title: Giordano Bruno

text: While Bruno was distinguished for outstanding ability, his taste for free thinking and forbidden books soon caused him difficulties. Given the controversy he caused in later life, it is surprising that he was able to remain within the monastic system for eleven years. In his testimony to Venetian inquisitors during his trial many years later, he says that proceedings were twice taken against him for having cast away images of the saints, retaining only a crucifix, and for having recommended controversial texts to a novice. Such behavior could perhaps be overlooked, but Bruno's situation became much more serious when he was reported to have defended the Arian heresy, and when a copy of the banned writings of Erasmus, annotated by him, was discovered hidden in the monastery latrine. When he learned that an indictment was being prepared against him in Naples he fled, shedding his religious habit, at least for a time.


title: Giordano Bruno

text: Bruno first went to the Genoese port of Noli, then to Savona, Turin and finally to Venice, where he published his lost work "On the Signs of the Times" with the permission (so he claimed at his trial) of the Dominican Remigio Nannini Fiorentino. From Venice he went to Padua, where he met fellow Dominicans who convinced him to wear his religious habit again. From Padua he went to Bergamo and then across the Alps to Chambéry and Lyon. His movements after this time are obscure.


title: Giordano Bruno

text: In 1579 he arrived in Geneva. As D.W. Singer, a Bruno biographer, notes, "The question has sometimes been raised as to whether Bruno became a Protestant,


title: Giordano Bruno

text: and there is evidence he joined a Calvinist church. During his Venetian trial he told inquisitors that while in Geneva he told the Marchese de Vico of Naples, who was notable for helping Italian refugees in Geneva, "I did not intend to adopt the religion of the city. I desired to stay there only that I might live at liberty and in security." Bruno had a pair of breeches made for himself, and the Marchese and others apparently made Bruno a gift of a sword, hat, cape and other necessities for dressing himself; in such clothing Bruno could no longer be recognized as a priest. Things apparently went well for Bruno for a time, as he entered his name in the Rector's Book of the University of Geneva in May 1579. But in keeping with his personality he could not long remain silent. In August he published an attack on the work of , a distinguished professor. Bruno and the printer, Jean Bergeon, were promptly arrested. Rather than apologizing, Bruno insisted on continuing to defend his publication. He was refused the right to take sacrament. Though this right was soon restored, he left Geneva.


title: Giordano Bruno

text: He went to France, arriving first in Lyon, and thereafter settling for a time (1580–1581) in Toulouse, where he took his doctorate in theology and was elected by students to lecture in philosophy. He also attempted at this time to return to Catholicism, but was denied absolution by the Jesuit priest he approached. When religious strife broke out in the summer of 1581, he moved to Paris. There he held a cycle of thirty lectures on theological topics and also began to gain fame for his prodigious memory. His talents attracted the benevolent attention of the king Henry III; Bruno subsequently reported "I got me such a name that King Henry III summoned me one day to discover from me if the memory which I possessed was natural or acquired by magic art. I satisfied him that it did not come from sorcery but from organized knowledge; and, following this, I got a book on memory printed, entitled "The Shadows of Ideas", which I dedicated to His Majesty. Forthwith he gave me an Extraordinary Lectureship with a salary."


title: Giordano Bruno

text: In Paris, Bruno enjoyed the protection of his powerful French patrons. During this period, he published several works on mnemonics, including "De umbris idearum" ("On the Shadows of Ideas", 1582), "" ("The Art of Memory", 1582), and "Cantus circaeus" ("Circe's Song", 1582; described at ). All of these were based on his mnemonic models of organized knowledge and experience, as opposed to the simplistic logic-based mnemonic techniques of Petrus Ramus then becoming popular. Bruno also published a comedy summarizing some of his philosophical positions, titled "Il Candelaio" ("The Torchbearer", 1582). In the 16th century dedications were, as a rule, approved beforehand, and hence were a way of placing a work under the protection of an individual. Given that Bruno dedicated various works to the likes of King Henry III, Sir Philip Sidney, Michel de Castelnau (French Ambassador to England), and possibly Pope Pius V, it is apparent that this wanderer had risen sharply in status and moved in powerful circles.


title: Giordano Bruno

text: In April 1583, Bruno went to England with letters of recommendation from Henry III as a guest of the French ambassador, Michel de Castelnau. Bruno lived at the French embassy with the lexicographer John Florio. There he became acquainted with the poet Philip Sidney (to whom he dedicated two books) and other members of the Hermetic circle around John Dee, though there is no evidence that Bruno ever met Dee himself. He also lectured at Oxford, and unsuccessfully sought a teaching position there. His views were controversial, notably with John Underhill, Rector of Lincoln College and subsequently bishop of Oxford, and George Abbot, who later became Archbishop of Canterbury. Abbot mocked Bruno for supporting "the opinion of Copernicus that the earth did go round, and the heavens did stand still; whereas in truth it was his own head which rather did run round, and his brains did not stand still", and found Bruno had both plagiarized and misrepresented Ficino's work, leading Bruno to return to the continent.


title: Giordano Bruno

text: Nevertheless, his stay in England was fruitful. During that time Bruno completed and published some of his most important works, the six "Italian Dialogues", including the cosmological tracts "La cena de le ceneri" ("The Ash Wednesday Supper", 1584), "De la causa, principio et uno" ("On Cause, Principle and Unity", 1584), "De l'infinito, universo et mondi" ("On the Infinite, Universe and Worlds", 1584) as well as "Lo spaccio de la bestia trionfante" ("The Expulsion of the Triumphant Beast", 1584) and "De gli eroici furori" ("On the Heroic Frenzies", 1585). Some of these were printed by John Charlewood. Some of the works that Bruno published in London, notably "The Ash Wednesday Supper", appear to have given offense. Once again, Bruno's controversial views and tactless language lost him the support of his friends. John Bossy has advanced the theory that, while staying in the French Embassy in London, Bruno was also spying on Catholic conspirators, under the pseudonym "Henry Fagot", for Sir Francis Walsingham, Queen Elizabeth's Secretary of State.


title: Giordano Bruno

text: Bruno is sometimes cited as being the first to propose that the universe is infinite, which he did during his time in England, but an English scientist, Thomas Digges, put forth this idea in a published work in 1576, some eight years earlier than Bruno. An infinite universe and the possibility of alien life had also been earlier suggested by German Catholic Cardinal Nicholas of Cusa in "On Learned Ignorance" published in 1440.


title: Giordano Bruno

text: In October 1585, Castelnau was recalled to France, and Bruno went with him. In Paris, Bruno found a tense political situation. Moreover, his 120 theses against Aristotelian natural science soon put him in ill favor. In 1586, following a violent quarrel over these theses, he left France for Germany.


title: Giordano Bruno

text: In Germany he failed to obtain a teaching position at Marburg, but was granted permission to teach at Wittenberg, where he lectured on Aristotle for two years. However, with a change of intellectual climate there, he was no longer welcome, and went in 1588 to Prague, where he obtained 300 taler from Rudolf II, but no teaching position. He went on to serve briefly as a professor in Helmstedt, but had to flee again in 1590 when he was excommunicated by the Lutherans.


title: Giordano Bruno

text: During this period he produced several Latin works, dictated to his friend and secretary Girolamo Besler, including "De Magia" ("On Magic"), "Theses De Magia" ("Theses on Magic") and "De Vinculis in Genere" ("A General Account of Bonding"). All these were apparently transcribed or recorded by Besler (or Bisler) between 1589 and 1590. He also published "De Imaginum, Signorum, Et Idearum Compositione" ("On the Composition of Images, Signs and Ideas", 1591).


title: Giordano Bruno

text: In 1591 he was in Frankfurt, where he received an invitation from the Venetian patrician Giovanni Mocenigo, who wished to be instructed in the art of memory, and also heard of a vacant chair in mathematics at the University of Padua. At the time the Inquisition seemed to be losing some of its strictness, and because the Republic of Venice was the most liberal state in the Italian Peninsula, Bruno was lulled into making the fatal mistake of returning to Italy.


title: Giordano Bruno

text: He went first to Padua, where he taught briefly, and applied unsuccessfully for the chair of mathematics, which was given instead to Galileo Galilei one year later. Bruno accepted Mocenigo's invitation and moved to Venice in March 1592. For about two months he served as an in-house tutor to Mocenigo, to whom he let slip some of his heterodox ideas. Mocenigo denounced him to the Venetian Inquisition, which had Bruno arrested on 22 May 1592. Among the numerous charges of blasphemy and heresy brought against him in Venice, based on Mocenigo's denunciation, was his belief in the plurality of worlds, as well as accusations of personal misconduct. Bruno defended himself skillfully, stressing the philosophical character of some of his positions, denying others and admitting that he had had doubts on some matters of dogma. The Roman Inquisition, however, asked for his transfer to Rome. After several months of argument, the Venetian authorities reluctantly consented and Bruno was sent to Rome in January 1593.


title: Giordano Bruno

text: During the seven years of his trial in Rome, Bruno was held in confinement, lastly in the Tower of Nona. Some important documents about the trial are lost, but others have been preserved, among them a summary of the proceedings that was rediscovered in 1940. The numerous charges against Bruno, based on some of his books as well as on witness accounts, included blasphemy, immoral conduct, and heresy in matters of dogmatic theology, and involved some of the basic doctrines of his philosophy and cosmology. Luigi Firpo speculates the charges made against Bruno by the Roman Inquisition were:


title: Giordano Bruno

text: Bruno defended himself as he had in Venice, insisting that he accepted the Church's dogmatic teachings, but trying to preserve the basis of his cosmological views. In particular, he held firm to his belief in the plurality of worlds, although he was admonished to abandon it. His trial was overseen by the Inquisitor Cardinal Bellarmine, who demanded a full recantation, which Bruno eventually refused. On 20 January 1600, Pope Clement VIII declared Bruno a heretic, and the Inquisition issued a sentence of death. According to the correspondence of Gaspar Schopp of Breslau, he is said to have made a threatening gesture towards his judges and to have replied: "Maiori forsan cum timore sententiam in me fertis quam ego accipiam" ("Perhaps you pronounce this sentence against me with greater fear than I receive it").


title: Giordano Bruno

text: He was turned over to the secular authorities. On 17 February 1600, in the Campo de' Fiori (a central Roman market square), with his "tongue imprisoned because of his wicked words", he was hung upside down naked before finally being burned at the stake. His ashes were thrown into the Tiber river.


title: Giordano Bruno

text: All of Bruno's works were placed on the "Index Librorum Prohibitorum" in 1603. The inquisition cardinals who judged Giordano Bruno were Cardinal Bellarmino (Bellarmine), Cardinal Madruzzo (Madruzzi), Camillo Cardinal Borghese (later Pope Paul V), Domenico Cardinal Pinelli, Pompeio Cardinal Arrigoni, Cardinal Sfondrati, Pedro Cardinal De Deza Manuel and Cardinal Santorio (Archbishop of Santa Severina, Cardinal-Bishop of Palestrina).


title: Giordano Bruno

text: The measures taken to prevent Bruno continuing to speak have resulted in his becoming a symbol for free thought and speech in present-day Rome, where an annual memorial service takes place close to the spot where he was executed.


title: Giordano Bruno

text: The earliest likeness of Bruno is an engraving published in 1715 and cited by Salvestrini as "the only known portrait of Bruno". Salvestrini suggests that it is a re-engraving made from a now lost original. This engraving has provided the source for later images.


title: Giordano Bruno

text: The records of Bruno's imprisonment by the Venetian inquisition in May 1592 describe him as a man "of average height, with a hazel-coloured beard and the appearance of being about forty years of age".


title: Giordano Bruno

text: Alternately, a passage in a work by George Abbot indicates that Bruno was of diminutive stature: "When that Italian Didapper, who intituled himselfe Philotheus Iordanus Brunus Nolanus, magis elaboratae Theologiae Doctor, &c. with a name longer than his body...". The word "didapper" used by Abbot is the derisive term which at the time meant "a small diving waterfowl".


title: Giordano Bruno

text: In the first half of the 15th century, Nicholas of Cusa challenged the then widely accepted philosophies of Aristotelianism, envisioning instead an infinite universe whose center was everywhere and circumference nowhere, and moreover teeming with countless stars. He also predicted that neither were the rotational orbits circular nor were their movements uniform.


title: Giordano Bruno

text: In the second half of the 16th century, the theories of Copernicus (1473–1543) began diffusing through Europe. Copernicus conserved the idea of planets fixed to solid spheres, but considered the apparent motion of the stars to be an illusion caused by the rotation of the Earth on its axis; he also preserved the notion of an immobile center, but it was the Sun rather than the Earth. Copernicus also argued the Earth was a planet orbiting the Sun once every year. However he maintained the Ptolemaic hypothesis that the orbits of the planets were composed of perfect circles—deferents and epicycles—and that the stars were fixed on a stationary outer sphere.


title: Giordano Bruno

text: Despite the widespread publication of Copernicus' work "De revolutionibus orbium coelestium", during Bruno's time most educated Catholics subscribed to the Aristotelian geocentric view that the Earth was the center of the universe, and that all heavenly bodies revolved around it. The ultimate limit of the universe was the "primum mobile", whose diurnal rotation was conferred upon it by a transcendental God, not part of the universe (although, as the kingdom of heaven, adjacent to it), a motionless prime mover and first cause. The fixed stars were part of this celestial sphere, all at the same fixed distance from the immobile Earth at the center of the sphere. Ptolemy had numbered these at 1,022, grouped into 48 constellations. The planets were each fixed to a transparent sphere.


title: Giordano Bruno

text: Few astronomers of Bruno's time accepted Copernicus's heliocentric model. Among those who did were the Germans Michael Maestlin (1550–1631), Christoph Rothmann, Johannes Kepler (1571–1630); the Englishman Thomas Digges (c. 1546–1595), author of "A Perfit Description of the Caelestial Orbes"; and the Italian Galileo Galilei (1564–1642).


title: Giordano Bruno

text: In 1584, Bruno published two important philosophical dialogues ("La Cena de le Ceneri" and "De l'infinito universo et mondi") in which he argued against the planetary spheres (Christoph Rothmann did the same in 1586 as did Tycho Brahe in 1587) and affirmed the Copernican principle.


title: Giordano Bruno

text: In particular, to support the Copernican view and oppose the objection according to which the motion of the Earth would be perceived by means of the motion of winds, clouds etc., in "La Cena de le Ceneri" Bruno anticipates some of the arguments of Galilei on the relativity principle. Note that he also uses the example now known as Galileo's ship.


title: Giordano Bruno

text: Theophilus – [...] air through which the clouds and winds move are parts of the Earth, [...] to mean under the name of Earth the whole machinery and the entire animated part, which consists of dissimilar parts; so that the rivers, the rocks, the seas, the whole vaporous and turbulent air, which is enclosed within the highest mountains, should belong to the Earth as its members, just as the air [does] in the lungs and in other cavities of animals by which they breathe, widen their arteries, and other similar effects necessary for life are performed. The clouds, too, move through accidents in the body of the Earth and are in its bowels as are the waters. [...]


title: Giordano Bruno

text: With the Earth move [...] all things that are on the Earth. If, therefore, from a point outside the Earth something were thrown upon the Earth, it would lose, because of the latter's motion, its straightness as would be seen on the ship [...] moving along a river, if someone on point C of the riverbank were to throw a stone along a straight line, and would see the stone miss its target by the amount of the velocity of the ship's motion. But if someone were placed high on the mast of that ship, move as it may however fast, he would not miss his target at all, so that the stone or some other heavy thing thrown downward would not come along a straight line from the point E which is at the top of the mast, or cage, to the point D which is at the bottom of the mast, or at some point in the bowels and body of the ship. Thus, if from the point D to the point E someone who is inside the ship would throw a stone straight up, it would return to the bottom along the same line however far the ship moved, provided it was not subject to any pitch and roll."


title: Giordano Bruno

text: Bruno's infinite universe was filled with a substance—a "pure air", aether, or "spiritus"—that offered no resistance to the heavenly bodies which, in Bruno's view, rather than being fixed, moved under their own impetus (momentum). Most dramatically, he completely abandoned the idea of a hierarchical universe.


title: Giordano Bruno

text: The universe is then one, infinite, immobile... It is not capable of comprehension and therefore is endless and limitless, and to that extent infinite and indeterminable, and consequently immobile.


title: Giordano Bruno

text: Bruno's cosmology distinguishes between "suns" which produce their own light and heat, and have other bodies moving around them; and "earths" which move around suns and receive light and heat from them. Bruno suggested that some, if not all, of the objects classically known as fixed stars are in fact suns. According to astrophysicist Steven Soter, he was the first person to grasp that "stars are other suns with their own planets."


title: Giordano Bruno

text: Bruno wrote that other worlds "have no less virtue nor a nature different from that of our Earth" and, like Earth, "contain animals and inhabitants".


title: Giordano Bruno

text: During the late 16th century, and throughout the 17th century, Bruno's ideas were held up for ridicule, debate, or inspiration. Margaret Cavendish, for example, wrote an entire series of poems against "atoms" and "infinite worlds" in "Poems and Fancies" in 1664. Bruno's true, if partial, vindication would have to wait for the implications and impact of Newtonian cosmology.


title: Giordano Bruno

text: Bruno's overall contribution to the birth of modern science is still controversial. Some scholars follow Frances Yates in stressing the importance of Bruno's ideas about the universe being infinite and lacking geocentric structure as a crucial crossing point between the old and the new. Others see in Bruno's idea of multiple worlds instantiating the infinite possibilities of a pristine, indivisible One, a forerunner of Everett's many-worlds interpretation of quantum mechanics.


title: Giordano Bruno

text: While many academics note Bruno's theological position as pantheism, several have described it as pandeism, and some also as panentheism. Physicist and philosopher Max Bernhard Weinstein in his "Welt- und Lebensanschauungen, Hervorgegangen aus Religion, Philosophie und Naturerkenntnis" ("World and Life Views, Emerging From Religion, Philosophy and Nature"), wrote that the theological model of pandeism was strongly expressed in the teachings of Bruno, especially with respect to the vision of a deity for which "the concept of God is not separated from that of the universe." However, Otto Kern takes exception to what he considers Weinstein's overbroad assertions that Bruno, as well as other historical philosophers such as John Scotus Eriugena, Anselm of Canterbury, Nicholas of Cusa, Mendelssohn, and Lessing, were pandeists or leaned towards pandeism. "Discover" editor Corey S. Powell also described Bruno's cosmology as pandeistic, writing that it was "a tool for advancing an animist or Pandeist theology", and this assessment of Bruno as a pandeist was agreed with by science writer Michael Newton Keas, and "The Daily Beast" writer David Sessions.


title: Giordano Bruno

text: The Vatican has published few official statements about Bruno's trial and execution. In 1942, Cardinal Giovanni Mercati, who discovered a number of lost documents relating to Bruno's trial, stated that the Church was perfectly justified in condemning him. On the 400th anniversary of Bruno's death, in 2000, Cardinal Angelo Sodano declared Bruno's death to be a "sad episode" but, despite his regret, he defended Bruno's prosecutors, maintaining that the Inquisitors "had the desire to serve freedom and promote the common good and did everything possible to save his life". In the same year, Pope John Paul II made a general apology for "the use of violence that some have committed in the service of truth".


title: Giordano Bruno

text: Some authors have characterized Bruno as a "martyr of science", suggesting parallels with the Galileo affair which began around 1610. "It should not be supposed," writes A. M. Paterson of Bruno and his "heliocentric solar system", that he "reached his conclusions via some mystical revelation...His work is an essential part of the scientific and philosophical developments that he initiated." Paterson echoes Hegel in writing that Bruno "ushers in a modern theory of knowledge that understands all natural things in the universe to be known by the human mind through the mind's dialectical structure".


title: Giordano Bruno

text: Ingegno writes that Bruno embraced the philosophy of Lucretius, "aimed at liberating man from the fear of death and the gods." Characters in Bruno's "Cause, Principle and Unity" desire "to improve speculative science and knowledge of natural things," and to achieve a philosophy "which brings about the perfection of the human intellect most easily and eminently, and most closely corresponds to the truth of nature."


title: Giordano Bruno

text: Other scholars oppose such views, and claim Bruno's martyrdom to science to be exaggerated, or outright false. For Yates, while "nineteenth century liberals" were thrown "into ecstasies" over Bruno's Copernicanism, "Bruno pushes Copernicus' scientific work back into a prescientific stage, back into Hermeticism, interpreting the Copernican diagram as a hieroglyph of divine mysteries."


title: Giordano Bruno

text: According to historian Mordechai Feingold, "Both admirers and critics of Giordano Bruno basically agree that he was pompous and arrogant, highly valuing his opinions and showing little patience with anyone who even mildly disagreed with him." Discussing Bruno's experience of rejection when he visited Oxford University, Feingold suggests that "it might have been Bruno's manner, his language and his self-assertiveness, rather than his ideas" that caused offence.


title: Giordano Bruno

text: In his "Lectures on the History of Philosophy" Hegel writes that Bruno's life represented "a bold rejection of all Catholic beliefs resting on mere authority."


title: Giordano Bruno

text: Alfonso Ingegno states that Bruno's philosophy "challenges the developments of the Reformation, calls into question the truth-value of the whole of Christianity, and claims that Christ perpetrated a deceit on mankind... Bruno suggests that we can now recognize the universal law which controls the perpetual becoming of all things in an infinite universe." A. M. Paterson says that, while we no longer have a copy of the official papal condemnation of Bruno, his heresies included "the doctrine of the infinite universe and the innumerable worlds" and his beliefs "on the movement of the earth".


title: Giordano Bruno

text: Michael White notes that the Inquisition may have pursued Bruno early in his life on the basis of his opposition to Aristotle, interest in Arianism, reading of Erasmus, and possession of banned texts. White considers that Bruno's later heresy was "multifaceted" and may have rested on his conception of infinite worlds. "This was perhaps the most dangerous notion of all... If other worlds existed with intelligent beings living there, did they too have their visitations? The idea was quite unthinkable."


title: Giordano Bruno

text: Frances Yates rejects what she describes as the "legend that Bruno was prosecuted as a philosophical thinker, was burned for his daring views on innumerable worlds or on the movement of the earth." Yates however writes that "the Church was... perfectly within its rights if it included philosophical points in its condemnation of Bruno's heresies" because "the philosophical points were quite inseparable from the heresies."


title: Giordano Bruno

text: According to the "Stanford Encyclopedia of Philosophy", "in 1600 there was no official Catholic position on the Copernican system, and it was certainly not a heresy. When [...] Bruno [...] was burned at the stake as a heretic, it had nothing to do with his writings in support of Copernican cosmology."


title: Giordano Bruno

text: The website of the Vatican Apostolic Archive, discussing a summary of legal proceedings against Bruno in Rome, states: "In the same rooms where Giordano Bruno was questioned, for the same important reasons of the relationship between science and faith, at the dawning of the new astronomy and at the decline of Aristotle's philosophy, sixteen years later, Cardinal Bellarmino, who then contested Bruno's heretical theses, summoned Galileo Galilei, who also faced a famous inquisitorial trial, which, luckily for him, ended with a simple abjuration."


title: Giordano Bruno

text: Following the 1870 Capture of Rome by the newly created Kingdom of Italy and the end of the Church's temporal power over the city, the erection of a monument to Bruno on the site of his execution became feasible. The monument was sharply opposed by the clerical party, but was finally erected by the Rome Municipality and inaugurated in 1889.


title: Giordano Bruno

text: A statue of a stretched human figure standing on its head, designed by Alexander Polzin and depicting Bruno's death at the stake, was placed in Potsdamer Platz station in Berlin on 2 March 2008.


title: Giordano Bruno

text: Retrospective iconography of Bruno shows him with a Dominican cowl but not tonsured. Edward Gosselin has suggested that it is likely Bruno kept his tonsure at least until 1579, and it is possible that he wore it again thereafter.


title: Giordano Bruno

text: An idealized animated version of Bruno appears in the first episode of the 2014 television series "". In this depiction, Bruno is shown with a more modern look, without tonsure and wearing clerical robes and without his hood. "Cosmos" presents Bruno as an impoverished philosopher who was ultimately executed due to his refusal to recant his belief in other worlds, a portrayal that was criticized by some as simplistic or historically inaccurate. Corey S. Powell, of "Discover" magazine, says of Bruno, "A major reason he moved around so much is that he was argumentative, sarcastic, and drawn to controversy...He was a brilliant, complicated, difficult man.


title: Giordano Bruno

text: Algernon Charles Swinburne wrote a poem honoring Giordano Bruno in 1889, when the statue of Bruno was constructed in Rome.


title: Giordano Bruno

text: Czeslaw Milosz evokes the story and image of Giordano Bruno in his poem "Campo Dei Fiori" (Warsaw 1943).


title: Giordano Bruno

text: Randall Jarell's poem "The Emancipators" addresses Bruno, along with Galileo and Newton, as an originator of the modern scientific-industrial world.


title: Giordano Bruno

text: Heather McHugh depicted Bruno as the principal of a story told (at dinner, by an "underestimated" travel guide) to a group of contemporary American poets in Rome. The poem (originally published in McHugh's collection of poems "Hinge & Sign", nominee for the National Book Award, and subsequently reprinted widely) channels the very question of "ars poetica", or meta-meaning itself, through the embedded narrative of the suppression of Bruno's words, silenced towards the end of his life both literally and literarily.


title: Giordano Bruno

text: Bruno and his theory of "the coincidence of contraries" ("coincidentia oppositorum") play an important role in James Joyce's 1939 novel "Finnegans Wake". Joyce wrote in a letter to his patroness, Harriet Shaw Weaver, "His philosophy is a kind of dualism – every power in nature must evolve an opposite in order to realise itself and opposition brings reunion". Amongst his numerous allusions to Bruno in his novel, including his trial and torture, Joyce plays upon Bruno's notion of "coincidentia oppositorum" through applying his name to word puns such as "Browne and Nolan" (the name of Dublin printers) and '"brownesberrow in nolandsland".


title: Giordano Bruno

text: Giordano Bruno features as the hero in a series of historical crime novels by S.J. Parris (a pseudonym of Stephanie Merritt). In order these are "Heresy", "Prophecy", "Sacrilege", "Treachery", "Conspiracy" and "Execution".


title: Giordano Bruno

text: "The Last Confession" by Morris West (posthumously published) is a fictional autobiography of Bruno, ostensibly written shortly before his execution.


title: Giordano Bruno

text: In 1963 soviet writer Alexander Volkov published "The Wandering", a novel about the childhood and youth of Bruno.


title: Giordano Bruno

text: In 1973 the biographical drama "Giordano Bruno" was released, an Italian/French movie directed by Giuliano Montaldo, starring Gian Maria Volonté as Bruno.


title: Giordano Bruno

text: Bruno is a central character, and his philosophy a central theme, in John Crowley’s "Aegypt" (1987), renamed "The Solitudes", and the ensuing series of novels: "Love & Sleep" (1994), "Daemonomania" (2000), and "Endless Things" (2007).


title: Giordano Bruno

text: Hans Werner Henze set his large scale cantata for orchestra, choir and four soloists, "Novae de infinito laudes" to Italian texts by Bruno, recorded in 1972 at the Salzburg Festival reissued on CD Orfeo C609 031B.


title: Giordano Bruno

text: Robert Ashley intones in his Perfect Lives (1983): "Giordano Bruno. I think they burned him. He was too... positive."


title: Giordano Bruno

text: The album "Numen Lumen" (2011) by neofolk group Hautville tracks Bruno's lyrics and is dedicated to the philosopher.


title: Giordano Bruno

text: In 2014 the Italian composer Francesco Filidei wrote an opera, based on a libretto by Stefano Busellato, titled "Giordano Bruno". The premiere took place on 12 September 2015 at the Casa da Música in Porto, Portugal.


title: Giordano Bruno

text: The Giordano Bruno Foundation (German: Giordano-Bruno-Stiftung) is a non-profit foundation based in Germany that pursues the "Support of Evolutionary Humanism". It was founded by entrepreneur Herbert Steffen in 2004. The Giordano Bruno Foundation is critical of religious fundamentalism and nationalism


title: Giordano Bruno

text: The SETI League makes an annual award honoring the memory of Giordano Bruno to a deserving person or persons who have made a significant contribution to the practice of SETI (the search for extraterrestrial intelligence). The award was proposed by sociologist Donald Tarter in 1995 on the 395th anniversary of Bruno's death. The trophy presented is called a Bruno.


title: Giordano Bruno

text: The 22 km impact crater Giordano Bruno on the far side of the Moon is named in his honor, as are the main belt Asteroids 5148 Giordano and 13223 Cenaceneri; the latter is named after his philosophical dialogue "La Cena de le Ceneri" ("The Ash Wednesday Supper") (see above).


title: Giordano Bruno

text: Radio broadcasting station 2GB in Sydney, Australia is named for Bruno. The two letters "GB" in the call sign were chosen to honor Bruno, who was much admired by Theosophists who were the original holders of the station's licence.


title: Humid subtropical climate

text: A humid subtropical climate is a zone of climate characterized by hot and humid summers, and cool to mild winters. These climates normally lie on the southeast side of all continents (except Antarctica), generally between latitudes 25° and 40° and are located poleward from adjacent tropical climates. It is also known as warm temperate climate in some climate classifications.


title: Humid subtropical climate

text: Under the Köppen climate classification, Cfa and Cwa climates are either described as humid subtropical climates or warm temperate climates. This climate features mean temperature in the coldest month between (or ) and and mean temperature in the warmest month or higher. However, while some climatologists have opted to describe this climate type as a "humid subtropical climate", Köppen himself never used this term. The humid subtropical climate classification was officially created under the Trewartha climate classification. In this classification, climates are termed humid subtropical when they have at least 8 months with a mean temperature above .


title: Humid subtropical climate

text: While many subtropical climates tend to be located at or near coastal locations, in some cases, they extend inland, most notably in China and the United States, where they exhibit more pronounced seasonal variations and sharper contrasts between summer and winter, as part of a gradient between the hotter tropical climates of the southern coasts and the colder continental climates to the north and further inland. As such, the climate can be said to exhibit somewhat different features depending on whether it is found inland, or in a maritime position.


title: Humid subtropical climate

text: In a humid subtropical climate, summers are typically long, hot and humid. Monthly mean summer temperatures are normally between . A deep current of tropical air dominates the humid subtropics at the time of high sun, and daily intense (but brief) convective thundershowers are common. Summer high temperatures are typically in the high 20s to mid-30s °C (80s or 90s °F), while overnight lows in the summer are typically in the lower 20s °C (70s °F). Monthly mean temperatures in winter are often mild, typically averaging . Daytime highs in winter normally are in the range, while overnight lows are from , though the poleward boundaries of this climate feature colder temperatures, possibly below freezing.


title: Humid subtropical climate

text: Rainfall often shows a summer peak especially where monsoons are well developed, as in Southeast Asia and South Asia. Other areas have a more uniform or varying rainfall cycles but consistently lack any predictably dry summer months unlike mediterranean climates. Most summer rainfall occurs during thunderstorms that build up due to the intense surface heating and strong subtropical sun angle. Weak tropical lows that move in from adjacent warm tropical oceans, as well as infrequent tropical storms often contribute to summer seasonal rainfall peaks. Winter rainfall is often associated with large storms in the westerlies that have fronts that reach down into subtropical latitudes. However, many subtropical climates such as southeast Asia and Florida in the United States have very dry winters, with frequent brush fires and water shortages.


title: Humid subtropical climate

text: Under the Holdridge life zones classification, the subtropical climates have a biotemperature between the frost or critical temperature line, (depending on locations in the world) and , and these climates are humid (or even perhumid or superhumid) when the potential evapotranspiration (PET) ratio (= PET / Precipitation) is less than 1. In the Holdridge classification, the humid subtropical climates coincide more or less with the warmest "Cfa" and "Cwa" climates and the less warm humid tropical "Köppen" climates ("Aw", "Am" and "Af").


title: Humid subtropical climate

text: In Africa, humid subtropical climates are primarily found in the southern part of the continent. The "Cwa" climate is found over a large portion of the interior of the Middle and Eastern African regions. This area includes central Angola, northeastern Zimbabwe, the Niassa, Manica and Tete provinces of Mozambique, the southern Congo provinces, southwest Tanzania, and the majority of Malawi, and Zambia. Some lower portions of the Ethiopian Highlands also have this climate.


title: Humid subtropical climate

text: The climate is also found in the narrow coastal sections of southern and eastern South Africa, primarily in KwaZulu-Natal and the Eastern Cape provinces. South Africa's version of this climate features heavy oceanic influences resulting in generally milder temperatures. This is particularly evident in its winters when temperatures do not drop as low as in many other regions within the humid subtropical category.


title: Humid subtropical climate

text: In East Asia, this climate type is found in the southeastern quarter of mainland China from Hong Kong north to Nanjing, the northern half of Taiwan, north through southern and central Japan (Kyushu, Shikoku and half of Honshu), and the most southern regions of Korea (the south coast and Jeju island). Cities near the equatorward boundary of this zone include Hong Kong and Taipei; while Tokyo, Busan and Qingdao are near the northern boundary.


title: Humid subtropical climate

text: The influence of the strong Siberian anticyclone in East Asia brings colder winter temperatures than in the humid subtropical zones in North America, South America, and Australia. The isotherm reaches as far south as the valleys of the Yellow and Wei rivers, roughly latitude 34° N. At Hainan Island and in Taiwan, the climate transitions from subtropical into tropical. In most of this region, the winter monsoon is very well developed, as such eastern Asian humid subtropical zones have a strong winter dry season and heavy summer rainfall.


title: Humid subtropical climate

text: Only in inland areas below the Yangtze River and coastal areas between approximately the Huai River and the beginning of the coast of Guangdong is there sufficient winter rainfall to produce a "Cfa" climate; even in these areas, rainfall and streamflow display a highly pronounced summer peak, unlike other regions of this climate type. Drought can be severe and often catastrophic to agriculture in the "Cwa" zone.


title: Humid subtropical climate

text: The only area where winter precipitation equals or even exceeds the summer rain is around the San'in region at the western coast of Japan, which during winter is on the windward side of the westerlies. The winter precipitation in these regions is usually produced by low-pressure systems off the east coast that develop in the onshore flow from the Siberian high. Summer rainfall comes from the East Asian Monsoon and from frequent typhoons. Annual rainfall is generally over , and in areas below the Himalayas can be much higher still.


title: Humid subtropical climate

text: In Southeast Asia, about 90% of the region has a tropical climate. But humid subtropical climates can also be seen in most of Northern Myanmar and Northern Laos; and in entire Northern Vietnam, including Hanoi.


title: Humid subtropical climate

text: Southeast Asian locations with these climates can feature cool temperatures, with lows reaching during the months of December, January, and February. Unlike a good portion of East Asian locations with this climate however, most of Southeast Asia seldom experiences snowfall. These areas tend to feature hot and humid summers and cool and wet winters, with mean temperatures varying between 25 to 30 °C (77 to 86 °F) in summer.


title: Humid subtropical climate

text: Humid subtropical climates can also be found in South Asia. However, the humid subtropical climates exhibited here typically differ markedly from those in East Asia (and for that matter a good portion of the globe). Winters here are typically mild, dry and relatively short. They also tend to be foggy. Summers tend to be long and very hot, starting from mid-April and peaking in May and early June with high temperatures often exceeding . They also tend to be extremely dry, complete with dust storms, traits usually associated with arid or semi-arid climates. During this period many native trees defoliate to save water. This is followed by the cooler monsoons, where the region experiences heavy rains on almost a daily basis. Average high temperatures decrease during the monsoon season, but humidity increases. This results in hot and humid conditions, similar to summers in other humid subtropical climates. Cities such as New Delhi, Lucknow, Kanpur and Patna exhibit this atypical version of the climate in India. In Pakistan, the twin cities of Islamabad and Rawalpindi, as well as the city of Swabi in Khyber Pakhtunkhwa, feature this weather pattern.


title: Humid subtropical climate

text: In Bangladesh, cities like Rangpur, Saidpur and Dinajpur in northern region features the monsoon variant ("Cwa"), where rainfall peaks at the monsoon season. Like neighboring Northern Indian plains, this region also shows a distinct three season pattern- relatively dry and very hot summer (March- early june), extremely wet, cooler Monsoon seson (June- September), and mild, foggy winter and autumn (Late October- February)


title: Humid subtropical climate

text: Humid subtropical climates can also be found in Nepal. However the Nepalese version of the climate generally do not feature the extreme hot spells that are commonplace for many other South Asian locations with this climate. In Nepal cities such as Kathmandu, Pokhara, Butwal, Birgunj and Biratnagar feature this iteration of the climate.


title: Humid subtropical climate

text: In South Asia, humid subtropical climates generally border on continental climates as altitude increases, or on winter-rainfall climates in western areas of Pakistan and northwestern India (e.g. Peshawar in northwestern Pakistan or Srinagar in the Kashmir Valley in India, where the primary precipitation peak occurs in March, not July or August). Further east, in highland areas with lengthier monsoons such as Nepal, seasonal temperature variation is lower than in the lowlands.


title: Humid subtropical climate

text: Although humid subtropical climates in Asia are mostly confined to the southeastern quarter of the continent, there are two narrow areas along the coast of the Caspian Sea and Black Sea with humid subtropical climates. Summers in these locations are cooler than typical humid subtropical climates and snowfall in winter is relatively common, but is usually of a short duration.


title: Humid subtropical climate

text: In Western Asia, the climate is prevalent in the Gilan, Māzandarān and Golestan Provinces of Iran, in parts of the Caucasus, in Azerbaijan and in Georgia wedged between the Caspian and Black seas and coastal (Black Sea) Turkey, albeit having more oceanic influence.


title: Humid subtropical climate

text: Annual rainfall ranges from around at Sari to over at Bandar-e Anzali, and is heavy throughout the year, with a maximum in October or November when Bandar-e Anzali can average 400 millimetres (16 inches). Temperatures are generally moderate in comparison with other parts of Western Asia. During winter, the coastal areas can receive snowfall, which is usually of a short duration.


title: Humid subtropical climate

text: In Rasht, the average maximum in July is around but with near-saturation humidity, whilst in January it is around . The heavy, evenly distributed rainfall extends north into the Caspian coastal strip of Azerbaijan up to its northern border but this climate in Azerbaijan is, however, a "Cfb"/"Cfa" ("Oceanic climate"/"Humid subtropical climate") borderline case.


title: Humid subtropical climate

text: Western Georgia (Batumi and Kutaisi) in the Kolkheti Lowland and the northeast coast of Turkey (Giresun), have a climate similar to that of Gilan and Mazandaran in Iran and very similar to that of southeastern and northern Azerbaijan. Temperatures range from in summer to in winter and rainfall is even heavier than in Caspian Iran, up to per year in Hopa (Turkey). These climates are a "Cfb"/"Cfa" ("Oceanic climate"/"Humid subtropical climate") borderline case.


title: Humid subtropical climate

text: In North America, humid subtropical climates are found in the American Gulf and lower East Coast states, including Texas, Oklahoma, Louisiana, Arkansas, Alabama, Mississippi, Tennessee, North Carolina, South Carolina, Georgia, and Florida. On the Florida peninsula, the humid subtropical climate gives way to the tropical climate of South Florida and the Florida Keys.


title: Humid subtropical climate

text: Under Köppen's climate classification, this zone includes locations further north, primarily Virginia, Kentucky, the lower elevations of West Virginia, Maryland, Delaware, Washington, D.C., New Jersey, southeastern Pennsylvania, most of Connecticut, and extreme southern New York around New York City and sections of Long Island. It can also be found in the lower Midwest, primarily in the central and southern portions of Kansas and Missouri, and the far southern portions of Illinois, Indiana and Ohio.


title: Humid subtropical climate

text: In Mexico, there are small areas of "Cfa" and "Cwa" climates. The climate can be found in small areas scattered around the northeastern part of the country, in proximity to the Gulf of Mexico. Other areas where the climate can be found is in the high elevations of Trans-Mexican Volcanic Belt and Sierra Madre Oriental. Despite being located at higher elevations, these locations have summers that are too warm to qualify as a subtropical highland climate. Guadalajara's climate is a major example of this.


title: Humid subtropical climate

text: Outside of isolated sections of Mexico, the southernmost limits of this climate zone in North America lie just north of South Florida and around southern coastal Texas. Cities at the southernmost limits, such as Tampa and Orlando and along the Texas coast around Corpus Christi down toward Brownsville generally feature warm weather year-round and minimal temperature differences between seasons. In contrast, cities at the northernmost limits of the climate zone such as New York, Philadelphia and Louisville feature hot, humid summers and chilly winters. These areas have average winter temperatures at the coldest limit of climates classed as humid subtropical.


title: Humid subtropical climate

text: Snowfall varies greatly in this climate zone. In locations at the southern limits of this zone and areas around the Gulf Coast, cities such as Orlando, Tampa, Houston, New Orleans, and Savannah rarely see snowfall, which occurs, at most, a few times per generation. In Southern cities farther north or inland, such as Atlanta, Birmingham, Charlotte, Dallas, Memphis, Nashville, Oklahoma City, and Raleigh, snow only occasionally falls and is usually three inches or less. However, for the majority of the winter here, temperatures remain above or well above freezing. At the northernmost limits of this zone, cities such as New York City, Philadelphia, and Baltimore typically see snowfall during the winter, with occasional heavy snowstorms. Still, average temperatures during a typical winter hover just above freezing at these locations.


title: Humid subtropical climate

text: Precipitation is plentiful in the humid subtropical climate zone in North America – but with significant variations in terms of wettest/driest months and seasons. Much of the interior South, including Tennessee, Kentucky, and the northern halves of Mississippi and Alabama, tends to have a winter or spring (not summer) precipitation maximum. Closer to the South Atlantic and Gulf coasts, there is a summer maximum, with July or August usually the wettest month – as at Norfolk, Cape Hatteras and Jacksonville, North Carolina, Charleston, South Carolina, Mobile, Alabama and New Orleans. A semblance of a monsoon pattern (dry winters/wet summers) is evident along the Atlantic coast from southern North Carolina (Wilmington, North Carolina area) south to Florida. The seasonal monsoon is much stronger on the Florida peninsula, as most locations in Florida have dry winters and wet summers.


title: Humid subtropical climate

text: In addition, areas in Texas that are slightly inland from the Gulf of Mexico, such as Austin and San Antonio that border the semi-arid climate zone, generally see a peak of precipitation in May, a drought-like nadir in mid-summer and a secondary, if not equal, precipitation peak in September or October. Areas further south along South Texas' Gulf Coast (Brownsville), which closely border tropical climate classification, typically have a strong September precipitation maximum, and a tendency toward dry conditions in winter with rain increasing in spring, with December or January often the driest months.


title: Humid subtropical climate

text: Humid subtropical climates are found in a sizable portion of South America. The climate extends over a few states of southern Brazil, including Paraná, into sections of Paraguay, all of Uruguay and central Argentina (Pampas region). Major cities such as São Paulo, Buenos Aires, Porto Alegre and Montevideo, have a humid subtropical climate, generally in the form of hot and humid summers, and mild to cool winters. These areas, which include the Pampas, generally feature a "Cfa" climate categorization. At 38°S, the Argentine city of Bahía Blanca lies on the southern limit of the humid subtropical zone.


title: Humid subtropical climate

text: The "Cwa" climate occurs in parts of tropical highlands of São Paulo state, Mato Grosso do Sul and near the Andean highland in northwestern Argentina. These highland areas feature summer temperatures that are warm enough to fall outside the subtropical highland climate category.


title: Humid subtropical climate

text: The humid subtropical climate zone predominantly lies in eastern Australia, which begins from the coastal strip of Mackay, Queensland and stretches down to the southern coast of Sydney, where it transitions into the cooler, oceanic climates.


title: Humid subtropical climate

text: From Newcastle, approximately northeast of Sydney, the "Cfa" zone would extend to inland New South Wales, excluding the highland regions (which have an oceanic climate), stretching towards Dubbo to the northwest and Wagga Wagga to the south, ending at the New South Wales/Victoria border (Albury–Wodonga). To note, these places would have characteristics of the semi-arid and/or Mediterranean climates. Furthermore, the inland "Cfa" climates generally have drier summers, or at least summers with low humidity.


title: Humid subtropical climate

text: Extreme heat is more often experienced in Sydney than in other large cities in Australia's "Cfa" zone, especially in the western suburbs, where highs over are not uncommon. Frost is prevalent in the more inland areas of Sydney, such as Richmond. Average annual rainfall in the Sydney region ranges between .


title: Humid subtropical climate

text: There is usually a distinct summer rainfall maximum that becomes more pronounced moving northwards. In Brisbane, the wettest month (February) receives five times the rainfall of the driest month (September). Temperatures are very warm to hot but are not excessive: the average maximum in February is usually around and in July around . Frosts are extremely rare except at higher elevations, but temperatures over are not common on the coast.


title: Humid subtropical climate

text: North of the "Cfa" climate zone there is a zone centred upon Rockhampton which extends north to the Köppen "Cwa" classified climate zone of the Atherton Tablelands region. This region has a very pronounced dry winter period, with often negligible rainfall between June and October. Winter temperatures generally only fall slightly below , which would classify the region as a tropical savanna, or "Aw", climate.


title: Humid subtropical climate

text: Annual rainfall within Australia's humid subtropical climate zone can reach as high as in coastal locations and is generally or above. The most intense 2-3 day rainfall periods that occur in this coastal zone however are the outcome of east coast lows forming to the north of a large high pressure system, there can be great variation in rainfall amounts from year to year as a result of these systems. As an example Lismore which lies in the centre of this zone, the annual rainfall can range from less than in 1915 to more than in 1950.


title: Humid subtropical climate

text: As the continent does not have a large ocean to its east as the case in many other continents within the climate zone, humid subtropical climate in Europe is limited to relatively small areas on the margins of the Mediterranean and Black Sea basins. "Cfa" zones are generally transitional between the Mediterranean climate zones along the coast and oceanic and humid continental zones to the west and north where rainfall in the warmer months is too high for a Mediterranean classification, while temperatures (either in the summer and/or winter) are too warm to qualify as oceanic or humid continental. Humidity is not as high here as in other continents within this climatic zone.


title: Humid subtropical climate

text: The Po Valley, in Northern Italy, including major cities such as Milan, Turin, Bologna, and Verona, has a humid subtropical climate, featuring hot, humid summers with frequent thunderstorms; winters are foggy, damp and chilly, with sudden bursts of frost. Some parts of the valley have a mild continental climate. Places along the shores of Lake Maggiore, Lake Lugano, Lake Como (Como and Verbania in Italy and Lugano in Switzerland) have a humid subtropical climate with a distinctive high amount of rainfall during summer. Due to recent warming, cities in the Pannonian Basin such as Belgrade, Novi Sad and Budapest, Hungary, are now just warm enough to be categorized as such. At 47°N, Budapest lies on the northern limit of the humid subtropical zone.


title: Humid subtropical climate

text: The coastal areas in the northern half of the Adriatic Sea also fall within this climate zone. The cities include Trieste, Venice, and Rimini in Italy, Rijeka and Split in Croatia, Koper in Slovenia and Kotor in Montenegro. Other Southern European areas in the "Cfa" zone include the central valleys and coast of Catalonia of Girona and Barcelona in Spain, some on the north-east of Spain (Huesca), West Macedonia in Greece (Kozani), the Garonne Valley (Toulouse) and Rhone Valley (Valence) in France.


title: Humid subtropical climate

text: Along the Black Sea coast of Bulgaria (Varna), coast of Romania (Constanta and Mamaia), Sochi, Russia and Crimea, have summers too warm (> in the warmest month) to qualify as oceanic, no freezing month, and enough summer precipitation and sometimes humid conditions, where they would be fit to be classed under "Cfa", though they closely border the humid continental zone. All these areas are subject to occasional, in some cases repeated snowfalls and freezes during winter.


title: Humid subtropical climate

text: In Central Europe, a small area of humid subtropical climates are located in transitional areas between the oceanic and continental climates in areas where higher summer temperature do not quite qualify it for inclusion in the Oceanic climate schema (but, in the Trewartha climate classification, this type of climate is included in that typical of the other areas of central-west Europe, classified as Do -the Temperate Oceanic climate-) and mild winters do not allow their inclusion into continental climates. Average summer temperatures in areas of Europe with this climate are generally not as hot as most other subtropical zones around the world.


title: Humid subtropical climate

text: In the Azores, some islands have this climate, with very mild and rainy winters (>) and no snowfall, warm summers (>) but with no dry season during the warmest period, which means that they can neither be classified as oceanic, nor as Mediterranean, only as humid subtropical, as with Corvo Island.


title: Humid subtropical climate

text: In many other climate classification systems outside of the Köppen, most of these locations would not be included in the humid subtropical grouping. The higher summer precipitation and poleward flow of tropical air-masses in summer are not present in Europe as they are in eastern Australia or the southern United States.


title: Bodhidharma

text: Bodhidharma was a semi-legendary Buddhist monk who lived during the 5th or 6th century CE. He is traditionally credited as the transmitter of Chan Buddhism to China, and regarded as its first Chinese patriarch. According to a 17th century apocryphal story found in a manual called Yijin Jing, he began the physical training of the monks of Shaolin Monastery that led to the creation of Shaolin kungfu. He is known as Dámó in China and as Daruma in Japan. His name means ""dharma" of awakening (bodhi)" in Sanskrit.


title: Bodhidharma

text: Little contemporary biographical information on Bodhidharma is extant, and subsequent accounts became layered with legend and unreliable details.


title: Bodhidharma

text: According to the principal Chinese sources, Bodhidharma came from the Western Regions, which typically refers to Central Asia but can also include the Indian subcontinent, and is described as either a "Persian Central Asian" or a "South Indian [...] the third son of a great Indian king."


title: Bodhidharma

text: Throughout Buddhist art, Bodhidharma is depicted as an ill-tempered, profusely-bearded, wide-eyed non-Chinese person. He is referred as "The Blue-Eyed Barbarian" () in Chan texts.


title: Bodhidharma

text: Aside from the Chinese accounts, several popular traditions also exist regarding Bodhidharma's origins.


title: Bodhidharma

text: The accounts also differ on the date of his arrival, with one early account claiming that he arrived during the Liu Song dynasty (420–479 CE) and later accounts dating his arrival to the Liang dynasty (502–557 CE). Bodhidharma was primarily active in the territory of the Northern Wei (386–534 CE). Modern scholarship dates him to about the early 5th century CE.


title: Bodhidharma

text: Bodhidharma's teachings and practice centered on meditation and the "Laṅkāvatāra Sūtra". The "Anthology of the Patriarchal Hall" (952) identifies Bodhidharma as the 28th Patriarch of Buddhism in an uninterrupted line that extends all the way back to the Gautama Buddha himself.


title: Bodhidharma

text: There are two known extant accounts written by contemporaries of Bodhidharma. According to these sources, Bodhidharma came from the Western Regions, and is described as either a "Persian Central Asian" or a "South Indian [...] the third son of a great Indian king." Later sources draw on these two sources, adding additional details, including a change to being descendent from a "Brahmin" king, which accords with the reign of the Pallavas, who "claim[ed] to belong to a brahmin lineage."


title: Bodhidharma

text: The "Western Regions" was a historical name specified in the Chinese chronicles between the 3rd century BC to the 8th century AD that referred to the regions west of Yumen Pass, most often Central Asia or sometimes more specifically the easternmost portion of it (e.g. Altishahr or the Tarim Basin in southern Xinjiang). Sometimes it was used more generally to refer to other regions to the west of China as well, such as the Indian subcontinent (as in the novel "Journey to the West").


title: Bodhidharma

text: The earliest text mentioning Bodhidharma is "The Record of the Buddhist Monasteries of Luoyang" ( "Luòyáng Qiélánjì") which was compiled in 547 by Yang Xuanzhi (), a writer and translator of Mahayana sutras into Chinese. Yang gave the following account:


title: Bodhidharma

text: The account of Bodhidharma in the Luoyan Record does not particularly associate him with meditation, but rather depicts him as a thaumaturge capable of mystical feats. This may have played a role in his subsequent association with the martial arts and esoteric knowledge.


title: Bodhidharma

text: The second account was written by Tanlin (曇林; 506–574). Tanlin's brief biography of the "Dharma Master" is found in his preface to the "Long Scroll of the Treatise on the Two Entrances and Four Practices", a text traditionally attributed to Bodhidharma and the first text to identify him as South Indian:


title: Bodhidharma

text: Tanlin's account was the first to mention that Bodhidharma attracted disciples, specifically mentioning Daoyu () and Dazu Huike (), the latter of whom would later figure very prominently in the Bodhidharma literature. Although Tanlin has traditionally been considered a disciple of Bodhidharma, it is more likely that he was a student of Huike.


title: Bodhidharma

text: The "Record of the Masters and Students of the Laṅka" (Léngqié Shīzī Jì 楞伽師資記), which survives both in Chinese and in Tibetan translation (although the surviving Tibetan translation is apparently of older provenance than the surviving Chinese version), states that Bodhidharma is not the first ancestor of Zen, but instead the second. This text instead claims that Guṇabhadra, the translator of the "Laṅkāvatāra Sūtra", is the first ancestor in the lineage. It further states that Bodhidharma was his student. The Tibetan translation is estimated to have been made in the late eighth or early ninth century, indicating that the original Chinese text was written at some point before that.


title: Bodhidharma

text: Tanlin's preface has also been preserved in Jingjue's (683–750) "Lengjie Shizi ji" "Chronicle of the "Laṅkāvatāra" Masters", which dates from 713–716./ca. 715 He writes,


title: Bodhidharma

text: In the 7th-century historical work "Further Biographies of Eminent Monks" (續高僧傳 "Xù gāosēng zhuàn"), Daoxuan () possibly drew on Tanlin's preface as a basic source, but made several significant additions:


title: Bodhidharma

text: Firstly, Daoxuan adds more detail concerning Bodhidharma's origins, writing that he was of "South Indian Brahman stock" (南天竺婆羅門種 "nán tiānzhú póluómén zhŏng").


title: Bodhidharma

text: Secondly, more detail is provided concerning Bodhidharma's journeys. Tanlin's original is imprecise about Bodhidharma's travels, saying only that he "crossed distant mountains and seas" before arriving in Wei. Daoxuan's account, however, implies "a specific itinerary": "He first arrived at Nan-yüeh during the Sung period. From there he turned north and came to the Kingdom of Wei" This implies that Bodhidharma had travelled to China by sea and that he had crossed over the Yangtze.


title: Bodhidharma

text: Thirdly, Daoxuan suggests a date for Bodhidharma's arrival in China. He writes that Bodhidharma makes landfall in the time of the Song, thus making his arrival no later than the time of the Song's fall to the Southern Qi in 479.


title: Bodhidharma

text: Finally, Daoxuan provides information concerning Bodhidharma's death. Bodhidharma, he writes, died at the banks of the Luo River, where he was interred by his disciple Dazu Huike, possibly in a cave. According to Daoxuan's chronology, Bodhidharma's death must have occurred prior to 534, the date of the Northern Wei's fall, because Dazu Huike subsequently leaves Luoyang for Ye. Furthermore, citing the shore of the Luo River as the place of death might possibly suggest that Bodhidharma died in the mass executions at Heyin () in 528. Supporting this possibility is a report in the Chinese Buddhist canon stating that a Buddhist monk was among the victims at Héyīn.


title: Bodhidharma

text: In the "Anthology of the Patriarchal Hall" (祖堂集 "Zǔtángjí") of 952, the elements of the traditional Bodhidharma story are in place. Bodhidharma is said to have been a disciple of Prajñātāra, thus establishing the latter as the 27th patriarch in India. After a three-year journey, Bodhidharma reached China in 527, during the Liang (as opposed to the Song in Daoxuan's text). The "Anthology of the Patriarchal Hall" includes Bodhidharma's encounter with Emperor Wu of Liang, which was first recorded around 758 in the appendix to a text by Shenhui (), a disciple of Huineng.


title: Bodhidharma

text: Finally, as opposed to Daoxuan's figure of "over 180 years," the "Anthology of the Patriarchal Hall" states that Bodhidharma died at the age of 150. He was then buried on Mount Xiong'er (熊耳山) to the west of Luoyang. However, three years after the burial, in the Pamir Mountains, Song Yun ()—an official of one of the later Wei kingdoms—encountered Bodhidharma, who claimed to be returning to India and was carrying a single sandal. Bodhidharma predicted the death of Song Yun's ruler, a prediction which was borne out upon the latter's return. Bodhidharma's tomb was then opened, and only a single sandal was found inside.


title: Bodhidharma

text: According to the "Anthology of the Patriarchal Hall", Bodhidharma left the Liang court in 527 and relocated to Mount Song near Luoyang and the Shaolin Monastery, where he "faced a wall for nine years, not speaking for the entire time", his date of death can have been no earlier than 536. Moreover, his encounter with the Wei official indicates a date of death no later than 554, three years before the fall of the Western Wei.


title: Bodhidharma

text: Subsequent to the "Anthology of the Patriarchal Hall", the only dated addition to the biography of Bodhidharma is in the "Jingde Records of the Transmission of the Lamp" (景德傳燈錄 "Jĭngdé chuándēng lù", published 1004 CE), by Daoyuan (), in which it is stated that Bodhidharma's original name had been Bodhitāra but was changed by his master Prajñātāra. The same account is given by the Japanese master Keizan's 13th-century work of the same title.


title: Bodhidharma

text: Several contemporary popular traditions also exist regarding Bodhidharma's origins. An Indian tradition regards Bodhidharma to be the third son of a Pallava king from Kanchipuram. This is consistent with the Southeast Asian traditions which also describe Bodhidharma as a former South Indian Tamil prince who had awakened his kundalini and renounced royal life to become a monk. The Tibetan version similarly characterises him as a dark-skinned siddha from South India. Conversely, the Japanese tradition generally regards Bodhidharma as Persian.


title: Bodhidharma

text: Several stories about Bodhidharma have become popular legends, which are still being used in the Ch'an, Seon and Zen-tradition.


title: Bodhidharma

text: The "Anthology of the Patriarchal Hall" says that in 527, Bodhidharma visited Emperor Wu of Liang, a fervent patron of Buddhism:


title: Bodhidharma

text: Failing to make a favorable impression in South China, Bodhidharma is said to have travelled to the Shaolin Monastery. After either being refused entry or being ejected after a short time, he lived in a nearby cave, where he "faced a wall for nine years, not speaking for the entire time".


title: Bodhidharma

text: The biographical tradition is littered with apocryphal tales about Bodhidharma's life and circumstances. In one version of the story, he is said to have fallen asleep seven years into his nine years of wall-gazing. Becoming angry with himself, he cut off his eyelids to prevent it from happening again. According to the legend, as his eyelids hit the floor the first tea plants sprang up, and thereafter tea would provide a stimulant to help keep students of Chan awake during zazen.


title: Bodhidharma

text: The most popular account relates that Bodhidharma was admitted into the Shaolin temple after nine years in the cave and taught there for some time. However, other versions report that he "passed away, seated upright"; or that he disappeared, leaving behind the "Yijin Jing"; or that his legs atrophied after nine years of sitting, which is why Daruma dolls have no legs.


title: Bodhidharma

text: In one legend, Bodhidharma refused to resume teaching until his would-be student, Dazu Huike, who had kept vigil for weeks in the deep snow outside of the monastery, cut off his own left arm to demonstrate sincerity.


title: Bodhidharma

text: "Jingde Records of the Transmission of the Lamp" (景德传灯录) of Daoyuan, presented to the emperor in 1004, records that Bodhidharma wished to return to India and called together his disciples:


title: Bodhidharma

text: Bodhidharma passed on the symbolic robe and bowl of dharma succession to Dazu Huike and, some texts claim, a copy of the "Laṅkāvatāra Sūtra". Bodhidharma then either returned to India or died.


title: Bodhidharma

text: Some Chinese myths and legends describe Bodhidharma as being disturbed by the poor physical shape of the Shaolin monks, after which he instructed them in techniques to maintain their physical condition as well as teaching meditation. He is said to have taught a series of external exercises called the Eighteen Arhat Hands and an internal practice called the Sinew Metamorphosis Classic. In addition, after his departure from the temple, two manuscripts by Bodhidharma were said to be discovered inside the temple: the "Yijin Jing" and the "Xisui Jing". Copies and translations of the "Yijin Jing" survive to the modern day. The "Xisui Jing" has been lost.


title: Bodhidharma

text: According to Southeast Asian folklore, Bodhidharma travelled from Jambudvipa by sea to Palembang, Indonesia. Passing through Sumatra, Java, Bali, and Malaysia, he eventually entered China through Nanyue. In his travels through the region, Bodhidharma is said to have transmitted his knowledge of the Mahayana doctrine and the martial arts. Malay legend holds that he introduced forms to silat.


title: Bodhidharma

text: Vajrayana tradition links Bodhidharma with the 11th-century south Indian monk Dampa Sangye who travelled extensively to Tibet and China spreading tantric teachings.


title: Bodhidharma

text: Three years after Bodhidharma's death, Ambassador Song Yun of northern Wei is said to have seen him walking while holding a shoe at the Pamir Mountains. Song asked Bodhidharma where he was going, to which Bodhidharma replied "I am going home". When asked why he was holding his shoe, Bodhidharma answered "You will know when you reach Shaolin monastery. Don't mention that you saw me or you will meet with disaster". After arriving at the palace, Song told the emperor that he met Bodhidharma on the way. The emperor said Bodhidharma was already dead and buried and had Song arrested for lying. At Shaolin Monastery, the monks informed them that Bodhidharma was dead and had been buried in a hill behind the temple. The grave was exhumed and was found to contain a single shoe. The monks then said "Master has gone back home" and prostrated three times: "For nine years he had remained and nobody knew him; Carrying a shoe in hand he went home quietly, without ceremony."


title: Bodhidharma

text: One of the fundamental Chán texts attributed to Bodhidharma is a four-line stanza whose first two verses echo the "Laṅkāvatāra Sūtra"s disdain for words and whose second two verses stress the importance of the insight into reality achieved through "self-realization":


title: Bodhidharma

text: Tanlin, in the preface to "Two Entrances and Four Acts", and Daoxuan, in the "Further Biographies of Eminent Monks", mention a practice of Bodhidharma's termed "wall-gazing" (壁觀 "bìguān"). Both Tanlin and Daoxuan associate this "wall-gazing" with "quieting [the] mind" ().


title: Bodhidharma

text: In the "Two Entrances and Four Acts", traditionally attributed to Bodhidharma, the term "wall-gazing" is given as follows: Daoxuan states, "The merits of Mahāyāna wall-gazing are the highest".


title: Bodhidharma

text: These are the first mentions in the historical record of what may be a type of meditation being ascribed to Bodhidharma.


title: Bodhidharma

text: Exactly what sort of practice Bodhidharma's "wall-gazing" was remains uncertain. Nearly all accounts have treated it either as an undefined variety of meditation, as Daoxuan and Dumoulin, or as a variety of seated meditation akin to the zazen () that later became a defining characteristic of Chan. The latter interpretation is particularly common among those working from a Chan standpoint.


title: Bodhidharma

text: There are early texts which explicitly associate Bodhidharma with the "Laṅkāvatāra Sūtra". Daoxuan, for example, in a late recension of his biography of Bodhidharma's successor Huike, has the sūtra as a basic and important element of the teachings passed down by Bodhidharma:


title: Bodhidharma

text: Another early text, the "Record of the Masters and Disciples of the Laṅkāvatāra Sūtra" () of Jingjue (淨覺; 683–750), also mentions Bodhidharma in relation to this text. Jingjue's account also makes explicit mention of "sitting meditation" or zazen:


title: Bodhidharma

text: In other early texts, the school that would later become known as Chan Buddhism is sometimes referred to as the "Laṅkāvatāra school" (楞伽宗 "Léngqié zōng").


title: Bodhidharma

text: The "Laṅkāvatāra Sūtra", one of the Mahayana sutras, is a highly "difficult and obscure" text whose basic thrust is to emphasize "the inner enlightenment that does away with all duality and is raised above all distinctions". It is among the first and most important texts for East Asian Yogācāra.


title: Bodhidharma

text: According to Suzuki, one of the recurrent emphases in the "Laṅkāvatāra Sūtra" is a lack of reliance on words to effectively express reality:


title: Bodhidharma

text: In contrast to the ineffectiveness of words, the sūtra instead stresses the importance of the "self-realization" that is "attained by noble wisdom" and, according to Suzuki, occurs "when one has an insight into reality as it is": "The truth is the state of self-realization and is beyond categories of discrimination". According to Suzuki, reflecting his own emphasis on kensho, the sūtra goes on to outline the ultimate effects of an experience of self-realization:


title: Bodhidharma

text: The idea of a patriarchal lineage in Ch'an dates back to the epitaph for Faru (), a disciple of the 5th patriarch Hongren (). In the "Long Scroll of the Treatise on the Two Entrances and Four Practices" and the "Continued Biographies of Eminent Monks", Daoyu and Dazu Huike are the only explicitly identified disciples of Bodhidharma. The epitaph gives a line of descent identifying Bodhidharma as the first patriarch.


title: Bodhidharma

text: In the 6th century biographies of famous monks were collected. From this genre the typical Chan lineage was developed:


title: Bodhidharma

text: D. T. Suzuki contends that Chan's growth in popularity during the 7th and 8th centuries attracted criticism that it had "no authorized records of its direct transmission from the founder of Buddhism" and that Chan historians made Bodhidharma the 28th patriarch of Buddhism in response to such attacks.


title: Bodhidharma

text: The earliest lineages described the lineage from Bodhidharma into the 5th to 7th generation of patriarchs. Various records of different authors are known, which give a variation of transmission lines:


title: Bodhidharma

text: Eventually these descriptions of the lineage evolved into a continuous lineage from Śākyamuni Buddha to Bodhidharma. The idea of a line of descent from Śākyamuni Buddha is the basis for the distinctive lineage tradition of Chan Buddhism.


title: Bodhidharma

text: According to the "Song of Enlightenment" (證道歌 "Zhèngdào gē") by Yongjia Xuanjue, one of the chief disciples of Huìnéng, was Bodhidharma, the 28th Patriarch of Buddhism in a line of descent from Gautama Buddha via his disciple Mahākāśyapa:


title: Bodhidharma

text: Bodhidharma has been the subject of critical scientific research, which has shed new light on the traditional stories about Bodhidharma.


title: Bodhidharma

text: According to John McRae, Bodhidharma has been the subject of a hagiographic process which served the needs of Chan Buddhism. According to him it is not possible to write an accurate biography of Bodhidharma:


title: Bodhidharma

text: McRae's standpoint accords with Yanagida's standpoint: "Yanagida ascribes great historical value to the witness of the disciple Tanlin, but at the same time acknowledges the presence of "many puzzles in the biography of Bodhidharma". Given the present state of the sources, he considers it impossible to compile a reliable account of Bodhidharma's life.


title: Bodhidharma

text: Several scholars have suggested that the composed image of Bodhidharma depended on the combination of supposed historical information on various historical figures over several centuries. Bodhidharma as a historical person may even never have actually existed.


title: Bodhidharma

text: Dumoulin comments on the three principal sources. The Persian heritage is doubtful, according to Dumoulin: "In the "Description of the Lo-yang temple", Bodhidharma is called a Persian. Given the ambiguity of geographical references in writings of this period, such a statement should not be taken too seriously." Dumoulin considers Tanlin's account of Bodhidharma being "the third son of a great Brahman king" to be a later addition, and finds the exact meaning of "South Indian Brahman stock" unclear: "And when Daoxuan speaks of origins from South Indian Brahman stock, it is not clear whether he is referring to roots in nobility or to India in general as the land of the Brahmans."


title: Bodhidharma

text: These Chinese sources lend themselves to make inferences about Bodhidharma's origins. "The third son of a Brahman king" has been speculated to mean "the third son of a Pallava king". Based on a specific pronunciation of the Chinese characters 香至 as Kang-zhi, meaning “fragrance extreme", Tsutomu Kambe identifies 香至 to be Kanchipuram, an old capital town in the state Tamil Nadu, India. According to Tsutomu Kambe, "Kanchi means 'a radiant jewel' or 'a luxury belt with jewels', and puram means a town or a state in the sense of earlier times. Thus, it is understood that the '香至-Kingdom' corresponds to the old capital 'Kanchipuram'."


title: Bodhidharma

text: Acharya Raghu, in his work 'Bodhidharma Retold', used a combination of multiple factors to identify Bodhidharma from the state of Andhra Pradesh in South India, specifically to the geography around Mt. Sailum or modern day Srisailam.


title: Bodhidharma

text: The Pakistani scholar Ahmad Hasan Dani speculated that according to popular accounts in Pakistan's northwest, Bodhidharma may be from the region around the Peshawar valley, or possibly around modern Afghanistan's eastern border with Pakistan.


title: Bodhidharma

text: In the context of the Indian caste system the mention of "Brahman king" acquires a nuance. Broughton notes that "king" implies that Bodhidharma was of caste of warriors and rulers. Brahman is, in western contexts, easily understood as Brahmana or Brahmin, which means "priest".


title: Bodhidharma

text: According to tradition Bodhidharma was given this name by his teacher known variously as Panyatara, Prajnatara, or Prajñādhara. His name prior to monkhood is said to be Jayavarman.


title: Bodhidharma

text: Bodhidharma is associated with several other names, and is also known by the name Bodhitara. Faure notes that:


title: Bodhidharma

text: Tibetan sources give his name as "Bodhidharmottara" or "Dharmottara", that is, "Highest teaching (dharma) of enlightenment".


title: Bodhidharma

text: Buswell dates Bodhidharma's abode in China approximately at the early 5th century. Broughton dates Bodhidharma's presence in Luoyang to between 516 and 526, when the temple referred to—Yongning Temple (), was at the height of its glory. Starting in 526, Yǒngníngsì suffered damage from a series of events, ultimately leading to its destruction in 534.


title: Bodhidharma

text: The idea that Bodhidharma founded martial arts at the Shaolin Temple was spread in the 20th century, however, martial arts historians have shown this legend stems from a 17th-century qigong manual known as the "Yijin Jing". The preface of this work says that Bodhidharma left behind the "Yi Jin Jing", from which the monks obtained the fighting skills which made them gain some fame.


title: Bodhidharma

text: The authenticity of the "Yijin Jing" has been discredited by some historians including Tang Hao, Xu Zhen and Matsuda Ryuchi. According to Lin Boyuan, "This manuscript is full of errors, absurdities and fantastic claims; it cannot be taken as a legitimate source."


title: Bodhidharma

text: The oldest available copy was published in 1827. The composition of the text itself has been dated to 1624. Even then, the association of Bodhidharma with martial arts only became widespread as a result of the 1904–1907 serialization of the novel "The Travels of Lao Ts'an" in "Illustrated Fiction Magazine". According to Henning, the "story is clearly a twentieth-century invention," which "is confirmed by writings going back at least 250 years earlier, which mention both Bodhidharma and martial arts but make no connection between the two."


title: Bodhidharma

text: In the Zen "kōan" tradition, Bodhidharma is mentioned as a significant figure. In Dogen's 13th century "kōan" collection, the "Shinji Shōbōgenzō", Bodhidharma is mentioned in fourteen different "kōans". In "The Gateless Gate" by Wumen Huikai:


title: Bodhidharma

text: In a short addendum from 1245 CE, the text refers to a motto attributed to Bodhidharma: "Bodhidharma coming from the west, unattached to any words, pointing directly to the mind of man, advocated seeing into one's nature and becoming Buddha." The legend of Dazu Huike and Bodhidharma is recounted in case no. 41 of "The Gateless Gate".


title: Bodhidharma

text: Bodhidharma's image became the inspiration for Japanese Daruma dolls, which originated in Meiwa-era Takasaki as good luck charms. A is held at the Shorinzan Daruma Temple in Takasaki every year, celebrating the city as the birthplace of the Daruma doll. Over 400,000 attendants come to purchase new dolls. The Japanese version of the children's game statues is named .


title: Bodhidharma

text: A 1989 South Korean film, "Why Has Bodhi-Dharma Left for the East?", derives its title from a "kōan" about Bodhidharma's legendary transmission of Chan Buddhism to China. The film screened at the 1989 Cannes Film Festival and was the first South Korean film to release theatrically in the United States. In 1994, the Hong Kong film "Master of Zen" (also known as "Bodhidharma") adapted the legends of Bodhidharma's life into martial arts drama film, partly inspired by the master's association with Shaolin Kung Fu. The 2011 Indian Tamil science fiction martial arts film "7aum Arivu" features a descendent of Bodhidharma as a main character and its plot focuses on the ancient monk's legendary skills and knowledge. The film was ultimately criticized for its historical inaccuracies in its portrayal of Bodhidharma (such as the monk's age upon entering China) and inappropriate emphasis of Bodhidharma as a Tamilian. The controversy caused hunger strikes among Indian followers of Bodhidharma.


title: Bodhidharma

text: According to Red Pine, many modern scholars agree that the "Two Entrances and Four Practices" can be attributed to Bodhidharma. However, there is a general disagreement about the other works which are commonly attributed to him. Red Pine further notes that "Yanagida, for example, attributes the Bloodstream Sermon to a member of the Oxhead Zen School, which flourished in the seventh and eighth centuries, and he thinks that the Wake-up Sermon was an eighth-century work of the Northern Zen School and the Break- through Sermon was by Shen-hsiu, the seventh-century patriarch of the Northern Zen School."


title: Bessel function

text: Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation


title: Bessel function

text: for an arbitrary complex number formula_2, the "order" of the Bessel function. Although formula_2 and formula_4 produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of formula_2.


title: Bessel function

text: The most important cases are when formula_2 is an integer or half-integer. Bessel functions for integer formula_2 are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer formula_2 are obtained when the Helmholtz equation is solved in spherical coordinates.


title: Bessel function

text: The Bessel function is a generalization of the sine function. It can be interpreted as the vibration of a string with variable thickness, variable tension (or both conditions simultaneously); vibrations in a medium with variable properties; vibrations of the disc membrane, etc.


title: Bessel function

text: Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (); in spherical problems, one obtains half-integer orders (). For example:


title: Bessel function

text: Bessel functions also appear in other problems, such as signal processing (e.g., see FM audio synthesis, Kaiser window, or Bessel filter).


title: Bessel function

text: Because this is a second-order linear differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient. Different variations are summarized in the table below and described in the following sections.


title: Bessel function

text: Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by and , respectively, rather than and .


title: Bessel function

text: Bessel functions of the first kind, denoted as , are solutions of Bessel's differential equation. For integer or positive , Bessel functions of the first kind are finite at the origin (); while for negative non-integer , Bessel functions of the first kind diverge as approaches zero. It is possible to define the function by its series expansion around , which can be found by applying the Frobenius method to Bessel's equation:


title: Bessel function

text: where is the gamma function, a shifted generalization of the factorial function to non-integer values. The Bessel function of the first kind is an entire function if is an integer, otherwise it is a multivalued function with singularity at zero. The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to formula_10 (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large . (The series indicates that is the derivative of , much like is the derivative of ; more generally, the derivative of can be expressed in terms of by the identities below.)


title: Bessel function

text: For non-integer , the functions and are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order , the following relationship is valid (the gamma function has simple poles at each of the non-positive integers):


title: Bessel function

text: This means that the two solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.


title: Bessel function

text: Another definition of the Bessel function, for integer values of , is possible using an integral representation:


title: Bessel function

text: This was the approach that Bessel used, and from this definition he derived several properties of the function. The definition may be extended to non-integer orders by one of Schläfli's integrals, for :


title: Bessel function

text: This expression is related to the development of Bessel functions in terms of the Bessel–Clifford function.


title: Bessel function

text: In terms of the Laguerre polynomials and arbitrarily chosen parameter , the Bessel function can be expressed as


title: Bessel function

text: The Bessel functions of the second kind, denoted by , occasionally denoted instead by , are solutions of the Bessel differential equation that have a singularity at the origin () and are multivalued. These are sometimes called Weber functions, as they were introduced by , and also Neumann functions after Carl Neumann.


title: Bessel function

text: In the case of integer order , the function is defined by taking the limit as a non-integer tends to :


title: Bessel function

text: When is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:


title: Bessel function

text: Both and are holomorphic functions of on the complex plane cut along the negative real axis. When is an integer, the Bessel functions are entire functions of . If is held fixed at a non-zero value, then the Bessel functions are entire functions of .


title: Bessel function

text: The Bessel functions of the second kind when is an integer is an example of the second kind of solution in Fuchs's theorem.


title: Bessel function

text: Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, and , defined as


title: Bessel function

text: where is the imaginary unit. These linear combinations are also known as Bessel functions of the third kind; they are two linearly independent solutions of Bessel's differential equation. They are named after Hermann Hankel.


title: Bessel function

text: These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, "simple" means an appearance of a factor of the form . For real formula_24 where formula_25, formula_26 are real-valued, the Bessel functions of the first and second kind are the real and imaginary parts, respectively, of the first Hankel function and the real and negative imaginary parts of the second Hankel function. Thus, the above formulae are analogs of Euler's formula, substituting , for formula_27 and formula_25, formula_26 for formula_30, formula_31, as explicitly shown in the asymptotic expansion.


title: Bessel function

text: The Hankel functions are used to express outward- and inward-propagating cylindrical-wave solutions of the cylindrical wave equation, respectively (or vice versa, depending on the sign convention for the frequency).


title: Bessel function

text: If is an integer, the limit has to be calculated. The following relationships are valid, whether is an integer or not:


title: Bessel function

text: where the integration limits indicate integration along a contour that can be chosen as follows: from to 0 along the negative real axis, from 0 to along the imaginary axis, and from to along a contour parallel to the real axis.


title: Bessel function

text: The Bessel functions are valid even for complex arguments , and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined as


title: Bessel function

text: when is not an integer; when is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments . The series expansion for is thus similar to that for , but without the alternating factor.


title: Bessel function

text: Using these two formulae the result to formula_39+formula_40, commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following


title: Bessel function

text: We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if ):


title: Bessel function

text: Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, and are exponentially growing and decaying functions respectively. Like the ordinary Bessel function , the function goes to zero at for and is finite at for . Analogously, diverges at with the singularity being of logarithmic type for , and otherwise.


title: Bessel function

text: Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example:


title: Bessel function

text: It can be proven by showing equality to the above integral definition for . This is done by integrating a closed curve in the first quadrant of the complex plane.


title: Bessel function

text: The modified Bessel function formula_48 is useful to represent the Laplace distribution as an Exponential-scale mixture of normal distributions.


title: Bessel function

text: The modified Bessel function of the second kind has also been called by the following names (now rare):


title: Bessel function

text: When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form


title: Bessel function

text: The two linearly independent solutions to this equation are called the spherical Bessel functions and , and are related to the ordinary Bessel functions and by


title: Bessel function

text: The zeroth spherical Bessel function is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are:


title: Bessel function

text: In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative integers :


title: Bessel function

text: The spherical Hankel functions appear in problems involving spherical wave propagation, for example in the multipole expansion of the electromagnetic field.


title: Bessel function

text: For example, this kind of differential equation appears in quantum mechanics while solving the radial component of the Schrödinger's equation with hypothetical cylindrical infinite potential barrier. This differential equation, and the Riccati–Bessel solutions, also arises in the problem of scattering of electromagnetic waves by a sphere, known as Mie scattering after the first published solution by Mie (1908). See e.g., Du (2004) for recent developments and references.


title: Bessel function

text: The Bessel functions have the following asymptotic forms. For small arguments formula_61, one obtains, when formula_2 is not a negative integer:


title: Bessel function

text: For large real arguments , one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless is half-integer) because they have zeros all the way out to infinity, which would have to be matched exactly by any asymptotic expansion. However, for a given value of one can write an equation containing a term of order :


title: Bessel function

text: (For the last terms in these formulas drop out completely; see the spherical Bessel functions above.) Even though these equations are true, better approximations may be available for complex . For example, when is near the negative real line is approximated better by


title: Bessel function

text: It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, is not asymptotic to the average of these two asymptotic forms when is negative (because one or the other will not be correct there, depending on the used). But the asymptotic forms for the Hankel functions permit us to write asymptotic forms for the Bessel functions of first and second kinds for "complex" (non-real) so long as goes to infinity at a constant phase angle (using the square root having positive real part):


title: Bessel function

text: an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by contour integration or other methods.)


title: Bessel function

text: which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series of a tone-modulated FM signal.


title: Bessel function

text: Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula:


title: Bessel function

text: Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by , the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that:


title: Bessel function

text: where , is the Kronecker delta, and is the th zero of . This orthogonality relation can then be used to extract the coefficients in the Fourier–Bessel series, where a function is expanded in the basis of the functions for fixed and varying .


title: Bessel function

text: (where is the rectangle function) then the Hankel transform of it (of any given order ), , approaches as approaches zero, for any given . Conversely, the Hankel transform (of the same order) of is :


title: Bessel function

text: which is zero everywhere except near 1. As approaches zero, the right-hand side approaches , where is the Dirac delta function. This admits the limit (in the distributional sense):