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http://platea.pntic.mec.es/~aperez4/html/grecia/grecia.html

figura de Pitágoras está envuelta en un halo de leyenda, misticismo y hasta de culto religioso. Y no es tan extraño si pensamos que fue contemporáneo de Buda, de Confucio y de Lao-Tse (los fundadores de las principales religiones orientales)

El término "matemática", al igual que el de filosofía, se le debemos a él.

¿Cuáles son las principales aportaciones matemáticas de la escuela pitagórica?...
La primera y quizás la más importante el introducir la necesidad de demostrar las proposiciones matemáticas de manera inmaterial e intelectual, al margen de su sentido práctico. Los pitagóricos dividieron el saber científico en cuatro ramas: la aritmética o ciencia de los números - su lema era "todo es número" -, la geometría, la música y la astronomía.

Pitágoras descubrió que existía una estrecha relación entre la armonía musical y la armonía de los números.

Si pulsamos una cuerda tirante obtenemos una nota. Cuando la longitud de la cuerda se reduce a la mitad, es decir en relación 1:2 obtenemos una octava.
Si la longitud era 3:4 obtenemos la cuarta y si es 2:3 tenemos la quinta.

Pero lo que colmó de gozo a Pitágoras, hasta el punto de mandar sacrificar un buey a los dioses, fue la demostración del famoso teorema. Por desgracia, el secreto que imponía las normas de la sociedad ha hecho imposible que esta demostración llegue a nuestro conocimiento, aunque podemos deducir que no sería muy distinta de la que Euclides nos brinda en sus Elementos.

Sin duda es el teorema que cuenta con más número de demostraciones.
Scott Loomis reunió y publicó a principios de este siglo 367 demostraciones.

Pythagoras is one of the most famous mathematicians. He was born in Samos, Greece around 582 B.C. and he died around 475 B.C. Although he made many important contributions to math we do not know much about him. We do not have anything that he wrote because he was the leader of a secret society where they didn’t write down what they did. All that we have about Pythagoras are a few biographies from a long time ago.

Some accounts state that Pythagoras went to Egypt in about 535 B.C. to learn more about mathematics and astronomy. In Egypt Pythagoras visited many of the temples and talked to the priests, but he was not allowed in any of the temples except for one. In 525 B.C. the king of Persia invaded Egypt and Polycrates sent 40 ships to help the Persians invade Egypt. During the invasion Pythagoras was captured and taken to Babylon. After Polycrates and the king of Persia died Pythagoras went back to Samos, but nobody knows how he was freed.

( http://library.thinkquest.org/J0110961/pyth.htm

 
Pythagoras also experimented with music. He found a relationship between music and math. Pythagoras found the relationship one day when he was walking past a blacksmith shop and heard the hammers hit the anvil. When Pythagoras heard the hammers hit the anvil he noticed that each hammer made a different sound according to the weight of the hammer. So, the weight of the hammer (i.e. number) decides the sound the hammer will make. What Pythagoras found out was that math controls the tone of music.

http://library.thinkquest.org/J0110961/pyth.htm

 
La estrella pitagórica

"Eso que es siempre lo mismo" o la esencia constante de la belleza podría consistir de por ejemplo las proporciones en las dimensiones. Esta idea se atribuye a Pitágoras (ca. 532 AC) del que se dice que había descubierto el hecho de que ciertas proporciones aritméticas en los instrumentos musicales, como las longitudes de las cuerdas, producen armonía de tonos (a la derecha, una ilustración de la Theorica Musice, de Gafurio, 1492). Sobre la base de estas armonías musicales los griegos intentaron explicar también la belleza en las proporciones del cuerpo humano, de la arquitectura y otros objetos.

Vitruvio (I:III:2) decía que un edificio es bello cuando la apariencia de la obra es agradable y de buen gusto, y cuando sus miembros son de las debidas proporciones de acuerdo a los principios correctos de "simetría" (donde "simetría" significa "una concordancia correcta entre los miembros de la obra misma, y relación entre las diferentes partes y el esquema general del conjunto en concordancia con una cierta parte elegida como estándar. -- La definición de simetría se encuentra en I:II:4).

Durante la Edad Media, las proporciones y relaciones numéricas fueron consideradas atributos importantes de los objetos, como se puede ver en el "libro de esbozos" de Villard de Honnecourt, del s. XIII, una ilustración de lo cual se ve a la izquierda. El documento no contiene explicación para estas figuras.

El Renacimiento reavivó de nuevo el estudio de las proporciones pitagóricas.

PYTHAGOREAN HARMONICS: FROM PYTHAGORAS TO NEWTON
Associated to Place: Hellas > articles -- by * DIonysia Xanthippos (50 Articles), Historical Article 1 Featured May 23 , 2005
by DIonysia Xanthippos and Sileinos Socrates

Musical chords and ratios, as demonstrated by Pythagoras. A woodcut from the THEORICA MUSICE, Naples, 1492, by Franchino Gafurio, music theorist and choirmaster of Milan cathedral.In this late medieval or early Renaissance woodcut, Pythagoras is shown discovering the physical and mathematical ratios that underlie the basic chords and the most harmonious musical intervals. Tradition says that after noticing the different sounds of blacksmiths’ hammers, Pythagoras embarked on the experiments shown here: striking bells, water glasses and strings; and blowing flute pipes. The hammers, bells, glasses, strings and pipes are labelled 4, 6, 8, 9, 12 and 16, to indicate their weights and sizes; and each experiment supposedly demonstrates the same interval or chord: the octave, whose ratio is 1/2 or 1:2, shown here as the ratio 8:16.

In the panel at upper left Pythagoras watches the blacksmiths take turns hitting an anvil. One strikes the base note, soon to be followed by its octave, supposedly sounded by the 8-pound hammer at the upper right. In a charming attempt at reconciling Christian piety with pagan wisdom, the artist has relabelled the Pythagoras figure as IV BAL — Latin lettering for JUBAL, whom the Bible declares to be the father of music.

In the panel at upper right Pythagoras is shown striking two bells at once: a size 8, or 8-pound bell, and a bell twice as big (#16), supposedly producing an octave, whose ratio is therefore 8/16 or 1/2. Just below he is shown performing the same experiment with glasses filled with varying amounts of water. Here 8 and l6 ounces also produce the high and low tones of an octave. By adding several more glasses, and rubbing their rims instead of striking them with sticks, Pythagoras could have invented the Glass Harmonica that Benjamin Franklin (re)invented and for which Mozart composed a delightful piece. In ancient Greece, however, even centuries after Pythagoras, even unblown glass was surprisingly rare and expensive. So it is doubtful that Pythagoras ever tinkered with glassware of any sort. Or with bells, for that matter.

In the panel at lower left Pythagoras is shown striking with sticks six stretched strings of ox sinews and sheep guts. Instead of a single string “stopped” at different lengths, or several strings of different lengths, the strings are "tuned" by different tensions — stretched by different weights. Pythagoras is striking the two strings whose weights are labelled 8 and l6, again supposedly producing an octave. The artist has been a bit careless, however, and it looks as if Pythagoras might be hitting instead the strings stretched by the 9 and l2 pound weights. In that case, he would be sounding the chord called a "fourth," whose ratio is 3:4 (= 9:12).

In the lower right panel Pythagoras, assisted by his disciple Philolaus, demonstrates the harmonic chords and ratios with flute pipes. Here also the chord being played is the octave, with Philolaus sounding the base note on a pipe l6 units long, while his master blows a pipe half its length, one labelled 8.

What these five experiments supposedly show is that the harmonic chord or interval called the octave always corresponds to a mathematical and physical difference whose ratio is 1:2.

 

By using other pairs of the instruments in each series Pythagoras and his followers could demonstrate the mathematical ratios underlying other basic chords or intervals. Thus, bells, strings or pipes sized, weighted, cut or stopped in the ratios 6 to 9 (= 2:3) were said to produce the chord or harmonic interval called a "fifth." And those sized or weighted 9 and 12 (= 3:4) would produce a chord or interval called the "fourth."

Why, then, does the series of numbers in the woodcut run from 4 to 16? For simplicity, why aren't there just four basic weights and lengths here, labelled 1, 2, 3 and 4? The reason is that theorists wanted to include the ratio for the interval between two whole tones on the musical scale, which has the seven whole notes plus the octave: Do re mi fa sol la ti do. The ratio for the physical difference between two successive whole notes is 8:9. In other words, a string has to vibrate along 8/9ths of its length to produce a tone that is one whole note above the base note produced by its entire unstopped length. Therefore the ratio for the interval between these two notes is 8:9. And the series of numbers from 4 to 16 is the smallest series that includes this ratio in addition to the ratios for the octave (1:2 = 8:16) and those for the fifth and fourth.

http://www.ancientworlds.net/aw/Article/568210

A la derecha, un estudio de Leonardo, que muestra la relación 1:3:1:2:1:2 en las proporciones del rostro humano.

(Img: www.eticayempresa.com/arteologia/255.htm)

Musical chords and ratios, as demonstrated by Pythagoras. A woodcut from the THEORICA MUSICE, Naples, 1492, by Franchino Gafurio, music theorist and choirmaster of Milan cathedral

Demostración del matemático inglés Henry Perigal al que se le atribuye una ingeniosa comprobación del teorema de Pitágoras. Sobre el mayor de los cuadrados construidos sobre los catetos se determina el centro (no necesariamente ha de ser este punto) y se trazan dos rectas paralela y perpendicular a la hipotenusa del triángulo. Con las cuatro piezas obtenidas más el cuadrado construido sobre el otro cateto podemos cubrir el cuadrado contruido sobre la hipotenusa.

( Img:www.arrakis.es/ ~mcj/teorema.htm)

Otra demostración del teorema de pitágoras. En este vemos cumo el cuadrado de la derecha coincide exactamente en el centro de otro cuadrado formado por las secciones resultado de la descomposición de otro cuadrado que tiene concretamente el doble de su área.

( Img:Enrique Rebasa. Escuela de Arquitectura de Madrid.)


Pythagoras of Samos (560BC - 480BC)


"Through Vibration comes Motion
Through Motion comes Color
Through Color comes Tone"

He was a Greek philosopher who was responsible for important developments in the history of mathematics, astronomy, and the theory of music. He founded the Pythagorean Brotherhood and formulated principles that influenced the thoughts of Plato and Aristotle. The influence of Pythagoras is so widespread, and coupled with the fact that no writings of Pythagoras exist today, this short article will attempt to guide the reader through the life of this most remarkable teacher.

He traveled widely in his youth with his father Mnesarchus, who was a gem merchant from Tyre. His family settled in the homeland of his mother, Pythais, on the island of Samos, where he studied with the philosopher Pherekydes. He was introduced to mathematical ideas and astronomy by Thales, and his pupil Anaximander in Miletus when he was between 18 and 20 years old. Thales advised Pythagoras to travel to Egypt to learn more of these subjects. Leaving Miletus, Pythagoras went first to Sidon, where he was initiated into the mysteries of Tyre and Byblos. It is claimed that Pythagoras went onto Egypt with a letter of introduction written by Polycrates, making the journey with some Egyptian sailors who believed that a god had taken passage on their ship. Arriving in Egypt, Pythagoras tried to gain entry into the Mystery Schools of that country. He applied again and again, but he was told that unless he goes through a particular training of fasting and breathing, he cannot be allowed to enter the school. Pythagoras is reported to have said, " I have come for knowledge, not any sort of discipline." But the school authorities said," we cannot give you knowledge unless you are different. And really, we are not interested in knowledge at all, we are interested in actual experience. No knowledge is knowledge unless it is lived and experienced. So you will have to go on a 40 day fast, continuously breathing in a certain manner, with a certain awareness on certain points." After 40 days of fasting and breathing, aware, attentive, he was allowed to enter the school at Diospolis. It is said that Pythagoras said,"You are not allowing Pythagoras in. I am a different man, I am reborn. You were right and I was wrong, because then my whole standpoint was intellectual. Through this purification, my center of being has changed. Before this training I could only understand through the intellect, through the head. Now I can feel. Now truth is not a concept to me, but a life."

He spent the next 22 years perfecting himself in mathematics, astronomy, music, and was initiated into the Egyptian Mysteries. When Cambyses II, the king of Persia invaded Egypt in 525BC, he made Pythagoras his prisoner and sent him to Babylon. He utilized this misfortune as an opportunity for growth, and for the next 12 years he studied with the Magi and was initiated into the Chaldean Mysteries. Leaving Babylon, he made his way through Persia to India, where he continued his education under the Brachmanes. At that time India was still feeling the effects of the spiritual revival brought about by Gautama the Buddha. Although Pythagoras arrived in India too late to come into personal contact with the Buddha, he was greatly influenced by his teachings. He went to India a student, he left it as a teacher, and even to this day he is known in that country as Pitar Guru, and as Yavanacharya, the Ionian Teacher.

Pythagoras was 56 years old when he finally returned to his homeland. When he arrived in Samos he found the island crushed and ruined, its temples and schools closed, its wise men fleeing from the tyranny and persecution of the Persian conquerors. Instead of being welcomed by his countrymen, Pythagoras found them indifferent to the wisdom he was eager to impart. He left Samos and went to southern Italy, settling in Crotona, a town situated on the Gulf of Tarentum. He was invited to speak before the Senate of Crotona, and so greatly impressed them with his wisdom, that they decided to build him an institute, which would serve as a school of philosophy and an academy of science. Although it was understood that it would be patterned after the Mystery Schools, there was nothing about the place suggesting secrecy save a statue of Hermes Trismegistus at the door of the inner school with the words on the pedestal: "Let no profane enter here."

The institute was comprised of three orders. The outer order was called the 'akoustici', who lived in their own houses only coming to the institute during the day. They were allowed their own possessions and were not required to be vegetarians. Acceptance into this outer society was granted after a 3 year probationary period. Both men and women were permitted to become members of the order, in fact 28 women were admitted to the institute. The inner order of the society was called the 'mathematikoi', who lived permanently with the society, and had no personal possessions and were vegetarians. They were taught by Pythagoras himself and obeyed strict rules. The third level of initiation within the institute was the 'electi', who were instructed in the secret processes of psychic transmutation, how to heal with sound, and lived a strict discipline in accordance with the code of the Great Mystery Schools.

The daily life of a student at Crotona followed a strict schedule. At sunrise they engaged in meditation, pronouncing a mantram on a certain tone. They reviewed all their actions of the previous day and planned the coming day in full detail. After breakfast they took a solitary walk and went to the gymnasium for exercise. The rest of the morning was spent in study. At noon the Order ate together in small groups dining on bread and honey. After lunch students could receive their relatives and friends in the gardens of the institute. This was followed by another walk in the company of other students. At the close of the day, they ate together and read aloud. Before retiring each student again meditated and chanted his evening mantram. Those who were unable to stand the discipline left the school and went out again into the world. Even in the higher degrees of the institute, some occasionally failed by breaking their pledge of secrecy or some other rule which bound them. These students were expelled from the institute, and a tomb bearing their name was erected in the garden. Pythagoras taught that such a student was dead."His body appears among men," he said, " but his soul is dead. Let us weep for it! "

In Astronomy Pythagoras taught that the Earth was a sphere at the center of the universe. He recognized that the orbit of the moon was inclined to the equator of the earth, and he was one of the first to realize that Venus as an evening star, was the same planet as the morning star. He taught that the movements of the planets traveling through the universe created sounds, and could be perceived by those who were trained to hear them. This music of the spheres could be replicated using a single stringed instrument called the monochord. Pythagoras used the monochord to explain musical intervals and harmonics to his students. He taught how harmony may be produced when tuning the high and low notes in the octave, thereby laying the foundation for many of the theories and teachings that have come down through the musical traditions.

Pythagoras observed that when a blacksmith struck his anvil, different notes were produced according to the weight of the hammer. That if you take 2 strings in the same degree of tension, and then divide one of them exactly in half, when they are plucked, the pitch of the shorter string is exactly one octave higher than the longer string. He also discovered that if the length of the 2 strings are in relation to each other 2:3, the difference in pitch is called a fifth. Pythagoras stressed that different musical modes have different effects on the person who hears them, and that music could be applied to healing illness both mental and physical. In 513BC he went to Delos to nurse his old teacher, Pherekydes who was dying. He remained at his bedside playing his lyre and feeding him until he died. In 508BC the Pythagoreon Society at Croton was attacked by Cylon, a noble of Croton itself. Pythagoras escaped to Metapontium and most authors say he died there. Evidence is unclear as to when and where the death of Pythagoras accurred.

The beliefs that Pythagoras held were:

1) that at its deepest level, reality is mathematical in nature
2) that philosophy can be used for spiritual purification
3) that the Soul can rise to union with the Divine
4) that certain symbols have a mystical significance
5) that all brothers of the Order should observe strict loyalty and secrecy

Pythagoras was the first to call the heavens a universe and the earth round. That the Soul was immortal, and that it changes from one body to another. The Pythagorean Brotherhood was one of the worlds earliest unpriestly cooperative scientific societies, if not the first, and that its members invented the multiplication table, and raised important scientific problems which were solved 15 centuries later.

http://9waysmysteryschool.tripod.com/sacredsoundtools/id13.html

 

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