Dungeons and Dragons Research Notes: Pythagoras

I'll be getting to the game mechanics as I'm still building the cultural foundation for Atlantis 600 B.C. Just be patient. Today, we are going to meet the famous, and inestimable, and mostly admired Pythagoras.

PYTHAGORAS AND THE PYTHAGOREAN THEOREM

In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (i.e. the two sides other than the hypotenuse).

Pythagoras is one of the most influentual Greek philosophers, and one of history's most influential mathematicians, he helped lay the ground work that Archimedes and other mathematicians would build on -- including Sir Isaac Newton, one of the Renassiance discoverers of Calculus. Pythagoras is also responsible for founding musicology and music theory.

Pythagoras was born in Samos, and later he would go on to found the Pythagorean school of Philosophy. The Pythagorean school was a religious and philosophic school which had a lasting influence on religion and philosophy -- including Gnostic and Aesthetic Christianity. Pythagoras was instructed in the Ionian school of thought and eventually made his way to Egypt where he was instructed by the Egyptian priesthood. Because of his disgust over the Tyranny of Polycrates, Pythagoras was exiled from Samos. Having moved to Italy, he founded the Pythagorean school in which men and women would flock to. However, Pythagoras' life on Earth came to an end when the Democratic party won out in Italy and the Aristocratic Pythagoreans attacked the Democrats and Pythagoras died in the struggle.

Accomplishments by Pythagoras:

• The Pythagorean Theorem is attributed to him. Necessarily, the history between Pythagoras and his theorem is quite complex.
• Pythagoras introduced the concept of reincarnation to the Greeks through the idea of mentempsychosis.
• A music theory that says that music is based on proportional intervals of four.
• The theory that the Universe is base 10. Including the postulation of a tenth planet, known as Gorea (see John Norman's novels).
• Pythagoras discovered the theory of mathematical proportion.
• Proposed that the Earth was a sphere that revolved on it's own axis.
• Killed a student for his discovery of irrational numbers.

Pythagorean Influence:

• Plato: Pythagoras or in a broader sense, the Pythagoreans, allegedly exercised an important influence on the work of Plato. According to R. M. Hare, his influence consists of three points: a) the platonic Republic might be related to the idea of "a tightly organized community of like-minded thinkers", like the one established by Pythagoras in Croton. b) there is evidence that Plato possibly took from Pythagoras the idea that mathematics and, generally speaking, abstract thinking is a secure basis for philosophical thinking as well as "for substantial theses in science and morals". c) Plato and Pythagoras shared a "mystical approach to the soul and its place in the material world". It is probable that both have been influenced by Orphism.
• Roman Influence: n the legends of ancient Rome, Numa Pompilius, the second King of Rome, is said to have studied under Pythagoras. This is unlikely, since the commonly accepted dates for the two lives do not overlap.

External Links:

• Pythagoras' biography on Hypatia-lovers.com.
• Wikipedia entry on Pythagoras.
• The Pythagorean Theorem on wikipedia and Apronus dot com.
• Pythagoras article on the Internet Encyclopedia of Philosophy.
• Pythagoras biography on MacTutor of Mathematics.
  

Research: Anaximander

The Father of recorded philosophy, Anaximander of Miletos was Thales' student. One of the pre-socratic philosophers, Anaximander was the philosopher who wrote his discoveries down while Thales did not. Anaximander succeeded Thales over the Milesian school of thought and counted Anaxamenes and Pythagoras as his pupils.

Though we know little of his life, Anaximander gave to the world such concepts as:

• The concept of the infinite Universe.
• The Obliquity of the eliptic.
• The first attempt to describe the motion of the Planets (which was eventually superceded by Sir Issac Newton by way of Calculus.)
• He tried to explain the Cosmos in a natural way, without using any mythological reference. A "godless" cosmology.
• Gave the first attempt to describe the Meteorological phenomenae such as thunder and lightning in natural process other than through the action of the god Zeus (pronounced "Zevs").
• Anaximander also speculated on a non-miraculous origin of Mankind. While still abiogenesis, his theories will lead to Louie Pasteur's discovery of Biogenesis (Life begets Life).
• Anaximander is said to pioneer the discipline of Cartography. Cartography is the discipline of drawing maps of the geosphere. Which is probably why Jason used an astrograph to sail to Colchis instead of a cartograph.
• Finally, Anaximander invented the Gnomon. This little invention consists of was simply a vertical pillar or rod mounted on a horizontal plane. The position of its shadow on the plane indicated the time of day. As it moves through its apparent course, the sun draws a curve with the tip of the projected shadow, which is shortest at noon, when pointing due south. The variation in the tip’s position at noon indicates the solar time and the seasons; the shadow is longest on the winter solstice and shortest on the summer solstice.

  However, the invention of the gnomon itself cannot be attributed to Anaximander because its use, as well as the division of days into twelve parts, came from the Babylonians. It is they, according to Herodotus' Histories (II, 109), who gave the Greeks the art of time measurement. It is likely that Anaximander was not the first to determine the solstices, because no calculation is necessary. On the other hand, equinoxes do not correspond to the middle point between the positions during solstices, as the Babylonians thought. As the Suda seems to suggest, it is very likely that Anaximander, with his knowledge of geometry, became the first Greek to accurately determine the equinoxes.

Anaximander's major contribution to Western thought was that he developed the framework with which early Greek Philosophy managed to develop from start to finish. This is something that Chinese philosophers like Mencius and Confucius would do in their own Civilization. This framework started Western Civilization on a course that would explain the World through basic observation and logical reasoning -- a search for the Truth without relying on superstition.

External Links:

• Anaximander, the philosopher who developed a non-mythical Cosmology.
• Wikipedia entry on Anaximander.
• Anaximander's biography at Science World.

Dungeons and Dragons Research: Thales

Thales is one of the first Natural Philosophers. He traveled extensively in Egypt and Mesopotamia and brought back Geometry and Astronomy with him. The Ancient Egyptians also taught him the principles of Astronomy they learned from a very wise man in the past. Thales is famous for laying down the basics of Geometry in Greek terms. Incuding a theorem that is still called Thales' Theorem today. It states that if the vertex of an angle moves in a semi-circle, and the end points of its arms remain at the end-points of the diameter, then the angle remains unaltered at the right angle.

The circle of this construction is known as the circle of Thales. Further, Thales also said that:

1. When two straight lines intersect one another the opposite angles are equal, and . . .
2. That two triangles having two angles and one side that are respectively equal are themselves equal; and .
3. that the sum of the angles in a triangle is equal to two right angles, and . . .
4. That is a circle is bisected by it's diameter, and . . .
5. that the angles at the base of an isoseles triangle are equal.

Thales influenced Greek Mathematics and Western Civilization by explaining the foundations of geometry and trigonometry. In his old age, he went by the name sophos, which means "sage." Thales spoke often of living a virtuous life and encouraged everyone to live to be true to themselves. His most famous quote: "The most difficult thing to do is to know thyself." -- Thales

External Links:

• Thales, the first philosopher
• Wikipedia entry on Thales.
• The internet encyclopedia of Philosophy's entry on Thales.
• A summary of Thales from the MacTutor of Mathemetical History.