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Bohuslav Martinů

Bohuslav Jan Martinů (December 8, 1890 – August 28, 1959) was a Czech composer of modern classical music. He wrote 6 symphonies, 15 operas, 14 ballet scores and a large body of orchestral, chamber, vocal and instrumental works. He became a violinist in the Czech Philharmonic Orchestra, and briefly studied under Czech composer and violinist Josef Suk. After leaving Czechoslovakia in 1923 for Paris, Martinů deliberately withdrew from the Romantic style in which he had been trained. During the 1920s he experimented with modern French stylistic developments, exemplified by his orchestral works Half-time and La Bagarre. He also adopted jazz idioms, for instance in his Kitchen Revue (Kuchyňská revue).

In the early 1930s he found his main fount for compositional style: neoclassicism, creating textures far denser than those found in composers treating Stravinsky as a model. He was prolific, quickly composing chamber, orchestral, choral and instrumental works. His Concerto Grosso and the Double Concerto for Two String Orchestras, Piano and Timpani are among his best-known works from this period. Among his operas, Juliette and The Greek Passion are considered the finest. He has been compared to Prokofiev and Bartók in his innovative incorporation of Czech folk elements into his music. He continued using Bohemian and Moravian folk melodies throughout his oeuvre, for instance in The Opening of the Springs (Otvírání studánek).

His symphonic career began when he emigrated to the United States in 1941, fleeing the German invasion of France. His six symphonies were performed by all the major US orchestras. Eventually Martinů returned to live in Europe for two years starting in 1953, then was back in New York until returning to Europe in May 1956. He died in Switzerland in August 1959.

1890–1923: Polička and Prague
The setting of Martinů's birth was unusual. He was born in the tower of the St. Jakub Church in Polička, a town in Bohemia, close to the Moravian border. His father, Ferdinand, a shoemaker, also worked as the church sexton and town fire watchman. For this, he and his family were allowed to live in the tower apartment. As a small boy Bohuslav was sickly, and frequently had to be carried up the 193 steps to the tower on the back of his father or his older sister. In school he was known to be very shy, and did not participate in the plays or pageants with his classmates. But as violinist, he excelled and developed a strong reputation, giving his first public concert in his hometown in 1905. The townspeople raised enough money to fund his schooling, and in 1906 he left the countryside to begin studies at the Prague Conservatory.

Whilst there he fared poorly as a student, showing little interest in the rigid pedagogy, nor the hours of violin practice required. He was far more interested in exploring Prague and learning on his own, attending concerts and reading books on many subjects. This was in contrast to his roommate, Stanislav Novák, who was an excellent student and a brilliant violinist. They frequently attended concerts together at which Martinů became engrossed in analysing new music, particularly French impressionist works. He could memorize much of it, to the extent that when back in their room, he could write out large parts of the score almost perfectly. Novák became astonished at how meticulously Martinů could do this. He became convinced that his roommate, while lacking in other subjects, possessed an incredible brain for analysing and memorizing music.

They became friends for life. Dropped from the violin program, Martinů was moved to the organ department that taught composition, but he was finally dismissed in 1910 for "incorrigible negligence".

Martinů spent the next several years living back home in Polička, attempting to gain some standing in the musical world. He had written several compositions by this time, including the Elegie for violin and piano, and the symphonic poems Angel of Death (Anděl smrti) and Death of Tintagiles (Smrt Tintagilova), and submitted samples of his work to Josef Suk, a leading Czech composer. Suk encouraged him to pursue formal composition training, but this would not be possible until years later. In the meantime, he passed the state teaching examination and maintained a studio in Polička throughout World War I, while continuing to compose and study on his own. It was during this time that he studied the ancient choral hymns of the Bohemian Brethren, which would influence his style and musical scope.

As World War I drew to a close and Czechoslovakia was declared an independent republic, Martinů composed the celebratory cantata Czech Rhapsody (Česká rapsodie), which was premiered in 1919 to great acclaim. He toured Europe as a violinist with the National Theatre Orchestra, and in 1920 became a full member of the Czech Philharmonic Orchestra that was led by the inspired young conductor Václav Talich, who was the first major conductor to promote Martinů. He also began formal composition study under Suk. During these last years in Prague he completed his first string quartet and two ballets: Who is the Most Powerful in the World? (Kdo je na světě nejmocnější?) and Istar.

1923–40: Paris
Martinů finally departed for Paris in 1923, having received a small scholarship from the Czechoslovak Ministry of Education. He sought out Albert Roussel, whose individualistic style he respected, and began a series of informal lessons with him. Roussel would teach Martinů until his death in 1937 by helping him focus and bring order to his compositions, rather than instructing him in a specific style. During his first years in Paris, Martinů incorporated many of the trends at the time, including jazz, neoclassicism, and surrealism. He was particularly attracted to Stravinsky, whose novel, angular, propulsive rhythms and sonorities reflected the industrial revolution, sports events and motorised transportation. Ballets were his favorite medium for experimentation, including The Revolt (1925), The Butterfly That Stamped (1926), Le raid merveilleux (1927), La revue de cuisine (1927), and Les larmes du couteau (1928). Martinů found friends in the Czechoslovak artistic community in Paris and would always retain close ties to his homeland, frequently returning during the summer. He continued to look to his Bohemian and Moravian roots for musical ideas. His best-known work from this time is the ballet Špalíček (1932–33), which incorporates Czech folk tunes and nursery rhymes.

The prime leader of new symphonic music in Paris at this time was Serge Koussevitzky, who presented the biannual Concerts Koussevitzsky (1921–29). He became the conductor of the Boston Symphony Orchestra in 1924, but still returned to Paris each summer to conduct his Concerts. In 1927, Martinů happened to see him at a café, introduced himself, and gave him the score of a symphonic triptych, La bagarre, that was inspired by Charles Lindbergh's recent landing. The maestro was impressed, and scheduled its premiere with the Boston Symphony in November 1927.

In 1926, Martinů met Charlotte Quennehen (1894–1978), a French seamstress from Picardy. She was employed at a large garment factory and, after their romance began, she moved into his small flat and helped to support him. She would become an important force in his life, handling the cuisine and business matters that he found trying. They married in 1931. Culturally, however, the two were quite different, a fact that would cause problems in their marriage over the years.

By 1930, Martinů had withdrawn from his seven years of experimentation to settle on a neo-classical style. In 1932, he won the Coolidge prize for the best of 145 chamber music works for his String Sextet with Orchestra. This was performed by Koussevitzsky with the Boston Symphony Orchestra in 1932. Martinu finished his opera Julietta in 1936; this was based upon a surrealistic play by Georges Neveux that he had seen in 1927. Its premiere was given in Prague under Václav Talich on 14 March 1938.

In 1937, Martinů became acquainted with a young Czech woman, Vítězslava Kaprálová, who was already a highly accomplished musician when she arrived in Paris, supported by a small Czech government grant to study conducting with Charles Munch and composition with Martinů. Their relationship soon developed beyond that of student-teacher as he fell madly in love with her. After she returned to Czechoslovakia, Martinů wrote her many long, passionate letters. In one of these, he proposed that he would divorce Charlotte and then take her to America. It was while he was in this distraught, frenzied state that Martinů composed one of his greatest works, the Double Concerto for two string orchestras, piano and timpani. It was finished just a few days before the Munich Agreement was sealed (30 September 1938).

After the Munich Agreement, President Edvard Beneš began to form a Czechoslovak government in exile set up in France and England. As a significant number of troops became organized into a Czech resistance force, Martinů tried to join them but was rejected because of his age. However, in 1939, he composed a tribute to this force, the Field Mass for baritone, chorus and orchestra. It was broadcast from England and was picked up in occupied Czechoslovakia. For this, Martinů was blacklisted by the Nazis and sentenced in absentia. In 1940, as the German army approached Paris, the Martinůs fled. They were sheltered by Charles Munch who had a place near Limoges. Soon, they journeyed on to Aix-en-Provence, where they stayed for six months while trying to find transit out of Vichy France. He was helped by the Czech artistic community, particularly Rudolf Kundera, along with Edmonde Charles-Roux and the Countess Lily Pastré. Despite the harsh conditions, he found inspiration in Aix and composed several works, notably the Sinfonietta giocosa. Charlotte wrote: "We fell in love with Aix: the delicate murmur of its fountains calmed our agitated feelings and later Bohus was inspired by them." Finally, on 8 January 1941, they left Marseilles for Madrid and Portugal, eventually reaching the United States in 1941 with the help of his friend, the diplomat Miloš Šafránek, and especially from Martinů's Swiss benefactor, Paul Sacher, the conductor of the Basel Chamber Orchestra, who arranged and paid for their passages.

1941–53: US
Life in the United States was difficult for him initially, just as it was for many other artist émigrés in similar circumstances. Lack of knowledge of English, of funds, and of opportunities to use their talents were common to them. When they first arrived in New York, the Martinůs rented a studio apartment at the Great Northern Hotel on 57th St. They were helped by several musician friends, including pianist Rudolf Firkušný, violinist Samuel Dushkin, cellist Frank Rybka, diplomat Miloš Šafránek, and multi-lingual lawyer Jan Löwenbach. Martinů soon found that he was unable to resume composing in noisy Manhattan, so for the following season they leased a small apartment in Jamaica Estates, Queens, close to the Rybkas. This leafy, residential neighborhood was conducive for him to take long solitary walks at night, during which he would work out music scores in his head. On several occasions he would "zone out" in deep concentration about the music, becoming oblivious of his surroundings and getting lost, and would then call a friend with a car to come find him and take him back home. Thereafter, he began to compose actively. When he contacted Serge Koussevitzsky, the conductor told him that his Concerto Grosso would receive its premiere in Boston the following season. One of the first compositions Martinů wrote in New York was the Concerto da Camera for violin and small orchestra, in fulfillment of a commission he had been awarded before the war by Paul Sacher. The following year, they moved back to Manhattan into an apartment in a brownstone on 58th St, across from the Hotel Plaza. That was where they lived for the rest of their years in America. Composer David Diamond, who sub-leased this apartment in 1954, has described it in an interview.

"As the War was coming to an end, the Martinůs encountered marital difficulties. Charlotte, who never did like America, wanted strongly to return to France. He did not, so when he accepted Koussevitzky's offer to teach at the Berkshire Music School for the summer of 1946, she went to France alone for a prolonged visit. In Great Barrington, Massachusetts, he was lodged with the students in Searles Castle, and his magnificent master bedroom opened onto a terrace. One night, Martinů took his customary walk on the terrace, a section of which had no railing, and he fell off, landing on concrete, and was hospitalized with a fractured skull and concussion. He drifted in and out of a coma, but survived. After several weeks he was released to recuperate with friends. By this time, Roe Barstow had entered his life. She was an attractive divorcee of independent means, who lived alone in Greenwich Village. With Charlotte away in France, she was at Martinů's side, assisting in his recovery, during which their relationship deepened. After Charlotte returned in the late fall, she found that her husband was a different man: gaunt, irritable, crippled and in pain from the accident. It required a few years before he was able to return to his former state as a solid composer."

Apart from his domestic problems, Martinů was unsure about which country he would live in. He had considered returning to Czechoslovakia as a teacher, despite having a powerful enemy there in the communist politician Zdeněk Nejedlý. Any plans to return were further hampered by the 1948 Czechoslovak coup d'état. With the communists' seizure of power, music, along with the other arts, became an instrument of propaganda along Soviet ideological lines. Martinů was branded as a formalist and émigré traitor, and he chose wisely not to pursue any kind of professional engagement in his native land from this time forward. Martinů became an American citizen in 1952.

Martinů was indeed reluctant to leave America which had been very supportive of him. He taught at the Mannes College of Music for most of the period from 1948 to 1956. He also taught at Princeton University and the Berkshire Music School (Tanglewood). At Princeton he was warmly received by faculty and students. His six symphonies were written in the eleven-year period 1942–1953, the first five being produced between 1942 and 1946. In addition, he composed the Violin Concerto No. 2, Memorial to Lidice for orchestra, Concerto for Two Pianos and Orchestra, Piano Concerto No. 3, Concerto da Camera for violin and small orchestra, Sinfonietta La Jolla for piano and small orchestra, Sonatas Nos. 2 and 3 for cello and piano, many chamber compositions, and a television opera, The Marriage (Ženitba). His symphonic scores were performed by most of the major orchestras: Boston, New York, Philadelphia, Cleveland, Chicago, and he generally received fine reviews from the leading critics.

Owing to the extraordinary volume of Martinů's oeuvre, some critics who never knew the man have stated that he composed too much, too fast, and therefore must have been careless in quality. However, he has been defended strongly by musicians and critics who did know him. Olin Downes knew Martinů better. For his interviews of Martinů, he had the benefit of having Jan Löwenbach, a friend of both men, present as an interpreter. Downes' defense of the composer came out in an article, "Martinu at 60". "Martinu […] is incapable of an unthorough or conscienceless job. He works very hard, systematically, scrupulously, modestly. He produces so much music because in the first place, his nature necessitates this. He has to write music. In the second place, he knows his business and loves it." The composer David Diamond knew Martinů both in Paris and New York. In an interview years later, he expressed amazement at how extraordinary Martinů's mind was in developing a whole orchestral score while taking a walk.

Martinů's notable students include Burt Bacharach, Alan Hovhaness, Vítězslava Kaprálová, Louis Lane, Jan Novák, H. Owen Reed, Howard Shanet and Chou Wen-chung.

1953–59: Europe
In 1953, Martinů left the United States for France and settled in Nice, and completed his Fantaisies symphoniques; the following year he composed Mirandolina and piano sonata, and met Nikos Kazantzakis, beginning work on The Greek Passion. During 1955 he created several key works: the oratorio Gilgames (The Epic of Gilgamesh), the Oboe Concerto, Les Fresques de Piero della Francesca, and the cantata Otvirani studanek (The Opening of the Wells); Charles Munch conducted the Fantaisies symphoniques premiere in Boston which gained the composer the annual New York critics' prize for the work. In 1956, he took up an appointment as composer-in-residence at the American Academy in Rome and composed Incantation (his fourth piano concerto) and much of The Greek Passion, which he completed in January the following year.

Jan Smaczny commented that in the compositions of Martinů's last years "we find the composer attempting through his music a vicarious homecoming", although he never returned to Czechoslovakia. His prolific output continued in 1958 with The Parables for orchestra and the opera Ariane. The following year he attended the first production of Julietta since the premiere in Prague, in Wiesbaden. Further composition continued up to his death: the second version of The Greek Passion, the Nonet, the Madrigaly, and the cantatas Mikeš z hor (Mikeš from the Mountains) and The Prophecy of Isaiah, "one of the most striking and most individual of Martinů's works".

He died of gastric cancer in Liestal, Switzerland, on 28 August 1959. His remains were moved and buried in Polička, Czechoslovakia, in 1979.

Music
Martinů was a prolific composer who wrote almost 400 pieces. Many of his works are regularly performed or recorded, among them his oratorio The Epic of Gilgamesh (1955, Epos o Gilgamešovi), his six symphonies, concertos (these number almost thirty – four violin concertos, eight compositions for solo piano, four cello concertos, one of each for harpsichord, viola, and oboe, five double concertos, two triple concertos, and two concertos for four solo instruments and orchestra), an anti-war opera Comedy on the Bridge (Veselohra na mostě), chamber music (including eight string quartets, three piano quintets, a piano quartet), a flute sonata, a clarinet sonatina and many others.

A characteristic feature of his orchestral writing is the near-omnipresent piano; many of his orchestral works include a prominent part for piano, including his small Concerto for harpsichord and chamber orchestra. The bulk of his writing from the 1930s into the 1950s was in a neoclassical vein, but with his last works he opened up his style to include more rhapsodic gestures and a looser, more spontaneous sense of form. This is easiest to hear by comparing his Fantaisies symphoniques (Symphony No. 6), H 343, with its five predecessors, all from the 1940s.

One of Martinů's lesser known works features the theremin. Martinů started working on his Fantasia for theremin, oboe, string quartet and piano in the summer of 1944, and finished it on October 1. He dedicated it to Lucie Bigelow Rosen, who had commissioned it and was the theremin soloist at its premiere at New York's Town Hall on 3 November 1945, joined by the Koutzen Quartet, Robert Bloom (oboe), and Carlos Salzedo (piano).

His opera The Greek Passion is based on the novel of the same name by Nikos Kazantzakis, and his orchestral work Memorial to Lidice (Památník Lidicím) was written in remembrance of the village of Lidice that was destroyed by the Nazis in reprisal for the assassination of Reinhard Heydrich in the late spring of 1942. It was completed in August 1943 whilst he was in New York, and premiered there in October of that year.

Personality
There have been many discussions about Martinů's personality, manners and possible Asperger syndrome. Frank James Rybka promoted the idea that Martinů suffered from this kind of autism spectrum disorder. He met Martinů in 1941, when the composer was 51 and Rybka was only six years old; later on he met him in 1951 and then in 1959, a month before the composer died.

According to Rybka, Martinů was quiet, introverted, and emotionally stolid when meeting persons he did not know well. He typically answered questions very slowly, even when conversing in his native Czech. He might fail to reciprocate socially when people would compliment his music, or do favors for him. Close friends found him to be a kind, gentle, self-effacing, unbiased person. In 2009, Rybka launched a retrospective study of the composer's unusual personality, based upon interviews of people who knew him, as well as a study of letters he had written to his family and friends. Evidence of his having an autism spectrum disorder was compiled and evaluated, using the established criteria found in the Diagnostic and Statistical Manual for Mental Disease (DSM-IV). This evidence was reviewed by a well-known autism neuroscientist who concurred that the composer had good evidence of having had an autistic spectrum disorder, most likely Asperger syndrome. This was described in their publication. In 2011, Rybka published a Martinů biography, in which such traits are reviewed, such as his failure of social reciprocity, his flat affect and stolidity, his phobias and extreme stage fright, his strict adherence to a ritualized schedule, and his zoning out into an aura, while walking with his mind deeply engrossed in composing. The biography concludes that there were both positive and negative ways Asperger's affected his life. It seems to have facilitated his extraordinary memory for music, and his ability to compose prolifically and skillfully, but it also left him unable to promote or showcase his music in public.

Against this, Erik Entwistle in his review of Rybka's publication emphasized three main points contradicting Rybka's conclusions. These are: firstly, that it is impossible to diagnose someone so many years after his death; secondly, that Asperger syndrome officially does not exist as a separate syndrome; and finally, that according to Entwistle, Rybka became obsessed with the idea of Martinů's disorder, finding evidence for his claim everywhere.

title: Bohuslav Martinů


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Daniel Massey (actor)

Daniel Raymond Massey (10 October 1933 – 25 March 1998) was an English actor and performer. He is possibly best known for his starring role in the British TV drama The Roads to Freedom, as Daniel, alongside Michael Bryant. He is also known for his role in the 1968 American film Star!, as Noël Coward (Massey's godfather), for which he won a Golden Globe Award and an Oscar nomination.

Early life
Massey was born in London in 1933. He was educated at Eton College and King's College, Cambridge. He was a member of the noted Massey family, which included his father, Raymond Massey, his sister, Anna Massey and his uncle Vincent Massey, the first Canadian-born Governor General of Canada. His mother was the actress Adrianne Allen.

Living with his mother after his parents' divorce, Massey rarely saw his father through most of his adult life; however, they were cast as father and son in The Queen's Guards (1961).

Career
Massey made his film debut as a child in his godfather Noël Coward's naval drama, In Which We Serve (1942). He would later play Coward in the 1968 Julie Andrews vehicle Star!, a performance for which he won a Golden Globe Award and received his sole Academy Award nomination for Best Supporting Actor.

He made a major impression as an adult as Laurence Olivier's son-in-law in the stage and screen versions of John Osborne's The Entertainer (film in 1960). Massey appeared in numerous British films from the 1950s onwards, including The Jokers (1967), Mary, Queen of Scots (1971), The Vault of Horror (1973, in which his character's sister was played by his real-life sister, Anna Massey),  The Cat and the Canary (1979), Victory! (1981) and In the Name of the Father (1993).

Other highlights of his career were his stage roles, especially that of the German conductor Wilhelm Furtwängler in Ronald Harwood's Taking Sides; Massey was nominated for the 1996 Olivier Award as Best Actor. He recreated the role for Broadway in 1996, earning a 1997 Drama Desk Award nomination for Outstanding Actor In A Play. His other Broadway stage appearances included musicals such as She Loves Me as Georg in 1963 and Gigi (as Gaston) in 1973.

He appeared in Stephen Sondheim's Follies as Benjamin Stone in the West End in 1987. In the 1980s and 1990s, he also appeared with the Royal Shakespeare Company in productions such as Love's Labour's Lost, Measure for Measure and The Time of Your Life, the latter alongside John Thaw.

In 1970 Massey played the role of the openly gay character Daniel, alongside a cast headed by Michael Bryant as Mathieu in the acclaimed multi-part BBC adaptation of Jean Paul Sartre's The Roads to Freedom.

Other television highlights of Massey's career include The Crucible on the BBC (1981) as Reverend Hale, The Golden Bowl (1972) as the Prince, in the Inspector Morse episode "Deceived by Flight" as Anthony Donn, again with John Thaw, and his performance as an AIDS patient in Intimate Contact (1987). With Jeremy Brett as Sherlock Holmes, he played a US Senator in "The Problem of Thor Bridge", series 3, episode 2, The Case-Book of Sherlock Holmes, Granada Television, 1991. Brett had once been married to Massey's sister, Anna, and was father to Massey's nephew by Anna, actor David Huggins. He also stole mostly every scene he appeared in the Alan Bleasdale classic G.B.H. (1991) as the awkward eccentric hotel owner, Grosvenor.

Personal life
Massey was married three times, two of his wives being well-known actresses:
 * Adrienne Corri (1961–1967)
 * Penelope Wilton (1975–1984); one daughter, Alice Massey and a stillborn son.
 * Linda Wilton (1986–1998) (a sister of Penelope)
 * Massey also had a close friendship/relationship with Marilu Henner. The two worked together in 1985 in Italy making the film, Love with a Perfect Stranger (1986). Henner wrote he fell for Massey.

Death
He died in London, on 25 March 1998 from Hodgkin's lymphoma, which had been diagnosed in 1992. His body was interred at Putney Vale Cemetery. Massey worked in theatre throughout his cancer treatments, rarely missing a performance.

Selected filmography

 * In Which We Serve (1942) – Bobby Kinross
 * Girls at Sea (1958) – Flag. Lt. Courtney
 * Operation Bullshine (1959) – Bombardier Peter Palmer
 * Upstairs and Downstairs (1959) – Wesley Cotes
 * The Entertainer (1960) – Graham
 * The Queen's Guards (1961) – John Fellowes
 * Go to Blazes (1962) – Harry
 * The Amorous Adventures of Moll Flanders (1965) – Elder Brother
 * The Jokers (1967) – Riggs
 * Star! (1968) – Noël Coward
 * Fragment of Fear (1970) – Maj. Ricketts
 * Mary, Queen of Scots (1971) – Robert Dudley
 * The Vault of Horror (1973) – Rogers (segment 1 "Midnight Mess")
 * The Incredible Sarah (1976) – Victorien Sardou
 * The Devil's Advocate (1977) – Black
 * Warlords of Atlantis (1978) – Atraxon
 * The Cat and the Canary (1978) – Dr. Harry Blythe
 * Love with a Perfect Stranger (1986) - Hugo DeLassey
 * Bad Timing (1980) – Foppish Man
 * Escape to Victory (1981) – Colonel Waldron
 * Scandal (1989) – Mervyn Griffith-Jones
 * In the Name of the Father (1993) – Prosecutor
 * The Miracle Maker (2000) – Cleopas (voice) (final film role)

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

title: Daniel Massey (actor)


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Hall–Héroult process

The Hall–Héroult process is the major industrial process for smelting aluminium. It involves dissolving aluminium oxide (alumina) (obtained most often from bauxite, aluminium's chief ore, through the Bayer process) in molten cryolite and electrolyzing the molten salt bath, typically in a purpose-built cell. The Hall–Héroult process applied at industrial scale happens at 940–980 °C (1700 to 1800°F) and produces 99.5–99.8% pure aluminium. Recycling aluminum requires no electrolysis, thus it is not treated in this way.

Because Hall–Héroult processing consumes copious electrical energy and its electrolysis stage creates much carbon dioxide and fluorocarbon compounds as byproducts, it contributes to air pollution as well as climate change.

Difficulties faced
Elemental aluminium cannot be produced by the electrolysis of an aqueous aluminium salt, because hydronium ions readily oxidize elemental aluminium. Although a molten aluminium salt could be used instead, aluminium oxide has a melting point of 2072 °C (3762°F) so electrolysing it is impractical. In the Hall–Héroult process, alumina, Al2O3, is dissolved in molten synthetic cryolite, Na3AlF6, to lower its melting point for easier electrolysis. The carbon source is generally a coke (fossil fuel).

Theory
In the Hall–Héroult process the following simplified reactions take place at the carbon electrodes:

Cathode:

Anode:

Overall:

In reality, much more CO2 is formed at the anode than CO:

Pure cryolite has a melting point of $1,009 degC$ (1848°F). With a small percentage of alumina dissolved in it, its melting point drops to about 1000 °C (1832°F). Besides having a relatively low melting point, cryolite is used as an electrolyte because, among other things, it also dissolves alumina well, conducts electricity, dissociates electrolytically at higher voltage than alumina, and also has a lower density than aluminum at the temperatures required by the electrolysis.

Aluminium fluoride (AlF3) is usually added to the electrolyte. The ratio NaF/AlF3 is called the cryolite ratio and it is 3 in pure cryolite. In industrial production, AlF3 is added so that the cryolite ratio is 2–3 to further reduce the melting point, so that the electrolysis can happen at temperatures between 940 and 980 °C (1700 to 1800°F). The density of liquid aluminum is 2.3 g/ml at temperatures between 950 and 1000 °C (1750° to 1830°F). The density of the electrolyte should be less than 2.1 g/ml, so that the molten aluminum separates from the electrolyte and settles properly to the bottom of the electrolysis cell. In addition to AlF3, other additives like lithium fluoride may be added to alter different properties (melting point, density, conductivity etc.) of the electrolyte.

The mixture is electrolysed by passing a low voltage (under 5 V) direct current at $100 kA$ through it. This causes liquid aluminium to be deposited at the cathode, while the oxygen from the alumina combines with carbon from the anode to produce mostly carbon dioxide.

The theoretical minimum energy requirement for this process is 6.23 kWh/(kg of Al), but the process commonly requires 15.37 kWh.

Cell operation
Cells in factories are operated 24 hours per day so that the molten material in them will not solidify. Temperature within the cell is maintained via electrical resistance. Oxidation of the carbon anode increases the electrical efficiency at a cost of consuming the carbon electrodes and producing carbon dioxide.

While solid cryolite is denser than solid aluminium at room temperature, liquid aluminium is denser than molten cryolite at temperatures around 1000 °C. The aluminium sinks to the bottom of the electrolytic cell, where it is periodically collected. The liquid aluminium is removed from the cell via a siphon every 1 to 3 days in order to avoid having to use extremely high temperature valves and pumps. Alumina is added to the cells as the aluminum is removed. Collected aluminium from different cells in a factory is finally melted together to ensure uniform product and made into metal sheets. The electrolytic mixture is sprinkled with coke to prevent the anode's oxidation by the oxygen evolved.

The cell produces gases at the anode. The exhaust is primarily CO2 produced from the anode consumption and hydrogen fluoride (HF) from the cryolite and flux (AlF3). In modern facilities, fluorides are almost completely recycled to the cells and therefore used again in the electrolysis. Escaped HF can be neutralized to its sodium salt, sodium fluoride. Particulates are captured using electrostatic or bag filters. The CO2 is usually vented into the atmosphere.

Agitation of the molten material in the cell increases its production rate at the expense of an increase in cryolite impurities in the product. Properly designed cells can leverage magnetohydrodynamic forces induced by the electrolysing current to agitate the electrolyte. In non-agitating static pool cells, the impurities either rise to the top of the metallic aluminium, or sink to the bottom, leaving high-purity aluminium in the middle area.

Electrodes
Electrodes in cells are mostly coke which has been purified at high temperatures. Pitch resin or tar is used as a binder. The materials most often used in anodes, coke and pitch resin, are mainly residues from the petroleum industry and need to be of high enough purity so no impurities end up into the molten aluminum or the electrolyte.

There are two primary anode technologies using the Hall–Héroult process: Söderberg technology and prebaked technology.

In cells using Söderberg or self-baking anodes, there is a single anode per electrolysis cell. The anode is contained within a frame and, as the bottom of the anode turns mainly into CO2 during the electrolysis, the anode loses mass and, being amorphous, it slowly sinks within its frame. More material to the top of the anode is continuously added in the form of briquettes made from coke and pitch. The lost heat from the smelting operation is used to bake the briquettes into the carbon form required for the reaction with alumina. The baking process in Söderberg anodes during electrolysis releases more carcinogenic PAHs and other pollutants than electrolysis with prebaked anodes and, partially for this reason, prebaked anode-using cells have become more common in the aluminium industry. More alumina is added to the electrolyte from the sides of the Söderberg anode after the crust on top of the electrolyte mixture is broken.

Prebaked anodes are baked in very large gas-fired ovens at high temperature before being lowered by various heavy industrial lifting systems into the electrolytic solution. There are usually 24 prebaked anodes in two rows per cell. Each anode is lowered vertically and individually by a computer, as the bottom surfaces of the anodes are eaten away during the electrolysis. Compared to Söderberg anodes, computer-controlled prebaked anodes can be brought closer to the molten aluminium layer at the bottom of the cell without any of them touching the layer and interfering with the electrolysis. This smaller distance decreases the resistance caused by the electrolyte mixture and increases the efficiency of prebaked anodes over Söderberg anodes. Prebake technology also has much lower risk of the anode effect (see below), but cells using it are more expensive to build and labor-intensive to use, as each prebaked anode in a cell needs to be removed and replaced once it has been used. Alumina is added to the electrolyte from between the anodes in prebake cells.

Prebaked anodes contain a smaller percentage of pitch, as they need to be more solid than Söderberg anodes. The remains of prebaked anodes are used to make more new prebaked anodes. Prebaked anodes are either made in the same factory where electrolysis happens, or are brought there from elsewhere.

The inside of the cell's bath is lined with cathode made from coke and pitch. Cathodes also degrade during electrolysis, but much more slowly than anodes do, and thus they need neither be as high in purity, nor be maintained as often. Cathodes are typically replaced every 2–6 years. This requires the whole cell to be shut down.

Anode effect
The anode effect is a situation where too many gas bubbles form at the bottom of the anode and join, forming a layer. This increases the resistance of the cell, because smaller areas of the electrolyte touch the anode. These areas of the electrolyte and anode heat up when the density of the electric current of the cell focuses to go through only them. This heats up the gas layer and causes it to expand, thus further reducing the surface area where electrolyte and anode are in contact with each other. The anode effect decreases the energy-efficiency and the aluminium production of the cell. It also induces the formation of tetrafluoromethane (CF4) in significant quantities, increases formation of CO and, to a lesser extent, also causes the formation of hexafluoroethane (C2F6). CF4 and C2F6 are not CFCs, and, although not detrimental to the ozone layer, are still potent greenhouse gases. The anode effect is mainly a problem in Söderberg technology cells, not in prebaked.

Existing need
Aluminium is the most abundant metallic element in the Earth's crust, but it is rarely found in its elemental state. It occurs in many minerals, but its primary commercial source is bauxite, a mixture of hydrated aluminium oxides and compounds of other elements such as iron.

Prior to the Hall–Héroult process, elemental aluminium was made by heating ore along with elemental sodium or potassium in a vacuum. The method was complicated and consumed materials that were in themselves expensive at that time. This meant that the cost to produce the small amount of aluminium made in the early 19th century was very high, higher than for gold or platinum. Bars of aluminium were exhibited alongside the French crown jewels at the Exposition Universelle of 1855, and Emperor Napoleon III of France was said to have reserved his few sets of aluminium dinner plates and eating utensils for his most honored guests.

Production costs using older methods did come down, but when aluminium was selected as the material for the cap/lightning rod to sit atop the Washington Monument in Washington, D.C., it was still more expensive than silver.

Independent discovery
The Hall–Héroult process was invented independently and almost simultaneously in 1886 by the American chemist Charles Martin Hall and by the Frenchman Paul Héroult —both 22 years old. Some authors claim Hall was assisted by his sister Julia Brainerd Hall; however, the extent to which she was involved has been disputed. In 1888, Hall opened the first large-scale aluminium production plant in Pittsburgh. It later became the Alcoa corporation.

In 1997, the Hall–Héroult process was designated a National Historic Chemical Landmark by the American Chemical Society in recognition of the importance of the process in the commercialization of aluminum.

Economic impact
Aluminium produced via the Hall–Héroult process, in combination with cheaper electric power, helped make aluminium (and incidentally magnesium) an inexpensive commodity rather than a precious metal.

This, in turn, helped make it possible for pioneers like Hugo Junkers to utilize aluminium and aluminium-magnesium alloys to make items like metal airplanes by the thousands, or Howard Lund to make aluminium fishing boats. In 2012 it was estimated that 12.7 tons of CO2 emissions are generated per ton of aluminium produced.

title: Hall–Héroult process


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Hurdegaryp

Hurdegaryp (Hardegarijp ) is a village in the northern part of the Netherlands, in the municipality of Tytsjerksteradiel. Its history dates back to at least the 13th century. It had a population of around 4,788 in January 2017.

In Dutch the village is called Hardegarijp, Hurdegaryp is its West Frisian and official name.

History
The village was first mentioned in 1401 as Herdegaryp. The etymology is unclear. Originally the centre of the village was located further south. In 1830, the road to Leeuwarden was built. The Hurdegaryp railway station followed in 1866, and the centre of the village moved towards to the road and station.

The Dutch Reformed church was built in 1711 and has a wooden tower. It was a replacement for a 13th century church. Up to 1829, the Grovestins, a fortified tower was located to the south of the village.

Hurdegaryp was home to 624 people in 1840. From the 1960s onwards, it became a suburb of Leeuwarden. In 2017, five tiny houses measuring 3 by 9 metres were built in Hurdegaryp as affordable houses at €50,000.- It was an initiative of a starting real estate agent who himself could not afford to exchange his rented student room with a house. They were immediately sold.

Transportation

 * Hurdegaryp railway station

title: Hurdegaryp


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Tattoo machine

A tattoo machine (colloquially referred to as a tattoo gun) is a hand-held device generally used to create a tattoo, a permanent marking of the skin with indelible ink. Modern tattoo machines use electromagnetic coils to move an armature bar up and down. Connected to the armature bar is a barred needle grouping that opens the skin for the ink to flow into. All electromagnetic coil machines are powered by a wired regulated DC power supplies which send an electric current through the copper coils wrapped around opposing magnets and then moves the armature bar up and down. In addition to coil tattoo machines, there are also rotary tattoo machines, which are operated with regulated rotary motors and are powered by a wired external RC power supply or a wireless battery pack attached to the machine. There are many types of rotary machines, some that look similar to coil machines and some that look more like "pens". Coil machines are usually each tuned for a single function, such as for shading, or lining or packing ink. Rotary machines are multifunctional, taking any size or type of needle or cartridge needle. "The basic machine is pretty much unchanged today, in recent years variations of the theme have crept into the market, namely Manfred Kohrs' rotary machine of 1978 or Carson Hill’s pneumatic machine that uses compressed air rather than electricity, but the principle is essentially the same."

History
The predecessor to the tattoo machine was Thomas Edison's electric pen, patented under the title Stencil-Pens in Newark, New Jersey, United States in 1876. It was originally intended to be used as a duplicating device, but in 1891, Samuel O'Reilly discovered that Edison's machine could be modified and used to introduce ink into the skin, and later patented a tube and needle system to provide an ink reservoir.

While O'Reilly's machine was based on the tattoo rotary technology of Edison's device, modern tattoo machines use electromagnets. The first machine based on this technology was a single coil machine patented by Thomas Riley of London, just twenty days after O'Reilly filed the patent for his rotary machine. For his machine, Riley placed a modified doorbell assembly in a brass box. The modern two-coil configuration was patented by Alfred Charles South, also of London. Because it was so heavy, a spring was often attached to the top of the machine and the ceiling to take most of the weight off the operator's hand.

″To move tattooing forward, German tattoo artist Manfred Kohrs had to take a look backward.″ In 1978, Kohrs "introduced the first new design for a rotary machine in nearly a century. His machine was functionally similar to O'Reilly's except an electric DC motor, rather than electrified magnets, drove the needles. This slimmer and streamlined version became lighter, quieter, and more portable. It also gave artists more control while ensuring the operator's hands and fingers cramped less. While some artists gravitated to this rotary revival, others preferred to stick with their trusty coil machines."

Most modern tattoo machines can control needle depth, speed, and force of application, which has allowed tattooing to become a very precise art form. Such advances in precision have also produced a style of facial tattooing that has attained mainstream popularity in America called dermapigmentation, or "permanent cosmetics" creating results such as addition/removal of freckles, beauty spots and scars.

Classification
There are many types of machines. Liners and shaders are the more common machines from a technical standpoint. Mechanically, there are coil tattoo machines; also pneumatic machines, and rotary, or linear, tattoo machines.

 * Rotary tattoo machines were the original machines, based on rotary technology, which was invented by Samuel O'Reilly and improved by the tattoo artists through the years. Rotary type machines use an electric motor to drive the needles. Some recent upgrades include using an armature bar to increase efficiency, a characteristic of coil machines. Recently, there have been improvements to make this type of machine pneumatic, in place of the electric motor used now.
 * Coil tattoo machines are the most commonly seen and used. These machines use an electromagnetic circuit to move the needle grouping. There are many variations, from single-coiled machines to triple-coiled machines. They can be made from many different materials and in many different sizes and shapes. Dual-coiled machines are considered to be standard. The coils generally range from 8 to 10 wrap. The coils create the impedance, or resistance, used to properly regulate the machine's speed and power. This causes less trauma to the skin.
 * Liner tattoo machine: The purpose of a liner machine is to lay the ink in the skin in one single pass to create a dominant line. It uses a short contact circuit (about 1.5mm–2mm), which causes the machine to cycle faster.
 * Shader tattoo machine: The shader machine is commonly used to shade black or variants of black ink, though colors other than black are also used in this type of machine. The saturation level of this machine is low. It uses a bigger contact gap than a liner (about 2mm–3.5mm) to make it cycle slightly slower. This machine is also used for sculpting lines. Some artists will use this type of machine for all lines, as it allows the lines to be retraced with less trauma to the skin.
 * Pneumatic tattoo machine: Tattoo artist Carson Hill in the year 2000 invented the first pneumatic tattoo machine and began the patent process. A pneumatic tattoo machine is powered by an air compressor and is extremely lightweight. Pneumatic tattoo machines use pressurized air to drive the needles up and down. These tattoo machines are easily sterilized as the entire machine can be placed in an autoclave without any major disassembly (unlike traditional coil machines which require complete disassembly before being placed in an autoclave).

Tattoo machines are not limited to just these types. A common variant is having a "cutback", which uses stiffer front springs. This is more commonly used in liners, but is known to be used on shader machines, more typically for portrait work. Machines are usually categorized into long stroke and short stroke varieties. The longer-stroked machines are good for coloring and shading, as well as sculpting lines, while doing less damage to clients' skin. Shorter-stroke machines are commonly used for lining in a single pass style, and also in a shader setup to achieve a more subtle gradation of black such as would be found in portraits. Length, width, tension, angle, and stiffness of the spring varies the functionality of the machine. The contact gaps, as well as capacitors and even the style of machine and its angles of deflection, can also all be variants in machine tuning. The proper tuning of the machine is essential for the type of machine being used, also for the type of tattoo the artist is doing.

Coil Machine Mechanisms
Coil tattoo machines function by passing current through two coils which alternate electromagnetic forces to move the tattoo needle up and down rapidly. Power supplies adapt AC current to power the machine. This allows artists to control the voltage at which their machines operate at and can be manipulated to achieve line variation. Coil machines consist of about fifteen different parts including but not limited to the needle, tube, tube grip, tube clamp, needle bar, armature bar, front spring, contact screw, o-ring, rear spring, rear spring mounting screw, and a yoke. These parts are held together with a frame. Frames are constructed from a wide variety of materials such as brass, iron, steel, zinc, and aluminum. The type of frame used heavily influences the overall operation of the tattoo machine. Some frames (such as those constructed from iron or brass) reduce vibration during operation. Other lighter frames (such as those constructed from aluminum and zinc) reduce the physical strain on artists during operation. Improper frame selection/alignment affect the longevity and quality of a coil machine’s overall function. Coil tattoo machines give artists a high degree of control over every aspect of the machine’s function. This requires artists to be intimately knowledgeable about their machines to efficiently utilize such a device.

Rotary Machine Mechanisms
Rotary Tattoo machines function by using the rotational motion of an electric motor to move a needle/bundle of tattoo needles up and down rapidly. Rotary machines, like coil machines, require a power supply that adapt AC current to power the machine. This voltage may be manipulated to achieve line variation. Rotary machines consist of about seven different parts including but not limited to the grip, tube, cam wheel, “adjustment mechanism”, and a power connection. Rotary machines vary in the type of needle they use. Common rotary machines use standard needles but can also use cartridge needles. This differs from the more traditional “coil tattoo machine” as one rotary machine can be used for both lining and shading. Rotary machines can be purchased pre-aligned and generally work “out of the box”. This makes them more beginner friendly. The electric motor that drives the needle up and down reduces the operational vibration of the machine and thusly reduces the physical strain of tattooing on artists. Rotary machines of all types typically produce less noise and vibration during operation. Coil machines produce a sort of buzzing noise that has commonly been associated with tattooing. Clients sometimes feel more comfortable being tattooed using a rotary machine due to how quiet the machines are. Rotary machines come in a variety of types. Direct drive rotary machines connect the needle bar directly to the motor. Eccentric rotary machines use an eccentric cam drive that converts the rotational movement of the motor to an oscillating motion which in turn drives the needle. Pneumatic rotary machines, as their name implies, utilize compressed air to directly oscillate the tattoo needle and do not use an electric motor whatsoever.

Selected bibliography

 * Victoria Groß: Hautgravuren – Zur Individualisierung des Körpers. GRIN Verlag 2006, ISBN 3-638-57827-5.
 * Anne Fuest: Die Tätowierung – Geschichte und Bedeutung in Afrika und Deutschland: Eine kulturanthropologische Untersuchung. GRIN Verlag 2008, ISBN 3-640-21092-1.
 * Matthias Friederich: Tätowierungen in Deutschland: eine kultursoziologische Untersuchung. Band 14 von Quellen und Forschungen zur europäischen Ethnologie, Königshausen & Neumann 1993, ISBN 3-884-79774-3.
 * Erick Alayon: The Craft of Tattooing. CreateSpace Independent Publishing Platform 2006, ISBN 1-419-62591-8.

title: Tattoo machine


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Nicolas Malebranche

Nicolas Malebranche (, ; 6 August 1638 – 13 October 1715) was a French Oratorian Catholic priest and rationalist philosopher. In his works, he sought to synthesize the thought of St. Augustine and Descartes, in order to demonstrate the active role of God in every aspect of the world. Malebranche is best known for his doctrines of vision in God, occasionalism and ontologism.

Early years
Malebranche was born in Paris in 1638, the youngest child of Nicolas Malebranche, secretary to King Louis XIII of France, and Catherine de Lauzon, sister of Jean de Lauson, a Governor of New France. Because of a malformed spine, Malebranche received his elementary education from a private tutor. He left home at the age of sixteen to pursue a course of philosophy at the Collège de la Marche, and subsequently to study theology at the Collège de Sorbonne, both colleges from the University of Paris. He eventually left the Sorbonne, having rejected scholasticism, and entered the Oratory in 1660. There, he devoted himself to ecclesiastical history, linguistics, the Bible, and the works of Saint Augustine. Malebranche was ordained a priest in 1664.

In 1664, Malebranche first read Descartes' Treatise on Man, an account of the physiology of the human body. Malebranche's biographer, Father Yves André reported that Malebranche was influenced by Descartes' book because it allowed him to view the natural world without Aristotelian scholasticism. Malebranche spent the next decade studying Cartesianism.

Philosophical career
In 1674–75, Malebranche published the two volumes of his first and most extensive philosophical work. Entitled ''Concerning the Search after Truth. In which is treated the nature of the human mind and the use that must be made of it to avoid error in the sciences'' (De la recherche de la vérité. Où l’on traite de la Nature de l’Esprit de l’homme, et de l’usage qu’il en doit faire pour éviter l’erreur dans les Sciences), the book laid the foundation for Malebranche's philosophical reputation and ideas. It dealt with the causes of human error and on how to avoid such mistakes. Most importantly, in the third book, which discussed pure understanding, he defended a claim that the ideas through which we perceive objects exist in God.

Malebranche's first critic was the Abbé Simon Foucher, who attacked the Search even before its second volume had been published. Malebranche replied in a short preface added to that second volume, and then, in the 1678 third edition, he added 50% to the already considerable size of the book with a sequence of (eventually) seventeen Elucidations. These responded to further criticisms, but they also expanded on the original arguments, and developed them in new ways. In the Tenth Elucidation, for instance, Malebranche introduced his theory of "intelligible extension", a single, archetypal idea of extension into which the ideas of all particular kinds of bodies could be jointly resolved. In others, Malebranche placed a greater emphasis than he had previously done on his occasionalist account of causation, and particularly on his contention that God acted for the most part through "general volitions" and only rarely, as in the case of miracles, through "particular volitions".

Malebranche expanded on this last point in 1680 when he published Treatise on Nature and Grace. Here, he made it explicit that the generality of the laws whereby God regulated His behaviour extended not only to His activity in the natural world but also applied to His gift of grace to human beings. The book was attacked by fellow Cartesian philosopher Antoine Arnauld, and, although Arnauld's initial concerns were theological ones, the bitter dispute which ensued very quickly branched out into most other areas of their respective systems. Over the next few years, the two men wrote enough polemics against one another to fill four volumes of Malebranche's collected works and three of Arnauld's. Arnauld's supporters managed to persuade the Roman Catholic Church to place Nature and Grace on its Index of Prohibited Books in 1690, and it was followed there by the Search nineteen years later. (Ironically, the Index already contained several works by the Jansenist Arnauld himself.) Other critics with whom Malebranche entered into significant discussion include another fellow Cartesian, Pierre Sylvain Regis, as well as Dortous de Mairan. De Mairan was sympathetic to the views of Baruch Spinoza, and felt that he had found similar views in his reading of Malebranche: Malebranche assiduously resisted such an association.

Timeline

 * 1638 - Born in Paris to Nicolas Malebranche and Catherine de Lauzon.
 * 1654 - Enters the Collège de la Marche and later the Sorbonne to study philosophy and theology.
 * 1660 - Ordained as a member of the French Oratory.
 * 1664 - First reads Descartes' Treatise on Man and spends the next ten years studying philosophy.
 * 1674–75 - Publishes The Search After Truth.
 * 1678 - Adds Elucidations to new edition of the Search.
 * 1680 - Publishes Treatise of Nature and Grace.
 * 1683 - Publishes Christian and Metaphysical Meditations. Arnauld publishes On True and False Ideas, the opening salvo in their dispute.
 * 1684 - Publishes Treatise on Ethics.
 * 1688 - Publishes  Dialogues on Metaphysics and Religion (Dialogues on Metaphysics and Religion).
 * 1690 - Treatise of Nature and Grace is placed on the Index of Prohibited Books.
 * 1694 - Death of Arnauld.
 * 1708 - Publishes Dialogue Between a Christian Philosopher and a Chinese Philosopher.
 * 1709 - The Search After Truth is also placed on the Index.
 * 1713–14 - Correspondence with Jean-Jacques d'Ortous de Mairan on Spinozism.
 * 1715 - Malebranche dies.

Vision in God
Just as all human action (along with the action of any other creature) is entirely dependent on God, so too is all human cognition. Malebranche argued that human knowledge is dependent on divine understanding in a way analogous to that in which the motion of bodies is dependent on divine will. Like René Descartes, Malebranche held that humans attain knowledge through ideas – immaterial representations present to the mind. But whereas Descartes believed ideas are mental entities, Malebranche argued that all ideas exist only in God. These ideas, therefore, are uncreated and independent of finite minds. When we access them intellectually, we apprehend objective truth. Malebranche defined "truth" as a relation between ideas: since these ideas are in God, they are eternal and immutable, and consequently the only truths worthy of the name will themselves be eternal and immutable. Malebranche divided these relations between ideas into two categories: relations of magnitude and relations of quality or perfection. The former constitute "speculative" truths, such as those of geometry, while the latter constitute the "practical" truths of ethics. Ethical principles, for Malebranche, are therefore divine in their foundation, universal in their application, and to be discovered by intellectual contemplation, just as geometrical principles are.

With regard to this account of intellectual knowledge, Malebranche was more or less following Saint Augustine. His great innovation was to explain how these same divine ideas could also serve as the immediate objects of human minds in sensual perception. The problem there is that the divine ideas are universal, whereas all perception seems to be of particulars. Malebranche's solution was to suggest that, whereas the mind's intellectual conception of these ideas is pure and direct, its sensual perception of them will be modified by "sensations". These sensations, unlike the ideas, are indeed proper to individual created minds, and subsist as modes thereof. The idea will represent only the geometrical or mechanical properties of bodies (size, shape, motion), while the sensation will consist in colour or some other sensible quality. The latter will limit the mind's apprehension of the former in such a manner as to make it represent a particular individual to that mind. To a different mind, one with a different sensation, the same idea could represent a different individual of the same general kind. In the Dialogues On Metaphysics And Religion (dialogue 1), Malebranche added that the same basic structure can also account for (the mental as opposed to the physiological element in) imagination, in this case where the idea only "lightly touches" the mind.

Malebranche was strongly influenced by Descartes but did not accept his philosophy uncritically. He is noted particularly for his view that we see all things in God and for his adoption of psycho-physical parallelism and 'occasionalism' to deal with the problem of interaction between mind and body. However, his attribution of epistemological and explanatory primacy to God leads to difficulties.

(1) If we see all things in God in the sense that He puts the ideas into our minds we can have no direct knowledge of the external world. We can appeal to clear and distinct ideas as a criterion for the veridicality of judgements about physical things, but it is God who is ultimately responsible for our ideas.

(2) If all things are under the direct control of God, subject to His will, what of human freedom? Malebranche's view that we have freedom to choose but only in relation to finite goods is not convincing, denying as it does the possibility of resistance of movement towards God as the universal good. (This may be a misrepresentation of Malebranche's view; see the first chapter of The Search for Truth, where he specifies that while we cannot but desire the good in general, we are free to apply that love to particulars, and can do so in a disordered fashion that leads to sin. His account is no different from St. Augustine's in this regard.)

(3) In so far as God is not to be identified with the archetypal eternal truths in his mind, Malebranche is not a pantheist. But, as in mediaeval philosophy, this gives rise to the problem of reconciling God's freedom with His supposed immutability.

Theodicy
Malebranche's theodicy is his solution to the problem of evil. Although he conceded that God had the power to create a more perfect world, free from all defects, such a world would have necessitated a greater complexity in divine ways. Thus, God produces the natural evils that follow from simple laws not because he wills those particular effects, but because he wills a world that best reflects his wisdom by achieving the best possible balance between the intrinsic perfection of the work and the simplicity and generality of its laws.

Malebranche's dualism
Whereas Malebranche followed Augustine in his description of intellectual knowledge, in his approach to mind–body problems he began as a follower of Descartes. But in contrast to Descartes, who considered it possible to form a clear and distinct idea of the mind, Malebranche argues in the Dialogues on Metaphysics, a dialogue between Theodore and Aristes, that we do not have a complete conception of the powers of the mind, and thus no clear conception of the nature of the mind.

I am unable, when I turn to myself, to recognize any of my faculties or my capacities. The inner sensation which I have of myself informs me that I am, that I think, that I will, that I have sensory awareness, that I suffer, and so on; but it provides me with no knowledge whatever of what I am – of the nature of my thought, my sensations, my passions, or my pain – or of the mutual relations that obtain between all these things. ...I have no idea whatever of my soul.

This leads Theodore to declare that 'I am not my own light to myself'; the nature of our own minds is highly obscure. What is more, with regard to psycho-physical interaction, Malebranche argues that body could not act on mind, nor mind on body. The only active power (hence the only efficient cause of change in the world) is God. When I will that my arm should rise, my volition is the "occasion" or the "occasional cause" of the movement of my arm; the efficient cause of both my volition and the movement of my arm is God. Malebranche's doctrine, which could be found in contemporary commentaries on Aristotle, and which first appeared in certain Arab philosophers, is therefore called "occasionalism".

Occasionalism
In general, occasionalism is the view that there are no efficient causes in the full sense other than God. Created things are at best "occasions" for divine activity. Bodies and minds act neither on themselves nor on each other; God alone brings about all the phenomena of nature and the mind. Changes occurring in created things will exhibit regularities (and will thus satisfy a Humean definition of causation) because God in creating the world observes what Malebranche calls "order": he binds himself to act according to laws of nature chosen in accordance with his general will that the world be as good as possible, and thus (for example) that the laws be simple and few in number.

In particular, there will be laws governing what we would customarily call the "interaction" of body and mind, so that similar movements in the body will "occasion" similar ideas in the mind. That relation has some features of the causal relation (it satisfies, for example, universal conditionals of the form "Whenever C occurs, E occurs"). But in reality both the idea in the mind and the movement in the body are caused by God.

Scientific contributions
Although better known for his philosophical work, Malebranche made some notable contributions to physics, working within a broadly Cartesian framework but nevertheless prepared to depart from Descartes where necessary. In 1699, he delivered an address to the Académie Royale des Sciences on the nature of light and color, wherein he argued that different colors resulted out of different frequencies in the pressure vibrations of subtle matter, much as different musical tones derived from different frequencies in the vibrations of air. His theory was presented as a corrective to Descartes' view, rather than a refutation thereof, but it has important parallels with the rival optical theory of Isaac Newton. Newton had already developed his position some thirty years earlier, but Malebranche probably would not have been aware of it until it was finally published in the Opticks of 1704, or, more likely, in its Latin translation of 1706. When Malebranche revised his 1699 paper for inclusion as the Sixteenth Elucidation of the 1712 edition of The Search After Truth, he inserted a number of references to "Newton's excellent work".

In addition, Malebranche wrote on the laws of motion, a topic he discussed extensively with Leibniz. He also wrote on mathematics and, although he made no major mathematical discoveries of his own, he was instrumental in introducing and disseminating the contributions of Descartes and Leibniz in France. Malebranche introduced l'Hôpital to Johann Bernoulli, with the ultimate result being the publication of the first textbook in infinitesimal calculus.

Malebranche also developed an original theory related to preformationism, postulating that each embryo probably contained even smaller embryos ad infinitum, like an idealized Matryoshka doll. According to Malebranche, "an infinite series of plants and animals were contained within the seed or the egg, but only naturalists with sufficient skill and experience could detect their presence" (Magner 158–9).

Legacy
Aside, perhaps, from John Norris (who, in any case, drew at least as much from Malebranche's own sources, primarily Saint Augustine, as he did from Malebranche himself), there are few if any philosophers who can be considered faithful followers of Malebranche in all matters. He was, however, held in widespread high regard within his own lifetime and for some time afterwards, and the influence of certain of his ideas can be discerned in the works of several important figures.

Pierre Bayle regarded Malebranche as "one of the greatest philosophers of this age" (though, admittedly, not as the greatest, as is often reported). In note H to his "Zeno of Elea" article, Bayle discussed Malebranche's views on material substance with particular approval. Occasionalism and the vision in God seem to make the real existence of material substance redundant. Not only is it unable to be directly perceived, but it cannot actually affect us or anything else in any way at all. Descartes had also maintained that matter was not directly perceivable, but he had argued that the veracity of God could support a proof of its certain existence. Malebranche, however, weakened Descartes' argument, concluding that, from a philosophical point of view, its existence could only be shown to be probable. Bayle pushed even further down this same path, thereby laying much of the ground work for the immaterialism of George Berkeley. Berkeley, influenced both by Bayle and directly by Malebranche himself, simply took the final step to a full denial of the existence of material substance. (Arthur Collier, who was also influenced directly by Malebranche, and by Norris, made the same move at around the same time as Berkeley did, but, it would appear, entirely independently of him.) Berkeley, admittedly, did reject the theory of vision in God. "It is evident", he insisted, "that the things I perceive are my own ideas." But he was influenced by Malebranche's occasionalism, even though he excluded the activity of created minds from its domain. In addition, Berkeley agreed with Malebranche, against Descartes, that we could not achieve a clear idea of the mind itself. John Locke had also argued for this, but he had made no distinction between minds and bodies on this point, whereas both Berkeley and Malebranche maintained (each in his own way) that we could have ideas of bodies but not of minds.

Gottfried Wilhelm Leibniz (who met Malebranche in Paris in about 1675 and corresponded with him thereafter) also rejected the vision in God, and his theory of pre-established harmony was designed as a new alternative to occasionalism as well as to the more traditional theory of efficient causal interaction. However, in his own theodicy, even if it was somewhat more elaborate than Malebranche's, he did at least agree with Malebranche's fundamental contention that the simplicity of God's ways had to be given as much regard as the world's perfection.

David Hume supported and drew upon Malebranche's negative arguments to show that no genuine causal connections could be conceived between distinct mundane entities. However, when it came to finding a positive replacement for such causal connections, he turned inwards to the workings of the human mind, instead of turning upwards to God. With regard to this second half of Malebranche's occasionalism, Hume wrote:

 * We are got into fairy land, long ere we have reached the last steps of our theory. ...Our line is too short to fathom such immense abysses.

Hume's empiricist epistemology led him to distrust Malebranche's confidence in discovering abstruse metaphysical truths through an intellectual union with God. Likewise, Locke felt that Malebranche's metaphysical speculations lacked a proper foundation, and, though ingenious, were ultimately unintelligible. In a somewhat similar manner, Arthur Schopenhauer regarded the theory of vision in God as "explaining something unknown by something even more unknown."

Locke withheld his "An Examination of P[ère] Malebranche's Opinion Of Seeing All Things In God" from publication, "because he looked upon it to be an opinion that would not spread but was like to die of itself, or at least to do no great harm." Much as Locke predicted, Malebranche's reputation outside France (where he always enjoyed high esteem) did begin to diminish during the 18th century, and remained low thereafter. However, over the last three or four decades, Malebranche's work has drawn renewed and ever-increasing interest. Several of his works have been translated into English for the first time, as scholars have been reassessing his ideas. Many have begun to argue that the originality and unity of his philosophical system merits him a place alongside such figures as Descartes, Spinoza, and Leibniz.

title: Nicolas Malebranche


text:
Best alternative to a negotiated agreement

In negotiation theory, the best alternative to a negotiated agreement or BATNA (no deal option) refers to the most advantageous alternative course of action a party can take if negotiations fail and an agreement cannot be reached. The BATNA could include diverse situations, such as suspension of negotiations, transition to another negotiating partner, appeal to the court's ruling, the execution of strikes, and the formation of other forms of alliances. BATNA is the key focus and the driving force behind a successful negotiator. A party should generally not accept a worse resolution than its BATNA. Care should be taken, however, to ensure that deals are accurately valued, taking into account all considerations, such as relationship value, time value of money and the likelihood that the other party will live up to their side of the bargain. These other considerations are often difficult to value since they are frequently based on uncertain or qualitative considerations rather than easily measurable and quantifiable factors.

Oftentimes, it is even more difficult to determine the BATNA of the other party. However, this information is crucial as the BATNA determines the other side's negotiation power. Sometimes, conclusions can be drawn by determining his/her main interests and the negotiation itself can be used to verify or falsify the assumptions. If, for example, it is assumed that a very early delivery date is of key importance to the negotiating partner, deliberately setting a later delivery date can be proposed. If this late delivery date is decidedly rejected, the desired delivery date is likely to be of great importance.

History
Best Alternative to a Negotiated Agreement was developed by negotiation researchers Roger Fisher and William Ury of the Harvard Program on Negotiation (PON), in their series of books on principled negotiation that started with Getting to YES, equivalent to the game theory concept of a disagreement point from bargaining problems pioneered by Nobel Laureate John Forbes Nash decades earlier. A Nash Equilibrium is reached among a group of players when no player can benefit from changing strategies if every other player sticks to their current strategy. For example, Amy and Phil are in Nash Equilibrium if Amy is making the best decision she can while taking into account Phil's decision, and if Phil is making the best decision he can while taking into account Amy's decision. Likewise, a group of players are in Nash Equilibrium if each one is making the best decision that he or she can while taking into account the decisions of others.

Overview
The BATNA is often seen by negotiators not as a safety net, but rather as a point of leverage in negotiations. Although a negotiator's alternative options should, in theory, be straightforward to evaluate, the effort to understand which alternative represents a party's BATNA is often not invested. Options need to be actual and actionable to be of value, however, without the investment of time, options will frequently be included that fail on one of these criteria. Most managers overestimate their BATNA whilst simultaneously investing too little time into researching their real options. This can result in poor or faulty decision making and negotiation outcomes. Negotiators also need to be aware of the other negotiator's BATNA and to identify how it compares to what they are offering.

Some people may adopt aggressive, coercive, threatening and/or deceptive techniques. This is known as a hard negotiation style; a theoretical example of this is adversarial approach style negotiation. Others may employ a soft style, which is friendly, trusting, compromising, and conflict avoiding. According to Fisher and Ury, when hard negotiators meet soft negotiators, the hard negotiators usually win their position, but at the cost of potentially damaging the long-term relationship between the parties.

Attractive alternatives are needed to develop a strong BATNA. In the book Getting to YES: Negotiating Agreement Without Giving In, the authors give three suggestions for how to accomplish this:

 * 1) Creating a list of actions one might take if no agreement is reached
 * 2) Converting some of the more promising ideas and transforming them into tangible and partial alternatives
 * 3) Selecting the alternative that sounds best

In negotiations involving different cultures, all parties need to account for cultural cognitive behaviours and should not let judgments and biases affect the negotiation. The individual should be separate from the objective.

The purpose here, as Philip Gulliver mentions, is for negotiation parties to be aware.

Preparation at all levels, including prejudice-free thoughts, emotion-free behaviour, bias-free behaviour are helpful according to Morris and Gelfand.

BATNA-based tactics and order of negotiations
Because of the importance of the BATNA for negotiation success, several tactics focus on weakening the opponent's BATNA. This may be achieved e.g. by striving for exclusive negotiations, delaying or accelerating the ongoing negotiations, or limiting the negotiation partner to technical systems. If a negotiator is faced with such tactics it is his/her task to examine the possible consequences for their own BATNA and to prevent or counteract any deterioration of the own party's BATNA.

The BATNA can also influence the order in which negotiations are taken up with potential contracting partners. A sequential approach is favourable. In a sequential approach, one starts negotiations with the less favoured partners and continues negotiations with the preferred option afterward. This way, there is the BATNA of a contract with the less favoured partner.

Difference between BATNA and reservation value
A BATNA represents one party's best option if negotiations fail, whereas a reservation value represents the worst deal they are willing to accept. A reservation value should never be less than the BATNA. When purchasing a bike, for example, the BATNA may represent the option of shopping at another dealer. Depending on the cost of finding other dealers, the reservation value would represent the highest price you are willing to pay.

Process to develop the best alternative to a negotiated agreement

 * 1) First, list out all possible alternatives to the current negotiation - what could you do if the negotiations fail?
 * 2) Determine the worth of each alternative - how much does each alternative mean to you?
 * 3) Go with the option that provides the most value to you (this is your best alternative to a negotiated agreement)
 * 4) Calculate the lowest valued-deal you're willing to accept after determining your BATNA.

Types of BATNA
There are three types of BATNA:
 * 1) Walk-away BATNA
 * 2) Interactive BATNA
 * 3) Third-party BATNA

Walk-away BATNA
If the seller does not want to drop the asking price to less than an alternative option, the buyer will walk away and buy the other alternative. Professional negotiators and researchers alike regard BATNA, or “walk away” outcome as the primary source of relative power for a negotiator. However, relying on alternatives can be risky. A party's relative power in a negotiation is their ability to use resources to influence the circumstances of another, and the role of a BATNA in this regard can range from significant to non-existent.

Interactive BATNA
Interactive BATNAs are sought when one or more parties in a negotiation are not cooperating with the other parties. There are many types of interactive BATNAs, but the three primary ones are:

 * 1) Economic: For example, a newspaper forgetting to replace an inappropriate ad in one issue resulting in 50% of their subscribers unsubscribing and them losing half of their ad revenue. In this case, the unsubscribers would be the non-cooperative party.
 * 2) Political: For example, a political party filibustering a piece of legislation another party is trying to pass. In this case, the party filibustering would be the non-cooperative one.
 * 3) Social: For example, a group of protestors not succumbing to the police's attempts at displacement. In this case, the protestors would be the non-cooperative party.

Third-party BATNA
Third-party BATNAs are sought when two parties in a negotiation are unable to come to a common conclusion on their own or the dispute between them is endless. So, a third party is required in the form of either:
 * 1) Mediation: A neutral third party is brought in to help the disputing parties resolve the dispute on their own. Mediation does not resolve the dispute, they just facilitate it.
 * 2) Arbitration: A neutral third party is brought in to arbitrate or resolve the negotiation through a BATNA. Unlike mediation, they resolve the issue.
 * 3) Litigation: When the negotiation takes a turn for the worst, an authoritative third party intervenes in the form of law and the issue is resolved in court with both parties having to comply with the court's decision.

In popular use
With tongue-in-cheek, American speculative fiction writer Neal Stephenson places the invention of BATNA in 17th century by putting the following words in his novel The Confusion: "“I learnt it from English traders in Surat,” said the befuddled Surendranath, “It stands for Best Alternative To a Negotiated Agreement.”" Another character of the novel responds: “To have a BATNA is good and wise, as Surendranath has pointed out. But the Negotiated Agreement is much better than this Best Alternative.”


text:
Spherical geometry

Spherical geometry or spherics  is the geometry of the two-dimensional surface of a sphere or the $n$-dimensional surface of higher dimensional spheres.

Long studied for its practical applications to astronomy, navigation, and geodesy, spherical geometry and the metrical tools of spherical trigonometry are in many respects analogous to Euclidean plane geometry and trigonometry, but also have some important differences.

The sphere can be studied either extrinsically as a surface embedded in 3-dimensional Euclidean space (part of the study of solid geometry), or intrinsically using methods that only involve the surface itself without reference to any surrounding space.

Principles
In plane (Euclidean) geometry, the basic concepts are points and (straight) lines. In spherical geometry, the basic concepts are point and great circle. However, two great circles on a plane intersect in two antipodal points, unlike coplanar lines in Elliptic geometry.

In the extrinsic 3-dimensional picture, a great circle is the intersection of the sphere with any plane through the center. In the intrinsic approach, a great circle is a geodesic; a shortest path between any two of its points provided they are close enough. Or, in the (also intrinsic) axiomatic approach analogous to Euclid's axioms of plane geometry, "great circle" is simply an undefined term, together with postulates stipulating the basic relationships between great circles and the also-undefined "points". This is the same as Euclid's method of treating point and line as undefined primitive notions and axiomatizing their relationships.

Great circles in many ways play the same logical role in spherical geometry as lines in Euclidean geometry, e.g., as the sides of (spherical) triangles. This is more than an analogy; spherical and plane geometry and others can all be unified under the umbrella of geometry built from distance measurement, where "lines" are defined to mean shortest paths (geodesics). Many statements about the geometry of points and such "lines" are equally true in all those geometries provided lines are defined that way, and the theory can be readily extended to higher dimensions. Nevertheless, because its applications and pedagogy are tied to solid geometry, and because the generalization loses some important properties of lines in the plane, spherical geometry ordinarily does not use the term "line" at all to refer to anything on the sphere itself. If developed as a part of solid geometry, use is made of points, straight lines and planes (in the Euclidean sense) in the surrounding space.

In spherical geometry, angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects; for example, the sum of the interior angles of a spherical triangle exceeds 180 degrees.

Relation to similar geometries
Because a sphere and a plane differ geometrically, (intrinsic) spherical geometry has some features of a non-Euclidean geometry and is sometimes described as being one. However, spherical geometry was not considered a full-fledged non-Euclidean geometry sufficient to resolve the ancient problem of whether the parallel postulate is a logical consequence of the rest of Euclid's axioms of plane geometry, because it requires another axiom to be modified. The resolution was found instead in elliptic geometry, to which spherical geometry is closely related, and hyperbolic geometry; each of these new geometries makes a different change to the parallel postulate.

The principles of any of these geometries can be extended to any number of dimensions.

An important geometry related to that of the sphere is that of the real projective plane; it is obtained by identifying antipodal points (pairs of opposite points) on the sphere. Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it is non-orientable, or one-sided, and unlike the sphere it cannot be drawn as a surface in 3-dimensional space without intersecting itself.

Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas.

Greek antiquity
The earliest mathematical work of antiquity to come down to our time is On the rotating sphere (Περὶ κινουμένης σφαίρας, Peri kinoumenes sphairas) by Autolycus of Pitane, who lived at the end of the fourth century BC.

Spherical trigonometry was studied by early Greek mathematicians such as Theodosius of Bithynia, a Greek astronomer and mathematician who wrote Spherics, a book on the geometry of the sphere, and Menelaus of Alexandria, who wrote a book on spherical trigonometry called Sphaerica and developed Menelaus' theorem.

Islamic world
The Book of Unknown Arcs of a Sphere written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle.

The book On Triangles by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe. However, Gerolamo Cardano noted a century later that much of its material on spherical trigonometry was taken from the twelfth-century work of the Andalusi scholar Jabir ibn Aflah.

Euler's work
Leonhard Euler published a series of important memoirs on spherical geometry:
 * L. Euler, Principes de la trigonométrie sphérique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9 (1753), 1755, p. 233–257; Opera Omnia, Series 1, vol. XXVII, p. 277–308.
 * L. Euler, Eléments de la trigonométrie sphéroïdique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9 (1754), 1755, p. 258–293; Opera Omnia, Series 1, vol. XXVII, p. 309–339.
 * L. Euler, De curva rectificabili in superficie sphaerica, Novi Commentarii academiae scientiarum Petropolitanae 15, 1771, pp. 195–216; Opera Omnia, Series 1, Volume 28, pp. 142–160.
 * L. Euler, De mensura angulorum solidorum, Acta academiae scientiarum imperialis Petropolitinae 2, 1781, p. 31–54; Opera Omnia, Series 1, vol. XXVI, p. 204–223.
 * L. Euler, Problematis cuiusdam Pappi Alexandrini constructio, Acta academiae scientiarum imperialis Petropolitinae 4, 1783, p. 91–96; Opera Omnia, Series 1, vol. XXVI, p. 237–242.
 * L. Euler, Geometrica et sphaerica quaedam, Mémoires de l'Académie des Sciences de Saint-Pétersbourg 5, 1815, p. 96–114; Opera Omnia, Series 1, vol. XXVI, p. 344–358.
 * L. Euler, Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta academiae scientiarum imperialis Petropolitinae 3, 1782, p. 72–86; Opera Omnia, Series 1, vol. XXVI, p. 224–236.
 * L. Euler, Variae speculationes super area triangulorum sphaericorum, Nova Acta academiae scientiarum imperialis Petropolitinae 10, 1797, p. 47–62; Opera Omnia, Series 1, vol. XXIX, p. 253–266.

Properties
Spherical geometry has the following properties:
 * Any two great circles intersect in two diametrically opposite points, called antipodal points.
 * Any two points that are not antipodal points determine a unique great circle.
 * There is a natural unit of angle measurement (based on a revolution), a natural unit of length (based on the circumference of a great circle) and a natural unit of area (based on the area of the sphere).
 * Each great circle is associated with a pair of antipodal points, called its poles which are the common intersections of the set of great circles perpendicular to it. This shows that a great circle is, with respect to distance measurement on the surface of the sphere, a circle: the locus of points all at a specific distance from a center.
 * Each point is associated with a unique great circle, called the polar circle of the point, which is the great circle on the plane through the centre of the sphere and perpendicular to the diameter of the sphere through the given point.

As there are two arcs determined by a pair of points, which are not antipodal, on the great circle they determine, three non-collinear points do not determine a unique triangle. However, if we only consider triangles whose sides are minor arcs of great circles, we have the following properties:
 * The angle sum of a triangle is greater than 180° and less than 540°.
 * The area of a triangle is proportional to the excess of its angle sum over 180°.
 * Two triangles with the same angle sum are equal in area.
 * There is an upper bound for the area of triangles.
 * The composition (product) of two reflections-across-a-great-circle may be considered as a rotation about either of the points of intersection of their axes.
 * Two triangles are congruent if and only if they correspond under a finite product of such reflections.
 * Two triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent).

Relation to Euclid's postulates
If "line" is taken to mean great circle, spherical geometry only obeys two of Euclid's five postulates: the second postulate ("to produce [extend] a finite straight line continuously in a straight line") and the fourth postulate ("that all right angles are equal to one another"). However, it violates the other three. Contrary to the first postulate ("that between any two points, there is a unique line segment joining them"), there is not a unique shortest route between any two points (antipodal points such as the north and south poles on a spherical globe are counterexamples); contrary to the third postulate, a sphere does not contain circles of arbitrarily great radius; and contrary to the fifth (parallel) postulate, there is no point through which a line can be drawn that never intersects a given line.

A statement that is equivalent to the parallel postulate is that there exists a triangle whose angles add up to 180°. Since spherical geometry violates the parallel postulate, there exists no such triangle on the surface of a sphere. The sum of the angles of a triangle on a sphere is 180°(1 + 4f), where f is the fraction of the sphere's surface that is enclosed by the triangle. For any positive value of f, this exceeds 180°.

title: Spherical geometry