One of the most famous of theorems from geometry (and/or trigonometry) is Pythagoras’ brain child: “The sum of the squares of the sides of a right triangle equals the square of the hypotenuse.”
One of the best examples of the theorem is the classic 3 - 4 - 5 triangle. Clearly,
32 + 42 = 9 + 16 = 25 = 52
But the theorem works for any right triangle (a right triangle being a triangle where one of the interior angles equals 90 degrees, and with the “hypotenuse” as the longest side, and opposite the 90 degree angle.) In general,
a2 + b2 = d2
Another special case is:
1 + (ÖF)2 = F2
Ha! The Golden Mean strikes again!
The Pythogorean Brotherhood
The theorem may not have been entirely the personal work of Phythagoras, but may have evolved from the Pythagoreans (the sum total of the members who followed the teaching of Pythagoras). As such, the Pythagoreans were a minor tradition during Classical Greece -- an apparent combination of Greek philosophy and eastern ideas. The philosophical school was founded by Pythagoras himself circa 530 B.C.E.
Many stories have been told about Pythagoras and his achievements, including his having competed and won prizes in the Olympic wrestling games at the age of eighteen, his having travelled to Egypt and Babylonia in order to learn the ancient wisdom of the priests there (thus the infusion of “eastern ideas”), and his alleged ability to tame a bear or stop an eagle in midair with a few magical words. He was also a masterful musician and physician, and had a Public Relations effort that way ahead of its time!
A bit of an independent, Pythagoras left his native Samos (when Polykrates became the local tyrant) and moved to Crotone in southern Italy. There he founded his school of philosophy and mysticism. He spent much of his time in a cave thinking, and allegedly discovered the hidden truths of the universe and wrote them down in legends or laws.
The Pythogorean sect was strict, ascetic (i.e. living in caves!) and centralized. Members swore an oath of silence to say nothing until having listened to the teachings of the Master for five years. The Master’s authority was total, his word unquestioned in any way whatsoever -- a tradition which gave birth to the idiom, “to swear on the word of the teacher”. The intellectual property of their discoveries were attributed to the sect or to the Master, not to their discoverers who remained anonymous -- so that it is not clear who the original discoverer of the famous theorem was. Strict vegetarians, the members of the sect were also prohibited from eating beans since the beans were thought to house the souls of the dead. Cooped up in a cave all day long would also justify this rule. The students were taught mathematics (mainly geometry, the supposedly “highest form of mathematics”), music, astronomy and magic.
The Pythagoreans believed in the eastern idea that the soul was divine and immortal, that it reincarnated after each death, that it was imprisoned in imperfect material bodies as a punishment, and that the goal of any rational person was to break free from this prison. The only way this could be achieved was by seeing and understanding the true reality. Or alternatively, to understand that all was illusion -- including the cave, beans, and so forth.
In the tradition of Greek philosophy, the Pythagoreans were more thinkers and mystics than practical magicians. They regarded the external use of magic (or mathematics) as filthy, and avoided it at all costs. They probably couldn’t even make change! Instead they turned inwards, studying the secrets of the universe. At the same time, they prefered not to reveal anything to what they considered to be the unworthy people outside their group, preferring instead to keep silent on the great truths only they knew. Which, unfortunately, can become very incestuous -- not to mention often wrong on certain subjects.
The Pythagoreans were primarily interested in the nature of space and geometry, but they studied all of reality with equal enthusiasm. They were theoreticians, attempting to under- stand the world, not manipulate or influence it to advantage. Most Pythagoreans turned inwards, trying to perfect their understanding of the cosmic harmony, and in order to focus their minds, they used geometry, meditation, and, of course, music (especially stringed instruments such as harps and lutes).
Obviously, the key (pardon the pun) is to go sit in a cave, meditate, study geometry, and listen to music. Perhaps even learn to play an harmonica. Or bagpipes (which would likely empty the cave quicker than the beans).
Alternatively, one can press on to A Graphics Description and Sacred Geometry or fall back to Sacred Mathematics or home, The Library of ialexandriah. Or take a ride on the A-train to Music, Geometry of Alphabets, or (if you feel ill at ease to make any definitive choice) Heisenberg’s Uncertainty Principle.
2003© Copyright Dan Sewell Ward, All Rights Reserved [Feedback]