth century BCE, the Pythagoreans as a distinct group no longer existed.
Though the group dispersed, the teachings of Pythagoras and much subsequent work have been attributed to his name. Because of the secrecy shrouding the beliefs of the brotherhood, it is not possible to accurately say which works are directly attributed to the philosopher and what were developed by his disciples. A man named Philolaus was one of his more eminent students and published Pythagorean views for the general public.
Transmigration of the Soul
One of the Pythagorean doctrines held that a soul could be transmigrated (metempsychosis); that is, one’s soul is immortal and after death it transfers into other animals. One account states that Pythagoras claimed to have lived previous lives and he could recall them in detail. A contemporary of Pythagoras, Xenophanes, recorded an incident, “Once when he [Pythagoras] was present at the beating of a puppy, he pitied it and said, ‘Stop, don’t keep hitting him, since it is the soul of a man who is dear to me, which I recognized when I heard it yelping.’ ” In addition to the immortality of the soul and reincarnation, he believed that “after certain periods of time the things that have happened once happen again and nothing is absolutely new.” It is possible he taught his followers to abstain from eating animal flesh on the grounds that there was kinship between human and animals.
Study of Sound and Music
Pythagoras was interested in the study of sound and musical instruments, a field of study for him that was closely tied to numerology and the cosmos. He noticed that the shorter the string on a stringed musical instrument, such as a lyre, the higher the pitch it would produce. He recognized that a string that was twice as long emitted a sound that was an octave lower. If the ratio of the length of the strings was three to two, the musical interval produced was a fifth. When the ratio was four to three, the interval called a fourth was produced. Additionally, he determined that the pitch of the sound became higher as the tension on the string increased. Many of his observations of sound and music are still relevant today.
Study of Astronomy
Pythagorean astronomy seems to have its roots in Babylon, which was where he presumably gained exposure to the subject during his early travels. The theories developed by Anaximander of Miletus, who was said to be his teacher, also influenced his views on the cosmos. The Pythagorean view of the stars and planets was tied to the master’s reverence for numbers and harmonies found in music. He used his concept of musical intervals to determine the position of the planets in relation to the position of Earth. Based on his theory, the order of the planets in increasing distance from Earth is: Moon, Mercury, Venus, the Sun, Jupiter, and Saturn. The theory was later refined by placing Mercury and Venus above the Sun, since no solar transits of the planets had been observed.
Using the perceived link to the orbits of the planets and musical intervals and the seven strings of the lyre, the Pythagoreans thought as the planets moved throughout the firmament they produced a celestial harmony called the music of the spheres. Supposedly, only Pythagoras could hear the music coming from the planets; common people could not hear it because they had grown accustomed to the sound. Another idea possibly adopted from the Babylonians was that of the “great year.” This concept used by Pythagoreans held that since the periods of revolution of the heavenly bodies were in integral ratios so that the same constellation of all stars must recur after some defined period of time, that is the great year.
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To satisfy the Pythagorean idea of beauty, the stars must accordingly move in the simplest of curves, which is the circle. This idea of circular orbits for planets permeated astronomy until Johannes Kepler showed in the seventeenth century that the planets move in elliptical orbits. The Pythagoreans did not believe in a flat Earth; rather, they adopted the notion of a spherical Earth. Their view of the solar system was that the Earth moves around an unseen central fire every 24 hours. The central fire cannot be seen because another object, a counter Earth, also rotates about the central fire every 24 hours and always remains between the Earth and the central fire. According to Aristotle, the Pythagoreans were also able to explain the cycle of night and day using this model.
The group’s mystical belief that numbers were associated with living and inanimate objects took them into many lines of thought. The members sought the number associated with a specific person or animal. As an example, for a horse they associated the number of small stones needed to outline a horse, thus this became the number of the horse. They associated even numbers with masculine traits and odd numbers with feminine traits; therefore, five was the number for marriage since it was the sum of two and three, an even and an odd. To many Pythagoreans this concept meant that objects are measurable or proportional in terms of numbers, a concept that would be significant in the development of Western culture.
The symbol tetractys or “perfect triangle,” a mystical figure consisting of 10 points arranged in a triangular fashion, became important to the Pythagoreans. The bottom of the four rows has four points, the next row up has three points, above that row are two points, and at the top of the triangle a single point. To the Pythagoreans the tetractys was a sacred symbol; they swore by it and referred to it in their oath: “By that pure, holy, four lettered name on high, nature's eternal fountain and supply, the parent of all souls that living be, by him, with faith find oath, I swear to thee.” Another type of numerical perfection came about when a number was equal to the sum of its factors; two such numbers are 6 and 28, because 1+2+3=6 and 1+2+4+7+14=28.
The early Pythagoreans used so-called gnomons, which in Greek means “carpenter’s squares,” to describe even and odd numbers. Based on the writing of Aristotle, gnomon numbers are represented by dots or pebbles arranged in triangular, square, or rectangular arrays.
The Pythagoreans carried the notion that all things could be reduced to numbers. They based this idea on three basic premises. First, the mathematical relationships of musical harmonies—that is, the ratio of lengths of sound producing instruments (e.g., the flute or lyre)—if extended to other instruments produce the same musical harmonies. Secondly, they noted that any triangle formed of three rods in the ratio 3:4:5 always produced a right triangle, one with one angle of 90 degrees. Their third important observation came from their belief derived from the fixed numerical relationships associated with the orbits of the heavenly bodies. Since the same musical harmonies and geometric shapes can be produced in different sizes or in different media, the numbers themselves must express the harmonies and shapes. Thus, the “essences” of things were actually numbers. The groups of numbers and their relationship to each other became known as “words” that embodied the essence of a thing, and the term evolved into “ratio.”
Aristotle wrote of Pythagoreans in book 1.5 of his work Metaphysics about 150 years later: “Contemporaneously with these philosophers and before them, the so-called Pythagoreans, who were the first to take up mathematics, not only advanced this study, but also having been brought up in it they thought its principles were the principles of all things. Since of these principles numbers are by nature the first, and in numbers they seemed to see many resemblances to the things that exist and come into being—more than in fire and earth and water…”
The Pythagorean Theorem
The mathematical theorem that every high school geometry student is familiar with is the Pythagorean Theorem, which states that the square of the hypotenuse (the longest side) of a right triangle is the sum of the squares of the two shorter sides. Written as an equation, the theorem becomes: a2 + b2 = c2. It is said that the discovery of the theorem by Pythagoras was celebrated with the sacrifice of an ox. As a simple example, take the right triangle with sides 3, 4, 5 units in length; the Pythagorean Theorem tells us that 32 +42 = 52 or 9 + 16 = 25. Though Pythagoras is credited with the discovery of the relationship, writings found on clay cuneiform tables from Babylon indicate that the theorem was known a thousand years before Pythagoras.
The Problem With Irrational Numbers
The Pythagoreans ran into a perplexing problem when they encountered irrational numbers; that is, a number that cannot be expressed as a ratio of two integers. A simple example comes from that of an isosceles right triangle with the lengths of the two sides one unit each. Per the Pythagorean Theorem, the length of the long side, the hypotenuse, will be the square root (1^2 +1^2) which is the square root of 2. One of the advanced students, Hippasus, attempted to calculate the square root of two and found it could not be expressed as the ratio of two whole numbers. This discovery sent shockwaves through the Pythagoreans as it threatened to overturn their world view of the relationship to whole numbers and reality.
As one story goes of the incident, Hippasus was loose with telling others outside the group and was drowned by members of the sect. Later Pythagorean mathematicians advanced the theory of irrational numbers, such as the square-root-of-n, where n is any rational number (quotient of two integers). They also developed a method for finding an approximate value of the square root of two by forming sets of so-called diagonal numbers.
Influence of the Pythagoreans
When the followers of Pythagoras dispersed in the middle of the fifth century BCE into the larger world, they carried with them the ideas and teachings of their master. These teachings, some profound and some absurd by today’s standards, affected the generations that came after them. The Greek philosopher Socrates apparently came into contact with some the Pythagoreans; thus, he passed the ideas to his students, including Plato, who in turn passed the teachings onto his star student Aristotle. The doctrines of the Pythagoreans inspired systematic study of mathematics and the numerical aspects of musical harmony. The influence of Pythagoras continued into the Middle Ages and his philosophy had an impact on such great scientists as Nicolaus Copernicus, Johannes Kepler, and Isaac Newton. Pythagoras remains one of the most important early Greek philosophers and is aptly credited with coining the term “philosopher.”
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- Huffman, Carl, “Pythagoras," The Stanford Encyclopedia of Philosophy (Winter 2018 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2018/entries/pythagoras/>. Accessed February 11, 2021.
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This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.
© 2021 Doug West
Doug West (author) from Missouri on February 19, 2021:
Thanks for the comment. Pythagoras and his followers were a strange lot but they did come up with some good and useful ideas.
Heidi Thorne from Chicago Area on February 19, 2021:
I think it's amazing that ancients like these could even come up with this stuff without computers or other sophisticated devices. This is a great review of Pythagoras' contributions to culture and science!