Nader Davoodi (born 9 January 1963) is an Iranian photojournalist. For the past two decades, he has been working to produce ethnographically detailed works that document a critical period in contemporary Iranian history. He is the first Iranian sports photographer to cover the FIFA World Cup (USA 1994), and his photo of  Yordan Lechkov's goal against Germany won him the best sports picture of the year award in the annual Iranian sport's photo contest.

Career
His coverage of the 12th Asian Games (Hiroshima 1994) is a precious collection of the games & sidelines of the most important Asian sports event, while 1994, 1998 & 2006 World Cups in the US, France & Germany were covered as the most important international sport occasions. In 2000, he covered the Sydney Olympic Games, publishing a book entitled Sydney 2000. The book includes the most amazing sidelines covering different nationalities in exceptional moments of happiness & passion.
He is a founding member of The International Soccer Cartoon Contest, held in 1999 for the first time.

He is one of the first Iranian photographers to have an exhibition in the month of photography in Toronto (Contact 2005 ), with the title Breaking the Rules at the University of Toronto. Currently residing in Toronto, Ontario, he is a Founding Member of the Iranian Photojournalists Association (1998) and one of the first to create an all English photoblog on Iran (July 2005). He is the publisher of Tamashagaran Weekly.

He has published four books titled Iran on its way to the World Cup, A simple Event, Iranian Artifacts in the Metropolitan Museum of Art, and Sydney 2000 as writer and editor. His documentary, 13 and a Half, co-directed with Abbas Ahmadi, premiered on October 1, 2006, at the 25th Vancouver International Film Festival.

See also
 Culture of Iran
 Islamic art
 Iranian art
 Iranian art and architecture
 List of Iranian artists

References

External links
Nader Davoodi's Collection

Iranian photojournalists
Sports photographers
1963 births
Living people

Nader Davoodi


Natural magic in the context of Renaissance magic is that part of the occult which deals with natural forces directly, as opposed to ceremonial magic which deals with the summoning of spirits. Natural magic sometimes makes use of physical substances from the natural world such as stones or herbs.

Natural magic so defined includes astrology, alchemy, and disciplines that we would today consider fields of natural science, such as astronomy and chemistry (which developed and diverged from astrology and alchemy, respectively, into the modern sciences they are today) or botany (from herbology). The Jesuit scholar Athanasius Kircher wrote that "there are as many types of natural magic as there are subjects of applied sciences".

Heinrich Cornelius Agrippa discusses natural magic in his Three Books of Occult Philosophy (1533), where he calls it "nothing else but the highest power of natural sciences". The Italian Renaissance philosopher Giovanni Pico della Mirandola, who founded the tradition of Christian Kabbalah, argued that natural magic was "the practical part of natural science" and was lawful rather than heretical.

See also

References

Further reading

History of science
Renaissance
Magic (supernatural)

Natural magic


Karel Václav Klíč (sometimes written Karl Klietsch, 30 May 1841, Hostinné – 16 November 1926, Vienna) was a Czech painter, photographer, early comics artist, caricaturist, lithographer and illustrator. He was one of the inventors of photogravure (heliogravura in Czech).

Klíč had such artistic talent that he was admitted into the Art Academy in Prague at the age of 14. For ridiculing school officials he was soon expelled, but eventually finished the school in 1862. Klíč worked as a photographer, caricaturist and illustrator in Brno, Budapest and Vienna, all the time trying to improve the technology of picture reproduction. During a long night in 1877, while working with zinc relief etching, he discovered, by chance, a process leading to photogravure. He further improved the process in 1890, when working in England.

Sources

External links

 Lambiek Comiclopedia article.
 Very short biography
 Biography (in Czech)
 Description of photogravure

1841 births
1926 deaths
People from Hostinné
Czech illustrators
Czech lithographers
Czech caricaturists
Czech comics artists
Czech inventors
19th-century Czech painters
Czech male painters
20th-century Czech painters
19th-century Czech male artists
20th-century Czech male artists

Karel Klíč


Beaumont was a make of mid-sized automobiles produced by General Motors of Canada from 1964 to 1969.  These cars were based on the Chevrolet Chevelle, but the line had its own logo and nameplate, and was neither marketed nor actively sold in the United States.  Its logo consisted of an arrow, similar to that of Pontiac, but with a maple leaf to signify its dual heritage from both sides of Lake Ontario.

1962-1965: As an Acadian

To promote automobile manufacturing in Canada, the Auto Pact (APTA) in the 1960s had provisions prohibiting sales of certain American-made cars.  General Motors responded by offering certain makes of cars manufactured in Canada primarily for the Canadian market such as Acadian and Beaumont.

Beaumont started out as a trim level of the Acadian line from 1962 to 1965.  The Acadian from 1962 to 1969 was based on the contemporary Chevrolet Chevy II (Nova).  Beaumonts were sold at Pontiac-Buick Dealers primarily for the Canadian market, but were also made and sold in some countries outside of North America.

In 1962, the Acadian was offered in eight models with either base Invader or deluxe Beaumont series.  The Beaumont provided extra quality trim, identifications, and luxury.  Items such as foam-cushioned rear seats, horn ring on steering wheel, rear armrests, and automatic front door dome light switches were standard Beaumont features.

In 1963, the Acadian was offered in four series: base Invader, midlevel Canso, deluxe Beaumont, or Beaumont Sport Deluxe.  The Beaumont again offered similar trim, identification, and luxury as the 1962 model did.  The Beaumont Sport Deluxe was equivalent to the Chevy II Nova Super Sport, added deluxe identification and a substantial number of luxury items - upgraded upholstery and trim in six possible colours, extra cushion padding in bucket seats and rear seats, deluxe door handles, glove box light, and chrome-plated heat control and instrument panel knobs.  With a Powerglide automatic transmission or three speed manual, a console and floor-mounted shift lever were standard on the Beaumont Sport Deluxe with the Powerglide only. Only a 6 cylinder was available.

In 1964 and 1965, GM of Canada sold an "Acadian Beaumont" based on the Chevelle A-body platform, and continued to sell the Chevy II-based Acadian.  By 1966, the Acadian was clearly Chevy II-based, and the Beaumont was Chevelle-based. Many people considered the cars to be Pontiacs, but GM marketed them as a separate marque. All model years are primarily Chevrolet, with only minor trim differences and (for Chevelle-based cars) Pontiac Tempest/LeMans-based instrument panels. This is because Canada required GM to have a certain percentage of Canadian content in the vehicles sold domestically. The popular Beaumont became available in three series on the Acadian - Beaumont Standard, Beaumont Custom, and Beaumont Sport Deluxe.  A base-trimmed Beaumont Standard convertible was also offered, a model also produced in the base Chevelle line in Canada, but not in the United States.

1966-1969: As a standalone marque

From 1966, the Acadian name was only used for the Nova-based car and Beaumont became a standalone marque, still sold by Pontiac-Buick dealers.  The interiors used the dash panel from the (U.S.) Pontiac Tempest/LeMans/GTO series. Exterior sheetmetal is shared with the Chevelle (the rear taillight housings are Beaumont-exclusive - taillight lenses have wraparound corners with the exception of the 1967 model, where its trapezoidal shape is similar in appearance to the one used on the later 1969 Plymouth Belvedere/Satellite). While full wheel covers came from the Pontiac LeMans, small hubcaps and rally wheels were similarly trimmed but not the same as on the Chevelle.

The cars sported an emblem based on Pontiac's arrowhead motif with two red maple leaves or fleur-de-lis added. They featured the same powerplants as the Chevrolet Chevelle, including the OHV inline six-cylinder engine, and a variety of small- and big-block V8s.  The V8 engine choices included small-block 283, 307, 327, and later 350 cubic-inch versions, while the Mark IV big-block could be ordered in its 396 cubic inch displacement from 1966 to 1969.  3- and 4-speed manual transmissions were available, as were the 2-speed Powerglide and 3-speed Turbo Hydra-Matic starting in 1967. A popular configuration for both the Beaumont and its sister, the Chevrolet Chevelle, were the six cylinder engines or the small block 283 V8, and to a lesser extent, the 327 V8 with the Powerglide transmission, allowing for fuel economy and reasonable performance at higher RPM. Models equipped with the 2 speed Powerglide were low on performance at the line and at low RPMs, but capable of exacting more torque and performance at higher speeds and RPM, simply due to the configuration of the transmission, and the fact there were only 2 forward gears.

The Sport Deluxe (SD) models were equivalent to the Chevrolet Chevelle Super Sport trim level, and featured optional bucket seats and center console, as well as SD body striping and trim.  The SD396 models are the most desirable Beaumonts today.  Only 451 were built in 1967, but most succumbed to the harsh Canadian winter climate, which makes them significantly more rare than equivalent Chevelles and desirable to some collectors.  The SD was actually the "Sports Option" package that was available on the Beaumont Custom Sport Coupe and convertible, rather than its own series early in 1967. Starting in December the model 738 Sport Deluxe model was separate, no longer the A51 Sport Option.  In addition to the SD series, the Beaumont line included base, custom and deluxe lines.  A convertible was available.  Other body styles were identical to those offered on the Chevelle for the given year, including a very rare four-door hardtop offered from 1966 to 1969. The SD 396 version of the Beaumont built in 1966, in particular with 4 speed manual transmissions are exceedingly rare, as only 41 were ever built.

Starting in 1970, GM Canada discontinued the Beaumont in favor of both Chevrolet Chevelle and Pontiac LeMans midsized cars, which were identical to U.S. models.

Production and sales out of Canada
Beaumonts were also made in a GM Overseas Operation factory in Arica, Chile, for the local market (the TECNA assembly plant assembled Beaumonts from CKD kits until 1971 using the 1968 bodystyle introduced during 1969), and the Canadian-made cars were sold in Puerto Rico for a time. Beaumonts were also locally assembled in South Africa at the GM Port Elizabeth assembly plant using CKD kits with a mirror-flipped 1968 Chevelle instrument cluster and dash (South African market cars are right-hand drive). Both Chile and South Africa commenced local assembly with CKD kits and local content components to avoid import tariffs.

Gallery

References

Further reading
Gunnell, John, "Filling in the Blanks: Canada's 1964-69 Beaumonts", Publication International, Ltd., Collectible Automobile, Volume 39 Number 3, October 2022, page 62.

External links

Tuerca.cl - Acadian y Beaumont made in Chile - History of chilean assembled Beaumont (in spanish)

General Motors marques
General Motors vehicles
Cars of Canada
Mid-size cars
Sedans
Coupés
Convertibles
Muscle cars
Station wagons
Rear-wheel-drive vehicles
Cars introduced in 1966

Beaumont (automobile)


Bruce Lester (6 June 1912 – 13 June 2008) was a South African-born English film actor with over 60 screen appearances to his credit between 1934 and his retirement from acting in 1958. Lester's career divided into two distinct periods. Between 1934 and 1938, billed as Bruce Lister, he appeared in upwards of 20 British films, mostly of the cheaply shot and quickly forgotten quota quickie variety. He then moved to the US, where he changed his surname to Lester, and found himself for a time appearing in some of the biggest prestige productions of their day, alongside stars such as Bette Davis, Joan Crawford, Tyrone Power and Errol Flynn. Lester himself never achieved star-billing, but was said to have remarked that this at least meant that if a film was a flop, no blame ever fell on his shoulders.

Early life
Born in Johannesburg to English parents, Lester was educated at Brighton College in Sussex. He dabbled in amateur dramatics, but also showed potential as an exceptionally talented tennis player, and was considering trying to make a living from the sport until friends convinced him that acting offered far greater earning potential.

Film career

1930s
After spending some time as a stage actor, Lester obtained his first minor film role in the 1934 quota quickie The Girl in the Flat. His performance was noticed and a good career opportunity presented itself straight away when he was offered the male romantic lead in To Be a Lady, opposite Chili Bouchier, one of Britain's top female stars of the time. Far from a quota quickie, this was a quality production with a generous budget, but failed to perform as well as expected at the box-office. This was followed by another romantic lead, this time with Valerie Hobson, in Badger's Green, a film in which cricket played a major part and which was praised for its "Englishness". Both of these films are believed lost, but are deemed of great interest by the British Film Institute and are both included on its "75 Most Wanted" list of missing British feature films.

Lester's career then began to stagnate. The most enduring British film in which he appeared was Death at Broadcasting House, also from 1934, although his role was relatively minor. The next few years were spent toiling mainly in low-budget programmes, often in insignificant roles. Things took an upturn in 1938, when he was signed to Teddington Studios, the British division of Warner Brothers. He was promptly sent to the US, where he was cast in the James Cagney vehicle Boy Meets Girl. He was then signed to a three-year Hollywood contract, reportedly at Cagney's urging.
One of his first Hollywood assignments was the high-profile historical drama If I Were King.

1940s and 1950s
Lester's film career hit its peak in the early 1940s, beginning with the role of Mr. Bingley in the 1940 adaptation of Pride and Prejudice, followed by landing a plum role in the Bette Davis classic The Letter. Lester and Davis shared some crucial scenes in the film, and it was soon being reported that Davis was very taken by the handsome young Englishman, going out of her way to help him and put him at his ease on set. It is generally believed that a brief relationship of some description developed between the two, although nothing came of it. Writer Jerry Asher, a long-time friend of Davis, recalled: "She found Bruce attractive and sweet, but he was a bit tame for her speed."

At this time Lester briefly returned to Britain, intending to join the army. However he was persuaded that his talents would be better employed in helping to promote the Allied cause in the US. He returned, and appeared in several films in which he played flying officers, including Man Hunt, A Yank in the RAF and Desperate Journey. In 1943 he played in the Joan Crawford and Fred MacMurray spy drama Above Suspicion. He did then decide to join the British services, serving through to the end of the war.

On his return to Hollywood, Lester's career faltered, with both the quality of film and size of role he was being offered tapering off. He did land a good part in 1947's Golden Earrings with Ray Milland and Marlene Dietrich, then returned for a time to Europe, where he starred in But Not in Vain (1948 – filmed in the Netherlands) and Celia: The Sinister Affair of Poor Aunt Nora (1949).

Returning to the US, Lester began to turn his hand to working as a story analyst for Paramount and Columbia. In the early 1950s he was playing minor roles in low-budget versions of English-set historical dramas such as Tales of Robin Hood and Lorna Doone. As the decade progressed film parts became ever-smaller, until he was mainly playing uncredited bit parts. Lester's final screen credit came in 1958's Tarzan and the Trappers, after which he decided to retire from acting and spend the rest of his life out of the spotlight.

Later life
In later years, Lester and his wife divided their time between Los Angeles and Palm Springs. He continued to play tennis, and was a keen member of the Hollywood Cricket Club. Lester died in Los Angeles on 13 June 2008, one week after his 96th birthday.

Filmography

1934: The Girl in the Flat - Bit Role (uncredited)
1934: To Be a Lady - Jerry Dean
1934: Badger's Green - Dickie Wetherby
1934: Death at Broadcasting House - Peter Ridgwell (uncredited)
1934: The Third Clue - Derek Clayton
1935: Old Faithful - Chris Morgan
1935: Old Roses - Alf Haines
1936: Ourselves Alone - 2nd Lieut. Lingard
1936: A Star Fell from Heaven - Winkler
1936: The Tenth Man - Edward O'Donnell
1936: Crime Over London - Ronald Martin
1936: Hail and Farewell - Peter Dove
1936: The Heirloom Mystery - Alf Dene
1936: Head Office - Halliday
1937: Mayfair Melody - Dickie
1937: The Windmill - Officer
1937: Ship's Concert (Short) - Purser
1937: Change for a Sovereign - Prince William
1938: Quiet Please - Dr. Faversham
1938: Thistledown - Lord James Dunfoyle
1938: Boy Meets Girl - Rodney Bowman
1938: If I Were King - Noel le Jolys
1939: Home from Home - Jim
1939: The Witness Vanishes - Lord Noel Stratton
1940: The Invisible Man Returns - Chaplain (uncredited)
1940: British Intelligence - Frank Bennett
1940: My Son, My Son! - Rory O'Riorden
1940: Pride and Prejudice - Mr. Bingley
1940: The Letter - John Withers
1941: Shadows on the Stairs - Hugh Bromilow
1941: Singapore Woman - Clyde
1941: Shining Victory - Dr. Bentley
1941: Man Hunt - Co-Pilot (uncredited)
1941: A Yank in the RAF - Flight Lt. Sterling Richardson
1942: Eagle Squadron - King (uncredited)
1942: Desperate Journey - English Officer
1943: The Mysterious Doctor - Lt. Christopher 'Kit' Hilton
1943: Above Suspicion - Thornley
1943: Flesh and Fantasy - Young Man (uncredited)
1946: The Wife of Monte Cristo - (uncredited)
1946: Without Reservations - Army Lieutenant (uncredited)
1946: Cloak and Dagger - British Officer (uncredited)
1946: Strange Journey - Roger Blythe
1947: Golden Earrings - Richard Byrd
1947: I Walk Alone - Charles (uncredited)
1948: But Not in Vain - Fred Van Nespen
1949: The Fool and the Princess - Harry Granville
1949: Celia: The Sinister Affair of Poor Aunt Nora - Larry Peters
1951: Tarzan's Peril - Warden (uncredited)
1951: Tales of Robin Hood - Alan-a-Dale
1951: Lorna Doone - Walt Snowe (uncredited)
1951: Secrets of Monte Carlo - British Inspector
1951: The Son of Dr. Jekyll - Daniels' Reporter (uncredited)
1952: The Pathfinder - Capt. Clint Bradford
1953: Rogue's March - Wilson (uncredited)
1953: Storm over Africa - Saxon
1954: King Richard and the Crusaders - Castelaine
1954: Charade - Capt. van Buren
1955: The Sea Chase - Petty Officer (uncredited)
1955: The Scarlet Coat - Col. Simcoe (uncredited)
1957: Something of Value - Physician - McKenzie Massacre (uncredited)
1957: The Seventh Sin - Allan (uncredited)
1958: Tarzan and the Trappers (TV Movie) - Commissioner Brandini (final film role)

References

External links

1912 births
2008 deaths
English male film actors
Actors from Johannesburg
English expatriates in the United States
People educated at Brighton College
Burials at Westwood Village Memorial Park Cemetery
South African emigrants to the United Kingdom

Bruce Lester


Melaleuca squamophloia is a plant in the myrtle family, Myrtaceae and is endemic to the black soil plains of south eastern Queensland in Australia. Like its close relative Melaleuca styphelioides, it is a small, erect tree with prickly leaves and spikes of cream or white flowers but its bark is hard rather than papery and the leaves have fewer veins than that species.

Description
Melaleuca squamophloia is a shrub or small tree growing to  high, with hard, scaly or fibrous bark. Its leaves are arranged alternately,  long,  wide, narrow egg-shaped with the end tapering to a sharp, prickly point and with 3 to 15 veins.

The flowers are white or cream-coloured and are arranged in spikes on the ends of branches which continue to grow after flowering and on the sides of the branches. Each spike contains 5 to 16 individual flowers and is up to  in diameter. The petals are  long and fall off as the flower opens or soon after. There are five bundles of stamens around the flower, each with 15 to 20 stamens. Flowering occurs in spring and from April to June, and is followed by fruit which are woody, almost spherical capsules,  long.

Taxonomy and naming
Melaleuca squamophloia was first named in 1997 by Lyndley Craven and Bryan Barlow in Novon. It was first formally described in 1984 as Melaleuca styphelioides var. squamophloia by Norman Byrnes. The specific epithet (squamophloia) is from the Latin word squama meaning "scale" and the Ancient Greek word φλοιός (phloiós) meaning "bark" referring to the bark of this species which is hard and scaly.

Distribution and habitat
This melaleuca occurs on the black soil plains in and between the Miles, Jandowae and Tara districts of south east Queensland. It grows in woodland and scrubland in clay and clay loam.

References

squamophloia
Myrtales of Australia
Flora of Queensland
Plants described in 1984

Melaleuca squamophloia


Clearpath Robotics, Inc. (also known as Clearpath) was founded in 2009 by a group of four University of Waterloo graduates, and remains headquartered in Waterloo Region, Canada. The original goal of Clearpath was to streamline field robotics research for universities and private corporations, but the company has since expanded and is now also manufacturing and selling the OTTO line of self-driving vehicles for industrial environments.

History
Clearpath Robotics was founded in 2009 by a group of four University of Waterloo graduates—Matthew Rendall, Ryan Gariepy, Pat Martinson and Bryan Webb—based on a growing demand for environmental monitoring and robotics research equipment. Unlike other robotics startups, they only raised a small amount of angel investment upon founding, and sought to achieve profitability before raising more.

In 2012, Open Robotics (then the Open Source Robotics Foundation) was established to ensure that there was continuity in the work being done with ROS, with Clearpath CTO Ryan Gariepy joining the founding board. Not too long after, original ROS creator Willow Garage announced a reduction in formal support. Due to concerns raised by the academic community regarding the future of robotics research, Clearpath issued a statement to reinforce that it would continue to do what it could to support ROS.

With the complete shutdown of Willow Garage in early 2014, Clearpath took over complete maintenance and support of the PR2 robot until through at least 2016.

In August 2014, Clearpath became the first robotics company in the world to pledge not to make 'killer robots' (lethal autonomous weapons), and continues to back initiatives to ensure AI and autonomous systems are developed in an ethical fashion.

By March 2015, Clearpath announced a $14 million Series A investment led by RRE Ventures meant to fund the development of products capable of automating dangerous jobs in industrial and manufacturing facilities. Shortly thereafter, they announced the first of their OTTO line of vehicles with General Electric as a first customer and strategic investor, as well as announced that Tesla, Inc. co-founder Marc Tarpenning had joined their advisory board. At the same time, it continued to develop and sell systems meant to fulfill its original mandate of helping researchers work more efficiently.

A Series B funding round was announced in October 2016 led by iNovia Capital, which also added Caterpillar Inc. as another strategic investor.

Clearpath is currently one of the most well-known Canadian robotics startups, and is regularly recognized as one of the "Top 50 Most Influential Companies in Robotics" by Robotics Business Review.

In September 2023, Clearpath Robotics was acquired by an American industrial automation company Rockwell Automation in a deal valued at over million.

Divisions

Clearpath Robotics
The original Clearpath Robotics brand encompasses the manufacture and sale of a variety of land and sea vehicles used for robotics research, as well as the sale of individual components for robotics prototyping through its online store. It also provides custom robotics integration as well as ROS consulting services, with a portion of the revenue of the latter going to support Open Robotics.

Clearpath was one of the original manufacturers and distributors of the TurtleBot series of open-source research robots, with the newest version now being produced in partnership with Open Robotics and iRobot.

OTTO Motors
The complete OTTO Motors brand was announced in 2016, and is Clearpath's first known entry into the large scale deployment of autonomous vehicles. Its first customer was General Electric, and other notable clients include John Deere and Toyota.

Litigation
In August 2016, Clearpath filed a complaint in the Northern district of California against Otto (company) with respect to Clearpath Robotics Inc.'s OTTO brand. The action was dismissed with prejudice on February 1, 2017. As of May 2017, Uber has ceased using the OTTO trademark, and Clearpath continues to operate the OTTO brand.

In light of Waymo LLC v. Uber Technologies, Inc. et al., Clearpath has issued a statement to clarify brand confusion.

References

External links
 Clearpath Robotics Official Site
 OTTO Motors Official Site

Canadian companies established in 2009
Companies based in Kitchener, Ontario
Technology companies of Canada
Robotics companies
2023 mergers and acquisitions

Clearpath Robotics


Diane Renay (born July 13, 1945), born Renee Diane Kushner, is an American pop singer, best known for her 1964 hit song, "Navy Blue".

Early life

Renay was born to a Jewish family in South Philadelphia, Pennsylvania. She started singing at an early age and took voice lessons from Artie Singer, a voice teacher who also managed Danny and the Juniors (of "At the Hop" fame). Singer encouraged Renay to pursue a recording career. Renay attended Northeast High School (Philadelphia, Pennsylvania).

Career
Record producer/songwriter Pete DeAngelis was a frequent customer at the Kushners' family jewelry store, and Renay's parents arranged for her to audition for him. DeAngelis, impressed with her talents, got Renay signed to the Atco Records label. Under the new stage name Diane Renay, she released her first single, "Little White Lies", in 1962, but it failed to chart nationally, as did the follow-up, "A Dime a Dozen", and Atco dropped her from the label.

However, Bob Crewe, who had written and produced material for Renay's second recording session, then signed her to a new recording contract whereby he would write and produce records for her. Under Crewe's guidance and signed to the 20th Century label, Renay, then 17 years old, released her biggest hit, "Navy Blue", in late 1963. The song told the story of a girl, lonely for her steady boyfriend away from home in the U.S. Navy and anxious to see him again. "Navy Blue", composed by Crewe with Bud Rehak and Eddie Rambeau, became a national smash, reaching No. 6 on the Hot 100 on 14–21 March 1964, and soaring to No. 1 on the Adult Contemporary singles chart.  The song was followed by Renay's debut album, also titled Navy Blue.

Renay's only other single release to crack the national Billboard chart was "Kiss Me Sailor", reaching number 29 later in 1964. Subsequent singles, including "Growin' Up Too Fast", "Watch Out Sally", "It's In Your Hands", and "Happy Birthday Broken Heart", were hits in certain local markets such as Salt Lake City, Las Vegas and Miami, but failed to break nationally. Renay moved to the Fontana label in 1969 and attempted a comeback with covers  of "Yesterday" and "Hold Me, Thrill Me, Kiss Me", but these also failed to chart. She did not record again until the early 1980s.

Renay remains active as a performer today and in 2001 released Diane Renay Sings Some Things Old and Some Things New, a double-CD compilation album of her work (including many previously unreleased tracks) from the 1960s through the 1990s.

Album

Singles

See also
List of artists who reached number one on the U.S. Adult Contemporary chart

Bibliography
 Larkin, Colin. The Encyclopedia of Popular Music, 3rd edition, Macmillan, 1998.
 Ruppli, Michel; Novitsky, Ed. The Mercury Labels. A Discography, Vol. V., record and artist indexes, Greenwood Press, 1993.
 Whitburn, Joel. The Billboard Book of Top 40 Hits, 5th edition, Watson-Guptill Publications, 1992.

References

External links

1945 births
American women pop singers
Living people
Singers from Philadelphia
20th-century American Jews
Jewish singers
MGM Records artists
21st-century American Jews
20th-century American women
21st-century American women
Jewish women singers
20th-century American women singers
20th-century American singers
21st-century American women singers

Diane Renay


Phillip Keating was a national correspondent for the Fox News Channel based in the Miami bureau. During his career at Fox, Keating has covered breaking news stories including the death of Anna Nicole Smith, Caylee Anthony, and the trial of Jose Padilla.

He covers NASA stories for Fox News, including NASA's Space Shuttle program, and the new Constellation program. Previously, he reported for the syndicated magazine show Geraldo Rivera on the Fox News channel-produced program Geraldo At Large. Before this, he was a correspondent in the Fox News Channel's Dallas bureau. Keating joined FOX after working at Denver's KUSA and KDVR.

References

1968 births
American television journalists
Living people
Place of birth missing (living people)
Fox News people
American male journalists

Phil Keating


Leyndardómar Reykjavíkur 2000 (The Reykjavík Mysteries 2000) is a crime novel which was written by multiple Icelandic authors, each author writing one chapter. The novel received negative reviews.

The first chapter was written by Viktor Arnar Ingólfsson and then the subsequent chapters by Hrafn Jökulsson, Birgitta Halldórsdóttir, Arnaldur Indriðason, Stella Blómkvist, Árni Þórarinsson and Gunnar Gunnarsson. The final chapter was composed by Kristinn Kristjánsson. The book was printed in 8000 copies and distributed promotionally at bookstores. It was written in less than two months.

Kolbrún Bergþórsdóttir commented that the beginning by Viktor Arnar Ingólfsson is fast-paced and successful and the following chapter by Hrafn Jökulsson has the funniest section of the book. This is followed by a clunky chapter
by Birgitta Halldórsdóttir and a successful one by Arnaldur Indriðason. At that point there is a chapter by Stella Blómkvist which steers the story into such a cliché-ridden form that it can never recover. The story is ultimately "yet another addition to the failed crime stories which have been written in this country in the past few years"

Katrín Jakobsdóttir commented that the initial chapter by Viktor Arnar Ingólfsson is skillfully written and also comments positively on the chapters by Arnaldur Indriðason and Stella Blómkvist. She describes the story as suffering from an overloaded plot and lack of verisimilitude. She is particularly critical of the characterization:

The Achilles heel of the work is the characterization. To begin with, the main character, Páll the policeman, is stuck in a cliché which Dashiell Hammett and Raymond Chandler popularized; he is a lecherous and drunk policeman with a constant hangover in his own private search for justice. The recipe for the character is one part Jack Daniel's stirred with one part sex. In Icelandic crime stories women have generally been even more clichéd and this book is very much in that spirit.

In a later work, Katrín observed that the book was an interesting experiment and even though it had not been particularly successful it was indicative of the rapidly increasing vitality of Icelandic crime literature at the time.

Notes

References

Sources cited

Icelandic novels
Crime novels
2000 novels
Novels set in Iceland
Collaborative novels
Icelandic-language novels

Leyndardómar Reykjavíkur 2000


Camp Blanding Joint Training Center is the primary military reservation and training base for the Florida National Guard, both the Florida Army National Guard and certain nonflying activities of the Florida Air National Guard. The installation is located in Clay County, Florida, near the city of Starke. The site measures about  and includes Kingsley Lake. It also hosts other Reserve, Army National Guard, Air National Guard, and some Active Component training for the U.S. Armed Forces.

Additionally, Camp Blanding serves as a training center for many ROTC units, both Army and Navy, as well as serving for the Florida Wing Summer Encampment for the Civil Air Patrol/US Air Force Auxiliary's Cadet Program each year. Jacksonville University and University of South Florida NROTC Battalions continue to conduct their week-long orientation at Camp Blanding each August before the college semester starts. Camp Blanding also hosts the Audie Murphy Field Training Exercise where Army ROTC units from more than a dozen Florida, Georgia, and Puerto Rican universities gather to conduct a five-day field problem focusing on small-unit tactics, land navigation, and leadership development every April.

Camp Blanding is the primary training site for most of the Florida National Guard's military units and the main combat arms brigade, the 53rd Infantry Brigade of the Florida Army National Guard. It is also home to the headquarters and support companies of the 3-20th Special Forces Group, the 211th Infantry Regiment, and the 2-111th Airfield Operations Battalion  of the 111th Aviation Regiment.

Camp Blanding also houses several non-flying units of the Florida Air National Guard, including the 202nd RED HORSE Squadron, 159th Weather Flight, 131st Training Flight, and the joint Army/Air Force 44th Civil Support Team. The base is also a training location for several counter-drug units and law enforcement agencies in Florida and functions as the alternate Emergency Operations Center  for Florida.

In 2008, Camp Blanding became host to the Army's latest air assault course in response to the growing need for air assault-trained individuals for the continuing operations in Iraq and Afghanistan. Also, a bombing and strafing target for military aircraft, primarily used by the Navy, Marine Corps, and Air Force, is located on the southern portion of the post.

Camp Blanding is home to the Multijurisdictional Counterdrug Task Force Training, which provides unique, tuition-free military and counterdrug training for local, state, federal, and military criminal justice professionals, as well as awareness training for community leaders.

Camp Blanding is also home to the Combating Transnational Organized Crime Center of Excellence, which provides unique, tuition-free training in support of Department of Defense strategies, and has campuses on Camp Blanding, at St. Petersburg College in St. Petersburg, Florida, and on Camp Murray near Tacoma, Washington.

Camp Blanding is also home to the annual FBI National Academy Associates Youth Leadership Academy (YLA). The YLA takes place during the summer and involves a rigorous selection process of 14-16 year olds throughout Florida. YLA exposes students to top-tier law enforcement professionals from across Florida.

History

Camp Blanding was established in 1939 on  as a training facility for the Florida National Guard after its previous training base (Camp Foster) on the St. Johns River near Jacksonville had been taken over by the Navy for Naval Air Station Jacksonville. The new camp was named for Albert H. Blanding, who had been commissioned in the Florida National Guard in 1899, and was then a Major General and Chief of the National Guard Bureau. In 1940, as the threat of war increased and the United States Army was built up, Camp Blanding became a Federal facility housing two infantry divisions plus auxiliary units. Between 1940 and 1943, nine US Army infantry divisions trained at Camp Blanding, including: 1st Infantry Division, 29th Infantry Division, 30th Infantry Division, 31st Infantry Division, 36th Infantry Division, 43rd Infantry Division, 63rd Infantry Division, 66th Infantry Division, and 79th Infantry Division. In 1943, Camp Blanding became an Infantry Replacement Center, training soldiers to be sent to existing infantry divisions as replacements, providing a high percentage of the replacements sent to Army combat units. During WWII more than 800,000 soldiers where trained at Camp Blanding.

The base was a holding center for 343 Japanese, German, and Italian immigrant residents of the United States. A small cemetery is located on the grounds of the former POW camp. In 1946 the actual bodies were removed to the Ft. Benning however the grave markers remain. Additionally five settler era cemeteries are located on Camp Blanding property. Most are not maintained and are heavily overgrown.

At one point during the war, the camp contained the population of the fourth-largest city in Florida. It had 10,000 buildings,  of paved roads, and the largest hospital in the state. It was one of the largest training bases in the country. There were several newspapers published at the camp during the war, which helped provide insights into daily life at the camp, such as The Bulletin, the Bayonet, and the Camp Blanding Report. After the war, Camp Blanding was the headquarters of the Military District of Florida and was commanded by Brigadier General Charles S. Kilburn.

An expeditionary airfield consisting of two gravel runways capable of accommodating C-130 Hercules aircraft was added in the 1970s along with the reactivation of the artillery training range and parachute drop zones.

From 2001 until 2008, Camp Blanding was used by the Southeast Region of the Civil Air Patrol to host their Southeast Region Encampment for cadets. The Florida Wing of Civil Air Patrol continues to use Camp Blanding for their wing-level summer cadet encampments.

Camp Blanding Museum and Memorial Park
Camp Blanding is also home to the Camp Blanding Museum and Memorial Park. Open to the public, the facility contains a history museum in one of Camp Blanding's restored World War II buildings, tracing the history of both Camp Blanding and the Florida National Guard. Outdoor exhibits and displays include equipment and Army, Navy, and Air Force aircraft from World War II, the Korean War, the Vietnam War, the Cold War, and Operation Desert Storm, including captured Soviet-manufactured Iraqi equipment from the last conflict.

References

External links
Globalsecurity.org page
Camp Blanding Museum and Memorial Park  - official site
Camp Blanding Museum  - Camp Blanding Museum and Historical Associates
U.S. Air Force 125th Fighter Wing's Camp Blanding History page
 Multijurisdictional Counterdrug Task Force Training website
 Combating Transnational Organized Crime Center of Excellence website

Blanding
Military installations in Florida
Buildings and structures in Clay County, Florida
Military and war museums in Florida
Installations of the United States Army National Guard
Museums in Clay County, Florida
Parks in Clay County, Florida
1940 establishments in Florida

Camp Blanding


Pierre Bureau (October 9, 1771 – June 6, 1836) was a businessman and political figure in Lower Canada.

He was born in L'Ancienne-Lorette, Quebec in 1771. Bureau operated an inn for travellers at Sainte-Anne-de-la-Pérade and a ferry service across the Sainte-Anne River. Around 1811, he moved to Trois-Rivières, where he became a merchant. He was elected to the Legislative Assembly of Canada for Saint-Maurice in an 1819 by-election and represented that region until his death at Trois-Rivières in 1836. He supported an elected Legisliative Council and tended to support the Parti canadien. Bureau voted in support of the Ninety-Two Resolutions.

His grandsons, Antoine-Aimé Dorion and Jean-Baptiste-Éric Dorion, both became members of the Legislative Assembly of the Province of Canada. Antoine-Aimé was also a lawyer, judge and Canadian cabinet minister; Jean-Baptiste-Éric was a journalist.

References

1771 births
1836 deaths
Members of the Legislative Assembly of Lower Canada
People from Capitale-Nationale

Pierre Bureau


Choix des plus belles fleurs ('Selection of the most beautiful flowers') is a book of watercolour paintings by French botanical artist Pierre-Joseph Redouté (10 July 1759 – 19 June 1840).  The full title is Choix des plus belles fleurs et de quelques branches des plus beaux fruits. Dédié à LL. AA. RR. les princesses Louise et Marie d'Orléans (1827) in French. English translation of the title is: "Selection of the most beautiful flowers and some branches of the finest fruits".

A folio edition with colour pages was printed in Paris in 1827.  From May 1827 to June 1833, 144 special pages with pictures were printed. The work was in 36 parts.  Each part had four pictures of flowers, flowering trees, or fruit.

Printing colour images
The printing method for such works in the 18th and early 19th century was engraving on copperplate. It is a form of intaglio printing.
Different colours could be got by running the plates through once for each colour, so building up the final picture. Fine colour works could also be hand-coloured by teams of women who were experts at the job. What this means is that every illustration in a book was coloured by hand. They had the artist's original in front of their eyes as they worked. The copper engraving gave the outline and any other black lines or areas.

In 1796, Redouté used a way to make prints which used both these methods. First, an engraver made an engraving from the painting, with small dots instead of lines. Then the colours were put on the engraved plate before each printing. After printing, each print was finished by hand with more colour. Redouté claimed to have invented this method. He was challenged in the courts, but he won the case.

Gallery

References

Other websites

Full text on Internet Archive
Full text on Biodiversity Heritage Library
Rare Book Room (photographed in very high resolution)

Biology books
1820s paintings
1820s books

Choix des plus belles fleurs


Panama Patrol is a 1939 American drama film. Directed by Charles Lamont, the film stars Leon Ames, Charlotte Wynters, and Adrienne Ames, it was released on May 20, 1939. The film was known during production by the working titles of Curio Cipher and Panama Cipher.

Plot
Major Phillip Waring, the head of the Cipher Bureau in Washington, D. C., is called away on the day of his wedding to secretary Helen Lane by an urgent request to decipher a diplomatic message intercepted from an international spy ring.

Waring and his assistant, Lieutenant Murdock, track down one of the spies, but he is shot before they are able to question him.

With the spy's death, their only clue is a letter written in Chinese, which they give to Bureau interpreter Artie Johnson to decipher. Unknown to Waring, Johnson is the clandestine leader of the spy ring, and the interpreter gives Waring information that leads him into a trap at the Maing curio shop.

When Waring fails to return from his mission, Murdock goes to the shop and rescues him. While at the shop, Waring discovers another letter, and when asked to interpret it, Johnson cleverly changes one of the letters in the code by means of a blotter.

Johnson's information sends Waring and Murdock to California on a wild goose chase. In their absence, however, Helen finds Johnson's blotter and realizes that he is involved in the spy ring. While confronting Johnson at his house, Helen is taken captive by the ring.

Meanwhile, Waring realizes that his mission is a frame-up, and returns to Washington to discover Helen's peril. Arriving at Johnson's house just after the spies have left with Helen, Waring discovers the final part of the code, which reveals the spy's plan to impede traffic traveling through the Panama Canal.

The cipher also discloses the location of the spy headquarters, and Waring and Murdock arrive just in time to rescue Helen and smash the spy ring.

Cast
 Leon Ames as Major Phillip Waring
 Charlotte Wynters as Helen Lane
 Adrienne Ames as Lia Maing
 Weldon Heyburn as Lt. Murdock
 Abner Biberman as Artie Johnson
 Sidney Miller as Jimmy
 Jack Smart as Eli Maing
 Donald Barry as Lt. Loring
 Hugh McArthur as Lt. Everett
 William Von Brincken as Marlin

References

External links

American drama films
1939 drama films
1939 films
American black-and-white films
1930s American films

Panama Patrol


Gyrwas (also called Gyrwe) was the name of an early Anglo-Saxon people. They lived mainly on the western edge of  the grassy wetlands called the Fens in eastern England. They were divided into two tribal groups; the northern Gyrwas and the southern Gyrwas. This is how they were recorded in the Tribal Hidage.

The name Gyrwe means 'fen-dweller'. The territory of the Gyrwas included Lindisfarne, Hatfield, Nottinghamshire, Northern Cambridgeshire, Huntingdonshire, and as far south as Peterborough in Northamptonshire. The area around Jarrow was in their territory as well. The Fens supplied the Gyrwas with fish and gamebirds but they lived on the drier lands and islands nearby.

The Gyrwas had their own leaders as late as the first half of the 7th century. Bede records a Tondbert, priceps of the South Gyrwas. He asked for the hand in marriage of Etheldreda, the daughter of King Anna of East Anglia. Tondbert died shortly after the wedding and Ethelreda was then given in marriage to Ecgfrith of Northumbria. Underkings and leaders of tribes like the Gyrwas probably did not claim descent from ancient Germanic gods, as many kings of the heptarchy did. But they were of high enough status to marry Anglo-Saxon royalty.

For a time the territory of the Gyrwas formed a buffer state between the Mercians and the East Anglians. The Gyrwas were later absorbed into Middle Anglia.

Notes

References

Anglo-Saxons
Petty kingdoms of England

Gyrwas


Francesco Lojacono or Lo Jacono (1838–1915) was an Italian painter, mainly of landscapes and seascapes, and mainly active in his native Palermo, Sicily.

Biography
He received his early training there with his father Luigi, a history and portrait painter, and subsequently with Salvatore Lo Forte. Francesco Lojacono won a gold medal for a small canvas presented at an exhibition in Palermo, and this gained him a stipend to move in 1856 to Naples, where attended the school of Filippo Palizzi. With the outbreak of revolution in 1860, he returned to Sicily,  he served in Garibaldi's expedition of one thousand volunteers in 1860 and participated in the march towards Rome in 1862, when he was taken prisoner on the Aspromonte. Once free, he devoted his energies to painting intensely lyrical landscapes drawing inspiration in particular from the area around Agrigento and later the coast near Palermo.

He obtained the post of professor of landscape at the Naples Institute of Fine Arts in 1872 and taught the courses of landscape and seascape painting at the Palermo Academy of Fine Arts from 1896 to 1914. He took part in the international exhibitions held in Vienna (1872) and Paris (1878) and presented work with unquestionable success at the major national events, including the Palermo exhibition of 1891–92. He was named Commendatore of the Order of the Crown of Italy.

One of his pupils was Michele Catti, who together with Lojacono and Antonio Leto forms the so-called canonical triad of the Sicilian landscape artists of the Belle Époque. Another pupil was Ettore De Maria Bergler.

Among his works are: Giorno di caldo; I pescatori d'ostriche (considered one of his masterworks); Una villa ai Colli; Dopo la pioggia; L'arrivo inatteso; Il riposo di maggio; Un giorno di caldo in Sicilia; L'ottobre; After il tramonto; La Conca d'oro; San Giovanni degli Eremiti; Presso Posillipo; Presso il Vesuvio; Le rovine del tempio di Castore e Polluce; Gli ultimi Saraceni; and La pesca dei vongoli.

Gallery of Works

References

19th-century Italian painters
Italian male painters
20th-century Italian painters
20th-century Italian male artists
Italian landscape painters
Painters from Palermo
1838 births
1915 deaths
19th-century Italian male artists

Francesco Lojacono


The Icemark Chronicles is a series of books consisting of The Cry of the Icemark, followed by Blade of Fire, and Last Battle of the Icemark.  The books were written by Stuart Hill from Leicester. Fox 2000 Pictures have bought the right to create a movie based on the books.  The first is planned for 2008.

Books
The Cry of the Icemark
Blade of Fire
Last Battle of the Icemark

References

The Icemark Chronicles


In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation:

The theorem is named for the Greek philosopher Pythagoras, born around 570 BC.  The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.

When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points.

The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps, and cartoons abound.

Proofs using constructed squares

Rearrangement proofs
In one rearrangement proof, two squares are used whose sides have a measure of    and which contain four right triangles whose sides are a, b and c, with the hypotenuse being c.  In the square on the right side, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square in the center whose sides are length c.  Each outer square has an area of   as well as , with  representing the total area of the four triangles.  Within the big square on the left side, the four triangles are moved to form two similar rectangles with sides of length a and b.  These rectangles in their new position have now delineated two new squares, one having side length a is formed in the bottom-left corner, and another square of side length b formed in the top-right corner.  In this new position, this left side now has a square of area    as well as  .  Since both squares have the area of  it follows that the other measure of the square area also equal each other such that  = .  With the area of the four triangles removed from both side of the equation what remains is

In another proof rectangles in the second box can also be placed such that both have one corner that correspond to consecutive corners of the square.  In this way they also form two boxes, this time in consecutive corners, with areas   and which will again lead to a second square of with the area .

English mathematician Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements, and mentions the proposals of German mathematicians Carl Anton Bretschneider and Hermann Hankel that Pythagoras may have known this proof.  Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him."  Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as a creator of mathematics, although debate about this continues.

Algebraic proofs

The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with side c, as shown in the lower part of the diagram. This results in a larger square, with side  and area . The four triangles and the square side c must have the same area as the larger square,

giving

A similar proof uses four copies of a right triangle with sides a, b and c, arranged inside a square with side c as in the top half of the diagram. The triangles are similar with area , while the small square has side  and area . The area of the large square is therefore

But this is a square with side c and area c2, so

Other proofs of the theorem
This theorem may have more known proofs than any other (the law of quadratic reciprocity being another contender for that distinction); the book The Pythagorean Proposition contains 370 proofs.

Proof using similar triangles

This proof is based on the proportionality of the sides of three similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.

Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e. The new triangle, ACH, is similar to triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as θ in the figure. By a similar reasoning, the triangle CBH is also similar to ABC. The proof of similarity of the triangles requires the triangle postulate: The sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides:

The first result equates the cosines of the angles θ, whereas the second result equates their sines.

These ratios can be written as

Summing these two equalities results in

which, after simplification, demonstrates the Pythagorean theorem:

The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another. One conjecture is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the Elements, and that the theory of proportions needed further development at that time.

Einstein's proof by dissection without rearrangement

Albert Einstein gave a proof by dissection in which the pieces do not need to be moved. Instead of using a square on the hypotenuse and two squares on the legs, one can use any other shape that includes the hypotenuse, and two similar shapes that each include one of two legs instead of the hypotenuse (see Similar figures on the three sides).  In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself.  The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts.  Those two parts have the same shape as the original right triangle, and have the legs of the original triangle as their hypotenuses, and the sum of their areas is that of the original triangle.  Because the ratio of the area of a right triangle to the square of its hypotenuse is the same for similar triangles, the relationship between the areas of the three triangles holds for the squares of the sides of the large triangle as well.

Trigonometric proof using Einstein's construction

Both the proof using similar triangles and Einstein's proof rely on constructing the height to the hypotenuse of the right triangle . The same construction provides a  trigonometric proof of the Pythagorean theorem using the definition of the  sine as a ratio inside a right triangle:

and thus

This proof is essentially the same as the above proof using similar triangles, where some ratios of lengths are replaced by sines.

Euclid's proof

In outline, here is how the proof in Euclid's Elements proceeds. The large square is divided into a left and right rectangle. A triangle is constructed that has half the area of the left rectangle. Then another triangle is constructed that has half the area of the square on the left-most side. These two triangles are shown to be congruent, proving this square has the same area as the left rectangle. This argument is followed by a similar version for the right rectangle and the remaining square. Putting the two rectangles together to reform the square on the hypotenuse, its area is the same as the sum of the area of the other two squares. The details follow.

Let A, B, C be the vertices of a right triangle, with a right angle at A. Drop a perpendicular from A to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs.

For the formal proof, we require four elementary lemmata:
 If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent (side-angle-side).
 The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.
 The area of a rectangle is equal to the product of two adjacent sides.
 The area of a square is equal to the product of two of its sides (follows from 3).

Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square.

The proof is as follows:
Let ACB be a right-angled triangle with right angle CAB.
On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in that order. The construction of squares requires the immediately preceding theorems in Euclid, and depends upon the parallel postulate.
From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE at K and L, respectively.
Join CF and AD, to form the triangles BCF and BDA.
Angles CAB and BAG are both right angles; therefore C, A, and G are collinear.
Angles CBD and FBA are both right angles; therefore angle ABD equals angle FBC, since both are the sum of a right angle and angle ABC.
Since AB is equal to FB, BD is equal to BC and angle ABD equals angle FBC, triangle ABD must be congruent to triangle FBC.
Since A-K-L is a straight line, parallel to BD, then rectangle BDLK has twice the area of triangle ABD because they share the base BD and have the same altitude BK, i.e., a line normal to their common base, connecting the parallel lines BD and AL. (lemma 2)
Since C is collinear with A and G, and this line is parallel to FB, then square BAGF must be twice in area to triangle FBC.
Therefore, rectangle BDLK must have the same area as square BAGF = AB2.
By applying steps 3 to 10 to the other side of the figure, it can be similarly shown that rectangle CKLE must have the same area as square ACIH = AC2.
Adding these two results, AB2 + AC2 = BD × BK + KL × KC
Since BD = KL, BD × BK + KL × KC = BD(BK + KC) = BD × BC
Therefore, AB2 + AC2 = BC2, since CBDE is a square.

This proof, which appears in Euclid's Elements as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares.
This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used.

Proofs by dissection and rearrangement
Another by rearrangement is given by the middle animation. A large square is formed with area c2, from four identical right triangles with sides a, b and c, fitted around a small central square. Then two rectangles are formed with sides a and b by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas a2 and b2, which must have the same area as the initial large square.

The third, rightmost image also gives a proof. The upper two squares are divided as shown by the blue and green shading, into pieces that when rearranged can be made to fit in the lower square on the hypotenuse – or conversely the large square can be divided as shown into pieces that fill the other two. This way of cutting one figure into pieces and rearranging them to get another figure is called dissection. This shows the area of the large square equals that of the two smaller ones.

Proof by area-preserving shearing

As shown in the accompanying animation, area-preserving shear mappings and translations can transform the squares on the sides adjacent to the right-angle onto the square on the hypotenuse, together covering it exactly. Each shear leaves the base and height unchanged, thus leaving the area unchanged too. The translations also leave the area unchanged, as they do not alter the shapes at all. Each square is first sheared into a parallelogram, and then into a rectangle which can be translated onto one section of the square on the hypotenuse.

Other algebraic proofs
A related proof was published by future U.S. President James A. Garfield (then a U.S. Representative) (see diagram). Instead of a square it uses a trapezoid, which can be constructed from the square in the second of the above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown in the diagram. The area of the trapezoid can be calculated to be half the area of the square, that is

The inner square is similarly halved, and there are only two triangles so the proof proceeds as above except for a factor of , which is removed by multiplying by two to give the result.

Proof using differentials
One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employing calculus.

The triangle ABC is a right triangle, as shown in the upper part of the diagram, with BC the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length y, the side AC of length x and the side AB of length a, as seen in the lower diagram part.

If x is increased by a small amount dx by extending the side AC slightly to D, then y also increases by dy. These form two sides of a triangle, CDE, which (with E chosen so CE is perpendicular to the hypotenuse) is a right triangle approximately similar to ABC. Therefore, the ratios of their sides must be the same, that is:

This can be rewritten as  , which is a differential equation that can be solved by direct integration:

giving

The constant can be deduced from x = 0, y = a to give the equation

This is more of an intuitive proof than a formal one: it can be made more rigorous if proper limits are used in place of dx and dy.

Converse
The converse of the theorem is also true:

Given a triangle with sides of length a, b, and c, if  then the angle between sides a and b is a right angle.

For any three positive real numbers a, b, and c such that , there exists a triangle with sides a, b and c as a consequence of the converse of the triangle inequality.

This converse appears in Euclid's Elements (Book I, Proposition 48):  "If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right."

It can be proved using the law of cosines or as follows:

Let ABC be a triangle with side lengths a, b, and c, with  Construct a second triangle with sides of length a and b containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length c = , the same as the hypotenuse of the first triangle. Since both triangles' sides are the same lengths a, b and c, the triangles are congruent and must have the same angles. Therefore, the angle between the side of lengths a and b in the original triangle is a right angle.

The above proof of the converse makes use of the Pythagorean theorem itself. The converse can also be proved without assuming the Pythagorean theorem.

A corollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Let c be chosen to be the longest of the three sides and  (otherwise there is no triangle according to the triangle inequality). The following statements apply:
 If  then the triangle is right.
 If  then the triangle is acute.
 If  then the triangle is obtuse.

Edsger W. Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language:

where α is the angle opposite to side a, β is the angle opposite to side b, γ is the angle opposite to side c, and sgn is the sign function.

Consequences and uses of the theorem

Pythagorean triples

A Pythagorean triple has three positive integers a, b, and c, such that  In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Such a triple is commonly written  Some well-known examples are  and

A primitive Pythagorean triple is one in which a, b and c are coprime (the greatest common divisor of a, b and c is 1).

The following is a list of primitive Pythagorean triples with values less than 100:

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

Inverse Pythagorean theorem
Given a right triangle with sides  and altitude  (a line from the right angle and perpendicular to the hypotenuse ). The Pythagorean theorem has,

while the inverse Pythagorean theorem relates the two legs  to the altitude ,

The equation can be transformed to,

where  for any non-zero real . If the  are to be integers, the smallest solution  is then

using the smallest Pythagorean triple . The reciprocal Pythagorean theorem is a special case of the optic equation

where the denominators are squares and also for a heptagonal triangle whose sides  are square numbers.

Incommensurable lengths

One of the consequences of the Pythagorean theorem is that line segments whose lengths are incommensurable (so the ratio of which is not a rational number) can be constructed using a straightedge and compass. Pythagoras' theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation.

The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras' theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is not a perfect square, it is a realization of a length incommensurable with the unit, such as , ,  . For more detail, see Quadratic irrational.

Incommensurable lengths conflicted with the Pythagorean school's concept of numbers as only whole numbers. The Pythagorean school dealt with proportions by comparison of integer multiples of a common subunit. According to one legend, Hippasus of Metapontum (ca. 470 B.C.) was drowned at sea for making known the existence of the irrational or incommensurable.
A careful discussion of Hippasus's contributions is found in Fritz.

Complex numbers

For any complex number

the absolute value or modulus is given by

So the three quantities, r, x and y are related by the Pythagorean equation,

Note that r is defined to be a positive number or zero but x and y can be negative as well as positive. Geometrically r is the distance of the z from zero or the origin O in the complex plane.

This can be generalised to find the distance between two points, z1 and z2 say. The required distance is given by

so again they are related by a version of the Pythagorean equation,

Euclidean distance

The distance formula in Cartesian coordinates is derived from the Pythagorean theorem. If  and  are points in the plane, then the distance between them, also called the Euclidean distance, is given by

More generally, in Euclidean n-space, the Euclidean distance between two points,  and , is defined, by generalization of the Pythagorean theorem, as:

If instead of Euclidean distance, the square of this value (the squared Euclidean distance, or SED) is used, the resulting equation avoids square roots and is simply a sum of the SED of the coordinates:

The squared form is a smooth, convex function of both points, and is widely used in optimization theory and statistics, forming the basis of least squares.

Euclidean distance in other coordinate systems

If Cartesian coordinates are not used, for example, if polar coordinates are used in two dimensions or, in more general terms, if curvilinear coordinates are used, the formulas expressing the Euclidean distance are more complicated than the Pythagorean theorem, but can be derived from it. A typical example where the straight-line distance between two points is converted to curvilinear coordinates can be found in the applications of Legendre polynomials in physics. The formulas can be discovered by using Pythagoras' theorem with the equations relating the curvilinear coordinates to Cartesian coordinates. For example, the polar coordinates  can be introduced as:

Then two points with locations  and  are separated by a distance s:

Performing the squares and combining terms, the Pythagorean formula for distance in Cartesian coordinates produces the separation in polar coordinates as:

using the trigonometric product-to-sum formulas. This formula is the law of cosines, sometimes called the generalized Pythagorean theorem. From this result, for the case where the radii to the two locations are at right angles, the enclosed angle  and the form corresponding to Pythagoras' theorem is regained:  The Pythagorean theorem, valid for right triangles, therefore is a special case of the more general law of cosines, valid for arbitrary triangles.

Pythagorean trigonometric identity

In a right triangle with sides a, b and hypotenuse c, trigonometry determines the sine and cosine of the angle θ between side a and the hypotenuse as:

From that it follows:

where the last step applies Pythagoras' theorem. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity. In similar triangles, the ratios of the sides are the same regardless of the size of the triangles, and depend upon the angles. Consequently, in the figure, the triangle with hypotenuse of unit size has opposite side of size sin θ and adjacent side of size cos θ in units of the hypotenuse.

Relation to the cross product

The Pythagorean theorem relates the cross product and dot product in a similar way:

This can be seen from the definitions of the cross product and dot product, as

with n a unit vector normal to both a and b. The relationship follows from these definitions and the Pythagorean trigonometric identity.

This can also be used to define the cross product. By rearranging the following equation is obtained

This can be considered as a condition on the cross product and so part of its definition, for example in seven dimensions.

Generalizations

Similar figures on the three sides

The Pythagorean theorem generalizes beyond the areas of squares on the three sides to  any similar figures. This was known by Hippocrates of Chios in the 5th century BC, and was included by Euclid in his Elements:

If one erects similar figures (see Euclidean geometry) with corresponding sides on the sides of a right triangle, then the sum of the areas of the ones on the two smaller sides equals the area of the one on the larger side.

This extension assumes that the sides of the original triangle are the corresponding sides of the three congruent figures (so the common ratios of sides between the similar figures are a:b:c). While Euclid's proof only applied to convex polygons, the theorem also applies to concave polygons and even to similar figures that have curved boundaries (but still with part of a figure's boundary being the side of the original triangle).

The basic idea behind this generalization is that the area of a plane figure is proportional to the square of any linear dimension, and in particular is proportional to the square of the length of any side. Thus, if similar figures with areas A, B and C are erected on sides with corresponding lengths a, b and c then:

But, by the Pythagorean theorem, a2 + b2 = c2, so A + B = C.

Conversely, if we can prove that A + B = C for three similar figures without using the Pythagorean theorem, then we can work backwards to construct a proof of the theorem. For example, the starting center triangle can be replicated and used as a triangle C on its hypotenuse, and two similar right triangles (A and B ) constructed on the other two sides, formed by dividing the central triangle by its altitude. The sum of the areas of the two smaller triangles therefore is that of the third, thus A + B = C and reversing the above logic leads to the Pythagorean theorem a2 + b2 = c2. (See also Einstein's proof by dissection without rearrangement)

Law of cosines

The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines, which states that

where  is the angle between sides  and .

When  is  radians or 90°, then , and the formula reduces to the usual Pythagorean theorem.

Arbitrary triangle

At any selected angle of a general triangle of sides a, b, c, inscribe an isosceles triangle such that the equal angles at its base θ are the same as the selected angle. Suppose the selected angle θ is opposite the side labeled c. Inscribing the isosceles triangle forms triangle CAD with angle θ opposite side b and with side r along c. A second triangle is formed with angle θ opposite side a and a side with length s along c, as shown in the figure. Thābit ibn Qurra stated that the sides of the three triangles were related as:

As the angle θ approaches /2, the base of the isosceles triangle narrows, and lengths r and s overlap less and less. When θ = /2, ADB becomes a right triangle, r + s = c, and the original Pythagorean theorem is regained.

One proof observes that triangle ABC has the same angles as triangle CAD, but in opposite order. (The two triangles share the angle at vertex A, both contain the angle θ, and so also have the same third angle by the triangle postulate.) Consequently, ABC is similar to the reflection of CAD, the triangle DAC in the lower panel. Taking the ratio of sides opposite and adjacent to θ,

Likewise, for the reflection of the other triangle,

Clearing fractions and adding these two relations:

the required result.

The theorem remains valid if the angle  is obtuse so the lengths r and s are non-overlapping.

General triangles using parallelograms

Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the original Pythagoras' theorem, and was considered a generalization by Pappus of Alexandria in 4 AD

The lower figure shows the elements of the proof. Focus on the left side of the figure. The left green parallelogram has the same area as the left, blue portion of the bottom parallelogram because both have the same base b and height h. However, the left green parallelogram also has the same area as the left green parallelogram of the upper figure, because they have the same base (the upper left side of the triangle) and the same height normal to that side of the triangle. Repeating the argument for the right side of the figure, the bottom parallelogram has the same area as the sum of the two green parallelograms.

Solid geometry

In terms of solid geometry, Pythagoras' theorem can be applied to three dimensions as follows. Consider a rectangular solid as shown in the figure. The length of diagonal BD is found from Pythagoras' theorem as:

where these three sides form a right triangle. Using horizontal diagonal BD and the vertical edge AB, the length of diagonal AD then is found by a second application of Pythagoras' theorem as:

or, doing it all in one step:

This result is the three-dimensional expression for the magnitude of a vector v (the diagonal AD) in terms of its orthogonal components {vk} (the three mutually perpendicular sides):

This one-step formulation may be viewed as a generalization of Pythagoras' theorem to higher dimensions. However, this result is really just the repeated application of the original Pythagoras' theorem to a succession of right triangles in a sequence of orthogonal planes.

A substantial generalization of the Pythagorean theorem to three dimensions is de Gua's theorem, named for Jean Paul de Gua de Malves: If a tetrahedron has a right angle corner (like a corner of a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. This result can be generalized as in the "n-dimensional Pythagorean theorem":

This statement is illustrated in three dimensions by the tetrahedron in the figure. The "hypotenuse" is the base of the tetrahedron at the back of the figure, and the "legs" are the three sides emanating from the vertex in the foreground. As the depth of the base from the vertex increases, the area of the "legs" increases, while that of the base is fixed. The theorem suggests that when this depth is at the value creating a right vertex, the generalization of Pythagoras' theorem applies. In a different wording:

Inner product spaces

The Pythagorean theorem can be generalized to inner product spaces, which are generalizations of the familiar 2-dimensional and 3-dimensional Euclidean spaces. For example, a function may be considered as a vector with infinitely many components in an inner product space, as in functional analysis.

In an inner product space, the concept of perpendicularity is replaced by the concept of orthogonality: two vectors v and w are orthogonal if their inner product  is zero. The inner product is a generalization of the dot product of vectors. The dot product is called the standard inner product or the Euclidean inner product. However, other inner products are possible.

The concept of length is replaced by the concept of the norm ‖v‖ of a vector v, defined as:

In an inner-product space, the Pythagorean theorem states that for any two orthogonal vectors v and w we have

Here the vectors v and w are akin to the sides of a right triangle with hypotenuse given by the vector sum v + w. This form of the Pythagorean theorem is a consequence of the properties of the inner product:

where  because of orthogonality.

A further generalization of the Pythagorean theorem in an inner product space to non-orthogonal vectors is the parallelogram law :

which says that twice the sum of the squares of the lengths of the sides of a parallelogram is the sum of the squares of the lengths of the diagonals. Any norm that satisfies this equality is ipso facto a norm corresponding to an inner product.

The Pythagorean identity can be extended to sums of more than two orthogonal vectors. If v1, v2, ..., vn are pairwise-orthogonal vectors in an inner-product space, then application of the Pythagorean theorem to successive pairs of these vectors (as described for 3-dimensions in the section on solid geometry) results in the equation

Sets of m-dimensional objects in n-dimensional space

Another generalization of the Pythagorean theorem applies to Lebesgue-measurable sets of objects in any number of dimensions.  Specifically, the square of the measure of an m-dimensional set of objects in one or more parallel m-dimensional flats in n-dimensional Euclidean space is equal to the sum of the squares of the measures of the orthogonal projections of the object(s) onto all m-dimensional coordinate subspaces.

In mathematical terms:

where:
  is a measure in m-dimensions (a length in one dimension, an area in two dimensions, a volume in three dimensions, etc.).
  is a set of one or more non-overlapping m-dimensional objects in one or more parallel m-dimensional flats in n-dimensional Euclidean space.
  is the total measure (sum) of the set of m-dimensional objects.
  represents an m-dimensional projection of the original set onto an orthogonal coordinate subspace.
  is the measure of the m-dimensional set projection onto m-dimensional coordinate subspace .  Because object projections can overlap on a coordinate subspace, the measure of each object projection in the set must be calculated individually, then measures of all projections added together to provide the total measure for the set of projections on the given coordinate subspace.
  is the number of orthogonal, m-dimensional coordinate subspaces in n-dimensional space () onto which the m-dimensional objects are projected (m ≤ n):

Non-Euclidean geometry
The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theorem implies, and is implied by, Euclid's Parallel (Fifth) Postulate. Thus, right triangles in a non-Euclidean geometry
do not satisfy the Pythagorean theorem. For example, in spherical geometry, all three sides of the right triangle (say a, b, and c) bounding an octant of the unit sphere have length equal to /2, and all its angles are right angles, which violates the Pythagorean theorem because
.

Here two cases of non-Euclidean geometry are considered—spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case for non-right triangles, the result replacing the Pythagorean theorem follows from the appropriate law of cosines.

However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition that the triangle be right is replaced with the condition that two of the angles sum to the third, say A+B = C. The sides are then related as follows: the sum of the areas of the circles with diameters a and b equals the area of the circle with diameter c.

Spherical geometry

For any right triangle on a sphere of radius  (for example, if  in the figure is a right angle), with sides  the relation between the sides takes the form:

This equation can be derived as a special case of the spherical law of cosines that applies to all spherical triangles:

For infinitesimal triangles on the sphere (or equivalently, for finite spherical triangles on a sphere of infinite radius), the spherical relation between the sides of a right triangle reduces to the Euclidean form of the Pythagorean theorem. To see how, assume we have a spherical triangle of fixed side lengths  and  on a sphere with expanding radius . As  approaches infinity the quantities  and  tend to zero and the spherical Pythagorean identity reduces to  so we must look at its asymptotic expansion.

The Maclaurin series for the cosine function can be written as  with the remainder term in big O notation. Letting  be a side of the triangle, and treating the expression as an asymptotic expansion in terms of  for a fixed ,

and likewise for  and . Substituting the asymptotic expansion for each of the cosines into the spherical relation for a right triangle yields

Subtracting 1 and then negating each side,

Multiplying through by  the asymptotic expansion for  in terms of fixed  and variable  is

The Euclidean Pythagorean relationship  is recovered in the limit, as the remainder vanishes when the radius R approaches infinity.

For practical computation in spherical trigonometry with small right triangles, cosines can be replaced with sines using the double-angle identity  to avoid loss of significance. Then the spherical Pythagorean theorem can alternately be written as

Hyperbolic geometry

In a hyperbolic space with uniform Gaussian curvature −1/R2, for a right triangle with legs a, b, and hypotenuse c, the relation between the sides takes the form:

where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles:

with γ the angle at the vertex opposite the side c.

By using the Maclaurin series for the hyperbolic cosine, , it can be shown that as a hyperbolic triangle becomes very small (that is, as a, b, and c all approach zero), the hyperbolic relation for a right triangle approaches the form of Pythagoras' theorem.

For small right triangles (a, b << R), the hyperbolic cosines can be eliminated to avoid loss of significance, giving

Very small triangles
For any uniform curvature K (positive, zero, or negative), in very small right triangles (|K|a2, |K|b2 << 1) with hypotenuse c, it can be shown that

Differential geometry

The Pythagorean theorem applies to infinitesimal triangles seen in differential geometry. In three dimensional space, the distance between two infinitesimally separated points satisfies

with ds the element of distance and (dx, dy, dz) the components of the vector separating the two points. Such a space is called a Euclidean space. However, in Riemannian geometry, a generalization of this expression useful for general coordinates (not just Cartesian) and general spaces (not just Euclidean) takes the form:

which is called the metric tensor. (Sometimes, by abuse of language, the same term is applied to the set of coefficients .) It may be a function of position, and often describes curved space. A simple example is Euclidean (flat) space expressed in curvilinear coordinates. For example, in polar coordinates:

History

There is debate whether the Pythagorean theorem was discovered once, or many times in many places, and the date of first discovery is uncertain, as is the date of the first proof.  Historians of Mesopotamian mathematics have concluded that the Pythagorean rule was in widespread use during the Old Babylonian period (20th to 16th centuries BC), over a thousand years before Pythagoras was born. The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system.

Written  1800BC, the Egyptian Middle Kingdom Berlin Papyrus 6619 includes a problem whose solution is the Pythagorean triple 6:8:10, but the problem does not mention a triangle. The Mesopotamian tablet Plimpton 322, also written  1800BC near Larsa, contains many entries closely related to Pythagorean triples.

In India, the Baudhayana Shulba Sutra, the dates of which are given variously as between the 8th and 5th century BC, contains a list of Pythagorean triples and a statement of the Pythagorean theorem, both in the special case of the isosceles right triangle and in the general case, as does the Apastamba Shulba Sutra ().

Byzantine Neoplatonic philosopher and mathematician Proclus, writing in the fifth century AD, states two arithmetic rules, "one of them attributed to Plato, the other to Pythagoras", for generating special Pythagorean triples.  The rule attributed to Pythagoras () starts from an odd number and produces a triple with leg and hypotenuse differing by one unit; the rule attributed to Plato (428/427 or 424/423 – 348/347 BC) starts from an even number and produces a triple with leg and hypotenuse differing by two units. According to Thomas L. Heath (1861–1940), no specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived. However, when authors such as Plutarch and Cicero attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted. Classicist Kurt von Fritz wrote, "Whether this formula is rightly attributed to Pythagoras personally ... one can safely assume that it belongs to the very oldest period of Pythagorean mathematics." Around 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented.

With contents known much earlier, but in surviving texts dating from roughly the 1st century BC, the Chinese text Zhoubi Suanjing (周髀算经), (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) gives a reasoning for the Pythagorean theorem for the (3, 4, 5) triangle — in China it is called the "Gougu theorem" (勾股定理). During the Han Dynasty (202 BC to 220 AD), Pythagorean triples appear in The Nine Chapters on the Mathematical Art, together with a mention of right triangles. Some believe the theorem arose first in China, where it is alternatively known as the "Shang Gao theorem" (商高定理), named after the Duke of Zhou's astronomer and mathematician, whose reasoning composed most of what was in the Zhoubi Suanjing.

See also

Addition in quadrature
At Dulcarnon
British flag theorem
Fermat's Last Theorem
Inverse Pythagorean theorem
Kepler triangle
Linear algebra
List of triangle topics
Lp space
Nonhypotenuse number
Parallelogram law
Parseval's identity
Ptolemy's theorem
Pythagoras in popular culture
Pythagorean expectation
Pythagorean tiling
Rational trigonometry in Pythagoras' theorem
Thales theorem

Notes and references

Notes

References

Works cited

  On-line text at archive.org

 This high-school geometry text covers many of the topics in this WP article.
  Reissued 1968 by the National Council of Teachers of Mathematics. A lower-quality scan was published online by the Education Resources Information Center, ERIC ED037335.

 Robson, Eleanor and Jacqueline Stedall, eds., The Oxford Handbook of the History of Mathematics, Oxford: Oxford University Press, 2009. pp. vii + 918. .
  Also .

External links

 In HTML with Java-based interactive figures.

 History topic: Pythagoras's theorem in Babylonian mathematics
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 Pythagorean theorem (more than 70 proofs from cut-the-knot)

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