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Helen Keller

Helen Adams Keller (June 27, 1880 – June 1, 1968) was an American author, disability rights advocate, political activist and lecturer. Born in West Tuscumbia, Alabama, she lost her sight and her hearing after a bout of illness when she was 19 months old. She then communicated primarily using home signs until the age of seven, when she met her first teacher and life-long companion Anne Sullivan. Sullivan taught Keller language, including reading and writing. After an education at both specialist and mainstream schools, Keller attended Radcliffe College of Harvard University and became the first deafblind person in the United States to earn a Bachelor of Arts degree.

Keller was also a prolific author, writing 14 books and hundreds of speeches and essays on topics ranging from animals to Mahatma Gandhi. Keller campaigned for those with disabilities, for women's suffrage, labor rights, and world peace. In 1909, she joined the Socialist Party of America (SPA). She was a founding member of the American Civil Liberties Union (ACLU).

Keller's autobiography, The Story of My Life (1903), publicized her education and life with Sullivan. It was adapted as a play by William Gibson, and this was also adapted as a film under the same title, The Miracle Worker. Her birthplace has been designated and preserved as a National Historic Landmark. Since 1954 it has been operated as a house museum and sponsors an annual "Helen Keller Day".

Early childhood and illness

Keller was born on June 27, 1880, in Tuscumbia, Alabama, the daughter of Arthur Henley Keller (1836–1896), and Catherine Everett (Adams) Keller (1856–1921), known as "Kate". Her family lived on a homestead, Ivy Green, that Helen's paternal grandfather had built decades earlier. She had four siblings: two full siblings, Mildred Campbell (Keller) Tyson and Phillip Brooks Keller; and two older half-brothers from her father's first marriage, James McDonald Keller and William Simpson Keller.

Keller's father worked for many years as an editor of the Tuscumbia North Alabamian. He had served as a captain in the Confederate Army. The family was part of the slaveholding elite before the American Civil War, but lost status later. Her mother was the daughter of Charles W. Adams, a Confederate general.

Keller's paternal lineage was traced to Casper Keller, a native of Switzerland. One of Helen's Swiss ancestors was the first teacher for the deaf in Zürich. Keller reflected on this fact in her first autobiography, asserting that "there is no king who has not had a slave among his ancestors, and no slave who has not had a king among his".

At 19 months old, Keller contracted an unknown illness described by doctors as "an acute congestion of the stomach and the brain". Contemporary doctors believe it might have been meningitis, caused by the bacterium Neisseria meningitidis (meningococcus), or possibly Haemophilus influenzae. (This could have caused the same symptoms, but is a less likely cause due to its 97% juvenile mortality rate at that time.). She was able to recover from her illness, but was left permanently blind and deaf, as she recalled in her autobiography, "at sea in a dense fog".

At that time, Keller was able to communicate somewhat with Martha Washington, who was two years older and the daughter of the family cook, and understood the girl's signs; by the age of seven, Keller had more than 60 home signs to communicate with her family, and could distinguish people by the vibration of their footsteps.

In 1886, Keller's mother, inspired by an account in Charles Dickens' American Notes of the successful education of Laura Bridgman, a deaf and blind woman, dispatched the young Keller and her father to consult physician J. Julian Chisholm, an eye, ear, nose and throat specialist in Baltimore, for advice. Chisholm referred the Kellers to Alexander Graham Bell, who was working with deaf children at the time. Bell advised them to contact the Perkins Institute for the Blind, the school where Bridgman had been educated. It was then located in South Boston. Michael Anagnos, the school's director, asked Anne Sullivan, a 20-year-old alumna of the school who was visually impaired, to become Keller's instructor. It was the beginning of a nearly 50-year-long relationship: Sullivan developed as Keller's governess and later her companion.

Sullivan arrived at Keller's house on March 5, 1887, a day Keller would forever remember as "my soul's birthday". Sullivan immediately began to teach Helen to communicate by spelling words into her hand, beginning with "d-o-l-l" for the doll that she had brought Keller as a present. Keller initially struggled with lessons since she could not comprehend that every object had a word identifying it. When Sullivan was trying to teach Keller the word for "mug", Keller became so frustrated she broke the mug. Keller remembered how she soon began imitating Sullivan's hand gestures: "I did not know that I was spelling a word or even that words existed. I was simply making my fingers go in monkey-like imitation."

The next month Keller made a breakthrough, when she realized that the motions her teacher was making on the palm of her hand, while running cool water over her other hand, symbolized the idea of "water". Writing in her autobiography, The Story of My Life, Keller recalled the moment:

"I stood still, my whole attention fixed upon the motions of her fingers. Suddenly I felt a misty consciousness as of something forgotten—a thrill of returning thought; and somehow the mystery of language was revealed to me. I knew then that w-a-t-e-r meant the wonderful cool something that was flowing over my hand. The living word awakened my soul, gave it light, hope, set it free!" Keller quickly demanded that Sullivan sign the names of all the other familiar objects in her world.

Helen Keller was viewed as isolated but was very in touch with the outside world. She was able to enjoy music by feeling the beat and she was able to have a strong connection with animals through touch. She was delayed at picking up language, but that did not stop her from having a voice.

Formal education
In May 1888, Keller started attending the Perkins Institute for the Blind. In 1893, Keller, along with Sullivan, attended William Wade House and Finishing School. In 1894, Keller and Sullivan moved to New York to attend the Wright-Humason School for the Deaf, and to learn from Sarah Fuller at the Horace Mann School for the Deaf. In 1896, they returned to Massachusetts, and Keller entered The Cambridge School for Young Ladies before gaining admittance, in 1900, to Radcliffe College of Harvard University, where she lived in Briggs Hall, South House. Her admirer, Mark Twain, had introduced her to Standard Oil magnate Henry Huttleston Rogers, who, with his wife Abbie, paid for her education. In 1904, at the age of 24, Keller graduated as a member of Phi Beta Kappa from Radcliffe, becoming the first deafblind person to earn a Bachelor of Arts degree. She maintained a correspondence with the Austrian philosopher and pedagogue Wilhelm Jerusalem, who was one of the first to discover her literary talent.

Determined to communicate with others as conventionally as possible, Keller learned to speak and spent much of her life giving speeches and lectures on aspects of her life. She learned to "hear" people's speech using the Tadoma method, which means using her fingers to feel the lips and throat of the speaker. She became proficient at using braille and using fingerspelling to communicate. Shortly before World War I, with the assistance of the Zoellner Quartet, she determined that by placing her fingertips on a resonant tabletop she could experience music played close by.

Companions
Anne Sullivan stayed as a companion to Helen Keller long after she taught her. Sullivan married John Macy in 1905, and her health started failing around 1914. Polly Thomson (February 20, 1885 – March 21, 1960) was hired to keep house. She was a young woman from Scotland who had no experience with deaf or blind people. She progressed to working as a secretary as well, and eventually became a constant companion to Keller.

Keller moved to Forest Hills, Queens, together with Sullivan and Macy, and used the house as a base for her efforts on behalf of the American Foundation for the Blind. "While in her thirties Helen had a love affair, became secretly engaged, and defied her teacher and family by attempting an elopement with the man she loved." He was the fingerspelling socialist "Peter Fagan, a young Boston Herald reporter who was sent to Helen's home to act as her private secretary when lifelong companion, Anne, fell ill." At the time, her father had died and Sullivan was recovering in Lake Placid and Puerto Rico. Keller had moved with her mother in Montgomery, Alabama.

Anne Sullivan died in 1936, with Keller holding her hand, after falling into a coma as a result of coronary thrombosis. Keller and Thomson moved to Connecticut. They traveled worldwide and raised funds for the blind. Thomson had a stroke in 1957 from which she never fully recovered and died in 1960. Winnie Corbally, a nurse originally hired to care for Thomson in 1957, stayed on after Thomson's death and was Keller's companion for the rest of her life.

Career, writing and political activities

On January 22, 1916, Keller and Sullivan traveled to the small town of Menomonie in western Wisconsin to deliver a lecture at the Mabel Tainter Memorial Building. Details of her talk were provided in the weekly Dunn County News on January 22, 1916:"A message of optimism, of hope, of good cheer, and of loving service was brought to Menomonie Saturday—a message that will linger long with those fortunate enough to have received it. This message came with the visit of Helen Keller and her teacher, Mrs. John Macy, and both had a hand in imparting it Saturday evening to a splendid audience that filled The Memorial. The wonderful girl who has so brilliantly triumphed over the triple afflictions of blindness, dumbness and deafness, gave a talk with her own lips on 'Happiness', and it will be remembered always as a piece of inspired teaching by those who heard it."

Keller became a world-famous speaker and author. She was an advocate for people with disabilities, amid numerous other causes. She traveled to twenty-five different countries giving motivational speeches about Deaf people's conditions. She was a suffragist, pacifist, radical socialist, birth control supporter, and opponent of Woodrow Wilson. In 1915, she and George A. Kessler founded the Helen Keller International (HKI) organization. This organization is devoted to research in vision, health, and nutrition. In 1916, she sent money to the NAACP, as she was ashamed of the Southern un-Christian treatment of "colored people".

In 1920, Keller helped to found the American Civil Liberties Union (ACLU). She traveled to over 40 countries with Sullivan, making several trips to Japan and becoming a favorite of the Japanese people. Keller met every U.S. president from Grover Cleveland to Lyndon B. Johnson and was friends with many famous figures, including Alexander Graham Bell, Charlie Chaplin and Mark Twain. Keller and Twain were both considered political radicals allied with leftist politics.

Keller, who believed that the poor were "ground down by industrial oppression", wanted children born into poor families to have the same opportunities to succeed that she had enjoyed. She wrote, "I owed my success partly to the advantages of my birth and environment. I have learned that the power to rise is not within the reach of everyone."

In 1909 Keller became a member of the Socialist Party; she actively campaigned and wrote in support of the working class from 1909 to 1921. Many of her speeches and writings were about women's right to vote and the effects of war; in addition, she supported causes that opposed military intervention. She had speech therapy to have her voice understood better by the public. When the Rockefeller-owned press refused to print her articles, she protested until her work was finally published.

She supported Socialist Party candidate Eugene V. Debs in each of his campaigns for the presidency. Before reading Progress and Poverty by Henry George, Helen Keller was already a socialist who believed that Georgism was a good step in the right direction. She later wrote of finding "in Henry George's philosophy a rare beauty and power of inspiration, and a splendid faith in the essential nobility of human nature".

Keller claimed that newspaper columnists who had praised her courage and intelligence before she expressed her socialist views now called attention to her disabilities. The editor of the Brooklyn Eagle wrote that her "mistakes sprung out of the manifest limitations of her development". Keller responded to that editor, referring to having met him before he knew of her political views: "At that time the compliments he paid me were so generous that I blush to remember them. But now that I have come out for socialism he reminds me and the public that I am blind and deaf and especially liable to error. I must have shrunk in intelligence during the years since I met him. ... Oh, ridiculous Brooklyn Eagle! Socially blind and deaf, it defends an intolerable system, a system that is the cause of much of the physical blindness and deafness which we are trying to prevent."

In 1912, Keller joined the Industrial Workers of the World (the IWW, known as the Wobblies), saying that parliamentary socialism was "sinking in the political bog". She wrote for the IWW between 1916 and 1918. In Why I Became an IWW, Keller explained that her motivation for activism came in part from her concern about blindness and other disabilities:

"I was appointed on a commission to investigate the conditions of the blind. For the first time I, who had thought blindness a misfortune beyond human control, found that too much of it was traceable to wrong industrial conditions, often caused by the selfishness and greed of employers. And the social evil contributed its share. I found that poverty drove women to a life of shame that ended in blindness."

The last sentence refers to prostitution and syphilis, the former a "life of shame" that women used to support themselves, which contributed to their contracting syphilis. Untreated, it was a leading cause of blindness. In the same interview, Keller also cited the 1912 strike of textile workers in Lawrence, Massachusetts for instigating her support of socialism.

Keller supported eugenics which had become popular with new understandings (as well as misapprehensions) of principles of biological inheritance. In 1915, she wrote in favor of refusing life-saving medical procedures to infants with severe mental impairments or physical deformities, saying that their lives were not worthwhile and they would likely become criminals. Keller also expressed concerns about human overpopulation.

From 1946 to 1957 Keller visited 35 countries. In 1948 she went to New Zealand and visited deaf schools in Christchurch and Auckland. She met Deaf Society of Canterbury Life Member Patty Still in Christchurch.

Works
Keller wrote a total of 12 published books and several articles.

One of her earliest pieces of writing, at age 11, was The Frost King (1891). There were allegations that this story had been plagiarized from The Frost Fairies by Margaret Canby. An investigation into the matter revealed that Keller may have experienced a case of cryptomnesia, which was that she had Canby's story read to her but forgot about it, while the memory remained in her subconscious.

At age 22, Keller published her autobiography, The Story of My Life (1903), with help from Sullivan and Sullivan's husband, John Macy. It recounts the story of her life up to age 21 and was written during her time in college.

In an article Keller wrote in 1907, she brought to public attention the fact that many cases of childhood blindness could be prevented by washing the eyes of every newborn baby with a disinfectant solution. At the time, only a fraction of doctors and midwives were doing this. Thanks to Keller's advocacy, this commonsense public health measure was swiftly and widely adopted.

Keller wrote The World I Live In in 1908, giving readers an insight into how she felt about the world. Out of the Dark, a series of essays on socialism, was published in 1913.

When Keller was young, Anne Sullivan introduced her to Phillips Brooks, who introduced her to Christianity, Keller famously saying: "I always knew He was there, but I didn't know His name!"

Her spiritual autobiography, My Religion, was published in 1927 and then in 1994 extensively revised by Ray Silverman and re-issued under the title Light in My Darkness. It advocates the teachings of Emanuel Swedenborg, the Christian theologian and mystic who gave a spiritual interpretation of the teachings of the Bible and who claimed that the Second Coming of Jesus Christ had already taken place.

Keller described the core of her belief in these words:

"But in Swedenborg's teaching it [Divine Providence] is shown to be the government of God's Love and Wisdom and the creation of uses. Since His Life cannot be less in one being than another, or His Love manifested less fully in one thing than another, His Providence must needs be universal ... He has provided religion of some kind everywhere, and it does not matter to what race or creed anyone belongs if he is faithful to his ideals of right living."
 * "The Frost King" (1891)
 * The Story of My Life (1903)
 * Optimism: an essay (1903) T. Y. Crowell and company
 * My Key of Life: Optimism (1904), Isbister
 * The World I Live In (1908)
 * The miracle of life (1909) Hodder and Stoughton
 * The song of the stone wall (1910) The Century co.
 * Out of the Dark, a series of essays on socialism (1913)
 * Uncle Sam Is Calling (set to music by Pauline B. Story) (1917)
 * My Religion (1927; also called Light in My Darkness)
 * Midstream: my later life (1929) Doubleday, Doran & company
 * We bereaved.(1929) L. Fulenwider, Inc
 * Peace at eventide (1932) Methuen & co. ltd
 * Helen Keller in Scotland: a personal record written by herself (1933) Methuen, 212pp
 * Helen Keller's journal (1938) M. Joseph, 296pp
 * Let us have faith (1940), Doubleday, & Doran & co., inc.
 * Teacher: Anne Sullivan Macy: a tribute by the foster-child of her mind. (1955), Doubleday (publisher)
 * The open door (1957), Doubleday, 140pp
 * The faith of Helen Keller (1967)
 * Helen Keller: her socialist years, writings and speeches (1967)

Archival material
The Helen Keller Archives in New York are owned by the American Foundation for the Blind. Archival material of Helen Keller stored in New York was lost when the Twin Towers were destroyed in the September 11 attacks.

Later life and death
Keller had a series of strokes in 1961 and spent the last years of her life at her home.

On September 14, 1964, President Lyndon B. Johnson awarded her the Presidential Medal of Freedom, one of the United States' two highest civilian honors. In 1965, she was elected to the National Women's Hall of Fame at the New York World's Fair.

Keller devoted much of her later life to raising funds for the American Foundation for the Blind. She died in her sleep on June 1, 1968, at her home, Arcan Ridge, located in Easton, Connecticut, at the age of 87. A service was held at the Washington National Cathedral in Washington, D.C., and her body was cremated in Bridgeport, Connecticut. Her ashes were buried at the Washington National Cathedral next to her constant companions, Anne Sullivan and Polly Thomson.

Portrayals

Keller's life has been interpreted many times. She and her companion Anne Sullivan appeared in a silent film, Deliverance (1919), which told her story in a melodramatic, allegorical style.

She was also the subject of the Academy Award-winning 1954 documentary Helen Keller in Her Story, narrated by her friend and noted theatrical actress Katharine Cornell. She was also profiled in The Story of Helen Keller, part of the Famous Americans series produced by Hearst Entertainment.

The Miracle Worker is a cycle of dramatic works ultimately derived from her autobiography, The Story of My Life. The various dramas each describe the relationship between Keller and Sullivan, depicting how the teacher led her from a state of almost feral wildness into education, activism, and intellectual celebrity. The common title of the cycle echoes Mark Twain's description of Sullivan as a "miracle worker". Its first realization was the 1957 Playhouse 90 teleplay of that title by William Gibson, starring Patty McCormack as Helen and Teresa Wright as Sullivan. He adapted it for a Broadway production in 1959 and an Oscar-winning feature film in 1962, starring Anne Bancroft and Patty Duke. It was remade for television in 1979 and 2000."

An anime movie called The Story of Helen Keller: Angel of Love and Light was made in 1981.

In 1984, Keller's life story was made into a TV movie called The Miracle Continues. This film, a semi-sequel to The Miracle Worker, recounts her college years and her early adult life. None of the early movies hint at the social activism that would become the hallmark of Keller's later life, although a Disney version produced in 2000 states in the credits that she became an activist for social equality.

The Bollywood movie Black (2005) was largely based on Keller's story, from her childhood to her graduation.

A documentary called Shining Soul: Helen Keller's Spiritual Life and Legacy was produced by the Swedenborg Foundation in the same year. The film focuses on the role played by Emanuel Swedenborg's spiritual theology in her life and how it inspired Keller's triumph over her triple disabilities of blindness, deafness and a severe speech impediment.

On March 6, 2008, the New England Historic Genealogical Society announced that a staff member had discovered a rare 1888 photograph showing Helen and Anne, which, although previously published, had escaped widespread attention. Depicting Helen holding one of her many dolls, it is believed to be the earliest surviving photograph of Anne Sullivan Macy.

Video footage showing Helen Keller speaking also exists.

A biography of Helen Keller was written by the German Jewish author Hildegard Johanna Kaeser.

A 10 by painting titled The Advocate: Tribute to Helen Keller was created by three artists from Kerala, India as a tribute to Helen Keller. The Painting was created in association with a non-profit organization Art d'Hope Foundation, artists groups Palette People and XakBoX Design & Art Studio. This painting was created for a fundraising event to help blind students in India and was inaugurated by M. G. Rajamanikyam, IAS (District Collector Ernakulam) on Helen Keller day (June 27, 2016). The painting depicts the major events of Helen Keller's life and is one of the biggest paintings done based on Helen Keller's life.

In 2020, the documentary essay Her Socialist Smile by John Gianvito evolves around Keller's first public talk in 1913 before a general audience, when she started speaking out on behalf of progressive causes.

Posthumous honors

In 1999, Keller was listed in Gallup's Most Widely Admired People of the 20th century.

In 1999, Keller was named one of Time magazine's 100 Most Important People of the 20th Century.

In 2003, Alabama honored its native daughter on its state quarter. The Alabama state quarter is the only circulating U.S. coin to feature braille.

The Helen Keller Hospital in Sheffield, Alabama, is dedicated to her.

Streets are named after Helen Keller in Zürich, Switzerland; in Alabama and New York the US; in Getafe, Spain; in Vienna, Austria; in Lod, Israel; in Lisbon, Portugal; in Caen, France and in Sao Paulo, Brazil.

A preschool for the deaf and hard of hearing in Mysore, India, was originally named after Helen Keller by its founder, K. K. Srinivasan.

In 1973, Helen Keller was inducted into the National Women's Hall of Fame. A stamp was issued in 1980 (pictured) by the United States Postal Service depicting Keller and Sullivan, to mark the centennial of Keller's birth. That year her birth was also recognized by a presidential proclamation from U.S. President Jimmy Carter. Pennsylvania annually commemorates her June 27 birthday as Helen Keller Day.

On October 7, 2009, the State of Alabama donated a bronze statue of Keller to the National Statuary Hall Collection, as a replacement for its 1908 statue of education reformer Jabez Lamar Monroe Curry.

Keller was posthumously inducted into the Alabama Women's Hall of Fame in 1971. She was one of twelve inaugural inductees to the Alabama Writers Hall of Fame on June 8, 2015.

title: Helen Keller


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Talk:Hydrogen atom

Density function image
I've made a newer version of the density function image, i think it should replace the old one. Yey or ney? —Preceding unsigned comment added by PoorLeno (talk • contribs) 22:52, 16 August 2008 (UTC)

Relativistic effects of electron
The discussion of the speed of the electron stated that it "moved" at 1/100 the speed of light. My research shows that it's 1/10, and doesn't apply to the innermost shell. The primer I linked to includes the math. —DÅ‚ugosz
 * The speed of the electron is not a precisely definable quantity in a given energy state. However, the low-energy states have v/c on the order of the fine structure constant, which is closer in order of magnitude to 1/100 than 1/10.--76.81.164.27 05:59, 13 April 2007 (UTC)

Deleted "Schrodinger's Paradox" Stuff
I'm a physics major at the University of Rochester, and came here to look up an equation for the ionization energy for the hydrogen atom for my statistical mechanics class. To my surprise, I found a section in the hydrogen atom article entitled "Schrodinger's Paradox". I deleted the section because it is utter pseudoscience and is misleading on a number of grounds. It's also original research, which has no business being on wiki. It was also biased towards the idea being correct. —Preceding unsigned comment added by 128.151.144.60 (talk) 01:55, 7 April 2008 (UTC)
 * User:200.222.237.108 has been going around adding fringe science junk to various articles. You just caught a chunk of it. Sorry. S  B Harris 02:55, 7 April 2008 (UTC)

Wave function
Although the wave function may be correct for a particular definition of the Generalized Laguerre polynomials, the expression in the article (before my edit) was not if we use the definition in the Laguerre polynomials. I think we should be coherent with the other articles, so I have changed the expression for the wave function to use those polynomials.

(I'm new at editing wikipedia, so if I haven't done anything properly, I would like you to tell me, please. Thank you). —Preceding unsigned comment added by John_C_PI (talk • contribs) 19:26, 19 December 2005

 * It seems to me that now the Laguerre polynomials in Wiki is consistent with the one with (n+l)! instead of (n+l)!3.But I want someone to confirm this before I edit the page.Send me a message if anyone agrees or disagrees with me.--Netheril96 (talk) 12:23, 10 September 2010 (UTC)

GA on hold failed
Some minor things to adjust before the GA is awarded : Nothing was changed, the article will be failed. Lincher 13:44, 2 July 2006 (UTC)
 * Needs just a bit more references.
 * The Mathematical summary of eigenstates of hydrogen atom section is really tough to understand by itself, it needs more text surrounding it. Lincher 15:48, 23 June 2006 (UTC)

Wavefunction formula with (n+l)!^3
The wavefuction formulas on Hydrogen atom and Hydrogen-like atom were recently changed ( and, respectively) to have (n+l)!^3 instead of (n+l)!. I have come across several instances with the (n+l)^3 form (e.g. ); this also seems to contain the (n+l)!^3 version, but the generalized Laguerre polynomials have subscripts of n+l, instead of n-l-1 as they are in Wikipedia's articles. I am guessing that maybe separate definitions of generalized Laguerre polynomials are being used, as suggested by a comment above by User:John C PI (cf. this edit)? This page has the (n+l)! version (I am assuming the use of (n+1)! is a typo) with Laguerre subscript of n-l-1. I tried a quick check in my head for n = 2, l = 1; based on Eq. 33 and 36 at, it seems that the use of (n+l)! with the n-l-1 degree generalized Laguerre would give the (presumably) correct result provided here, whereas the n+l degree version would result in a polynomial in r of at least degree 3. (Also, the use of (n+l)!^3 instead of (n+l)! would seem to give a different constant muliplier than provided in the previous link.) I am going to revert the changes based on my limited investigation into this issue...if anyone is able to confirm the validity of my assessment or clarify the seemingly contradictory results that I found, that would be great.--GregRM 20:39, 29 January 2007 (UTC)

 * Yes, this is true. The problem is that different sources use different definitions for laguerre polynomials, and we expect Wikipedia to be consistent. In fact, when I studied the quantum physics subject (I'm a student of physics), it was very confusing that the two professors we had used different definitions! Anyway, the reversion you did is correct if we want to be consistent with the definitions in the Generalized Laguerre Polynomials article.
 * I don't remember which recognised books use which definition, and which is more widespread, since my references are my professor's notes, which are correct. But at the time I first dealt with this for some reason I thought the definition in the Generalized Laguerre polynomials article was more appropiate (at least, in this last article there is no history of doubt, and this is a good signal).
 * To clarify further doubts, this is a correct group of formulas and polynomials:
 * Wavefunction:
 * $$ \psi_{nlm}(r,\theta,\phi) = \sqrt {{\left ( \frac{2}{n a_0} \right )}^3\frac{(n-l-1)!}{2n[(n+l)!]} } e^{- \rho / 2} \rho^{l} L_{n-l-1}^{2l+1}(\rho) \cdot Y_{l,m}(\theta, \phi ) $$
 * Polynomials:
 * $$L_\nu^\beta ( \rho ) = \sum_{m=0}^{\nu} (-1)^{m} \frac{(\nu + \beta)!}{m!(\nu - m)! (\beta + m)!} {\rho}^m \ ; \ \beta > -1$$
 * $$L_0^\beta ( \rho ) = 1$$
 * $$L_1^\beta ( \rho ) = - \rho + \beta +1$$
 * $$L_2^\beta ( \rho ) = \frac{\rho^2}{2} - ( \beta + 2 ) \rho + \frac{(\beta + 2)(\beta+1)}{2}$$
 * I hope this makes it a little more clear. John C PI 23:03, 29 January 2007 (UTC)
 * I just corrected this error (again) from another (well-intentioned) anonymous editor. We need to keep an eye on this one.  —Preceding unsigned comment added by Stephenedie (talk • contribs) 02:22, 20 January 2010 (UTC)
 * I hope this makes it a little more clear. John C PI 23:03, 29 January 2007 (UTC)
 * I just corrected this error (again) from another (well-intentioned) anonymous editor. We need to keep an eye on this one.  —Preceding unsigned comment added by Stephenedie (talk • contribs) 02:22, 20 January 2010 (UTC)

I edited the subscript (degree) of the Laguerre polynomial appearing in the wave function and cited a couple of references that use this convention (n-l-1). I am happy to see the degree displayed as n+l, but please give a reference for this convention if you change it back. Cheers! — Preceding unsigned comment added by Micah.prange (talk • contribs) 15:26, 17 August 2011 (UTC)

The formula for the radial part is incorrect. If the generalized Laguerre polynomials are defined as in the corresponding article, than the correct factor should be (n+l)!, not [(n+l)!]^3. Just check the normalization condition. — Preceding unsigned comment added by 129.100.61.12 (talk) 17:05, 19 September 2011 (UTC)

A simple check in math program, e.g. Mathematica, confirms the correct normalization factor with (n+l)! in the denominator (not cubed). Here is the output:

Radial part
 * In[1]:= $$R[n_{-},l_{-},r_{-}] := \sqrt {{\left ( \frac{2}{n a0} \right )}^3\frac{(n-l-1)!}{2n((n+l)!)} } \textrm{Exp}[- \frac{r}{n a0}] (\frac{2 r}{n a0})^{l} \textrm{LaguerreL}[n-l-1,2 l + 1, \frac{2 r}{n a0}]$$

Spherical part
 * In[2]:= $$Y[n_{-},l_{-},\Theta_{-},\phi_{-}] := \textrm{SphericalHarmonicY}[n,l,\Theta,\phi] $$

Full wavefunction
 * In[3]:= $$\Psi[n_{-},l_{-},m_{-},r_{-},\Theta_{-},\phi_{-}] := R[n,l,r] Y[l,m,\Theta,\phi] $$

Check normalization (3D volume integration in spherical coordinates, here for specific quantum numbers)
 * In[4]:= $$\textrm{Simplify}[\int_{0}^{\infty}(\int_{0}^{\pi}(\int_{0}^{2\pi}\textrm{Conjugate}[\Psi[n,l,m,r,\Theta,\phi]]\Psi[n,l,m,r,\Theta,\phi]r\textrm{Sin}[\Theta] d\phi)r d\theta)dr $$
 * $$ /.\{n->2,l->1,m->1\}, a0>0] $$
 * Out[4]= $$1$$

Thomas.fernholz (talk) 14:52, 7 May 2012 (UTC)
 * In that case, I'm getting rid of the square brackets because I really thought someone removed the "^3" as vandalism, since the square brackets are completely redundant anyways. --Freiddie (talk) 23:54, 7 May 2012 (UTC)

The issue is that the article cites Griffith as the source of the equation but Griffith uses a different definition of the Laguerre polynomials, his are a factor of $$(n+l)!$$ larger. So people keep coming along seeing it's different from Griffith and adding the ^3 thinking it is a typo. I removed the reference to Griffith and added a note below pointing out this difference in definition. Now there is no reference for the equation, but none of the text books I have looked at use this definition of the Laguerre polynomials and give a statement of the general hydrogen wave function, even Messiah as far as I can tell. Timothyduignan (talk) 03:38, 4 December 2012 (UTC)

Apparently that same error has found its way back into the article. Was just thrown off by it, took an hour until I figured it out. Corrected it. 88.215.115.26 (talk) 10:20, 12 May 2018 (UTC)

The definition of the nomalized energy eigenfunction given here agrees with that given in other Wikipedia articles and in common textbooks. The normalization factor used here differs from that given in Messiah, who replaces the present (n+l)! by [(n+l)!]^3 (Ch. 11, Sec. 6). The definition of the associated Laguerre polynomials used by Messiah (Appendix B2) differs from that given by eg. Abramowitz and Stegun by a factor of (n+l)!, which I believe agrees with that given in other Wikipedia articles. I have numerically checked the orthonormality of the present definition of the wave functions together with the definition of the associated Laguerre polynomials given by Abramowitz and Stegun. I suggest that the statement saying that the present definition is consistent with Messiah be removed. — Preceding unsigned comment added by 49.180.73.107 (talk) 05:17, 28 May 2019 (UTC)

 * I agree with this comment that Messiah differs from the convention in the article. See my comment below. If my comment below "sticks" I can clean up the text regarding the Messiah reference. Twistar48 (talk) 11:12, 16 November 2021 (UTC)

I've added a section at Laguerre polynomials that I hope will help clarify the discussion regarding the appropriate normalization factor. It seems that throughout Wikipedia (and the numerical computation and math worlds) the convention in Abramowitz is taken. In the physics literature the convention is taken which differs by a factor of $$(n+l)!$$. I find the physics convention most clearly expressed in. This gives us a hard time on Wikipedia because we want to stay consistent with the "standard" convention, but we cannot use that convention and directly cite the physics literature without giving an explicitly caveat as to why the formula on Wikipedia differs from that found in the referred textbooks. It seems to me the best strategy is to be as clear as possible about the different convention choices and which is taken where.

The convention difference regarding the factor of $$(n+l)!$$ seems to be well understood, but there seems to be another convention difference in which $$L_{n-l-1}^{2l+1}$$ is replaced by $$L_{n+l}^{2l+1}$$. This apparently regards a difference in how the Associated Laguerre polynomials can be defined as derivatives of the Laguerre polynomials. This is discussed in. One reference I've seen that seems to do this is which is of note because this reference is directly cited by Griffiths in the context of Laguerre polynomial conventions. Twistar48 (talk) 11:12, 16 November 2021 (UTC)

poorly written
The level of this article is wildly inappropriate for the general reader. Much of the section "Features going beyond the Schrödinger solution" doesn't belong in this article. --76.81.164.27 05:56, 13 April 2007 (UTC)
 * Hear, hear! David F (talk) 03:19, 10 March 2013 (UTC)

Atomic hydrogen
Atomic hydrogen and Hydrogen atom do NOT have overlapping meanings. This article is junk. Was it written by a non-native English speaker?71.31.145.210 (talk) 18:41, 16 March 2012 (UTC)
 * They can be used interchangeably (say, for isolated hydrogen atoms), or can have different meanings (as in compounds), thus I disagree. Materialscientist (talk) 00:32, 17 March 2012 (UTC)

Eccess and Binding energy
"Eccess energy" and "binding energy" links to the same article, What is the diference? 80.24.186.207 22:42, 2 September 2007 (UTC)
 * The sign. 149.217.1.6 16:09, 27 October 2007 (UTC)

 * The explanation (the difference between an isotope's mass number and its actual mass in AMU (times c2, of course)) can be found at "Mass excess"; I have fixed the redirect. --DWIII (talk) 01:54, 28 September 2008 (UTC)

Wavefunction and normalization
As a follow-up to the discussion above about the different definitions of the associated Laguerre polynomials, I have come across the following problem: The radial part of the wavefunction as given in the article
 * $$ R(r) = \sqrt {{\left ( \frac{2}{n a_0} \right )}^3\frac{(n-l-1)!}{2n[(n+l)!]} } e^{- \rho / 2} \rho^{l} L_{n-l-1}^{2l+1}(\rho) $$

doesn't (quite) seem to be correctly normalized. The integral over r2|R|2 yields the value 2 for n = 1, 1 for n = 2, 2/3 for n = 3, 0.5 for n = 4, etc. Could someone double-check this? 149.217.1.6 16:08, 27 October 2007 (UTC)
 * Follow-up: I found the source of the discrepancy. When making the change of radial variable from $$r$$ to $$\rho$$, a factor $$dr/d\rho$$ is introduced in the integration. Hence,
 * $$\int r^{2} \left|R(\rho)\right|^{2} dr = \int r^{2} \left|R(\rho)\right|^{2} \frac{dr}{d\rho} d\rho$$
 * with
 * $$\frac{dr}{d\rho} = \frac{na_{0}}{2}$$.
 * With this additional factor, the integral over all space yields unity and the normalization condition holds. 149.217.1.6 20:18, 28 October 2007 (UTC)

H-1
Is H-1 a proper notation for the hydrogen atom? The only article in which I've seen it used is Big Bang nucleosynthesis. It is also not mentioned in the disambiguation page H-1. Should it be added to this page? --George100 (talk) 13:05, 12 July 2008 (UTC)
 * You usually write either "1H" or "hydrogen-1". BTW, why does it say "This article primarily concerns hydrogen-1", and why does hydrogen-1 redirect here? I see absolutely no reason why the stuff in this article can't apply to deuterium or tritium. -- Army1987 (t — c) 10:29, 30 October 2008 (UTC)

Incomplete and incorrect definition of j
I've chaged the definition of j: IIUC j is not an integer (it is an integer +/- 1/2). I've also added to the definition to clarify the meaning of 'total angular momentum'. I'm a bit rusty on this so please check! --Kevin Cowtan (talk) 12:11, 19 November 2008 (UTC)

electron spin -- schrodinger equation / pauli equation /dirac equation
there seems to be a bit of a muddle here concerning the electron spin and the schrodinger equation. the normal unmodified schrodinger equation describes  the behaviour of a spin zero particle, and as such will predict that a hydrogen atom is possessed of zero angular momentum in the ground state, contrary to what is observed. it is only when one steps up to the pauli a.k.a schrodinger-pauli equation that this defect is corrected. (or, of course, the fully relativistic dirac equation). this is not at all made clear in the present state of the article (march 2010), and will i fear need a deal of work to unravel. —Preceding unsigned comment added by 62.56.52.132 (talk) 22:51, 30 March 2010 (UTC)

Wrong diameter for illustration
The diameter is given as 2.4 Angstroms, or "twice the Bohr radius." In reality the correct diameter is about 1.1 Angstroms-- about half this. I realize these things are not exact, but 1.1 A is much closer than 2.4 Angstroms. The original artist and uploader apparently is no longer active. I wonder if anybody would like to re-do this illustration? S B Harris 04:58, 19 May 2010 (UTC)

Eigenstate 4,3,1, figure caption
There's a figure in the text showing a constant probability surface for the 4,3,1 state, the caption for which says, "The solid body contains 45% of the electron's probability." This is pretty ambiguous language, and it seems like it would be easy to misunderstand unless you already knew what it's supposed to mean; I mean, it contains "45% of the probability" for position measurements. I'm going to be bold and change it to say s/t like "3D illustration of the eigenstate $$\psi_{4,3,1}$$. Electrons in this state are 45% likely to be found within the solid body shown." I think this is a bit more precise, but it doesn't quite capture the fact that electrons aren't inside that space "45% of the time"; rather, position measurements are 45% likely to find them in that space. I haven't come up with a way to emphasize this subtle but key point in any kind of succinct way. I invite further suggestions for improved language. Flies 1 (talk) 16:27, 21 April 2012 (UTC)

Apparent editorial comments moved here from the article
In this edit I removed the following text which had just been added by an anon to the Quantum theoretical analysis section of the article:

"Emission spectrum of hydrogen. When excited, hydrogen gas gives off light in four distinct colours (spectral lines) in the visible spectrum, as well as a number of lines in the infra-red and ultra-violet. Bohr (and others) were aware of this and discovered orbitals happened to co-incide with wavelengths. From this exciting realization they began using electric force / motion functions and wave functions (Einstein) to explain the coincidence they saw in what they could only first imagine to be a classical orbit situation.

Before reading Schrödinger it's HIGHLY advised to check the acrticles concerning the spectrum of hydrogen and the articles concerning the simpler Bohr and Einstein models (which became quantum calculation) to see why these complexities became tried and relevant. (note also brownian motion and other experiments in the experiments listsing - these show allot too - and as well how often scientists move from bulk observation to particular causes and values underpinning)"

Wtmitchell (talk) (earlier Boracay Bill) 05:25, 16 July 2012 (UTC)

Dark matter
I removed the sentence about dark matter and dark energy from the lead paragraph because it was distracting. I left a link to the word "baryonic" in the preceding sentence, and that article mentions dark (non-baryonic) matter. If someone feels strongly about pointing out the difference between hydrogen and dark matter, I suggest it belongs somewhere other than the opening paragraph. - LesPaul75 talk 18:08, 14 January 2014 (UTC)

Wrong binding energy
The binding energy of the hydrogen atom is listed as "0.000± 0.0000 keV". It's actually about 13.6 eV, although I don't know the uncertainty. Comment by 128.111.18.19

 * That part of the infobox is called Nuclide data, and what is meant is the Nuclear binding energy to hold the nucleus together. Since there is only one nucleon, this is automatically zero. The atomic binding energy is 13.6 eV as you say, and is normally called the ionization energy.
 * I agree that this is confusing, and the entry really should be changed to say Nuclear binding energy. But I can't figure out how to make that change since the infobox only accepts labels that are on the preprogrammed template. Perhaps we should just delete that line from the infobox. Dirac66 (talk) 22:59, 12 August 2015 (UTC)

Error in Bohr-Sommerfeld Description
In the Bohr-Sommerfeld Model description, the symbol ε0 is referred to as "permeability." It should be "permittivity of free space." The symbol μ0 would be "permeability." — Preceding unsigned comment added by Nick0927 (talk • contribs) 02:34, 27 October 2016 (UTC)

Wavefunction
how can this be correct, $$ \sqrt {{\left ( \frac{2}{n a_0} \right )}^3 ...}$$ since a_0 is of dimension meters? Isn't there a factor like "r" missing like in $$ \rho = {2r \over {na_0}} $$? Ra-raisch (talk) 21:03, 4 December 2016 (UTC)

 * This factor implies that the wavefunction has units m-3/2 which is correct. The probability of finding the electron in a volume element dV is |Ψ|2dV which is dimensionless as required for a probability. And the integral of this probability over all space is 1, which is also dimensionless.

 * It would probably be helpful to explain this in the article, however. Dirac66 (talk) 22:14, 4 December 2016 (UTC)

 * is this cgs or SI? Ra-raisch (talk) 21:03, 18 December 2016 (UTC)
 * Well, I have written m-3/2 which assumes length in m so I have implied SI units. If you want to consider cgs with length units cm, then by a similar argument the units of Ψ would be cm-3/2. Dirac66 (talk) 21:23, 18 December 2016 (UTC)

Hydrogen atom mass - 1.007825 or 1.00794 ?
If you google 'hydrogen atom mass', Google gives result 1.00794 u at the top of page. That means most of the sites on internet mention 1.00794 u. I googled for 'hydrogen atom mass 1.007825'. Only 2 pages of results appeared and I think those sites are not reliable. Also in this article, 1.007825 is mentioned without any reference. Before removing this entry, I just want to confirm from you that I am not making any mistake. Thank you. — Preceding unsigned comment added by AbhiRiksh (talk • contribs) 18:47, 2 April 2018 (UTC)

 * Hydrogen is a mixture of 2 isotopes: 99.9885% H-1 and 0.000115 H-2 (terrestrial abundances). The mass of H-1 is 1.007825 u. The average mass of H is (1.007825 x 0.999885) + (2.014102 x 0.000115) = 1.007941 u. See data at Isotopes of hydrogen.
 * So the infobox is now technically correct because it specifies that the value 1.007825 is for the isotope H-1 (protium). However I think it would be clearer to give both values and explain the significance of each as well as the relationship between them. Dirac66 (talk) 23:47, 2 April 2018 (UTC)

Atomic hydrogen in relation to solvated electron and standard electrode potential
Some reactions of the solvated electron give atomic hydrogen and have electrochemical importance. I think it should be mentioned in article.--5.2.200.163 (talk) 15:42, 12 June 2018 (UTC)
 * Step 1: a strong reference supporting the assertion. --Smokefoot (talk) 18:26, 25 June 2018 (UTC)

Could it be that atomic hydrogen has identified identified in the past as nascent hydrogen?--5.2.200.163 (talk) 16:23, 25 June 2018 (UTC)

Reduced mass in solution to Schrödinger eq?
I've seen some texts saying that the µ letter refers to the reduced mass of the electron and the nucleus. I wonder why some texts (including Wikipedia) says that. In no moment during the formulation of the problem in Schrödinger equation appears the mass of the nucleus. The only reference of the nucleus is in the form of a Coulomb Potential. The letter µ refers to the electron mass, not to the reduced mass. — Preceding unsigned comment added by Heitorpb (talk • contribs) 00:55, 29 April 2019 (UTC)

 * Reduced mass is correct in this problem. Since the nucleus has a finite mass (even if it is much heavier the electron), it can move and has some kinetic energy which must be included in the total energy of the atom. Introducing the reduced mass allows the two-body problem (electron plus nucleus) to be transformed into an equivalent one-body problem, since the total (electron plus nuclear) kinetic energy is equal to an equivalent kinetic energy calculated with the reduced mass and the relative velocity. This is discussed in the article on reduced mass.Dirac66 (talk) 18:35, 2 May 2019 (UTC)

Unified Location and Treatment of Hydrogen Orbitals
Right now if someone wants information about hydrogen orbitals they may have to look at https://en.wikipedia.org/wiki/Hydrogen_atom#Wavefunction on "Hydrogen Atom" page and https://en.wikipedia.org/wiki/Hydrogen-like_atom#Non-relativistic_wavefunction_and_energy on "Hydrogen-Like Atom" page. Apart from already being inconvenient, it seems like these 2 pages sometimes end up differing in their normalization factor due to issues discussed above. It is a pain to keep both pages correct and consistent.

It would be convenient if there was one location where a reader could get all information about the hydrogen orbitals. In particular, I'd like to expand such a section with some helpful intuition and understandings to help readers predict and decipher the shapes of the hydrogen orbitals. However, I don't want to go through that effort until there's a clear dedicated location where this topic could be discussed.

My suggestion would be as follows: I don't think an entire article is needed for the hydrogen like atom. I think that article could be renamed to "Solutions to the Hydrogen Atom" or something like that without much change anywhere except the intro. Most of the article would then be about the mathematical solution to the Hydrogen atom (at varying levels of approximation) and there could be a small section on Hydrogen-like atoms (basically defining what they are and why they're useful). That article could then include the authoritative solution to the Hydrogen Schrodinger equation.

What do others think? Twistar48 (talk) 14:26, 27 October 2021 (UTC)

 * It is true that the articles Hydrogen-like atom could be combined with this one, following the procedure known in Wikipedia as merging. I would suggest retaining the name Hydrogen atom for the combined article, with one section called Hydrogen-like atom. We can explain that for hydrogen-like atoms, the Schrödinger equation and the orbitals contain a factor Z which is the nuclear charge. (There was one factor Z in the Wavefunction section, but I have just removed it since the rest of the section assumes Z = 1. If we carry out a merger, then the section on H-like atoms could include a factor Z consistently.)

 * The Hydrogen-like atom article now includes not only the Schrödinger equation but also a section on the Dirac equation which is not in the Hydrogen atom article. A merged article should include this section also. Dirac66 (talk) 18:20, 29 November 2021 (UTC)

Picture in info box should show the atom
I suggest to replace the picture in the info box (excerpt from table of nuclides) by something that shows the atom, e.g. file "Bohr model.jpg". The table of nuclides shows nuclear properties (and the isotopes separately); but here we are dealing with the atom. -- Wassermaus (talk) 21:15, 15 March 2022 (UTC)

Momentum representation of the Hydrogen atom is unphysical
The formulation of $$\varphi(p,\theta_p,\varphi_p)$$ involving a normalised product of a Gegenbauer polynomial and a spherical harmonic could actually be missleading. According to J.R. Lombardi (Chemical Physics 538 (2020) 110886, https://doi.org/10.1016/j.chemphys.2020.110886), the missconception stems from the fact that the momentum representation of a position-space wave function is its Fourier transform if and only if it is expressed in Cartesian coordinates. It is true that the Fourier transform of the spherical-coordinate wave function (as also performed in Bransden et al.) gives a mathematically correct expression, but these variables can not be interpreted as representations of momentum in quantum mechanics: they do not fulfill the standard commutation relation w.r.t. their position operator counterparts in curvlinear coordinates. As elaborated by Lombardi et al., the deWitt-transformation gives properly transformed and physically more reasonable expressions (where the spherical part no longer can be represented by spherical harmonics) with variables fulfilling the commutation relations. This missconception unfortunately is deeply rooted now in QM teaching and got replicated in quite some books, partly also because Fock, Pauling and Hylleraas fell for it. 2003:F4:8F20:4A00:FDA1:8FE7:ED46:8A30 (talk) 21:57, 25 July 2022 (UTC)

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

title: Talk:Hydrogen atom


text:
Haddocks' Eyes

"Haddocks' Eyes" is the nickname of the name of a song sung by The White Knight from Lewis Carroll's 1871 novel Through the Looking-Glass, chapter VIII.

"Haddocks' Eyes" is an example used to elaborate on the symbolic status of the concept of "name": a name as identification marker may be assigned to anything, including another name, thus introducing different levels of symbolization. It has been discussed in several works on logic and philosophy.

Naming
The White Knight explains to Alice a confusing nomenclature for the song.

"You are sad," the Knight said in an anxious tone: "let me sing you a song to comfort you." "Is it very long?" Alice asked, for she had heard a good deal of poetry that day. "It's long," said the Knight, "but very, beautiful. Everybody that hears me sing it—either it brings the into their eyes, or else—" "Or else what?" said Alice, for the Knight had made a sudden pause. "Or else it doesn't, you know. The name of the song is called Haddocks' Eyes." "Oh, that's the name of the song, is it?" Alice said, trying to feel interested. "No, you don't understand," the Knight said, looking a little vexed. "That's what the name is . The name really The Aged Aged Man." "Then I ought to have said 'That's what the is called'?" Alice corrected herself. "No, you oughtn't: that's quite another thing! The is called Ways And Means: but that's only what it's, you know!" "Well, what the song, then?" said Alice, who was by this time completely bewildered. "I was coming to that," the Knight said. "The song really A-sitting On A Gate: and the tune's my own invention."

To summarize:

 * The song's name is called Haddocks' Eyes
 * The song's name is The Aged Aged Man
 * The song is called Ways and Means
 * The song is A-sitting on a Gate

The complicated terminology distinguishing between 'the song, what the song is called, the name of the song, and what the name of the song is called' both uses and mentions the use–mention distinction.

The song
The White Knight sings the song to a tune he claims as his own invention, but which Alice recognises as "I give thee all, I can no more". By the time Alice heard it, she was already tired of poetry.

The song parodies the plot, but not the style or metre, of "Resolution and Independence" by William Wordsworth.

I'll tell thee everything I can: There's little to relate. I saw an aged aged man, A-sitting on a gate. "Who are you, aged man?" I said, "And how is it you live?" And his answer trickled through my head, Like water through a sieve.

He said "I look for butterflies    That sleep among the wheat: I make them into mutton-pies,     And sell them in the street. I sell them unto men," he said, "Who sail on stormy seas; And that's the way I get my bread –    A trifle, if you please."

But I was thinking of a plan To dye one's whiskers green, And always use so large a fan That they could not be seen. So, having no reply to give To what the old man said, I cried "Come, tell me how you live!" And thumped him on the head.

His accents mild took up the tale: He said "I go my ways, And when I find a mountain-rill,    I set it in a blaze; And thence they make a stuff they call     Rowlands' Macassar-Oil – Yet twopence-halfpenny is all     They give me for my toil."

But I was thinking of a way To feed oneself on batter, And so go on from day to day Getting a little fatter. I shook him well from side to side, Until his face was blue: "Come, tell me how you live," I cried, "And what it is you do!"

He said "I hunt for haddocks' eyes    Among the heather bright, And work them into waistcoat-buttons     In the silent night. And these I do not sell for gold     Or coin of silvery shine, But for a copper halfpenny,     And that will purchase nine.

"I sometimes dig for buttered rolls,    Or set limed twigs for crabs: I sometimes search the grassy knolls     For wheels of Hansom-cabs. And that's the way" (he gave a wink) "By which I get my wealth-- And very gladly will I drink    Your Honour's noble health."

I heard him then, for I had just Completed my design To keep the Menai bridge from rust By boiling it in wine. I thanked him much for telling me    The way he got his wealth, But chiefly for his wish that he    Might drink my noble health.

And now, if e'er by chance I put My fingers into glue, Or madly squeeze a right-hand foot Into a left-hand shoe,

Or if I drop upon my toe A very heavy weight, I weep, for it reminds me so Of that old man I used to know-- Whose look was mild, whose speech was slow Whose hair was whiter than the snow, Whose face was very like a crow, With eyes, like cinders, all aglow, Who seemed distracted with his woe, Who rocked his body to and fro, And muttered mumblingly and low, As if his mouth were full of dough, Who snorted like a buffalo-- That summer evening long ago, A-sitting on a gate.

Upon the Lonely Moor
Like "Jabberwocky," another poem published in Through the Looking Glass, "Haddocks’ Eyes" appears to have been revised over the course of many years. In 1856, Carroll published the following poem anonymously under the name Upon the Lonely Moor. It bears an obvious resemblance to "Haddocks' Eyes."

I met an aged, aged man Upon the lonely moor: I knew I was a gentleman, And he was but a boor. So I stopped and roughly questioned him, "Come, tell me how you live!" But his words impressed my ear no more Than if it were a sieve.

He said, "I look for soap-bubbles, That lie among the wheat, And bake them into mutton-pies, And sell them in the street. I sell them unto men," he said, "Who sail on stormy seas; And that's the way I get my bread – A trifle, if you please."

But I was thinking of a way To multiply by ten, And always, in the answer, get The question back again. I did not hear a word he said, But kicked that old man calm, And said, "Come, tell me how you live!" And pinched him in the arm.

His accents mild took up the tale: He said, "I go my ways, And when I find a mountain-rill, I set it in a blaze. And thence they make a stuff they call Rowland's Macassar Oil; But fourpence-halfpenny is all They give me for my toil."

But I was thinking of a plan To paint one's gaiters green, So much the color of the grass That they could ne'er be seen. I gave his ear a sudden box, And questioned him again, And tweaked his grey and reverend locks, And put him into pain.

He said, "I hunt for haddock's eyes Among the heather bright, And work them into waistcoat-buttons In the silent night. And these I do not sell for gold, Or coin or silver-mine, But for a copper-halfpenny, And that will purchase nine.

"I sometimes dig for buttered rolls, Or set limed twigs for crabs; I sometimes search the flowery knolls For wheels of hansom cabs. And that's the way" (he gave a wink) "I get my living here, And very gladly will I drink Your Honour's health in beer."

I heard him then, for I had just Completed my design To keep the Menai bridge from rust By boiling it in wine. I duly thanked him, ere I went, For all his stories queer, But chiefly for his kind intent To drink my health in beer.

And now if e'er by chance I put My fingers into glue, Or madly squeeze a right-hand foot Into a left-hand shoe; Or if a statement I aver Of which I am not sure, I think of that strange wanderer Upon the lonely moor.

title: Haddocks' Eyes


text:
Hoosier

Hoosier is the official demonym for the people of the U.S. state of Indiana. The origin of the term remains a matter of debate, but "Hoosier" was in general use by the 1840s, having been popularized by Richmond resident John Finley's 1833 poem "The Hoosier's Nest". Indiana adopted the nickname "The Hoosier State" more than 150 years ago.

"Hoosier" is used in the names of numerous Indiana-based businesses and organizations. "Hoosiers" is also the name of the Indiana University athletic teams. As there is no accepted embodiment of a Hoosier, the IU schools are represented through their letters and colors alone. In addition to universal acceptance by residents of Indiana, the term is also the official demonym according to the U.S. Government Publishing Office (GPO) as of 2017 (the GPO previously identified Indiana residents as "Indianians"), the only GPO-recommended demonym of any US state that is not directly formed from the state's name.

Origin
In addition to "The Hoosier's Nest," the term also appeared in the Indianapolis Journal ' s "Carrier's Address" on January 1, 1833. There are many suggestions for the derivation of the word but none is universally accepted. In 1833 the Pittsburgh Statesman said the term had been in use for "some time past" and suggested it originated from census workers calling "Who's here?". Also in 1833, former Indiana Governor James B. Ray began publishing a newspaper titled The Hoosier.

Scholarship
In 1900, Meredith Nicholson wrote The Hoosiers, an early attempt to study the etymology of the word as applied to Indiana residents. Jacob Piatt Dunn, longtime secretary of the Indiana Historical Society, published The Word Hoosier, a similar attempt, in 1907. Both chronicled some of the popular and satirical etymologies circulating at the time and focused much of their attention on the use of the word in the Upland South to refer to woodsmen, yokels, and rough people. Dunn traced the word back to the Cumbrian hoozer, meaning anything unusually large, derived from the Old English hoo (as at Sutton Hoo), meaning "high" and "hill". The importance of immigrants from northern England and southern Scotland was reflected in numerous placenames including the Cumberland Mountains, the Cumberland River, and the Cumberland Gap. Nicholson defended the people of Indiana against such an association, while Dunn concluded that the early settlers had adopted the nickname self-mockingly and that it had lost its negative associations by the time of Finley's poem.

Johnathan Clark Smith subsequently showed that Nicholson and Dunn's earliest sources within Indiana were mistaken. A letter by James Curtis cited by Dunn and others as the earliest known use of the term was actually written in 1846, not 1826. Similarly, the use of the term in an 1859 newspaper item quoting an 1827 diary entry by Sandford Cox was more likely an editorial comment and not from the original diary. Smith's earliest sources led him to argue that the word originated as a term along the Ohio River for flatboatmen from Indiana and did not acquire its pejorative meanings until 1836, after Finley's poem.

William Piersen, a history professor at Fisk University, argued for a connection to the Methodist minister Rev. Harry Hosier (c. 1750–May 1806), who evangelized the American frontier at the beginning of the 19th century as part of the Second Great Awakening. "Black Harry" had been born a slave in North Carolina and sold north to Baltimore, Maryland, before gaining his freedom and beginning his ministry around the end of the American Revolution. He was a close associate and personal friend of Bishop Francis Asbury, the "Father of the American Methodist Church". Benjamin Rush said of him that "making allowances for his illiteracy, he was the greatest orator in America". His sermons called on Methodists to reject slavery and to champion the common working man. Piersen proposed that Methodist communities inspired by his example took or were given a variant spelling of his name (possibly influenced by the "yokel" slang ) during the decades after his ministry.

According to Washington County newspaper reports of the time, Abraham Stover was Colonel of the Indiana Militia. He was a colorful figure in early Washington County history. Along with his son-in-law, John B. Brough, he was considered one of the two strongest men in Washington County. He was always being challenged to prove his might, and seems to have won several fights over men half his age. After whipping six or eight men in a fist fight in Louisville, Kentucky, he cracked his fists and said, "Ain't I a husher", which was changed in the news to "Hoosier", and thus originated the name of Hoosier in connection with Indiana men.

Jorge Santander Serrano, a PhD student from Indiana University, has also suggested that Hoosier might come from the French words for 'redness', rougeur, or 'red-faced', rougeaud. According to this hypothesis, the early pejorative use of the word Hoosier may have a link to the color red ("rouge" in French) which is associated with indigenous peoples, pejoratively called "red men" or "red-skins", and also with poor white people by calling them "red-necks".

"Who'sh 'ere?"
Humorous folk etymologies for the term "hoosier" have a long history, as recounted by Dunn in The Word Hoosier.

One account traces the word to the necessary caution of approaching houses on the frontier. In order to avoid being shot, a traveler would call out from afar to let themselves be known. The inhabitants of the cabin would then reply "Who's here?" which – in the Appalachian English of the early settlers – slurred into "Who'sh 'ere?" and thence into "Hoosier?" A variant of this account had the Indiana pioneers calling out "Who'sh 'ere?" as a general greeting and warning when hearing someone in the bushes and tall grass, to avoid shooting a relative or friend in error.

The poet James Whitcomb Riley facetiously suggested that the fierce brawling that took place in Indiana involved enough biting that the expression "Whose ear?" became notable. This arose from or inspired the story of two 19th-century French immigrants brawling in a tavern in the foothills of southern Indiana. One was cut and a third Frenchman walked in to see an ear on the dirt floor of the tavern, prompting him to slur out "Whosh ear?"

Mr. Hoosier's men
Two related stories trace the origin of the term to gangs of workers from Indiana under the direction of a Mr. Hoosier.

The account related by Dunn is that a Louisville contractor named Samuel Hoosier preferred to hire workers from communities on the Indiana side of the Ohio River like New Albany rather than Kentuckians. During the excavation of the first canal around the Falls of the Ohio from 1826 to 1833, his employees became known as "Hoosier's men" and then simply "Hoosiers". The usage spread from these hard-working laborers to all of the Indiana boatmen in the area and then spread north with the settlement of the state. The story was told to Dunn in 1901 by a man who had heard it from a Hoosier relative while traveling in southern Tennessee. Dunn could not find any family of the given name in any directory in the region or anyone else in southern Tennessee who had heard the story and accounted himself dubious. This version was subsequently retold by Gov. Evan Bayh and Sen. Vance Hartke, who introduced the story into the Congressional Record in 1975, and matches the timing and location of Smith's subsequent research. However, the U.S. Army Corps of Engineers has been unable to find any record of a Hoosier or Hosier in surviving canal company records.

Other uses
The word "hoosier" has been used in Greater St. Louis as a pejorative for an unintelligent or uncultured person. The word is also encountered in sea shanties. In the book Shanties from the Seven Seas by Stan Hugill, in reference to its former use to denote cotton-stowers, who would move bales of cotton to and from the holds of ships and force them in tightly by means of jackscrews.

A Hoosier cabinet, often shortened to "hoosier", is a type of free-standing kitchen cabinet popular in the early decades of the twentieth century. Almost all of these cabinets were produced by companies located in Indiana and the name derives from the largest of them, the Hoosier Manufacturing Co. of New Castle, Indiana. Other Indiana businesses include Hoosier Racing Tire and the Hoosier Bat Company, manufacturer of wooden baseball bats.

The RCA Dome, former home of the Indianapolis Colts, was known as the "Hoosier Dome" before RCA purchased the naming rights in 1994. The RCA Dome was replaced by Lucas Oil Stadium in 2008.

In popular culture

 * Indiana native Kurt Vonnegut's book Cat's Cradle describes this identification as an example of a granfalloon.
 * In the movie The Outlaw Josey Wales (starring Clint Eastwood), a shopkeeper states "I'm a Hoosier" to the disgust of an elderly customer.
 * The HBO miniseries The Pacific refers to PFC Bill Smith by the nickname "Hoosier", as do the two Marine memoirs on which the series is based.
 * Adam Savage, host of the Discovery Channel series MythBusters, often refers to co-host Jamie Hyneman as a Hoosier, the latter having been raised on a farm in Indiana, and attended Indiana University.
 * Serial killer Carl Panzram's last words were reportedly, "Hurry it up, you Hoosier bastard! I could kill 10 men while you're fooling around!"
 * In the movie We're No Angels, Sean Penn's character says when asked to wear work clothes as a disguise, "Whaddya think I am, a Hoosier or something?"
 * In the book Donnie Brasco: My Undercover Life in the Mafia, gangster Benjamin "Lefty Guns" Ruggiero uses Hoosier as an epithet.
 * Hoosiers, a 1986 sports film about a small-town Indiana high school basketball team that wins the state championship.
 * The Frugal Hoosier is a fictional discount grocery store depicted in the ABC sitcom The Middle, based in the fictional Indiana town of Orson.
 * In the NBC sitcom Parks and Recreation episode "Soulmates," a fictional online dating site called "hoosiermate.com" was the main subject. The series is set in Indiana.
 * The US Secret Service has designated the code name "Hoosier" for former US Vice President, former Indiana Governor, and Indiana native Mike Pence.
 * On his podcast, retired Indianapolis Colts punter Pat McAfee defined the term as "a human who is willing to stand up in the face of adversity, chug two beers, and do anything he can to make America a better place. That's what a Hoosier is."
 * In 1987, then-United States Senator Dan Quayle of Indiana requested of the Merriam-Webster dictionary to redefine the term "Hoosier" to mean "someone who is smart, resourceful, skillful, a winner, unique and brilliant." The dictionary denied the request.

title: Hoosier


text:
Horner's method

In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. After the introduction of computers, this algorithm became fundamental for computing efficiently with polynomials.

The algorithm is based on Horner's rule, in which a polynomial is written in nested form: $$\begin{align} &a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots + a_nx^n \\ ={} &a_0 + x \bigg(a_1 + x \Big(a_2 + x \big(a_3 + \cdots + x(a_{n-1} + x \, a_n) \cdots \big) \Big) \bigg). \end{align}$$

This allows the evaluation of a polynomial of degree $n$ with only $$n$$ multiplications and $$n$$ additions. This is optimal, since there are polynomials of degree $n$ that cannot be evaluated with fewer arithmetic operations.

Alternatively, Horner's method also refers to a method for approximating the roots of polynomials, described by Horner in 1819. It is a variant of the Newton–Raphson method made more efficient for hand calculation by the application of Horner's rule. It was widely used until computers came into general use around 1970.

Polynomial evaluation and long division
Given the polynomial $$p(x) = \sum_{i=0}^n a_i x^i = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n,$$ where $$a_0, \ldots, a_n$$ are constant coefficients, the problem is to evaluate the polynomial at a specific value $$x_0$$ of $$x.$$

For this, a new sequence of constants is defined recursively as follows:

Then $$b_0$$ is the value of $$p(x_0)$$.

To see why this works, the polynomial can be written in the form $$p(x) = a_0 + x \bigg(a_1 + x \Big(a_2 + x \big(a_3 + \cdots + x(a_{n-1} + x \, a_n) \cdots \big) \Big) \bigg) \ .$$

Thus, by iteratively substituting the $$b_i$$ into the expression, $$\begin{align} p(x_0) & = a_0 + x_0\Big(a_1 + x_0\big(a_2 + \cdots + x_0(a_{n-1} + b_n x_0) \cdots \big)\Big) \\ & = a_0 + x_0\Big(a_1 + x_0\big(a_2 + \cdots + x_0 b_{n-1}\big)\Big) \\ & \vdots \\ & = a_0 + x_0 b_1 \\ & = b_0. \end{align}$$

Now, it can be proven that;

This expression constitutes Horner's practical application, as it offers a very quick way of determining the outcome of; $$p(x) / (x-x_0) $$ with $$b_0$$ (which is equal to $$p(x_0)$$) being the division's remainder, as is demonstrated by the examples below. If $$x_0$$ is a root of $$p(x)$$, then $$b_0 = 0$$ (meaning the remainder is $$0$$), which means you can factor $$p(x)$$ as $$x-x_0$$.

To finding the consecutive $$b$$-values, you start with determining $$b_n$$, which is simply equal to $$a_n$$. Then you then work recursively using the formula: $$ b_{n-1} = a_{n-1} + b_{n}x_0 $$ till you arrive at $$b_0$$.

Examples
Evaluate $$f(x)=2x^3-6x^2+2x-1$$ for $$x=3$$.

We use synthetic division as follows:

x$$│  x$$    x$0$    x$3$    x$2$ 3 │  2    −6     2    −1    │         6     0     6    └────────────────────────        2     0     2     5

The entries in the third row are the sum of those in the first two. Each entry in the second row is the product of the $1$-value ($0$ in this example) with the third-row entry immediately to the left. The entries in the first row are the coefficients of the polynomial to be evaluated. Then the remainder of $$f(x)$$ on division by $$x-3$$ is $x$.

But by the polynomial remainder theorem, we know that the remainder is $$f(3) $$. Thus, $$f(3) = 5$$.

In this example, if $$a_3 = 2, a_2 = -6, a_1 = 2, a_0 = -1$$ we can see that $$b_3 = 2, b_2 = 0, b_1 = 2, b_0 = 5 $$, the entries in the third row. So, synthetic division (which was actually invented and published by Ruffini 10 years before Horner's publication) is easier to use; it can be shown to be equivalent to Horner's method.

As a consequence of the polynomial remainder theorem, the entries in the third row are the coefficients of the second-degree polynomial, the quotient of $$f(x)$$ on division by $$ x-3 $$. The remainder is $3$. This makes Horner's method useful for polynomial long division.

Divide $$x^3-6x^2+11x-6$$ by $$x-2$$:

2 │  1    −6    11    −6    │         2    −8     6    └────────────────────────        1    −4     3     0

The quotient is $$x^2-4x+3$$.

Let $$f_1(x)=4x^4-6x^3+3x-5$$ and $$f_2(x)=2x-1$$. Divide $$f_1(x)$$ by $$f_2\,(x)$$ using Horner's method.

0.5 │ 4 −6   0   3  −5       │     2  −2  −1   1       └───────────────────────         2  −2  −1   1  −4

The third row is the sum of the first two rows, divided by $5$. Each entry in the second row is the product of $5$ with the third-row entry to the left. The answer is $$\frac{f_1(x)}{f_2(x)}=2x^3-2x^2-x+1-\frac{4}{2x-1}.$$

Efficiency
Evaluation using the monomial form of a degree $$n$$ polynomial requires at most $$n$$ additions and $$(n^2+n)/2$$ multiplications, if powers are calculated by repeated multiplication and each monomial is evaluated individually. The cost can be reduced to $$n$$ additions and $$2n-1$$ multiplications by evaluating the powers of $$x$$ by iteration.

If numerical data are represented in terms of digits (or bits), then the naive algorithm also entails storing approximately $$2n$$ times the number of bits of $$x$$: the evaluated polynomial has approximate magnitude $$x^n$$, and one must also store $$x^n$$ itself. By contrast, Horner's method requires only $$n$$ additions and $$n$$ multiplications, and its storage requirements are only $$n$$ times the number of bits of $$x$$. Alternatively, Horner's method can be computed with $$n$$ fused multiply–adds. Horner's method can also be extended to evaluate the first $$k$$ derivatives of the polynomial with $$kn$$ additions and multiplications.

Horner's method is optimal, in the sense that any algorithm to evaluate an arbitrary polynomial must use at least as many operations. Alexander Ostrowski proved in 1954 that the number of additions required is minimal. Victor Pan proved in 1966 that the number of multiplications is minimal. However, when $$x$$ is a matrix, Horner's method is not optimal.

This assumes that the polynomial is evaluated in monomial form and no preconditioning of the representation is allowed, which makes sense if the polynomial is evaluated only once. However, if preconditioning is allowed and the polynomial is to be evaluated many times, then faster algorithms are possible. They involve a transformation of the representation of the polynomial. In general, a degree-$$n$$ polynomial can be evaluated using only $2$+2 multiplications and $$n$$ additions.

Parallel evaluation
A disadvantage of Horner's rule is that all of the operations are sequentially dependent, so it is not possible to take advantage of instruction level parallelism on modern computers. In most applications where the efficiency of polynomial evaluation matters, many low-order polynomials are evaluated simultaneously (for each pixel or polygon in computer graphics, or for each grid square in a numerical simulation), so it is not necessary to find parallelism within a single polynomial evaluation.

If, however, one is evaluating a single polynomial of very high order, it may be useful to break it up as follows: $$\begin{align} p(x) & = \sum_{i=0}^n a_i x^i \\[1ex] & = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n \\[1ex] & = \left( a_0 + a_2 x^2 + a_4 x^4 + \cdots\right) + \left(a_1 x + a_3 x^3 + a_5 x^5 + \cdots \right) \\[1ex] & = \left( a_0 + a_2 x^2 + a_4 x^4 + \cdots\right) + x \left(a_1 + a_3 x^2 + a_5 x^4 + \cdots \right) \\[1ex] & = \sum_{i=0}^{\lfloor n/2 \rfloor} a_{2i} x^{2i} + x \sum_{i=0}^{\lfloor n/2 \rfloor} a_{2i+1} x^{2i} \\[1ex] & = p_0(x^2) + x p_1(x^2). \end{align}$$

More generally, the summation can be broken into k parts: $$p(x) = \sum_{i=0}^n a_i x^i = \sum_{j=0}^{k-1} x^j \sum_{i=0}^{\lfloor n/k \rfloor} a_{ki+j} x^{ki} = \sum_{j=0}^{k-1} x^j p_j(x^k)$$ where the inner summations may be evaluated using separate parallel instances of Horner's method. This requires slightly more operations than the basic Horner's method, but allows k-way SIMD execution of most of them. Modern compilers generally evaluate polynomials this way when advantageous, although for floating-point calculations this requires enabling (unsafe) reassociative math.

Application to floating-point multiplication and division
Horner's method is a fast, code-efficient method for multiplication and division of binary numbers on a microcontroller with no hardware multiplier. One of the binary numbers to be multiplied is represented as a trivial polynomial, where (using the above notation) $$a_i = 1$$, and $$x = 2$$. Then, x (or x to some power) is repeatedly factored out. In this binary numeral system (base 2), $$x = 2$$, so powers of 2 are repeatedly factored out.

Example
For example, to find the product of two numbers (0.15625) and m: $$\begin{align} (0.15625) m & = (0.00101_b) m = \left( 2^{-3} + 2^{-5} \right) m = \left( 2^{-3})m + (2^{-5} \right)m \\ & = 2^{-3} \left(m + \left(2^{-2}\right)m\right) = 2^{-3} \left(m + 2^{-2} (m)\right). \end{align}$$

Method
To find the product of two binary numbers d and m:
 * 1) A register holding the intermediate result is initialized to d.
 * 2) Begin with the least significant (rightmost) non-zero bit in m. 1. Count (to the left) the number of bit positions to the next most significant non-zero bit. If there are no more-significant bits, then take the value of the current bit position.

2. Using that value, perform a left-shift operation by that number of bits on the register holding the intermediate result
 * 1) If all the non-zero bits were counted, then the intermediate result register now holds the final result.  Otherwise, add d to the intermediate result, and continue in step 2 with the next most significant bit in m.

Derivation
In general, for a binary number with bit values ($$ d_3 d_2 d_1 d_0 $$) the product is $$ (d_3 2^3 + d_2 2^2 + d_1 2^1 + d_0 2^0)m = d_3 2^3 m + d_2 2^2 m + d_1 2^1 m + d_0 2^0 m. $$ At this stage in the algorithm, it is required that terms with zero-valued coefficients are dropped, so that only binary coefficients equal to one are counted, thus the problem of multiplication or division by zero is not an issue, despite this implication in the factored equation: $$ = d_0\left(m + 2 \frac{d_1}{d_0} \left(m + 2 \frac{d_2}{d_1} \left(m + 2 \frac{d_3}{d_2} (m)\right)\right)\right). $$

The denominators all equal one (or the term is absent), so this reduces to $$ = d_0(m + 2 {d_1} (m + 2 {d_2} (m + 2 {d_3} (m)))),$$ or equivalently (as consistent with the "method" described above) $$ = d_3(m + 2^{-1} {d_2} (m + 2^{-1}{d_1} (m + {d_0} (m)))). $$

In binary (base-2) math, multiplication by a power of 2 is merely a register shift operation. Thus, multiplying by 2 is calculated in base-2 by an arithmetic shift. The factor (2−1) is a right arithmetic shift, a (0) results in no operation (since 20 = 1 is the multiplicative identity element), and a (21) results in a left arithmetic shift. The multiplication product can now be quickly calculated using only arithmetic shift operations, addition and subtraction.

The method is particularly fast on processors supporting a single-instruction shift-and-addition-accumulate. Compared to a C floating-point library, Horner's method sacrifices some accuracy, however it is nominally 13 times faster (16 times faster when the "canonical signed digit" (CSD) form is used) and uses only 20% of the code space.

Other applications
Horner's method can be used to convert between different positional numeral systems – in which case x is the base of the number system, and the ai coefficients are the digits of the base-x representation of a given number – and can also be used if x is a matrix, in which case the gain in computational efficiency is even greater. However, for such cases faster methods are known.

Polynomial root finding
Using the long division algorithm in combination with Newton's method, it is possible to approximate the real roots of a polynomial. The algorithm works as follows. Given a polynomial $$p_n(x)$$ of degree $$n$$ with zeros $$ z_n < z_{n-1} < \cdots < z_1,$$ make some initial guess $$ x_0 $$ such that $$ z_1 < x_0 $$. Now iterate the following two steps:

 * 1) Using Newton's method, find the largest zero $$z_1$$ of $$p_n(x)$$ using the guess $$x_0$$.
 * 2) Using Horner's method, divide out $$(x-z_1)$$ to obtain $$p_{n-1}$$. Return to step 1 but use the polynomial $$p_{n-1}$$ and the initial guess $$z_1$$.

These two steps are repeated until all real zeros are found for the polynomial. If the approximated zeros are not precise enough, the obtained values can be used as initial guesses for Newton's method but using the full polynomial rather than the reduced polynomials.

Example

Consider the polynomial $$p_6(x) = (x+8)(x+5)(x+3)(x-2)(x-3)(x-7)$$ which can be expanded to $$p_6(x) = x^6 + 4x^5 - 72x^4 -214x^3 + 1127x^2 + 1602x -5040.$$

From the above we know that the largest root of this polynomial is 7 so we are able to make an initial guess of 8. Using Newton's method the first zero of 7 is found as shown in black in the figure to the right. Next $$p(x)$$ is divided by $$(x-7)$$ to obtain $$p_5(x) = x^5 + 11x^4 + 5x^3 - 179x^2 - 126x + 720$$ which is drawn in red in the figure to the right. Newton's method is used to find the largest zero of this polynomial with an initial guess of 7. The largest zero of this polynomial which corresponds to the second largest zero of the original polynomial is found at 3 and is circled in red. The degree 5 polynomial is now divided by $$(x-3)$$ to obtain $$p_4(x) = x^4 + 14x^3 + 47x^2 - 38x - 240$$ which is shown in yellow. The zero for this polynomial is found at 2 again using Newton's method and is circled in yellow. Horner's method is now used to obtain $$p_3(x) = x^3 + 16x^2 + 79x + 120$$ which is shown in green and found to have a zero at −3. This polynomial is further reduced to $$p_2(x) = x^2 + 13x + 40$$ which is shown in blue and yields a zero of −5. The final root of the original polynomial may be found by either using the final zero as an initial guess for Newton's method, or by reducing $$p_2(x)$$ and solving the linear equation. As can be seen, the expected roots of −8, −5, −3, 2, 3, and 7 were found.

Divided difference of a polynomial
Horner's method can be modified to compute the divided difference $$(p(y) - p(x))/(y - x).$$ Given the polynomial (as before) $$p(x) = \sum_{i=0}^n a_i x^i = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n,$$ proceed as follows $$\begin{align} b_n & = a_n,                &\quad d_n &= b_n, \\ b_{n-1} & = a_{n-1} + b_n x, &\quad d_{n-1} &= b_{n-1} + d_n y, \\ & {}\ \  \vdots             &\quad &  {}\ \ \vdots\\ b_1 & = a_1 + b_2 x,        &\quad d_1 &= b_1 + d_2 y,\\ b_0 & = a_0 + b_1 x. \end{align}$$

At completion, we have $$\begin{align} p(x) &= b_0, \\ \frac{p(y) - p(x)}{y - x} &= d_1, \\ p(y) &= b_0 + (y - x) d_1. \end{align}$$ This computation of the divided difference is subject to less round-off error than evaluating $$p(x)$$ and $$p(y)$$ separately, particularly when $$ x \approx y$$. Substituting $$y = x$$ in this method gives $$d_1 = p'(x)$$, the derivative of $$p(x)$$.

History
Horner's paper, titled "A new method of solving numerical equations of all orders, by continuous approximation", was read before the Royal Society of London, at its meeting on July 1, 1819, with a sequel in 1823. Horner's paper in Part II of Philosophical Transactions of the Royal Society of London for 1819 was warmly and expansively welcomed by a reviewer in the issue of The Monthly Review: or, Literary Journal for April, 1820; in comparison, a technical paper by Charles Babbage is dismissed curtly in this review. The sequence of reviews in The Monthly Review for September, 1821, concludes that Holdred was the first person to discover a direct and general practical solution of numerical equations. Fuller showed that the method in Horner's 1819 paper differs from what afterwards became known as "Horner's method" and that in consequence the priority for this method should go to Holdred (1820).

Unlike his English contemporaries, Horner drew on the Continental literature, notably the work of Arbogast. Horner is also known to have made a close reading of John Bonneycastle's book on algebra, though he neglected the work of Paolo Ruffini.

Although Horner is credited with making the method accessible and practical, it was known long before Horner. In reverse chronological order, Horner's method was already known to:

 * Paolo Ruffini in 1809 (see Ruffini's rule)
 * Isaac Newton in 1669
 * the Chinese mathematician Zhu Shijie in the 14th century
 * the Chinese mathematician Qin Jiushao in his Mathematical Treatise in Nine Sections in the 13th century
 * the Persian mathematician Sharaf al-Dīn al-Ṭūsī in the 12th century (the first to use that method in a general case of cubic equation)
 * the Chinese mathematician Jia Xian in the 11th century (Song dynasty)
 * The Nine Chapters on the Mathematical Art, a Chinese work of the Han dynasty (202 BC – 220 AD) edited by Liu Hui (fl. 3rd century).

Qin Jiushao, in his Shu Shu Jiu Zhang (Mathematical Treatise in Nine Sections; 1247), presents a portfolio of methods of Horner-type for solving polynomial equations, which was based on earlier works of the 11th century Song dynasty mathematician Jia Xian; for example, one method is specifically suited to bi-quintics, of which Qin gives an instance, in keeping with the then Chinese custom of case studies. Yoshio Mikami in Development of Mathematics in China and Japan (Leipzig 1913) wrote:"'... who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe ... We of course don't intend in any way to ascribe Horner's invention to a Chinese origin, but the lapse of time sufficiently makes it not altogether impossible that the Europeans could have known of the Chinese method in a direct or indirect way.'" Ulrich Libbrecht concluded: It is obvious that this procedure is a Chinese invention ... the method was not known in India. He said, Fibonacci probably learned of it from Arabs, who perhaps borrowed from the Chinese. The extraction of square and cube roots along similar lines is already discussed by Liu Hui in connection with Problems IV.16 and 22 in Jiu Zhang Suan Shu, while Wang Xiaotong in the 7th century supposes his readers can solve cubics by an approximation method described in his book Jigu Suanjing.

title: Horner's method


text:
Hapworth 16, 1924

"Hapworth 16, 1924" is an uncollected work of short fiction by J. D. Salinger that appeared in the June 19, 1965, issue of The New Yorker.

The story is the last original work Salinger published during his lifetime, and filled almost the entire magazine. It is the "youngest" of his Glass family stories, in the sense that the narrated events happen chronologically before those in the rest of the series.

Plot
46-year-old Buddy Glass reproduces the contents of a letter written by his older brother Seymour, who died by suicide 17 years earlier in 1948. Seymour wrote the letter to their parents while he and Buddy (two years his junior) were attending Camp Simon Hapworth, Maine, in 1924. The literary voice conveyed in the letter is that of a highly articulate and strikingly precocious boy of seven. The letter, written from the camp infirmary (Seymour has injured his leg) is a wide-ranging commentary on the camp personnel, the camp attendees, and his relationships with his family, humanity and God. Seymour and Buddy largely prefer to occupy themselves writing poems and short stories rather than participate in group activities. They therefore meet with some hostility.

Seymour devotes a large part of the letter to enumerating his reading list and requests for further reading material from his parents. He offers critical appraisals of a number of major literary figures. The letter closes with a lengthy discourse on the significance of God.

Publishing history
The circumstances and considerations that led chief fiction editor William Shawn at The New Yorker to devote virtually the entire June 19, 1965, edition to "Hapworth 16, 1924" are obscure. Biographer Kenneth Slawenki writes, "the files of The New Yorker are unusually silent on the details of the novella's reception by the editorial staff and its eventual reception by William Shawn." The correspondence between Salinger and Shawn chronicling the decision may have been deliberately suppressed. Slawenski speculates that the appearance of Salinger's piece in the journal was "a fait accompli rather than a topic of debate".

After the story's appearance in The New Yorker, Salinger—who had already withdrawn to his New Hampshire home—stopped publishing altogether. Since the story never appeared in book form, readers had to seek out that issue or find it on microfilm. Finally, with the release of The Complete New Yorker on DVD in 2005, the story was once again widely available.

In 1996, Orchises Press, a small Virginia publishing house, started a process of publishing "Hapworth" in book form. Orchises Press owner Roger Lathbury has described the effort in The Washington Post and, three months after Salinger's death, in New York magazine. According to Lathbury, Salinger was deeply concerned with the proposed book's appearance, even visiting Washington to examine the cloth for the binding. Salinger also sent Lathbury numerous "infectious and delightful and loving" letters.

Following publishing norms, Lathbury applied for Library of Congress Cataloging in Publication data, unaware of how publicly available the information would be. A writer in Seattle, researching an article on Jeff Bezos, came across the "Hapworth" publication date, and told his sister, a journalist for the Washington Business Journal, who wrote an article about the upcoming book. This led to substantial coverage in the press. Shortly before the books were to be shipped, Salinger changed his mind, and Orchises withdrew the book. New publication dates were repeatedly announced, but it never appeared. Lathbury said, "I never reached back out. I thought about writing some letters, but it wouldn't have done any good."

Reception and assessment
Both contemporary and later literary critics harshly panned "Hapworth 16, 1924"; writing in The New York Times, Michiko Kakutani called it "a sour, implausible and, sad to say, completely charmless story .... filled with digressions, narcissistic asides and ridiculous shaggy-dog circumlocutions."

Calling it "virtually unreadable" and "an enigma", critic John Wenke compares "Hapworth" to viewing a neighbor's unedited family home movies. He writes:

"Possibly the least structured and most tedious piece of fiction ever produced by an important writer, 'Hapworth' seems designed to bore, to tax patience, as if Salinger was trying to torment his readers."

Wenke adds that the story is a striking departure from the "urbane, pithy and wry" short fiction The New Yorker's editors and readership favored.

Biographer Kenneth Slawenski considers the piece "professionally, a disaster" and ponders what may have motivated Salinger to submit the work for publication:

"Questions regarding 'Hapworth' have plagued Salinger fans ever since. Did he intentionally write the story as his final publication? Why is 'Hapworth' so unreadable? The story fostered a suspicion that...Salinger attempted to release himself from the affections of average readers by feeding them a work that was completely indigestible."

Biographer Ian Hamilton concurs that Salinger appears to abandon his loyal readership and retreat into the exclusive realm of his characters. He writes, "The Glass family has, in this last story, become Salinger's subject and his readership, his creatures and his companions. His life is finally made one with art."

Salinger is said to have considered the story a "high point of his writing" and made tentative steps to have it reprinted, though those came to nothing.

title: Hapworth 16, 1924