Essay, Research Paper: Pythagoras

Our generation receives an abundance of credit for the technology and innovations we, as creators, invent. We probably receive too much recognition for the sciences that the geniuses of our era bring into existence. Without the help of previous generations, the modern day world would look somewhat like it did 200 years ago. There have been countless contributors toward our high-tech, busy lives that are not acknowledged and do not get the credit that they deserve.
Pythagoras was both a mathematician and scientist. The time, effort, and ideas that he produced, allow us to use his achievements for our own purposes. He deserves the honor that we give to others.
Pythagoras of Samos was not only a Greek philosopher, but he was also a deeply religious man who was responsible for the important developments in the history of mathematics, astronomy, and the theory of music! The basis of Pythagoras? philosophy was number. Number ruled the universe. Number was the basic description of everything. He had a theory that every number had a special character or trait. For instance, one is reason; two is opinion; four is justice. Odd numbers were masculine. Evens were feminene. Pythagoras then came to the conclusion that odds represented good, and evens, bad. Today we have our own unlucky 13 which is a remnant of Pythagoras? ideas.
Pythagoras was born around the year 560 B.C. He died close to 480 B.C. As a young man he migrated to Croton and founded a philosophical and religious school that attracted many followers. These followers, Pythagoreans, were dedicated and committed to Pythagoras. They believed that the soul is immortal and separable from the body. It is reincarnated in different animal bodies until it completes the cycle of all creatures. For this reason they practiced vegetarianism. By leading a pure life, an individual might secure the release of his or her soul from all flesh.
In astronomy, the Pythagoreans were well aware of the periodic numerical relations of heavenly bodies. Pythagoreans believed that the earth itself was in motion. The most important discovery of this school, which upset Greek mathematics as well as the Pythagoreans? own beliefs, was that whole numbers and their ratios could account for geometrical properties. This result showed the existence of irrational numbers.

Geometry is a very complex subject made up of postulates, theorems, and corollaries. Every proof and reason given to a specific problem has philosophy and vitality to it. A major advance in geometry came when Pythagoras uncovered what came to be known as the ?Pythagorean theorem?. The theorem is what made Pythagoras a known mathematician as well as a philosopher and astronomer. The Pythagorean theorem is a statement that the square of the hypotenuse of a right triangle is equal to the sum of the squares of its other two sides. At the time of Pythagoras, the square of a number ?n? was represented by the area of a square with side of length ?n?. Using this representation, the Pythagorean theorem may be stated: The area of the square on the hypotenuse of a right triangle is equal to the sum of the areas of the squares on the legs. Pythagoras, or perhaps one of his students, proved that if triangle ABC is a right triangle with a right angle at C, then c2 = a2 + b2.
The theorem was not the only great achievement of Pythagoras. Before him, geometry was full of rules of thumb and postulates. Proof is now taken for granted so much as to the fact that we can not imagine what could have possibly proceeded it. Pythagoras made a science out of a collection of rules that had been found by observation of trial and error.

Although many names of musicians are recorded in ancient sources, none played a more important role in the development of Greek musical thought than Pythagoras. Pythagoras through his wisdom discovered the mathematical rationale of musical tones from the weights of hammers used by smiths. He is thus given credit for discovering that the interval of an octave is rooted in the ratio 2:1. Followers of Pythagoras applied these ratios to lengths of a string on an instrument called a canon, or monochord, and thereby were able to determine mathematically the ?intonation? of an entire musical system. The Pythagoreans saw these ratios as governing forces in the cosmos as well as in sounds, and Plato's Timaeus describes the soul of the world as structured according to these same musical ratios. For the Pythagoreans, as well as for Plato, music consequently became a branch of mathematics as well as an art. This tradition of musical thought flourished throughout antiquity in such theorists as Nicomachus of Gerasa (2d century AD) and Ptolemy (2d century AD) and was transmitted into the Middle Ages by Boethius (6th century AD). The mathematics and ?intonation? of the Pythagorean tradition therefore became a crucial influence in the development of music in medieval Europe. The most basic musical interval is the octave, which occurs when the frequency of any tone is doubled or halved. Two tones set one octave apart create a feeling of identity, or the duplication of a single pitch in a higher or lower register. Pythagoras discovered similar simple ratios for other important intervals. These acoustically "pure" intervals, however, lead to mistunings and other problems in chromatic music. Various tuning systems in which minor adjustments are made in the size of Pythagorean intervals have been devised to deal with this problem. The interval between any two adjacent tones is called a half-step. An interval equal to two half-steps (such as between two white keys separated by a black key) is termed a whole-step. This contribution to music was appreciated first by the Greeks, but now, it is recogized all over the world as being the most vital step in music history.
Philosophy is related to most disciplines, yet it also differs from them. On one hand, philosophers were strongly impressed by the degree of certainty and rigor that appeared to exist in mathematics as compared to any other subject. Some, like Pythagoras, felt that mathematics must be the key to understanding reality. Plato claimed that mathematics provided the forms out of which everything was made. Aristotle, on the other hand, held that mathematics was about ideal objects rather than real ones; he held that mathematics could be certain without telling us anything about reality. He believed in his philsophies with such devotion that he was restricted in his own lifestyle. There was no one telling him what to do, except for his conscience and heart which brought him total peace within himself.
We should model ourselves to live like Pythagoras. Maybe not with his specicific beliefs and philsophies, but rather with the eagerness to learn and grow and be at harmony with nautre and those around us. Every indivual?s extent of knowledge only stretches to a certain point. For Pythagoras, that boundry was endless. His stamina and capaicty to possess the information that he did, was unprecednt to anyone else. None of us may ever be as smart, famous, or peaceful as Pythagoras, but we can try. His contributions to so many subjects is unthinkable in our times. Our focus is generated at one target or subject. We mater that subject and use it in our livlihoods. Pythagoras mastered several subjects and related them all to his everyday life. He tied in music with math and astronomy with philsophy.
Pythagoras and the Pythagoreans were the lowest wrung on the scientific ladder. Their attributes toward people and religion made such a great impression on the rest of the world, that here, 15 centuries later, we are still using their work in our classrooms and research rooms as if it is new information. The Pythagorean theory that deals with triangles is so vital to students of math, it is almost unheard of to go through high school having not learned it.