When Pythagoras was asked [why humans exist], he said, "to observe the heavens," and he used to claim that he himself was an observer of nature, and it was for the sake of this that he had passed over into life.
In his preface to his book On the Revolution of the Heavenly Spheres (1543), Nicolaus Copernicus cites various Pythagoreans as the most important influences on the development of his heliocentric model of the universe, deliberately omitting mention of Aristarchus of Samos, a non-Pythagorean Astronomer who had developed a fully heliocentric model in the fourth century BC, in effort to portray his model as fundamentally Pythagorean. Johannes Kepler considered himself to be a Pythagorean. He believed in the Pythagorean doctrine of musica universalis and it was his search for the mathematical equations behind this doctrine that led to his discovery of the laws of planetary motion. Kepler titled his book on the subject Harmonices Mundi (Harmonics of the World), after the Pythagorean teaching that had inspired him. Near the conclusion of the book, Kepler describes himself falling asleep to the sound of the heavenly music, "warmed by having drunk a generous draught... from the cup of Pythagoras."
A fictionalized portrayal of Pythagoras appears in Book XV of Ovid's Metamorphoses, in which he delivers a speech imploring his followers to adhere to a strictly vegetarian diet. It was through Arthur Golding's 1567 English translation of Ovid's Metamorphoses that Pythagoras was best known to English-speakers throughout the early modern period. John Donne's Progress of the Soul discusses the implications of the doctrines expounded in the speech and Michel de Montaigne quoted the speech no less than three times in his treatise Of Cruelty to voice his moral objections against eating meat. william Shakespeare references the speech in his play The Merchant of Venice. John Dryden included a translation of the scene with Pythagoras in his 1700 work Fables, Ancient and Modern and John Gay's 1726 fable "Pythagoras and the Countryman" reiterates its major themes, linking carnivorism with tyranny. Lord Chesterfield records that his conversion to vegetarianism had been motivated by reading Pythagoras's speech in Ovid's Metamorphoses. Until the word vegetarianism was coined in the 1840s, vegetarians were referred to in English as "Pythagoreans". Percy Bysshe Shelley wrote an ode entitled "To the Pythagorean Diet" and Leo Tolstoy adopted the Pythagorean diet himself.
Heinrich Cornelius Agrippa's popular and influential three-volume treatise De Occulta Philosophia cites Pythagoras as a "religious magi" and indicates that Pythagoras's mystical numerology operates on a supercelestial level. The freemasons deliberately modeled their society on the community founded by Pythagoras at Croton. Rosicrucianism used Pythagorean symbolism, as did Robert Fludd (1574–1637), who believed his own musical writings to have been inspired by Pythagoras. John Dee was heavily influenced by Pythagorean ideology, particularly the teaching that all things are made of numbers. Adam Weishaupt, the founder of the Illuminati, was a strong admirer of Pythagoras and, in his book Pythagoras (1787), he advocated that society should be reformed to be more like Pythagoras's commune at Croton. Wolfgang Amadeus Mozart incorporated Masonic and Pythagorean symbolism into his opera The Magic Flute. Sylvain Maréchal, in his six-volume 1799 biography The Voyages of Pythagoras, declared that all Revolutionaries in all time periods are the "heirs of Pythagoras".
Both Plato and Isocrates affirm that, above all else, Pythagoras was famous for leaving behind him a way of life. Carl B. Boyer (1968) characterizes the Pythagorean school as "politically conservative and with a strict code of conduct." Leonid Zhmud (2012) identifies two camps within the early Pythagoreans: the scientific mathematikoi and the religious akousmatikoi, who engaged in politics. The study of mathematics and music may have been connected to the worship of Apollo.
Following a similar logic, the Egyptians are said to have taught him geometry, the Phoenicians arithmetic, the Chaldeans astronomy, and the Magi the principles of religion and practical maxims for the conduct of life. According to Diogenes Laërtius, Pythagoras not only visited Egypt and learnt the Egyptian language (as reported by Antiphon in his On Men of Outstanding Merit), but also "journeyed among the Chaldaeans and Magi." Later in Crete, he went to the Cave of Ida with Epimenides, and entered Egyptian sanctuaries for the purpose to learn information concerning the secret lore of the different gods. The Middle Platonist biographer Plutarch (c. 46 – c. 120 AD) writes in his treatise On Isis and Osiris that, during his visit to Egypt, Pythagoras received instruction from the Egyptian priest Oenuphis of Heliopolis (meanwhile Solon received lectures from a Sonchis of Sais). Other ancient Writers also mention Pythagoras's visit to Egypt. According to the Christian theologian Clement of Alexandria (c. 150 – c. 215 AD), "Pythagoras was a disciple of Soches, an Egyptian archprophet, as well as Plato of Sechnuphis of Heliopolis."
Dante Alighieri was fascinated by Pythagorean numerology and based his descriptions of Hell, Purgatory, and Heaven on Pythagorean numbers. Dante wrote that Pythagoras saw Unity as Good and Plurality as Evil and, in Paradiso XV, 56–57, he declares: "five and six, if understood, ray forth from unity." The number eleven and its multiples are found throughout the Divine Comedy, each book of which has thirty-three cantos, except for the Inferno, which has thirty-four, the first of which serves as a general introduction. Dante describes the ninth and tenth bolgias in the Eighth Circle of Hell as being twenty-two miles and eleven miles respectively, which correspond to the fraction 22/7, which was the Pythagorean approximation of pi. Hell, Purgatory, and Heaven are all described as circular and Dante compares the wonder of God's majesty to the mathematical puzzle of squaring the circle. The number three also features prominently: the Divine Comedy has three parts and Beatrice is associated with the number nine, which is equal to three times three.
The Pythagorean theorem was known and used by the Babylonians and Indians centuries before Pythagoras, but it is possible that he may have been the first one to introduce it to the Greeks. Some historians of mathematics have even suggested that he—or his students—may have constructed the first proof. Burkert rejects this suggestion as implausible, noting that Pythagoras was never credited with having proved any theorem in antiquity. Furthermore, the manner in which the Babylonians employed Pythagorean numbers implies that they knew that the principle was generally applicable, and knew some kind of proof, which has not yet been found in the (still largely unpublished) cuneiform sources. Pythagoras's biographers state that he also was the first to identify the five regular solids and that he was the first to discover the Theory of Proportions.
