thumb|upright=1.3|In [[Raphael's fresco The School of Athens, Pythagoras is shown writing in a book as a young man presents him with a tablet showing a diagrammatic representation of music theory on a lyre above a drawing of the sacred tetractys.]]
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Croton, in modern Calabria (Italy) circa 530 BC. Early Pythagorean communities spread throughout Magna Graecia.
Already during Pythagoras's life it is likely that the distinction between the akousmatikoi ("those who listen"), who are conventionally regarded as more concerned with religious, and ritual elements, and associated with the oral tradition, and the mathematikoi ("those who learn") existed. The ancient biographers of Pythagoras, Iamblichus () and his master Porphyry () seem to make the distinction between the two as that of 'beginner' and 'advanced'. As the Pythagorean cenobites practiced an esoteric path, like the mystery schools of antiquity, the adherents, akousmatikoi, following initiation, became mathematikoi. It is wrong to say that the Pythagoreans were superseded by the Cynics in the 4th century BC, but it seems to be a distinctive mark of the Cynics to disregard the hierarchy and protocol, ways of initiatory proceedings significant for the Pythagorean community. The Greek philosophical traditions subsequently became more diverse. The Platonic Academy was arguably a Pythagorean cenobitic institution, outside the city walls of Athens in the 4th century BC; a sacred grove dedicated to Athena, and Hecademos (Academos). As contemporaries seem to have believed, the academy may have existed since the Bronze Age, even prior to the time of the Trojan War. Yet according to Plutarch it was the Athenian strategos (general) Kimon () who converted this, "waterless and arid spot into a well-watered grove, which he provided with clear running-tracks and shady walks". Plato lived almost a hundred years later, circa 427 to 348 BC. On the other hand, it seems likely that this was a part of the rebuilding of Athens led by Kimon and Themistocles, following the Achaemenid destruction of Athens in 480–479 BC during the war with Persia. Kimon is at least associated with the building of the southern Wall of Themistocles, the city walls of ancient Athens. It seems likely that the Athenians saw this as a rejuvenation of the sacred grove of Academos.
Following political instability in Magna Graecia, some Pythagorean philosophers moved to mainland Greece while others regrouped in Rhegium. By about the majority of Pythagorean philosophers had left Italy. Pythagorean ideas exercised a marked influence on Plato and through him, on all of Western philosophy. Many of the surviving sources on Pythagoras originate with Aristotle and the philosophers of the Peripatetic school.
As a philosophic tradition, Pythagoreanism was revived in the giving rise to Neopythagoreanism. The worship of Pythagoras continued in Italy and as a religious community Pythagoreans appear to have survived as part of, or deeply influenced, the Bacchic cults and Orphism.
History
thumb|The [[Plimpton 322|Plimpton 322 tablet records Pythagorean triples from Babylonian times.]]
thumb|Animation demonstrating the smallest whole number Pythagorean triple,
thumb|Bust of [[Pythagoras, Musei Capitolini, Rome.]]
Pythagoras was already well known in ancient times for his supposed mathematical achievement of the Pythagorean theorem. Pythagoras had been credited with discovering that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. In ancient times Pythagoras was also noted for his discovery that music had mathematical foundations. Antique sources that credit Pythagoras as the philosopher who first discovered music intervals also credit him as the inventor of the monochord, a straight rod on which a string and a movable bridge could be used to demonstrate the relationship of musical intervals.
Much of the surviving sources on Pythagoras originated with Aristotle and the philosophers of the Peripatetic school, which founded historiographical academic traditions such as biography, doxography and the history of science. The surviving 5th century BC sources on Pythagoras and early Pythagoreanism are void of supernatural elements, while surviving 4th century BC sources on Pythagoras's teachings introduced legend and fable. Philosophers who discussed Pythagoreanism, such as Anaximander, Andron of Ephesus, Heraclides and Neanthes had access to historical written sources as well as the oral tradition about Pythagoreanism, which by the 4th century BC was in decline. Neopythagorean philosophers, who authored many of the surviving sources on Pythagoreanism, continued the tradition of legend and fantasy.
The earliest surviving ancient source on Pythagoras and his followers is a satire by Xenophanes, on the Pythagorean beliefs on the transmigration of souls. Xenophanes wrote of Pythagoras that:
In a surviving fragment from Heraclitus, Pythagoras and his followers are described as follows:
Two other surviving fragments of ancient sources on Pythagoras are by Ion of Chios and Empedocles. Both were born in the 490s, after Pythagoras's death. By that time, he was known as a sage and his fame had spread throughout Greece. According to Ion, Pythagoras was:
Empedocles described Pythagoras as "a man of surpassing knowledge, master especially of all kinds of wise works, who had acquired the upmost wealth of understanding". In the 4th century BC the Sophist Alcidamas wrote that Pythagoras was widely honored by Italians.
Today scholars typically distinguish two periods of Pythagoreanism: early-Pythagoreanism, from the 6th until the 5th century BC, and late-Pythagoreanism, from the 4th until the 3rd century BC. The Spartan colony of Taranto in Italy became the home for many practitioners of Pythagoreanism and later for Neopythagorean philosophers. Pythagoras had also lived in Crotone and Metaponto, both of which were Achaean colonies. Early-Pythagorean sects lived in Croton and throughout Magna Graecia. They espoused to a rigorous life of the intellect and strict rules on diet, clothing and behavior. Their burial rites were tied to their belief in the immortality of the soul.
Early-Pythagorean sects were closed societies and new Pythagoreans were chosen based on merit and discipline. Ancient sources record that early-Pythagoreans underwent a five-year initiation period of listening to the teachings (akousmata) in silence. Initiates could through a test become members of the inner circle. However, Pythagoreans could also leave the community if they wished. Iamblichus listed 235 Pythagoreans by name, among them 17 women whom he described as the "most famous" women practitioners of Pythagoreanism. It was customary that family members became Pythagoreans, as Pythagoreanism developed into a philosophic tradition that entailed rules for everyday life and Pythagoreans were bound by secrets. The home of Pythagoras was known as the site of mysteries.
Pythagoras had been born on the island of Samos at around 570 BC and left his homeland at around 530 BC in opposition to the policies of Polycrates. Before settling in Croton, Pythagoras had traveled throughout Egypt and Babylonia. In Croton, Pythagoras established the first Pythagorean community, described as a secret society, and attained political influence. In the early 5th century BC Croton acquired great military and economic importance. Pythagoras emphasised moderation, piety, respect for elders and of the state, and advocated a monogamous family structure. The Croton Council appointed him to official positions. Among others Pythagoras was in charge of education in the city. His influence as political reformer reputedly extended to other Greek colonies in southern Italy and in Sicily. Pythagoras died shortly after an arson attack on the Pythagorean meeting place in Croton.
The anti-Pythagorean attacks in were headed by Cylon of Croton. Pythagoras escaped to Metapontium. After these initial attacks and the death of Pythagoras, Pythagorean communities in Croton and elsewhere continued to flourish. At around 450 BC attacks on Pythagorean communities were carried out across Magna Graecia. In Croton, a house where Pythagoreans gathered was set on fire and all but two of the Pythagorean philosophers burned alive. Pythagorean meeting places in other cities were also attacked and philosophic leaders killed. These attacks occurred in the context of widespread violence and destruction in Magna Graecia. Following the political instability in the region, some Pythagorean philosophers fled to mainland Greece while others regrouped in Rhegium. By about 400 BC the majority of Pythagorean philosophers had left Italy. Archytas remained in Italy and ancient sources record that he was visited there by young Plato in the early 4th century BC. The Pythagorean schools and societies died out from the 4th century BC. Pythagorean philosophers continued to practice, albeit no organised communities were established. According to Cicero (de Orat. III 34.139), Philolaus was teacher of Archytas. According to the Neoplatonist philosopher Iamblichus, Archytas in turn became the head of the Pythagorean school about a century after the Pythagoras's death. Philolaus, Eurytus and Xenophilus are identified by Aristoxenus as the teachers of the last generation of Pythagoreans.
Philosophic traditions
Following Pythagoras's death, disputes about his teachings led to the development of two philosophical traditions within Pythagoreanism in Italy: akousmatikoi and mathēmatikoi. The mathēmatikoi recognised the akousmatikoi as fellow Pythagoreans, but because the mathēmatikoi allegedly followed the teachings of Hippasus, the akousmatikoi philosophers did not recognise them. Despite this, both groups were regarded by their contemporaries as practitioners of Pythagoreanism.
The akousmatikoi were superseded in the 4th century BC as a significant mendicant school of philosophy by the Cynics. Mathēmatikoi philosophers were in the 4th century BC absorbed into the Platonic school of Speusippus, Xenocrates and Polemon. As a philosophic tradition, Pythagoreanism was revived in the 1st century BC, giving rise to Neopythagoreanism. The worship of Pythagoras continued in Italy in the two intervening centuries. As a religious community Pythagoreans appear to have survived as part of, or deeply influenced, the Bacchic cults and Orphism.
Akousmatikoi
thumb|Pythagoreans celebrate sunrise, 1869 painting by [[Fyodor Bronnikov. ]]
The akousmatikoi believed that humans had to act in appropriate ways. The Akousmata (translated as "oral saying") was the collection of all the sayings of Pythagoras as divine dogma. The tradition of the akousmatikoi resisted any reinterpretation or philosophical evolution of Pythagoras's teachings. Individuals who strictly followed most akousmata were regarded as wise. The akousmatikoi philosophers refused to recognise that the continuous development of mathematical and scientific research conducted by the mathēmatikoi was in line with Pythagoras's intention. Until the demise of Pythagoreanism in the 4th century BC, the akousmatikoi continued to engage in a pious life by practicing silence, dressing simply and avoiding meat, for the purpose of attaining a privileged afterlife. The akousmatikoi engaged deeply in questions of Pythagoras's moral teachings, concerning matters such as harmony, justice, ritual purity, and moral behavior.
Mathēmatikoi
thumb|The Archytas curve
The mathēmatikoi acknowledged the religious underpinning of Pythagoreanism and engaged in mathēma (translated as "learning" or "studying") as part of their practice. While their scientific pursuits were largely mathematical, they also promoted other fields of scientific study in which Pythagoras had engaged during his lifetime. A sectarianism developed between the dogmatic akousmatikoi and the mathēmatikoi, who in their intellectual activism became regarded as increasingly progressive. This tension persisted until the 4th century BC, when the philosopher Archytas engaged in advanced mathematics as part of his devotion to Pythagoras's teachings.
Early-Pythagorean philosophers such as Philolaus and Archytas held the conviction that mathematics could help in addressing important philosophical problems. In Pythagoreanism numbers became related to intangible concepts. The one was related to the intellect and being, the two to thought, the number four was related to justice because 2 * 2 = 4 and equally even. A dominant symbolism was awarded to the number three, Pythagoreans believed that the whole world and all things in it are summed up in this number, because end, middle and beginning give the number of the whole. The triad had for Pythagoreans an ethical dimension, as the goodness of each person was believed to be threefold: prudence, drive and good fortune.
Pythagoreans thought numbers existed "outside of [human] minds" and separate from the world. They had many mystical and magical interpretations of the roles of numbers in governing existence.
Music
right|thumb|Medieval woodcut by [[Franchino Gaffurio, depicting Pythagoras and Philolaus conducting musical investigations.]]
Pythagoras pioneered the mathematical and experimental study of music. He objectively measured physical quantities, such as the length of a string and discovered quantitative mathematical relationships of music through arithmetic ratios. Pythagoras attempted to explain subjective psychological and aesthetic feelings, such as the enjoyment of musical harmony. Pythagoras and his students experimented systematically with strings of varying length and tension, with wind instruments, with brass discs of the same diameter but different thickness, and with identical vases filled with different levels of water. Early Pythagoreans established quantitative ratios between the length of a string or pipe and the pitch of notes and the frequency of string vibration.
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions."
The fact that mathematics could explain the human sentimental world had a profound impact on the Pythagorean philosophy. Pythagoreanism became the quest for establishing the fundamental essences of reality. Pythagorean philosophers advanced the unshakable belief that the essence of all things are numbers and that the universe was sustained by harmony. Pythagoras had in his teachings named numbers and the symmetries of them as the first principle and called these numeric symmetries harmony. This numeric harmony could be discovered in rules throughout nature. Numbers governed the properties and conditions of all beings and were regarded the causes of being in everything else. Pythagorean philosophers believed that numbers were the elements of all beings and the universe as a whole was composed of harmony and numbers.
Cosmology
thumb|According to a collection of ancient philosophical texts by [[Stobaeus in the 5th century AD, Philolaus believed there was a "Counter-Earth" (Antichthon) orbiting a "central fire" but not visible from Earth. ]]
The philosopher Philolaus, one of the most prominent figures in Pythagoreanism,
It is not known whether Philolaus believed Earth to be round or flat, but he did not believe the earth rotated, so that the Counter-Earth and the Central Fire were both not visible from Earth's surface, or at least not from the hemisphere where Greece was located. Early-Pythagoreans believed that after the death of the body, the soul would be punished or rewarded. Humans could, through their conduct, ensure that their soul was admitted to another world. The reincarnation in this world equated to a punishment. In Pythagoreanism life in this world is social and in the realm of society justice existed when each part of society received its due. The Pythagorean tradition of universal justice was later referenced by Plato. For Pythagorean philosophers the soul was the source of justice and through the harmony of the soul, divinity could be achieved. Injustice inverted the natural order. According to the 4th century BC philosopher Heraclides Ponticus, Pythagoras taught that "happiness consists in knowledge of the perfection of the numbers of the soul.
Body and soul
Pythagoreans believed that body and soul functioned together, and a healthy body required a healthy psyche. Early Pythagoreans conceived of the soul as the seat of sensation and emotion. They regarded the soul as distinct from the intellect. However, only fragments of the early Pythagorean texts have survived, and it is not certain whether they believed the soul was immortal. The surviving texts of the Pythagorean philosopher Philolaus indicate that while early Pythagoreans did not believe that the soul contained all psychological faculties, the soul was life and a harmony of physical elements. As such the soul died when certain arrangements of these elements ceased to exist.
However, the teaching most securely identified with Pythagoras is metempsychosis, or the "transmigration of souls", which holds that every soul is immortal and, upon death, enters into a new body. Pythagorean metempsychosis resembles the teachings of the Orphics, although its version contains substantial differences. Unlike the Orphics, who considered metempsychosis a cycle of grief that could be escaped by attaining liberation from it, Pythagoras seems to postulate an eternal, endless reincarnation where subsequent lives would not be conditioned by any action done in the previous.
Vegetarianism
thumb|right|Pythagoras and [[Vicia faba|faba beans, French, 1512/1514. Pythagoreans refused to eat beans. Already in antique times there was much speculation about the reason for this custom.]]
Some medieval authors refer to a "Pythagorean diet", entailing the abstention from eating meat, beans or fish. Pythagoreans believed that a vegetarian diet fostered a healthy body and enhanced the search for Arete. The purpose of vegetarianism in Pythagoreanism was not self-denial; instead, it was regarded as conducive to the best in a human being. The prohibition on beans may be related to older Athenian ideas associating beans with Hades, as in the cult of Cyamites. Pythagoreans advanced a grounded theory on the treatment of animals. They believed that any being that experienced pain or suffering should not have pain inflicted on it unnecessarily. Because it was not necessary to inflict pain on animals for humans to enjoy a healthy diet, they believed that animals should not be killed for the purpose of eating them. The Pythagoreans advanced the argument that unless an animal posed a threat to a human, it was not justifiable to kill an animal and that doing so would diminish the moral status of a human. By failing to show justice to the animal, humans diminish themselves. The arguments advanced by Pythagoreans convinced numerous of their philosopher contemporaries to adopt a vegetarian diet.
thumb|upright=1.4|Illustration from 1913 showing Pythagoras teaching a class of women
Many of the surviving texts of women Pythagorean philosophers are part of a collection, known as pseudoepigrapha Pythagorica, which was compiled by Neopythagoreans in the 1st or 2nd century. Some surviving fragments of this collection are by early-Pythagorean women philosophers, while the bulk of surviving writings are from late-Pythagorean women philosophers who wrote in the 4th and 3rd century BC. Female Pythagoreans are some of the first female philosophers from which texts have survived.
Theano of Croton, the wife of Pythagoras, is considered a major figure in early-Pythagoreanism. She was noted as distinguished philosopher and in the lore that surrounds her, is said to have taken over the leadership of the school after his death. Text fragments have also survived from women philosophers of the late-Pythagorean period. These include Perictione I, Perictione II, Aesara of Lucania and Phintys of Sparta. The woman Pythagorean philosopher Phyntis was Spartan and is believed to have been the daughter of a Spartan admiral killed in the battle of Arginusae in 406 BC. The influence of Pythagoreanism extended throughout and beyond antiquity because the Pythagorean doctrine of reincarnation was recounted in Plato's Gorgias, Phaedo, and Republic, while the Pythagorean cosmology was discussed in Plato's Timaeus. The possible influence of Pythagoreanism on Plato's concept of harmony and the Platonic solids has been discussed extensively. The belief in transmigration or reincarnation found its way into Plato's dialogue teaching as well. Plato's dialogues have become an important surviving source of Pythagorean philosophic arguments. Plato referenced Philolaus in Phaedo and wrote a Platonic adaptation of Philolaus' metaphysical system of limiters and unlimiteds. Plato also quoted from one of the surviving Archytas fragments in the Republic. However, Plato's views that the primary role of mathematics was to turn the soul towards the world of forms, as expressed in Timaeus, is regarded as Platonic philosophy, rather than Pythagorean. Nevertheless, he wrote a treatise on the Pythagoreans of which only fragments survive, in which he treats Pythagoras as a wonder-working religious teacher.
Neopythagoreanism
The Neopythagoreans were a school and a religious community. The revival of Pythagoreanism has been attributed to Publius Nigidius Figulus, Eudorus of Alexandria and Arius Didymus. In the 1st century AD Moderatus of Gades and Nicomachus of Gerasa emerged as leading teachers of Neopythagoreanism. The most significant Neopythagorean teacher was Apollonius of Tyana in the 1st century AD, who was regarded as a sage and lived as an ascetic. The last Neopythagorean philosopher was Numenius of Apamea in the 2nd century. Neopythagoreanism remained an elite movement which in the 3rd century merged into Neoplatonism. In the Middle Ages this numerological division of the universe was credited to the Pythagoreans, while early on it was regarded as an authoritative source of Christian doctrine by Photius and John of Sacrobosco. The Corpus Areopagiticum or Corpus Dionysiacum was to be referenced in the late Middle Ages by Dante and in the Renaissance a new translation of it was produced by Marsilio Ficino.
Early Christian theologians, such as Clement of Alexandria, adopted the ascetic doctrines of the neopythagoreans.
On numerology
thumb|Pythagoras is credited with having devised the [[tetractys, an important sacred symbol in later Pythagoreanism.]]
1st century treatises of Philo and Nicomachus popularised the mystical and cosmological symbolism Pythagoreans attributed to numbers. This interest in Pythagorean views on the importance of numbers was sustained by mathematicians such as Theon of Smyrna, Anatolius and Iamblichus. These mathematicians relied on Plato's Timaeus as their source for Pythagorean philosophy.
In the Middle Ages studies and adaptations of Timaeus solidified the view that there was a numerical explanation for proportion and harmony among learned men. Pythagoreanism, as mediated in Plato's Timaeus, spurred increasingly detailed studies of symmetry and harmony. Intellectuals pondered how knowledge of the geometry in which God had arranged the universe could be applied to life. By the 12th century Pythagorean numerological concepts had become a universal language in medieval Europe and were no longer recognised as Pythagorean.
The 11th-century Byzantine professor of philosophy Michael Psellus popularised Pythagorean numerology in his treatise on theology, arguing that Plato was the inheritor of the Pythagorean secret. Psellus also attributed arithmetical inventions by Diophantus to Pythagoras. Psellus thought to reconstruct Iamblichus' 10 book encyclopedia on Pythagoreanism from surviving fragments, leading to the popularisation of Iamblichus' description of Pythagorean physics, ethics and theology at the Byzantine court. Psellus was reputably in the possession of the Hermetica, a set of texts thought to be genuinely antique and which would be prolifically reproduced in the late Middle Ages. Manuel Bryennios introduced Pythagorean numerology to Byzantine music with his treatise Harmonics. He argued that the octave was essential in attaining perfect harmony.
In the Jewish communities the development of the Kabbalah as esoteric doctrine became associated with numerology. It was only in the 1st century that Philo of Alexandria, developed a Jewish Pythagoreanism. In the 3rd century Hermippus popularised the belief that Pythagoras had been the basis for establishing key dates in Judaism. In the 4th century this assertion was further developed by Aristobulus. The Jewish Pythagorean numerology developed by Philo held that God as the unique One was the creator of all numbers, of which seven was the most divine and ten the most perfect. The medieval edition of the Kabbalah focused largely on a cosmological scheme of creation, in reference to early Pythagorean philosophers Philolaus and Empedocles and helped to disseminate Jewish Pythagorean numerology.
On mathematics
thumb|A page of [[Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Roman numerals and the value in Eastern Arabic numerals.]]
Nicomachus' treatises were well known in Greek, Latin and the Arabic worlds. In the 9th century an Arabic translation of Nicomachus' Introduction to Arithmetic was published. The Arabic translations of Nicomachus' treatises were in turn translated into Latin by Gerard of Cremona, making them part of the Latin tradition of numerology. The Pythagorean theorem was referenced in Arabic manuscripts.
Besides the enthusiasm that developed in the Latin and Byzantine worlds in the Middle Ages for Pythagorean numerology, the Pythagorean tradition of perfect numbers inspired profound scholarship in mathematics. In the 13th century Leonardo of Pisa, better known as Fibonacci, published the Libre quadratorum (The Book of Squares). Fibonacci had studied scripts from Egypt, Syria, Greece and Sicily, and was learned in Hindu, Arabic and Greek methodologies. Using the Hindu–Arabic numeral system instead of the Roman numerals, he explored numerology as it had been set forth by Nicomachus. Fibonacci observed that square numbers always arise through the addition of consecutive odd numbers starting with unity. Fibonacci put forward a method of generating sets of three square numbers that satisfied the relationship first attributed to Pythagoras by Vitruvius, that . This equation is now known as the Pythagorean triple.
In the Middle Ages
In the Middle Ages, from the 5th until the 15th century, Pythagorean texts remained popular. Late antique writers had produced adaptions of the Sentences of Sextus as The golden verses of Pythagoras. The Golden Verses gained popularity and Christian adaptations of it appeared. These Christian adaptations were adopted by monastic orders, such as Saint Benedict, as authoritative Christian doctrine. In the Latin medieval western world, the Golden Verses became a widely reproduced text.
In the early 6th century the Roman philosopher Boethius popularised Pythagorean and Platonic conceptions of the universe and expounded the supreme importance of numerical ratios. The 7th century Bishop Isidore of Seville expressed his preference for the Pythagorean vision of a universe governed by the mystical properties of certain numbers, over the newly emerging Euclidean notion that knowledge could be built through deductive proofs. Isidore relied on the arithmetic of Nicomachus, who had fashioned himself as heir of Pythagoras, and took things further by studying the etymology of the name of each number. in Kitab al-Musiqa al-Kabir Al-Farabi rejected the notion of celestial harmony on the grounds that it was "plainly wrong" and that it was not possible for the heavens, orbs and stars to emit sounds through their motions. Later I also discovered in Plutarch that others were of this opinion. I have decided to set his words down here, so that they may be available to everybody: "Some think that the earth remains at rest. But Philolaus the Pythagorean believes that, like the sun and moon, it revolves around the fire in an oblique circle. Heraclides of Pontus and Ecphantus the Pythagorean make the earth move, not in a progressive motion, but like a wheel in a rotation from west to east about its own center."
In the 16th century Vincenzo Galilei challenged the prevailing Pythagorean wisdom about the relationship between pitches and weights attached to strings. Vincenzo Galilei, the father of Galileo Galilei, engaged in an extended public exchange with his former teacher Zarlino. Zarlino supported the theory that if two weights in the ratio of 2 to 1 were attached to two strings, the pitches generated by the two strings would produce the octave. Vincenzo Galilei proclaimed that he had been a committed Pythagorean, until he "ascertained the truth by means of experiment, the teacher of all things". He devised an experiment which showed that the weights attached to the two strings needed to increase as the square of the string length. This public challenge to prevailing numerology in musical theory triggered an experimental and physical approach to acoustics in the 17th century. Acoustics emerged as a mathematical field of music theory and later an independent branch of physics. In the experimental investigation of sound phenomena, numbers had no symbolic meaning and were merely used to measure physical phenomena and relationships such as frequency and vibration of a string.
Many of the most eminent 17th century natural philosophers in Europe, including Francis Bacon, Descartes, Beeckman, Kepler, Mersenne, Stevin and Galileo, had a keen interest in music and acoustics. By the late 17th century it was accepted that sound travels like a wave in the air at a finite speed and experiments to establish the speed of sound were carried out by philosophers attached to the French Academy of Sciences, the Accademia del Cimento and the Royal Society.
At the height of the Scientific Revolution, as Aristotelianism declined in Europe, the ideas of early-Pythagoreanism were revived. Mathematics regained importance and influenced philosophy as well as science. Mathematics was used by Kepler, Galileo, Descartes, Huygens and Newton to advance physical laws that reflected the inherent order of the universe. Twenty-one centuries after Pythagoreas had taught his disciples in Italy, Galileo announced to the world that "the great book of nature" could only be read by those who understood the language of mathematics. He set out to measure whatever is measurable, and to render everything measurable that is not. The Pythagorean concept of cosmic harmony deeply influenced western science. It served as the basis for Kepler's harmonices mundi and Leibniz's pre-established harmony.