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His Dark Materials

His Dark Materials is a trilogy of fantasy novels by Philip Pullman consisting of Northern Lights (1995; published as The Golden Compass in North America), The Subtle Knife (1997), and The Amber Spyglass (2000). It follows the coming of age of two children, Lyra Belacqua and Will Parry, as they wander through a series of parallel universes. The novels have won a number of awards, including the Carnegie Medal in 1995 for Northern Lights and the 2001 Whitbread Book of the Year for The Amber Spyglass. In 2003, the trilogy was ranked third on the BBC's The Big Read poll.

Although His Dark Materials has been marketed as young adult fiction, and the central characters are children, Pullman wrote with no target audience in mind. The fantasy elements include witches and armoured polar bears; the trilogy also alludes to concepts from physics, philosophy, and theology. It functions in part as a retelling and inversion of John Milton's epic Paradise Lost, with Pullman commending humanity for what Milton saw as its most tragic failing, original sin. The trilogy has attracted controversy for its criticism of religion.

The London Royal National Theatre staged a two-part adaptation of the trilogy in 2003–2004. New Line Cinema released a film adaptation of Northern Lights, The Golden Compass, in 2007. A HBO/BBC television series based on the novels was broadcast between November 2019 and February 2023.

Pullman followed the trilogy with three novellas set in the Northern Lights universe: Lyra's Oxford (2003), Once Upon a Time in the North (2008), and Serpentine (2020). La Belle Sauvage, the first book in a new trilogy titled The Book of Dust, was published on 19 October 2017; the second book of the new trilogy, The Secret Commonwealth, was published in October 2019. Both are set in the same universe as Northern Lights.

Setting
The trilogy takes place across a multiverse, moving between many parallel worlds. In Northern Lights, the story takes place in a world with some similarities to our own: dress-style resembles that of the UK's Edwardian era; the technology does not include cars or fixed-wing aircraft, but zeppelins feature as a mode of transport.

The dominant religion has parallels with Christianity. The Church (governed by the "Magisterium", the same name as the authority of the Catholic Church) exerts a strong control over society and has some of the appearance and organisation of the Catholic Church, but one in which the centre of power had been moved from Rome to Geneva, moved there by Pullman's fictional "Pope John Calvin" (Geneva was the home of the historical John Calvin).

In The Subtle Knife, the story moves between our own world, the world of the first novel, and a third world containing the city of Cittàgazze. In The Amber Spyglass, several other worlds appear alongside those three.

Titles
The title of the series comes from 17th-century poet John Milton's Paradise Lost: Into this wilde Abyss, The Womb of nature and perhaps her Grave, Of neither Sea, nor Shore, nor Air, nor Fire, But all these in their pregnant causes mixt Confus'dly, and which thus must ever fight, Unless th' Almighty Maker them ordain His dark materials to create more Worlds, Into this wilde Abyss the warie fiend Stood on the brink of Hell and look'd a while, Pondering his Voyage; for no narrow frith He had to cross. — Paradise Lost, Book 2, lines 910–920 Pullman chose this particular phrase from Milton because it echoed the dark matter of astrophysics.

Pullman earlier proposed to name the series The Golden Compasses, also a reference to Paradise Lost, where they denote God's circle-drawing instrument used to establish and set the bounds of all creation:

Then staid the fervid wheels, and in his hand He took the golden compasses, prepared In God's eternal store, to circumscribe This universe, and all created things: One foot he centred, and the other turned Round through the vast profundity obscure... — Paradise Lost, Book 7, lines 224–229

Despite the confusion with the other common meaning of compass (the navigational instrument), The Golden Compass became the title of the American edition of Northern Lights (the book features an "alethiometer", a rare truth-telling device that one might describe as a "golden compass").

Northern Lights (or The Golden Compass)
In Jordan College, Oxford, 11-year-old Lyra Belacqua and her dæmon Pantalaimon witness the Master attempt to poison Lord Asriel, Lyra's rebellious and adventuring uncle. She warns Asriel, then spies on his lecture about Dust, mysterious elementary particles. Lyra's friend Roger is kidnapped by child abductors known as Gobblers. Lyra is adopted by a charming socialite, Mrs Coulter. The Master secretly entrusts Lyra with an alethiometer, a truth-telling device. Lyra discovers that Mrs Coulter is the leader of the Gobblers, and that it is a project secretly funded by the Church. Lyra flees to the Gyptians, canal-faring nomads, whose children have also been abducted. They reveal to Lyra that Asriel and Mrs Coulter are actually her parents.

The Gyptians form an expedition to the Arctic with Lyra to rescue the children. Lyra recruits Iorek Byrnison, an armoured bear, and his human aeronaut friend, Lee Scoresby. She also learns that Lord Asriel has been exiled, guarded by the bears on Svalbard.

Near Bolvangar, the Gobbler research station, Lyra finds an abandoned child who has been cut from his dæmon; the Gobblers are experimenting on children by severing the bond between human and dæmon, a procedure called intercision.

Lyra is captured and taken to Bolvangar, where she is reunited with Roger. Mrs Coulter tells Lyra that the intercision prevents the onset of troubling adult emotions. Lyra and the children are rescued by Scoresby, Iorek, the Gyptians, and Serafina Pekkala's flying witch clan. Lyra falls out of Scoresby's balloon and is taken by the panserbjørne to the castle of their usurping king, Iofur Raknison. She tricks Iofur into fighting Iorek, who arrives with the others to rescue Lyra. Iorek kills Iofur and takes his place as the rightful king.

Lyra, Iorek, and Roger travel to Svalbard, where Asriel has continued his Dust research in exile. He tells Lyra that the Church believes Dust is the basis of sin, and plans to visit the other universes and destroy its source. He severs Roger from his dæmon, killing him and releasing enough energy to create an opening to a parallel universe. Lyra resolves to stop Asriel and discover the source of Dust for herself.

The Subtle Knife
Lyra journeys through Asriel's opening between worlds to Cittàgazze, a city whose denizens discovered a way to travel between worlds. Cittàgazze's reckless use of the technology has released Spectres which destroy adult souls but to which children are immune, rendering the world empty of adults. Here Lyra meets and befriends Will Parry, a twelve-year-old boy from our world's Oxford. Will, who recently killed a man to protect his ailing mother, has stumbled into Cittàgazze in an effort to locate his long-lost father. Venturing into Will's (our) world, Lyra meets Dr. Mary Malone, a physicist who is researching dark matter, which is analogous to Dust in Lyra's world. Lyra encourages Dr. Malone to attempt to communicate with the particles, and when she does they tell her to travel into the Cittàgazze world. Lyra's alethiometer is stolen by Lord Boreal alias Sir Charles Latrom, an ally of Mrs Coulter who has found a way to Will's Oxford and established a home there.

Will becomes the bearer of the Subtle Knife, a tool forged three hundred years before by Cittàgazze's scientists from the same alloy used to make the guillotine in Bolvangar. One edge of the knife can divide subatomic particles and form subtle divisions in space, creating portals between worlds; the other edge easily cuts through any form of matter. Using the knife's portal-creating powers, Will and Lyra are able to retrieve her alethiometer from Latrom's mansion in Will's world.

Meanwhile, in Lyra's world, Lee Scoresby seeks out the Arctic explorer Stanislaus Grumman, who years before entered Lyra's world through a portal in Alaska. Scoresby finds him living as a shaman under the name Jopari and he turns out to be Will's father, John Parry. Parry insists on being taken through the opening into the Cittàgazze world in Scoresby's balloon, since he has foreseen that he should meet the wielder of the Subtle Knife there. In that world, Scoresby dies defending Parry from the forces of the Church, while Parry succeeds in reuniting with his son moments before being murdered by Juta Kamainen, a witch whose love John had once rejected. After his father's death, Will discovers that Lyra has been kidnapped by Mrs Coulter, and he is approached by two angels requesting his aid.

The Amber Spyglass
At the beginning of The Amber Spyglass, Lyra has been kidnapped by her mother, Mrs Coulter, an agent of the Magisterium who has learned of the prophecy identifying Lyra as the next Eve. A pair of angels, Balthamos and Baruch, tell Will that he must travel with them to give the Subtle Knife to Lyra's father, Lord Asriel, as a weapon against The Authority. Will ignores the angels; with the help of a local girl named Ama, the Bear King Iorek Byrnison, and Lord Asriel's Gallivespian spies, the Chevalier Tialys and the Lady Salmakia, he rescues Lyra from the cave where her mother has hidden her from the Magisterium, which has become determined to kill her before she yields to temptation and sin like the original Eve.

Will, Lyra, Tialys and Salmakia journey to the Land of the Dead, temporarily parting with their dæmons to release the ghosts from their captivity. Mary Malone, a scientist from Will's world interested in "shadows" (or Dust in Lyra's world), travels to a land populated by strange sentient creatures called Mulefa. There, she comes to understand the true nature of Dust, which is both created by and nourishes life that has become self-aware. Lord Asriel and the reformed Mrs Coulter work to destroy the Authority's Regent Metatron. They succeed, but themselves suffer annihilation in the process by pulling Metatron into the abyss.

The Authority himself dies of his own frailty when Will and Lyra free him from the crystal prison wherein Metatron had trapped him, able to do so because an attack by cliff-ghasts kills or drives away the prison's protectors. When Will and Lyra emerge from the land of the dead, they find their dæmons.

The book ends with Will and Lyra falling in love but realising they cannot live together in the same world, because all windows – except one from the underworld to the world of the Mulefa – must be closed to prevent the loss of Dust, because with every window opening, a Spectre would be created and that means Will must never use the knife again. They must also be apart because both of them can only live full lives in their native worlds. During the return, Mary Malone learns how to see her own dæmon, who takes the form of a black Alpine chough. Lyra loses her ability to intuitively read the alethiometer and determines to learn how to use her conscious mind to achieve the same effect.

Characters
All humans in Lyra's world, including witches, have a dæmon. It is the physical manifestation of a person's 'inner being', soul or spirit. It takes the form of a creature (moth, bird, dog, monkey, snake, etc.) and is usually the opposite sex to its human counterpart. The dæmons of children have the ability to change form - from one creature to another - but during a child's puberty, their dæmon "settles" into a permanent form, which reflects the person's personality. When a person dies, the dæmon dies too. Armoured bears, cliff ghasts, and other creatures do not have dæmons. An armoured bear's armour is his soul.

 * Lyra Belacqua, a wild 12-year-old girl, has grown up in the fictional Jordan College, Oxford. She prides herself on her capacity for mischief, especially her ability to lie, earning her the epithet "Silvertongue" from Iorek Byrnison. Lyra has a natural ability to use the alethiometer, which is capable of answering any question when properly manipulated and read.
 * Pantalaimon is Lyra's dæmon. Like all children's dæmons, he changes form from one creature to another frequently. When Lyra reaches puberty, he assumes the permanent form of a pine marten.
 * Will Parry, a sensible, morally conscious, assertive 12-year-old boy from our world. He becomes the bearer of the subtle knife. Will is independent and responsible for his age, having looked after his mentally ill mother for several years. Will's dæmon is Kirjava.
 * The Authority is the first angel to have emerged from Dust. He told the later-arriving angels that he created them and the universe, but this is a lie. Although he is the overarching antagonist of the series, the Authority remains in the background; he makes his only appearance late in The Amber Spyglass. The Authority has grown weak and transferred most of his powers to his regent, Metatron.
 * Lord Asriel, ostensibly Lyra's uncle, is actually her father. He opens a rift between the worlds in his pursuit of Dust. His dream of establishing a Republic of Heaven leads him to use his power to raise a grand army from across the multiverse to rise up in rebellion against the forces of the Church.
 * Marisa Coulter is the beautiful and manipulative mother of Lyra, and the former lover of Lord Asriel. She serves the Church by kidnapping children for research into the nature of Dust, in the course of which she separates them from their dæmons - a procedure known as intercision. Initially hostile to Lyra, she realises that she loves her daughter and seeks to protect her from agents of the Church who want to kill Lyra. Her dæmon is a golden monkey with a cruel streak.
 * Metatron, Asriel's principal adversary, was a human, Enoch, in biblical times, but was later transfigured into an angel. The Authority, his health declining, appointed Metatron his Regent. As Regent, Metatron has implanted the monotheistic religions across the universes.
 * Lord Carlo Boreal - or Sir Charles Latrom, CBE, as he is known as in Will Parry's world-is a minor character in Northern Lights, but the main antagonist in The Subtle Knife. He is an old Englishman, appearing to be in his sixties.
 * Mary Malone, is a physicist and former nun from Will's world. She meets Lyra during Lyra's first visit to Will's world. Lyra provides Mary with insight into the nature of Dust. Agents of the Church force Mary to flee to the world of the Mulefa. There she constructs the amber spyglass, which enables her to see the otherwise invisible Dust. Her purpose is to learn why Dust, which Mulefa civilisation depends on, is flowing out of the universe.
 * Iorek Byrnison is a massive armoured bear. An armoured bear's armour is his soul. Iorek's armour is stolen, so he becomes despondent. With Lyra's help he regains his armour, his dignity, and his kingship over the armoured bears. In gratitude, and impressed by her cunning, he dubs her "Lyra Silvertongue". A powerful warrior and blacksmith, Iorek repairs the Subtle Knife when it shatters. He later goes to war against The Authority and Metatron.
 * Lee Scoresby, a rangy Texan, is a balloonist. He helps Lyra in an early quest to reach Asriel's residence in the North, and he later helps John Parry reunite with his son Will.
 * Serafina Pekkala is the beautiful queen of a clan of Northern witches. Her snow-goose dæmon Kaisa, like all witches' dæmons, can travel much farther apart from her than the dæmons of humans, without feeling the pain of separation.
 * The Master of Jordan heads Jordan College, part of Oxford University in Lyra's world. Helped by other Jordan College employees, he is raising the supposedly orphaned Lyra. Faced with difficult choices that only later become apparent, he tries unsuccessfully to poison Lord Asriel.
 * Roger Parslow is the kitchen boy at Jordan College and Lyra's best friend.
 * John Parry is Will's father. He is an explorer from our world who discovered a portal to Lyra's world and became the shaman known as Stanislaus Grumman or Jopari, a variation of his original name.
 * The Four Gallivespians—Lord Roke, Madame Oxentiel, Chevalier Tialys, and Lady Salmakia—are tiny people (a hand-span tall) with poisonous heel spurs.
 * Ma Costa: A Gyptian woman whose son, Billy Costa, is abducted by the Gobblers. She rescues Lyra from Mrs Coulter and takes her to John Faa. Ma Costa nursed Lyra when she was a baby.
 * John Faa: The King of all the Gyptians. He journeys with Lyra to the North with his companion Farder Coram. Faa and Costa rescue Lyra when she runs away from Mrs Coulter. Then they take her to Iorek Byrnison.
 * Father Gomez is a priest sent by the Church to assassinate Lyra.
 * Fra Pavel Rašek is an alethiometrist of the Consistorial Court of Discipline, but a sluggish reader of the device.
 * Balthamos is a rebel angel who, with his lover Baruch, join in Will's journey to find the captured Lyra. Near the end of the story, he saves their lives by killing Father Gomez.
 * Mulefa are four-legged wheeled animals; they have one leg in front, one in back, and one on each side. Their "wheels" are large, round, hard seed-pods from trees; an axle-like claw at the end of each leg grips a seed-pod.

Dæmons

One distinctive aspect of Pullman's story is the presence of "dæmons" (pronounced "demon"). In the birth-universe of the story's protagonist Lyra Belacqua, a human individual's inner-self manifests itself throughout life as an animal-shaped "dæmon" that almost always stays near its human counterpart. During the childhood of its associated human, a dæmon can change its animal shape at will, but with the onset of adolescence it settles into a fixed, final animal form.

Influences
Pullman has identified three major literary influences on His Dark Materials: the essay On the Marionette Theatre by Heinrich von Kleist, the works of William Blake, and, most important, John Milton's Paradise Lost, from which the trilogy derives its title. In his introduction, he adapts a famous description of Milton by Blake to quip that he (Pullman) "is of the Devil's party and does know it".

Critics have compared the trilogy with C. S. Lewis's The Chronicles of Narnia, which Pullman despises, and also with such fantasy books as Bridge to Terabithia by Katherine Paterson and A Wrinkle in Time by Madeleine L'Engle.

Awards and recognition
The first volume, Northern Lights, won the Carnegie Medal for children's fiction in the UK in 1995. In 2007, the judges of the CILIP Carnegie Medal for children's literature selected it as one of the ten most important children's novels of the previous 70 years. In an online June 2007 poll, it was voted the best Carnegie Medal winner in the 70-year history of the award, the Carnegie of Carnegies. The Amber Spyglass won the 2001 Whitbread Book of the Year award, the first time that such an award has been bestowed on a book from their "children's literature" category.

The trilogy came third in the 2003 BBC's Big Read, a national poll of viewers' favourite books, after The Lord of the Rings and Pride and Prejudice.

On 19 May 2005, Pullman attended the British Library in London to receive formal congratulations for his work from culture secretary Tessa Jowell "on behalf of the government". On 25 May 2005, Pullman received the Swedish government's Astrid Lindgren Memorial Award for children's and youth literature (sharing it with Japanese illustrator Ryōji Arai). Swedes regard this prize as second only to the Nobel Prize in Literature; it has a value of 5 million Swedish Kronor or approximately £385,000. In 2008, The Observer cites Northern Lights as one of the 100 best novels. Time magazine in the US included Northern Lights (The Golden Compass) in its list of the 100 Best Young-Adult Books of All Time. In November 2019, the BBC listed His Dark Materials on its list of the 100 most influential novels.

Christian opposition
His Dark Materials has occasioned controversy, primarily among some Christian groups.

Cynthia Grenier, in the Catholic Culture, said: "In the world of Pullman, God Himself (the Authority) is a merciless tyrant. His Church is an instrument of oppression, and true heroism consists of overthrowing both". William A. Donohue of the Catholic League described Pullman's trilogy as "atheism for kids". Pullman said of Donohue's call for a boycott, "Why don't we trust readers? [...] Oh, it causes me to shake my head with sorrow that such nitwits could be loose in the world".

In a November 2002 interview, Pullman was asked to respond to the Catholic Herald calling his books "the stuff of nightmares" and "worthy of the bonfire". He replied: "My response to that was to ask the publishers to print it in the next book, which they did! I think it's comical, it's just laughable". The original remark in Catholic Herald (which was "there are numerous candidates that seem to me to be far more worthy of the bonfire than Harry Potter") was written in the context of parents in South Carolina pressing their Board of Education to ban the Harry Potter books.

Pullman expressed surprise over what he considered to be a relatively low level of criticism for His Dark Materials on religious grounds, saying "I've been surprised by how little criticism I've got. Harry Potter's been taking all the flak... Meanwhile, I've been flying under the radar, saying things that are far more subversive than anything poor old Harry has said. My books are about killing God". Others support this interpretation, arguing that the series, while clearly anticlerical, is also anti-theological because the death of God is represented as a fundamentally unimportant question.

Pullman found support from some other Christians, most notably from Rowan Williams, the former archbishop of Canterbury (spiritual head of the Anglican Communion), who argued that Pullman's attacks focus on the constraints and dangers of dogmatism and the use of religion to oppress, not on Christianity itself. Williams also recommended the His Dark Materials series of books for inclusion and discussion in Religious Education classes, and stated that "To see large school-parties in the audience of the Pullman plays at the National Theatre is vastly encouraging". Pullman and Williams took part in a National Theatre platform debate a few days later to discuss myth, religious experience, and its representation in the arts.

Lyra's Oxford
The 2003 novella Lyra's Oxford takes place two years after the timeline of The Amber Spyglass. A witch who seeks revenge for her son's death in the war against the Authority draws Lyra, now 15, into a trap. Birds mysteriously rescue her and Pan, and she makes the acquaintance of an alchemist, formerly the witch's lover.

Once Upon a Time in the North
This 2008 novella serves as a prequel to His Dark Materials and focuses on the Texan aeronaut Lee Scoresby as a young man. After winning his hot-air balloon, Scoresby heads to the North, landing on the Arctic island Novy Odense, where he is pulled into a conflict between the oil tycoon Larsen Manganese, the corrupt mayoral candidate Ivan Poliakov, and his longtime enemy from the Dakota Country, Pierre McConville. The story tells of Lee and Iorek's first meeting and of how they overcame these enemies.

The Collectors
A short story originally released exclusively as an audiobook by Audible in December 2014, narrated by actor Bill Nighy. The story refers to the early life of Mrs Coulter and is set in the senior common room of an Oxford college. The story was released by Penguin Books as a physical book in September 2022.

The Book of Dust
The Book of Dust is a second trilogy of novels set before, during and after His Dark Materials. The first book, La Belle Sauvage, was published on 19 October 2017. The second book, The Secret Commonwealth, was published on 3 October 2019.

Serpentine
A novella that was released in October 2020. Set after the events of The Amber Spyglass and before The Secret Commonwealth, Lyra and Pantalaimon journey back to the far North to meet with the Consul of Witches.

The Imagination Chamber
In January 2022, Pullman announced the release of the book The Imagination Chamber: Cosmic Rays from Lyra's Universe, which would include new scenes set during the events of His Dark Materials and The Book of Dust. It was published on 28 April 2022.

Radio
BBC Radio 4 broadcast a radio play adaptation of His Dark Materials in 3 episodes, each lasting 2.5 hours. It was first broadcast in 2003, and re-broadcast in both 2008-9 and in 2017, and was released by the BBC on CD and cassette. Cast included Terence Stamp as Lord Asriel and Lulu Popplewell as Lyra.

Also in 2003 a radio dramatisation of Northern Lights was made by RTÉ (Irish public radio).

Theatre
Nicholas Hytner directed a theatrical version of the books as a two-part, six-hour performance for London's Royal National Theatre in December 2003, running until March 2004. It starred Anna Maxwell-Martin as Lyra, Dominic Cooper as Will, Timothy Dalton as Lord Asriel, Patricia Hodge as Mrs Coulter and Niamh Cusack as Serafina Pekkala, with dæmon puppets designed by Michael Curry. The play was successful and was revived (with a different cast and a revised script) for a second run between November 2004 and April 2005. It has since been staged by several other theatres in the UK and elsewhere.

A new production was staged at Birmingham Repertory Theatre in March and April 2009, directed by Rachel Kavanaugh and Sarah Esdaile and starring Amy McAllister as Lyra. This version toured the UK and included a performance in Pullman's hometown of Oxford; Pullman made a cameo appearance.

Film
New Line Cinema released a film adaptation, titled The Golden Compass, on 7 December 2007. Directed by Chris Weitz, the production had a mixed reception, and though worldwide sales were strong, its U.S. earnings were not as high as the studio had hoped.

The filmmakers obscured the explicitly Biblical character of the Authority to avoid offending viewers. Weitz declared that he would not do the same for the planned sequels. "Whereas The Golden Compass had to be introduced to the public carefully", he said, "the religious themes in the second and third books can't be minimised without destroying the spirit of these books. ...I will not be involved with any 'watering down' of books two and three, since what I have been working towards the whole time in the first film is to be able to deliver on the second and third".

The Golden Compass film stars Dakota Blue Richards as Lyra, Nicole Kidman as Mrs Coulter, and Daniel Craig as Lord Asriel. Eva Green plays Serafina Pekkala, Ian McKellen voices Iorek Byrnison, and Freddie Highmore voices Pantalaimon. While Sam Elliott blamed the Catholic Church's opposition for forcing the cancellation of any adaptations of the rest of the trilogy, The Guardian's film critic Stuart Heritage believed disappointing reviews may have been the real reason.

Television
In November 2015, the BBC commissioned a television adaptation of His Dark Materials. The eight-part adaptation had a planned premiere date in 2017. By July 2018, Dafne Keen had been provisionally cast as Lyra Belacqua, Ruth Wilson as Marisa Coulter, James McAvoy as Lord Asriel, Lin-Manuel Miranda as Lee Scoresby and Clarke Peters as the Master of Jordan College. The series received its premiere in London on 15 October 2019. Broadcast began on BBC One in the United Kingdom and in Ireland on 3 November and on HBO in the United States on 4 November 2019. In 2020 the second series of His Dark Materials began streaming on BBC One in the United Kingdom on 8 November and on HBO Max in the United States on 16 November. The third and final eight-episode series premiered first on HBO on 5 December 2022, and on 18 December 2022 in the UK.

Audiobooks
Random House produced unabridged audiobooks of each His Dark Materials novel, read by Pullman, with parts read by actors including Jo Wyatt, Steven Webb, Peter England, Stephen Thorne and Douglas Blackwell.

title: His Dark Materials


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Talk:Civil and political rights

Civil and political rights organizations
"These organizations serve a variety of causes one being the AFL–CIO," - I don't see how the previously mentioned organisations can be said to serve the cause of the AFL-CIO. Masonmilan (talk) 15:55, 18 July 2023 (UTC)

 * True. Could it be that two sentences got combined by accident? Maybe they were trying to say: "Another one being the AFL–CIO, which is America's union that represents the working-class people nationwide. These organizations serve a variety of causes." 23impartial (talk) 18:22, 5 October 2023 (UTC)

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

title: Talk:Civil and political rights


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La Brabançonne

"La Brabançonne" ( (La Brabançonne); "De Brabançonne"; "Das Lied von Brabant") is the national anthem of Belgium. The originally French title refers to the Duchy of Brabant; the name is usually untranslated in Belgium's other two official languages, Dutch and German.

History

According to legend, the Belgian national anthem was written in September 1830, during the Belgian Revolution, by a young revolutionary called "Jenneval", who read the lyrics during a meeting at the Aigle d'Or café.

Jenneval, a Frenchman whose real name was Alexandre Dechet (sometimes known as Louis-Alexandre Dechet), wrote the Brabançonne. At the time, he was an actor at the theatre where, in August 1830, the revolution started, which led to independence from the Netherlands. Jenneval died in the war of independence. François van Campenhout composed the accompanying score, based on the tune of a French song called "L'Air des lanciers polonais" ("the tune of the Polish Lancers"), written by the French poet Eugène de Pradel, whose tune was itself an adaptation of the tune of a song, "L'Air du magistrat irréprochable", found in a popular collection of drinking songs called La Clé du caveau (The Key to the cellar) and it was first performed in September 1830.

In 1860, Belgium formally adopted the song and music as its national anthem, although the then prime minister, Charles Rogier edited out lyrics attacking the Dutch House of Orange, inspired by the version written by Louis Hymans.

The Brabançonne is also a monument (1930) by the sculptor Charles Samuel on the Surlet de Chokier square in Brussels. The monument contains partial lyrics of both the French and Dutch versions of the anthem. Like many elements in Belgian folklore, this is mainly based on the French "La Marseillaise" which is also both an anthem and the name of a monument – the sculptural group Departure of the Volunteers of 1792, commonly called La Marseillaise, at the base of the Arc de Triomphe in Paris.

Current version
Various committees were charged with reviewing the text and tune of the Brabançonne and establishing an official version. A ministerial circular of the Ministry of the Interior on 8 August 1921 decreed that only the fourth verse of the text by Charles Rogier should be considered official for all three, French, German and in Dutch. Here below:

Modern short trilingual version
In recent years, an unofficial short version of the anthem is sung during Belgian National Day on 21 July each year, combining the words of the anthem in all three of Belgium's official languages, similar to the bilingual version of "O Canada". The lyrics are from the 4th verse of the anthem.

2007 Yves Leterme incident
On the 2007 Belgian national day (21 July), Flemish politician Yves Leterme, who would become Prime Minister two years later, was asked by a French-speaking reporter if he also knew the French lyrics of the Belgian national anthem, whereupon he began to sing La Marseillaise, the French national anthem, instead of La Brabançonne. The lyrics are not taught in Belgian schools and many people do not know them. In 2018, the Minister of Education of Wallonia and Brussels (the French speaking community) proposed to make it mandatory for students to be taught the lyrics at school.

title: La Brabançonne


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Jan Kjærstad

Jan Kjærstad (born 6 March 1953 in Oslo) is a Norwegian author. Kjærstad is a theology graduate from MF Norwegian School of Theology and the University of Oslo (cand. theol.). He has written a string of novels, short stories and essays and was editor of the literary magazine Vinduet ("The Window"). He has received a number of prizes, the most important being the Nordic Council Literature Prize, which he received for the perspectivist trilogy about the TV personality Jonas Wergeland (The Seducer, The Conqueror and The Discoverer).

With a string of ingenious novels Kjærstad is established as one of the leading and most original writers in contemporary Norwegian literature. His writing in the 1980s was postmodern works that put special emphasis on the importance of the information society. Kjærstad has often been called an encyclopedic writer. His main work, the trilogy about Jonas Wergeland, a fictive biography that tells three different versions about the protagonists life, combines several different genres.

His books have been translated to English, French, German, Danish, Swedish, and Hungarian, among others.

Prizes and recognition

 * 1984 – Mads Wiel Nygaards Endowment (a prize awarded to promising authors in memory of Mads Wiel Nygaard by the publisher Aschehoug)
 * 1984 – Norwegian Critics Prize for Literature for Homo Falsus
 * 1993 – Aschehougprisen (the primary literature prize awarded annually by the publisher Aschehoug)
 * 1998 – Henrik Steffens-prisen (the Henrik Steffens literature prize is awarded by the university system in Hamburg)
 * 2000 – Doblougprisen (Awarded by the Swedish Academy)
 * 2001 – Nordic Council's Literature Prize for Oppdageren

title: Jan Kjærstad


text:
Talk:Lithium

Semi-protected edit request on 6 July 2023
223.227.20.35 (talk) 18:50, 6 July 2023 (UTC) Mania
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate. Troutfarm27  (Talk) 18:51, 6 July 2023 (UTC)

 * No change, per se. I stated that "Under "pricing' this should be added."  This is additional, timely information.
 * Reuters is listed in Wikipedia's reliable sources (https://en.wikipedia.org/wiki/Wikipedia:Reliable_sources): "Most newspapers also reprint items from news agencies such as Reuters, Interfax, Agence France-Presse, United Press International or the Associated Press, which are responsible for accuracy." MkenneyMF (talk) 18:50, 10 August 2023 (UTC)

Should this topic reference use of lithium citrate in 7Up soda?
https://en.wikipedia.org/wiki/7_Up Liturgist (talk) 04:10, 25 September 2023 (UTC)

Possibly slightly incorrect petalite discovery history
The article states that the mineral petalite was discovered in 1800 on the island of Utö. It seems however, that it could be found in various places, and that we do not know where it was first found. We would probably need a travel diary from the discoverer to find out where he went first.

I think that it would be more correct to say something like that the first recorded mention of petalite was in 1800, stating that it had could found in three places: Utö, Sala and Finngruvan, in Sweden. (It seems correct that the actual piece of petalite that was analyzed 1817 with the discovery of lithium came from Utö and not from Sala or Finngruvan.)

I've looked at the references for the claim that petalite was discovered 1800 on the island of Utö, references 68-71:

Ref. 68, which is the original reference, from 1800 mentions three different places in Sweden where petalite can be found (not where it was first found): "On le trouve à Utoen (= Utö), Sala et Fingrufan (= Finngruvan), près de Niakoparberg (Nya Kopparberg) en Suède." Ref. 69 refers to 3 sources: -1.Refers to ref. 68. -2.Refers to "Tschermaks MPM (1983), 31, 81-96": which mentions 5 different localities of pegmatite, including Utö, but nothing regarding where it was first found. -3.Langhof, Jörgen, Jonsson, Erik, Gustafsson, Lars, Otter, Bertil (1998) Utö- en klassisk svensk mineralfyndort [Utö - a classical Swedish mineral locality]. Norsk Bergverksmuseum Skrift, 14. 29-31: States that Utö is the locality - but no specific source to back this up (there are 14 references to the chapter). [So this is the only real source of the claim - but it is from 1998 - and it does not mention the 2 other places mentioned in the 1800 paper, it does not mention the discoverer - so I do not know why this source should take precedence. If it was known that d'Andrade mentions 3 places, but that one actually was first, it would be probably have been mentioned.] Ref. 70 does not say that petalite was discovered on Utö. It does say that lithium was discovered in petalite from Utö (which refers to 17 years after the mentioned discovery of petalite.) Ref. 71 mentions the same 3 localities from ref 68, and also refers to ref 68. (The ref. also seems wrong: “p. 124”, the word petalite first appears on page 484. The Swedish version of the Wikipedia lithium page has another reference for the same claim that petalite was discovered 1800 on Utö: Per Enghag (2000), "Jordens grundämnen och deras upptäckt. Byggstenar för marken och vattnet - luften och livet", Industrilitteratur, sidan 216, ISBN 91-7548-590-7.

However this book states that petalite had been found on Utö among other places, and also citing a Swedish text from 1914 describing Petalite having been found in all of the three places (which must come from the 1800 paper).

It might of course be true that petalite was first found in Utö, but the current references do not support the claim. Wikfredd (talk) 16:26, 23 December 2023 (UTC)

Semi-protected edit request on 18 January 2024
Arfwedson is misspelled as Arwedson several times Dzimmer15 (talk) 09:15, 18 January 2024 (UTC)
 * ✅ Tollens (talk) 10:33, 18 January 2024 (UTC)

Vague article that might report a fact
What are we to make of this? It seems that there is an extraction process for lithium using the common mineral "pyrite" (FeS2), presumably (I guess) by having the Lithium bind to the Sulfur. The article presents it as the Next Big Thing since we need more Lithium. But it also sets off some alarm bells because it lacks detail. I dislike fake science and if anyone can shed some light on this and determine if this should be included here, I'd appreciate it. Wastrel Way (talk) 19:49, 19 April 2024 (UTC)Eric

Semi-protected edit request on 6 May 2024
i like fortnite

quantum degenerate Bose–Fermi mixture"
"Bose-Fermi" is linked to 2 separate wiki articles and the phrase 'Bose-Fermi mixture' is no where to be found. 142.163.195.114 (talk) 03:16, 18 May 2024 (UTC)

"places such as Australia and North Carolina"
from Production/extraction. what do these places have in common? The grammatical structure implies I should be able to figure out other places like them. 142.163.195.114 (talk) 03:28, 18 May 2024 (UTC)

title: Talk:Lithium


text:
13 (number)

13 (thirteen) is the natural number following 12 and preceding 14.

Strikingly folkloric aspects of the number 13 have been noted in various cultures around the world: one theory is that this is due to the cultures employing lunar-solar calendars (there are approximately 12.41 lunations per solar year, and hence 12 "true months" plus a smaller, and often portentous, thirteenth month). This can be witnessed, for example, in the "Twelve Days of Christmas" of Western European tradition.

In mathematics
The number 13 is the sixth prime number. It is a twin prime with 11, as well as a cousin prime with 17. It is the second Wilson prime, of three known (the others being 5 and 563), and the smallest emirp in decimal.

13 is:

 * The second star number:
 * The third centered square number:
 * A happy number and a lucky number.
 * A Fibonacci number, preceded by 5 and 8.
 * The smallest number whose fourth power can be written as a sum of two consecutive square numbers (1192 + 1202).
 * The sum and the difference of 2 consecutive squares: 13 = (22 + 32) = (72 - 62).

Also,

 * There are 13 ways for the three fastest horses in a horse race to finish, allowing for ties, a fact that can be expressed mathematically by 13 being the third ordered Bell number.
 * Since 52 + 122 = 132, (5, 12, 13) forms a Pythagorean triple, and as such represent the sides of a right triangle.
 * A 13-sided regular polygon is called a tridecagon.
 * There are 13 hexagonal isohedral tilings.
 * There are 13 distinct cubic distance-regular graphs, of which the Platonic Tetrahedral graph and dodecahedral graph are part.
 * There are 13 Archimedean solids without counting enantiomorphic forms. Some also include the elongated square gyrobicupola as another Archimedean solid.
 * There are 13 architectonic honeycombs: 8 originate from the $${\tilde{C}}_3$$ cubic group and 5 originate from the $${\tilde{B}}_3$$ alternated cubic group and $${\tilde{A}}_3$$ cyclic group. The 13 dual honeycombs to these are the 13 catoptric honeycombs, which are made of 13 types of stereohedra.
 * A standard torus can be sliced into 13 pieces with just 3 plane cuts.
 * There are 13 particular Coxeter groups that generate uniform prismatic tessellations in Euclidean 4-space.
 * There are 13 collections of subsets of {1, 2} that are closed under union and intersection.
 * It is the smallest non-trivial hyperfactorion i.e., it is equal to sum of hyperfactorials of its digits.

H(1) + H(3) = (1!) + (1! * 2! * 3!) = 1 + (1 * 2 * 6) = 1 + 12 = 13

In science

 * The atomic number of aluminium is 13.

Grammar

 * In all Germanic languages, 13 is the first compound number; the numbers 11 and 12 have their own names.
 * The Romance languages use different systems: In Italian, 11 is the first compound number (undici), as in Romanian (unsprezece), while in Spanish and Portuguese, the numbers up to and including 15 (Spanish quince, Portuguese quinze), and in French and Italian up to and including 16 (French seize, Italian sedici) have their own names. This is also the case in most Slavic languages, Hindi-Urdu and other South Asian languages.

Folklore
In Germany, according to an old tradition, 13 (dreizehn), as the first compound number, was the first number written in digits; the numbers 0 (null) through 12 (zwölf) were spelled out. The Duden (the German standard dictionary) now calls this tradition (which was actually never written down as an official rule) outdated and no longer valid, but many writers still follow it.

In English
Thirteen is one of two numbers within the teen numerical range (13–19), along with fifteen, not derived by cardinal numeral (three) and the teen suffix; instead, it is derived from the ordinal numeral (third).

Islam
In Shia, 13 signifies the 13th day of the month of Rajab (the Lunar calendar), which is the birth of Imam Ali. 13 also is a total of 1 Prophet and 12 Shia Imams in the Islamic School of Thought.

 Christianity  In all branches of Christianity it's said that Christ had 12 disciples, thus forming a group of 13. It's best known by the representations of the last supper.

Catholicism
The apparitions of the Virgin of Fátima in 1917 were claimed to occur on the 13th day of six consecutive months.

In Catholic devotional practice, the number thirteen is also associated with Saint Anthony of Padua, since his feast day falls on June 13. A traditional devotion called the Thirteen Tuesdays of St. Anthony involves praying to the saint every Tuesday over a period of thirteen weeks. Another devotion, St. Anthony's Chaplet, consists of thirteen decades of three beads each.

Sikhism
According to famous Sakhi (evidence) or the story of Guru Nanak Dev Ji, when he was an accountant at a town of Sultanpur Lodhi, he was distributing groceries to people. When he gave groceries to the 13th person, he stopped because in Gurmukhi and Hindi the word 13 is called terah, which means yours. And Guru Nanak Dev Ji kept saying, "Yours, yours, yours..." remembering God. People reported to the emperor that Guru Nanak Dev Ji was giving out free food to the people. When treasures were checked, there was more money than before.

Judaism

 * In Judaism, 13 signifies the age at which a boy matures and becomes a Bar Mitzvah, i.e., a full member of the Jewish faith (counts as a member of Minyan).
 * The number of principles of Jewish faith according to Maimonides.
 * According to Rabbinic commentary on the Torah, God has 13 Attributes of Mercy.

Zoroastrianism
Since beginning of the Nowruz tradition, the 13th day of each new Iranian year is called Sizdah Be-dar, a festival dedicated to pranks and spending time outdoors.

Confucianism

 * The Thirteen Classics is considered to be a part of the Chinese classics.

Aztec mythology
According to Aztec mythology, the Thirteen Heavens were formed by the gods from the head of Cipactli during creation.

Wicca
A common tradition in the religion Wicca holds that the number of members for a coven is ideally thirteen, though this tradition is not universal.

Bad
The number 13 is considered an unlucky number in some countries. The end of the Mayan calendar's 13th Baktun was superstitiously feared as a harbinger of the apocalyptic 2012 phenomenon. Fear of the number 13 has a specifically recognized phobia, triskaidekaphobia, a word first recorded in 1911. The superstitious sufferers of triskaidekaphobia try to avoid bad luck by keeping away from anything numbered or labelled thirteen. As a result, companies and manufacturers use another way of numbering or labelling to avoid the number, with hotels and tall buildings being conspicuous examples (thirteenth floor). It is also considered unlucky to have thirteen guests at a table. Friday the 13th has been considered an unlucky day.

There are a number of theories as to why the number thirteen became associated with bad luck, but none of them have been accepted as likely.
 * The Last Supper: At Jesus Christ's Last Supper, there were thirteen people around the table, counting Christ and the twelve apostles. Some believe this is unlucky because one of those thirteen, Judas Iscariot, was the betrayer of Jesus Christ. From the 1890s, a number of English language sources relate the "unlucky" thirteen to an idea that at the Last Supper, Judas, the disciple who betrayed Jesus, was the 13th to sit at the table.
 * Knights Templar: On Friday, 13 October 1307, King Philip IV of France ordered the arrest of the Knights Templar, and most of the knights were tortured and killed.
 * Full moons: A year with 13 full moons instead of 12 posed problems for the monks in charge of the calendars. "This was considered a very unfortunate circumstance, especially by the monks who had charge of the calendar of thirteen months for that year, and it upset the regular arrangement of church festivals. For this reason, thirteen came to be considered an unlucky number." However, a typical century has about 37 years that have 13 full moons, compared to 63 years with 12 full moons, and typically every third or fourth year has 13 full moons.
 * A suppressed lunar cult: In ancient cultures, the number 13 represented femininity, because it corresponded to the number of lunar (menstrual) cycles in a year ($13 × 28 = 364 days$). The theory is that, as the solar calendar triumphed over the lunar, the number thirteen became anathema.
 * Hammurabi's code: There is a myth that the earliest reference to thirteen being unlucky or evil is in the Babylonian Code of Hammurabi (circa 1780 BC), where the thirteenth law is said to be omitted. In fact, the original Code of Hammurabi has no numeration. The translation by L.W. King (1910), edited by Richard Hooker, omitted one article: If the seller have gone to (his) fate (i. e., have died), the purchaser shall recover damages in said case fivefold from the estate of the seller. Other translations of the Code of Hammurabi, for example the translation by Robert Francis Harper, include the 13th article.

Good

 * France: 13 was traditionally considered a lucky number in France prior to the First World War, and was used in numerical form as a good luck symbol on postcards and charms.
 * Italy: 13 was the lucky number in the Italian football pools (Totocalcio). The Italian expression "fare tredici" (literally, "make thirteen") means to hit the jackpot.
 * United States: Colgate University also considers 13 a lucky number. They were founded in 1819 by 13 men with 13 dollars, 13 prayers and 13 articles. (To this day, members of the Colgate community consider the number 13 a good omen.) In fact, the campus address is 13 Oak Drive in Hamilton, New York, and the male a cappella group is called the Colgate 13.

Other
In the Mayan Tzolk'in calendar, trecenas mark cycles of 13-day periods. The pyramids are also set up in 9 steps divided into 7 days and 6 nights, 13 days total.

In the standard 52-card deck of playing cards there are four suits, each of 13 ranks.

A baker's dozen, devil's dozen, long dozen, or long measure is 13, one more than a standard dozen. The thirteenth loaf is called the vantage loaf because it is considered advantageous overall to get 13 loaves for the price of 12.

In Arthurian legend, which was recorded in Medieval texts, King Arthur is resting in Avalon with the twelve greatest knights of the Round Table, totalling 13, and will return when his country is in peril.

The Thirteen Treasures of Britain are a series of magical items listed in late Medieval texts.

The Thirteen Postures of Tai Chi are thirteen postures (consisting of Eight Gates and Five Steps) which are considered to be of fundamental importance in the practice of tai chi.

In astronomy there are 13 star constellations in the zodiac (including Ophiuchus); this can be compared with astrology where there are 12 signs of the zodiac.

In Judaism, 13 signifies the age at which a boy matures and becomes a Bar Mitzvah, i.e., a full member of the Jewish faith (counts as a member of Minyan). The number of principles of Jewish faith according to Maimonides. According to Rabbinic commentary on the Torah, God has 13 Attributes of Mercy.

In a tarot card deck, XIII is the card of Death, usually picturing the Pale Horse with its rider.

Age 13

 * In Judaism, thirteen signifies the age at which a boy matures and becomes a Bar Mitzvah, i.e., a full member of the Jewish faith (is qualified to be counted as a member of Minyan).
 * Thirteen is the minimum age of consent in Argentina, Burkina Faso, Japan, Niger, and two Mexican states.
 * On many social media platforms in the United States, 13 is the standard minimum age to be allowed to create an account in compliance with the Children's Online Privacy Protection Act (COPPA).
 * This is the age at which the Entertainment Software Rating Board (ESRB) assesses for T-rated games and the Motion Picture Association recommends movies with a PG-13 rating.

History

 * The United States of America was created from thirteen British colonies and as such, the number thirteen is a commonly recurring motif in American heraldry. For example, there are thirteen stars on the Great Seal of the United States and there are thirteen stripes on the American flag as well as thirteen rays on the Flag of Arizona.
 * The first flag of the United States bore thirteen stripes, alternating red and white, and thirteen white stars in the blue union. The thirteen stripes represented the Thirteen Colonies from which the United States was created, and the thirteen stars represented the number of states in the new nation. When two new states were added to the Union in 1795, the flag bore fifteen stars and fifteen stripes.  With the addition of five new states in 1818, the number of stripes was re-set and permanently fixed at thirteen.
 * The Great Seal of the United States bears many images of the number thirteen, representing the Thirteen Colonies from which the United States was created. On the Seal's obverse, the overhead glory bears thirteen stars. The chest shield in front of the spread eagle bears thirteen stripes (seven white and six red). The eagle's right talon holds the Olive Branch of Peace, bearing thirteen olives and thirteen olive leaves. The eagle's left talon holds the Weapons of War, consisting of thirteen arrows. The eagle's mouth holds a scroll bearing the national motto "E Pluribus Unum" (which, by coincidence, consists of thirteen letters).  On the Seal's reverse, the unfinished pyramid consists of thirteen levels.
 * The Thirteenth Amendment to the United States Constitution abolished slavery and involuntary servitude (except as a punishment for crime).
 * Apollo 13 was a NASA Moon mission famous for being a "successful failure" in 1970.

In sports

 * The number 13 was not used in the Indianapolis 500 from 1915 to 2002. It was not permitted for use between 1926 and 2002. In 2009, E. J. Viso, driving for HVM Racing in the 2009 IndyCar Series season, drove a green number 13 car full-time, despite terrible superstitions about it in motorsports.
 * The number 13 was not used in Formula One from 1977 to 2013.
 * In rugby league:
 * Each side has 13 players on the field at any given time.
 * The jersey number 13 is worn by the starting loose forward or lock forward in most competitions. An exception is in the Super League, which uses static squad numbering.
 * In rugby union, the jersey number 13 is worn by the outside centre.
 * In triathlon, the number 13 is not used. As such, the numbering goes 11, 12, 14, 15 under the current numbering system.
 * In Petanque, standard games are won when a team reaches the score of 13 points. A 13–0 score is called Fanny.
 * Effective with the 2020 season, Major League Baseball teams are allowed no more than 13 pitchers on their active rosters at any time during the season. The main exception is from September 1 to the end of the regular season, when this number increases to 14.
 * In NCAA Division I men's college basketball, teams are allowed to provide scholarships to no more than 13 players at any given time.
 * In pool:
 * The pool ball 13 is the 13th in pool and its color is orange (pink in the Brazilian pool).

In TV, films and literature

 * 13th, a 2016 documentary.
 * 13 (musical), a 2007 musical.
 * 13 (film), an English-language remake of the 2005 French film 13 Tzameti.
 * Thirteen (film), a 2003 American film.
 * Thirteen (TV series), a 2016 British five-part police drama.
 * Number 13 (film), an uncompleted Hitchcock 1922 film.
 * 13 Tzameti, a 2005 French film ("Tzameti" means "13" in Georgian).
 * 13 Ghosts is a 1960 horror film.
 * Thirteen Ghosts is a 2001 remake.
 * Thirteen is the nickname of Dr. Remy Hadley on the American medical drama House played by Olivia Wilde.
 * 13 Assassins is a 2010 film by Japanese director Takashi Miike.
 * The Thirteen is a 1936 Soviet war film by Mikhail Romm.
 * The 13th Warrior is a 1999 historical fiction action film starring Antonio Banderas.
 * District 13 is a 2004 French film (with David Belle).
 * Apollo 13 is a 1995 American film.
 * Warehouse 13 is a television show about the 13th warehouse in the line of warehouses that store supernatural artifacts.
 * Friday the 13th is a horror film series involving a mass murderer named Jason Voorhees.
 * Friday the 13th: The Series is a syndicated American-Canadian horror television series, that originally ran from 1987 to 1990.
 * The 13 Ghosts of Scooby-Doo: Seventh incarnation of the Hanna-Barbera Scooby-Doo cartoon franchise, first run 1985–1986.
 * The Thirteenth Floor is a 1999 sci-fi film.
 * Golgo 13, centered around a self-titled assassin, is the oldest manga still in publication. It is the second or third best selling manga in history.
 * 13B/Yaavarum Nalam is a 2009 Hindi/Tamil Horror movie starring R Madhavan.
 * Thirteen Reasons Why is a novel containing 13 tapes received by a student learning of his classmate's suicide.
 * The Thirteenth Doctor, the 13th incarnation of the titular character of Doctor Who, played by Jodie Whittaker.
 * In The Hobbit, the Company of Thorin Oakenshield consists of 13 Dwarves trying to reclaim their homeland from the dragon Smaug (Thorin Oakenshield, Fíli, Kíli, Óin, Glóin, Balin, Dwalin, Bifur, Bofur, Bombur, Dori, Nori and Ori).
 * In The Wheel of Time series, the 13 Forsaken are sorcerers imprisoned for 3,000 years before escaping: Aginor, Asmodean, Balthamel, Be'lal, Demandred, Graendal, Ishamael, Lanfear, Mesaana, Moghedien, Rahvin, Sammael and Semirhage.
 * 13 Going on 30 is a 2004 romantic comedy film.

title: 13 (number)


text:
Fermat's principle

Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. Fermat's principle states that the path taken by a ray between two given points is the path that can be traveled in the least time.

First proposed by the French mathematician Pierre de Fermat in 1662, as a means of explaining the ordinary law of refraction of light (Fig.1), Fermat's principle was initially controversial because it seemed to ascribe knowledge and intent to nature. Not until the 19th century was it understood that nature's ability to test alternative paths is merely a fundamental property of waves. If points A and B are given, a wavefront expanding from A sweeps all possible ray paths radiating from A, whether they pass through B or not. If the wavefront reaches point B, it sweeps not only the ray path(s) from A to B, but also an infinitude of nearby paths with the same endpoints. Fermat's principle describes any ray that happens to reach point B; there is no implication that the ray "knew" the quickest path or "intended" to take that path.

In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the path that can be traveled in the least time. In order to be true in all cases, this statement must be weakened by replacing the "least" time with a time that is "stationary" with respect to variations of the path – so that a deviation in the path causes, at most, a second-order change in the traversal time. To put it loosely, a ray path is surrounded by close paths that can be traversed in very close times. It can be shown that this technical definition corresponds to more intuitive notions of a ray, such as a line of sight or the path of a narrow beam.

For the purpose of comparing traversal times, the time from one point to the next nominated point is taken as if the first point were a point-source. Without this condition, the traversal time would be ambiguous; for example, if the propagation time from $P$ to $P′$ were reckoned from an arbitrary wavefront W containing $P$ (Fig.2), that time could be made arbitrarily small by suitably angling the wavefront.

Treating a point on the path as a source is the minimum requirement of Huygens' principle, and is part of the explanation of Fermat's principle. But it can also be shown that the geometric construction by which Huygens tried to apply his own principle (as distinct from the principle itself) is simply an invocation of Fermat's principle. Hence all the conclusions that Huygens drew from that construction – including, without limitation, the laws of rectilinear propagation of light, ordinary reflection, ordinary refraction, and the extraordinary refraction of "Iceland crystal" (calcite) – are also consequences of Fermat's principle.

Sufficient conditions
Let us suppose that:
 * 1) A disturbance propagates sequentially through a medium (a vacuum or some material, not necessarily homogeneous or isotropic), without action at a distance;
 * 2) During propagation, the influence of the disturbance at any intermediate point P upon surrounding points has a non-zero angular spread (as if P were a source), so that a disturbance originating from any point A arrives at any other point B via an infinitude of paths, by which B receives an infinitude of delayed versions of the disturbance at A; and
 * 3) These delayed versions of the disturbance will reinforce each other at B if they are synchronized within some tolerance.

Then the various propagation paths from A to B will help each other, or interfere constructively, if their traversal times agree within the said tolerance. For a small tolerance (in the limiting case), the permissible range of variations of the path is maximized if the path is such that its traversal time is stationary with respect to the variations, so that a variation of the path causes at most a second-order change in the traversal time.

The most obvious example of a stationarity in traversal time is a (local or global) minimum – that is, a path of least time, as in the "strong" form of Fermat's principle. But that condition is not essential to the argument.

Having established that a path of stationary traversal time is reinforced by a maximally wide corridor of neighboring paths, we still need to explain how this reinforcement corresponds to intuitive notions of a ray. But, for brevity in the explanations, let us first define a ray path as a path of stationary traversal time.

A ray as a signal path (line of sight)
If the corridor of paths reinforcing a ray path from A to B is substantially obstructed, this will significantly alter the disturbance reaching B from A – unlike a similar-sized obstruction outside any such corridor, blocking paths that do not reinforce each other. The former obstruction will significantly disrupt the signal reaching B from A, while the latter will not; thus the ray path marks a signal path. If the signal is visible light, the former obstruction will significantly affect the appearance of an object at A as seen by an observer at B, while the latter will not; so the ray path marks a line of sight.

In optical experiments, a line of sight is routinely assumed to be a ray path.

A ray as an energy path (beam)

If the corridor of paths reinforcing a ray path from A to B is substantially obstructed, this will significantly affect the energy reaching B from A – unlike a similar-sized obstruction outside any such corridor. Thus the ray path marks an energy path – as does a beam.

Suppose that a wavefront expanding from point A passes point P, which lies on a ray path from point A to point B. By definition, all points on the wavefront have the same propagation time from A. Now let the wavefront be blocked except for a window, centered on P, and small enough to lie within the corridor of paths that reinforce the ray path from A to B. Then all points on the unobstructed portion of the wavefront will have, nearly enough, equal propagation times to B, but not to points in other directions, so that B will be in the direction of peak intensity of the beam admitted through the window. So the ray path marks the beam. And in optical experiments, a beam is routinely considered as a collection of rays or (if it is narrow) as an approximation to a ray (Fig.3).

Analogies
According to the "strong" form of Fermat's principle, the problem of finding the path of a light ray from point A in a medium of faster propagation, to point B in a medium of slower propagation (Fig.1), is analogous to the problem faced by a lifeguard in deciding where to enter the water in order to reach a drowning swimmer as soon as possible, given that the lifeguard can run faster than (s)he can swim. But that analogy falls short of explaining the behavior of the light, because the lifeguard can think about the problem (even if only for an instant) whereas the light presumably cannot. The discovery that ants are capable of similar calculations does not bridge the gap between the animate and the inanimate.

In contrast, the above assumptions (1) to (3) hold for any wavelike disturbance and explain Fermat's principle in purely mechanistic terms, without any imputation of knowledge or purpose.

The principle applies to waves in general, including (e.g.) sound waves in fluids and elastic waves in solids. In a modified form, it even works for matter waves: in quantum mechanics, the classical path of a particle is obtainable by applying Fermat's principle to the associated wave – except that, because the frequency may vary with the path, the stationarity is in the phase shift (or number of cycles) and not necessarily in the time.

Fermat's principle is most familiar, however, in the case of visible light: it is the link between geometrical optics, which describes certain optical phenomena in terms of rays, and the wave theory of light, which explains the same phenomena on the hypothesis that light consists of waves.

Equivalence to Huygens' construction

In this article we distinguish between Huygens' principle, which states that every point crossed by a traveling wave becomes the source of a secondary wave, and Huygens' construction, which is described below.

Let the surface $P′$ be a wavefront at time $P$, and let the surface $P$ be the same wavefront at the later time $t + Δt$ (Fig.4). Let $P′$ be a general point on $P$. Then, according to Huygens' construction,   $P′$ is the envelope (common tangent surface), on the forward side of $PP′$, of all the secondary wavefronts each of which would expand in time $Δt$ from a point on $P$, and  if the secondary wavefront expanding from point $P′$ in time $Δt$ touches the surface $P$ at point $W′$, then $W$ and $P$ lie on a ray.  The construction may be repeated in order to find successive positions of the primary wavefront, and successive points on the ray.

The ray direction given by this construction is the radial direction of the secondary wavefront, and may differ from the normal of the secondary wavefront (cf. Fig.2), and therefore from the normal of the primary wavefront at the point of tangency. Hence the ray velocity, in magnitude and direction, is the radial velocity of an infinitesimal secondary wavefront, and is generally a function of location and direction.

Now let $W$ be a point on $W$ close to $t$, and let $W′$ be a point on $P$ close to $W$. Then, by the construction,   the time taken for a secondary wavefront from $W′$ to reach $W$ has at most a second-order dependence on the displacement $W$, and  the time taken for a secondary wavefront to reach $P$ from $W′$ has at most a second-order dependence on the displacement $P′$.  By (i), the ray path is a path of stationary traversal time from $P$ to $P′$; and by (ii), it is a path of stationary traversal time from a point on $Q$ to $W$.

So Huygens' construction implicitly defines a ray path as a path of stationary traversal time between successive positions of a wavefront, the time being reckoned from a point-source on the earlier wavefront. This conclusion remains valid if the secondary wavefronts are reflected or refracted by surfaces of discontinuity in the properties of the medium, provided that the comparison is restricted to the affected paths and the affected portions of the wavefronts.

Fermat's principle, however, is conventionally expressed in point-to-point terms, not wavefront-to-wavefront terms. Accordingly, let us modify the example by supposing that the wavefront which becomes surface $P$ at time $Q′$, and which becomes surface $W′$ at the later time $t + Δt$, is emitted from point $P′$ at time $0$. Let $P$ be a point on $Q′$ (as before), and $P′Q′$ a point on $P′$. And let $Q$, $PQ$, $P$, and $W′$ be given, so that the problem is to find $W$.

If $P′$ satisfies Huygens' construction, so that the secondary wavefront from $Δt$ is tangential to $W$ at $t$, then $W′$ is a path of stationary traversal time from $A$ to $P$. Adding the fixed time from $W$ to $B$, we find that $W′$ is the path of stationary traversal time from $A$ to $W$ (possibly with a restricted domain of comparison, as noted above), in accordance with Fermat's principle. The argument works just as well in the converse direction, provided that $W′$ has a well-defined tangent plane at $B$. Thus Huygens' construction and Fermat's principle are geometrically equivalent.

Through this equivalence, Fermat's principle sustains Huygens' construction and thence all the conclusions that Huygens was able to draw from that construction. In short, "The laws of geometrical optics may be derived from Fermat's principle". With the exception of the Fermat–Huygens principle itself, these laws are special cases in the sense that they depend on further assumptions about the media. Two of them are mentioned under the next heading.

Isotropic media: rays normal to wavefronts
In an isotropic medium, because the propagation speed is independent of direction, the secondary wavefronts that expand from points on a primary wavefront in a given infinitesimal time are spherical, so that their radii are normal to their common tangent surface at the points of tangency. But their radii mark the ray directions, and their common tangent surface is a general wavefront. Thus the rays are normal (orthogonal) to the wavefronts.

Because much of the teaching of optics concentrates on isotropic media, treating anisotropic media as an optional topic, the assumption that the rays are normal to the wavefronts can become so pervasive that even Fermat's principle is explained under that assumption, although in fact Fermat's principle is more general.

Homogeneous media: rectilinear propagation
In a homogeneous medium (also called a uniform medium), all the secondary wavefronts that expand from a given primary wavefront $P$ in a given time $Δt$ are congruent and similarly oriented, so that their envelope $P$ may be considered as the envelope of a single secondary wavefront which preserves its orientation while its center (source) moves over $P$. If $W′$ is its center while $B$ is its point of tangency with $PB$, then $W$ moves parallel to $B$, so that the plane tangential to $A$ at $W$ is parallel to the plane tangential to $APB$ at $A$. Let another (congruent and similarly orientated) secondary wavefront be centered on $B$, moving with $W′$, and let it meet its envelope $B$ at point $W$. Then, by the same reasoning, the plane tangential to $APP′P$ at $P′$ is parallel to the other two planes. Hence, due to the congruence and similar orientations, the ray directions $P′$ and $PP′$ are the same (but not necessarily normal to the wavefronts, since the secondary wavefronts are not necessarily spherical). This construction can be repeated any number of times, giving a straight ray of any length. Thus a homogeneous medium admits rectilinear rays.

Formulation in terms of refractive index
Let a path $Γ$ extend from point $W$ to point $W′$. Let $W$ be the arc length measured along the path from $P$, and let $P′$ be the time taken to traverse that arc length at the ray speed $$v_{\mathrm{r}}$$ (that is, at the radial speed of the local secondary wavefront, for each location and direction on the path). Then the traversal time of the entire path $Γ$ is

(where $W′$ and $P′$ simply denote the endpoints and are not to be construed as values of $P$ or $W′$). The condition for $Γ$ to be a ray path is that the first-order change in $P′$ due to a change in $Γ$ is zero; that is, $$\delta T =\, \delta\int_A^B\frac{ds}{\,v_{\mathrm{r}}\,} \,=\, 0\,.$$

Now let us define the optical length of a given path (optical path length, OPL) as the distance traversed by a ray in a homogeneous isotropic reference medium (e.g., a vacuum) in the same time that it takes to traverse the given path at the local ray velocity. Then, if $W$ denotes the propagation speed in the reference medium (e.g., the speed of light in vacuum), the optical length of a path traversed in time $P$ is $dS = c dt$, and the optical length of a path traversed in time $P′$ is $S = cT$. So, multiplying equation (1) through by $P$, we obtain $$S \,= \int_A^B\!dS \,= \int_A^B\frac{\,c\,}{v_{\mathrm{r}}}\,ds = \int_A^B\!n_{\mathrm{r}}\,ds\,,$$ where $$\,n_{\mathrm{r}}=c/v_{\mathrm{r}}\,$$ is the ray index – that is, the refractive index calculated on the ray velocity instead of the usual phase velocity (wave-normal velocity). For an infinitesimal path, we have $$dS=n_{\mathrm{r}\,}ds\,,\,$$ indicating that the optical length is the physical length multiplied by the ray index: the OPL is a notional geometric quantity, from which time has been factored out. In terms of OPL, the condition for $Γ$ to be a ray path (Fermat's principle) becomes

This has the form of Maupertuis's principle in classical mechanics (for a single particle), with the ray index in optics taking the role of momentum or velocity in mechanics.

In an isotropic medium, for which the ray velocity is also the phase velocity, we may substitute the usual refractive index $W″$ for $n_{r}$.

Relation to Hamilton's principle
If $x$, $y$, $z$ are Cartesian coordinates and an overdot denotes differentiation with respect to $P″$, Fermat's principle (2) may be written $$\begin{align} \delta S &=\,\delta\int_A^B\!n_{\mathrm{r}}\,\sqrt{dx^2+dy^2+dz^2}\\ &=\,\delta\int_A^B\!n_{\mathrm{r}}\,\sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}~ds \,=\,0\,. \end{align}$$ In the case of an isotropic medium, we may replace $n_{r}$ with the normal refractive index $n(x, y, z)$, which is simply a scalar field. If we then define the optical Lagrangian as $$L(x,y,z,\dot{x},\dot{y},\dot{z}) \,=\, n(x,y,z)\,\sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}\,,$$ Fermat's principle becomes $$\delta S =\, \delta\int_A^B\!L(x,y,z,\dot{x},\dot{y},\dot{z})\,ds\,=\,0\,.$$ If the direction of propagation is always such that we can use $W″$ instead of $P″$ as the parameter of the path (and the overdot to denote differentiation w.r.t. $PP′$ instead of $P′P″$), the optical Lagrangian can instead be written $$L\big(x(z),y(z),\dot{x}(z),\dot{y}(z),z\big) = n(x,y,z)\,\sqrt{1+\dot{x}^2+\dot{y}^2}\,,$$ so that Fermat's principle becomes $$\delta S =\, \delta\int_A^B\! L\big(x(z),y(z),\dot{x}(z),\dot{y}(z),z\big)\,dz\,=\,0.$$ This has the form of Hamilton's principle in classical mechanics, except that the time dimension is missing: the third spatial coordinate in optics takes the role of time in mechanics. The optical Lagrangian is the function which, when integrated w.r.t. the parameter of the path, yields the OPL; it is the foundation of Lagrangian and Hamiltonian optics.

History
If a ray follows a straight line, it obviously takes the path of least length. Hero of Alexandria, in his Catoptrics (1st century CE), showed that the ordinary law of reflection off a plane surface follows from the premise that the total length of the ray path is a minimum. Ibn al-Haytham, an 11th-century polymath later extended this principle to refraction, hence giving an early version of the Fermat's principle.

Fermat vs. the Cartesians

In 1657, Pierre de Fermat received from Marin Cureau de la Chambre a copy of newly published treatise, in which La Chambre noted Hero's principle and complained that it did not work for refraction.

Fermat replied that refraction might be brought into the same framework by supposing that light took the path of least resistance, and that different media offered different resistances. His eventual solution, described in a letter to La Chambre dated 1 January 1662, construed "resistance" as inversely proportional to speed, so that light took the path of least time. That premise yielded the ordinary law of refraction, provided that light traveled more slowly in the optically denser medium.

Fermat's solution was a landmark in that it unified the then-known laws of geometrical optics under a variational principle or action principle, setting the precedent for the principle of least action in classical mechanics and the corresponding principles in other fields (see History of variational principles in physics). It was the more notable because it used the method of adequality, which may be understood in retrospect as finding the point where the slope of an infinitesimally short chord is zero, without the intermediate step of finding a general expression for the slope (the derivative).

It was also immediately controversial. The ordinary law of refraction was at that time attributed to René Descartes (d.1650), who had tried to explain it by supposing that light was a force that propagated instantaneously, or that light was analogous to a tennis ball that traveled faster in the denser medium, either premise being inconsistent with Fermat's. Descartes' most prominent defender, Claude Clerselier, criticized Fermat for apparently ascribing knowledge and intent to nature, and for failing to explain why nature should prefer to economize on time rather than distance. Clerselier wrote in part: 1. The principle that you take as the basis of your demonstration, namely that nature always acts in the shortest and simplest ways, is merely a moral principle and not a physical one; it is not, and cannot be, the cause of any effect in nature .... For otherwise we would attribute knowledge to nature; but here, by "nature", we understand only this order and this law established in the world as it is, which acts without foresight, without choice, and by a necessary determination.

2. This same principle would make nature irresolute ... For I ask you ... when a ray of light must pass from a point in a rare medium to a point in a dense one, is there not reason for nature to hesitate if, by your principle, it must choose the straight line as soon as the bent one, since if the latter proves shorter in time, the former is shorter and simpler in length? Who will decide and who will pronounce?

Fermat, being unaware of the mechanistic foundations of his own principle, was not well placed to defend it, except as a purely geometric and kinematic proposition. The wave theory of light, first proposed by Robert Hooke in the year of Fermat's death, and rapidly improved by Ignace-Gaston Pardies and (especially) Christiaan Huygens, contained the necessary foundations; but the recognition of this fact was surprisingly slow.

Huygens's oversight

In 1678, Huygens proposed that every point reached by a luminous disturbance becomes a source of a spherical wave; the sum of these secondary waves determines the form of the wave at any subsequent time. Huygens repeatedly referred to the envelope of his secondary wavefronts as the termination of the movement, meaning that the later wavefront was the outer boundary that the disturbance could reach in a given time, which was therefore the minimum time in which each point on the later wavefront could be reached. But he did not argue that the direction of minimum time was that from the secondary source to the point of tangency; instead, he deduced the ray direction from the extent of the common tangent surface corresponding to a given extent of the initial wavefront. His only endorsement of Fermat's principle was limited in scope: having derived the law of ordinary refraction, for which the rays are normal to the wavefronts, Huygens gave a geometric proof that a ray refracted according to this law takes the path of least time. He would hardly have thought this necessary if he had known that the principle of least time followed directly from the same common-tangent construction by which he had deduced not only the law of ordinary refraction, but also the laws of rectilinear propagation and ordinary reflection (which were also known to follow from Fermat's principle), and a previously unknown law of extraordinary refraction – the last by means of secondary wavefronts that were spheroidal rather than spherical, with the result that the rays were generally oblique to the wavefronts. It was as if Huygens had not noticed that his construction implied Fermat's principle, and even as if he thought he had found an exception to that principle. Manuscript evidence cited by Alan E.Shapiro tends to confirm that Huygens believed the principle of least time to be invalid "in double refraction, where the rays are not normal to the wave fronts".

Shapiro further reports that the only three authorities who accepted "Huygens' principle" in the 17th and 18th centuries, namely Philippe de La Hire, Denis Papin, and Gottfried Wilhelm Leibniz, did so because it accounted for the extraordinary refraction of "Iceland crystal" (calcite) in the same manner as the previously known laws of geometrical optics. But, for the time being, the corresponding extension of Fermat's principle went unnoticed.

Laplace, Young, Fresnel, and Lorentz

On 30 January 1809, Pierre-Simon Laplace, reporting on the work of his protégé Étienne-Louis Malus, claimed that the extraordinary refraction of calcite could be explained under the corpuscular theory of light with the aid of Maupertuis's principle of least action: that the integral of speed with respect to distance was a minimum. The corpuscular speed that satisfied this principle was proportional to the reciprocal of the ray speed given by the radius of Huygens' spheroid. Laplace continued: According to Huygens, the velocity of the extraordinary ray, in the crystal, is simply expressed by the radius of the spheroid; consequently his hypothesis does not agree with the principle of the least action: but it is remarkable that it agrees with the principle of Fermat, which is, that light passes, from a given point without the crystal, to a given point within it, in the least possible time; for it is easy to see that this principle coincides with that of the least action, if we invert the expression of the velocity.

Laplace's report was the subject of a wide-ranging rebuttal by Thomas Young, who wrote in part: The principle of Fermat, although it was assumed by that mathematician on hypothetical, or even imaginary grounds, is in fact a fundamental law with respect to undulatory motion, and is explicitly [sic] the basis of every determination in the Huygenian theory... Mr. Laplace seems to be unacquainted with this most essential principle of one of the two theories which he compares; for he says, that "it is remarkable" that the Huygenian law of extraordinary refraction agrees with the principle of Fermat; which he would scarcely have observed, if he had been aware that the law was an immediate consequence of the principle. In fact Laplace was aware that Fermat's principle follows from Huygens' construction in the case of refraction from an isotropic medium to an anisotropic one; a geometric proof was contained in the long version of Laplace's report, printed in 1810.

Young's claim was more general than Laplace's, and likewise upheld Fermat's principle even in the case of extraordinary refraction, in which the rays are generally not perpendicular to the wavefronts. Unfortunately, however, the omitted middle sentence of the quoted paragraph by Young began "The motion of every undulation must necessarily be in a direction perpendicular to its surface ..." (emphasis added), and was therefore bound to sow confusion rather than clarity.

No such confusion subsists in Augustin-Jean Fresnel's "Second Memoir" on double refraction (Fresnel, 1827), which addresses Fermat's principle in several places (without naming Fermat), proceeding from the special case in which rays are normal to wavefronts, to the general case in which rays are paths of least time or stationary time. (In the following summary, page numbers refer to Alfred W.Hobson's translation.) Thus Fresnel showed, even for anisotropic media, that the ray path given by Huygens' construction is the path of least time between successive positions of a plane or diverging wavefront, that the ray velocities are the radii of the secondary "wave surface" after unit time, and that a stationary traversal time accounts for the direction of maximum intensity of a beam. However, establishing the general equivalence between Huygens' construction and Fermat's principle would have required further consideration of Fermat's principle in point-to-point terms.
 * For refraction of a plane wave at parallel incidence on one face of an anisotropic crystalline wedge (pp.291–2), in order to find the "first ray arrived" at an observation point beyond the other face of the wedge, it suffices to treat the rays outside the crystal as normal to the wavefronts, and within the crystal to consider only the parallel wavefronts (whatever the ray direction). So in this case, Fresnel does not attempt to trace the complete ray path.
 * Next, Fresnel considers a ray refracted from a point-source M inside a crystal, through a point A on the surface, to an observation point B outside (pp.294–6). The surface passing through B and given by the "locus of the disturbances which arrive first" is, according to Huygens' construction, normal to "the ray AB of swiftest arrival". But this construction requires knowledge of the "surface of the wave" (that is, the secondary wavefront) within the crystal.
 * Then he considers a plane wavefront propagating in a medium with non-spherical secondary wavefronts, oriented so that the ray path given by Huygens' construction – from the source of the secondary wavefront to its point of tangency with the subsequent primary wavefront – is not normal to the primary wavefronts (p.296). He shows that this path is nevertheless "the path of quickest arrival of the disturbance" from the earlier primary wavefront to the point of tangency.
 * In a later heading (p.305) he declares that "The construction of Huygens, which determines the path of swiftest arrival" is applicable to secondary wavefronts of any shape. He then notes that when we apply Huygens' construction to refraction into a crystal with a two-sheeted secondary wavefront, and draw the lines from the two points of tangency to the center of the secondary wavefront, "we shall have the directions of the two paths of swiftest arrival, and consequently of the ordinary and of the extraordinary ray."
 * Under the heading "Definition of the word Ray" (p.309), he concludes that this term must be applied to the line which joins the center of the secondary wave to a point on its surface, whatever the inclination of this line to the surface.
 * As a "new consideration" (pp.310–11), he notes that if a plane wavefront is passed through a small hole centered on point E, then the direction ED of maximum intensity of the resulting beam will be that in which the secondary wave starting from E will "arrive there the first", and the secondary wavefronts from opposite sides of the hole (equidistant from E) will "arrive at D in the same time" as each other. This direction is not assumed to be normal to any wavefront.

Hendrik Lorentz, in a paper written in 1886 and republished in 1907, deduced the principle of least time in point-to-point form from Huygens' construction. But the essence of his argument was somewhat obscured by an apparent dependence on aether and aether drag.

Lorentz's work was cited in 1959 by Adriaan J. de Witte, who then offered his own argument, which "although in essence the same, is believed to be more cogent and more general". De Witte's treatment is more original than that description might suggest, although limited to two dimensions; it uses calculus of variations to show that Huygens' construction and Fermat's principle lead to the same differential equation for the ray path, and that in the case of Fermat's principle, the converse holds. De Witte also noted that "The matter seems to have escaped treatment in textbooks."

In popular culture
The short story Story of Your Life by the speculative fiction writer Ted Chiang contains visual depictions of Fermat's Principle along with a discussion of its teleological dimension. Keith Devlin's The Math Instinct contains a chapter, "Elvis the Welsh Corgi Who Can Do Calculus" that discusses the calculus "embedded" in some animals as they solve the "least time" problem in actual situations.

title: Fermat's principle