Joseph A. Montagna
Pythagoras established a school in southern Italy. What we know about this school is through the writings of his students, or his students’ students. Pythagoras wrote nothing of a permanent nature. Philolaus is a figure who is generally connected with the beginnings of Pythagorean literature. He was a student of Pythagoras, and probably a teacher himself. According to tradition Philolaus was either commissioned by Plato to write three books on the teachings of Pythagoras, or to buy three books on the subject. The latter is probably closer to the truth since it is said of Philolaus, ejhyeg¸e, meaning “he brought out,” i.e., “he published.” There is a great deal of conjecture connected with the life of Pythagoras, since the evidence is not complete. Any information that is available about the man amounts to little more than a coloring book outline, the details to be filled in by legend and the imaginations of authors who came later.
Pythagoras of Samos was born around 570 B.C. Samos was an island in the Ionian group, along the coast of what is presently called Turkey. Pythagoras’ family was wealthy, his father being a merchant.
Pythagoras’ quest for knowledge is legendary. At the time the major sources of knowledge were the temples. Their bodies of knowledge were not told to the general populace, but shared only with those who were their initiates, usually the wealthy. Pythagoras was a student of the local Ionian temples. It is generally accepted that Pythagoras was a student of Thales of Miletus. After absorbing as much knowledge as the local temples could impart, Pythagoras looked elsewhere for new knowledge. He was probably encouraged to do so by a fellow brother of the temples of Ionia, as this was the accepted method of sharing knowledge between temples. Pythagoras is said to have enlisted the help of the tyrant king, Polycrates. Polycrates is said to have contacted the brethren of the Temple of Memphis in Egypt through an ally of his at the time, Amasis, the ruler of Egypt. Pythagoras was accepted to the Temple of Memphis. Egyptian temples had long been regarded as possessing the richest of all knowledge that man possessed. Pythagoras, thirsting for new knowledge, must have been overwhelmed with joy at this occasion.
Being admitted to the Temple of Memphis was the first and the easiest of all obstacles for Pythagoras to overcome in his pursuit of the higher degrees of the Temple of Memphis brotherhood. The customary length of time for an initiate to remain at the Temple to learn the sacred secrets was twenty-two years. Upon completion, the student was free to leave. There was, of course, the oath of silence, wherein a brother swears never to reveal the Temple’s secrets. Legend has it that it was in Egypt that Pythagoras developed his understanding of the properties of right triangles and proportions. Supposedly, Pythagoras, while contemplating the inner secrets of his new brotherhood, discovered the concept that the sides of one right triangle are proportional to corresponding sides of another right triangle. This revelation might have come to him as he sat on the Temple grounds contemplating its surroundings, and discovering that objects in the sunlight cast shadows that are proportional to their actual height. The objects, a tree, an obelisk, a statue, standing perpendicular to the earth, formed right triangles; the object being the vertical side, the shadow being the horizontal side, and the sun’s rays forming the hypotenuse. Pythagoras learned well.
Having completed his studies in Egypt Pythagoras longed to return to his homeland to impart some of his newly acquired knowledge. It was permissible at the time to teach certain things to outsiders of the temples, as long as one did not reveal the inner secrets. One was also permitted to teach that which was one’s own creations and discoveries. It was also customary to share the knowledge with those brothers of one’s homeland temple. Pythagoras’ departure was delayed by the war with Persia (Ionian Revolt). Pythagoras was taken captive and brought back to Babylon. He and his store of knowledge were considered to be spoils of war, as would be precious metals, or slaves. In Babylon, Pythagoras was well received. The brothers of the temples there were anxious to talk with him and learn from him. These brothers also shared their own secrets with Pythagoras in return. He was given the freedom to move about the temples. Characteristically, Pythagoras delved into this new body of knowledge and absorbed as much of it as they would permit.
Twelve years later he was allowed to leave. He returned to Samos, only to leave again for southern Italy, where he would soon start a school. Supposedly, Pythagoras was accompanied to southern Italy by his mother and a single disciple of his. There is a legend about Pythagoras and a young man who challenged him to explain the utility and benefit of his teachings. Pythagoras responded with an offer to the boy. He would give the boy a quantity of silver for every lesson that he learned, since it was a material gain that he sought from his learning. The boy agreed. When Pythagoras taught the boy a certain number of lessons and doled out the stipulated amount of silver, he was left with none. By now the boy’s appetite for learning had grown to a point that he wanted to learn more lessons. He offered to pay Pythagoras an equal amount of silver for each lesson he taught him. This legend illustrates a belief in the intrinsic value of the love of learning. The boy in this tale could very well have been the disciple who accompanied Pythagoras to southern Italy. It is also possible that this person was Philolaus.
Pythagoras formed his secret society, teaching the knowledge he learned in Egypt and Babylon, as well as some of his own discoveries. As mentioned above, Pythagoras wrote nothing of a permanent nature. He probably did all of his teaching by talking to his students arranged in a semi-circle, resorting occasionally to sand drawings as a means of illustrating lessons, as would a modern teacher using a blackboard. One might almost imagine Pythagoras in front of his students. Using little more than a straightedge, two sticks joined by a string as a compass, and perhaps the shadows created by the sun, Pythagoras shared his discoveries with them. Off to the side was an initiate holding a device similar to a casino croupier’s stick for the purpose of smoothing out the sand drawing after the Master no longer needed it. Pythagoras was often referred to as the Master by his students. In some of their writings one finds the Greek, ayto§ ewa, meaning “the Master has said.”
The teachings of the Pythagorean Society were more than information to be learned or taught, they were a way of life, a philosophy that sought to guide and direct the Pythagoreans and explain nature. According to the Pythagoreans, “ . . . all things are number.” They believed that number had special qualities that could help to explain the cosmos. They believed that number was the key to unlocking nature’s secrets.
No one can say for sure what happened to the Pythagorean meeting place. Some accounts relate that it was destroyed by fire, set by some of the local people. Another account explains that the Pythagorean Society literally fell apart when their philosophy was dealt a serious and fatal attack, when a follower of Pythagoras, Hippasus, illustrated the existence of irrational numbers. Supposedly, he did so by taking the famous Pythagorean Theorem. According to this theorem, if one squared the sides of a right triangle and added the results, this sum would equal the square of the hypotenuse. Hippasus did not attack the theorem’s validity. What he did was to assign a measure of one unit to the sides of the right triangle, resulting in the measure of the hypotenuse to be the square root of two (Ã2). This is, of course, an irrational number, for it results in a nonrepeating decimal. Supposedly, Hippasus was drowned at sea for this revelation. Whether this is true is of little importance. One could believe that the Pythagorean Society whose philosophy was inextricably entwined with rational numbers could have fallen apart because of this.
The Pythagorean Society had ceased to exist, but not without first making mathematics part of a liberal education. The Pythagoreans divided the mathematical subjects into four main parts:
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Numbers absolute (arithmetic)
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Numbers applied (music)
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Magnitudes at rest (geometry)
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Magnitudes in motion (astronomy)
This so-called “quadrivium”, in the belief of the Pythagoreans, was what constituted the necessary course of studies for a liberal education.
Geometry is the main focus of this unit. It is a field of study that rests squarely on a foundation of axioms, from the Greek “axioma”, meanings things that are worthy. These axioms are premises held to be so self-evident that one does not need a proof. A certain amount of faith is needed to give one a point of departure, since these axioms cannot, in fact, be proved. The entire system of geometry is built upon these simplest of concepts.
The basic concepts of geometry are point, line, angle and surface, or plane surface. The definitions of these concepts that follow are also discussed in the lesson plan portion of this unit. We shall examine them here for the purpose of identifying those aspects that require a measure of faith in order to be accepted.
POINT
A point is the location of a point in space, yet it does not occupy space. A point has neither length, nor width, nor thickness, and it is indivisible. It is impossible to see a point, yet we use a dot to represent its existence, even though, by definition, there are an infinite number of points contained in that dot. The dot on this “i” is clearly visible. It has length, width and thickness; therefore, it is composed of a infinite number of points.
A point is an abstract concept. Clearly, a certain amount of faith is needed to support its existence. It is similar to the belief in the existence of atoms before one could actually see them.
LINE
If one were to set a point in motion the result would be a line. A line has length, yet it has no width or thickness. It, too, is an abstract concept.
ANGLE
Two lines which emanate from the same point produce an angle. The lines are called the sides of the angle, and the common point is called the vertex of the angle.
SURFACE or PLANE
If one were to move a line at right angles to its own direction the result is a surface, or plane surface. It is often referred to simply as a plane. Tabletops, walls, a pane of glass, or a floor are all examples of what a plane is, with one important distinction—a plane has no edges. A plane extends indefinitely in all directions. It has length and width, yet no thickness. Planes are two dimensional.
