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Begging the question

In classical rhetoric and logic, begging the question or assuming the conclusion (Latin: petītiō principiī) is an informal fallacy that occurs when an argument's premises assume the truth of the conclusion. Historically, begging the question refers to a fault in a dialectical argument in which the speaker assumes some premise that has not been demonstrated to be true. In modern usage, it has come to refer to an argument in which the premises assume the conclusion without supporting it. This makes it an example of circular reasoning.

Some examples are:

 * "People have known for thousands of years that the earth is round. Therefore, the earth is round."
 * "Drugs are illegal so they must be bad for you. Therefore, we ought not legalize drugs because drugs are bad for you."

Colloquial misuse of the phrase "begs the question" also occurs with an entirely dissimilar sense in place of "prompts a question" or "raises a question".

History

The original phrase used by Aristotle from which begging the question descends is τὸ ἐξ ἀρχῆς αἰτεῖν, or sometimes ἐν ἀρχῇ αἰτεῖν, asking for the initial thing. Aristotle's intended meaning is closely tied to the type of dialectical argument he discusses in his Topics, book VIII: a formalized debate in which the defending party asserts a thesis that the attacking party must attempt to refute by asking yes-or-no questions and deducing some inconsistency between the responses and the original thesis.

In this stylized form of debate, the proposition that the answerer undertakes to defend is called the initial thing (τὸ ἐξ ἀρχῆς, τὸ ἐν ἀρχῇ) and one of the rules of the debate is that the questioner cannot simply ask (beg) for it (that would be trivial and uninteresting). Aristotle discusses this in Sophistical Refutations and in Prior Analytics book II, (64b, 34–65a 9, for circular reasoning see 57b, 18–59b, 1).

The stylized dialectical exchanges Aristotle discusses in the Topics included rules for scoring the debate, and one important issue was precisely the matter of asking for the initial thing—which included not just making the actual thesis adopted by the answerer into a question, but also making a question out of a sentence that was too close to that thesis (for example, PA II 16).

The term was translated into English from Latin in the 16th century. The Latin version, petitio principii asking for the starting point, can be interpreted in different ways. Petitio (from peto), in the post-classical context in which the phrase arose, means assuming or postulating, but in the older classical sense means petition, request or beseeching. Principii, genitive of principium, means beginning, basis or premise (of an argument). Literally petitio principii means assuming the premise or assuming the original point.

The Latin phrase comes from the Greek τὸ ἐν ἀρχῇ αἰτεῖσθαι (tò en archêi aiteîsthai asking the original point) in Aristotle's Prior Analytics II xvi 64b28–65a26:

"Begging or assuming the point at issue consists (to take the expression in its widest sense) [in] failing to demonstrate the required proposition. But there are several other ways in which this may happen; for example, if the argument has not taken syllogistic form at all, he may argue from premises which are less known or equally unknown, or he may establish the antecedent utilizing its consequents; for demonstration proceeds from what is more certain and is prior. Now begging the question is none of these.[...] If, however, the relation of B to C is such that they are identical, or that they are clearly convertible, or that one applies to the other, then he is begging the point at issue.... [B]egging the question is proving what is not self-evidently employing itself... either because identical predicates belong to the same subject, or because the same predicate belongs to identical subjects."

Aristotle's distinction between apodictic science and other forms of nondemonstrative knowledge rests on an epistemology and metaphysics wherein appropriate first principles become apparent to the trained dialectician: "Aristotle's advice in S.E. 27 for resolving fallacies of Begging the Question is brief. If one realizes that one is being asked to concede the original point, one should refuse to do so, even if the point being asked is a reputable belief. On the other hand, if one fails to realize that one has conceded the point at issue and the questioner uses the concession to produce the apparent refutation, then one should turn the tables on the sophistical opponent by oneself pointing out the fallacy committed. In dialectical exchange, it is a worse mistake to be caught asking for the original point than to have inadvertently granted such a request. The answerer in such a position has failed to detect when different utterances mean the same thing. The questioner, if he did not realize he was asking the original point, has committed the same error. But if he has knowingly asked for the original point, then he reveals himself to be ontologically confused: he has mistaken what is non-self-explanatory (known through other things) to be something self-explanatory (known through itself). In pointing this out to the false reasoner, one is not just pointing out a tactical psychological misjudgment by the questioner. It is not simply that the questioner falsely thought that the original point is placed under the guise of a semantic equivalent, or a logical equivalent, or a covering universal, or divided up into exhaustive parts, would be more persuasive to the answerer. Rather, the questioner falsely thought that a non-self-explanatory fact about the world was an explanatory first principle. For Aristotle, that certain facts are self-explanatory while others are not is not a reflection solely of the cognitive abilities of humans. It is primarily a reflection of the structure of noncognitive reality. In short, a successful resolution of such a fallacy requires a firm grasp of the correct explanatory powers of things. Without a knowledge of which things are self-explanatory and which are not the reasoner is liable to find a question-begging argument persuasive."

Thomas Fowler believed that petitio principii would be more properly called petitio quæsiti, which is literally begging the question.

Definition
To beg the question (also called petitio principii) is to attempt to support a claim with a premise that itself restates or presupposes the claim. It is an attempt to prove a proposition while simultaneously taking the proposition for granted.

When the fallacy involves only a single variable, it is sometimes called a hysteron proteron (Greek for later earlier), a rhetorical device, as in the statement: "Opium induces sleep because it has a soporific quality."

Reading this sentence, the only thing one can learn is a new word (soporific) that refers to a more common action (inducing sleep); it does not explain why opium causes that effect. A sentence that explains why opium induces sleep (or the same, why opium has soporific quality) could be the following one: "Opium induces sleep because it contains Morphine-6-glucuronide, which inhibits the brain's receptors for pain, causing a pleasurable sensation that eventually induces sleep."

A less obvious example from Fallacies and Pitfalls of Language: The Language Trap by S. Morris Engel:

"Free trade will be good for this country. The reason is patently clear. Isn't it obvious that unrestricted commercial relations will bestow on all sections of this nation the benefits which result when there is an unimpeded flow of goods between countries?"

This form of the fallacy may not be immediately obvious. Linguistic variations in syntax, sentence structure, and the literary device may conceal it, as may other factors involved in an argument's delivery. It may take the form of an unstated premise which is essential but not identical to the conclusion, or is "controversial or questionable for the same reasons that typically might lead someone to question the conclusion":

"...[S]eldom is anyone going to simply place the conclusion word-for-word into the premises... Rather, an arguer might use phraseology that conceals the fact that the conclusion is masquerading as a premise. The conclusion is rephrased to look different and is then placed in the premises."

For example, one can obscure the fallacy by first making a statement in concrete terms, then attempting to pass off an identical statement, delivered in abstract terms, as evidence for the original. One could also "bring forth a proposition expressed in words of Saxon origin, and give as a reason for it the very same proposition stated in words of Norman origin", as here:

"To allow every man an unbounded freedom of speech must always be, on the whole, advantageous to the State, for it is highly conducive to the interests of the community that each individual should enjoy a liberty perfectly unlimited of expressing his sentiments.'"

When the fallacy of begging the question is committed in more than one step, some authors dub it circulus in probando reasoning in a circle, or more commonly, circular reasoning.

Begging the question is not considered a formal fallacy (an argument that is defective because it uses an incorrect deductive step). Rather, it is a type of informal fallacy that is logically valid but unpersuasive, in that it fails to prove anything other than what is already assumed.

Related fallacies
Closely connected with begging the question is the fallacy of circular reasoning (circulus in probando), a fallacy in which the reasoner begins with the conclusion. The individual components of a circular argument can be logically valid because if the premises are true, the conclusion must be true, and does not lack relevance. However, circular reasoning is not persuasive because a listener who doubts the conclusion also doubts the premise that leads to it.

Begging the question is similar to the complex question (also known as trick question or fallacy of many questions): a question that, to be valid, requires the truth of another question that has not been established. For example, "Which color dress is Mary wearing?" may be fallacious because it presupposes that Mary is wearing a dress. Unless it has previously been established that her outfit is a dress, the question is fallacious because she could be wearing pants instead.

Another related fallacy is ignoratio elenchi or irrelevant conclusion: an argument that fails to address the issue in question, but appears to do so. An example might be a situation where A and B are debating whether the law permits A to do something. If A attempts to support his position with an argument that the law to allow him to do the thing in question, then he is guilty of ignoratio elenchi.

Vernacular
In vernacular English,  begging the question (or equivalent rephrasing thereof) sometimes occurs in place of "raises the question", "invites the question", "suggests the question", "leaves unanswered the question" etc. Such preface is then followed with the question, as in:
 * "[...]personal letter delivery is at an all-time low... Which begs the question: are open letters the only kind the future will know?"
 * "Hopewell's success begs the question: why aren't more companies doing the same?"
 * "Spending the summer traveling around India is a great idea, but it does beg the question of how we can afford it."

Sometimes it is further confused with "dodging the question", an attempt to avoid it, or perhaps more often begging the question is simply used to mean leaving the question unanswered.

title: Begging the question


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Teresa of Ávila

Teresa of Ávila, OCD (Teresa Sánchez de Cepeda Dávila y Ahumada; 28 March 1515 – 4 or 15 October 1582), also called Saint Teresa of Jesus, was a Carmelite nun and prominent Spanish mystic and religious reformer.

Active during the Counter-Reformation, Teresa became the central figure of a movement of spiritual and monastic renewal, reforming the Carmelite Orders of both women and men. The movement was later joined by the younger Carmelite friar and mystic Saint John of the Cross, with whom she established the Discalced Carmelites. A formal papal decree adopting the split from the old order was issued in 1580.

Her autobiography, The Life of Teresa of Jesus, and her books, The Interior Castle and The Way of Perfection, are prominent works on Christian mysticism and Christian meditation practice. In her autobiography, written as a defense of her ecstatic mystical experiences, she discerns four stages in the ascent of the soul to God: mental prayer and meditation; the prayer of quiet; absorption-in-God; ecstatic consciousness. The Interior Castle, written as a spiritual guide for her Carmelite sisters, uses the illustration of seven mansions within the castle of the soul to describe the different states one's soul can be in during life.

Forty years after her death, in 1622, Teresa was canonized by Pope Gregory XV. On 27 September 1970 Pope Paul VI proclaimed Teresa the first female Doctor of the Church in recognition of her centuries-long spiritual legacy to Catholicism.

Early life
Teresa Sánchez de Cepeda Dávila y Ahumada was born on 28 March 1515. Her birthplace was either Ávila or Gotarrendura. Her paternal grandfather, Juan Sánchez de Toledo, was a marrano or converso, a Jew forced to convert to Christianity or emigrate. When Teresa's father was a child, Juan was condemned by the Spanish Inquisition for allegedly returning to Judaism, but he was later able to assume a Catholic identity. Her father, Alonso Sánchez de Cepeda, was a successful wool merchant and one of the wealthiest men in Ávila. He bought a knighthood and assimilated successfully into Christian society. Previously married to Catalina del Peso y Henao, with whom he had three children, in 1509, Sánchez de Cepeda married Teresa's mother, Beatriz de Ahumada y Cuevas, in Gotarrendura.

Teresa's mother brought her up as a dedicated Christian. Fascinated by accounts of the lives of the saints, she ran away from home at age seven, with her brother Rodrigo, to seek martyrdom in the fight against the Moors. Her uncle brought them home, when he spotted them just outside the town walls.

When Teresa was eleven years old, her mother died, leaving her grief-stricken. This prompted her to embrace a deeper devotion to the Virgin Mary as her spiritual mother. Teresa was also enamored of popular fiction, which at the time consisted primarily of medieval tales of knighthood and works about fashion, gardens and flowers. Teresa was sent to the Augustinian nuns' school in Ávila.

Ascetic and mystical practice
After completing her education, she initially resisted the idea of a religious vocation, but after a stay with her uncle and other relatives, she relented. In 1534, aged 20, much to the disappointment of her pious and austere father, she decided to enter the local easy-going Carmelite Convent of the Incarnation, significantly built on top of land that had been used previously as a burial ground for Jews. She took up religious reading on contemplative prayer, especially Osuna's Abecedario espiritual ("Third Spiritual Alphabet," 1527), a guide on examination of conscience and "spiritual self-concentration and inner contemplation, known in mystical nomenclature as oratio recollectionis". She also dipped into other mystical ascetical works such as the Tractatus de oratione et meditatione of Peter of Alcantara.

Her zeal for mortification caused her to become ill again and she spent almost a year in bed, causing huge worry to her community and family. She nearly died but she recovered, attributing her recovery to the miraculous intercession of Saint Joseph. She began to experience bouts of religious ecstasy. She reported that, during her illness, she had progressed from the lowest stage of "recollection", to the "devotions of silence" and even to the "devotions of ecstasy", in which was one of perceived in "perfect union with God" (see ). She said she frequently experienced the rich "blessing of tears" during this final stage. As the Catholic distinction between mortal and venial sin became clear to her, she came to understand the awful horror of sin and the inherent nature of original sin. She also became conscious of her own natural impotence in confronting sin and the need for absolute surrender to God.

Around the same time, she received a copy of the full Spanish translation of Augustine of Hippo's autobiographical work Confessions, which helped her resolve and to tend to her own bouts of religious scruples. The text helped her realize that holiness was indeed possible and she found solace in the idea that such a great saint was once an inveterate sinner. In her autobiography, she wrote that she "was very fond of St. Augustine[...] for he was a sinner too".

Transverberation
Around 1556, friends suggested that her newfound knowledge could be of diabolical rather than divine origin. She had begun to inflict mortifications of the flesh upon herself. But her confessor, the Jesuit Francis Borgia, reassured her of the divine inspiration of her thoughts. On St. Peter's Day in 1559, Teresa became firmly convinced that Jesus Christ had presented himself to her in bodily form, though invisible. These visions lasted almost uninterruptedly for more than two years. In another vision, the famous transverberation, a seraph drove the fiery point of a golden lance repeatedly through her heart, causing her an ineffable spiritual and bodily pain: "I saw in his hand a long spear of gold, and at the point there seemed to be a little fire. He appeared to me to be thrusting it at times into my heart, and to pierce my very entrails; when he drew it out, he seemed to draw them out also, and to leave me all on fire with a great love of God. The pain was so great, that it made me moan; and yet so surpassing was the sweetness of this excessive pain, that I could not wish to be rid of it ..."

The account of this vision was the inspiration for one of Bernini's most famous works, the Ecstasy of Saint Teresa at Santa Maria della Vittoria in Rome. Although based in part on Teresa's description of her mystical transverberation in her autobiography, Bernini's depiction of the event is considered by some to be highly eroticized, especially when compared to the entire preceding artistic Teresian tradition.

The memory of this episode served as an inspiration throughout the rest of her life, and motivated her lifelong imitation of the life and suffering of Jesus, epitomized in the adage often associated with her: "Lord, either let me suffer or let me die"

Teresa, who became a celebrity in her town dispensing wisdom from behind the convent grille, was known for her raptures, which sometimes involved levitation. It was a source of embarrassment to her and she bade her sisters hold her down when this occurred. Subsequently, historians, neurologists and psychiatrists like Peter Fenwick and Javier Álvarez-Rodríguez, among others, have taken an interest in her symptomatology. The fact that she wrote down virtually everything that happened to her during her religious life means that an invaluable and exceedingly rare medical record from the 16th century has been preserved. Examination of this record has led to the speculative conclusion that she may have suffered from temporal lobe epilepsy.

Monastic reformer
Over time, Teresa found herself increasingly at odds with the spiritual malaise prevailing in her convent of the Incarnation. Among the 150 nuns living there, the observance of cloister, designed to protect and strengthen spiritual practice and prayer, became so lax that it appeared to lose its purpose. The daily invasion of visitors, many of high social and political rank, disturbed the atmosphere with frivolous concerns and vacuous conversation. Such intrusions in the solitude essential to develop and sustain contemplative prayer so grieved Teresa that she longed to intervene.

The incentive to take the practical steps inspired by her inward motivation was supported by the Franciscan priest, Peter of Alcantara, who met her early in 1560 and became her spiritual adviser. She resolved to found a "reformed" Carmelite convent, correcting the laxity which she had found at the Incarnation convent and elsewhere besides. Doña Guiomar of Ulloa, a friend, was granted permission for the project.

The abject poverty of the new convent, established in 1562 and named St. Joseph's (San José), at first caused a scandal among the citizens and authorities of Ávila, and the small house with its chapel was in peril of suppression. However, powerful patrons, including the local bishop, coupled with the impression of well ordered subsistence and purpose, turned animosity into approval.

In March 1563, after Teresa had moved to the new convent house, she received papal sanction for her primary principles of absolute poverty and renunciation of ownership of property, which she proceeded to formulate into a "constitution". Her plan was the revival of the earlier, stricter monastic rules, supplemented by new regulations including the three disciplines of ceremonial flagellation prescribed for the Divine Office every week, and the discalceation of the religious. For the first five years, Teresa remained in seclusion, mostly engaged in prayer and writing.

Extended travels
In 1567, Teresa received a patent from the Carmelite General, Rubeo de Ravenna, to establish further houses of the new order. This process required many visitations and long journeys across nearly all the provinces of Spain. She left a record of the arduous project in her Libro de las Fundaciones. Between 1567 and 1571, reformed convents were established at Medina del Campo, Malagón, Valladolid, Toledo, Pastrana, Salamanca, and Alba de Tormes.

As part of the original patent, Teresa was given permission to set up two houses for men who wished to adopt the reforms. She convinced two Carmelite friars, John of the Cross and Anthony of Jesus to help with this. They founded the first monastery of Discalced Carmelite brothers in November 1568 at Duruelo. Another friend of Teresa, Jerónimo Gracián, the Carmelite visitator of the older observance of Andalusia and apostolic commissioner, and later provincial of the Teresian order, gave her powerful support in founding monasteries at Segovia (1571), Beas de Segura (1574), Seville (1575), and Caravaca de la Cruz (Murcia, 1576). Meanwhile, John of the Cross promoted the inner life of the movement through his power as a teacher and preacher.

Opposition to reforms
In 1576, unreformed members of the Carmelite order began to persecute Teresa, her supporters and her reforms. Following a number of resolutions adopted at the general chapter at Piacenza, the governing body of the order forbade all further founding of reformed convents. The general chapter instructed her to go into "voluntary" retirement at one of her institutions. She obeyed and chose St. Joseph's at Toledo. Meanwhile, her friends and associates were subjected to further attacks.

Several years later, her appeals by letter to King Philip II of Spain secured relief. As a result, in 1579, the cases before the inquisition against her, Gracián and others, were dropped. This allowed the reform to resume. An edict from Pope Gregory XIII allowed the appointment of a special provincial for the newer branch of the Carmelite religious, and a royal decree created a "protective" board of four assessors for the reform.

During the last three years of her life, Teresa founded convents at Villanueva de la Jara in northern Andalusia (1580), Palencia (1580), Soria (1581), Burgos, and Granada (1582). In total, seventeen convents, all but one founded by her, and as many men's monasteries, were owed to her reforms over twenty years.

Last days
Her final illness overtook her on one of her journeys from Burgos to Alba de Tormes. She died in 1582, just as Catholic Europe was making the switch from the Julian to the Gregorian calendar, which required the excision of the dates of 5–14 October from the calendar. She died either before midnight of 4 October or early in the morning of 15 October, which is celebrated as her feast day. According to the liturgical calendar then in use, she died on the 15th. Her last words were: "My Lord, it is time to move on. Well then, may your will be done. O my Lord and my Spouse, the hour that I have longed for has come. It is time to meet one another."

Holy relics
She was buried at the Convento de la Anunciación in Alba de Tormes. Nine months after her death the coffin was opened and her body was found to be intact but the clothing had rotted. Before the body was re-interred one of her hands was cut off, wrapped in a scarf and sent to Ávila. Gracián cut the little finger off the hand and – according to his own account – kept it with him until it was taken by the occupying Ottoman Turks, from whom he had to redeem it with a few rings and 20 reales. The body was exhumed again on 25 November 1585 to be moved to Ávila and found to be incorrupt. An arm was removed and left in Alba de Tormes at the nuns' request, to compensate for losing the main relic of Teresa, but the rest of the body was reburied in the Discalced Carmelite chapter house in Ávila. The removal was done without the approval of the Duke of Alba de Tormes and he brought the body back in 1586, with Pope Sixtus V ordering that it remain in Alba de Tormes on pain of excommunication. A grander tomb on the original site was raised in 1598 and the body was moved to a new chapel in 1616.

The body still remains there, except for the following parts:
 * Rome – right foot and part of the upper jaw
 * Lisbon – hand
 * Ronda, Spain – left eye and left hand (the latter was kept by Francisco Franco until his death, after Francoist troops captured it from Republican troops during the Spanish Civil War)
 * Museum of the Church of the Annunciation, Alba de Tormes – left arm and heart
 * Church of Our Lady of Loreto, Paris, France – one finger
 * Sanlúcar de Barrameda – one finger

Canonization

In 1622, forty years after her death, she was canonized by Pope Gregory XV. The Cortes exalted her to patroness of Spain in 1627. The University of Salamanca had granted her the title Doctor ecclesiae (Latin for "Doctor of the Church") with a diploma in her lifetime, but that title is distinct from the papal honour of Doctor of the Church, which is always conferred posthumously. The latter was finally bestowed upon her by Pope Paul VI on 27 September 1970, along with Catherine of Siena, making them the first women to be awarded the distinction. Teresa is revered as the Doctor of Prayer. The mysticism in her works exerted a formative influence upon many theologians of the following centuries, such as Francis of Sales, Fénelon, and the Port-Royalists. In 1670, her coffin was plated in silver.

Teresa of Avila is honored in the Church of England and in the Episcopal Church on 15 October.

Patron saint
In 1626, at the request of Philip IV of Spain, the Castilian parliament elected Teresa "without lacking one vote" as copatron saint of Castile. This status was affirmed by Pope Urban VIII in a brief issued on 21 July 1627 in which he stated: "For these reasons [the king's and Cortes's elections] and for the great devotion which they have for Teresa, they elected her for patron and advocate of these kingdoms in the last Cortes of the aforementioned kingdoms ... And because ... the representatives in the Cortes desired it so greatly that their vote be firm and perpetual, we grant it our patronage and the approval of the Holy Apostolic See."

More broadly, the 1620s, the entirety of Spain (Castile and beyond) debated who should be the country's patron saint; the choices were either the current patron, James Matamoros, or a pairing of him and the newly canonised Saint Teresa of Ávila. Teresa's promoters said Spain faced newer challenges, especially the threat of Protestantism and societal decline at home, thus needing a more contemporary patron who understood those issues and could guide the Spanish nation. Santiago's supporters (Santiaguistas) fought back and eventually won the argument, but Teresa of Ávila remained far more popular at the local level. James the Great kept the title of patron saint for the Spanish people, and the most Blessed Virgin Mary under the title Immaculate Conception as the sole patroness for the entire Spanish Kingdom.

Legacy regarding the Infant Jesus of Prague

The Spanish nuns who established Carmel in France brought a devotion to the Infant Jesus with them, and it became widespread in France.

Though there are no written historical accounts establishing that Teresa of Ávila ever owned the famous Infant Jesus of Prague statue, according to tradition, such a statue is said to have been in her possession and Teresa is reputed to have given it to a noblewoman travelling to Prague. The age of the statue dates to approximately the same time as Teresa. It has been thought that Teresa carried a portable statue of the Child Jesus wherever she went; the idea circulated by the early 1700s.

Autobiography
The autobiography La Vida de la Santa Madre Teresa de Jesús (The Life of the Holy Mother Teresa of Jesus) was written at Avila between 1562 and 1565, but published posthumously. Editions include:
 * The Life of St. Teresa of Jesus ... Written by herself. Translated from the Spanish by D. Lewis, 1870. London: Burns, Oates, & Co
 * The Autobiography, written before 1567, under the direction of her confessor, Fr. Pedro Ibáñez, 1882
 * The Life of Saint Teresa of Avila by herself. J. M. Cohen, 1957. Penguin Classics
 * Life of St. Teresa of Jesus. Translated by Benedict Zimmerman, 1997. Tan Books, ISBN 978-0-89555-603-5
 * The Life of Teresa of Jesus: The Autobiography of Teresa of Avila. Translated by E. Allison Peers, 1991. Doubleday, ISBN 978-0-385-01109-9
 * The Book of Her Life, translated, with notes, by Kieran Kavanaugh, OCD and Otilio Rodriguez, OCD, 2008. Introduction by Jodi Bilinkoff. Indianapolis/Cambridge: Hackett Publishing Company, ISBN 978-0-87220-907-7
 * The Book of My Life. Mirabai Starr, 2008. Boston, Massachusetts: Shambhala Publications, ISBN 978-1-59030-573-7

The Way of Perfection
The Way of Perfection (Camino de Perfección) was published in 1566. Teresa called this a "living book" and in it set out to teach her nuns how to progress through prayer and Christian meditation. She discusses the rationale for being a Carmelite, and the rest deals with the purpose of and approaches to spiritual life. The title was inspired by the devotional book The Imitation of Christ (1418) which had become a favourite expression of Teresa much before she wrote this work, as it appeared at several places in her autobiography, The Life of Teresa of Jesus. Like her other books, The Way of Perfection was written on the advice of her counsellors to describe her experiences in prayer during the period when the Reformation was spreading through Europe. Herein she describes ways of attaining spiritual perfection through prayer and its four stages, as in meditation, quiet, repose of soul and finally perfect union with God, which she equates with rapture.

 * Editions:
 * The Way of Perfection. Translated and Edited by E. Allison Peers, Doubleday, 1991. ISBN 978-0-385-06539-9
 * The Way of Perfection, TAN Books, 1997. ISBN 978-0-89555-602-8
 * Way of Perfection, London, 2012. limovia.net ISBN 978-1-78336-025-3
 * El Camino de Perfección (The Way of Perfection), written also before 1567, at the direction of her confessor.

Interior Castle
The Interior Castle, or The Mansions, (El Castillo Interior or Las Moradas) was written in 1577, and published in 1588. It contained the basis for what she felt should be the ideal journey of faith, comparing the contemplative soul to a castle with seven successive interior courts, or chambers, analogous to the seven mansions. The work was inspired by her vision of the soul as a diamond in the shape of a castle containing seven mansions, which she interpreted as the journey of faith through seven stages, ending with union with God. Fray Diego, one of Teresa's former confessors wrote that God revealed to Teresa: "...a most beautiful crystal globe, made in the shape of a castle, and containing seven mansions, in the seventh and innermost of which was the King of Glory, in the greatest splendour, illumining and beautifying them all. The nearer one got to the centre, the stronger was the light; outside the palace limits everything was foul, dark and infested with toads, vipers and other venomous creatures.'"

Christia Mercer, Columbia University philosophy professor, claims that the seventeenth-century Frenchman René Descartes lifted some of his most influential ideas from Teresa of Ávila, who, fifty years before Descartes, wrote popular books about the role of philosophical reflection in intellectual growth. She describes a number of striking similarities between Descartes's seminal work Meditations on First Philosophy and Teresa's Interior Castle.

Translations

 * The first English translation was published in 1675.
 * Fr. John Dalton (1852). John Dalton’s translation of The Interior Castle contains an interesting preface and translations of other letters by St. Teresa.
 * Benedictines of Stanbrook, edited by Fr. Zimmerman (1921). The translation of The Interior Castle by the Benedictines of Stanbrook also has an excellent introduction and includes many cross-references to other works by St. Teresa.
 * E. Allison Peers (1946). E. Allison Peers’ translation of The Interior Castle is another popular public domain version translated by a professor and scholar of Hispanic studies.
 * Fr. Kieran Kavanaugh (1979). This translation also stays true to the text and contains many useful cross-references. An updated study edition contains comprehensive notes, reflection questions and a glossary.
 * The Interior Castle – The Mansions, TAN Books, 1997. ISBN 978-0-89555-604-2
 * Mirabai Starr (2004). Described as "free of religious dogma, this modern translation renders St. Teresa's work a beautiful and practical set of teachings for seekers of all faiths in need of spiritual guidance". Starr’s interpretive version of The Interior Castle eliminates Teresa’s use of words such as "sin", which results in a translation which is more paraphrased than accurate translation and departs significantly from the original's meaning.
 * The Interior Castle – Modern update of the spiritual guide by Teresa of Avila. by M.B. Anderson, Root Classics (publisher), 2022. ISBN 978-1-956314-01-4.

In popular culture
St. Teresa's mystical experiences have inspired several authors in modern times, but not necessarily from Teresa's Christian theological perspective.
 * She is mentioned in Elizabeth Goudge's play, The Brontës of Haworth (in Three Plays, Duckworth, London, 1939), as one of the authors included by Emily Brontë when she and her sister Charlotte are packing to go to Brussels. In the play, Emily is depicted as very interested in mysticism, and is also packing a book by Saint John of the Cross, and another by John Ruysbroeck (John of Ruusbroec or Jan van Ruusbroec: 1293/94-1381: a Medieval mystic from the Low Countries).
 * The 2006 book Eat, Pray, Love by Elizabeth Gilbert recognizes St. Teresa as "that most mystical of Catholic figures" and alludes to St. Teresa's Interior Castle as the "mansions of her being" and her journey as one of "divine meditative bliss".
 * The 2007 book by American spiritual author Caroline Myss Entering the Castle was inspired by St. Teresa's Interior Castle, but still has a New Age approach to mysticism.
 * St. Teresa also inspired American author R. A. Lafferty in his novel Fourth Mansions (1969), which was nominated for the Nebula Award for Best Novel in 1970.
 * Brooke Fraser's song "Orphans, Kingdoms" was inspired by St. Teresa's Interior Castle.
 * Jean Stafford's short story 'The Interior Castle' relates the intense preoccupation of an accident victim with her own brain, which she sees variously as a jewel, a flower, a light in a glass and a set of envelopes within envelopes.
 * Jeffrey Eugenides' 2011 novel The Marriage Plot refers to St. Teresa's Interior Castle when recounting the religious experience of Mitchell Grammaticus, one of the main characters of the book.
 * Teen Daze's 2012 release The Inner Mansions refers to St. Teresa's Interior Castle in the album's title as well as in the first track. "... have mercy on yourselves! If you realize your pitiable condition, how can you refrain from trying to remove the darkness from the crystal of your souls? Remember, if death should take you now, you would never again enjoy the light of this Sun". This line appears dubbed over the musical introduction to "New Life".
 * In Mark Williamson's ONE: a memoir (2018), the metaphor of the Interior Castle is used to describe an inner world of introspective reflection on past events, a set of "memory loci" based on the ancient system of recall for rhetorical purposes.

Other

 *  Relaciones (Relationships), an extension of the autobiography giving her inner and outer experiences in epistolary form.
 * Her rare poems (Todas las poesías, Munster, 1854) are distinguished for tenderness of feeling and rhythm of thought.
 * The Complete Poetry of St. Teresa of Avila. A Bilingual Edition – Edición y traducción de Eric W. Vogt. New Orleans University Press of the South, 1996. Second edition, 2015. xl, 116 p. ISBN 978-1-937030-52-0
 * "Meditations on Song of Songs", 1567, written nominally for her daughters at the convent of Our Lady of Mount Carmel.
 * Conceptos del Amor ("Concepts of Love") and
 * Exclamaciones.
 * Las Cartas (Saragossa, 1671), or her correspondence, of which there are 342 extant letters and 87 fragments of others. The first edition of Teresa's letters was published in 1658 with the comment of Juan de Palafox y Mendoza, Roman Catholic bishop of Osma and an opponent to the Company of Jesus.
 * The Complete Works of St Teresa of Jesus, in five volumes, translated and edited by E. Allison Peers, including 2 volumes of correspondence. London: Sheed and Ward, 1982.

Mysticism
The ultimate preoccupation of Teresa's mystical thought, as consistently reflected in her writings, is the ascent of the soul to God. Aumann notes that, "the grades of prayer described in The Life do not correspond to the division of prayer commonly given in the manuals of spiritual life", due to the fact that "St. Teresa did not write a systematic theology of prayer". According to Zimmerman, "In all her writings on this subject she deals with her personal experiences [...] there is no vestige in her writings of any influence of the Areopagite, the Patristic, or the Scholastic Mystical schools, as represented among others, by the German Dominican Mystics. She is intensely personal, her system going exactly as far as her experiences, but not a step further."

Teresa describes in the Interior Castle that the treasure of heaven lies buried within our hearts, and that there is an interior part of the heart which is the centre of the soul.

Four stages as described in the autobiography
In her autobiography she describes four stages, in which she uses the image of watering one's garden as a metaphor for mystical prayer:
 * The first, Devotion of the Heart, consists of mental prayer and meditation. It means the withdrawal of the soul from without, penitence and especially the devout meditation on the passion of Christ (Autobiography 11.20).
 * The second, Devotion of Peace, is where human will is surrendered to God. This occurs by virtue of an uplifted awareness granted by God, while other faculties, such as memory, reason, and imagination, are not yet safe from worldly distraction. Although a partial distraction can happen, due to outer activity such as repetition of prayers or writing down spiritual things, the prevailing state is one of quietude (Autobiography 14.1).
 * The third, Devotion of Union, concerns the absorption-in-God. It is not only a heightened, but essentially, an ecstatic state. At this level, reason is also surrendered to God, and only the memory and imagination are left to ramble. This state is characterized by a blissful peace, a sweet slumber of at least the higher soul faculties, that is a consciousness of being enraptured by the love of God.
 * The fourth, Devotion of Ecstasy, is where the consciousness of being in the body disappears. Sensory faculties cease to operate. Memory and imagination also become absorbed in God, as though intoxicated. Body and spirit dwell in the throes of exquisite pain, alternating between a fearful fiery glow, in complete unconscious helplessness, and periods of apparent strangulation. Sometimes such ecstatic transports literally cause the body to be lifted into space. This state may last as long as half an hour and tends to be followed by relaxation of a few hours of swoon-like weakness, attended by the absence of all faculties while in union with God. The subject awakens from this trance state in tears. It may be regarded as the culmination of mystical experience. Indeed, Teresa was said to have been observed levitating during Mass on more than one occasion.

The seven mansions of the Interior castle
The Interior Castle is divided into seven mansions (also called dwelling places), each level describing a step to get closer to God. In her work, Teresa already assumed entrance into the first mansions by prayer and meditation.

The purgative stage, involving active prayer and asceticism:
 * The first mansion begin with a soul's state of grace, but the souls are surrounded by sin and only starting to seek God's grace through humility in order to achieve perfection.
 * The second mansions are also called the Mansion of the Practice of Prayer because the soul seeks to advance through the castle by daily thoughts of God, humble recognition of God's work in the soul and ultimately daily prayer.
 * The third mansions are the Mansions of Exemplary Life characterized through divine grace and a love for God that is so great that the soul has an aversion to both mortal and venial sin and a desire to do works of charitable service to man for the ultimate glory of God. The prayer of acquired recollection belongs to the third mansion.

The illuminative stage, the beginning of mystical or contemplative or supernatural prayer:
 * The fourth mansions are a departure from the soul actively acquiring what it gains as God increases his role. In this mansion, the soul begins to experience two types of supernatural prayer, namely the Prayer of Supernatural (or passive) Recollection and The Prayer of Quiet;
 * The fifth mansion is The Prayer of Union, in which the soul prepares itself to receive gifts from God;

Unitive stage:
 * The sixth mansion is the betrothal (engagement) of the soul with God can be compared to lovers. The soul spends increasing amounts of time torn between favors from God and from outside afflictions.
 * The seventh mansion is the spiritual marriage with God, in which the soul achieves clarity in prayer

Some scholars have compared the seven mansions to the seven chakras in Hindu culture.

Overview
The first four grades of Teresa's classifications of prayer belong to the ascetical stage of spiritual life. These are vocal prayer, meditational or mental prayer, affective prayer, and acquired or natural recollection.

According to Augustin Poulain and Robert Thouless, Teresa described four degrees or stages of mystical union, namely the prayer of quiet, full or semi-ecstatic union, ecstatic union or ecstasy, and transforming or deifying union, or spiritual marriage (properly) of the soul with God. While Augustin Poulain and Robert Thouless do not mention the Prayer of Supernatural (or passive) Recollection as a separate stage, Aumann discerns infused contemplation as a separate stage in the fourth mansion of the Interior Castle. Together, these "five grades are infused prayer and belong to the mystical phase of spiritual life".

Thomas Merton disagrees on a fine-cut distinction between acquired contemplation and the prayer of quiet, noticing the Carmelite tendency of systematization, whereas Teresa herself was just describing her personal experiences. Commenting on Teresa's writings and the scholarly discussions on the precise stages, Thomas Merton comments: "with all these divisions and distinctions, comings and goings and varieties of terms, one tends to become impatient with the saint".

Aumann synthesizes Teresa's writings into nine grades of prayer:

Mental or meditational prayer
Mental prayer is a form of prayer "performed without aid of any particular formula." It is distinguished from vocal prayers, "prayers performed by means of a given formula", Prayer is mental when the thoughts and affections of the soul are not expressed in a previously determined formula. According to Teresa of Ávila, mental prayer is meditational prayer, in which the person is like a gardener, who, with much labour, draws the water up from the depths of the well to water the plants and flowers. According to Teresa of Avila, mental prayer can proceed by using vocal prayers in order to improve dialogue with God. According to Lehodey, mental prayer can be divided into meditation, more active in reflections, and contemplation, more quiet and gazeful.

Natural or acquired contemplation – prayer of simplicity
For Teresa of Avila, in natural or acquired contemplation, also called the prayer of simplicity there is one dominant thought or sentiment which recurs constantly and easily (although with little or no development) amid many other thoughts, beneficial or otherwise. The prayer of simplicity often has a tendency to simplify itself even in respect to its object, leading one to think chiefly of God and of his presence, but in a confused manner.

In the words of Saint Alphonsus Maria de Liguori, acquired contemplation "consists in seeing at a simple glance the truths which could previously be discovered only through prolonged discourse": reasoning is largely replaced by intuition and affections and resolutions, though not absent, are only slightly varied and expressed in a few words. Similarly, Saint Ignatius of Loyola, in his 30-day retreat or Spiritual Exercises beginning in the "second week" with its focus on the life of Jesus, describes less reflection and more simple contemplation on the events of Jesus' life. These contemplations consist mainly in a simple gaze and include an "application of the senses" to the events, to further one's empathy for Jesus' values, "to love him more and to follow him more closely".

Definitions similar to that of Saint Alphonsus Maria de Liguori are given by Adolphe Tanquerey ("a simple gaze on God and divine things proceeding from love and tending thereto") and Saint Francis de Sales ("a loving, simple and permanent attentiveness of the mind to divine things").

Natural or acquired contemplation has been compared to the attitude of a mother watching over the cradle of her child: she thinks lovingly of the child without reflection and amid interruptions. The Catechism of the Catholic Church states: "What is contemplative prayer? St. Teresa answers: 'Contemplative [sic] prayer [oración mental] in my opinion is nothing else than a close sharing between friends; it means taking time frequently to be alone with him who we know loves us.' Contemplative prayer seeks him 'whom my soul loves'. It is Jesus, and in him, the Father. We seek him, because to desire him is always the beginning of love, and we seek him in that pure faith which causes us to be born of him and to live in him. In this inner prayer we can still meditate, but our attention is fixed on the Lord himself."

Infused or higher contemplation – mystical union
According to Hardon, infused contemplation is "A supernatural gift by which a person's mind and will become totally centered on God. Under this influence the intellect receives special insights into things of the spirit, and the affections are extraordinarily animated with divine love. Infused contemplation assumes the free co-operation of the human will." According to Poulain, it is a form of mystical union with God, a union characterized by the fact that it is God, and God only, who manifests Himself. According to Poulain, mystical grace may also manifest as visions of the humanity of Christ or an angel or revelations of a future event, and include miraculous bodily phenomena sometimes observed in ecstatics.

In Teresa's mysticism, infused contemplation is described as a "divinely originated, general, non-conceptual, loving awareness of God". According to Dubay: "It is a wordless awareness and love that we of ourselves cannot initiate or prolong. The beginnings of this contemplation are brief and frequently interrupted by distractions. The reality is so unimposing that one who lacks instruction can fail to appreciate what exactly is taking place. Initial infused prayer is so ordinary and unspectacular in the early stages that many fail to recognize it for what it is. Yet with generous people, that is, with those who try to live the whole Gospel wholeheartedly and who engage in an earnest prayer life, it is common."

According to Thomas Dubay, infused contemplation is the normal, ordinary development of discursive prayer (mental prayer, meditative prayer), which it gradually replaces. Dubay considers infused contemplation as common only among "those who try to live the whole Gospel wholeheartedly and who engage in an earnest prayer life". Other writers view contemplative prayer in its infused supernatural form as far from common. John Baptist Scaramelli, reacting in the 17th century against quietism, taught that asceticism and mysticism are two distinct paths to perfection, the former being the normal, ordinary end of the Christian life, and the latter something extraordinary and very rare. Jordan Aumann considered that this idea of the two paths was "an innovation in spiritual theology and a departure from the traditional Catholic teaching". And Jacques Maritain proposed that one should not say that every mystic necessarily enjoys habitual infused contemplation in the mystical state, since the gifts of the Holy Spirit are not limited to intellectual operations.

The Prayer of Quiet
For Teresa of Avila, the Prayer of Quiet is a state in which the soul experiences an extraordinary peace and rest, accompanied by delight or pleasure in contemplating God as present. According to Poulain, "Mystical union will be called spiritual quiet when the Divine action is still too weak to prevent distractions: in a word, when the imagination still retains a certain liberty". According to Poulain, in incomplete mystical union, or the prayer of quiet or supernatural recollection, the action of God is not strong enough to prevent distractions, and the imagination still retains a certain liberty.

Full or semi-ecstatic union
According to Poulain, "Mystical union will be called [...] full union when its strength is so great that the soul is fully occupied with the Divine object, whilst, on the other hand, the senses continue to act (under these conditions, by making a greater or less effort, one can cease from prayer".

Ecstatic union
According to Poulain, "Mystical union will be called [...] ecstasy when communications with the external world are severed or nearly so (in this event one can no longer make voluntary movement nor energy from the state at will)."

Transforming union
The transforming union differs from the other three specifically and not merely in intensity. According to Poulain, "It consists in the habitual consciousness of a mysterious grace which all shall possess in heaven: the anticipation of the Divine nature. The soul is conscious of the Divine assistance in its superior supernatural operations, those of the intellect and the will. Spiritual marriage differs from spiritual espousals inasmuch as the first of these states is permanent and the second only transitory."

Portrayals
Portrayals of Teresa include the following:

Literature

 * Simone de Beauvoir singles out Teresa as a woman who truly lived life for herself (and perhaps the only woman to do so) in her book The Second Sex.
 * She is mentioned prominently in Kathryn Harrison's novel Poison. The main character, Francisca De Luarca, is fascinated by her life.
 * Don DeLillo in End Zone depicted Teresa as a saint who eats from a human skull to remind herself of final things.
 * R. A. Lafferty was strongly inspired by El Castillo Interior when he wrote his novel Fourth Mansions. Quotations from St. Teresa's work are frequently used as chapter headings.
 * Pierre Klossowski prominently features Saint Teresa of Ávila in his metaphysical novel The Baphomet.
 * George Eliot compared Dorothea Brooke to St. Teresa in Middlemarch (1871–1872) and wrote briefly about the life and works of St. Teresa in the "Prelude" to the novel.
 * Thomas Hardy took Saint Teresa as the inspiration for much of the characterisation of the heroine Tess (Teresa) Durbeyfield, in Tess of the d'Urbervilles (1891), most notably the scene in which she lies in a field and senses her soul ecstatically above her.
 * The contemporary poet Jorie Graham features Saint Teresa in the poem Breakdancing in her volume The End of Beauty.
 * Bárbara Mujica's novel Sister Teresa, while not strictly hagiographical, is based upon Teresa's life.
 * Timothy Findley's 1999 novel Pilgrim features Saint Teresa as a minor character.
 * Vita Sackville-West wrote a double biography contrasting the two Carmelites Teresa of Avila and Thérèse of Lisieux, The Eagle and the Dove, re-issued in 2018.

Painting and sculpture

 * Saint Teresa of Ávila's Vision of the Holy Spirit is a 1612-1614 painting by Peter Paul Rubens and is exhibited in the Museum Boijmans Van Beuningen in Rotterdam, and Rubens' c. 1614 painting of the same subject is in the Fitzwilliam Museum, Cambridge.
 * Another Rubens portrait of Teresa, from 1615, is now in the collection of the Kunsthistorisches Museum, Vienna.
 * Saint Teresa was the inspiration for one of Bernini's most famous sculptures, The Ecstasy of St. Teresa (mid-17th century) in Santa Maria della Vittoria, Rome.
 * St. Teresa was painted in 1819–20 by François Gérard, a French neoclassical painter.
 * St Theresa of Avila is a 1754-1755 painting by Joseph-Marie Vien and is exhibited in the New Orleans Museum of Art in New Orleans, Louisiana.

Iconography
Theresa is usually shown in the habit of the Discalced Carmelites, and writing in a book with a quill pen. Sometimes there is a dove, symbolizing the Holy Spirit.

Drama and film

 * Hugh Ross Williamson wrote a play, Teresa of Avila, about her life, which premiered in London in 1961.
 * Performance artist Linda Montano has cited Teresa of Ávila as one of the most important influences on her work and since her return to Catholicism in the 2000s has done performances of her life.
 * Teresa de Jesús (1984), directed by Josefina Molina and starring Concha Velasco, is a Spanish made-for-TV mini-series. In it, Teresa is portrayed as the determined foundress of new Carmelite houses while protecting the infant Jesus statue on her many arduous journeys. The devotion to the Child Jesus spread quickly in Spain, possibly due to her mystical reputation, and then to other places, including France.
 * Nigel Wingrove's 1989 short film Visions of Ecstasy was based on Teresa of Ávila. The film features fantasised sexualised scenes of Teresa with the body of Jesus on the cross. It is the only work to be refused certification by the British Board of Film Classification (BBFC) on the grounds of blasphemy.
 * Dževad Karahasan. The Delighted Angel drama about Teresa of Ávila and Rabija al-Adavija, Vienna-Salzburg-Klagenfurt, ARBOS 1995.
 * Paz Vega stars as Teresa in Teresa, el cuerpo de Cristo, a 2007 Spanish biopic directed by Ray Loriga.
 * St. Teresa also features prominently in the 2009 Ron Howard film, Angels and Demons, where the Bernini sculpture, "The Ecstasy of St. Teresa", is an important clue in helping Robert Langdon (Tom Hanks) find an anti-matter bomb that is hidden in and set to destroy the Vatican.
 * Marian Álvarez portrays Teresa in the 2015 television film of the same name directed by Jorge Dorado and made for the 500th anniversary of her birth.
 * Blanca Portillo, Greta Fernández, and Ainet Jounou portray Teresa in the 2023 film of the same name directed by Paula Ortiz.

Music

 * Marc-Antoine Charpentier composed two motets for the feast of Saint Teresa: Flores, flores o Gallia for two voices, two flutes and continuo (H.374), c. 1680 and the other, for three voices and continuo (H.342), in 1686–87.
 * She is a principal character of the opera Four Saints in Three Acts by the composer Virgil Thomson with a libretto by Gertrude Stein.
 * Saint Teresa is the subject of the song "Theresa's Sound-World" by Sonic Youth off the 1992 album Dirty, lyrics by Thurston Moore.
 * "Saint Teresa" is a track on Joan Osborne's Relish album, nominated for a Grammy Award in 1996.
 * References to Saint Theresa and her visions appear in several songs across multiple albums by the Hold Steady, including "Chicago Seemed Tired Last Night", "The Sweet Part of the City", and "Our Whole Lives"

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Teresa of Ávila


text:
753 BC

Europe

 * April 21: Romulus and Remus legendarily found the city of Rome (according to the calculations of the Roman scholar Varro Reatinus). According to the legend, Romulus and Remus are the sons of Rhea Silvia, daughter of Numitor, king of Alba Longa, and descended from Aeneas. Alba Longa is an ancient Latin city, located in the Alban Hills in Central Italy. Before the birth of the twin brothers, Numitor is deposed by his younger brother, Amulius, who forces Rhea to become a vestal virgin, so that she will not give birth to rival claimants to his title. However, Rhea is impregnated (raped) by the war god Mars and gives birth to Romulus and Remus. Amulius orders the infants to be drowned in the Tiber River, but they survive and wash ashore at the foot of the Palatine Hill – where they are suckled by a she-wolf until they are found by the shepherd Faustulus. Reared by Faustulus and his wife, the twins later become leaders of a band of young shepherd warriors. After learning their true identity, they attack Alba Longa, killing the wicked Amulius, and restore their grandfather to the throne. The twins decide to found a town on the site where they had been saved as infants. Romulus and Remus soon become involved in a quarrel, however, Remus is slain by his brother. Romulus then becomes ruler of the settlement, which is named Rome after him. To populate his town, Romulus offers asylum to fugitives and exiles.

Chronology

 * Beginning of the Roman Ab urbe condita calendar.

title: 753 BC


text:
Fluorite

Fluorite (also called fluorspar) is the mineral form of calcium fluoride, CaF2. It belongs to the halide minerals. It crystallizes in isometric cubic habit, although octahedral and more complex isometric forms are not uncommon.

The Mohs scale of mineral hardness, based on scratch hardness comparison, defines value 4 as fluorite.

Pure fluorite is colourless and transparent, both in visible and ultraviolet light, but impurities usually make it a colorful mineral and the stone has ornamental and lapidary uses. Industrially, fluorite is used as a flux for smelting, and in the production of certain glasses and enamels. The purest grades of fluorite are a source of fluoride for hydrofluoric acid manufacture, which is the intermediate source of most fluorine-containing fine chemicals. Optically clear transparent fluorite has anomalous partial dispersion, that is, its refractive index varies with the wavelength of light in a manner that differs from that of commonly used glases, so fluorite is useful in making apochromatic lenses, making it valuable, particularly in photographic optics. Fluorite optics are also usable in the far-ultraviolet and mid-infrared ranges, where conventional glasses are too opaque for use. Fluorite also has low dispersion, and a high refractive index for its density, which can make it useful for some specialized purposes in optics.

History and etymology
The word fluorite is derived from the Latin verb fluere, meaning to flow. The mineral is used as a flux in iron smelting to decrease the viscosity of slag. The term flux comes from the Latin adjective fluxus, meaning flowing, loose, slack. The mineral fluorite was originally termed fluorspar and was first discussed in print in a 1530 work Bermannvs sive de re metallica dialogus [Bermannus; or dialogue about the nature of metals], by Georgius Agricola, as a mineral noted for its usefulness as a flux. Agricola, a German scientist with expertise in philology, mining, and metallurgy, named fluorspar as a Neo-Latinization of the German Flussspat from Fluss (stream, river) and Spat (meaning a nonmetallic mineral akin to gypsum, spærstān, spear stone, referring to its crystalline projections).

In 1852, fluorite gave its name to the phenomenon of fluorescence, which is prominent in fluorites from certain locations, due to certain impurities in the crystal. Fluorite also gave the name to its constitutive element fluorine. Currently, the word "fluorspar" is most commonly used for fluorite as an industrial and chemical commodity, while "fluorite" is used mineralogically and in most other senses.

In archeology, gemmology, classical studies, and Egyptology, the Latin terms murrina and myrrhina refer to fluorite. In book 37 of his Naturalis Historia, Pliny the Elder describes it as a precious stone with purple and white mottling, whose objects carved from it, the Romans prize.

Structure

Fluorite crystallizes in a cubic motif. Crystal twinning is common and adds complexity to the observed crystal habits. Fluorite has four perfect cleavage planes that help produce octahedral fragments. The structural motif adopted by fluorite is so common that the motif is called the fluorite structure. Element substitution for the calcium cation often includes strontium and certain rare-earth elements (REE), such as yttrium and cerium.

Occurrence and mining
Fluorite forms as a late-crystallizing mineral in felsic igneous rocks typically through hydrothermal activity. It is particularly common in granitic pegmatites. It may occur as a vein deposit formed through hydrothermal activity particularly in limestones. In such vein deposits it can be associated with galena, sphalerite, barite, quartz, and calcite. Fluorite can also be found as a constituent of sedimentary rocks either as grains or as the cementing material in sandstone.

It is a common mineral mainly distributed in South Africa, China, Mexico, Mongolia, the United Kingdom, the United States, Canada, Tanzania, Rwanda and Argentina.

The world reserves of fluorite are estimated at 230 million tonnes (Mt) with the largest deposits being in South Africa (about 41 Mt), Mexico (32 Mt) and China (24 Mt). China is leading the world production with about 3 Mt annually (in 2010), followed by Mexico (1.0 Mt), Mongolia (0.45 Mt), Russia (0.22 Mt), South Africa (0.13 Mt), Spain (0.12 Mt) and Namibia (0.11 Mt).

One of the largest deposits of fluorspar in North America is located on the Burin Peninsula, Newfoundland, Canada. The first official recognition of fluorspar in the area was recorded by geologist J.B. Jukes in 1843. He noted an occurrence of "galena" or lead ore and fluoride of lime on the west side of St. Lawrence harbour. It is recorded that interest in the commercial mining of fluorspar began in 1928 with the first ore being extracted in 1933. Eventually, at Iron Springs Mine, the shafts reached depths of 970 ft. In the St. Lawrence area, the veins are persistent for great lengths and several of them have wide lenses. The area with veins of known workable size comprises about 60 sqmi.

In 2018, Canada Fluorspar Inc. commenced mine production again in St. Lawrence; in spring 2019, the company was planned to develop a new shipping port on the west side of Burin Peninsula as a more affordable means of moving their product to markets, and they successfully sent the first shipload of ore from the new port on July 31, 2021. This marks the first time in 30 years that ore has been shipped directly out of St. Lawrence.

Cubic crystals up to 20 cm across have been found at Dalnegorsk, Russia. The largest documented single crystal of fluorite was a cube 2.12 meters in size and weighing approximately 16 tonnes. In Asturias (Spain) there are several fluorite deposits known internationally for the quality of the specimens they have yielded. In the area of Berbes, Ribadesella, fluorite appears as cubic crystals, sometimes with dodecahedron modifications, which can reach a size of up to 10 cm of edge, with internal colour zoning, almost always violet in colour. It is associated with quartz and leafy aggregates of baryte. In the Emilio mine, in Loroñe, Colunga, the fluorite crystals, cubes with small modifications of other figures, are colourless and transparent. They can reach 10 cm of edge. In the Moscona mine, in Villabona, the fluorite crystals, cubic without modifications of other shapes, are yellow, up to 3 cm of edge. They are associated with large crystals of calcite and barite.

"Blue John"
One of the most famous of the older-known localities of fluorite is Castleton in Derbyshire, England, where, under the name of "Derbyshire Blue John", purple-blue fluorite was extracted from several mines or caves. During the 19th century, this attractive fluorite was mined for its ornamental value. The mineral Blue John is now scarce, and only a few hundred kilograms are mined each year for ornamental and lapidary use. Mining still takes place in Blue John Cavern and Treak Cliff Cavern.

Recently discovered deposits in China have produced fluorite with coloring and banding similar to the classic Blue John stone.

Fluorescence
George Gabriel Stokes named the phenomenon of fluorescence from fluorite, in 1852.

Many samples of fluorite exhibit fluorescence under ultraviolet light, a property that takes its name from fluorite. Many minerals, as well as other substances, fluoresce. Fluorescence involves the elevation of electron energy levels by quanta of ultraviolet light, followed by the progressive falling back of the electrons into their previous energy state, releasing quanta of visible light in the process. In fluorite, the visible light emitted is most commonly blue, but red, purple, yellow, green, and white also occur. The fluorescence of fluorite may be due to mineral impurities, such as yttrium and ytterbium, or organic matter, such as volatile hydrocarbons in the crystal lattice. In particular, the blue fluorescence seen in fluorites from certain parts of Great Britain responsible for the naming of the phenomenon of fluorescence itself, has been attributed to the presence of inclusions of divalent europium in the crystal. Natural samples containing rare earth impurities such as erbium have also been observed to display upconversion fluorescence, in which infrared light stimulates emission of visible light, a phenomenon usually only reported in synthetic materials.

One fluorescent variety of fluorite is chlorophane, which is reddish or purple in color and fluoresces brightly in emerald green when heated (thermoluminescence), or when illuminated with ultraviolet light.

The color of visible light emitted when a sample of fluorite is fluorescing depends on where the original specimen was collected; different impurities having been included in the crystal lattice in different places. Neither does all fluorite fluoresce equally brightly, even from the same locality. Therefore, ultraviolet light is not a reliable tool for the identification of specimens, nor for quantifying the mineral in mixtures. For example, among British fluorites, those from Northumberland, County Durham, and eastern Cumbria are the most consistently fluorescent, whereas fluorite from Yorkshire, Derbyshire, and Cornwall, if they fluoresce at all, are generally only feebly fluorescent.

Fluorite also exhibits the property of thermoluminescence.

Color
Fluorite is allochromatic, meaning that it can be tinted with elemental impurities. Fluorite comes in a wide range of colors and has consequently been dubbed "the most colorful mineral in the world". Every color of the rainbow in various shades is represented by fluorite samples, along with white, black, and clear crystals. The most common colors are purple, blue, green, yellow, or colorless. Less common are pink, red, white, brown, and black. Color zoning or banding is commonly present. The color of the fluorite is determined by factors including impurities, exposure to radiation, and the absence of voids of the color centers.

Source of fluorine and fluoride
Fluorite is a major source of hydrogen fluoride, a commodity chemical used to produce a wide range of materials. Hydrogen fluoride is liberated from the mineral by the action of concentrated sulfuric acid:
 * CaF2(s) + H2SO4 → CaSO4(s) + 2 HF(g)

The resulting HF is converted into fluorine, fluorocarbons, and diverse fluoride materials. As of the late 1990s, five billion kilograms were mined annually.

There are three principal types of industrial use for natural fluorite, commonly referred to as "fluorspar" in these industries, corresponding to different grades of purity. Metallurgical grade fluorite (60–85% CaF2), the lowest of the three grades, has traditionally been used as a flux to lower the melting point of raw materials in steel production to aid the removal of impurities, and later in the production of aluminium. Ceramic grade fluorite (85–95% CaF2) is used in the manufacture of opalescent glass, enamels, and cooking utensils. The highest grade, "acid grade fluorite" (97% or more CaF2), accounts for about 95% of fluorite consumption in the US where it is used to make hydrogen fluoride and hydrofluoric acid by reacting the fluorite with sulfuric acid.

Internationally, acid-grade fluorite is also used in the production of AlF3 and cryolite (Na3AlF6), which are the main fluorine compounds used in aluminium smelting. Alumina is dissolved in a bath that consists primarily of molten Na3AlF6, AlF3, and fluorite (CaF2) to allow electrolytic recovery of aluminium. Fluorine losses are replaced entirely by the addition of AlF3, the majority of which react with excess sodium from the alumina to form Na3AlF6.

Lapidary uses
Natural fluorite mineral has ornamental and lapidary uses. Fluorite may be drilled into beads and used in jewelry, although due to its relative softness it is not widely used as a semiprecious stone. It is also used for ornamental carvings, with expert carvings taking advantage of the stone's zonation.

Optics
In the laboratory, calcium fluoride is commonly used as a window material for both infrared and ultraviolet wavelengths, since it is transparent in these regions (about 0.15 μm to 9 μm) and exhibits an extremely low change in refractive index with wavelength. Furthermore, the material is attacked by few reagents. At wavelengths as short as 157 nm, a common wavelength used for semiconductor stepper manufacture for integrated circuit lithography, the refractive index of calcium fluoride shows some non-linearity at high power densities, which has inhibited its use for this purpose. In the early years of the 21st century, the stepper market for calcium fluoride collapsed, and many large manufacturing facilities have been closed. Canon and other manufacturers have used synthetically grown crystals of calcium fluoride components in lenses to aid apochromatic design, and to reduce light dispersion. This use has largely been superseded by newer glasses and computer-aided design. As an infrared optical material, calcium fluoride is widely available and was sometimes known by the Eastman Kodak trademarked name "Irtran-3", although this designation is obsolete.

Fluorite should not be confused with fluoro-crown (or fluorine crown) glass, a type of low-dispersion glass that has special optical properties approaching fluorite. True fluorite is not a glass but a crystalline material. Lenses or optical groups made using this low dispersion glass as one or more elements exhibit less chromatic aberration than those utilizing conventional, less expensive crown glass and flint glass elements to make an achromatic lens. Optical groups employ a combination of different types of glass; each type of glass refracts light in a different way. By using combinations of different types of glass, lens manufacturers are able to cancel out or significantly reduce unwanted characteristics; chromatic aberration being the most important. The best of such lens designs are often called apochromatic (see above). Fluoro-crown glass (such as Schott FK51) usually in combination with an appropriate "flint" glass (such as Schott KzFSN 2) can give very high performance in telescope objective lenses, as well as microscope objectives, and camera telephoto lenses. Fluorite elements are similarly paired with complementary "flint" elements (such as Schott LaK 10). The refractive qualities of fluorite and of certain flint elements provide a lower and more uniform dispersion across the spectrum of visible light, thereby keeping colors focused more closely together. Lenses made with fluorite are superior to fluoro-crown based lenses, at least for doublet telescope objectives; but are more difficult to produce and more costly.

The use of fluorite for prisms and lenses was studied and promoted by Victor Schumann near the end of the 19th century. Naturally occurring fluorite crystals without optical defects were only large enough to produce microscope objectives.

With the advent of synthetically grown fluorite crystals in the 1950s - 60s, it could be used instead of glass in some high-performance optical telescope and camera lens elements. In telescopes, fluorite elements allow high-resolution images of astronomical objects at high magnifications. Canon Inc. produces synthetic fluorite crystals that are used in their better telephoto lenses. The use of fluorite for telescope lenses has declined since the 1990s, as newer designs using fluoro-crown glass, including triplets, have offered comparable performance at lower prices. Fluorite and various combinations of fluoride compounds can be made into synthetic crystals which have applications in lasers and special optics for UV and infrared.

Exposure tools for the semiconductor industry make use of fluorite optical elements for ultraviolet light at wavelengths of about 157 nanometers. Fluorite has a uniquely high transparency at this wavelength. Fluorite objective lenses are manufactured by the larger microscope firms (Nikon, Olympus, Carl Zeiss and Leica). Their transparence to ultraviolet light enables them to be used for fluorescence microscopy. The fluorite also serves to correct optical aberrations in these lenses. Nikon has previously manufactured at least one fluorite and synthetic quartz element camera lens (105 mm f/4.5 UV) for the production of ultraviolet images. Konica produced a fluorite lens for their SLR cameras – the Hexanon 300 mm f/6.3.

Source of fluorine gas in nature
In 2012, the first source of naturally occurring fluorine gas was found in fluorite mines in Bavaria, Germany. It was previously thought that fluorine gas did not occur naturally because it is so reactive, and would rapidly react with other chemicals. Fluorite is normally colorless, but some varied forms found nearby look black, and are known as 'fetid fluorite' or antozonite. The minerals, containing small amounts of uranium and its daughter products, release radiation sufficiently energetic to induce oxidation of fluoride anions within the structure, to fluorine that becomes trapped inside the mineral. The color of fetid fluorite is predominantly due to the calcium atoms remaining. Solid-state fluorine-19 NMR carried out on the gas contained in the antozonite, revealed a peak at 425 ppm, which is consistent with F2.

title: Fluorite


text:
Flux

Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface.

Terminology
The word flux comes from Latin: fluxus means "flow", and fluere is "to flow". As fluxion, this term was introduced into differential calculus by Isaac Newton.

The concept of heat flux was a key contribution of Joseph Fourier, in the analysis of heat transfer phenomena. His seminal treatise Théorie analytique de la chaleur (The Analytical Theory of Heat), defines fluxion as a central quantity and proceeds to derive the now well-known expressions of flux in terms of temperature differences across a slab, and then more generally in terms of temperature gradients or differentials of temperature, across other geometries. One could argue, based on the work of James Clerk Maxwell, that the transport definition precedes the definition of flux used in electromagnetism. The specific quote from Maxwell is: "In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the surface integral of the flux. It represents the quantity which passes through the surface."

According to the transport definition, flux may be a single vector, or it may be a vector field / function of position. In the latter case flux can readily be integrated over a surface. By contrast, according to the electromagnetism definition, flux is the integral over a surface; it makes no sense to integrate a second-definition flux for one would be integrating over a surface twice. Thus, Maxwell's quote only makes sense if "flux" is being used according to the transport definition (and furthermore is a vector field rather than single vector). This is ironic because Maxwell was one of the major developers of what we now call "electric flux" and "magnetic flux" according to the electromagnetism definition. Their names in accordance with the quote (and transport definition) would be "surface integral of electric flux" and "surface integral of magnetic flux", in which case "electric flux" would instead be defined as "electric field" and "magnetic flux" defined as "magnetic field". This implies that Maxwell conceived of these fields as flows/fluxes of some sort.

Given a flux according to the electromagnetism definition, the corresponding flux density, if that term is used, refers to its derivative along the surface that was integrated. By the Fundamental theorem of calculus, the corresponding flux density is a flux according to the transport definition. Given a current such as electric current—charge per time, current density would also be a flux according to the transport definition—charge per time per area. Due to the conflicting definitions of flux, and the interchangeability of flux, flow, and current in nontechnical English, all of the terms used in this paragraph are sometimes used interchangeably and ambiguously. Concrete fluxes in the rest of this article will be used in accordance to their broad acceptance in the literature, regardless of which definition of flux the term corresponds to.

Flux as flow rate per unit area
In transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as the rate of flow of a property per unit area, which has the dimensions [quantity]·[time]−1·[area]−1. The area is of the surface the property is flowing "through" or "across". For example, the amount of water that flows through a cross section of a river each second divided by the area of that cross section, or the amount of sunlight energy that lands on a patch of ground each second divided by the area of the patch, are kinds of flux.

General mathematical definition (transport)
Here are 3 definitions in increasing order of complexity. Each is a special case of the following. In all cases the frequent symbol j, (or J) is used for flux, q for the physical quantity that flows, t for time, and A for area. These identifiers will be written in bold when and only when they are vectors.

First, flux as a (single) scalar: $$j = \frac{I}{A},$$ where $$I = \lim_{\Delta t \to 0}\frac{\Delta q}{\Delta t} = \frac{\mathrm{d}q}{\mathrm{d}t}.$$ In this case the surface in which flux is being measured is fixed and has area A. The surface is assumed to be flat, and the flow is assumed to be everywhere constant with respect to position and perpendicular to the surface.

Second, flux as a scalar field defined along a surface, i.e. a function of points on the surface: $$j(\mathbf{p}) = \frac{\partial I}{\partial A}(\mathbf{p}),$$ $$I(A,\mathbf{p}) = \frac{\mathrm{d}q}{\mathrm{d}t}(A, \mathbf{p}).$$ As before, the surface is assumed to be flat, and the flow is assumed to be everywhere perpendicular to it. However the flow need not be constant. q is now a function of p, a point on the surface, and A, an area. Rather than measure the total flow through the surface, q measures the flow through the disk with area A centered at p along the surface.

Finally, flux as a vector field: $$\mathbf{j}(\mathbf{p}) = \frac{\partial \mathbf{I}}{\partial A}(\mathbf{p}),$$ $$\mathbf{I}(A,\mathbf{p}) = \underset{\mathbf{\hat{n}}}{\operatorname{arg\,max}} \mathbf{\hat{n}}_{\mathbf p} \frac{\mathrm{d}q}{\mathrm{d}t}(A,\mathbf{p}, \mathbf{\hat{n}}).$$ In this case, there is no fixed surface we are measuring over. q is a function of a point, an area, and a direction (given by a unit vector $$\mathbf{\hat{n}}$$), and measures the flow through the disk of area A perpendicular to that unit vector. I is defined picking the unit vector that maximizes the flow around the point, because the true flow is maximized across the disk that is perpendicular to it. The unit vector thus uniquely maximizes the function when it points in the "true direction" of the flow. (Strictly speaking, this is an abuse of notation because the "argmax" cannot directly compare vectors; we take the vector with the biggest norm instead.)

Properties
These direct definitions, especially the last, are rather unwieldy. For example, the argmax construction is artificial from the perspective of empirical measurements, when with a weathervane or similar one can easily deduce the direction of flux at a point. Rather than defining the vector flux directly, it is often more intuitive to state some properties about it. Furthermore, from these properties the flux can uniquely be determined anyway.

If the flux j passes through the area at an angle θ to the area normal $$\mathbf{\hat{n}}$$, then the dot product $$\mathbf{j} \cdot \mathbf{\hat{n}} = j\cos\theta.$$ That is, the component of flux passing through the surface (i.e. normal to it) is jcosθ, while the component of flux passing tangential to the area is jsinθ, but there is no flux actually passing through the area in the tangential direction. The only component of flux passing normal to the area is the cosine component.

For vector flux, the surface integral of j over a surface S, gives the proper flowing per unit of time through the surface: $$\frac{\mathrm{d}q}{\mathrm{d}t} = \iint_S \mathbf{j} \cdot \mathbf{\hat{n}}\, dA = \iint_S \mathbf{j} \cdot d\mathbf{A},$$ where A (and its infinitesimal) is the vector area – combination $$\mathbf{A} = A \mathbf{\hat{n}}$$ of the magnitude of the area A through which the property passes and a unit vector $$\mathbf{\hat{n}}$$ normal to the area. Unlike in the second set of equations, the surface here need not be flat.

Finally, we can integrate again over the time duration t1 to t2, getting the total amount of the property flowing through the surface in that time (t2 − t1): $$q = \int_{t_1}^{t_2}\iint_S \mathbf{j}\cdot d\mathbf A\, dt.$$

Transport fluxes
Eight of the most common forms of flux from the transport phenomena literature are defined as follows:

 * 1) Momentum flux, the rate of transfer of momentum across a unit area (N·s·m−2·s−1). (Newton's law of viscosity)
 * 2) Heat flux, the rate of heat flow across a unit area (J·m−2·s−1). (Fourier's law of conduction) (This definition of heat flux fits Maxwell's original definition.)
 * 3) Diffusion flux, the rate of movement of molecules across a unit area (mol·m−2·s−1). (Fick's law of diffusion)
 * 4) Volumetric flux, the rate of volume flow across a unit area (m3·m−2·s−1). (Darcy's law of groundwater flow)
 * 5) Mass flux, the rate of mass flow across a unit area (kg·m−2·s−1). (Either an alternate form of Fick's law that includes the molecular mass, or an alternate form of Darcy's law that includes the density.)
 * 6) Radiative flux, the amount of energy transferred in the form of photons at a certain distance from the source per unit area per second (J·m−2·s−1). Used in astronomy to determine the magnitude and spectral class of a star.  Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the electromagnetic spectrum.
 * 7) Energy flux, the rate of transfer of energy through a unit area (J·m−2·s−1). The radiative flux and heat flux are specific cases of energy flux.
 * 8) Particle flux, the rate of transfer of particles through a unit area ([number of particles] m−2·s−1)

These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the divergence of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For incompressible flow, the divergence of the volume flux is zero.

Chemical diffusion
As mentioned above, chemical molar flux of a component A in an isothermal, isobaric system is defined in Fick's law of diffusion as: $$\mathbf{J}_A = -D_{AB} \nabla c_A$$ where the nabla symbol ∇ denotes the gradient operator, DAB is the diffusion coefficient (m2·s−1) of component A diffusing through component B, cA is the concentration (mol/m3) of component A.

This flux has units of mol·m−2·s−1, and fits Maxwell's original definition of flux.

For dilute gases, kinetic molecular theory relates the diffusion coefficient D to the particle density n = N/V, the molecular mass m, the collision cross section $$\sigma$$, and the absolute temperature T by $$D = \frac{2}{3 n\sigma}\sqrt{\frac{kT}{\pi m}}$$ where the second factor is the mean free path and the square root (with the Boltzmann constant k) is the mean velocity of the particles.

In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient.

Quantum mechanics
In quantum mechanics, particles of mass m in the quantum state ψ(r, t) have a probability density defined as $$\rho = \psi^* \psi = |\psi|^2. $$ So the probability of finding a particle in a differential volume element d3r is $$ dP = |\psi|^2 \, d^3\mathbf{r}. $$ Then the number of particles passing perpendicularly through unit area of a cross-section per unit time is the probability flux; $$\mathbf{J} = \frac{i \hbar}{2m} \left(\psi \nabla \psi^* - \psi^* \nabla \psi \right). $$ This is sometimes referred to as the probability current or current density, or probability flux density.

General mathematical definition (surface integral)
As a mathematical concept, flux is represented by the surface integral of a vector field,

 * $$\Phi_F=\iint_A\mathbf{F}\cdot\mathrm{d}\mathbf{A}$$
 * $$\Phi_F=\iint_A\mathbf{F}\cdot\mathbf{n}\,\mathrm{d}A$$

where F is a vector field, and dA is the vector area of the surface A, directed as the surface normal. For the second, n is the outward pointed unit normal vector to the surface.

The surface has to be orientable, i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative.

The surface normal is usually directed by the right-hand rule.

Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density.

Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive divergence (sources) and end at areas of negative divergence (sinks).

See also the image at right: the number of red arrows passing through a unit area is the flux density, the curve encircling the red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the inner product of the vector field with the surface normals.

If the surface encloses a 3D region, usually the surface is oriented such that the influx is counted positive; the opposite is the outflux.

The divergence theorem states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence).

If the surface is not closed, it has an oriented curve as boundary. Stokes' theorem states that the flux of the curl of a vector field is the line integral of the vector field over this boundary. This path integral is also called circulation, especially in fluid dynamics. Thus the curl is the circulation density.

We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.

Electric flux
An electric "charge," such as a single proton in space, has a magnitude defined in coulombs. Such a charge has an electric field surrounding it. In pictorial form, the electric field from a positive point charge can be visualized as a dot radiating electric field lines (sometimes also called "lines of force"). Conceptually, electric flux can be thought of as "the number of field lines" passing through a given area. Mathematically, electric flux is the integral of the normal component of the electric field over a given area. Hence, units of electric flux are, in the MKS system, newtons per coulomb times meters squared, or N m2/C. (Electric flux density is the electric flux per unit area, and is a measure of strength of the normal component of the electric field averaged over the area of integration. Its units are N/C, the same as the electric field in MKS units.)

Two forms of electric flux are used, one for the E-field:

and one for the D-field (called the electric displacement):

This quantity arises in Gauss's law – which states that the flux of the electric field E out of a closed surface is proportional to the electric charge QA enclosed in the surface (independent of how that charge is distributed), the integral form is:

where ε0 is the permittivity of free space.

If one considers the flux of the electric field vector, E, for a tube near a point charge in the field of the charge but not containing it with sides formed by lines tangent to the field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any cross-sectional surface of the tube will be the same. The total flux for any surface surrounding a charge q is q/ε0.

In free space the electric displacement is given by the constitutive relation D = ε0 E, so for any bounding surface the D-field flux equals the charge QA within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow", since nothing actually flows along electric field lines.

Magnetic flux
The magnetic flux density (magnetic field) having the unit Wb/m2 (Tesla) is denoted by B, and magnetic flux is defined analogously:
 * $$\Phi_B=\iint_A\mathbf{B}\cdot\mathrm{d}\mathbf{A}$$

with the same notation above. The quantity arises in Faraday's law of induction, where the magnetic flux is time-dependent either because the boundary is time-dependent or magnetic field is time-dependent. In integral form:

 * $$- \frac{{\rm d} \Phi_B}{ {\rm d} t} =

\oint_{\partial A} \mathbf{E} \cdot d \boldsymbol{\ell}$$

where d$\ell$ is an infinitesimal vector line element of the closed curve $$\partial A$$, with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve $$\partial A$$, with the sign determined by the integration direction.

The time-rate of change of the magnetic flux through a loop of wire is minus the electromotive force created in that wire. The direction is such that if current is allowed to pass through the wire, the electromotive force will cause a current which "opposes" the change in magnetic field by itself producing a magnetic field opposite to the change. This is the basis for inductors and many electric generators.

Poynting flux
Using this definition, the flux of the Poynting vector S over a specified surface is the rate at which electromagnetic energy flows through that surface, defined like before:

The flux of the Poynting vector through a surface is the electromagnetic power, or energy per unit time, passing through that surface. This is commonly used in analysis of electromagnetic radiation, but has application to other electromagnetic systems as well.

Confusingly, the Poynting vector is sometimes called the power flux, which is an example of the first usage of flux, above. It has units of watts per square metre (W/m2).

title: Flux


text:
Édouard Roche

Édouard Albert Roche (17 October 1820 – 27 April 1883) was a French astronomer and mathematician, who is best known for his work in the field of celestial mechanics. His name was given to the concepts of the Roche sphere, Roche limit, and Roche lobe. He also was the author of works in meteorology.

Biography
He was born in Montpellier, and studied at the University of Montpellier, receiving his D.Sc. in 1844 and later becoming a professor at the same institution, where he served in the Faculté des Sciences starting in 1849. Roche made a mathematical study of Laplace's nebular hypothesis and presented his results in a series of papers to the Academy of Montpellier from his appointment until 1877. The most important were on comets (1860) and the nebular hypothesis itself (1873). Roche's studies examined the effects of strong gravitational fields upon swarms of tiny particles.

He is perhaps most famous for his theory that the planetary rings of Saturn were formed when a large icy moon came too close to Saturn and was pulled apart by gravitational forces. He described a method of calculating the distance at which an object held together only by gravity would break up due to tidal forces; this distance became known as the Roche limit.

His other best known works also involved orbital mechanics. The Roche sphere describes the limits at which an object which is in orbit around two other objects will be captured by one or the other, and the Roche lobe approximates the gravitational sphere of influence of one astronomical body in the face of perturbations from another heavier body around which it orbits.

Works
Roche's works are in French, his vernacular language.

Lists of works

 * List of works, on the site of the Académie des sciences (31 items) (Includes—unnumbered—works commenting that of Roche. Also includes works in meteorology)
 * , in Mémoires de la Société des sciences, de l'agriculture et des arts de Lille, 1885 (34 items)

title: Édouard Roche