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2009-10-08T04:47:00.000-07:00

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Niccolò Fontana Tartaglia

2009-10-08T04:02:00.000-07:00

Niccolò Fontana Tartaglia (1499/1500, Brescia, Italy – December 13, 1557, Venice, Italy) was a mathematician, an engineer (designing fortifications), a surveyor (of topography, seeking the best means of defense or offense) and a bookkeeper from the then-Republic of Venice (now part of Italy). He published many books, including the first Italian translations of Archimedes and Euclid, and an acclaimed compilation of mathematics. Tartaglia was the first to apply mathematics to the investigation of the paths of cannonballs; his work was later validated by Galileo's studies on falling bodies.

Niccolò Fontana was the son of Michele Fontana, a rider and deliverer. In 1505, Michele was murdered and Niccolò, his two siblings, and his mother were impoverished. Niccolò experienced further tragedy in 1512 when the French invaded Brescia during the War of the League of Cambrai. The militia of Brescia defended their city for seven days. When the French finally broke through, they took their revenge by massacring the inhabitants of Brescia. By the end of battle, over 45,000 residents were killed. During the massacre, a French soldier sliced Niccolò's jaw and palate. This made it impossible for Niccolò to speak normally, prompting the nickname "Tartaglia" (stammerer).

There is a story that Tartaglia learned only half the alphabet from a private tutor before funds ran out, and he had to learn the rest for himself. Be that as it may, he was essentially self-taught. He and his contemporaries, working outside the academies, were responsible for the spread of classic works in modern languages among the educated middle class.

Tartaglia was self taught in mathematics but, having an extraordinary ability, his mother was able to find him a patron. Ludovico Balbisonio took him to Padua to study there, but when he returned with his patron to Brescia he made himself unpopular by having an inflated opinion of himself. He left Brescia to earn his living teaching mathematics at Verona which he did between 1516 and 1518. Later, still in Verona, he taught at a school in the Palazzo Mizzanti but it is recorded that at that time he was married with a family, yet was very poor. He moved to Venice in 1534. As a lowly mathematics teacher in Venice, Tartaglia gradually acquired a reputation as a promising mathematician by participating successfully in a large number of debates.

His edition of Euclid in 1543, the first translation of the Elements into any modern European language, was especially significant. For two centuries Euclid had been taught from two Latin translations taken from an Arabic source; these contained errors in Book V, the Eudoxian theory of proportion, which rendered it unusable. Tartaglia's edition was based on Zamberti's Latin translation of an uncorrupted Greek text, and rendered Book V correctly. He also wrote the first modern and useful commentary on the theory. Later, the theory was an essential tool for Galileo, just as it had been for Archimedes.

Solution to cubic equations

Tartaglia is perhaps best known today for his conflicts with Gerolamo Cardano. Cardano nagged Tartaglia into revealing his solution to the cubic equations, by promising not to publish them. Several years later, Cardano happened to see unpublished work by Scipione del Ferro who independently came up with the same solution as Tartaglia. As the unpublished work was dated before Tartaglia's, Cardano decided his promise could be broken and included Tartaglia's solution in his next publication. Since Cardano credited his discovery, Tartaglia was extremely upset. He responded by publicly insulting Cardano.

The first person known to have solved cubic equations algebraically was del Ferro but he told nobody of his achievement. On his deathbed, however, del Ferro passed on the secret to his (rather poor) student Fior. For mathematicians of this time there was more than one type of cubic equation and Fior had only been shown by del Ferro how to solve one type, namely 'unknowns and cubes equal to numbers' or (in modern notation) x3 + ax = b. As negative numbers were not used this led to a number of other cases, even for equations without a square term. Fior began to boast that he was able to solve cubics and a challenge between him and Tartaglia was arranged in 1535. In fact Tartaglia had also discovered how to solve one type of cubic equation since his friend Zuanne da Coi had set two problems which had led Tartaglia to a general solution of a different type from that which Fior could solve, namely 'squares and cubes equal to numbers' or (in modern notation) x3 + ax2 = b. For the contest between Tartaglia and Fior, each man was to submit thirty questions for the other to solve. Fior was supremely confident that his ability to solve cubics would be enough to defeat Tartaglia but Tartaglia submitted a variety of different questions, exposing Fior as an, at best, mediocre mathematician. Fior, on the other hand, offered Tartaglia thirty opportunities to solve the 'unknowns and cubes' problem since he believed that he would be unable to solve this type, as in fact had been the case when the contest was set up. However, in the early hours of 13 February 1535, inspiration came to Tartaglia and he discovered the method to solve 'squares and cubes equal to numbers'. Tartaglia was then able to solve all thirty of Fior's problems in less than two hours. As Fior had made little headway with Tartaglia's questions, it was obvious to all who was the winner. Tartaglia did not take his prize for winning from Fior, however, the honour of winning was enough.

Volume of a tetrahedron

Tartaglia is also known for having given an expression (Tartaglia's formula) for the volume of a tetrahedron (incl. any irregular tetrahedra) as the Cayley–Menger determinant of the distance values measured pairwise between its four corners:

where d ij is the distance between vertices i and j. This is a generalization of Heron's formula for the area of a triangle

Triangle
Tartaglia is known for having devised a method to obtain binomial coefficients called Tartaglia's Triangle (also called Pascal's Triangle).

FAMOUS TARTAGLIA PUZZLE
1.A man dies leaving 17 horses to be divide amongst his heirs in the ratio 1/2:1/3:1/9.How can this be done.

Fontana's solution involved borrowing an extra horse to calculate the distribution,politely returning it after the calculation.in fact one need only multiply the solution by 18 to arrive at the solution 9,6,2.

2.How to put 3 quarts of liquid in two 10 quarts container ,using a five quart and four quart measures.

This problem is is similar to the "tower of hanoi type problems",which is originally said to be put forward by chienese around 100 b.c,these kind of problems are of utmost philosiphical value,and is often referred to the class of problem the describes the working of nature.Acyually satisfaction of human need and also depicts the principle of least action.

Warren Buffett

2009-04-12T03:53:00.000-07:00

Warren Edward Buffett (born August 30, 1930 in Omaha, Nebraska, United States) is an American investor, businessman, and philanthropist. He is one of the world's most successful investors and the largest shareholder and CEO of Berkshire Hathaway. He is constantly ranked by Forbes as the second richest person in the world after Bill Gates with an estimated net worth of approximately $40.0 billion.

Buffet is often called the "Oracle of Omaha" or the "Sage of Omaha"and is noted for his adherence to the value investing philosophy and for his personal frugality despite his immense wealth. His 2006 annual salary was about $100,000, which is small compared to senior executive remuneration in comparable companies. In 2007, he earned a total compensation of $175,000, which included a base salary of just $100,000. He lives in the same house in the central Dundee neighborhood of Omaha that he bought in 1958 for $31,500, today valued at around $700,000. When Buffett spent $9.7 million of Berkshire's funds on a private jet in 1989, he jokingly named it "The Indefensible" because of his past criticisms of such purchases by other CEOs.

Buffet is also a notable philanthropist, having pledged to give away 85% of his fortune, then roughly $31 billion. He also serves as a member of the board of trustees at Grinnell College.

In 1999, Buffett was named the top money manager of the twentieth century in a survey by the Carson Group, ahead of Peter Lynch and John Templeton, and in 2007, he was listed among Time's 100 Most Influential People in the world.

Biography

Early life

Warren Buffett was born in Omaha, Nebraska on August 30, 1930, the son of Howard Buffett. He worked at his grandfather's grocery store. In 1943, Buffett filed his first income tax return, deducting his bicycle and watch as a work expense for $35 for his work as newspaper delivery boy. After his father was elected to Congress, Buffett was educated at Woodrow Wilson High School, Washington, D.C., where he graduated in 1947. In 1945, in his freshman year of high school, Buffett and a friend spent $25.00 to purchase a used pinball machine, which they placed in a barber shop. Within months, they owned three machines in different locations.

Buffett first enrolled at The Wharton School, University of Pennsylvania, (1947–1949) where he joined the Alpha Sigma Phi Fraternity. His father and uncles were Alpha Sigma Phi brothers from the chapter in Nebraska. In 1950 he transferred to the University of Nebraska where he received a B.S. in Economics.

Buffett then enrolled at Columbia Business School after learning that Benjamin Graham, (the author of The Intelligent Investor), and David Dodd, two well-known securities analysts, taught there. He then received a M.S. in Economics, Columbia University, in 1951.

In Buffett’s own words:

"I’m 15 percent Fisher and 85 percent Benjamin Graham.”

The basic ideas of investing are to look at stocks as business, use the market's fluctuations to your advantage, and seek a margin of safety. That’s what Ben Graham taught us. A hundred years from now they will still be the cornerstones of investing.

Career
Buffett was employed at 1951–1954 Buffett-Falk & Co., Omaha—Investment Salesman, then 1954–1956 Graham-Newman Corp., New York—Securities Analyst, then 1956–1969 Buffett Partnership, Ltd., Omaha—General Partner, then 1970–Present Berkshire Hathaway Inc, Omaha—Chairman, CEO.

In 1951, Buffett discovered Graham was on the board of GEICO insurance at the time. After taking a train to Washington, D.C. on a Saturday, Buffett knocked on the door of GEICO's headquarters until a janitor allowed him in. There, he met Lorimer Davidson, the vice president, who was to become a lasting influence on him and life-long friend. They talked for four hours about the insurance business. Davidson recalled that he found Buffett to be an “extraordinary man” after fifteen minutes. Buffett graduated from Columbia and wanted to work on Wall Street. Both his father and Ben Graham urged him not to. Buffett offered to work for Graham for free, but Graham refused.

Returning to Omaha, he purchased a Sinclair Texaco gas station as a side investment, but that venture did not work out as well as he had hoped. Meanwhile, he worked as a stockbroker. During that time, Buffett also took a Dale Carnegie public speaking course. Using what he learned, he felt confident enough to teach a night class at the University of Nebraska, "Investment Principles." The average age of the students he taught was more than twice his own.

In 1952, Buffett married Susan Thompson. In 1953, Susan and Warren Buffett had their first child, Susan Alice Buffett. In 1954, Benjamin Graham offered Buffett a job at his partnership with a starting salary of $12,000 a year. Here, he worked closely with Walter Schloss. Graham, who was a tough man to work for, was adamant that a stock provide a wide margin of safety after weighting the trade-off between its price and intrinsic value. Graham’s demand that a stock be worth more than its price made sense to Buffett, but it also made him question whether the criteria were too stringent, causing them to miss out on some big winners that had more qualitative values. Susan and Warren Buffett had their second child, Howard Graham Buffett. In 1956, Benjamin Graham retired, and closed his partnership. Buffett's personal savings were now over $140,000. Buffett returned home to Omaha and created Buffett Partnership Ltd., an investment partnership.

In 1957, Buffett had three partnerships operating the entire year. Buffett purchased a five-bedroom stucco house in Omaha, in which he still lives, for $31,500. In 1958, Susan and Warren Buffett had their third child, Peter Andrew Buffett. Buffett had five partnerships operating the entire year. In 1959, Buffett had six partnerships operating the entire year. Buffett was introduced to Charlie Munger. In 1960, Buffett had seven partnerships operating the entire year. The partnerships were: Buffett Associates, Buffett Fund, Dacee, Emdee, Glenoff, Mo-Buff, and Underwood. Buffett asked one of his partners, a doctor, to find ten other doctors who would be willing to invest $10,000 each in his partnership. Eventually, eleven doctors agreed to invest. In 1961, Buffett revealed that Sanborn Map Company accounted for 35% of the partnerships' assets. Buffett explained that in 1958, Sanborn sold at $45 per share when the value of the Sanborn investment portfolio was $65 per share. This meant buyers valued Sanborn at "minus $20" per share, and buyers were unwilling to pay more than 70 cents on the dollar for an investment portfolio with a map business thrown in for nothing. Buffett revealed that he earned a spot on the board of Sanborn.

Becoming a millionaire
In 1962, Buffett became a millionaire, because of Buffett's partnerships, which in January 1962, had in excess of $7,178,500, of which over $1,025,000 belonged to Buffett. Buffett merged all partnerships into one partnership. Buffett discovered a textile manufacturing firm, Berkshire Hathaway. Buffett's partnerships began purchasing shares at $7.60 per share. In 1965, when Buffett's partnerships aggressively began purchasing Berkshire, they paid $14.86 per share while the company had working capital (current assets minus current liabilities) of $19 per share. This did not include the value of fixed assets (factory and equipment). Buffett took control of Berkshire Hathaway at the board meeting and named a new president, Ken Chace, to run the company. In 1966, Buffett closed the partnership to new money. Buffett wrote in his letter:

unless it appears that circumstances have changed (under some conditions added capital would improve results) or unless new partners can bring some asset to the partnership other than simply capital, I intend to admit no additional partners to BPL.

In a second letter, Buffett announced his first investment in a private business — Hochschild, Kohn and Co, a privately owned Baltimore department store. In 1967, Berkshire paid out its first and only dividend of 10 cents. In 1969, following his most successful year, Buffett liquidated the partnership and transferred their assets to his partners. Among the assets paid out were shares of Berkshire Hathaway. In 1970, as chairman of Berkshire Hathaway, Buffett began writing his now-famous annual letters to shareholders.

However, he lived solely on his salary of $50,000 per year, and his outside investment income. In 1979, Berkshire began the year trading at $775 per share, and ended at $1,310. Buffett's net worth reached $620 million, placing him on the Forbes 400 for the first time.

In 2006, Buffett announced in June that he gradually would give away 85% of his Berkshire holdings to five foundations in annual gifts of stock, starting in July 2006. The largest contribution would go to the Bill and Melinda Gates Foundation.

In 2007, in a letter to shareholders, Buffett announced that he was looking for a younger successor, or perhaps successors, to run his investment business. Buffett had previously selected Lou Simpson, who runs investments at Geico, to fill that role. However, Simpson is only six years younger than Buffett.

In 2008, Buffett became the richest man in the world dethroning Bill Gates, worth $62 billion according to Forbes, and $58 billion according to Yahoo. Bill Gates had been number 1 on the Forbes list for 13 consecutive years.

Business

Acquisitions

In 1973, Berkshire began to acquire stock in the Washington Post Company. Buffett became close friends with Katharine Graham, who controlled the company and its flagship newspaper, and became a member of its board of directors.

In 1974, the SEC opened a formal investigation into Warren Buffett and Berkshire's acquisition of WESCO, due to possible conflict of interest. No charges were brought.

In 1977, Berkshire indirectly purchased the Buffalo Evening News for $32.5 million. Antitrust charges started, instigated by its rival, the Buffalo Courier-Express. Both papers lost money, until the Courier-Express folded in 1982.

In 1979, Berkshire began to acquire stock in ABC. With the stock trading at $290 per share, Buffett's net worth neared $140 million. Capital Cities' announced $3.5 billion purchase of ABC on March 18, 1985, surprising the media industry, as ABC was some four times bigger than Capital Cities was at the time. Berkshire Hathaway chairman Warren Buffett helped financed the deal in return for a 25 percent stake in the combined company. The newly merged company, known as Capital Cities/ABC (or CapCities/ABC), was forced to sell off some stations due to FCC ownership rules. Also, the two companies owned several radio stations in the same markets.

In 1987, Berkshire Hathaway purchased 12% stake in Salomon Inc., making it the largest shareholder and Buffett the director. In 1990, a scandal involving John Gutfreund (former CEO of Salomon Brothers) surfaced. A rogue trader, Paul Mozer, was submitting bids in excess of what was allowed by the Treasury rules. When this was discovered and brought to the attention of Gutfreund, he did not immediately suspend the rogue trader. Gutfreund left the company in August 1991. Buffett became CEO of Salomon until the crisis passed; on September 4 1991, he testified before Congress.

In 1988, Buffett began buying stock in Coca-Cola Company, eventually purchasing up to 7 percent of the company for $1.02 billion. It would turn out to be one of Berkshire's most lucrative investments, and one which it still holds. In 2002, Buffett entered in $11 billion worth of forward contracts to deliver U.S. dollars against other currencies. By April 2006, his total gain on these contracts was over $2 billion.

In 1998, he acquired General Re, (in a rare move, for stock). In 2002, Buffett became involved with Maurice R. Greenberg at AIG, with General Re providing reinsurance. On March 15, 2005, AIG's board forced Greenberg to resign from his post as Chairman and CEO under the shadow of criticism from Eliot Spitzer, attorney general of the state of New York. On February 9, 2006, AIG and the New York State Attorney General's office agreed to a settlement in which AIG would pay a fine of $1.6 billion.

In 2009, Warren Buffett invested $2.6 billion as a part of Swiss Re's raising equity captal. Berkshire Hathaway already owns a 3% stake, with rights to own more than 20%.

Late 2000s recession
Buffett ran into criticism, during the subprime crisis of 2007–2008, part of the late 2000s recession, that he has allocated capital too early resulting in suboptimal deals.

Buffett has called the 2007—present downturn in the financial sector "poetic justice".

Buffett's Berkshire Hathaway suffered a 77% drop in earnings during Q3 2008 and several of his recent deals appear to be running into large mark-to-market losses.

Berkshire Hathaway acquired 10% perpetual preferred stock of Goldman Sachs at $123 only for it to fall to below $60. Furthermore some of Buffett's Index put options (European exercise at expiry only) that he wrote (sold) are currently running around $6.73 billion mark-to-market losses.[41] The scale of the potential loss prompted the SEC to demand that Berkshire produce, "a more robust disclosure" of factors used to value the contracts.

Buffett also helped Dow Chemical pay for its $18.8 billion takeover of Rohm & Haas. He thus became the single largest shareholder in the enlarged group with his Berkshire Hathaway, which provided $3 billion, underlining his instrumental role during the current crisis in debt and equity markets.

In October 2008, the media reported that Warren Buffett had agreed to buy General Electric (GE) preferred stock, when it was trading in the mid 20s of dollar. The operation included extra special incentives: he received an option to buy 3 billion GE at $22.25 in the next five years and also received a 10% dividend (callable within three years). However, shortly after, GE gave up tens of billions in market capitalization and just bounced off a low of $8.80 in February 23, 2009, a price that has not been seen in over a decade. GE's stock price continued to fall after that point, and by early March, for example, it had declined to an 18 year low. Events like these have prompted a wave of criticism against Berkshire Hathaway and Warren Buffett. In February 2009, Warren Buffett unloaded part of Procter & Gamble Co and Johnson & Johnson shares from his portfolio..

Some have claimed that there is a financial incentive for Berkshire Hathaway to keep the myth that Buffett is an “oracle” alive and that the company is dependent on the Warren Buffett myth: that exaggerated sense of comfort investors share when it comes to Buffett’s beliefs and recommendations. In addition to suggestions of mistiming, questions have been raised as to the wisdom in keeping some of Berkshire's major holdings including The Coca-Cola Company (NYSE:KO) which in 1998 peaked at $86. Buffett discussed the difficulties of knowing when to sell in the company's 2004 annual report: "That may seem easy to do when one looks through an always-clean, rear-view mirror. Unfortunately, however, it’s the windshield through which investors must peer, and that glass is invariably fogged." In March 2009, Buffett stated in a cable television interview that the economy had "fallen off a cliff... Not only has the economy slowed down a lot, but people have really changed their habits like I haven't seen." Additionally, Buffett fears we may revisit a 1970s level of inflation, which led to a painful stagflation that lasted many years.

Personal life
Buffett married Susan Thompson in 1952. They had three children, Susie, Howard, and Peter. The couple began living separately in 1977, although they remained married until her death in July 2004. Their daughter Susie lives in Omaha and does charitable work through the Susan A. Buffett Foundation and is a national board member of Girls, Inc. In 2006, on his seventy-sixth birthday, he married his never-married longtime-companion, Astrid Menks, who was then sixty years old. She had lived with him since his wife's departure in 1977 to San Francisco. It was Susan Buffett who arranged for the two to meet before she left Omaha to pursue her singing career. All three were close and holiday cards to friends were signed "Warren, Susie and Astrid". Susan Buffett briefly discussed this relationship in an interview on the Charlie Rose Show shortly before her death, in a rare glimpse into Buffett's personal life.

He remains an avid player of the card game bridge, which he learned from Sharon Osberg, and plays with her and Bill Gates. He spends twelve hours a week playing the game. In 2006, he sponsored a bridge match for the Buffett Cup. Modeled on the Ryder Cup in golf, held immediately before it, and in the same city, a team of twelve bridge players from the United States took on twelve Europeans in the event.

Warren Buffett works with Christopher Webber on an animated series with DiC Entertainment chief Andy Heyward. According to information presented by Buffett at the Berkshire Hathaway annual meeting on May 6, 2006, the series will feature Buffett and Munger in roles and the series will teach children healthy financial habits for life. Cartoon drawings of Buffett and Munger were displayed throughout the events during the weekend as well as in a special animated movie from Heyward, displayed before the meeting.

Buffett has described himself as agnostic when it comes to religious beliefs. In December 2006 it was reported that Buffett does not carry a cell phone, does not have a computer at his desk, and drives his own automobile, a Cadillac DTS.

Buffett's DNA report revealed that his paternal ancestors hail from northern Scandinavia, while his maternal ancestors most likely have roots in Iberia or Estonia. Despite widespread suggestions to the contrary, and the casual friendship which has developed between their families, Warren Buffett has no clear relation to the well-known singer Jimmy Buffett.

Politics
In addition to other political contributions over the years, Buffett has formally endorsed and made campaign contributions to Barack Obama's presidential campaign. On July 2, 2008, Buffett attended a $28,500 per plate fundraiser for Mr. Obama's campaign in Chicago hosted by Mr. Obama's National Finance Chair, Penny Pritzker and her husband, as well as Obama advisor Valerie Jarrett.Buffett backed Obama for president, and intimated that John McCain's views on social justice were so far from his own that McCain would need a "lobotomy" for Buffett to change his endorsement. During the second 2008 U.S. presidential debate, candidates John McCain and Barack Obama, after being asked first by presidential debate mediator Tom Brokaw, both mentioned Buffett as a possible future Secretary of the Treasury. Later, in the third and final presidential debate, Obama mentioned Buffett as a potential economic advisor. Buffett was also finance advisor to California Republican Governor Arnold Schwarzenegger during his 2003 election campaign.

Writings
Warren Buffett's writings include his annual reports and various articles.

He warned about the pernicious effects of inflation:

“ The arithmetic makes it plain that inflation is a far more devastating tax than anything that has been enacted by our legislatures. The inflation tax has a fantastic ability to simply consume capital. It makes no difference to a widow with her savings in a 5 percent passbook account whether she pays 100 percent income tax on her interest income during a period of zero inflation, or pays no income taxes during years of 5 percent inflation. ”

In his article The Superinvestors of Graham-and-Doddsville, Buffett refuted the academic Efficient-market hypothesis, that beating the S&P 500 was "pure chance", by highlighting a number of students of the Graham and Dodd value investing school of thought. In addition to himself, Buffett named Walter J. Schloss, Tom Knapp, Ed Anderson (Tweedy, Brown Inc.), Bill Ruane (Sequoia Fund, Inc.), Charles Munger (Buffett's own business partner at Berkshire), Rick Guerin (Pacific Partners, Ltd.), and Stan Perlmeter (Perlmeter Investments)

In his November, 1999 Fortune article, he warned of investors' unrealistic expectations:

“ Let me summarize what I've been saying about the stock market: I think it's very hard to come up with a persuasive case that equities will over the next 17 years perform anything like--anything like--they've performed in the past 17. If I had to pick the most probable return, from appreciation and dividends combined, that investors in aggregate--repeat, aggregate--would earn in a world of constant interest rates, 2% inflation, and those ever hurtful frictional costs, it would be 6%. ”

Philanthropy
The following quotation from 1988, respectively, highlights Warren Buffett's thoughts on his wealth and why he long planned to re-allocate it:

“ I don't have a problem with guilt about money. The way I see it is that my money represents an enormous number of claim checks on society. It's like I have these little pieces of paper that I can turn into consumption. If I wanted to, I could hire 10,000 people to do nothing but paint my picture every day for the rest of my life. And the GNP would go up. But the utility of the product would be zilch, and I would be keeping those 10,000 people from doing AIDS research, or teaching, or nursing. I don't do that though. I don't use very many of those claim checks. There's nothing material I want very much. And I'm going to give virtually all of those claim checks to charity when my wife and I die. (Lowe 1997:165–166) ”

From a NY Times article: "I don't believe in dynastic wealth," Warren Buffett said, calling those who grow up in wealthy circumstances "members of the lucky sperm club." Buffett has written several times of his belief that, in a market economy, the rich earn outsized rewards for their talents:

“ A market economy creates some lopsided payoffs to participants. The right endowment of vocal chords, anatomical structure, physical strength, or mental powers can produce enormous piles of claim checks (stocks, bonds, and other forms of capital) on future national output. Proper selection of ancestors similarly can result in lifetime supplies of such tickets upon birth. If zero real investment returns diverted a bit greater portion of the national output from such stockholders to equally worthy and hardworking citizens lacking jackpot-producing talents, it would seem unlikely to pose such an insult to an equitable world as to risk Divine Intervention. ”

His children will not inherit a significant proportion of his wealth. These actions are consistent with statements he has made in the past indicating his opposition to the transfer of great fortunes from one generation to the next. Buffett once commented, "I want to give my kids just enough so that they would feel that they could do anything, but not so much that they would feel like doing nothing".

In 2006, he auctioned his 2001 Lincoln Town Car on eBay to raise money for Girls, Inc.

In 2007, he auctioned a luncheon with himself that raised a final bid of $650,100 for a charity.

In 2006, he announced a plan to give away his fortune to charity, with 83% of it going to the Bill & Melinda Gates Foundation. In June 2006, Buffett gave approximately 10 million Berkshire Hathaway Class B shares to the Bill & Melinda Gates Foundation (worth approximately USD 30.7 billion as of 23 June 2006)making it the largest charitable donation in history and Buffett one of the leaders in the philanthrocapitalism revolution. The foundation will receive 5% of the total donation on an annualised basis each July, beginning in 2006. Buffett also will join the board of directors of the Gates Foundation, although he does not plan to be actively involved in the foundation's investments.

He also announced plans to contribute additional Berkshire stock valued at approximately $6.7 billion to the Susan Thompson Buffett Foundation and to other foundations headed by his three children. This is a significant shift from previous statements Buffett has made, having stated that most of his fortune would pass to his Buffett Foundation. The bulk of the estate of his wife, valued at $2.6 billion, went to that foundation when she died in 2004.

He also pledged $50-million to the Nuclear Threat Initiative, in Washington, where he has served as an adviser since 2002.

On 27 June 2008, Zhao Danyang, a general manager at Pure Heart China Growth Investment Fund, won the 2008 5-day online "Power Lunch with Warren Buffett" charity auction with a bid of $2,110,100. Auction proceeds benefit the San Francisco Glide Foundation.

Public positions
Buffett's speeches are known for mixing business discussions with humor. Each year, Buffett presides over Berkshire Hathaway's annual shareholder meeting in the Qwest Center in Omaha, Nebraska, an event drawing over 20,000 visitors from both United States and abroad, giving it the nickname "Woodstock of Capitalism". Berkshire's annual reports and letters to shareholders, prepared by Buffett, frequently receive coverage by the financial media. Buffett's writings are known for containing literary quotes ranging from the Bible to Mae West, as well as Midwestern advice, and numerous jokes. Various websites extol Buffett's virtues while others decry Buffett’s business models or dismiss his investment advice and decisions.

Buffett and tobacco
During the RJR Nabisco, Inc. hostile takeover fight in 1987, Buffett was quoted as telling John Gutfreund:

“ I’ll tell you why I like the cigarette business. It costs a penny to make. Sell it for a dollar. It’s addictive. And there’s fantastic brand loyalty. ”

However, by 1994, Buffett had changed his stance on tobacco. Speaking at Berkshire Hathaway Inc.'s 1994 annual meeting, Buffett said investments in tobacco are:

“ fraught with questions that relate to societal attitudes and those of the present administration. I would not like to have a significant percentage of my net worth invested in tobacco businesses. The economy of the business may be fine, but that doesn't mean it has a bright future. ”

Buffett and coal
In 2007, Buffett's PacifiCorp, a subsidiary of his MidAmerican Energy Company, cancelled six proposed coal-fired power plants. These included Utah's Intermountain Power Project Unit 3, Jim Bridger Unit 5, and four proposed plants previously included in PacifiCorp's Integrated Resource Plan. The cancellations came in the wake of pressure from regulators and citizen groups, including a petition drive organized by Salt Lake City commercial real estate broker Alexander Lofft and directed at Buffett personally. The 1,600 petitioners, who described themselves in a letter to Buffett as "a collection of citizens, business owners and managers, service professionals, public servants, and organization representatives ... your friends and new customers here in Utah," explained that, in their view, any further expansion of coal generation in Utah would "compromise our health, obscure our viewsheds, shrink and contaminate our watersheds, and thin out our most beloved snowpack," concluding that "our attractiveness as a place to live and work is also threatened, and so is our economic competitiveness as a major metro area and a state, compromising our recent gains in income and property values."

Klamath river
American Indian tribes, and salmon fisherman sought to win support from Warren Buffett, for a proposal to remove four hydroelectric dams from the Klamath River. He had David Sokol respond that the FERC would decide the question.

Trade deficit
Buffett views the United States' expanding trade deficit as an alarming trend that will devalue the U.S. dollar and U.S. assets. He believes that the U.S. dollar will lose value in the long run, as a result of putting a larger portion of ownership of U.S. assets, in the hands of foreigners.

In his letter to shareholders in March, 2005, Warren Buffett predicted that in another ten years’ time the net ownership of the U.S. by outsiders would amount to $11 trillion. “Americans … would chafe at the idea of perpetually paying tribute to their creditors and owners abroad. A country that is now aspiring to an ‘ownership society’ will not find happiness in—and I’ll use hyperbole here for emphasis—a 'sharecropping society’.” Author Ann Pettifor has adopted the image in her writings and has stated: "He is right. And so the thing we must fear most now, is not just the collapse of banks and investment funds, or of the international financial architecture, but of a 'sharecropper society, angry at its downfall."

Dollar and gold
This induced Buffett to enter the foreign currency market for the first time in 2002. However, he substantially reduced his stake in 2005 as changing interest rates increased the costs of holding currency contracts. Buffett continues to be bearish on the dollar, and says he is looking to make acquisitions of companies which derive a substantial portion of their revenues from outside the United States.

Buffett emphasized the non-productive aspect of gold in 1998 at Harvard:

“ It gets dug out of the ground in Africa, or someplace. Then we melt it down, dig another hole, bury it again and pay people to stand around guarding it. It has no utility. Anyone watching from Mars would be scratching their head. ”

In 1977 Buffett was also quoted as saying about stocks, gold, farmland, and inflation:

“ stocks are probably still the best of all the poor alternatives in an era of inflation—at least they are if you buy in at appropriate prices. ”

Taxes
Buffett stated that he only paid 19% of his income for 2006 ($48.1 million) in total federal taxes, while his employees paid 33% of theirs, despite making much less money. Buffett favors the inheritance tax, saying that repealing it would be like "choosing the 2020 Olympic team by picking the eldest sons of the gold-medal winners in the 2000 Olympics". In 2007, Buffett testified before the Senate and urged them to preserve the estate tax so as to avoid a plutocracy.

Buffett believes government should not be in the business of gambling, or legalize casinos, believing it to be a tax on ignorance.

Expensing of stock options
He has been a strong proponent of stock option expensing, on the Income Statement. At the 2004 annual meeting, he compared the United States Congress, and the Securities and Exchange Commission’s decision to override FASB, who wanted to consider company-issued stock-option compensation as an expense, to a bill proposed in the Indiana House of Representatives to change Pi from 3.14159 to 3.2 .

When a company gives something of value to its employees in return for their services, it is clearly a compensation expense. And if expenses don't belong in the earnings statement, where in the world do they belong?

Investment in China
Buffett invested in PetroChina Company Limited and in a rare move, posted a commentary on Berkshire Hathaway's website stating why he would not divest from the company despite calls from some activists to do so. (He did, however, sell this stake, apparently for purely financial reasons.) Buffett believes that the world is nearing its maximum capacity of oil production and that gradually depleted oil fields could reduce the amount produced.

Books about Warren Buffett
Numerous books have been written about Warren Buffett and his investment strategies. In October 2008, USA Today reported that there were at least 47 books in print with Buffett's name in the title. The article quoted the CEO of Borders Books, George Jones, as saying that the only other living persons named in as many book titles were U.S. presidents, major world political figures, and the Dalai Lama. Buffett said that his own personal favorite is a collection of his essays called The Essays of Warren Buffett, which he described as "a coherent rearrangement of ideas from my annual report letters" as edited by Larry Cunningham.

Best-selling or otherwise notable books about Buffett include the following:

Robert Hagstrom, The Warren Buffett Way. (As of 2008, the bestselling book about Buffett.)
Alice Schroeder, The Snowball: Warren Buffett and the Business of Life. (Written with Buffett's cooperation.)
Mary Buffett and David Clark, Buffettology and four subsequent books. (Combined sales of more than 1.5 million copies.)
Janet Lowe, Warren Buffett Speaks: Wit and Wisdom from the Word's Greatest Investor.

John Train, The Midas Touch: The Strategies That Have Made Warren Buffett 'America's Preeminent Investor'.
Andrew Kilpatrick, Of Permanent Value: The Story of Warren Buffett. (The longest of the books about Buffett, with 330 chapters, 1,874 pages and 1,400 photos, weighing 10.2 pounds.)
Warren Buffett, Lawrence Cunningham (editor), The Essays of Warren Buffett. (A rearrangement of the Chairman's letters by topic.)

Barack h. obama

2009-01-25T00:02:00.000-08:00

Barack Hussein Obama II (pronounced /bəˈrɑːk hʊˈseɪn oʊˈbɑːmə/; born August 4, 1961) is the forty-fourth and current President of the United States. He is the first African American to hold the office. Obama was the junior United States Senator from Illinois from January 3, 2005 until his resignation on November 16, 2008, following his election to the presidency. He was sworn into office on January 20, 2009 in an inaugural ceremony at the United States Capitol in Washington D.C..

Obama is a graduate of Columbia University and Harvard Law School, where he was the first African-American president of the Harvard Law Review. He worked as a community organizer, and practiced as a civil rights attorney in Chicago before serving three terms in the Illinois Senate from 1997 to 2004. He also taught Constitutional Law at the University of Chicago Law School from 1992 to 2004. Following an unsuccessful bid for a seat in the U.S. House of Representatives in 2000, Obama was elected to the Senate in November 2004. Obama delivered the keynote address at the Democratic National Convention in July 2004.

As a member of the Democratic minority in the 109th Congress, Obama helped create legislation to control conventional weapons and to promote greater public accountability in the use of federal funds. He also made official trips to Eastern Europe, the Middle East, and Africa. During the 110th Congress, he helped create legislation regarding lobbying and electoral fraud, climate change, nuclear terrorism, and care for U.S. military personnel returning from combat assignments in Iraq and Afghanistan.

President Barack Obama's Inaugural Address

My fellow citizens:

I stand here today humbled by the task before us, grateful for the trust you have bestowed, mindful of the sacrifices borne by our ancestors. I thank President Bush for his service to our nation, as well as the generosity and cooperation he has shown throughout this transition.

Forty-four Americans have now taken the presidential oath. The words have been spoken during rising tides of prosperity and the still waters of peace. Yet, every so often the oath is taken amidst gathering clouds and raging storms. At these moments, America has carried on not simply because of the skill or vision of those in high office, but because We the People have remained faithful to the ideals of our forbearers, and true to our founding documents.

So it has been. So it must be with this generation of Americans.

That we are in the midst of crisis is now well understood. Our nation is at war, against a far-reaching network of violence and hatred. Our economy is badly weakened, a consequence of greed and irresponsibility on the part of some, but also our collective failure to make hard choices and prepare the nation for a new age. Homes have been lost; jobs shed; businesses shuttered. Our health care is too costly; our schools fail too many; and each day brings further evidence that the ways we use energy strengthen our adversaries and threaten our planet.

These are the indicators of crisis, subject to data and statistics. Less measurable but no less profound is a sapping of confidence across our land - a nagging fear that America's decline is inevitable, and that the next generation must lower its sights.

Today I say to you that the challenges we face are real. They are serious and they are many. They will not be met easily or in a short span of time. But know this, America - they will be met.

On this day, we gather because we have chosen hope over fear, unity of purpose over conflict and discord.

On this day, we come to proclaim an end to the petty grievances and false promises, the recriminations and worn out dogmas, that for far too long have strangled our politics.

We remain a young nation, but in the words of Scripture, the time has come to set aside childish things. The time has come to reaffirm our enduring spirit; to choose our better history; to carry forward that precious gift, that noble idea, passed on from generation to generation: the God-given promise that all are equal, all are free, and all deserve a chance to pursue their full measure of happiness.

In reaffirming the greatness of our nation, we understand that greatness is never a given. It must be earned. Our journey has never been one of short-cuts or settling for less. It has not been the path for the faint-hearted - for those who prefer leisure over work, or seek only the pleasures of riches and fame. Rather, it has been the risk-takers, the doers, the makers of things - some celebrated but more often men and women obscure in their labor, who have carried us up the long, rugged path towards prosperity and freedom.

For us, they packed up their few worldly possessions and traveled across oceans in search of a new life.

For us, they toiled in sweatshops and settled the West; endured the lash of the whip and plowed the hard earth.

For us, they fought and died, in places like Concord and Gettysburg; Normandy and Khe Sahn.

Time and again these men and women struggled and sacrificed and worked till their hands were raw so that we might live a better life. They saw America as bigger than the sum of our individual ambitions; greater than all the differences of birth or wealth or faction.

This is the journey we continue today. We remain the most prosperous, powerful nation on Earth. Our workers are no less productive than when this crisis began. Our minds are no less inventive, our goods and services no less needed than they were last week or last month or last year. Our capacity remains undiminished. But our time of standing pat, of protecting narrow interests and putting off unpleasant decisions - that time has surely passed. Starting today, we must pick ourselves up, dust ourselves off, and begin again the work of remaking America.

For everywhere we look, there is work to be done. The state of the economy calls for action, bold and swift, and we will act - not only to create new jobs, but to lay a new foundation for growth. We will build the roads and bridges, the electric grids and digital lines that feed our commerce and bind us together. We will restore science to its rightful place, and wield technology's wonders to raise health care's quality and lower its cost. We will harness the sun and the winds and the soil to fuel our cars and run our factories. And we will transform our schools and colleges and universities to meet the demands of a new age. All this we can do. And all this we will do.

Now, there are some who question the scale of our ambitions - who suggest that our system cannot tolerate too many big plans. Their memories are short. For they have forgotten what this country has already done; what free men and women can achieve when imagination is joined to common purpose, and necessity to courage.

What the cynics fail to understand is that the ground has shifted beneath them - that the stale political arguments that have consumed us for so long no longer apply. The question we ask today is not whether our government is too big or too small, but whether it works - whether it helps families find jobs at a decent wage, care they can afford, a retirement that is dignified. Where the answer is yes, we intend to move forward. Where the answer is no, programs will end. And those of us who manage the public's dollars will be held to account - to spend wisely, reform bad habits, and do our business in the light of day - because only then can we restore the vital trust between a people and their government.

Nor is the question before us whether the market is a force for good or ill. Its power to generate wealth and expand freedom is unmatched, but this crisis has reminded us that without a watchful eye, the market can spin out of control - and that a nation cannot prosper long when it favors only the prosperous. The success of our economy has always depended not just on the size of our Gross Domestic Product, but on the reach of our prosperity; on our ability to extend opportunity to every willing heart - not out of charity, but because it is the surest route to our common good.

As for our common defense, we reject as false the choice between our safety and our ideals. Our Founding Fathers, faced with perils we can scarcely imagine, drafted a charter to assure the rule of law and the rights of man, a charter expanded by the blood of generations. Those ideals still light the world, and we will not give them up for expedience's sake. And so to all other peoples and governments who are watching today, from the grandest capitals to the small village where my father was born: know that America is a friend of each nation and every man, woman, and child who seeks a future of peace and dignity, and that we are ready to lead once more.

Recall that earlier generations faced down fascism and communism not just with missiles and tanks, but with sturdy alliances and enduring convictions. They understood that our power alone cannot protect us, nor does it entitle us to do as we please. Instead, they knew that our power grows through its prudent use; our security emanates from the justness of our cause, the force of our example, the tempering qualities of humility and restraint.

We are the keepers of this legacy. Guided by these principles once more, we can meet those new threats that demand even greater effort - even greater cooperation and understanding between nations. We will begin to responsibly leave Iraq to its people, and forge a hard-earned peace in Afghanistan. With old friends and former foes, we will work tirelessly to lessen the nuclear threat, and roll back the specter of a warming planet. We will not apologize for our way of life, nor will we waver in its defense, and for those who seek to advance their aims by inducing terror and slaughtering innocents, we say to you now that our spirit is stronger and cannot be broken; you cannot outlast us, and we will defeat you.

For we know that our patchwork heritage is a strength, not a weakness. We are a nation of Christians and Muslims, Jews and Hindus - and non-believers. We are shaped by every language and culture, drawn from every end of this Earth; and because we have tasted the bitter swill of civil war and segregation, and emerged from that dark chapter stronger and more united, we cannot help but believe that the old hatreds shall someday pass; that the lines of tribe shall soon dissolve; that as the world grows smaller, our common humanity shall reveal itself; and that America must play its role in ushering in a new era of peace.

To the Muslim world, we seek a new way forward, based on mutual interest and mutual respect. To those leaders around the globe who seek to sow conflict, or blame their society's ills on the West - know that your people will judge you on what you can build, not what you destroy. To those who cling to power through corruption and deceit and the silencing of dissent, know that you are on the wrong side of history; but that we will extend a hand if you are willing to unclench your fist.

To the people of poor nations, we pledge to work alongside you to make your farms flourish and let clean waters flow; to nourish starved bodies and feed hungry minds. And to those nations like ours that enjoy relative plenty, we say we can no longer afford indifference to suffering outside our borders; nor can we consume the world's resources without regard to effect. For the world has changed, and we must change with it.

As we consider the road that unfolds before us, we remember with humble gratitude those brave Americans who, at this very hour, patrol far-off deserts and distant mountains. They have something to tell us today, just as the fallen heroes who lie in Arlington whisper through the ages. We honor them not only because they are guardians of our liberty, but because they embody the spirit of service; a willingness to find meaning in something greater than themselves. And yet, at this moment - a moment that will define a generation - it is precisely this spirit that must inhabit us all.

For as much as government can do and must do, it is ultimately the faith and determination of the American people upon which this nation relies. It is the kindness to take in a stranger when the levees break, the selflessness of workers who would rather cut their hours than see a friend lose their job which sees us through our darkest hours. It is the firefighter's courage to storm a stairway filled with smoke, but also a parent's willingness to nurture a child, that finally decides our fate.

Our challenges may be new. The instruments with which we meet them may be new. But those values upon which our success depends - hard work and honesty, courage and fair play, tolerance and curiosity, loyalty and patriotism - these things are old. These things are true. They have been the quiet force of progress throughout our history. What is demanded then is a return to these truths. What is required of us now is a new era of responsibility - a recognition, on the part of every American, that we have duties to ourselves, our nation, and the world, duties that we do not grudgingly accept but rather seize gladly, firm in the knowledge that there is nothing so satisfying to the spirit, so defining of our character, than giving our all to a difficult task.

This is the price and the promise of citizenship.

This is the source of our confidence - the knowledge that God calls on us to shape an uncertain destiny.

This is the meaning of our liberty and our creed - why men and women and children of every race and every faith can join in celebration across this magnificent mall, and why a man whose father less than sixty years ago might not have been served at a local restaurant can now stand before you to take a most sacred oath.

So let us mark this day with remembrance, of who we are and how far we have traveled. In the year of America's birth, in the coldest of months, a small band of patriots huddled by dying campfires on the shores of an icy river. The capital was abandoned. The enemy was advancing. The snow was stained with blood. At a moment when the outcome of our revolution was most in doubt, the father of our nation ordered these words be read to the people:

"Let it be told to the future world...that in the depth of winter, when nothing but hope and virtue could survive...that the city and the country, alarmed at one common danger, came forth to meet [it]."

America. In the face of our common dangers, in this winter of our hardship, let us remember these timeless words. With hope and virtue, let us brave once more the icy currents, and endure what storms may come. Let it be said by our children's children that when we were tested we refused to let this journey end, that we did not turn back nor did we falter; and with eyes fixed on the horizon and God's grace upon us, we carried forth that great gift of freedom and delivered it safely to future generations.

VIVEK OBEROI

2008-09-02T10:28:00.001-07:00

VIVEK OBEROI was born on 3^rd September 1976 in Hyderabad, India. His father Suresh Oberoi was a Popular Actor of his times and his mother name is Shikha Oberoi (Yashodhara) . Vivek has an elder brother Prithviraj, younger Sister Meghna Oberoi and younger brother Anand Oberoi. Vivek is also called as ‘Tiger’ as pet name by his Family. Vivek was a student of Mayo College in Ajmer and later moved to London at an actor’s workshop where he was spotted by the director of New York University (NYU). From there Vivek was taken to New York City in order to complete his Masters Degree in Film Acting. Before making an entry in Film Industry as an Actor, he first wrote some of the Scripts and made his debut with Ram Gopal Varma’s directed ‘Company’ in 2002. The film as well as Vivek, both critically received good reviews. Vivek performance was appreciated with a great round applauded. Its is said that after Hrithik Roshan, the son of great producer Rakesh Roshan Vivek Oberoi was the only son of a Super-Star Actor Suresh Oberoi who Successfully managed to attract all the media and his fans attention towards him, few months before his debut film hitting out the theatres. For him it was an unconventional debut in "Company" that won him accolades from both the New-commers and the connoisseurs alike. Vivek Kept on winning hearts with his continuous super hit films as ‘Road’ opposite with Antara Mali, ‘Saathiya’ with Beautiful Rani Mukherjee and ‘Dum’ casting with Gorgeous Dia Mirza in 2002 and in all the hits films his mind-blowing performance was unforgettable for his fans and was appreciated with great respect. Then came a comedy film named ‘Masti’ in 2004 with C0-Stars like Aftab Shivdasani, Ritesh Deshmukh, Amrita Rao and Ajay Devgan. Followed by ‘Kyun Ho Gaya Na’ and Mani Ratnam’s directed film ‘Yuva’, both the roles were different form each other but Vivek managed to act very well in both the character. Then came Karan Johar’s ‘Kaal’ that was a suspense and thriller action movie casting opposite John Abraham, Esha Deol, Lara Dutta and Ajay Devgan. He had also participated in Mani Ratnam’s Stage Show at Netru, Indru in Naalai. Vivek was previously engaged to model Gurpreet Gill but the engagement didn’t work out and was broken for an unknown reason. Very Soon Vivek was seen dating the Most – talented and Gorgeous Actress Aishwarya Rai, but even their relationship failed to carry and was broken down. Vivek has a very humane side to him, he adopted and helped re-build a village, which was hit severely by the Tsunami. Vivek has many films in his kitty and he is one of the costliest stars in Bollywood Industry.

AWARDS:

Vivek Oberoi had received many Awards for his Debut film ‘Company’ directed by Ram Gopal Varma in the year 2003. Firstly, he was awarded Filmfare Best Debut Award and Filmfare Best Supporting Actor Award for the Film Company. Nextly its was in Zee Cine Awards where he received Zee Cine Best Male Debut Award and Zee Cine Best Actor Award in a Supporting role – Male for Company. And at last, he was honored in Star Screen Awards as Most Promising New Comer – Male Award for Company. He was later nominated for Shaad Ali’s ‘Saathiya’ in the category of Filmfare Best Actor Award in 2004.

EMRAAN HASHMI

2008-09-02T10:25:00.002-07:00

EMRAAN HASHMI was born on 23^rd March 1979 in Mumbai (India). His full name is taken as Emraan Anwar Hashmi, but the people closed to him call him ‘Emmy". His parents have a mix religious background. His father Anwar Hashmi belongs to a Muslim race while is mother Maherahh is a Catholic. His father’s name was later changed from Anwar Hashmi to Farhaan Hashmi. But he decided to keep the extra “A” as he received some good signs and benefits form numerology about that letter.

Just at the age of five years Emraan had perform in many commercials like Rasna, Goodnight etc. Emraan was then known as a child model. Emraan graduated from Sydenham College in Mumbai where he took up graphics and animations studies. He is a nephew to Mahesh Bhatt, the famous Producer as well as a Director in the film Industry and Actress turned Director Pooja Bhatt is one of his Cousins.

After completing his graduation he started working as an assistant director with Vikram Bhatt and there his interest started losing in other monotonous work. As he was an assistant to Sir Vikram Bhatt, he got a golden opportunity to work in films like ‘Raj’ and ‘Kasoor’. Emraan’s work was then noticed by brother Mahesh Bhatt and one day suddenly Mahesh Bhatt asked him to act in his next film and Emraan agreed in Acting in his film.

Emraan made his debut with the film Footpath in 2003. However the movie did not perform well at the box office and his role was missing somewhere in the movie lack his critical acclaim and his talent too. His Hard work, Dashing looks and Mind-blowing performance was realized by the film industry in his next super hit film ‘Murder’ directed by Mahesh Bhatt in 2004. This movie was Emraan's first hit film in his career. His next Average movies at the Box- office were ‘Tum sa Nahi Dekha’ with Dia Mirza, ‘Zeher’ with Shamita Shetty and Udita Goswami, ‘Aashiq Banaya Aapne’ with Tanushree Dutta, ‘Chocolate’ with Tanushree Duta, ‘Kalyug’ with Deepal Shaw and Newcomer Actor Kunal, ‘Jawaani Diwaani’ with Celina Jaitley and Hrishita Bhatt, ‘Aksar’ with Udita Goswami, ‘Gangster’ with Kangana Raut , ‘The Killer’ with Nisha Kothari and ‘Dil Diya Hai’ with Geeta Basra but the Soundtrack of all these respective movies had given birth to Emraan's innumerable fans list.

All the songs from his movies had become an instant hits in the Musical Market. His Role has titled him as a serial – kisser for his kissing scenes in all of his movies. This new trend was not seen in the Indian movies three to four years before, it started first from the movie ‘Murder’ and kept on continuing in his lastest film ‘Gangster’. Emraan, as said in an Interview describes his nervousness during his first on screen kissing scene. He had almost opted out of the scene before shooting the scene with a Female Co–star. Emraan Hashmi and his pheonsic Parveen Sahani are further planning to get married in this December 2006.

Its is heard that, Emraan has stop on screen kissing after the movie ‘Gangster’. The reason behind this decision was a surprise for the Bollywood Industry. He says that his Pheonsic had a personal request and then Emraan approved her request and quit out by saying ‘No More On - Screen Kissing Please’. His further realeses film like ‘Good Boy Baad Boy’ has a Question to his fans –

Will there be any Kissing Scene in the film or not ?

FILMOGRAPHY :

Awaarapan
Train (2007)
Good Boy, Bad Boy (2007)
Dil Diya Hai Saahil Khanna (2006)
The Killer (2006)
Gangster (2006)
Aksar (2006)
Jawani Diwani: A Youthful Joyride (2006)
Kalyug (2005)
Chocolate (2005)
Aashiq Banaya Aapne (2005)
Zeher (2005)
Tumsa Nahin Dekha (2004)
Murder (2004)
Footpath (2003)

ZAYED KHAN

2008-09-02T10:25:00.001-07:00

Zayed Khan was born on 27^th April 1980 in Mumbai (India). His father SANJAY KHAN was a great Actor and his mother name is ZARINE KHAN. He has three sisters in his family named as, SUZANNE KHAN, SIMONE KHAN and FARAH KHAN. In which, SUZZANE KHAN had got married to the handsome and mind-blowing Actor HRITHIK ROSHAN. Not only this, he have many more other family connections in the Bollywood industry. His Uncle FIROZ KHAN, A well-known film producer and director and FARDEEN KHAN, a talented actor is one of his cousins.

Zayed was a student of Welham Boys Boarding School in Dehradun. After completing his Secondary Examination from Boarding school, Zayed moved to an International School at Kodaikanal that taught and build up a strong desire in him to study abroad. He stood in front for such strange but comprehensive lessons of filmmaking and learnt all aspects, needs and efforts to be put during a film. Zayed is Intelligent, Mature and a Perfect Actor for young generation age groups. His Mind blowing On-screen presence denotes him the looks of a Greek God. When Zayed was just 8 yrs old, he was taken on the sets of ‘Kaala Dhanda Gore Log’ to see the actual making of a film.

And That experience had inspired him a lot and he then thought of becoming a Star at his childhood age. And his dreams of becoming a Super-Star which he buried deep in his heart during his childhood days came true with his debut film ‘Chura Liya Hai Tumne’ with Esha Deol in 2000. Though the film didn’t go well at the box-office and failed to win the hearts of the viewer’s but Zayed performance was realized and appreciated at a very high level. His second film actually made him a Super – Star and he received critically acclaim from the viewer's. ‘Main Hoon Na’ directed by FARAH KHAN and supported by SHAH RUKH KHAN was a Blockbuster hit.

This film gave him his own platform in the Bollywood Industry. Later Zayed Khan was tied into knots and got married with his longtime girlfriend Malaika Parekh on 20^th November 2005. His looks and styles made him the ‘REAL DUDE’ of Bollywood. His next releases in 2005 like ‘Shabd’ with Sanjay Dutt and Aishwarya Rai and ‘Vaada’ starring Arjun Rampal and Amisha Patel, both were Averages. But in 2005 it was VENKY’S ROCK ‘Dus’ casting with Sanjay Dutt, Shilpa Shetty, Suneil Shetty, Raima Sen, Abhishek Bachchan, Esha Deol, Dia Mirza and Pankaj Kapoor which turned on the Zayed fans once again and resulted in order of a Super hit film.

And then his friends and C0-stars represented him with many names like; BRO, BUDDY, DUDE, TECHNO, COOL GUY etc. Then came ‘Shaadi No. 1’ which was a comedy movie and Zayed did justice with this role also. In ‘Fight Club’ his fans were crazy and loved his acting very much. His talent and hardwork won many hearts of Bollywood Industry. His fans is desperately waiting for his up-coming movies like Idiot casting Celina Jaitley and Esha Deol with Bisaat. Zayed has signed an International Touring Contact of stage show, ‘ROCK STARS’.

SHAHID KAPOOR

2008-09-02T10:24:00.001-07:00

Shahid Kapoor with the innocence look throbs the millions heart by his debut film “Ishq Vishq Pyaar Vyaar”. Shahid was born on 25^th February 1981 in Mumbai. He completed his early education from Gyan Bharathi (Delhi) and his further studies were completed in Rajhans Vidyalaya (Mumbai). When Shahid Debuted his age was just 22 years. This younger hero did just few films but has offer from Boney Kapoor, Nitin Manmohan and Abbas Mastan.

Shahid believes in achieving all by him, though he has a film family tag. His father Pankaj Kapoor, a versatile actor and mother Neelima Azmi, an actress and marvellous classical dancer. Shahid has inherited from his parent’s great acting talent. He had never told his parents that he want to be an actor before signing his first film. This charming actor, who started his career with a video album “Aryans”. Shahid made his film “Ishq Vishq Pyaar Vyaar” in 2003 and was a hit. He received the “ Hero Cycles Stardust Award” and “Filmfare” for the most promising newcomer of the year 2003.

Shahid went for the audition of Ishq Vishq first time and was rejected, as he looked too young. After years later Ken Ghosh saw him in a public place, he was totally changed. Ghosh asked him to come to the audition again and he was selected. He wasn’t offer much film after the released of “Ishq Vishq….”for about few months though the film was hit and his acting was also appreciated. The main reason behind this was that producers and directors had no subjects for a 22 years old guy. After that people started writing subjects for him and he started getting lots of offers. Shahid signed his second film after waiting for the six months of his first release. He did Ken Ghosh’s “Fida” (2004) with Fardeen Khan and his girlfriend Kareena Kapoor, which was a thriller movie. Then he was seen in Nitin Manmohan’s “ Dil Maange More” (2004), which was a hit. It was a light romantic comedy about which of the three girls get him.

Shahid also did “Deewane Huye Paagal” and Mahesh Manjrekar’s “Vaah! Life Ho To Aisi” (2005), this movie became popular among the children and was appreciated. Then came “Shikar” (2006) casting Shahid Kapoor, Ajay Devgan, Amrita Roa and Bipasha Basu, though the movie didn’t do well at the box – office but Shahid’s role as Jaidev Vardhan was appreciated and praised a lot. His efforts in this movie made him to be nominated in the category of Screen Best Actor Award for Shikar in 2006. His famous movies are ”Vaah! Life Ho To Aise”, ”Dil Maange More”, “Fida”, “Ishq Vishq Pyaar Vyaar”. In 2006 at the mid of summer, Shahid moved out on a World Tour along with his close and fellow Bollywood co-stars starting with Salman Khan to Kareena Kapoor followed John Abraham, Esha Deol, Mallika Sherawat, and Zayed Khan.

This activity also shows that Shahid also like traveling and visiting places. Shahid is continuing with Boney Kapoor’s “Milenge Milenge”, which is been directing by Satish Kaushik, starring with the gorgeous Kareena Kapoor. His upcoming movie is “Jeevansh” and Vivaah, casting Shahid and Amrita Roa. He had also planned a comedy film named ‘Fool and Final’ with Ayesha Takia. His biggest fear is flying. He likes drinking energy drinks. His admires Pankaj Kapoor as his favourite actor. In Hollywood he likes actress Julia Roberts. He loves to dance and act. He also enjoys playing cricket.

ABHISHEK BACHCHAN

2008-09-02T10:22:00.000-07:00

This Dashing, Hot and Innocent Personality was born on 5^th February 1976. Abhishek has done his Debut and enters into the Bollywood Film Industry in the year 2000. Abhishek is the son of the Bollywood Icon ‘Mr.Amitab Bachchan’ and renowned Actress ‘Mrs. Jaya Bachchan’ who is also a MP in Rajya Sabha, Samajwadi Party. Abhishek has only one Sister name ‘Shweta Bachchan’ who is married to ‘Nikhil Nanda’. Abhishek has completed his schooling from ‘Bombay Scottish School’ a Modern School in New Delhi. Abhishek has also gone to ‘Aiglon College’ in ‘Switzerland’. He than went to United States to complete his education from ‘Boston University’ but unfortunately Abhishek had not finish his Business Degree as he had made a decision to pursue in an Acting Career. His grandfather was a very well known poet of Urdu and Hindi, Mr. Harivansh Rai Bachchan.

The Original last name of the Bachchan’s family was ‘Srivastava’. Bachchan was a pen name, which called by Abhishekh’s grandfather ‘Mr. Harivansh Rai Bachchan’. So as his father Mr. Amitabh Bachchan entered into the world of Indian Cinema and Bollywood Industry even he was also under his father’s pen name. Abhishek has enter into the film industry by debuting in J.P. Dutta’s film ‘Refugee’ along with the co-star Kareena Kapoor in the year 2000. This film has not achieved a great success in the box office. Abhishek went on for many movies in a span of four years without any major box- office successes but after Mani Ratnam’s directed film ‘YUVA’, he proved himself as a Mettle Actor.

After the success of ‘YUVA’ Abhishek has received many film in which he has gone to do his best. The year 2000 and onwards was very lucky for him. From the year 2000 he had starred in many hit films. He had portrayed a character of ‘ Lallan’, which earned him the Award with every other supporting Actor at the major film award functions. In the year 2004 he had starred in ‘Yash Chopra’s’ films “DHOOM” which was produced by ‘Yash Chopra’ only. This film was also a hit in Abhishek film Career. He has broken the record of all ‘Bollywood Actor and Actress’ by shooting to fame with four consecutive hits. He has shot in the film ‘‘BUNTY AUR BABLI” with the co-star ‘Rani Mukherjee’, In the film “SARKAR”he has worked with his dad ‘Mr. Amitabh Bachchan’, In the film “DUS” he has worked with the pillar of Bollywood ‘Mr.Sanjay Dutt’ and In the film “BLUFFMASTER” he has worked with ‘Ritesh Deshmukh’, Miss. World ‘Priyanka Chopra’ and ‘Nana Patekar’.

Abhishek has recently won Awards for ‘Best Supporting Actor’. In the year 2006 he has shot a film with Bollywood King ‘Shah Rukh Khan’ named ‘Kabhi Alvida Naa Kehna’ which was co-starred with ‘Amitabh Bachchan’, ‘Rani Mukherjee’ and ‘Pretty Zinta’. This film is having a good result in the box-office and doing extremely well. He was also a part of ‘Mani Ratnam’s’ stage show named ‘Netru, Indru and Naalai’ along with many co-stars. Abhishek was engaged with the Kapoor’s Family girl named ‘Karishma Kapoor’ in the year 2002 but unfortunately they broke up. However, now much media has their intentioned that Abhishek has focused towards alleged romantic involvement with fellow Bollywood celebrity and Miss. World of the year 1994, Aishwarya Rai.

Bollywood Awards Achieved by Indian Cinema :

2001: Nomination, ‘Best Actor’ for “REFUGEE”

2004: Nomination, ‘Best Supporting Actor’ for “MAIN PREM KI DIWAANI”

2005: Nomination, ‘Best Villain Award’ for “YUVA” & Winner, ‘Best Supporting Actor’ for “YUVA”.

2006: Nomination, ‘Best Actor’ for “BUNTY AUR BABLI” & Winner, ‘Best Supporting Actor for SARKAR.

IIFA Awards :

2001: Nomination, ‘Best Upcoming Talent’.

2005: Winner, ‘Best Supporting Actor’ for “YUVA”.

2006: Winner, ‘Best Supporting Actor’ for “SARKAR”.

Star Screen Awards :

2001: Nomination, ‘Most Promising New Comer – Male’ for “REFUGEE”.

2004: Winner, ‘Stardust Star of the Year Award – Male’ for “YUVA”.

2005: Winner, ‘Best Supporting Actor for “YUVA” & Nomination, for ‘JODI No.1 along with ‘Rani Mukherjee’ for “YUVA”

2005: Nomination, ‘Best Performance in a Villainous Role’ for “YUVA” & Winner, ‘Zee Cine Award for Best Actor in a Supporting Role – Male’ for “PHIR MILENGE”.

2005: Winner, ‘Best Actor in a Supporting’ for “SARKAR” for ‘Stardust Awards’.

2006: Winner, ‘Best Comedian’ for “BUNTY AUR BABLI” & Winner, ‘Jodi No.1’ along with ‘Rani Mukherjee’ for “BUNTY AUR BABLI” and ‘Zee Cine Awards’.

Abhishek Bachchan’s Profile :

Full Name: Abhishek Bachchan.
Nick Name: Abhi.

Star: Aquarius.

Date Of Birth: 5^th February 1976
Height: 6'3" or 1.91 m
Weight: 75 Kgs
Country: India
Mother: Jaya Bhaduri
Father: Amitabh Bachchan
Brother-in-law: Nikhil Nanda

Grand father: Lt. Harivanshrai Bachchan
Siblings: sister Shweta Nanda; niece Navya Naveli, nephew Agastye
Marital status: single, engaged to Aishwarya Rai

Other facts: Abhishek was engaged to Karisma Kapoor but the engagement was broken off
Education: Bombay Scottish School, studied in Switzerland and the US (Boston University) but didn't finish studying business in the United States since he wanted to become an actor
Eyes: Black
Hair: Black

First big hit: Dhoom
Favorite food: anything indian
Favourite Perfume: eternity
Favourite Holiday Spot: home
Dream Role: Agnipath
His Hero: Dad, Manoj Bajpai, Akshaye Khanna, Sanjay Dutt, Jackie Shroff and Govinda
His Heroine: Kareena, Keerthi Reddy and Aishwarya Rai
Something about yourself you would like to change: Maybe my physique
Director you would like to work with: Whoever wants to work with me
Actors I admire: Dad, Manoj Bajpai, Akshay Khanna, Sanjay Dutt, Jackie Shroff and Govinda
Favourite Co-stars: Kareena, Keerthi Reddy and Aishwarya Rai

Languages: hindi, french, english
Address: Pratiksha, 10th Road, J.V.P.D Scheme. Mumbai - 400049

FILMOGRAPHY :

2007 Untitled Tarun Mansukhani Project
2007 Sarkar 2
2007 Laaga Chunari Mein Daag
2007 Jhoom Barabar Jhoom
2007 Drona
2007 Shootout at Lokhandwala
2007 Guru
2006 Dhoom 2
2006 Umrao Jaan
2006 Lage Raho Munna Bhai
2006 Kabhi Alvida Naa Kehna
2006 Alag
2005 Bluffmaster
2005 Neal n' Nikki
2005 Home Delivery: Aapko... Ghar Tak
2005 Ek Ajnabee
2005 Salaam Namaste
2005 Antar Mahal
2005 Dus
2005 Sarkar
2005 Bunty Aur Babli
2004 Hum Tum
2004 Naach
2004 Dhoom
2004 Yuva
2004 Phir Milenge
2004 Run
2004 Rakht: What If You Can See the Future
2003 Zameen
2003 LOC Kargil
2003 Kuch Naa Kaho
2003 Mumbai Se Aaya Mera Dost
2003 Main Prem Ki Diwani Hoon
2002 Om Jai Jagadish
2002 Desh
2002 Haan Maine Bhi Pyaar Kiya
2001 Shararat
2001 Bas Itna Sa Khwaab Hai
2000 Tera Jadoo Chal Gaya
2000 Dhai Akshar Prem Ke
2000 Refugee

JOHN ABRAHAM

2008-09-02T10:21:00.000-07:00

John Abraham was born on 17^th December 1973 in Mumbai (INDIA). According to his dad John was born in a Malayalee Christian family of Aluva (Kerala) but his mother is an Irani from the family of Parsi. His father was an architect while his mother was a basketball teacher. Abraham has a younger brother named as Alan Abraham. However, being a Catholic his father added the name ‘Abraham’ to his son John, who was named by his grandparents as a mark of respect and give honor to the first patriarch and progenitor of the Hebrew people in the Bible. Later on John moved to Mumbai and he studied from Bombay Scottish School continuing with Jai Hind College, where he became the captain of the college football team.

After completing his studies of MBA in marketing, he started working with an ad agency as a media planner for the Enterprises-Nexus as well as being captain of Bombay's soccer team. His revolutionary looks, macho body, baby smiles and nice character made him to move in modeling field. And there was the time when Abraham’s luck stroked out with many hearts started beating just for him. He then became the highest paid supermodel in India.

John Abraham won the Gladrags Manhunt Contest in the year 1999 and further moved on to Singapore for International Manhunt Contest. He secured second place there made proved his talent in that contest. He then continued his modeling career and started working with branded companies like Provogue, Wardrobes etc.

John Abraham became India’s best-paid supermodel. As his modeling world made him the real star of his life he moved on to acting field and entered in the Bollywood Acting world with his debut film Jism in 2003. This movie was not based on any action or any thriller scenes but was an erotic emotional. Though this movie didn’t work out for him but the songs and his acting talent were appreciated by a lot from the Bollywood viewers. He then engaged to Actress Bipasha Basu who had acted with him in his debut film. In the same year later came Saaya (2003) and romantic Paap (2003) but unfortunately none of them did well. Then came his first smash hit Dhoom (2004) in which he played a negative role but this negativity really turned his life in Dhoom. Followed by Garam Masala (2005) in which he performed very well in a comic role. His work was appreciated at a higher level and made up many minds. Recently he was featured in the Indian PETA campaign. This young actor is the owner of great mischievous smile and rule the million hearts all over the world.

AWARDS

John was little behind in the category of Awards. Though he received is first award in the second year of his acting career i.e. in 2004 but the award didn’t mean a lot for him as he was nominated for Most Promising New-comer Male in the Weekly Screen Awards. Then in the up-coming acting years he was then nominated for the super-duper hit Yash Raj films ‘DHOOM’ in which he played the role of a villain - Kabir. He received two awards for this film, one was for the nomination of Best Actor in a Villainous Role for the year 2005 and secondly Zee Cine Awards in the category of Best Performance in a Villainous Role for the same year. Both this awards raised his morale and the fans started calling John Abraham as ‘ SMART RIDER’. His work was appreciated a lot in the film by his fans and the industry.

SUSHMITA SEN

2008-09-02T10:18:00.000-07:00

SUSHMITA SEN – The Girl with Gorgeous Looks won the Heart’s in the Miss Universe Contest.

She was born on November 19, 1975 in Hyderbad, Andhra pradesh (India),but grew up in Delhi. Her father Shubeer Sen is a former Indian Air Force officer; Mother Subhra Sen worked as a fashion artist and jewellery designer. She has one brother, Rajiv Sen and a sister, Neelam Sen.She belongs to Bengali family as her mother tongue is Bengali. She attended school at Air Force Golden Jubilee Institute and Air Force Silver in Delhi, also studied English and planned a career in journalism, her later schools were in Nagpur and Jorhat.In 1994 at age of eighteen,Shusmita won the title of Miss India (beating out Ashwaraya rai),she also became the India’s first Miss Universe (’94) in the Miss Universe Pageant held in the Philippines and even one of the youngest woman to achieve this title.She then stepped her career in Bollywood, working as an actress.Her first movie Ratchakan in Tamil,was a huge flop.Being careful about her career, her subsequent movies like Dastak (’96), where she played the victim of the stalker, that even didn’t do well in cinema. But her appearance as Rupali in David Dhawan’s movie Biwi No.1 gave her the Filmfare Best Supporting Actress Award in(’99).In 2001, she adopted a girl named,Ranee.So far, her biggest hits,in which she starred as Shah Rukh Khan’s love interest,movie Main Hoon Na (’04).Later on,she played a lawyer in Main Aisa Hi Hoon oppisite Ajay Devgan.She even starred in a remake of Cactus Flower, called in Maine Pyaar Kyon Kiya 2005):with a wonderful performance,she played the lead oppisite Salman Khan and Katrina Kaif and eassayed the role of Ingrid Bergman. Her most recent movies include Chingaari 2005 as a prostitute, Gulel (’06), Karma, Confessions and Holi (’06) oppisite Drena DeNiro and Naomi Campbell and her latest movie Zindagi Rocks 2006. She is also a well-known popular actress in Bollywood Industry.

Award’s

Sushmita Sen had received four awards in the similar category of Best Actress in a supporting Role for the film ‘Biwi No.1’ in the year 1999. This Award was presented to her first in the Filmfare Awards then in the Star Screen Awards, following by Zee Cine Awards and last came the biggest Award ceremony i.e. IIFA Awards in the year of 2000. She again repeated this incident with her acting talent and efforts for the film ‘Filhaal’ in 2002. She was awarded with three awards in different Awards Ceremony starting from the Filmfare’s moving on to Star Screen Awards then finally at Zee Cine Awards in the year 2003. In 2004, she was being nominated in the category of Best Actress for the film ‘Samay – When time strikes’ (2003) at Stars Screen Awards. Her last achievement till now was for the film ‘Main Hoon Na’ (2004), where she was nominated once again in the category of Best Actress in a Supporting Role in the Zee Cine Awards for the year 2005.

FILMOGRAPHY:

Ram Gopal Varma Ke Sholay
The Expat
Karma, Confessions and Holi
Zindaggi Rocks
Alag
Gulel
Chingari
Maine Pyar Kyun Kiya?
D
Main Aisa Hi Hoon
Bewafaa
Kisna: The Warrior Poet
It Was Raining That Night
Vaastu Shastra
Main Hoon Na
Paisa Vasool
Samay: When Time Strikes
Pran Jaaye Par Shaan Na Jaaye
Aankhen
Leela
Tumko Na Bhool Paayenge
Filhaal...
Kyo Kii... Main Jhuth Nahin Bolta
Nayak: The Real Hero
Bas Itna Sa Khwaab Hai
Aaghaaz
Fiza
Hindustan Ki Kasam
Biwi No. 1
Mudhalvan
Sirf Tum
Zor
Dastak

Salman Khan

2008-09-02T10:17:00.001-07:00

This Bollywood Actor has a habit to walk in trouble which is also a positive feature for his fans or loved ones. He has been named ‘The Bad Guy’ in the Bollywood Career innumerable time’s then too his friends / relatives, fans , the Co-stars who had worked with him finds him a Real Hero of Life with a Heart which is made-up with PURE GOLD.

This Macho – Hunk was born on December 27, 1965. Salman was born in a Muslim Family of the Legendary Writer of Sholay, Deewar and Don yes it is SALIM KHAN.

SALMAN was the eldest among brothers but younger in the family. The family of SALIM KHAN is completed with 5 children’s in which three sons and two daughters as like (starting from the elder than to younger) ARPITA KHAN, SALMAN KHAN, ARBAAZ KHAN, ALVIRA KHAN and then the youngest and the cutest actor in the Bollywood SOHAIL KHAN.

MALIAKA ARORA now (MALIAKA ARORA KHAN) is Salman sister-in-law. ATUL AGNIHOTRI which had started his career as an actor now changed to Directorship is Salman brother-in-law. Salman Khan best friends place is been honored to SANJAY DUTT. Salman also has a strong friendship with Actress Sushmita Sen. He founds Sylvester Stallone his favorite Actor.

On Looking Salman our eyes will be on his body arms, chest and shoulders. That means he is Gymnasia and workouts and exercises every day. He feels his day incomplete with out exercise.
He frequently visits sick children in hospitals and also goes often to donate blood. He also loves to initiate people who aren't as health-conscious as himself into workouts. He forced co-star Anil Kapoor to work out at the gymnasium every day. While Saif Ali Khan and Hrithik Roshan are his early trainees.

He was an excellent swimmer in his school days and was even tipped for representing the country one day. In his early days as a struggling actor he never used his father's name and influence like many other heroes with relatives in the industry. He used to visit producers on his own and audition for roles. Always believes in giving newcomers a chance. So he spares time in between shots to listen to their scrip ideas, and credits this to his memory of when he was a newcomer.

SALMAN KHAN started acting with a supportive role in the film ‘BIWI HO TO AISI’ which was featured in the year 1988. This supporting role improved his acting work and confidence level too which had seen in the up-coming year with his leading role in the film ‘MAINE PYAAR KIYA’ which had been a Romantic hit in the box-office in the featured year 1989. This was the actual time when people started knowing him and he put on his first step towards his fame. His on screen acting career was realized by the directors and other actors after his other box-offices hit in SAAJAN in the year 1991, ANDAAZ APNA APNA which was a super hit Comedy film featuring Aamir Khan ,Salman Khan, Karishma Kapoor, Raveena Tondon and Paresh Rawal in the year 1994 ; Followed by HUM AAPKE HAIN KAUN …….! which another a big hit from Salman Khan and Madhuri Dixit now (Madhuri D. Nene) in the year 1994.

The next big hit was KARAN ARJUN in the year 1995 casting Salman Khan with Shah Rukh Khan, Kajol, (late) Amrish Puri. Salman his movie had developed a new positive feature \full of emotions and love. This movie has scribbled the thoughts of people believing that Salman can
Do only actions movies. KHAMOSHI: The Musical casting Salman Khan with Manisha Koirala and Nana Pathekar in the year 1996. Kuch Kuch Hota Hai (1998). His transformations can be sensitive, vulnerable, funny, aggressive and charming as his role demands.

In 2003 he gave his emotionally charged performance as playing an obsessed lover in "Tere Naam" that translated into good reviews and a good run at the box office. He has not only managed to revive his career, but also to restore the confidence of his producers and distributors alike. His work was noticed internationally in the movie Phir Milenge (2004) where he played the role of an AIDS patient. It was well appreciated by the World Health Organization (WHO) for presenting the problems of AIDS patient in today's

He had a well publicized affair with Aishwarya Rai which lasted 2 years. Apparently his endless tantrums traumatized her enough to give a press conference in spring 2003 where she declared that she would never work with him again for personal reasons. He was accused in another press conference in spring 2003 by Vivek Oberoi to have called the latter 41 times during one night and threatened him with death as well. But above all he is one of the best Person one could ever meet in his or her life.

ABOUT HIM SELF :

Nickname: Sallu
Date of Birth: 27 December 1965
Sign: Capricorn
Height: 5'7" or 1.70 m
City: Indore
Country: Madhya Pradesh, India
Mother: Salma Khan
Stepmother: actress Helen
Father: screen writer Salim Khan
Siblings: two brothers - Arbaaz Khan and Sohail Khan and two sisters Alvira Khan and Arpita Khan
Sisters-in-law: Malaika & Seema
Brother-in-law: Atul Agnihotri
Niece: Alizeh
Nephews: Nirvaan, Ayaan & Arhaan
Children: none
Marital status: single
Spouse: girlfriend Katrina Kaif
Girlfriend: dated Sangeeta Bijlani, Aishwarya Rai
First Film: Biwi Ho To Aisi (1988) and Maine Pyar Kiya (1989 - as the main lead)
Likes: travel to London
Favorite drink: Iced Tea
Favorite clothes: Loves tight fitting 501 jeans
His hero: Sylvester Stallone
Other facts: known as Bollywood's Bad boy, dedicated bodybuilder
Hobbies: exercises daily

FILMOGRAPHY :

The Mahabharata (2007)
Amar Akbar Anthony (2006)
London Dreams (2007)
Partner (2006)
Saawariya (2006)
Salaam E Ishq: A Tribute to Love (2006)
Baabul (2006)
God Tussi Great Ho (2006)
Jaan-E-Mann (2006)
Marigold (2006)
Saawan: The Love Season (2006)
Shaadi Karke Phas Gaya Yaar (2006)
Kyon Ki (2005)
No Entry (2005)
Maine Pyar Kyun Kiya? (2005)
Lucky: No Time for Love (2005)
"Indian Idol" (2004)
Dil Ne Jise Apna Kaha (2004)
Phir Milenge (2004)
Mujhse Shaadi Karogi (2004)
Garv: Pride and Honour (2004)
Baghban (2003)
Tere Naam (2003)
Stumped (2003)
Love at Times Square (2003)
Yeh Hai Jalwa (2002)
Hum Tumhare Hain Sanam (2002)
Tumko Na Bhool Paayenge (2002)
Chori Chori Chupke Chupke (2001)
Kahin Pyaar Na Ho Jaaye (2000)
Dhaai Akshar Prem Ke (2000)
Har Dil Jo Pyar Karega... (2000)
Chal Mere Bhai (2000)
Dulhan Hum Le Jayenge (2000)
Hello Brother (1999)
Hum Dil De Chuke Sanam (1999)
Biwi No. 1 (1999)
Jaanam Samjha Karo (1999)
Hum Saath-Saath Hain: We Stand United (1999)
Sirf Tum (1999)
Kuch Kuch Hota Hai (1998)
Bandhan (1998)
Sar Utha Ke Jiyo (1998)
Jab Pyaar Kisise Hota Hai (1998)
Pyaar Kiya To Darna Kya (1998)
Deewana Mastana (1997)
Auzaar (1997)
Dus (1997)
Judwaa (1997)
Dushman Duniya Ka (1996)
Jeet (1996)
Khamoshi: The Musical (1996)
Majhdhaar (1996)
Veergati (1995)
Karan Arjun (1995)
Sangdil Sanam (1994)
Chaand Kaa Tukdaa (1994)
Andaz Apna Apna (1994)
Hum Aapke Hain Koun...! (1994)
Chandra Mukhi (1993)
Dil Tera Aashiq (1993)
Jaagruti (1993)
Nishchaiy (1992)
Ek Ladka Ek Ladki (1992)
Suryavanshi (1992)
Love (1991)
Saajan (1991)
Kurbaan (1991)
Sanam Bewafa (1991)
Patthar Ke Phool (1991)
Baaghi (1990)
Maine Pyar Kiya (1989)
Biwi Ho To Aisi (1988)

Awards :

1990: Filmfare Best Debut Award win for the role of Prem in Maine Pyar Kiya
1998: Filmfare Awards - Best Supporting Actor win for the role of Aman Mehra in Kuch Kuch Hota Hai

Bollywood Movie Awards :

He won the Best Sensational Actor award for his role in Chori Chori Chupke Chupke at the 2002 Z-Gold- Bollywood Movie Awards.

DIRECTORS HOLLYWOOD

2008-09-01T12:56:00.002-07:00

INGMAR BERGMAN
Martin scorcesse
AKIRA KUROSAWA

PHYSICIST

2008-09-01T12:56:00.001-07:00

BUSSINESS

2008-09-01T12:54:00.002-07:00

INDIAN CINEMA DIRECTORS

2008-09-01T12:54:00.001-07:00

Biography of Satyajit Ray (1921-1992

INDIAN WRITERS

2008-09-01T12:53:00.000-07:00

INDIAN MUSIC

2008-09-01T12:52:00.001-07:00

A.R rahmann

CRICKETERS

2008-09-01T12:48:00.000-07:00

Sachin tendulkar
Mahendra singh dhoni
RAHUL DRAVID
Wasim Akram

HOLLYWOOD ACTORS ,ACTRESS

2008-09-01T12:43:00.000-07:00

ROBERT DENERO

Nicole Kidman

POLITICIANS

2008-09-01T12:41:00.000-07:00

MAHATMA GANDHI
Atal bihari vajpayee
Benazir Bhutto
ADOLF HITLER
Barack Obama

MATHEMATICIANS

2008-09-01T12:38:00.000-07:00

- Niels Henrik Abel
- Anna Galiena

- Niccolo fontana tartaglia

ACTORS,ACTRESS BOLLYWOOD

2008-09-01T12:35:00.000-07:00

H R I T H I K R O S H A N

Rani mukherjee

LIST OF ENGLISH WRITERS

2008-09-01T12:30:00.000-07:00

- - Daniel Defoe
  - Walter de la Mare
- - John Galsworthy
  - Elizabeth Gaskell

- - David Icke
  - William Ralph Inge
- - Christopher Isherwood

RUSSIAN WRITERS

2008-09-01T12:19:00.000-07:00

Nikolai Gogol

Francis Bacon

Franz Kafka

2008-03-10T07:46:00.000-07:00

Franz Kafka (IPA: [ˈfʀanʦ ˈkafka]) (3 July 1883 - 3 June 1924) was one of the major German-language fiction writers of the 20th century. He was born to a middle-class Jewish family in Prague, then part of Austria-Hungary. His unique body of writing - much of which is incomplete and was published posthumously - is among the most influential in Western literature.[1]

His stories, such as The Metamorphosis (1915), and novels, including The Trial (1925) and The Castle (1926), concern troubled individuals in a nightmarishly impersonal and bureaucratic world.

Family

Kafka was born into a middle-class, German-speaking Jewish family in Prague, the capital of Bohemia. His father, Hermann Kafka (1852–1931), was described as a "huge, selfish, overbearing businessman"[2] and by Kafka himself as "a true Kafka in strength, health, appetite, loudness of voice, eloquence, self-satisfaction, worldly dominance, endurance, presence of mind, [and] knowledge of human nature".[3] Hermann was the fourth child of Jacob Kafka, a shochet, and came to Prague from Osek, a Czech-speaking Jewish village near Písek in southern Bohemia. After working as a traveling sales representative, he established himself as an independent retailer of men's and women's fancy goods and accessories, employing up to 15 people and using a jackdaw (kavka in Czech) as his business logo. Kafka's mother, Julie (1856—1934), was the daughter of Jakob Löwy, a prosperous brewer in Poděbrady, and was better educated than her husband.

Kafka was the eldest of six children.[5] He had two younger brothers, Georg and Heinrich, who died at the ages of fifteen months and six months, respectively, before Kafka was seven, and three younger sisters, Gabriele ("Elli") (1889–1941), Valerie ("Valli") (1890–1942), and Ottilie ("Ottla") (1891–1943). On business days, both parents were absent from the home. His mother helped to manage her husband's business and worked in it as much as 12 hours a day. The children were largely reared by a series of governesses and servants.

Kafka's sisters were sent with their families to the Łódź Ghetto and died there or in concentration camps. Ottla was sent to the concentration camp at Theresienstadt and then on October 7, 1943 to the death camp at Auschwitz, where 1267 children and 51 guardians, including Ottla, were gassed to death on their arrival.[6]

Education
Kinsky Palace where Kafka attended gymnasium and where his father later owned a shop
Kinsky Palace where Kafka attended gymnasium and where his father later owned a shop

Kafka learned German as his first language, but he was also fluent in Czech. Later, Kafka acquired some knowledge of French language and culture; one of his favorite authors was Flaubert. From 1889 to 1893, he attended the Deutsche Knabenschule, the boys' elementary school at the Masný trh/Fleischmarkt (meat market), the street now known as Masná street. His Jewish education was limited to his Bar Mitzvah celebration at 13 and going to the synagogue four times a year with his father.[7] After elementary school, he was admitted to the rigorous classics-oriented state gymnasium, Altstädter Deutsches Gymnasium, an academic secondary school with eight grade levels, where German was also the language of instruction, at Old Town Square, within the Kinsky Palace. He completed his Maturita exams in 1901.

Admitted to the German Charles-Ferdinand University of Prague, Kafka first studied chemistry, but switched after two weeks to law. This offered a range of career possibilities, which pleased his father, and required a longer course of study that gave Kafka time to take classes in German studies and art history. At the university, he joined a student club, named Lese- und Redehalle der Deutschen Studenten, which organized literary events, readings and other activities. In the end of his first year of studies, he met Max Brod, who would become a close friend of his throughout his life, together with the journalist Felix Weltsch, who also studied law. Kafka obtained the degree of Doctor of Law on June 18, 1906 and performed an obligatory year of unpaid service as law clerk for the civil and criminal courts.[1]

Work

On November 1, 1907, he was hired at the Assicurazioni Generali, a huge Italian insurance company, where he worked for nearly a year. His correspondence, during that period, witnesses that he was unhappy with his working time schedule - from 8 p.m. (20:00) until 6 a.m. (06:00) - as it made it extremely difficult for him to concentrate on his writing. On July 15, 1908, he resigned, and two weeks later found more congenial employment with the Worker's Accident Insurance Institute for the Kingdom of Bohemia. His father often referred to his son's job as insurance officer as a "Brotberuf", literally "bread job", a job done only to pay the bills. However, he did not show any signs of indifference towards his job, as the several promotions that he received during his career suggest that he was a hardworking employee. A little-known fact about this period, reported by Peter Drucker in Managing in the Next Society, is that Kafka invented the first civilian hard hat. He received a medal for this invention in 1912 because it reduced Bohemian steel mill deaths to fewer than 25 per thousand employees. He was also given the task of compiling and composing the annual report and was reportedly quite proud of the results, sending copies to friends and family. In parallel, Kafka was also committed to his literary work. Together with his close friends Max Brod and Felix Weltsch, these three were called "Der enge Prager Kreis", the close Prague circle, which was part of a broader Prague Circle, "a loosely knit group of German-Jewish writers who contributed to the culturally fertile soil of Prague during the 1880s until after World War I."[8]

In 1911, Karl Hermann, spouse of his sister Elli, proposed Kafka collaborate in the operation of an asbestos factory known as Prager Asbestwerke Hermann and Co. Kafka showed a positive attitude at first, dedicating much of his free time to the business. During that period, he also found interest and entertainment in the performances of Yiddish theatre, despite the misgivings of even close friends such as Max Brod, who usually supported him in everything else. Those performances also served as a starting point for his growing relationship with Judaism.

Later years

In 1912, at Max Brod's home, Kafka met Felice Bauer, who lived in Berlin and worked as a representative for a dictaphone company. Over the next five years they corresponded a great deal, met occasionally, and twice were engaged to be married. Their relationship finally ended in 1917.

In 1917, Kafka began to suffer from tuberculosis, which would require frequent convalescence during which he was supported by his family, most notably his sister Ottla. Despite his fear of being perceived as both physically and mentally repulsive, he impressed others with his boyish, neat, and austere good looks, a quiet and cool demeanor, obvious intelligence and dry sense of humor.[9]

In the early 1920s he developed an intense relationship with Czech journalist and writer Milena Jesenská. In 1923, he briefly moved to Berlin in the hope of distancing himself from his family's influence to concentrate on his writing. In Berlin, he lived with Dora Diamant, a 25-year-old kindergarten teacher from an orthodox Jewish family, who was independent enough to have escaped her past in the ghetto. Dora became his lover, and influenced Kafka's interest in the Talmud.[10]

It is generally agreed that Kafka suffered from clinical depression and social anxiety throughout his entire life[citation needed]. He also suffered from migraines, insomnia, constipation, boils, and other ailments, all usually brought on by excessive stresses and strains. He attempted to counteract all of this by a regimen of naturopathic treatments, such as a vegetarian diet and the consumption of large quantities of unpasteurized milk. However, Kafka's tuberculosis worsened; he returned to Prague, then went to Dr. Hoffmann sanatorium in Kierling near Vienna for treatment, where he died on June 3, 1924, apparently from starvation. The condition of Kafka's throat made eating too painful for him, and since intravenous therapy had not been developed, there was no way to feed him (a fate resembling that of Gregor in the Metamorphosis and the main character of A Hunger Artist). His body was ultimately brought back to Prague where he was interred on June 11, 1924, in the New Jewish Cemetery (sector 21, row 14, plot 33) in Prague-Žižkov.

Personal views

Kafka maintained his indifference to formal religion throughout most of his life. While he had a sense of Jewish identity, this identity was complicated by a sense of alienation from Judaism and Jewish life: "What have I in common with Jews? I have hardly anything in common with myself and should stand very quietly in a corner, content that I can breathe."[11] Kafka wrote that even his father attended synagogue only four times per year; when he accompanied his father, he reported that he "almost suffocated from the terrible boredom and pointlessness of the hours in the synagogue."[11]

During the later years of his life, Kafka suggested an interest in moving to Palestine. At this stage of his life, he was in quite ill health, and Palestine seemed to represent something metaphysical to him: "If I'm never going to leave my bed why shouldn't I go at least as far as Palestine?"[12]

Literary work
Franz Kafka's grave in Prague-Žižkov.
Franz Kafka's grave in Prague-Žižkov.

Kafka published only a few short stories during his lifetime, a small part of his work, and never finished any of his novels (with the possible exception of The Metamorphosis, which some consider to be a short novel). His writing attracted little attention until after his death. Prior to his death, he instructed his friend and literary executor Max Brod to destroy all of his manuscripts. His lover, Dora Diamant, partially executed his wishes, secretly keeping up to 20 notebooks and 35 letters until they were confiscated by the Gestapo in 1933. An ongoing international search is being conducted for these missing Kafka papers. Brod overrode Kafka's instructions and instead oversaw the publication of most of the work in his possession, which soon began to attract attention and high critical regard.

All of Kafka's published works, except several letters he wrote in Czech to Milena Jesenská, were written in German.

Style of writing

Kafka often made extensive use of a trait special to the German language allowing for long sentences that sometimes can span an entire page. Kafka's sentences then deliver an unexpected impact just before the period - that being the finalizing meaning and focus. This is achieved due to the construction of certain sentences in German which require that the verb be positioned at the end of the sentence. Such constructions cannot be duplicated in English, so it is up to the translator to provide the reader with the same effect found in the original text.[13] One such instance of a Kafka translator's quandary is demonstrated in the first sentence of The Metamorphosis.

Another virtually insurmountable problem facing the translator is how to deal with the author's intentional use of ambiguous terms or of words that have several meanings. An example is Kafka's use of the German noun Verkehr in the final sentence of The Judgment. The sentence can be translated as: "At that moment an unending stream of traffic crossed over the bridge."[14] What gives added weight to the obvious double meaning of Verkehr is Kafka's confession to his friend and biographer Max Brod that when he wrote that final line, he was thinking of "a violent ejaculation." In the English translation, of course, what can Verkehr be but "traffic"?[15]

Critical interpretation
Bronze statue of Franz Kafka in Prague.
Bronze statue of Franz Kafka in Prague.

Critics have interpreted Kafka's works in the context of a variety of literary schools, such as modernism, magical realism, and so on.[16] The apparent hopelessness and absurdity that seem to permeate his works are considered emblematic of existentialism. Others have tried to locate a Marxist influence in his satirization of bureaucracy in pieces such as In the Penal Colony, The Trial, and The Castle,[16] whereas others point to anarchism as an inspiration for Kafka's anti-bureaucratic viewpoint. Still others have interpreted his works through the lens of Judaism (Borges made a few perceptive remarks in this regard), through Freudianism[16] (because of his familial struggles), or as allegories of a metaphysical quest for God (Thomas Mann was a proponent of this theory).

Themes of alienation and persecution are repeatedly emphasized, and the emphasis on this quality, notably in the work of Marthe Robert, partly inspired the counter-criticism of Gilles Deleuze and Felix Guattari, who argued that there was much more to Kafka than the stereotype of a lonely figure writing out of anguish, and that his work was more deliberate, subversive, and more "joyful" than it appears to be.

Furthermore, an isolated reading of Kafka's work — focusing on the futility of his characters' struggling without the influence of any studies on Kafka's life was worthless — reveals the humor of Kafka. Kafka's work, in this sense, is not a written reflection of any of his own struggles, but a reflection of how people invent struggles.

Biographers have said that it was common for Kafka to read chapters of the books he was working on to his closest friends, and that those readings usually concentrated on the humorous side of his prose. Milan Kundera refers to the essentially surrealist humour of Kafka as a main predecessor of later artists such as Federico Fellini, Gabriel García Márquez, Carlos Fuentes and Salman Rushdie. For García Márquez, it was as he said the reading of Kafka's The Metamorphosis that showed him "that it was possible to write in a different way".

Publications and dates

Much of Kafka's work was unfinished, or prepared for publication posthumously by Max Brod. The novels The Castle (which stopped mid-sentence and had ambiguity on content), The Trial (chapters were unnumbered and some were incomplete) and Amerika (Kafka's original title was The Man who Disappeared) were all prepared for publication by Brod. It appears Brod took a few liberties with the manuscript (moving chapters, changing the German and cleaning up the punctuation), and thus the original German text was altered prior to publication. The editions by Brod are generally referred to as the Definitive Editions.

According to the publisher's note[17] for The Castle,[18] Malcolm Pasley was able to get most of Kafka's original handwritten work into the Oxford Bodleian Library in 1961. The text for The Trial was later acquired through auction and is stored at the German literary archives[19] at Marbach, Germany.[20]

Subsequently, Pasley headed a team (including Gerhard Neumann, Jost Schillemeit, and Jürgen Born) in reconstructing the German novels and S. Fischer Verlag republished them.[21] Pasley was the editor for Das Schloß (The Castle), published in 1982, and Der Proceß (The Trial), published in 1990. Jost Schillemeit was the editor of Der Verschollene (Amerika) published in 1983. These are all called the 'Critical Editions' or the 'Fischer Editions'. The German critical text of these, and Kafka's other works, may be found online at The Kafka Project.[22]

There is another Kafka Project based at San Diego State University, which began in 1998 as the official international search for Kafka's last writings. Consisting of 20 notebooks and 35 letters to Kafka's last companion, Dora Diamant (later, Dymant-Lask), this missing literary treasure was confiscated from her by the Gestapo in Berlin 1933. The Kafka Project's four-month search of government archives in Berlin in 1998 uncovered the confiscation order and other significant documents. In 2003, the Kafka Project discovered three original Kafka letters, written in 1923. Building on the search conducted by Max Brod and Klaus Wagenbach in the mid-1950s, the Kafka Project at SDSU has an advisory committee of international scholars and researchers, and is calling for volunteers who want to help solve a literary mystery.[23]

[edit] Translations

There are two primary sources for the translations based on the two German editions. The earliest English translations were by Edwin and Willa Muir and published by Alfred A. Knopf. These editions were widely published and spurred the late-1940's surge in Kafka's popularity in the United States. Later editions (notably the 1954 editions) had the addition of the deleted text translated by Eithne Wilkins and Ernst Kaiser. These are known 'Definitive Editions'. They translated both The Trial, Definitive and The Castle, Definitive among other writings. Definitive Editions are generally accepted to have a number of biases and to be dated in interpretation.

After Pasley and Schillemeit completed their recompilation of the German text, the new translations were completed and published -- The Castle, Critical by Mark Harman (Schocken Books, 1998), The Trial, Critical by Breon Mitchell (Schocken Books, 1998) and Amerika: The Man Who Disappeared by Michael Hoffman (New Directions Publishing, 2004). These editions are often noted as being based on the restored text.

Legacy
The entrance to the Franz Kafka museum in Prague.
The entrance to the Franz Kafka museum in Prague.

Franz Kafka has a museum dedicated to his work in Prague, Czech Republic. The term "Kafkaesque" is widely used and misused to describe concepts, situations, and ideas which are reminiscent of Kafka's works, particularly The Trial and "The Metamorphosis".

In Mexico, the phrase "Si Franz Kafka fuera mexicano, sería costumbrista" (If Franz Kafka were Mexican, he would be a Costumbrista writer) is commonly used in newspapers, blogs, and online forums to tell how hopeless and absurd the situation in the country is.[24]

It has been noted that "from the Czech point of view, Kafka was German, and from the German point of view he was, above all, Jewish" and that this was a common "fate of much of Western Jewry."[8]

[edit] Kafka in literature

* Nobel Prize winner Isaac Bashevis Singer wrote a short story called "A Friend of Kafka," which was about a Yiddish actor called Jacques Kohn who said he knew Franz Kafka. In this story, according to Jacques Kohn, Kafka believed in the Golem, a legendary creature from Jewish folklore.[25]
* Kafka Americana by Jonathan Lethem and Carter Scholz is a collection of stories based on Kafka's life and works.
* Kafka on the Shore by Haruki Murakami
* Kafka was the Rage, a Greenwich Village Memoir by Anatole Broyard
* Kafka's Curse by Achmat Dangor

Film
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The fat man on a litter, as depicted in a short film adaptation of Description of a Struggle.
The fat man on a litter, as depicted in a short film adaptation of Description of a Struggle.

For a full list of films The IMDb filmography
Kafka's Life

* Kafka (1990) Jeremy Irons stars as the eponymous author. Written by Lem Dobbs and directed by Steven Soderbergh, the movie mixes his life and fiction providing a semi-biographical presentation of Kafka's life and works. The story concerns Kafka investigating the disappearance of one of his work colleagues. The plot takes Kafka through many of the writer's own works, most notably The Castle and The Trial.
* Franz Kafka (1992) at the Internet Movie Database : an animated film by Piotr Dumała

Novels

* The Trial (1962) Orson Welles wrote and directed this adaptation of the novel starring Anthony Perkins. In a 1962 BBC Interview with Huw Wheldon, Orson Welles noted, "Say what you like, but The Trial is the best film I have ever made".
* Klassenverhältnisse Class Relations (1984) Directed by the experimental filmmaking duo of Jean-Marie Straub and Danièle Huillet based on Kafka's novel Amerika.
* The Trial (1993) Starring Kyle MacLachlan as Joseph K. with Anthony Hopkins in a cameo role as the priest as a strictly faithful adaptation with a screenplay by playwright Harold Pinter.
* Das Schloß (1997) at the Internet Movie Database by Michael Haneke

Metamorphosis

* Die Verwandlung (1975) at the Internet Movie Database
* Förvandlingen (1976/I) at the Internet Movie Database
* The Metamorphosis of Mr. Samsa (1977) at the Internet Movie Database : an animated short by Caroline Leaf
* Metamorphosis (1987) at the Internet Movie Database
* Franz Kafka's 'It's a Wonderful Life' (1993) is an Oscar-winning short film written and directed by Peter Capaldi and starring Richard E. Grant as Kafka. The film blends "Metamorphosis" with Frank Capra's It's a Wonderful Life.
* The Metamorphosis of Franz Kafka (1993) by Carlos Atanes, at YouTube.
* Prevrashcheniye (2002) at the Internet Movie Database
* Metamorfosis (2004) at the Internet Movie Database
* Video "Human Cockroach" (2007) - A comical look at a man who insists he's undergone The Metamorphosis .

Short stories

* Zoetrope : an experimental avant-garde short film by Charlie Deaux, Zoetrope (1999) at the Internet Movie Database. Adaptation of "In the Penal Colony".
* The wish to be a red indian, inspired in Kafka's short story, written and directed by Pablo Di Luozzo(2006)
* The Hunger Artist (2002) at the Internet Movie Database : an animated feature by Tom Gibbons
* Menschenkörper (2004) at the Internet Movie Database[26] Adaptation of "A Country Doctor".

Theatre

* Alan Bennett, Kafka's Dick, 1986, a play in which the ghosts of Kafka, his father Hermann, and Max Brod arrive at the home of an English insurance clerk (and Kafka aficionado) and his wife.
* Milan Richter, Kafka's Hell-Paradise, 2006, a play with 5 characters, using Kafka's aphorisms, dreams and re-telling his relations to his father and to the women. Translated from the Slovak by Ewald Osers.
* Milan Richter, Kafka's Second Life, 2007, a play with 17 characters, starting in Kierling where Kafka is dying and ending in Prague in 1961. Translated from the Slovak by Ewald Osers.
* Tadeusz Różewicz, Pułapka (The Trap), 1982, a play loosely based on Kafka's diaries and letters

Bibliography

See also: Bibliography of Franz Kafka

Short stories

* Description of a Struggle (Beschreibung eines Kampfes, 1904-1905)
* Wedding Preparations in the Country (Hochzeitsvorbereitungen auf dem Lande, 1907-1908)
* Contemplation (Betrachtung, 1904-1912)
* The Judgment (Das Urteil - September 22-23, 1912)
* The Stoker
* In the Penal Colony (In der Strafkolonie, October 1914)
* The Village Schoolmaster (The Giant Mole) (Der Dorfschullehrer or Der Riesenmaulwurf, 1914-1915)
* Blumfeld, an Elderly Bachelor (Blumfeld, ein älterer Junggeselle, 1915)
* The Warden of the Tomb (Der Gruftwächter, 1916-1917), the only play Kafka wrote
* The Hunter Gracchus (Der Jäger Gracchus, 1917)
* The Great Wall of China (Beim Bau der Chinesischen Mauer, 1917)
* A Report to an Academy (Ein Bericht für eine Akademie, 1917)
* Jackals and Arabs (Schakale und Araber, 1917)
* A Country Doctor (Ein Landarzt, 1919)
* A Message from the Emperor (Eine kaiserliche Botschaft, 1919)
* An Old Leaf (Ein altes Blatt, 1919)
* The Refusal (Die Abweisung, 1920)
* A Hunger Artist (Ein Hungerkünstler, 1924)
* Investigations of a Dog (Forschungen eines Hundes, 1922)
* A Little Woman (Eine kleine Frau, 1923)
* First Sorrow
* The Burrow (Der Bau, 1923-1924)
* Josephine the Singer, or The Mouse Folk (Josephine, die Sängerin, oder Das Volk der Mäuse, 1924)

Many collections of the stories have been published, and they include:

* The Penal Colony: Stories and Short Pieces. New York: Schocken Books, 1948.
* The Complete Stories, (ed. Nahum N. Glatzer). New York: Schocken Books, 1971.
* The Basic Kafka. New York: Pocket Books, 1979.
* The Sons. New York: Schocken Books, 1989.
* The Metamorphosis, In the Penal Colony, and Other Stories. New York: Schocken Books, 1995.
* Contemplation. Twisted Spoon Press, 1998.
* Metamorphosis and Other Stories. Penguin Classics, 2007

Novellas

* The Metamorphosis (Die Verwandlung - November-December 1915)
* The Metamorphosis

Novels

* The Trial (Der Prozeß - 1925) (includes short story Before the Law)
* The Castle (Das Schloß - 1926)
* Amerika (Amerika or Der Verschollene - 1927)
Diaries and notebooks

* Diaries 1910-1923 [1]
* The Blue Octavo Notebooks

Letters

* Letter to His Father
* Letters to Felice
* Letters to Ottla
* Letters to Milena
* Letters to Family, Friends, and Editors

Works about Kafka

* Brod, Max. Franz Kafka: A Biography. New York: Da Capo Press, 1995. ISBN 0-306-80670-3
* Brod, Max. The Biography of Franz Kafka, tr. from the German by G. Humphreys Roberts. London: Secker & Warburg, 1947. OCLC 2771397
* Calasso, Roberto. K. Knopf, 2005. ISBN 1-4000-4189-9
* Citati, Pietro, Kafka, 1987. ISBN 0-7859-2173-7
* Coots, Steve. Franz Kafka (Beginner's Guide). Headway, 2002, ISBN 0-340-84648-8
* Deleuze, Gilles & Félix Guattari. Kafka: Toward a Minor Literature (Theory and History of Literature, Vol 30). Minneapolis, University of Minnesota, 1986. ISBN 0-8166-1515-2
* Glatzer, Nahum N., The Loves of Franz Kafka. New York: Schocken Books, 1986. ISBN 0-8052-4001-2
* Greenberg, Martin, The Terror of Art: Kafka and Modern Literature. New York, Basic Books, 1968. ISBN 0-465-08415-X
* Gordimer, Nadine (1984). "Letter from His Father" in Something Out There, London, Penguin Books. ISBN 0-14-007711-1
* Hayman, Ronald. K, a Biography of Kafka. London: Phoenix Press, 2001.ISBN 1-84212-415-3
* Janouch, Gustav. Conversations with Kafka. New York: New Directions Books, second edition 1971. (Translated by Goronwy Rees.)ISBN 0-8112-0071-X
* Kwinter, Sanford. Architectures of Time: Toward a Theory of the Event in Modernist Culture. Cambridge, MIT Press, 2002. ISBN 0-262-11260-4
* Murray, Nicholas. Kafka. New Haven: Yale, 2004.
* Pawel, Ernst. The Nightmare of Reason: A Life of Franz Kafka. New York: Vintage Books, 1985. ISBN 0-374-52335-5
* Thiher, Allen (ed.). Franz Kafka: A Study of the Short Fiction (Twayne's Studies in Short Fiction, No. 12). ISBN 0-8057-8323-7
* Wagenbach, Klaus, Kafka. London: Haus Publishing, 2003. (Paperback: ISBN 978-1-904341-02-4).

[edit] See also

* Asteroid 3412 Kafka, named after the author.
* Max Brod
* Felix Weltsch
* Kafka on the Shore
* Franz Kafka Prize

Notes

1. ^ a b (Spanish)Contijoch, Francesc Miralles (2000) "Franz Kafka". Oceano Grupo Editorial, S.A. Barcelona. ISBN 84-494-1811-9.
2. ^ Corngold 1973
3. ^ Franz Kafka's Letter to his father www.kafka-franz.com
4. ^ Gilman, Sander L. (2005) Franz Kafka. Reaktion Books Ltd. London, UK. p.20-21. ISBN 1-88187-264-5.
5. ^ Hamalian ([1975], 3).
6. ^ Danuta Czech: Kalendarz wydarzeń w KL Auschwitz, Oświęcim 1992, p. 534. In the archives of the camp a list with the names of the guardians was preserved.
7. ^ Franz Kafka Biography www.kafka-franz.com
8. ^ a b The Metamorphosis and Other Stories, notes. Herberth Czermak. Lincoln, Nebraska: Cliffs Notes 1973, 1996.
9. ^ Ryan McKittrick speaks with director Dominique Serrand and Gideon Lester about Amerika www.amrep.org
10. ^ Lothar Hempel www.atlegerhardsen.com
11. ^ a b Elon, Amos. The Pity of it All: A History of the Jews in Germany, 1743-1933. 2002, page 229.
12. ^ Zilcosky, John. Kafka's Travels: Exoticism, Colonialism, and the Traffic of Writing. 2003, page 188.
13. ^ Kafka (1996, xi).
14. ^ Kafka (1996, 75).
15. ^ Kafka (1996, xii).
16. ^ a b c Franz Kafka 1883 – 1924 www.coskunfineart.com
17. ^ A Kafka For The 21st Century by Arthur Samuelson, publisher, Schocken Books www.jhom.com
18. ^ Schocken Books, 1998
19. ^ (German) Herzlich Willkommen www.dla-marbach.de
20. ^ (publisher's note, The Trial, Schocken Books, 1998
21. ^ Stepping into Kafka’s head, Jeremy Adler, Times Literary Supplement, October 13, 1995 (http://www.textkritik.de/rezensionen/kafka/einl_04.htm)
22. ^ The Kafka Project - Kafka's Works in German According to the Manuscript www.kafka.org
23. ^ Sources: Kafka, by Nicolas Murray, pages 367, 374; Kafka's Last Love, by Kathi Diamant; "Summary of the Results of the Kafka Project Berlin Research June 1-September 1998" published in December 1998 Kafka Katern, quarterly of the Kafka Circle of the Netherlands. More information is available at http://www.kafkaproject.com
24. ^ Aquella, Daniel (2006-11-22). México kafkiano y costumbrista. Daquella manera:Paseo personal por inquietudes culturales, sociales y lo que tengamos a bien obrar.. Retrieved on 2007-2-16.
25. ^ Bashevis Singer, Isaac (1970). A Friend of Kafka, and Other Stories. Farrar, Straus and Giroux, 311. ISBN 0-37415-880-0.
26. ^ (German) Menschenkörper movie website www.menschenkoerper.de

References

* Adorno, Theodor. "Prisms." The MIT Press: Cambridge, MA. 1967.

* Corngold, Stanley (1972). Introduction to The Metamorphosis, reissue edition. Bantam Classics. ISBN 0-553-21369-5.

* Hamalian, Leo (Ed.). [1974]. Franz Kafka: A Collection of Criticism. New York: McGraw-Hill. ISBN 0-07-025702-7.

* Kafka, Franz (1996). The Metamorphosis and Other Stories, trans. Donna Freed. New York: Barnes & Noble. ISBN 1-56619-969-7.

* Paul Heller: Franz Kafka. Wissenschaft und Wissenschaftskritik. Tuebingen: Stauffenburg 1989. ISBN 3-923-72140-4.

Nikolai Gogol

2008-02-23T07:30:00.000-08:00

Nikolai Vasilievich Gogol' (Russian: Никола́й Васи́льевич Го́голь; pronounced [nʲɪkɐˈlaj vɐˈsʲilʲjɪvʲɪtɕ ˈgogəlʲ]; Ukrainian: Микола Васильович Гоголь, Mykola Vasylovych Hohol) (April 1, 1809 – March 4, 1852) was a Russian writer of Ukrainian ethnicity and birth . Often called the “father of modern Russian realism,” Nikolai Gogol was one of the first Russian authors to criticize his country's way of life. Although his early works were heavily influenced by his Ukrainian upbringing, he wrote in Russian and his works belong to the tradition of Russian literature. The novel Dead Souls (1842), the play The Government Inspector (1836, 1842), and the short story The Overcoat (1842) are among his masterpieces.

Provenance and early life
Gogol was born in the Cossack village of Sorochyntsi, Poltava guberniya (now Ukraine). His father was Vasily Gogol-Yanovsky, a small squire and an amateur Ukrainian playwright who died when the boy was 15 years old. Some of his ancestors culturally associated themselves with Polish szlachta.

In 1820 Gogol went to a school of higher art in Nezhin (currently Nizhyn) and remained there until 1828. It was there that he began writing. He was not very popular among his school-fellows who called him their "mysterious dwarf", but with two or three of them he formed lasting friendships. Very early he developed a dark and secretive disposition, mingled of painful self-consciousness and boundless ambition. Equally early he developed an extraordinary mimic talent which later on made him a matchless reader of his own works and induced him to toy with the idea of becoming an actor.

In 1828, on leaving school, Gogol came to Petersburg, full of vague but glowingly ambitious hopes. He had hoped for literary fame and brought with him a Romantic poem of German idyllic life — Hanz Küchelgarten. He had it published, at his own expense, under the name of "V. Alov". Unfortunately it was met by the magazines with deserved derision. He bought all the copies and destroyed them, swearing never to write poetry again.

Gogol was one of the first masters of short prose, alongside Pushkin, Mérimée, Hoffmann, and Hawthorne. He was in touch with the "literary aristocracy", had a story published in Delvig's Northern Flowers, was taken up by Vasily Zhukovsky and Pyotr Pletnyov, and (in 1831) was introduced to Pushkin.

Literary development

Cover of the first edition of The Government Inspector (1836).In 1831, he brought out the first volume of his Ukrainian stories (Evenings on a Farm Near Dikanka), which met immediate success. It was followed in 1832 by a second volume, and in 1835 by two volumes of stories entitled Mirgorod, as well as by two volumes of miscellaneous prose entitled Arabesques. At this time, Gogol developed a passion for Ukrainian history and tried to obtain an appointment to the history department at Kiev University. Despite the support of Alexander Pushkin and the Russian minister of education Sergey Uvarov, his appointment was blocked by a Kievan bureaucrat on the grounds that he was unqualified. His fictional story Taras Bulba, based on the history of Ukrainian cossacks, was the result of this phase in his interests. During this time he also developed a close and life-long friendship with another Ukrainian then living in Russia, the historian and naturalist Mykhaylo Maksymovych. Indeed, throughout his life Gogol maintained close contact with his fellow countrymen. According to the poet Nikolai Berg, in his interactions with fellow Ukrainians Gogol demonstrated a joyfullness and passion that contrasted with usual morose and quiet demeanor.

In 1834 Gogol was made Professor of Medieval History at the University of St. Petersburg, a job for which "he had no qualifications. He turned in a performance ludicrous enough to warrant satiric treatment in one of his own stories. After an introductory lecture made up of brilliant generalizations which the 'historian' had prudently prepared and memorized, he gave up all pretense at erudition and teaching, missed two lectures out of three, and when he did appear, muttered unintelligibly through his teeth and displayed small photographs of classic monuments. At the final examination, he sat in utter silence with a black handkerchief wrapped around his head, simulating a toothache, while another professor interrogated the students." This academic venture proved a failure and he resigned his chair in 1835.

Between 1832 and 1836 Gogol worked at his imaginative creations with great energy, and though almost all his work has in one way or another its sources in these four years of contact with Pushkin, he had not yet decided that his ambitions were to be fulfilled by success in literature. It was only after the presentation, on April 19, 1836, of his comedy The Government Inspector (Revizor) that he finally came to believe in his literary vocation. The comedy, a violent satire of Russian provincial bureaucracy, was able to be staged thanks only to the personal intervention of Nicholas I.

Commemorative plaque in his house in RomeFrom 1836 to 1848 he lived abroad, travelling throughout Germany and Switzerland. Gogol spent the winter of 1836-1837 in Paris, where he spent time among Russian expatriates and Polish exiles, frequently meeting with the Polish poets Adam Mickiewicz and Bohdan Zaleski. After having chosen Rome for his headquarters, he became enamoured with the Eternal City, which answered to his highly developed sense of the magnificent, and where even the visions that always obsessed him of vulgar and animal humanity assumed picturesque and poetical appearances that fitted harmoniously into the beautiful whole.

The death of Pushkin produced a strong impression on Gogol. His principal work during years following poet's death was the great satirical epic (poema, or an epic poem, as the Russian subheading goes) — Dead Souls. Concurrently, he worked at other tasks — recast Taras Bulba and The Portrait, completed his second comedy, Marriage (Zhenitba), wrote the fragment Rome and his greatest short story, The Overcoat.

In 1841 the first part of Dead Souls was ready, and Gogol took it to Russia to supervise its printing. It appeared in Moscow in 1842, under the title, imposed by the censorship, of The Adventures of Chichikov. The book instantly established his reputation as the greatest prose writer in the language. Nobody could have expected that it would be the last work of fiction published during his lifetime.

Creative decline and death

Gogol in the 1840s.After the triumph of Dead Souls, Gogol came to be regarded by his contemporaries as a great satirist who lampooned the unseemly sides of Imperial Russia. Little did they know that the 33-year-old author viewed himself primarily as a prophet and preacher, for whom Dead Souls was but the first part of a modern-day counterpart to The Divine Comedy. The first part represented the Inferno; the second part was to depict the gradual purification and transformation of the rogue Chichikov under the influence of virtuous publicans and governors — Purgatory.

A church on New Arbat in Moscow, in which the great writer was mourned before his burial.From Palestine he returned to Russia and passed his last years in restless movement throughout the country. While visiting the capitals, he stayed with various friends such as Mikhail Pogodin and Sergei Aksakov. During this period of his life he also spent much time with his old Ukrainian friends, Maksymovych and Osyp Bodiansky. More importantly, he intensified his relationship with a church elder, Matvey Konstantinovsky, whom he had known for several years. Konstantinovsky seems to have strengthened in Gogol the fear of perdition by insisting on the sinfulness of all his imaginative work. His health was undermined by exaggerated ascetic practices and he fell into a state of black melancholy. On the night of February 24, 1852, he burnt some of his manuscripts, which contained most of the second part of Dead Souls. He explained this as a mistake — a practical joke played on him by the Devil. Soon thereafter he took to bed, refused all food, and died in great pain nine days later.

Gogol was buried at the Danilov Monastery, close to his fellow Slavophile Aleksey Khomyakov. In 1931, when Moscow authorities decided to demolish the monastery, his remains were transferred to the Novodevichy Cemetery.

Gogol's grave at the Novodevichy ConventHis body was discovered lying face down, which gave rise to the story that Gogol had been buried alive. A Soviet critic even cut a part of his jacket to use as a binding for his copy of Dead Souls. A piece of rock which used to stand on his grave at the Danilov was reused for the tomb of Gogol's admirer Mikhail Bulgakov.

The first Gogol monument in Moscow was a striking Symbolist statue on Arbat Square, which represented the sculptor Nikolai Andreyev's idea of Gogol, rather than the real man (picture). Unveiled in 1909, the statue was praised by Ilya Repin and Leo Tolstoy as an outstanding projection of Gogol's tortured personality. Stalin did not like it, however; and the statue was replaced by a more orthodox Socialist Realism monument in 1952. It took enormous efforts to save Andreyev's original work from destruction; it now stands in front of the house where Gogol died.

Style

Among the illustrators of Dead Souls were Pyotr Sokolov and Marc Chagall.D.S. Mirsky characterized Gogol's universe as "one of the most marvellous, unexpected — in the strictest sense, original — worlds ever created by an artist of words". The enormous potency of his imagination stands at a strange contrast (or complement) to his physical sterility. He seems to have never had a sexual contact with a woman (or a man). Woman was to him a terrible, fascinating, but unapproachable obsession, and he is known never to have loved. This makes the women of his imagination either strange, inhuman visions of form and color that are redeemed from melodramatic banality only by the force of the rhetoric they are enshrined in, or entirely unsexed, even dehumanized, caricatures.

The main and most persistent characteristic of Gogol's style is its verbal expressiveness. He wrote with a view not so much to the acoustic effect on the ears of the listener as to the sensuous effect on the vocal apparatus of the reciter. This makes his prose ornate and agitated. It is all alive with the vibration of actual speech. This makes it hopelessly untranslatable — more untranslatable than any other Russian prose of the 19th century.

The other main characteristic of Gogol's genius is the extraordinary intensity and vividness of impressionist vision, sometimes skirting expressionism. He saw the outer world romantically metamorphosed, a singular gift particularly evident from the fantastic spatial transformations in his Gothic stories, A Terrible Vengeance and A Bewitched Place. His pictures of nature are strange mounds of detail heaped on detail, resulting in an unconnected chaos of things. His people are caricatures, drawn with the method of the caricaturist — which is to exaggerate salient features and to reduce them to geometrical pattern. But these cartoons have a convincingness, a truthfulness, and inevitability — attained as a rule by slight but definitive strokes of unexpected reality — that seems to beggar the visible world itself.

The aspect under which the mature Gogol sees reality is expressed by the untranslatable Russian word poshlost', which is perhaps best rendered as "self-satisfied inferiority", moral and spiritual. Like Sterne before him, Gogol was a great destroyer of prohibitions and romantic illusions. It was he who undermined Russian Romanticism by making vulgarity reign where only the sublime and the beautiful had reigned. "Characteristic of Gogol is a sense of boundless superfluity that is soon revealed as utter emptiness and a rich comedy that suddenly turns into metaphysical horror".His stories often interweave pathos and mockery, while the most comic of them all begins as a merry farce and ends with the famous dictum: It is dull in this world, gentlemen!

Influence and interpretations

A statue of Gogol at Villa Borghese, Rome.Even before the publication of Dead Souls, Belinsky recognized Gogol as the first realist writer in the language and the head of the Natural School, to which he also assigned such minor or young authors as Aksakov, Turgenev, Dmitry Grigorovich, Vladimir Dahl, and Vladimir Sollogub. Gogol himself seemed to be skeptical about the existence of such literary movement. Although he recognized "several young writers" who "have shown a particular desire to observe real life", he upbraided the deficient composition and style of their works. Nevertheless, subsequent generations of radical critics celebrated Gogol (the author in whose world a nose roams the streets of the Russian capital) as a great realist, a reputation decried by the Encyclopaedia Britannica as "the triumph of Gogolesque irony".

The period of modernism saw a revival of interest in and a change of attitude towards Gogol's work. One of the pioneering works of Russian formalism was Eichenbaum's reappraisal of The Overcoat. In the 1920s, a group of Russian short story writers, known as the Serapion Brothers, placed Gogol among their precursors and consciously sought to imitate his techniques. The leading novelists of the period — notably Yevgeny Zamyatin and Mikhail Bulgakov — also admired Gogol and followed in his footsteps. In 1926, Vsevolod Meyerhold staged The Government Inspector as a "comedy of the absurd situation", revealing to his fascinated spectators a corrupt world of endless self-deception. In 1934, Andrei Bely published the most meticulous study of Gogol's literary techniques up to that date, in which he analyzed the colours prevalent in Gogol's work depending on the period, his impressionistic use of verbs, expressive discontinuity of his syntax, complicated rhythmical patterns of his sentences, and many other secrets of his craft. Based on this work, Vladimir Nabokov published a summary account of Gogol's masterpieces in 1944.

Gogol had a huge and enduring impact on Russian literature, but his works were appreciated differently depending on the background of the reader. Belinsky, for instance, berated his horror stories as "moribund, monstrous works", while Andrei Bely counted them among his most stylistically daring creations. Nabokov singled out Dead Souls, The Government Inspector, and The Overcoat as the works of genius and dismissed the remainder as puerile essays. The latter story has been traditionally interpreted as a masterpiece of "humanitarian realism", but Nabokov and some other attentive readers argued that "holes in the language" make the story susceptible to another interpretation, as a supernatural tale about a ghostly double of a "small man". Of all Gogol's stories, The Nose has stubbornly defied all abstruse interpretations: D.S. Mirsky declared it "a piece of sheer play, almost sheer nonsense".

Gogol's oeuvre has also had a large impact on Russia's non-literary culture, and his stories have been adapted numerous times into opera and film. Russian Composer Alfred Schnittke wrote the eight part Gogol Suite as incidental music to the The Government Inspector performed as a play, and composer Dmitri Shostakovich set The Nose as his first opera in 1930, despite the peculiar choice of subject for what was meant to initiate the great tradition of Soviet opera.

In Marathi, P. L. Deshpande adapted his play "The Government Inspector" as "Ammaldar" (literally 'the Government Inspector') in late 1950s, skillfully cladding it with all indigenous politico-cultural robe of Maharashtra, while maintaining the comic satire of the original.

Gogol in Pop Culture
Gypsy punk band Gogol Bordello is named after Gogol. Lead singer Eugene Hütz is Ukrainian; he sees himself as another Ukrainian infiltrating western artistic culture.
In the book The Namesake (also adapted into a film of the same name), the protagonist's name is Gogol. Gogol's 'namesake' is explained in the book.
In the episode 'Charlie' from the 1st series of the comedy series The Mighty Boosh, Howard Moon uses Gogol's Dead Souls to spy on his fellow zookeeper, Mrs Gideon, through eyeholes inside the two 'O's in the author's name, on the book's front cover.
His satirical work The Nose inspired the song Detachable Penis by King Missile.
In the anime movie Ghost in the Shell 2: Innocence, Batou quotes Gogol early on saying "It is no use to blame the looking glass if your face is awry."

Xu Yue

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Born: about 160 in Donglai, Shandong province, China
Died: about 227 in China
We know a little of Xu Yue but the main text which bears his name, the Shushu jiyi (Notes on Traditions of Arithmetic Methods), is probably the work of a later author trying to claim a certain respectability for his writings.

Xu Yue was a pupil of Liu Hong and he studied mathematics under the famous calendar expert. Liu Hong worked at the Imperial Observatory and it was there that Xu Yue held discussions with him and also with the head of the Astronomical Bureau. Mathematics was used by Liu Hong and others at the Observatory in their studies of astronomy and the related work on the calendar which, of course, was based on the apparent motion of the sun and the moon. It was natural, therefore that Xu Yue would gain expertise from these men in astronomy and calendar science.

It is reported that Xu Yue wrote a commentary on the Nine Chapters on the Mathematical Art. This preceded the major commentary written by Liu Hui in the second half of the third century, and it would appear that Liu Hui commented on a version of the text which did not include Xu Yue's comments. As a result Xu Yue's commentary has not come down to us.

Whether Xu Yue wrote the Shushu jiyi (Notes on Traditions of Arithmetic Methods) is uncertain. It is a rather strange work filled with ideas from Buddhism and from Taoism and describes arithmetical systems intermingled with religious nuances. The version which has come down to us today has a commentary by Zhen Luan written around 566, but that does little to help understand the difficulties in the text.

Fourteen old methods of calculation are mentioned in the text. One of these uses a device resembling the abacus called ball-arithmetic. Three others also uses balls, one involving balls in columns, one involving two balls of different colours which move at right angles to each other suggesting almost the idea of Cartesian coordinates. Needham writes in [4] that Xu Yue:-

... shows an interesting appreciation of coordinate relationships.

Unfortunately there is no great clarity in the descriptions.

There is no doubt that one of the main aims of the text is to introduce a notation which will allow the representation of large numbers. Most historians believe that the aim of the author was to suggest that it was possible to represent any number, no matter how large. Three systems of powers of 10 are given. The lower system is based on the sequence of powers of 10

10, 102, 103, 104, 105, ...

the middle system on powers of 104

104, 108, 1012, 1016, ...

the upper system being based on powers of 10, each being the square of its predecessor

104, 108, 1016, 1032, ...

Finally, Xu Yue talks about calculations of the nine balls. Again what is intended here is unclear, but the commentator Zhen Luan is in no doubt that this refers to a 3 by 3 magic square.

Despite being an obscure text, after editing by Li Chunfeng, the Shushu jiyi (Notes on Traditions of Arithmetic Methods) was selected as a text for the Imperial examinations in 656 and became one of The Ten Classics in 1084.

Liu Hong

2008-02-15T00:59:00.002-08:00

Born: 129 in Lo-yang, China
Died: 210 in China
Liu Hong was of noble birth, descended from the Imperial family of the Eastern Han Dynasty. This dynasty was established in 25 AD after the brief 15 year reign of Wang Mang's Hsin dynasty. The capital was Lo-yang where a large ornate palace was built. In Encyclopaedia Britannica the aims and achievements of the Han rulers are described:-

... the Han came to require cultural accomplishment from their public servants, making mastery of classical texts a condition of employment. The title list of the enormous imperial library is China's first bibliography. Its text included works on practical matters such as mathematics and medicine, as well as treatises on philosophy and religion and the arts. Advancement in science and technology was also sought by the rulers, and the Han invented paper, used water clocks and sundials, and developed a seismograph. Calendars were published frequently during the period.

Liu Hong became interested in astronomy as a young boy. He was appointed to the Imperial Observatory in 160 and was involved in making astronomical observations. Two works which he wrote, namely the Qi Yao Shu (The Art of Seven Planets) and a new version entitled the Ba Yuan Shu (The Art of Eight Elements) have been lost so we know little of their contents. We do know that Liu presented these works to the Emperor in 174 or 175, and we also know that they were based on Buddhist beliefs.

Perhaps the greatest of Liu's achievements was his work which led to a new calendar. This calendar was published in 187 and described the motion of the moon far more accurately than any previous Chinese calendar. His measurements of the length of the shadow of a pole at the summer and at the winter solstices give results which are accurate to within 1% of their true value.

Yavanesvara

2008-02-15T00:59:00.001-08:00

Born: about 120 in Western India
Died: about 180 in India
Indian astrology was originally known as Jyotisha, which means "science of the stars". Until around the first century AD no real distinction was made between astrology and astronomy and in fact most astronomical theories were propounded to support the theory that the positions of the heavenly bodies directly influenced human events.

The Indian methods of computing horoscopes all date back to the translation of a Greek astrology text into Sanskrit prose by Yavanesvara in 149 AD. Yavanesvara (or Yavanaraja) literally means "Lord of the Greeks" and it was a name given to many officials in western India during the period 130 AD - 390 AD. During this period the Ksatrapas ruled Gujarat (or Madhya Pradesh) and these "Lord of the Greeks" officials acted for the Greek merchants living in the area.

The particular "Lord of the Greeks" official Yavanesvara who we are interested in here worked under Rudradaman. Rudradaman became ruler of the Ksatrapas in around 130 AD and it was during the period of his rule that Yavanesvara worked as an official and made his translation. We know of Rudradaman because information is recorded in a lengthy Sanskrit inscription at Junagadh written around 150 AD.

The Greek astrology text in question was written in Alexandria some time round about 120 BC. Yavanesvara did far more than just translate the Greek text for such a translation would have had little relevance to the Indians. He therefore not only translated the language but he translated the context too. Instead of the Greek gods who appear in the original, Yavanesvara used Hindu images. Again he worked the Indian caste system into the work and made the work one which would fit well with the Indian thought.

The work was written with the aim of letting Indians became astrologers so it had to present astronomy in a form in which it could be used for astrology. In order to do this Yavanesvara put into his work an explanation of the Greek version of the Babylonian theory of the motions of the planets. All this he wrote in Sanskrit prose but sadly the original has not survived. We do have, however, a version written in Sanskrit verse 120 years after Yavanesvara's work appeared.

Yavanesvara had an important influence on the whole of astrology in India for centuries after he made his popular translation. Although the influence was more than on astrology, as the science of astronomy split from astrology, the influence of Yavanesvara's work reached into astronomy too.

Claudius Ptolemy

2008-02-15T00:57:00.000-08:00

Born: about 85 in Egypt
Died: about 165 in Alexandria, Egypt
One of the most influential Greek astronomers and geographers of his time, Ptolemy propounded the geocentric theory in a form that prevailed for 1400 years. However, of all the ancient Greek mathematicians, it is fair to say that his work has generated more discussion and argument than any other. We shall discuss the arguments below for, depending on which are correct, they portray Ptolemy in very different lights. The arguments of some historians show that Ptolemy was a mathematician of the very top rank, arguments of others show that he was no more than a superb expositor, but far worse, some even claim that he committed a crime against his fellow scientists by betraying the ethics and integrity of his profession.

We know very little of Ptolemy's life. He made astronomical observations from Alexandria in Egypt during the years AD 127-41. In fact the first observation which we can date exactly was made by Ptolemy on 26 March 127 while the last was made on 2 February 141. It was claimed by Theodore Meliteniotes in around 1360 that Ptolemy was born in Hermiou (which is in Upper Egypt rather than Lower Egypt where Alexandria is situated) but since this claim first appears more than one thousand years after Ptolemy lived, it must be treated as relatively unlikely to be true. In fact there is no evidence that Ptolemy was ever anywhere other than Alexandria.

His name, Claudius Ptolemy, is of course a mixture of the Greek Egyptian 'Ptolemy' and the Roman 'Claudius'. This would indicate that he was descended from a Greek family living in Egypt and that he was a citizen of Rome, which would be as a result of a Roman emperor giving that 'reward' to one of Ptolemy's ancestors.

We do know that Ptolemy used observations made by 'Theon the mathematician', and this was almost certainly Theon of Smyrna who almost certainly was his teacher. Certainly this would make sense since Theon was both an observer and a mathematician who had written on astronomical topics such as conjunctions, eclipses, occultations and transits. Most of Ptolemy's early works are dedicated to Syrus who may have also been one of his teachers in Alexandria, but nothing is known of Syrus.

If these facts about Ptolemy's teachers are correct then certainly in Theon he did not have a great scholar, for Theon seems not to have understood in any depth the astronomical work he describes. On the other hand Alexandria had a tradition for scholarship which would mean that even if Ptolemy did not have access to the best teachers, he would have access to the libraries where he would have found the valuable reference material of which he made good use.

Ptolemy's major works have survived and we shall discuss them in this article. The most important, however, is the Almagest which is a treatise in thirteen books. We should say straight away that, although the work is now almost always known as the Almagest that was not its original name. Its original Greek title translates as The Mathematical Compilation but this title was soon replaced by another Greek title which means The Greatest Compilation. This was translated into Arabic as "al-majisti" and from this the title Almagest was given to the work when it was translated from Arabic to Latin.

The Almagest is the earliest of Ptolemy's works and gives in detail the mathematical theory of the motions of the Sun, Moon, and planets. Ptolemy made his most original contribution by presenting details for the motions of each of the planets. The Almagest was not superseded until a century after Copernicus presented his heliocentric theory in the De revolutionibus of 1543. Grasshoff writes in [8]:-

Ptolemy's "Almagest" shares with Euclid's "Elements" the glory of being the scientific text longest in use. From its conception in the second century up to the late Renaissance, this work determined astronomy as a science. During this time the "Almagest" was not only a work on astronomy; the subject was defined as what is described in the "Almagest".

Ptolemy describes himself very clearly what he is attempting to do in writing the work (see for example [15]):-

We shall try to note down everything which we think we have discovered up to the present time; we shall do this as concisely as possible and in a manner which can be followed by those who have already made some progress in the field. For the sake of completeness in our treatment we shall set out everything useful for the theory of the heavens in the proper order, but to avoid undue length we shall merely recount what has been adequately established by the ancients. However, those topics which have not been dealt with by our predecessors at all, or not as usefully as they might have been, will be discussed at length to the best of our ability.

Ptolemy first of all justifies his description of the universe based on the earth-centred system described by Aristotle. It is a view of the world based on a fixed earth around which the sphere of the fixed stars rotates every day, this carrying with it the spheres of the sun, moon, and planets. Ptolemy used geometric models to predict the positions of the sun, moon, and planets, using combinations of circular motion known as epicycles. Having set up this model, Ptolemy then goes on to describe the mathematics which he needs in the rest of the work. In particular he introduces trigonometrical methods based on the chord function Crd (which is related to the sine function by sin a = (Crd 2a)/120).

Ptolemy devised new geometrical proofs and theorems. He obtained, using chords of a circle and an inscribed 360-gon, the approximation

π = 3 17/120 = 3.14166

and, using √3 = chord 60,

√3 = 1.73205.

He used formulae for the Crd function which are analogous to our formulae for sin(a + b), sin(a - b) and sin a/2 to create a table of the Crd function at intervals of 1/2 a degree.

This occupies the first two of the 13 books of the Almagest and then, quoting again from the introduction, we give Ptolemy's own description of how he intended to develop the rest of the mathematical astronomy in the work (see for example [15]):-

[After introducing the mathematical concepts] we have to go through the motions of the sun and of the moon, and the phenomena accompanying these motions; for it would be impossible to examine the theory of the stars thoroughly without first having a grasp of these matters. Our final task in this way of approach is the theory of the stars. Here too it would be appropriate to deal first with the sphere of the so-called 'fixed stars', and follow that by treating the five 'planets', as they are called.

In examining the theory of the sun, Ptolemy compares his own observations of equinoxes with those of Hipparchus and the earlier observations Meton in 432 BC. He confirmed the length of the tropical year as 1/300 of a day less than 365 1/4 days, the precise value obtained by Hipparchus. Since, as Ptolemy himself knew, the accuracy of the rest of his data depended heavily on this value, the fact that the true value is 1/128 of a day less than 365 1/4days did produce errors in the rest of the work. We shall discuss below in more detail the accusations which have been made against Ptolemy, but this illustrates clearly the grounds for these accusations since Ptolemy had to have an error of 28 hours in his observation of the equinox to produce this error, and even given the accuracy that could be expected with ancient instruments and methods, it is essentially unbelievable that he could have made an error of this magnitude. A good discussion of this strange error is contained in the excellent article [19].

Based on his observations of solstices and equinoxes, Ptolemy found the lengths of the seasons and, based on these, he proposed a simple model for the sun which was a circular motion of uniform angular velocity, but the earth was not at the centre of the circle but at a distance called the eccentricity from this centre. This theory of the sun forms the subject of Book 3 of the Almagest.

In Books 4 and 5 Ptolemy gives his theory of the moon. Here he follows Hipparchus who had studied three different periods which one could associate with the motion of the moon. There is the time taken for the moon to return to the same longitude, the time taken for it to return to the same velocity (the anomaly) and the time taken for it to return to the same latitude. Ptolemy also discusses, as Hipparchus had done, the synodic month, that is the time between successive oppositions of the sun and moon. In Book 4 Ptolemy gives Hipparchus's epicycle model for the motion of the moon but he notes, as in fact Hipparchus had done himself, that there are small discrepancies between the model and the observed parameters. Although noting the discrepancies, Hipparchus seems not to have worked out a better model, but Ptolemy does this in Book 5 where the model he gives improves markedly on the one proposed by Hipparchus. An interesting discussion of Ptolemy's theory of the moon is given in [24].

Having given a theory for the motion of the sun and of the moon, Ptolemy was in a position to apply these to obtain a theory of eclipses which he does in Book 6. The next two books deal with the fixed stars and in Book 7 Ptolemy uses his own observations together with those of Hipparchus to justify his belief that the fixed stars always maintain the same positions relative to each other. He wrote (see for example [15]):-

If one were to match the above alignments against the diagrams forming the constellations on Hipparchus's celestial globe, he would find that the positions of the relevant stars on the globe resulting from the observations made at the time of Hipparchus, according to what he recorded, are very nearly the same as at present.

In these two book Ptolemy also discusses precession, the discovery of which he attributes to Hipparchus, but his figure is somewhat in error mainly because of the error in the length of the tropical year which he used. Much of Books 7 and 8 are taken up with Ptolemy's star catalogue containing over one thousand stars.

The final five books of the Almagest discuss planetary theory. This must be Ptolemy's greatest achievement in terms of an original contribution, since there does not appear to have been any satisfactory theoretical model to explain the rather complicated motions of the five planets before the Almagest. Ptolemy combined the epicycle and eccentric methods to give his model for the motions of the planets. The path of a planet P therefore consisted of circular motion on an epicycle, the centre C of the epicycle moving round a circle whose centre was offset from the earth. Ptolemy's really clever innovation here was to make the motion of C uniform not about the centre of the circle around which it moves, but around a point called the equant which is symmetrically placed on the opposite side of the centre from the earth.

The planetary theory which Ptolemy developed here is a masterpiece. He created a sophisticated mathematical model to fit observational data which before Ptolemy's time was scarce, and the model he produced, although complicated, represents the motions of the planets fairly well.

Toomer sums up the Almagest in [1] as follows:-

As a didactic work the "Almagest" is a masterpiece of clarity and method, superior to any ancient scientific textbook and with few peers from any period. But it is much more than that. Far from being a mere 'systemisation' of earlier Greek astronomy, as it is sometimes described, it is in many respects an original work.

We will return to discuss some of the accusations made against Ptolemy after commenting briefly on his other works. He published the tables which are scattered throughout the Almagest separately under the title Handy Tables. These were not merely lifted from the Almagest however but Ptolemy made numerous improvements in their presentation, ease of use and he even made improvements in the basic parameters to give greater accuracy. We only know details of the Handy Tables through the commentary by Theon of Alexandria but in [76] the author shows that care is required since Theon was not fully aware of Ptolemy's procedures.

Ptolemy also did what many writers of deep scientific works have done, and still do, in writing a popular account of his results under the title Planetary Hypothesis. This work, in two books, again follows the familiar route of reducing the mathematical skills needed by a reader. Ptolemy does this rather cleverly by replacing the abstract geometrical theories by mechanical ones. Ptolemy also wrote a work on astrology. It may seem strange to the modern reader that someone who wrote such excellent scientific books should write on astrology. However, Ptolemy sees it rather differently for he claims that the Almagest allows one to find the positions of the heavenly bodies, while his astrology book he sees as a companion work describing the effects of the heavenly bodies on people's lives.

In a book entitled Analemma he discussed methods of finding the angles need to construct a sundial which involves the projection of points on the celestial sphere. In Planisphaerium he is concerned with stereographic projection of the celestial sphere onto a plane. This is discussed in [48] where it is stated:-

In the stereographic projection treated by Ptolemy in the "Planisphaerium" the celestial sphere is mapped onto the plane of the equator by projection from the south pole. Ptolemy does not prove the important property that circles on the sphere become circles on the plane.

Ptolemy's major work Geography, in eight books, attempts to map the known world giving coordinates of the major places in terms of latitude and longitude. It is not surprising that the maps given by Ptolemy were quite inaccurate in many places for he could not be expected to do more than use the available data and this was of very poor quality for anything outside the Roman Empire, and even parts of the Roman Empire are severely distorted. In [19] Ptolemy is described as:-

... a man working [on map-construction] without the support of a developed theory but within a mathematical tradition and guided by his sense of what is appropriate to the problem.

Another work on Optics is in five books and in it Ptolemy studies colour, reflection, refraction, and mirrors of various shapes. Toomer comments in [1]:-

The establishment of theory by experiment, frequently by constructing special apparatus, is the most striking feature of Ptolemy's "Optics". Whether the subject matter is largely derived or original, "The Optics" is an impressive example of the development of a mathematical science with due regard to physical data, and is worthy of the author of the "Almagest".

An English translation, attempting to remove the inaccuracies introduced in the poor Arabic translation which is our only source of the Optics is given in [14].

The first to make accusations against Ptolemy was Tycho Brahe. He discovered that there was a systematic error of one degree in the longitudes of the stars in the star catalogue, and he claimed that, despite Ptolemy saying that it represented his own observations, it was merely a conversion of a catalogue due to Hipparchus corrected for precession to Ptolemy's date. There is of course definite problems comparing two star catalogues, one of which we have a copy of while the other is lost.

After comments by Laplace and Lalande, the next to attack Ptolemy vigorously was Delambre. He suggested that perhaps the errors came from Hipparchus and that Ptolemy might have done nothing more serious than to have failed to correct Hipparchus's data for the time between the equinoxes and solstices. However Delambre then goes on to say (see [8]):-

One could explain everything in a less favourable but all the simpler manner by denying Ptolemy the observation of the stars and equinoxes, and by claiming that he assimilated everything from Hipparchus, using the minimal value of the latter for the precession motion.

However, Ptolemy was not without his supporters by any means and further analysis led to a belief that the accusations made against Ptolemy by Delambre were false. Boll writing in 1894 says [4]:-

To all appearances, one will have to credit Ptolemy with giving an essentially richer picture of the Greek firmament after his eminent predecessors.

Vogt showed clearly in his important paper [77] that by considering Hipparchus's Commentary on Aratus and Eudoxus and making the reasonable assumption that the data given there agreed with Hipparchus's star catalogue, then Ptolemy's star catalogue cannot have been produced from the positions of the stars as given by Hipparchus, except for a small number of stars where Ptolemy does appear to have taken the data from Hipparchus. Vogt writes:-

This allows us to consider the fixed star catalogue as of his own making, just as Ptolemy himself vigorously states.

The most recent accusations of forgery made against Ptolemy came from Newton in [12]. He begins this book by stating clearly his views:-

This is the story of a scientific crime. ... I mean a crime committed by a scientist against fellow scientists and scholars, a betrayal of the ethics and integrity of his profession that has forever deprived mankind of fundamental information about an important area of astronomy and history.

Towards the end Newton, having claimed to prove every observation claimed by Ptolemy in the Almagest was fabricated, writes [12]:-

[Ptolemy] developed certain astronomical theories and discovered that they were not consistent with observation. Instead of abandoning the theories, he deliberately fabricated observations from the theories so that he could claim that the observations prove the validity of his theories. In every scientific or scholarly setting known, this practice is called fraud, and it is a crime against science and scholarship.

Although the evidence produced by Brahe, Delambre, Newton and others certainly do show that Ptolemy's errors are not random, this last quote from [12] is, I [EFR] believe, a crime against Ptolemy (to use Newton's own words). The book [8] is written to study validity of these accusations and it is a work which I strongly believe gives the correct interpretation. Grasshoff writes:-

... one has to assume that a substantial proportion of the Ptolemaic star catalogue is grounded on those Hipparchan observations which Hipparchus already used for the compilation of the second part of his "Commentary on Aratus". Although it cannot be ruled out that coordinates resulting from genuine Ptolemaic observations are included in the catalogue, they could not amount to more than half the catalogue.

... the assimilation of Hipparchan observations can no longer be discussed under the aspect of plagiarism. Ptolemy, whose intention was to develop a comprehensive theory of celestial phenomena, had no access to the methods of data evaluation using arithmetical means with which modern astronomers can derive from a set of varying measurement results, the one representative value needed to test a hypothesis. For methodological reason, then, Ptolemy was forced to choose from a set of measurements the one value corresponding best to what he had to consider as the most reliable data. When an intuitive selection among the data was no longer possible ... Ptolemy had to consider those values as 'observed' which could be confirmed by theoretical predictions.

As a final comment we quote the epigram which is accepted by many scholars to have been written by Ptolemy himself, and it appears in Book 1 of the Almagest, following the list of contents (see for example [11]):-

Well do I know that I am mortal, a creature of one day.
But if my mind follows the winding paths of the stars
Then my feet no longer rest on earth, but standing by
Zeus himself I take my fill of ambrosia, the divine dish.

Zhang Heng

2008-02-15T00:56:00.000-08:00

Born: 78 in Nan-yang, China
Died: 139 in China
Zhang Heng was born at the time of the Eastern Han (sometimes called Later Han) dynasty, the second half of the longest lasting Chinese dynasty. The Eastern Han was established in 25 AD after the brief 15 year reign of Wang Mang's Hsin dynasty had replaced the Western Han dynasty. At the time of Zhang Heng's birth the Emperor was Chang-ti, the third of the Eastern Han emperors. The capital of the country had moved to Lo-yang where a large ornate palace had been built. In Encyclopaedia Britannica the aims and achievements of the Han rulers are described as follows:-

... the Han came to require cultural accomplishment from their public servants, making mastery of classical texts a condition of employment. The title list of the enormous imperial library is China's first bibliography. Its text included works on practical matters such as mathematics and medicine, as well as treatises on philosophy and religion and the arts. Advancement in science and technology was also sought by the rulers, and the Han invented paper, used water clocks and sundials, and developed a seismograph. Calendars were published frequently during the period.

Over the years that Zhang Heng grew up, Chinese influence and prestige were growing rapidly and reached their peak in around 90 when he was about 12 years old. This was the period during which Ho-ti was the emperor and the court began to be influenced by family members seeking to extend their own power. Zhang, who had been born into an important family, was educated in the moral and political philosophy of Confucianism. For ten years he studied literature and trained as a writer. He published a number of literary works which gained him considerable fame. We shall give more information below on these aspects of Zhang's achievements as well as examples of his poetry. Zhang was thirty years old before his interests turned from literature to scientific matters, and at that time he became particularly interested in astronomy.

In around 116 he was appointed an official at the Emperor's court in Lo-yang. The court, however, was beginning to provide a less efficient government due to the weakness of successive emperors who were manipulated by those around them seeking advantage for themselves. This was hardly surprising since many emperors came to the throne as children. If China began to suffer due to ambitious people seeking to further their own influence, this was certainly not the case for Zhang. His biography in The History of the Eastern Han Dynasty (see [5]) suggests that he was not as successful an official as he might have been precisely because of a lack of ambition. This seemed to stem, at least in part, from his strong moral beliefs based in Confucianism. He refused advancement in his career on several occasions when he turned down posts that were offered to him, and he also spent periods away from the capital when he lived in isolation and thought about the nature of the universe and about a wide variety of scientific topics. His highest position at court was when he became chief astrologer and minister under the emperor An-ti. He held the position of chief astrologer on a number of occasions.

We will describe below some of Zhang's outstanding scientific achievements. However, as we indicated above, he first achieved fame as a poet and writer of over twenty works, and in this capacity he had a lasting influence on Chinese culture. His works Si Chou Shi (Four Chapters of Distressed Poems) and Gui Tian Fu (To Live in Seclusion) are considered literary masterpieces. Zhang's poem, in his highly influential style of prose poetry, which we now quote comes from [1]. It is a telling criticism of the last rulers of the Western Han dynasty:-

Those who won this territory were strong;
Those who depended on it endured.
When a stream is long, its water is not easily exhausted.
When roots are deep, they do not rot easily.
Therefore, as extravagance and ostentation were given free reign,
The odour became pungent and increasingly fulsome.

Zhang wrote the Four Stanzas of Sorrow which is the first seven-syllabic poem which we know of in China. We quote (in translation of course) only the first of its four stanzas:-

In Taishan stays my dear sweetheart,
But Liangfu keeps us long apart;
Looking east, I find tears start.
She gives me a sword to my delight;
A jade I give her as requite.
I'm at a loss as she is out of sight;
Why should I trouble myself all night?

In ancient China there was a belief that an emperor received his right to rule from heaven. Changing the calendar was seen as one of the duties of the office, establishing the emperor's heavenly link on earth. After a change of ruler, and even more significantly after a change of dynasty, the new Chinese emperor would seek a new official calendar thus establishing a new rule with new celestial influences. It was natural therefore that Zhang having become an expert in astronomy should become involved in calendar reform by the year 123. In that year he corrected the calendar to bring it into line with his accurate astronomical observations.

In 132 Zhang invented the first seismograph for measuring earthquakes. One has to understand how significant earthquakes were in China at this time, not only for the destructive power which they unleashed but also because they were seen as punishment from the gods for poor governance of the country. In his role as chief astrologer he was responsible for detecting signs of bad government which were indicated by earthquakes. Zhang's device, which he called Hou Feng Di Dong Yi, was made of copper. It was in the shape of an egg with eight dragon heads around the top, each with a copper ball in its mouth, and a pendulum in the centre. Around the bottom were eight frogs, each directly under a dragon head. When an earthquake occurred, a ball fell out of a the dragon's mouth into a frog's mouth, making a noise. In fact the seismograph detected an earthquake in February of 138 and Zhang reported this fact to the Emperor despite no other evidence of the earthquake being felt in the capital Lo-yang. He was even able to indicate that the earthquake was to the west of the capital. He achieved fame when reports of an earthquake more than a thousand kilometres north west reached the capital several days later.

Zhang appears to have been the first person in China to construct an equatorial armillary sphere. It consisted of a system of rings corresponding to the great circles of the celestial sphere with a central tube which was used to line up stars and planets. With this instrument Zhang was able to make more accurate star maps than earlier Chinese astronomers. He wrote about his instrument in the work Hun-i chu where he described his version of the universe as follows:-

The sky is like a hen's egg, and is as round as a crossbow pellet, the Earth is like the yolk of the egg, lying alone at the centre. The sky is large and the Earth small.

In another work, Ling Xian (Mystical Laws), he describes the stars:-

North and south of the equator there are 124 groups which are always brightly shining. 320 stars can be named. There are in all 2500, not including those which the sailors observe. Of the very small stars there are 11520.

Only the first part of this text by Zhang has survived.

In mathematics Zhang studied 3 by 3 magic squares. He also proposed, in a treatise on inscribed and circumscribed circles of a square, that π = √10 or approximately 3.162. Although this is not particularly accurate the significance of his work is pointed out by S K Mo in [8]. As Mo notes, the significance here is that all earlier attempts to calculate were based on practical measurement, whereas the work by Zhang was based on a theoretical calculation.

Zhang also gave formulae for the volume of a sphere in terms of the volume of the circumscribing cube. These results are not very accurate and Liu Hui in his commentary on the Nine Chapters on the Mathematical Art writes:-

In an attempt to make his statement consistent and harmonise his philosophy of yin and yang, and the doctrine of odd and even, [Zhang] neglected the precision of the data.

Li Chunfeng wrote in his commentary on the Nine Chapters on the Mathematical Art:-

Zhang Heng followed the ancients blindly, making himself a laughing stock for later generations.

One interesting point to note in some of Zhang's mathematical work is that he leaves square roots as unevaluated. Some historians believe that Zhang understood the difference between rational and irrational numbers but this seems to be stretching things a bit too far.

Menelaus of Alexandria

2008-02-15T00:55:00.000-08:00

Born: about 70 in (possibly) Alexandria, Egypt
Died: about 130
Although we know little of Menelaus of Alexandria's life Ptolemy records astronomical observations made by Menelaus in Rome on the 14th January in the year 98. These observation included that of the occultation of the star Beta Scorpii by the moon.

He also makes an appearance in a work by Plutarch who describes a conversation between Menelaus and Lucius in which Lucius apologises to Menelaus for doubting the fact that light, when reflected, obeys the law that the angle of incidence equals the angle of reflection. Lucius says (see for example [1]):-

In your presence, my dear Menelaus, I am ashamed to confute a mathematical proposition, the foundation, as it were, on which rests the subject of catoptrics. Yet it must be said that the proposition, "All reflection occurs at equal angles" is neither self evident nor an admitted fact.

This conversation is supposed to have taken place in Rome probably quite a long time after 75 AD, and indeed if our guess that Menelaus was born in 70 AD is close to being correct then it must have been many years after 75 AD.

Very little else is known of Menelaus's life, except that he is called Menelaus of Alexandria by both Pappus and Proclus. All we can deduce from this is that he spent some time in both Rome and Alexandria but the most likely scenario is that he lived in Alexandria as a young man, possibly being born there, and later moved to Rome.

An Arab register of mathematicians composed in the 10th century records Menelaus as follows (see [1]):-

He lived before Ptolemy, since the latter makes mention of him. He composed: "The Book of Spherical Propositions", "On the Knowledge of the Weights and Distribution of Different Bodies" ... Three books on the "Elements of Geometry", edited by Thabit ibn Qurra, and "The Book on the Triangle". Some of these have been translated into Arabic.

Of Menelaus's many books only Sphaerica has survived. It deals with spherical triangles and their application to astronomy. He was the first to write down the definition of a spherical triangle giving the definition at the beginning of Book I:-

A spherical triangle is the space included by arcs of great circles on the surface of a sphere ... these arcs are always less than a semicircle.

In Book I of Sphaerica he set up the basis for treating spherical triangles as Euclid treated plane triangles. He used arcs of great circles instead of arcs of parallel circles on the sphere. This marks a turning point in the development of spherical trigonometry. However, Menelaus seems unhappy with the method of proof by reductio ad absurdum which Euclid frequently uses. Menelaus avoids this way of proving theorems and, as a consequence, he gives proofs of some of the theorems where Euclid's proof could be easily adapted to the case of spherical triangles by quite different methods.

It is also worth commenting that [3]:-

In some respects his treatment is more complete than Euclid's treatment of the analogous plane case.

Book 2 applies spherical geometry to astronomy. It largely follows the propositions given by Theodosius in his Sphaerica but Menelaus give considerably better proofs.

Book 3 deals with spherical trigonometry and includes Menelaus's theorem. For plane triangles the theorem was known before Menelaus:-

... if a straight line crosses the three sides of a triangle (one of the sides is extended beyond the vertices of the triangle), then the product of three of the nonadjacent line segments thus formed is equal to the product of the three remaining line segments of the triangle.

Menelaus produced a spherical triangle version of this theorem which is today also called Menelaus's Theorem, and it appears as the first proposition in Book III. The statement is given in terms of intersecting great circles on a sphere.

Many translations and commentaries of Menelaus Sphaerica were made by the Arabs. Some of these survive but differ considerably and make an accurate reconstruction of the original quite difficult. On the other hand we do know that some of the works are commentaries on earlier commentaries so it is easy to see how the original becomes obscured. There are detailed discussions of these Arabic translations in [6], [9], and [10].

There are other works by Menelaus which are mentioned by Arab authors but which have been lost both in the Greek and in their Arabic translations. We gave a quotation above from the 10th century Arab register which records a book called Elements of Geometry which was in three volumes and was translated into Arabic by Thabit ibn Qurra. It also records another work by Menelaus was entitled Book on Triangles and although this has not survived fragments of an Arabic translation have been found.

Proclus referred to a geometrical result of Menelaus which does not appear in the work which has survived and it is thought that it must come from one of the texts just mentioned. This was a direct proof of a theorem in Euclid's Elements and given Menelaus's dislike for reductio ad absurdum in his surviving works this seems a natural line for him to follow. The new proof which Proclus attributes to Menelaus is of the theorem (in Heath's translation of Euclid):-

If two triangles have the two sides equal to two sides respectively, but have the base of one greater than the base of the other, it will also have the angle contained by the equal straight lines of the first greater than that of the other.

Another Arab reference to Menelaus suggests that his Elements of Geometry contained Archytas's solution of the problem of duplicating the cube. Paul Tannery in [8] argues that this make it likely that a curve which it is claimed by Pappus that Menelaus discussed at length was the Viviani's curve of double curvature. Bulmer-Thomas in [1] comments that:-

It is an attractive conjecture but incapable of proof on present evidence.

Menelaus is believed by a number of Arab writers to have written a text on mechanics. It is claimed that the text studied balances studied by Archimedes and those devised by Menelaus himself. In particular Menelaus was interested in specific gravities and analysing alloys.

Theon of Smyrna

2008-02-15T00:54:00.001-08:00

Born: about 70 in Smyrna (now Izmir), Turkey
Died: about 135
Little is known of Theon of Smyrna's life. He was called 'the old Theon' by Theon of Alexandria and 'Theon the mathematician' by Ptolemy. The date of his birth is little better than a guess, but we do have some firm data about dates in his life. We know that he was making astronomical observations of Mercury and Venus between 127 and 132 since Ptolemy lists four observations which Theon made in 127, 129, 130 and 132. From these observations Theon made estimates of the greatest angular distance that Mercury and Venus can reach from the Sun. The style of his bust, dedicated by his son 'Theon the priest', gives us the date of his death to within 10 years and it is placed within the period 130-140 (hence our midpoint guess of 135).

Theon's most important work is Expositio rerum mathematicarum ad legendum Platonem utilium. This work is a handbook for philosophy students to show how prime numbers, geometrical numbers such as squares, progressions, music and astronomy are interrelated. Its rather curious title means that it was intended as an introduction to a study of the works of Plato, but this is rather fanciful. As Huxley writes in [1]:-

... the book has little to offer the specialist student of Plato's mathematics. It is, rather, a handbook for philosophy students, written to illustrate how arithmetic, geometry, stereometry, music, and astronomy are interrelated.

The most important feature of the work is the wide range of citations of earlier sources. Its worst feature is its lack of originality. Heath writes [2]:-

Theon's work is a curious medley, valuable, not intrinsically, but for the numerous historical notices which it contains.

In the introduction Theon gives his reason for writing the work:-

Everyone would agree that he could not understand the mathematical arguments used by Plato unless he were practised in this science... One who had become skilled in all geometry and all music and astronomy would be reckoned most happy on making acquaintance with the writings of Plato, but this cannot be come by easily or readily, for it calls for a very great deal of application from youth upwards. In order that those who have failed to become practised in these studies, but aim at a knowledge of his writings, should not wholly fail in their desires, I shall make a summary and concise sketch of the mathematical theorems which are specially necessary for readers of Plato....

The work begins with a collection of theorems which Theon says will be useful for the study of arithmetic, music, geometry, and astronomy in Plato. However his coverage of geometry is none too good and later in the book he makes an excuse for this saying that anyone who reads his book, or the works of Plato, will have already studied elementary geometry.

In the section on numbers Theon adopts a Pythagorean approach, writing about odd numbers, even numbers, prime numbers, composite numbers, square numbers, oblong numbers, triangular numbers, polygonal numbers, circular numbers, spherical numbers, solid numbers with three factors, pyramidal numbers, perfect numbers, deficient numbers and abundant numbers.

The best section of Expositio rerum mathematicarum is the astronomy section which teaches that the Earth is spherical, that mountains are negligible in height compared with the Earth etc. It includes knowledge of conjunctions, eclipses, occultations and transits. However, Neugebauer writes in [3]:-

It is clear that Theon's treatise does not pretend to make original contributions to astronomy. Unfortunately it is also clear that Theon has not fully digested the material he is presenting to his readers.

Theon also wrote commentaries on the main authorities of mathematics and astronomy. In particular he wrote an important work on Ptolemy and another on Plato's Republic which he refers to himself in work which survives. Whether his work on the ancestry of Plato is a separate work or a section of one of his commentaries on Plato's work, it is impossible to say.

Nicomachus of Gerasa

2008-02-15T00:52:00.000-08:00

Born: about 60 in Gerasa, Roman Syria (now Jarash, Jordan)
Died: about 120
Nicomachus of Gerasa is mentioned in a small number of sources and we can date him fairly accurately from the information given. Nicomachus himself refers to Thrasyllus who died in 36 AD so this gives lower limits on his dates. On the other hand Apuleius, the Platonic philosopher, rhetorician and author whose dates are 124 AD to about 175 AD, translated Nicomachus's Introduction to Arithmetic into Latin so this gives an upper limit on his dates. One of the most interesting references is by Lucian, the rhetorician, pamphleteer and satirist who was born about 120 AD, who makes one of his characters say:-

You calculate like Nicomachus.

Clearly Nicomachus had achieved fame for his arithmetical work!

In the paper [7] Dillon argues that Nicomachus died in 196 AD. His argument is based on the fact that Marinus claimed that Proclus believed that he was the reincarnation of Nicomachus. Since Proclus was born in 412 AD and there was a belief among Pythagoreans that reincarnations occurred with an interval of 216 years, the date fits. Although 196 AD is not ruled out by his translator dying in 175 AD (although it comes close) the most serious objection to Dillon's theory seems to be the lack of evidence that Proclus himself believed in the 216 year interval.

Let us move from conjectures to more certain ground, and record that Nicomachus was a Pythagorean. This is obvious from his writings on numbers and music, but we are also told this by Porphyry who says that he was one of the leading members of the Pythagorean School.

Nicomachus wrote Arithmetike eisagoge (Introduction to Arithmetic) which was the first work to treat arithmetic as a separate topic from geometry. Unlike Euclid, Nicomachus gave no abstract proofs of his theorems, merely stating theorems and illustrating them with numerical examples.

However Introduction to Arithmetic does contain quite elementary errors which show that Nicomachus chose not to give proofs of his results because he did not in general have such proofs. Many of the results were known by Nicomachus to be true since they appeared with proofs in Euclid, although in a geometrical formulation. Sometimes Nicomachus stated a result which is simply false and then illustrated it with an example which happens to have the properties described in the result. We must deduce from this that some of the results are merely guesses based on the evidence of the numerical examples (and in some cases perhaps even based on one example!).

An example of this we look more closely at the results which Nicomachus quotes on perfect numbers. He states that the nth perfect number has n digits, and that all perfect numbers end in 6 and 8 alternately. These statements must be merely false deductions from the fact that there were four perfect numbers known to Nicomachus, namely 6, 28, 496 and 8128.

The work contains the first multiplication table in a Greek text. It is also remarkable in that it contains Arabic numerals, not Greek ones. However, in many respects the book is old fashioned in its style since it appears more in tune with the number theoretic ideas of Pythagoras with his mystical approach, rather than a true mathematical approach. To illustrate Nicomachus's rather strange approach to numbers, giving the moral properties, we look at his description of abundant numbers and deficient numbers. An abundant number has the sum of its proper divisors greater than the number, while a deficient number has the sum of its proper divisors less than the number. Nicomachus writes of these numbers in Introduction to Arithmetic (see [6], or [3] for a different translation):-

In the case of the too much, is produced excess, superfluity, exaggerations and abuse; in the case of too little, is produced wanting, defaults, privations and insufficiencies. And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort - of which the most exemplary form is that type of number which is called perfect.

He then continues his description of abundant numbers as resembling an animal:-

... with ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands....

while a deficient number is like an animal:-

... with a single eye, ... one armed or one of his hands has less than five fingers, or if he does not have a tongue...

For over 1000 years Introduction to Arithmetic was the standard arithmetic text. In view of the comments we have made regarding the work, this may seem a surprising fact. Mathematicians disliked the work, in particular Pappus is said to have despised it. However, several people including Boethius translated Introduction to Arithmetic into Latin and it was used as a school book. How then could a poor book become so popular. Heath tries to explain the apparent contradiction in [4], suggesting that:-

... it was at first read by philosophers rather than mathematicians, and afterwards became generally popular at a time when there were no mathematicians left, but only philosophers who incidentally took an interest in mathematics.

Arab translations of Nicomachus's Introduction to Arithmetic were important and in [5] Brentjes studies the influence of these Arabic translations. She concludes that most Arabic texts on number theory written by mathematicians were influenced by both Euclid and Nicomachus, but were mainly influenced by Euclid. However, texts by non-mathematicians were most strongly influenced by Nicomachus. This research in [5] tends to support the views of Heath on this subject.

Nicomachus also wrote two volumes Theologoumena arithmetikes (The Theology of Numbers) which was completely concerned with mystic properties of numbers. However Heath writes [4]:-

The curious farrago which has come down to us under that title and which was edited by Ast [published in Leipzig in 1817] is, however, certainly not by Nicomachus; for among the authors from whom it gives extracts is Anatolius, Bishop of Laodicaea (270 AD); but it contains quotations from Nicomachus which appear to come from the genuine work.

Another work by Nicomachus which has survived is Manual of Harmonics which is a work on music. Again Nicomachus shows the influence of Pythagoras but also Aristotle's theories of music. The work looks at musical notes and the octave. The principles of tuning a stretched string are studied as is an extension of the octave to the two-octave range.

The influences of Pythagoras's theory of music are seen from Nicomachus' (see [1]):-

... assignment of number and numerical ratios to notes and intervals, his recognition of the indivisibility of the octave and the whole tone... But, unlike Euclid, who attempts to prove musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of the lengths of strings.

Both Porphyry and Iamblichus wrote biographies of Pythagoras which quote from Nicomachus. From this evidence some historians have conjectured that Nicomachus also wrote a biography of Pythagoras and, although there is no direct evidence, it is indeed quite possible.

Cleomedes

2008-02-15T00:51:00.000-08:00

Born: 1st century AD in possibly Lysimachia, Hellespont, Greece
Died: 1st century AD
Cleomedes is known only through his book On the Circular Motions of the Celestial Bodies which is an uninspiring astronomy textbook. There are a number of points of interest in this book, however, as we shall discuss below.

We should first discuss the perplexing question of the period in which Cleomedes lived. The only certainty here is that On the Circular Motions of the Celestial Bodies discusses the work of Posidonius at length and so is clearly written after the middle of the first century BC. In fact On the Circular Motions of the Celestial Bodies ends with the words (see for example [3]):-

The preceding teachings are not the author's own opinion but collected from older or more recent summaries; much of it is taken from Posidonius.

It is hard to estimate from these words how long after Posidonius the author, Cleomedes, is writing.

Heath [2] favours a date in the middle of the first century BC. He also points out:-

As [Cleomedes] seems to know nothing of the works of Ptolemy, he can hardly... have lived later than the beginning of the second century AD.

Neugebauer, however, disagrees with these conclusions of Heath and proposes that Cleomedes wrote his text around 370 AD. His argument is based on a comment by Cleomedes in the text where he remarks than there are two bright stars (Aldebaran and Antares) such that the rising of one and the setting of the other take place at the same time. These stars Cleomedes claims lie at 15 of their sign. Using Ptolemy's positions for the stars at the time the Almagest was written and Ptolemy's value of 1 per 100 years for precession, Neugebauer gets his date of 371 AD for Cleomedes writings, to which Neugebauer estimates a maximum error of 50 years on either side.

Astronomically Neugebauer's calculations are of course perfectly correct. However they are suspect for a number of other reasons. Firstly the data in On the Circular Motions of the Celestial Bodies does not seem to be due to Cleomedes but to a variety of sources. Of course accepting this argument would make Cleomedes dates later still. Secondly the data in Cleomedes is of widely differing degrees of accuracy. Some is very good, while other data has errors of 20%. Thirdly the actual astronomical event of Aldebaran setting and Antares rising at the same instant can never be observed.

Heath's comment that Cleomedes knows nothing of the works of Ptolemy is also less certain than it might at first appear. Cleomedes is writing an elementary textbook and it is certainly not always the case that one mentions recent research in a low level textbook. For example many elementary textbooks on applied mathematics still use Newton's gravitational methods (and for good reason) 100 years after Einstein gave an improved theory of gravitation.

In [1] Dicks suggests that the most likely date for Cleomedes is the first century AD and we have taken that as the best available compromise from what is known. That is not to say that Neugebauer's date is impossible. In fact there are other features of the text which would tend to support the fourth century AD as a date, despite the lack of references to Ptolemy. Not least of these is the fact that this was a period when many second rate textbooks of this nature were written and the style is not unlike that of other fourth century AD texts, some of which give the same astronomical data as Cleomedes.

On the Circular Motions of the Celestial Bodies is a work in two volumes. We commented above that it was important for a number of reasons. The most important certainly is that it gives us the best indication that we have of the contents and the style of a text by Posidonius. As Heath comments [2]:-

... the very long first chapter of Book II (nearly half of the Book) ... seems for the most part to be copied bodily from Posidonius.

But this is not Cleomedes' main aim in writing the text. It is written to attack the Epicureans who believed among other odd beliefs, that the sun was as large as it looked, namely one foot across. Cleomedes spends much time in his text showing that this is false, but it does seem as if he is going to extremes when he compares Epicureans unfavourably with rats, reptiles and worms. Cleomedes' own philosophical views show that he is a Stoic.

Since On the Circular Motions of the Celestial Bodies is compiled from a number of sources, there is a fair variation in the quality, and in many places the book fails to be consistent where the various sources disagree. Whenever a piece of text is thought to be due to Cleomedes himself, there is much evidence that his understanding of the topic was very limited and, but for the quality of his sources, one feels that Cleomedes would not fare any better than the Epicureans for naiveté.

Lunar eclipses are described well in the text, and the conical shape of the earth's shadow shows an interesting depth of understanding (at least of Cleomedes' source). He also correctly explains the reports of lunar eclipses seen when both the sun and moon are above the horizon as being due to refraction.

Neugebauer gives samples of the sense and nonsense which are mixed in Cleomedes [3]:-

Cleomedes states that no fixed star has an apparent diameter less than one finger (a rather absurd exaggeration) while the apparent diameter of Venus should be two fingers, i.e. 1/6 of the lunar or solar diameter. Of some interest is the remark that the absolute size of fixed stars may reach, or even surpass, the sun ... it is [also] said that the earth, seen from the sun, would appear at best a very small star.

One further interest in Cleomedes' work is that it is in On the Circular Motions of the Celestial Bodies that we learn of Eratosthenes method of measuring the circumference of the earth. This is one of the best known of the achievements of early mathematical astronomy and we are indebted to Cleomedes for relating the method. Of course not everyone believes that the story of Eratosthenes' measurement is authentic but, despite this, it is widely accepted.

Finally we should remark that Neugebauer suggests from a study of certain astronomical data given in On the Circular Motions of the Celestial Bodies that Cleomedes lived in the Hellespont on the Black Sea, suggesting the city of Lysimachia. Neugebauer admits that the city of Lysimachia was destroyed in 144 BC which seems at odds with his own date of 370 AD for Cleomedes but he is able to show that despite the disaster of 144 BC records of the city certainly extend up to the fourth century AD. The weakness of Neugebauer's argument must surely be that almost all of Cleomedes' text and data is taken from the works of others so Neugebauer's arguments seem only to give strong evidence for one of Cleomedes' sources having written in Lysimachia.

Heron of Alexandria

2008-02-15T00:50:00.001-08:00

Born: about 10 in (possibly) Alexandria, Egypt
Died: about 75
Sometimes called Hero, Heron of Alexandria was an important geometer and worker in mechanics. Perhaps the first comment worth making is how common the name Heron was around this time and it is a difficult problem in the history of mathematics to identify which references to Heron are to the mathematician described in this article and which are to others of the same name. There are additional problems of identification which we discuss below.

A major difficulty regarding Heron was to establish the date at which he lived. There were two main schools of thought on this, one believing that he lived around 150 BC and the second believing that he lived around 250 AD. The first of these was based mainly on the fact that Heron does not quote from any work later than Archimedes. The second was based on an argument which purported to show that he lived later that Ptolemy, and, since Pappus refers to Heron, before Pappus.

Both of these arguments have been shown to be wrong. There was a third date proposed which was based on the belief that Heron was a contemporary of Columella. Columella was a Roman soldier and farmer who wrote extensively on agriculture and similar subjects, hoping to foster in people a love for farming and a liking for the simple life. Columella, in a text written in about 62 AD [5]:-

... gave measurements of plane figures which agree with the formulas used by Heron, notably those for the equilateral triangle, the regular hexagon (in this case not only the formula but the actual figures agree with Heron's) and the segment of a circle which is less than a semicircle ...

However, most historians believed that both Columella and Heron were using an earlier source and claimed that the similarity did not prove any dependence. We now know that those who believed that Heron lived around the time of Columella were in fact correct, for Neugebauer in 1938 discovered that Heron referred to a recent eclipse in one of his works which, from the information given by Heron, he was able to identify with one which took place in Alexandria at 23.00 hours on 13 March 62.

From Heron's writings it is reasonable to deduce that he taught at the Museum in Alexandria. His works look like lecture notes from courses he must have given there on mathematics, physics, pneumatics, and mechanics. Some are clearly textbooks while others are perhaps drafts of lecture notes not yet worked into final form for a student textbook.

Pappus describes the contribution of Heron in Book VIII of his Mathematical Collection. Pappus writes (see for example [8]):-

The mechanicians of Heron's school say that mechanics can be divided into a theoretical and a manual part; the theoretical part is composed of geometry, arithmetic, astronomy and physics, the manual of work in metals, architecture, carpentering and painting and anything involving skill with the hands.

... the ancients also describe as mechanicians the wonder-workers, of whom some work by means of pneumatics, as Heron in his Pneumatica, some by using strings and ropes, thinking to imitate the movements of living things, as Heron in his Automata and Balancings, ... or by using water to tell the time, as Heron in his Hydria, which appears to have affinities with the science of sundials.

A large number of works by Heron have survived, although the authorship of some is disputed. We will discuss some of the disagreements in our list of Heron's works below. The works fall into several categories, technical works, mechanical works and mathematical works. The surviving works are:

On the dioptra dealing with theodolites and surveying. It contains a chapter on astronomy giving a method to find the distance between Alexandria and Rome using the difference between local times at which an eclipse of the moon is observed at each cities. The fact that Ptolemy does not appear to have known of this method led historians to mistakenly believe Heron lived after Ptolemy;

The pneumatica in two books studying mechanical devices worked by air, steam or water pressure. It is described in more detail below;

The automaton theatre describing a puppet theatre worked by strings, drums and weights;

Belopoeica describing how to construct engines of war. It has some similarities with work by Philon and also work by Vitruvius who was a Roman architect and engineer who lived in the 1st century BC;

The cheirobalistra about catapults is thought to be part of a dictionary of catapults but was almost certainly not written by Heron;

Mechanica in three books written for architects and described in more detail below;

Metrica which gives methods of measurement. We give more details below;
Definitiones contains 133 definitions of geometrical terms beginning with points, lines etc. In [15] Knorr argues convincingly that this work is in fact due to Diophantus;

Geometria seems to be a different version of the first chapter of the Metrica based entirely on examples. Although based on Heron's work it is not thought to be written by him;

Stereometrica measures three-dimensional objects and is at least in part based on the second chapter of the Metrica again based on examples. Again it is though to be based on Heron's work but greatly changed by many later editors;

Mensurae measures a whole variety of different objects and is connected with parts of Stereometrica and Metrica although it must be mainly the work of a later author;

Catoprica deals with mirrors and is attributed by some historians to Ptolemy although most now seem to believe that this is a genuine work of Heron. In this work, Heron states that vision results from light rays emitted by the eyes. He believes that these rays travel with infinite velocity.

Let us examine some of Heron's work in a little more depth. Book I of his treatise Metrica deals with areas of triangles, quadrilaterals, regular polygons of between 3 and 12 sides, surfaces of cones, cylinders, prisms, pyramids, spheres etc. A method, known to the Babylonians 2000 years before, is also given for approximating the square root of a number. Heron gives this in the following form (see for example [5]):-

Since 720 has not its side rational, we can obtain its side within a very small difference as follows. Since the next succeeding square number is 729, which has 27 for its side, divide 720 by 27. This gives 26 2/3. Add 27 to this, making 53 2/3, and take half this or 26 5/6. The side of 720 will therefore be very nearly 26 5/6. In fact, if we multiply 26 5/6 by itself, the product is 720 1/36, so the difference in the square is 1/36. If we desire to make the difference smaller still than 1/36, we shall take 720 1/36 instead of 729 (or rather we should take 26 5/6 instead of 27), and by proceeding in the same way we shall find the resulting difference much less than 1/36.

Heron also proves his famous formula in Book I of the Metrica :

if A is the area of a triangle with sides a, b and c and s = (a + b + c)/2 then
A2 = s (s - a)(s - b)(s - c).

In Book II of Metrica, Heron considers the measurement of volumes of various three dimensional figures such as spheres, cylinders, cones, prisms, pyramids etc. His preface is interesting, partly because knowledge of the work of Archimedes does not seem to be as widely known as one might expect (see for example [5]):-

After the measurement of surfaces, rectilinear or not, it is proper to proceed to solid bodies, the surfaces of which we have already measured in the preceding book, surfaces plane and spherical, conical and cylindrical, and irregular surfaces as well. The methods of dealing with these solids are, in view of their surprising character, referred to Archimedes by certain writers who give the traditional account of their origin. But whether they belong to Archimedes or another, it is necessary to give a sketch of these results as well.

Book III of Metrica deals with dividing areas and volumes according to a given ratio. This was a problem which Euclid investigated in his work On divisions of figures and Heron's Book III has a lot in common with the work of Euclid. Also in Book III, Heron gives a method to find the cube root of a number. In particular Heron finds the cube root of 100 and the authors of [9] give a general formula for the cube root of N which Heron seems to have used in his calculation:

a + b d/(b d + aD)(b - a),
where a3 < N < b3, d = N - a3, D = b3 - N.

In [9] it is remarked that this is a very accurate formula, but, unless a Byzantine copyist is to be blamed for an error, they conclude that Heron might have borrowed this accurate formula without understanding how to use it in general.

The Pneumatica is a strange work which is written in two book, the first with 43 chapters and the second with 37 chapters. Heron begins with a theoretical consideration of pressure in fluids. Some of this theory is right but, not surprisingly, some is quite wrong. Then there follows a description of a whole collection of what might best be described as mechanical toys for children [1]:-

Trick jars that give out wine or water separately or in constant proportions, singing birds and sounding trumpets, puppets that move when a fire is lit on an altar, animals that drink when they are offered water ...

Although all this seems very trivial for a scientist to be involved with, it would appear that Heron is using these toys as a vehicle for teaching physics to his students. It seems to be an attempt to make scientific theories relevant to everyday items that students of the time would be familiar with.

There is, rather remarkably, descriptions of over 100 machines such as a fire engine, a wind organ, a coin-operated machine, and a steam-powered engine called an aeolipile. Heron's aeolipile, which has much in common with a jet engine, is described in [2] as follows:-

The aeolipile was a hollow sphere mounted so that it could turn on a pair of hollow tubes that provided steam to the sphere from a cauldron. The steam escaped from the sphere from one or more bent tubes projecting from its equator, causing the sphere to revolve. The aeolipile is the first known device to transform steam into rotary motion.

Heron wrote a number of important treatises on mechanics. They give methods of lifting heavy weights and describe simple mechanical machines. In particular the Mechanica is based quite closely on ideas due to Archimedes. Book I examines how to construct three dimensional shapes in a given proportion to a given shape. It also examines the theory of motion, certain statics problems, and the theory of the balance.

In Book II Heron discusses lifting heavy objects with a lever, a pulley, a wedge, or a screw. There is a discussion on centres of gravity of plane figures. Book III examines methods of transporting objects by such means as sledges, the use of cranes, and looks at wine presses.

Other works have been attributed to Heron, and for some of these we have fragments, for others there are only references. The works for which fragments survive include one on Water clocks in four books, and Commentary on Euclid's Elements which must have covered at least the first eight books of the Elements. Works by Heron which are referred to, but no trace survives, include Camarica or On vaultings which is mentioned by Eutocius and Zygia or On balancing mentioned by Pappus. Also in the Fihrist, a tenth century survey of Islamic culture, a work by Heron on how to use an astrolabe is mentioned.

Finally it is interesting to look at the opinions that various writers have expressed as to the quality and importance of Heron. Neugebauer writes [7]:-

The decipherment of the mathematical cuneiform texts made it clear that much of the "Heronic" type of Greek mathematics is simply the last phase of the Babylonian mathematical tradition which extends over 1800 years.

Some have considered Heron to be an ignorant artisan who copied the contents of his books without understanding what he wrote. This in particular has been levelled against the Pneumatica but Drachmann, writing in [1], says:-

... to me the free flowing, rather discursive style suggests a man well versed in his subject who is giving a quick summary to an audience that knows, or who might be expected to know, a good deal about it.

Some scholars have approved of Heron's practical skills as a surveyor but claimed that his knowledge of science was negligible. However, Mahony writes in [1]:-

In the light of recent scholarship, he now appears as a well-educated and often ingenious applied mathematician, as well as a vital link in a continuous tradition of practical mathematics from the Babylonians, through the Arabs, to Renaissance Europe.

Finally Heath writes in [5]:-

The practical utility of Heron's manuals being so great, it was natural that they should have great vogue, and equally natural that the most popular of them at any rate should be re-edited, altered and added to by later writers; this was inevitable with books which, like the "Elements" of Euclid, were in regular use in Greek, Byzantine, Roman, and Arabian education for centuries.

Bryson of Heraclea

2008-02-15T00:48:00.000-08:00

Born: about 450 BC in Heraclea (now Taranto, Italy)
Died: ?

Plato and Aristotle both mention a mathematician called Bryson, but as is often the case, there is not complete agreement among scholars as to whether these refer to the same person or to two different people.

Aristotle mentions Bryson of Heraclea, who was the son of Herodorus of Heraclea. Bryson was a Sophist and Aristotle criticises him both for his assertion that there is no such thing as indecent language, and also for his method of squaring the circle. We do know some details of this methods of squaring the circle and, despite the criticisms of Aristotle, it was an important step forward in the development of mathematics. Aristotle's criticism appears to have been based on the fact that Bryson's proof used general principles rather than on geometric ones, but it is somewhat unclear exactly what Aristotle meant by this.

Diogenes Laertius gives some other biographical details of Bryson, but these cannot all be correct since Bryson's interaction with a number of philosophers is stated, yet certain of these are impossible due to the dates during which these men lived. Perhaps the most likely of the details preserved by Diogenes Laertius is that Bryson was either a pupil of Socrates or of Euclid of Megara.

It is a little difficult to reconstruct exactly what Bryson's method of squaring the circle was. According to Alexander Aphrodisiensis, writing in about 210 AD, Bryson inscribed a square in the circle and circumscribed a second square. Bryson then constructed a third square between the inscribed and circumscribed square (but Alexander does not tell us how this third square was constructed).

Alexander then claims that Bryson's argument was that the circle was intermediate between the inscribed and circumscribed squares, the third square is also intermediate between the inscribed and circumscribed squares and therefore the third square equals the circle. Alexander then rightly points out that this argument of nonsense since, to use Alexander's example, 8 and 9 are both greater than 7 and less than 10 but 8 does not equal 9.

If indeed Alexander is right in what he attributes to Bryson then his contribution would not merit inclusion in this archive. However, other commentators attribute a much more significant argument to Bryson. Themistius, another ancient commentator, writes that Bryson claimed that the circle was greater than all inscribed polygons and less than all circumscribed polygons. It is unclear how quite how Bryson continued the argument but it seems likely that he was saying that by taking polygons with larger and larger numbers of sides then the difference the inscribed and circumscribed polygons could be made as small as we like so that a polygon intermediate between them will equal the circle to whatever degree of accuracy we chose.

This would be an improvement on Antiphon's argument and Bryson is getting close to the method of exhaustion as rigorously applied by Archimedes.

We know little else of Bryson. He wrote Diatribes which some accused Plato of stealing and indeed Bryson is claimed to have associated with Polyxenus who put forward philosophical arguments which appear in Plato's Pramenides and could well be the arguments which were claimed stolen from Bryson's Diatribes.

Democritus of Abdera

2008-02-15T00:46:00.000-08:00

Born: about 460 BC in Abdera, Thrace, Greece
Died: about 370 BC
Democritus of Abdera is best known for his atomic theory but he was also an excellent geometer. Very little is known of his life but we know that Leucippus was his teacher.

Democritus certainly visited Athens when he was a young man, principally to visit Anaxagoras, but Democritus complained how little he was known there. He said, according to Diogenes Laertius writing in the second century AD [5]:-

I came to Athens and no one knew me.

Democritus was disappointed by his trip to Athens because Anaxagoras, then an old man, had refused to see him.

As Brumbaugh points out in [3]:-

How different he would find the trip today, where the main approach to the city from the northeast runs past the impressive "Democritus Nuclear Research Laboratory".

Certainly Democritus made many journeys other than the one to Athens. Russell in [9] writes:-

He travelled widely in southern and eastern lands in search of knowledge, he perhaps spent a considerable time in Egypt, and he certainly visited Persia. He then returned to Abdera, where he remained.

Democritus himself wrote (but some historians dispute that the quote is authentic) (see [5]):-

Of all my contemporaries I have covered the most ground in my travels, making the most exhaustive inquiries the while; I have seen the most climates and countries and listened to the greatest number of learned men.

His travels certainly took him to Egypt and Persia, as Russell suggests, but he almost certainly also travelled to Babylon, and some claim he travelled to India and Ethiopia. Certainly he was a man of great learning. As Heath writes in [7]:-

... there was no subject to which he did not notably contribute, from mathematics and physics on the one hand to ethics and poetics on the other; he even went by the name of 'wisdom'.

Although little is known of his life, quite a lot is known of his physics and philosophy. There are two main sources for our knowledge of his of physical and philosophical theories. Firstly Aristotle discusses Democritus's ideas thoroughly because he strongly disagreed with his ideas of atomism. The second source is in the work of Epicurus but, in contrast to Aristotle, Epicurus is a strong believer in Democritus's atomic theory. This work of Epicurus is preserved by Diogenes Laertius in his second century AD book [5].

Certainly Democritus was not the first to propose an atomic theory. His teacher Leucippus had proposed an atomic system, as had Anaxagoras of Clazomenae. In fact traces of an atomic theory go back further than this, perhaps to the Pythagorean notion of the regular solids playing a fundamental role in the makeup of the universe. However Democritus produced a much more elaborate and systematic view of the physical world than had any of his predecessors. His view is summarised in [2]:-

Democritus asserted that space, or the Void, had an equal right with reality, or Being, to be considered existent. He conceived of the Void as a vacuum, an infinite space in which moved an infinite number of atoms that made up Being (i.e. the physical world). These atoms are eternal and invisible; absolutely small, so small that their size cannot be diminished (hence the name atomon, or "indivisible"); absolutely full and incompressible, as they are without pores and entirely fill the space they occupy; and homogeneous, differing only in shape, arrangement, position, and magnitude.

With this as a basis to the physical world, Democritus could explain all changes in the world as changes in motion of the atoms, or changes in the way that they were packed together. This was a remarkable theory which attempted to explain the whole of physics based on a small number of ideas and also brought mathematics into a fundamental physical role since the whole of the structure proposed by Democritus was quantitative and subject to mathematical laws. Another fundamental idea in Democritus's theory is that nature behaves like a machine, it is nothing more than a highly complex mechanism.

There are then questions for Democritus to answer. Where do qualities such as warmth, colour, and taste fit into the atomic theory? To Democritus atoms differ only in quantity, and all qualitative differences are only apparent and result from impressions of an observer caused by differing configurations of atoms. The properties of warmth, colour, taste are only by convention - the only things that actually exist are atoms and the Void.

Democritus's philosophy contains an early form of the conservation of energy. In his theory atoms are eternal and so is motion. Democritus explained the origin of the universe through atoms moving randomly and colliding to form larger bodies and worlds. There was no place in his theory for divine intervention. Instead he postulated a world which had always existed, and would always exist, and was filled with atoms moving randomly. Vortex motions occurred due to collisions of the atoms and in resulting vortex motion created differentiation of the atoms into different levels due only to their differing mass. This was not a world which came about through the design or purpose of some supernatural being, but rather it was a world which came about through necessity, that is from the nature of the atoms themselves.

Democritus built an ethical theory on top of his atomist philosophy. His system was purely deterministic so he could not admit freedom of choice to individuals. To Democritus freedom of choice was an illusion since we are unaware of all the causes for a decision. Democritus believed that [3]:-

... the soul will either be disturbed, so that its motion affects the body in a violent way, or it will be at rest in which case it regulates thoughts and actions harmoniously. Freedom from disturbance is the condition that causes human happiness, and this is the ethical goal.

Democritus describes the ultimate good, which he identifies with cheerfulness, as:-

... a state in which the soul lives peacefully and tranquilly, undisturbed by fear or superstition or any other feeling.

He wanted to remove the belief in gods which were, he believed, only introduced to explain phenomena for which no scientific explanation was then available.

Very little is known for certainty about Democritus's contributions to mathematics. As stated in the Oxford Classical Dictionary :-

Little is known (although much is written) about the mathematics of Democritus.

We do know that Democritus wrote many mathematical works. Diogenes Laertius (see [5]) lists his works and gives Thrasyllus as the source of this information. He wrote On numbers, On geometry, On tangencies, On mappings, On irrationals but none of these works survive. However we do know a little from other references. Heath [7] writes:-

In the Method of Archimedes, happily discovered in 1906, we are told that Democritus was the first to state the important propositions that the volume of a cone is one third of that of a cylinder having the same base and equal height, and that the volume of a pyramid is one third of that of a prism having the same base and equal height; that is to say, Democritus enunciated these propositions some fifty years or more before they were first scientifically proved by Eudoxus.

There is another intriguing piece of information about Democritus which is given by Plutarch in his Common notions against the Stoics where he reports on a dilemma proposed by Democritus as reported by the Stoic Chrysippus (see [7], [10] or [11]).

If a cone were cut by a plane parallel to the base [by which he means a plane indefinitely close to the base], what must we think of the surfaces forming the sections? Are they equal or unequal? For, if they are unequal, they will make the cone irregular as having many indentations, like steps, and unevennesses; but, if they are equal, the sections will be equal, and the cone will appear to have the property of the cylinder and to be made up of equal, not unequal, circles, which is very absurd.

There are important ideas in this dilemma. Firstly notice, as Heath points out in [7], that Democritus has the idea of a solid being the sum of infinitely many parallel planes and he may have used this idea to find the volumes of the cone and pyramid as reported by Archimedes. This idea of Democritus may have led Archimedes later to apply the same idea to great effect. This idea would eventually lead to theories of integration.

There is much discussion in [7], [8], [10] and [11] as to whether Democritus distinguished between the geometrical continuum and the physical discrete of his atomic system. Heath points out that if Democritus carried over his atomic theory to geometrical lines then there is no dilemma for him since his cone is indeed stepped with atom sized steps. Heath certainly believed that to Democritus lines were infinitely divisible. Others, see for example [10], have come to the opposite conclusion, believing that Democritus made contributions to problems of applied mathematics but, because of his atomic theory, he could not deal with the infinitesimal questions arising.

Hippias of Elis

2008-02-15T00:44:00.000-08:00

Born: about 460 BC in Elis, Peloponnese, Greece
Died: about 400 BC
Hippias of Elis was a statesman and philosopher who travelled from place to place taking money for his services. He lectured on poetry, grammar, history, politics, archaeology, mathematics and astronomy. Plato describes him as a vain man being both arrogant and boastful, having a wide but superficial knowledge. Heath tells us something of this character when he writes in [3]:-

He claimed ... to have gone once to the Olympian festival with everything that he wore made by himself, ring and sandal (engraved), oil-bottle, scraper, shoes, clothes, and a Persian girdle of expensive type; he also took poems, epics, tragedies, dithyrambs, and all sorts of prose works.

As to Hippias's academic achievements, Heath writes:-

He was a master of the science of calculation, geometry, astronomy, 'rhythms and harmonies and correct writing'. He also had a wonderful system of mnemonics enabling him, if he once heard a string of fifty names to remember them all.

A rather nice story, which says more of the Spartans than it does of Hippias, is that it was reported that he received no payment for the lectures he gave in Sparta since [3]:-

... the Spartans could not endure lectures on astronomy or geometry or calculation; it was only a small minority of them who could even count; what they liked was history and archaeology.

Since Hippias was reported to give lectures on archaeology, he seems to have chosen the wrong topics when he lectured in Sparta!

Hippias's only contribution to mathematics seems to be the quadratrix which may have been used by him for trisecting an angle and squaring the circle. The curve may be used for dividing an angle into any number of equal parts. Perhaps the highest compliment that we can pay to Hippias is to report on the arguments of certain historians of mathematics who have claimed that the Hippias who discovered the quadratrix cannot be Hippias of Elis since geometry was not far enough advanced at this time to have allowed him to make these discoveries. However, their arguments are not generally accepted and there is ample evidence to attribute the discovery of the quadratrix to Hippias of Elis.

Heath [3] writes:-

It was probably about 420 BC that Hippias of Elis invented the curve known as the quadratrix for the purpose of trisecting any angle.

However this is far from certain and there is some evidence to suggest that Geminus, writing in the first century BC, had in his possession a treatise by Hippias of Elis on the quadratrix which indicated how it could be used to square the circle. If this is indeed the case then the treatise by Hippias must have been lost between this time and that of Sporus in the third century AD.

Pappus wrote his major work on geometry Synagoge in 340. It is a collection of mathematical writings in eight books. Book IV contains a description of the quadratrix of Hippias.

Look at the diagram of the quadratrix.

ABCD is a square and BED is part of a circle, centre A radius AB. As the radius AB rotates about A to move to the position AD then the line BC moves at the same rate parallel to itself to end at AD. Then the locus of the point of intersection F of the rotating radius AB and the moving line BC is the quadratrix. Hence

angle EAD/angle BAD = arc ED/arc BED = FH/AB,

so, taking AB = 1,

angle EAD = arc ED = FH π/2.

To divide the angle FAD in a given ratio, say p : q, then draw a point P on the line FH dividing it in the ratio p : q.
Draw a line through P parallel to AD to meet the quadratrix at Q. Then AQ divides angle FAD in the ratio p : q.

Pappus also gives the rather more complicated version of the construction necessary to square the circle. However, Pappus reports that Sporus had two criticisms of Hippias's method with which he agrees. The second is specifically related to the construction necessary for squaring the circle which we have not described. The first however relates to the construction of the quadratrix itself. Pappus reports that Sporus writes (see [3]):-

The very thing for which the construction is thought to serve is actually assumed in the hypothesis. For how is it possible, with two points starting from B, to make one of them move along a straight line to A and the other along a circumference to D in an equal time, unless you first know the ratio of the straight line AB to the circumference BED? In fact this ratio must also be that of the speeds of motion. For, if you employ speeds not definitely adjusted to this ratio, how can you make the motions end at the same moment, unless this should sometime happen by pure chance? Is not the thing thus shown to be absurd?

The point here seems to be a question of what exactly Hippias is trying to show with his quadratrix. Certainly he knew perfectly well that he was not providing a ruler and compass construction for squaring the circle. Exactly what he has proved concerning squaring the circle is, as Pappus and Sporus suggest, far from clear.

Theodorus of Cyrene

2008-02-15T00:42:00.000-08:00

Born: 465 BC in Cyrene (now Shahhat, Libya)
Died: 398 BC in Cyrene (now Shahhat, Libya)
Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato, teaching him mathematics, and also the tutor of Theaetetus. Plato travelled to and from Egypt and on such occasions he spent time with Theodorus in Cyrene. Theodorus, however, did not spend his whole life in Cyrene for he was certainly in Athens at a time when Socrates was alive.

Theodorus, in addition to his work in mathematics, was [5]:-

... distinguished ... in astronomy, arithmetic, music and all educational subjects.

A member of the society of Pythagoras, Theodorus was one of the main philosophers in the Cyrenaic school of moral philosophy. He believed that pleasures and pains are neither good nor bad. Cheerfulness and wisdom, he believed, were sufficient for happiness.

Our knowledge of Theodorus comes through Plato who wrote about him in his work Theaetetus. Theodorus is remembered by mathematicians for his contribution to the development of irrational numbers and it is this aspect of his work which Plato describes (see for example [5]):-

[Theodorus] was proving to us a certain thing about square roots, I mean the side (i.e. root) of a square of three square units and of five square units, that these roots are not commensurable in length with the unit length, and he went on in this way, taking all the separate cases up to the root of seventeen square units, at which point, for some reason, he stopped.

Our whole knowledge of Theodorus's mathematical achievements are given by this passage from Plato. Yet there are points of interest which immediately arise. The first point is that Plato does not credit Theodorus with a proof that the square root of two was irrational. This must be because √2 was proved irrational before Theodorus worked on the problem, some claim this was proved by Pythagoras himself.

There is no doubt that Theodorus would have constructed lines of length √3, √5 etc. using Pythagoras's theorem. It is also clear that Theodorus had no general result here, for Plato goes on to describe how Theodorus's results inspired Theaetetus and Socrates to look at generalisations:-

The idea occurred to the two of us (Theaetetus and Socrates), seeing that these square roots appeared to be unlimited in multitude, to try to arrive at one collective term by which we could designate all these roots....

So the question which naturally comes next is how did Theodorus prove that √3, √5, ..., √17 were irrational without giving a proof which would clearly prove that any non-square number was irrational. The usual proof that √2 is irrational, namely the one which supposes that √2 = p/q where p/q is a rational in its lowest terms and derives a contradiction by showing that p and q are both even, would have been known to Theodorus. This proof generalises easily (for a modern mathematicians thinking in terms of numbers rather than lengths) to show √n is irrational for any non-square n. It is almost impossible to conceive that Theodorus would have used this proof on each of √3, √5, ..., √17 without obtaining a general theorem long before he got to 17.

An interesting proposal was made by Zeuthen in 1915. He suggested that Theodorus may have used the result which would later appear in Euclid's Elements namely:-

If, when the lesser of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incommensurable.

Heath [5] illustrates the use of this result to show that √5 is irrational. Start with 1 and √5.

√5/1 = 2 +(√5-2)
1/(√5-2) = 4 + (√5-2)2
(√5-2)/(√5-2)2= 1/(√5-2) = 4 + (√5-2)2
.......

The process now clearly fails to terminate since the ratio 1 : (√5-2) is the same as (√5-2) : (√5-2)2. Heath [5] gives a geometric version of this, starting with a right-angled triangle with sides 1, 2 and √5 which may be close to the method that Theodorus used. However there is little chance to do more than guess at Theodorus's method.

Hippocrates of Chios

2008-02-14T06:22:00.000-08:00

Born: about 470 BC in Chios (now Khios), Greece
Died: about 410 BC
Hippocrates of Chios taught in Athens and worked on the classical problems of squaring the circle and duplicating the cube. Little is known of his life but he is reported to have been an excellent geometer who, in other respects, was stupid and lacking in sense. Some claim that he was defrauded of a large sum of money because of his naiveté. Iamblichus [4] writes:-

One of the Pythagoreans [Hippocrates] lost his property, and when this misfortune befell him he was allowed to make money by teaching geometry.

Heath [6] recounts two versions of this story:-

One version of the story is that [Hippocrates] was a merchant, but lost all his property through being captured by a pirate vessel. He then came to Athens to persecute the offenders and, during a long stay, attended lectures, finally attaining such proficiency in geometry that he tried to square the circle.

Heath also recounts a different version of the story as told by Aristotle:-

... he allowed himself to be defrauded of a large sum by custom-house officers at Byzantium, thereby proving, in Aristotle's opinion, that, though a good geometer, he was stupid and incompetent in the business of ordinary life.

The suggestion is that this 'long stay' in Athens was between about 450 BC and 430 BC.

In his attempts to square the circle, Hippocrates was able to find the areas of lunes, certain crescent-shaped figures, using his theorem that the ratio of the areas of two circles is the same as the ratio of the squares of their radii. We describe this impressive achievement more fully below.

Hippocrates also showed that a cube can be doubled if two mean proportionals can be determined between a number and its double. This had a major influence on attempts to duplicate the cube, all efforts after this being directed towards the mean proportionals problem.

He was the first to write an Elements of Geometry and although his work is now lost it must have contained much of what Euclid later included in Books 1 and 2 of the Elements. Proclus, the last major Greek philosopher, who lived around 450 AD wrote:-

Hippocrates of Chios, the discoverer of the quadrature of the lune, ... was the first of whom it is recorded that he actually compiled "Elements".

Hippocrates' book also included geometrical solutions to quadratic equations and included early methods of integration.

Eudemus of Rhodes, who was a pupil of Aristotle, wrote History of Geometry in which he described the contribution of Hippocrates on lunes. This work has not survived but Simplicius of Cilicia, writing in around 530, had access to Eudemus's work and he quoted the passage about the lunes of Hippocrates 'word for word except for a few additions' taken from Euclid's Elements to make the description clearer.

We will first quote part of the passage of Eudemus about the lunes of Hippocrates, following the historians of mathematics who have disentangled the additions from Euclid's Elements which Simplicius added. See [6] both for the translation which we give and for a discussion of which parts are due to Eudemus:-

The quadratures of lunes, which were considered to belong to an uncommon class of propositions on account of the close relation of lunes to the circle, were first investigated by Hippocrates, and his exposition was thought to be correct; we will therefore deal with them at length and describe them. He started with, and laid down as the first of the theorems useful for the purpose, the proposition that similar segments of circles have the same ratio to one another as the squares on their bases. And this he proved by first showing that the squares on the diameters have the same ratio as the circles.

Before continuing with the quote we should note that Hippocrates is trying to 'square a lune' by which he means to construct a square equal in area to the lune. This is precisely what the problem of 'squaring the circle' means, namely to construct a square whose area is equal to the area of the circle. Again following Heath's translation in [6]:-

After proving this, he proceeded to show in what way it was possible to square a lune the outer circumference of which is that of a semicircle. This he affected by circumscribing a semicircle about an isosceles right-angled triangle and a segment of a circle similar to those cut off by the sides. Then, since the segment about the base is equal to the sum of those about the sides, it follows that, when the part of the triangle above the segment about the base is added to both alike, the lune will be equal to the triangle. Therefore the lune, having been proved equal to the triangle, can be squared.

To follow Hippocrates' argument here, look at the diagram.

ABCD is a square and O is its centre. The two circles in the diagram are the circle with centre O through A, B, C and D, and the circle with centre D through A and C.

Notice first that the segment marked 1 on AB subtends a right angle at the centre of the circle (the angle AOB) while the segment 2 on AC also subtends a right angle at the centre (the angle ADC).

Therefore the segment 1 on AB and the segment 2 on AC are similar. Now
segment 1/segment 2 = AB2/AC2 = 1/2 since AB2+ BC2= AC2 by Pythagoras's theorem, and AB = BC so AC2= 2AB2.

Now since segment 2 is twice segment 1, the segment 2 is equal to the sum of the two segments marked 1.

Then Hippocrates argues that the semicircle ABC with the two segments 1 removed is the triangle ABC which can be squared (it was well known how to construct a square equal to a triangle).

However, if we subtract the segment 2 from the semicircle ABC we get the lune shown in the second diagram. Thus Hippocrates has proved that the lune can be squared.

However, Hippocrates went further than this in studying lunes. The proof we have examined in detail is one where the outer circumference of the lune is the arc of a semicircle. He also studied the cases where the outer arc was less than that of a semicircle and also the case where the outer arc was greater than a semicircle, showing in each case that the lune could be squared. This was a remarkable achievement and a major step in attempts to square the circle. As Heath writes in [6]:-

... he wished to show that, if circles could not be squared by these methods, they could be employed to find the area of some figures bounded by arcs of circles, namely certain lunes, and even of the sum of a certain circle and a certain lune.

There is one further remarkable achievement which historians of mathematics believe that Hippocrates achieved, although we do not have a direct proof since his works have not survived. In Hippocrates' study of lunes, as described by Eudemus, he uses the theorem that circles are to one another as the squares on their diameters. This theorem is proved by Euclid in the Elements and it is proved there by the method of exhaustion due to Eudoxus. However, Eudoxus was born within a few years of the death of Hippocrates, and so there follows the intriguing question of how Hippocrates proved this theorem. Since Eudemus seems entirely satisfied that Hippocrates does indeed have a correct proof, it seems almost certain from this circumstantial evidence that we can deduce that Hippocrates himself developed at least a variant of the method of exhaustion.

Antiphon the Sophist

2008-02-14T06:21:00.000-08:00

Born: 480 BC in (possibly) Athens, Greece
Died: 411 BC in Athens, Greece
Antiphon was an orator and statesman who took up rhetoric as a profession. He was a Sophist and a contemporary of Socrates. These definite assertions are, however, disputed by some historians. The problem seems to revolve round whether there was one Sophist philosopher named Antiphon who lived around this time or whether there are two, or as some experts claim, three distinct Antiphons.

In what follows we shall assume that at least the orator named Antiphon was the same person as the Sophist who made the mathematical advances. This is the same line as taken in [1] while in [2] only Antiphon as an orator is discussed without reference to the philosophical or mathematical works. In [7] the hypothesis that Antiphon is one, or several different men is discussed without any definite view being preferred either way.

A number of speeches which were written by Antiphon have been preserved. Three of these speeches were real speeches made by Antiphon as the prosecutor in murder trials. Twelve speeches are specimen speeches written by Antiphon for use in teaching students the skills of prosecuting and defending clients in cases. The speeches come as three collections of four; two prosecution speeches and two defence speeches for each of three different cases.

Antiphon published a number of works on philosophy which have been lost except for a small number of fragments which have been discovered together with some quotations from the works in the writings of other authors. These works include On Truth, On Concord, The Statesman, and On Interpretation of Dreams. The work On Truth is written to support the views of Parmenides who believed that there was a single sole reality and that the apparent world of many things was unreal. In this work Antiphon is defending the same philosophical ideas which Zeno of Elea supported with his paradoxes.

In On Concord Antiphon [1]:-

... defends the authority of the community as a safeguard against anarchy and recommends the ideals of concord and self-restraint both within communities and within the individual soul. Most probably he was only concerned to criticise the laws of a city by asking whether or not they satisfy the "natural" needs of the individual.

Hobbs in [7] notes that:-

... some have doubted whether the same man could have written "On Truth" and the conventional gnomic utterances of "On Concord".

In [7] three reasons are given to support at least the same author for these two philosophical works:-

(1) "On Truth" is not as radical as it appears, but simply a plea for legal reform;

(2) its doctrines, although radical, are not endorsed by Antiphon;

(3) Antiphon changed his mind.

Leucippus of Miletus

2008-02-14T06:20:00.000-08:00

Born: about 480 BC in (possibly) Miletus, Asia Minor
Died: about 420 BC
Leucippus of Miletus carried on the scientific philosophy which had begun to become associated with Miletus. We know little of his life but it is thought that he founder the School at Abdera on the coast of Thrace near the mouth of the Nestos River. Today the town is in Greece and is called Avdhira. At the time that Leucippus would have lived in Abdera it was a prosperous town which politically was a member of the Delian League.

The philosopher Protagoras was born in Abdera and he was a contemporary of Leucippus bu

Oenopides of Chios

2008-02-14T06:19:00.000-08:00

Born: about 490 BC in Chios (now Khios), Greece
Died: about 420 BC
Very little is known about the life of Oenopides of Chios except that his place of birth was the island of Chios. We believe that Oenopides was in Athens when a young man but there is only circumstantial evidence for this.

In Plato's Erastae Oenopides is described as (see for example [1]):-

... having acquired a reputation for mathematics...

and Plato also describes a scene where Socrates comes across two young men in the school of Dionysius who was Plato's teacher. The young men were discussing a question in mathematical astronomy which had been tackled by Oenopides and Anaxagoras. This question was certainly that of the angle that the ecliptic makes with the celestial equator. Bulmer-Thomas writes in [1]:-

... it was probably [Oenopides] who settled on the value of 24, which was accepted in Greece until refined by Eratosthenes. Indeed, if Oenopides did not fix on this or some other figure it is difficult to know in what his achievement consisted, for the Babylonians no less than the Pythagoreans and Egyptians must have realised from early days that the apparent path of the sun was inclined to the celestial equator.

However, in contrast to these claims, Heath writes [2]:-

It does not appear that Oenopides made any measurement of the obliquity of the ecliptic.

Another major contribution to mathematical astronomy made by Oenopides was his discovery of the period of the Great Year. Originally the "Great Year" was the period after which the motions of the sun and moon came to repeat themselves. Later it came to mean the period after which the motions of the sun, moon and planets all repeated themselves so in the period of one Great Year all should have returned to their positions at the beginning of the Great Year.

Oenopides gave a value of the Great Year as 59 years. Heath writes [2]:-

His Great Year clearly had reference to the sun and moon only; he merely sought to find the least integral number of complete years which would contain an exact number of lunar months.

Based on this Paul Tannery [5] showed that Oenopides' result leads to a lunar month of 29.53013 days which is remarkably close to the modern value of 29.53059 days. However many historians doubt whether Oenopides could have collected sufficient good quality data to enable him to obtain a value as accurate as this. To collect the data for even one period requires 59 years and this makes it almost impossible for someone to gather the data in their own lifetime.

Toomer believed that in fact despite Oenopides' Great Year of 59 years, he did not have this accurate value for the length of the month, and later calculations were made using better data than would have been available to Oenopides to give this very accurate value for the length of the month, more accurate than Oenopides could ever have known.

Paul Tannery in [5] makes another claim however when he states that Oenopides considered some of the planets as well as the sun and moon as part of his 59 year Great Year. The data works well for some of the planets, for example Saturn is only 2 from its starting position at the end of the 59 year cycle. Tannery is forced to conclude that not all the planets could have been taken into account by Oenopides, however, as some of the planets would be in the wrong sign of the Zodiac after the period ended.

Proclus attributes two theorems which appear in Euclid's Elements to Oenopides. These are to draw a perpendicular to a line from a given point not on the line, and to construct on a line from a given point a line at a given angle to the first line. These are elementary results but Heath believes that their significance might be that Oenopides set out for the first time the explicit 'ruler and compass' type of allowable construction. He writes [2]:-

... [Oenopides] may have been the first to lay down the restriction of the means permissible in constructions t the ruler and compasses which became a canon of Greek geometry for all plane constructions...

Oenopides also developed a theory to account for the Nile floods. He suggested that heat stored in the ground during the winter dries up the underground water so that the river shrinks. In the summer the heat disappears, as testing the temperature of deep wells suggests, and water flows up into the river so causing floods. This theory, which of course is false, did not prove popular as other rivers in Libya were subject to similar conditions but did not behave in the same way.

We have some other indications of the philosophy of Oenopides. He is said to have believed in fire and air as basic elements and thought of the world as a living being with God as its soul.

Zeno of Elea

2008-02-14T06:14:00.000-08:00

Born: about 490 BC in Elea, Lucania (now southern Italy)
Died: about 425 BC in Elea, Lucania (now southern Italy)

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Very little is known of the life of Zeno of Elea. We certainly know that he was a philosopher, and he is said to have been the son of Teleutagoras. The main source of our knowledge of Zeno comes from the dialogue Parmenides written by Plato.

Zeno was a pupil and friend of the philosopher Parmenides and studied with him in Elea. The Eleatic School, one of the leading pre-Socratic schools of Greek philosophy, had been founded by Parmenides in Elea in southern Italy. His philosophy of monism claimed that the many things which appear to exist are merely a single eternal reality which he called Being. His principle was that "all is one" and that change or non-Being are impossible. Certainly Zeno was greatly influenced by the arguments of Parmenides and Plato tells us that the two philosophers visited Athens together in around 450 BC.

Despite Plato's description of the visit of Zeno and Parmenides to Athens, it is far from universally accepted that the visit did indeed take place. However, Plato tells us that Socrates, who was then young, met Zeno and Parmenides on their visit to Athens and discussed philosophy with them. Given the best estimates of the dates of birth of these three philosophers, Socrates would be about 20, Zeno about 40, and Parmenides about 65 years of age at the time, so Plato's claim is certainly possible.

Zeno had already written a work on philosophy before his visit to Athens and Plato reports that Zeno's book meant that he had achieved a certain fame in Athens before his visit there. Unfortunately no work by Zeno has survived, but there is very little evidence to suggest that he wrote more than one book. The book Zeno wrote before his visit to Athens was his famous work which, according to Proclus, contained forty paradoxes concerning the continuum. Four of the paradoxes, which we shall discuss in detail below, were to have a profound influence on the development of mathematics.

Diogenes Laertius [10] gives further details of Zeno's life which are generally thought to be unreliable. Zeno returned to Elea after the visit to Athens and Diogenes Laertius claims that he met his death in a heroic attempt to remove a tyrant from the city of Elea. The stories of his heroic deeds and torture at the hands of the tyrant may well be pure inventions. Diogenes Laertius also writes about Zeno's cosmology and again there is no supporting evidence regarding this, but we shall give some indication below of the details.

Zeno's book of forty paradoxes was, according to Plato [8]:-

... a youthful effort, and it was stolen by someone, so that the author had no opportunity of considering whether to publish it or not. Its object was to defend the system of Parmenides by attacking the common conceptions of things.

Proclus also described the work and confirms that [1]:-

... Zeno elaborated forty different paradoxes following from the assumption of plurality and motion, all of them apparently based on the difficulties deriving from an analysis of the continuum.

In his arguments against the idea that the world contains more than one thing, Zeno derived his paradoxes from the assumption that if a magnitude can be divided then it can be divided infinitely often. Zeno also assumes that a thing which has no magnitude cannot exist. Simplicius, the last head of Plato's Academy in Athens, preserved many fragments of earlier authors including Parmenides and Zeno. Writing in the first half of the sixth century he explained Zeno's argument why something without magnitude could not exist [1]:-

For if it is added to something else, it will not make it bigger, and if it is subtracted, it will not make it smaller. But if it does not make a thing bigger when added to it nor smaller when subtracted from it, then it appears obvious that what was added or subtracted was nothing.

Although Zeno's argument is not totally convincing at least, as Makin writes in [25]:-

Zeno's challenge to simple pluralism is successful, in that he forces anti-Parmenideans to go beyond common sense.

The paradoxes that Zeno gave regarding motion are more perplexing. Aristotle, in his work Physics, gives four of Zeno's arguments, The Dichotomy, The Achilles, The Arrow, and The Stadium. For the dichotomy, Aristotle describes Zeno's argument (in Heath's translation [8]):-

There is no motion because that which is moved must arrive at the middle of its course before it arrives at the end.

In order the traverse a line segment it is necessary to reach its midpoint. To do this one must reach the 1/4 point, to do this one must reach the 1/8 point and so on ad infinitum. Hence motion can never begin. The argument here is not answered by the well known infinite sum

1/2 + 1/4 + 1/8 + ... = 1

On the one hand Zeno can argue that the sum 1/2 + 1/4 + 1/8 + ... never actually reaches 1, but more perplexing to the human mind is the attempts to sum 1/2 + 1/4 + 1/8 + ... backwards. Before traversing a unit distance we must get to the middle, but before getting to the middle we must get 1/4 of the way, but before we get 1/4 of the way we must reach 1/8 of the way etc. This argument makes us realise that we can never get started since we are trying to build up this infinite sum from the "wrong" end. Indeed this is a clever argument which still puzzles the human mind today.

Zeno bases both the dichotomy paradox and the attack on simple pluralism on the fact that once a thing is divisible, then it is infinitely divisible. One could counter his paradoxes by postulating an atomic theory in which matter was composed of many small indivisible elements. However other paradoxes given by Zeno cause problems precisely because in these cases he considers that seemingly continuous magnitudes are made up of indivisible elements. Such a paradox is 'The Arrow' and again we give Aristotle's description of Zeno's argument (in Heath's translation [8]):-

If, says Zeno, everything is either at rest or moving when it occupies a space equal to itself, while the object moved is in the instant, the moving arrow is unmoved.

The argument rests on the fact that if in an indivisible instant of time the arrow moved, then indeed this instant of time would be divisible (for example in a smaller 'instant' of time the arrow would have moved half the distance). Aristotle argues against the paradox by claiming:-

... for time is not composed of indivisible 'nows', no more than is any other magnitude.

However, this is considered by some to be irrelevant to Zeno's argument. Moreover to deny that 'now' exists as an instant which divides the past from the future seems also to go against intuition. Of course if the instant 'now' does not exist then the arrow never occupies any particular position and this does not seem right either. Again Zeno has presented a deep problem which, despite centuries of efforts to resolve it, still seems to lack a truly satisfactory solution. As Frankel writes in [20]:-

The human mind, when trying to give itself an accurate account of motion, finds itself confronted with two aspects of the phenomenon. Both are inevitable but at the same time they are mutually exclusive. Either we look at the continuous flow of motion; then it will be impossible for us to think of the object in any particular position. Or we think of the object as occupying any of the positions through which its course is leading it; and while fixing our thought on that particular position we cannot help fixing the object itself and putting it at rest for one short instant.

Vlastos (see [32]) points out that if we use the standard mathematical formula for velocity we have v = s/t, where s is the distance travelled and t is the time taken. If we look at the velocity at an instant we obtain v = 0/0, which is meaningless. So it is fair to say that Zeno here is pointing out a mathematical difficulty which would not be tackled properly until limits and the differential calculus were studied and put on a proper footing.

As can be seen from the above discussion, Zeno's paradoxes are important in the development of the notion of infinitesimals. In fact some authors claim that Zeno directed his paradoxes against those who were introducing infinitesimals. Anaxagoras and the followers of Pythagoras, with their development of incommensurables, are also thought by some to be the targets of Zeno's arguments (see for example [10]). Certainly it appears unlikely that the reason given by Plato, namely to defend Parmenides' philosophical position, is the whole explanation of why Zeno wrote his famous work on paradoxes.

The most famous of Zeno's arguments is undoubtedly the Achilles. Heath's translation from Aristotle's Physics is:-

... the slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead.

Most authors, starting with Aristotle, see this paradox to be essentially the same as the Dichotomy. For example Makin [25] writes:-

... as long as the Dichotomy can be resolved, the Achilles can be resolved. The resolutions will be parallel.

As with most statements about Zeno's paradoxes, there is not complete agreement about any particular position. For example Toth [29] disputes the similarity of the two paradoxes, claiming that Aristotle's remarks leave much to be desired and suggests that the two arguments have entirely different structures.

Both Plato and Aristotle did not fully appreciate the significance of Zeno's arguments. As Heath says [8]:-

Aristotle called them 'fallacies', without being able to refute them.

Russell certainly did not underrate Zeno's significance when he wrote in [13]:-

In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance ....

Here Russell is thinking of the work of Cantor, Frege and himself on the infinite and particularly of Weierstrass on the calculus. In [2] the relation of the paradoxes to mathematics is also discussed, and the author comes to a conclusion similar to Frankel in the above quote:-

Although they have often been dismissed as logical nonsense, many attempts have also been made to dispose of them by means of mathematical theorems, such as the theory of convergent series or the theory of sets. In the end, however, the difficulties inherent in his arguments have always come back with a vengeance, for the human mind is so constructed that it can look at a continuum in two ways that are not quite reconcilable.

It is difficult to tell precisely what effect the paradoxes of Zeno had on the development of Greek mathematics. B L van der Waerden (see [31]) argues that the mathematical theories which were developed in the second half of the fifth century BC suggest that Zeno's work had little influence. Heath however seems to detect a greater influence [8]:-

Mathematicians, however, ... realising that Zeno's arguments were fatal to infinitesimals, saw that they could only avoid the difficulties connected with them by once and for all banishing the idea of the infinite, even the potentially infinite, altogether from their science; thenceforth, therefore, they made no use of magnitudes increasing or decreasing ad infinitum, but contented themselves with finite magnitudes that can be made as great or as small as we please.

We commented above that Diogenes Laertius in [10] describes a cosmology that he believes is due to Zeno. According to his description, Zeno proposed a universe consisting of several worlds, composed of "warm" and "cold, "dry" and "wet" but no void or empty space. Because this appears to have nothing in common with his paradoxes, it is usual to take the line that Diogenes Laertius is in error. However, there is some evidence that this type of belief was around in the fifth century BC, particularly associated with medical theory, and it could easily have been Zeno's version of a belief held by the Eleatic School.

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List of References (35 books/articles)

A Poster of Zeno of Elea Mathematicians born in the same country

Cross-references in MacTutor

History Topics: An overview of the history of mathematics
History Topics: The rise of the calculus
History Topics: Infinity
History Topics: A history of time: Classical time
Chronology: 500BC to 1AD

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Other Web sites
Astroseti (A Spanish translation of this biography)
The Catholic Encyclopedia
Kevin Brown
The big view
Internet Encyclopedia of Philosophy
S M Cohen (Zeno's paradoxes)
Stanford Encyclopedia of Philosophy (Zeno's paradoxes)

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Empedocles of Acragas

2008-02-14T06:10:00.000-08:00

Born: about 492 BC in Acragas (now Agrigento, Sicily,Italy)
Died: about 432 BC in Peloponnese, Greece
Empedocles was born in Acragas on the south coast of Sicily. The name Acragas is Greek, while the Latin name for the town was Agrigentum. Later the town was called Girgenti and more recently it became known by its present name of Agrigento. It was one of the most beautiful cities of the ancient world up to the time it was destroyed by the Carthaginians in 406 BC. It was, in Empedocles time, a rich city containing the finest Greek culture. Some of the Pythagoreans had come there after being attacked in their centre at Croton.

Empedocles was born into a rich aristocratic family. He travelled throughout the Greek world participating fully in the extraordinary desire for learning and understanding which gripped that part of the world. He is described as follows by Sarton [5]:-

He was not only a philosopher but a poet, a seer, a physicist, a social reformer, a man of so much enthusiasm that he would easily be considered a charlatan by some people, or become a legendary hero in the eyes of others.

There are many legends regarding Empedocles life. He wrote poetry and 450 lines of such had been preserved by later writers such as Simplicius, Aristotle, Plutarch and others. It is not difficult to see the source of most of the legends about Empedocles for these are built on the poems that he wrote himself. In these he claims god-like powers, but whether this was simply a poetic style or whether he really did believe that he had such powers it is hard to say. Certainly his poems were much appreciated, for example Lucretius admired his hexametric poetry.

If we are to gather anything about the character of the man then it will come from the lines of poetry which have been preserved: 400 lines from his poem Peri physeos (On Nature) and the remainder from his poem Katharmoi (Purifications). These [1]:-

... reveal a man of fervid imagination, versatility, and eloquence, with a touch of theatricality.

Some details of his travels appear accurate. He went to Italy and was in the town of Thurii, Lucania shortly after 445 BC. From there he went to the Peloponnese and he was in Olympia in 440 BC. His songs were sung at the Olympic games in that year. He had a young friend, Pausanias the son of Anchitos, who went with him on his travels. Of the many legends regarding his death, the most likely would appear to be that he died following a feast in the Peloponnese. Sarton writes [5]:-

Empedocles was so great and rare a man that he left no school; none of his admirers or disciples, not even the faithful Pausanias, was able to continue the master's work.

Certainly Empedocles was attributed with many "firsts". Aristotle is said to have considered him the inventor of rhetoric while Galen regarded him as the founder of the science of medicine in Italy. He is best known, however, for his belief that all matter was composed of four elements: fire, air, water, and earth.

The reason for his four element theory was to argue a modification of the belief of the Eleatic School, one of the leading pre-Socratic schools of Greek philosophy, which had been founded by Parmenides in Elea in southern Italy. The philosophy of this school, which included Zeno of Elea, was the claim that the many things which appear to exist are merely a single eternal reality. Empedocles did not go for the "all is one" version, but his "all is composed of the four elements" is extremely important in the development of science since it was adopted by Plato and Aristotle. As Sarton writes [5]:-

In spite of its arbitrariness, that hypothesis had a singular fortune, for it dominated Western thought in one form or another almost until the eighteenth century.

We should also note an important feature of the hypothesis. It, like the ideas of Pythagoras, tried to explain the multitude of complexity seen in the world as being the consequence of a small number of simple underlying properties. Although we no longer believe in Empedocles' four element theory, we do still look for simple mathematics which will explain the complex phenomena that surround us.

Empedocles did not base his four element hypothesis on any experimental evidence. He did base some other scientific ideas on experiment, however, and he showed by experiment that air existed and was not empty space. He did this with a clepsydra, a vessel with a hole in the bottom and one in the top. Placing the bottom hole of the vessel under water, Empedocles observed that the vessel filled up with water. If, however, he put his finger over the top hole, then the water did not enter the hole at the bottom but it did once he removed his finger. Empedocles correctly deduced that the air in the container prevented the water entering.

Empedocles believed that light travelled with a finite velocity, not through any experimental evidence, of course, but simply through reasoning. Aristotle writes in De sensu :-

Empedocles says that the light from the Sun arrives first in the intervening space before it comes to the eye, or reaches the Earth. This might plausibly seem to be the case. For whatever is moved through space, is moved from one place to another; hence, there must be a corresponding interval of time also in which it is moved from the one place to the other. But any given time is divisible into parts; so that we should assume a time when the sun's ray was not as yet seen, but was still travelling in the middle space.

It is remarkable how many of Empedocles' ideas have turned out to be correct. In addition to his belief in the finite velocity of light he also developed a crude evolutionary theory based on the survival of the fittest. He also had a form of the law of conservation of energy and had a theory of constant proportions in chemical reactions. His ideas, although they had little influence on the development of science, can be seen in the light of our current scientific knowledge to be quite incredible. If we have to explain how such prophetically correct ideas could have such little influence we have to agree with the philosopher Hans Reichenbach who, in a book published in 1957, said (see [1]):-

... a good idea stated within an insufficient theoretical frame loses its explanatory power and is forgotten.

Anaxagoras of Clazomenae

2008-01-23T00:52:00.000-08:00

Born: 499 BC in Clazomenae (30 km west of Izmir), Lydia (now Turkey)
Died: 428 BC in Lampsacus, Mysia (now Turkey)
Anaxagoras of Clazomenae was described by Proclus, the last major Greek philosopher, who lived around 450 AD as (see for example [4]):-

After [Pythagoras] Anaxagoras of Clazomenae dealt with many questions in geometry...

Anaxagoras was an Ionian, born in the neighbourhood of Smyrna in what today is Turkey. We know few details of his early life, but certainly he lived the first part of his life in Ionia where he learnt about the new studies that were taking place there in philosophy and the new found enthusiasm for a scientific study of the world. He came from a rich family but he gave up his wealth. As Heath writes in [4]:-

He neglected his possessions, which were considerable, in order to devote himself to science.

Although Ionia had produced philosophers such as Pythagoras, up to the time of Anaxagoras this new study of knowledge had not spread to Athens. Anaxagoras is famed as the first to introduce philosophy to the Athenians when he moved there in about 480 BC. During Anaxagoras's stay in Athens, Pericles rose to power. Pericles, who was about five years younger than Anaxagoras, was a military and political leader who was successful in both developing democracy and building an empire which made Athens the political and cultural centre of Greece. Anaxagoras and Pericles became friends but this friendship had its drawbacks since Pericles' political opponents also set themselves against Anaxagoras.

In about 450 BC Anaxagoras was imprisoned for claiming that the Sun was not a god and that the Moon reflected the Sun's light. This seems to have been instigated by opponents of Pericles. Russell in [6] writes:-

The citizens of Athens ... passed a law permitting impeachment of those who did not practice religion and taught theories about 'the things on high'. Under this law they persecuted Anaxagoras, who was accused of teaching that the sun was a red-hot stone and the moon was earth.

We should examine this teaching of Anaxagoras about the sun more closely for, although it was used as a reason to put him in prison, it is a most remarkable teaching. It was based on his doctrine of "nous" which is translated as "mind" or "reason". Initially "all things were together" and matter was some homogeneous mixture. The nous set up a vortex in this mixture. The rotation [4]:-

... began in the centre and then gradually spread, taking in wider and wider circles. The first effect was to separate two great masses, one consisting of the rare, hot, dry, called the "aether", the other of the opposite categories and called "air". The aether took the outer, the air the inner place. From the air were next separated clouds, water, earth and stones. The dense, the moist, the dark and cold, and all the heaviest things, collected in the centre as a result of the circular motion, and it was from these elements when consolidated that the earth was formed; but after this, in consequence of the violence of the whirling motion, the surrounding fiery aether tore stones away from the earth and kindled them into stars.

There are remarkable insights in this description. The idea of differentiation of matter which plays a large role in modern theories of creation of the solar system is present. Anaxagoras also shows an understanding of centrifugal force which again shows the major scientific insights that he possessed.

Anaxagoras proposed that the moon shines by reflected light from the "red-hot stone" which was the sun, the first such recorded claim. Showing great genius he was also then able to take the next step and become the first to explain correctly the reason for eclipses of the sun and moon. His explanation of eclipses of the sun is completely correct but he did spoil his explanation of eclipses of the moon by proposing that in addition to being caused by the shadow of the earth, there were other dark bodies between the earth and the moon which also caused eclipses of the moon. It is a little unclear why he felt it necessary to postulate the existence of these bodies but it does not detract from this major breakthrough in mathematical astronomy. There is also other evidence to suggest that Anaxagoras had applied geometry to the study of astronomy.

As to the structure of matter, Anaxagoras postulated an infinite number of elements, or basic building blocks. He claimed:-

... there is a portion of every thing, i.e. of every elemental stuff, in every thing...[but] each is and was most manifestly those things of which there is most in it.

However, it was the power of nous, or mind, that not only created the world but also was the driving force in its day to day processes. For example [2]:-

The growth of living things, according to Anaxagoras, depends on the power of mind within the organisms that enables them to extract nourishment from surrounding substances.

Aristotle both found much to praise in Anaxagoras's theory of nous. Both Plato and Aristotle, however, were critical of the fact that the driving force of the nous as proposed by Anaxagoras was not ethical. They wanted nous to always act in the best interests of the world. In fact the nous of Anaxagoras does provide a mechanical explanation of the world after the non-mechanical start when the vortex is produced. It is worth noting that Newton's mechanical universe would have more in common with Anaxagoras's views than the continuing ethical intelligence proposed by Plato and Aristotle.

We can obtain some clues to the mathematics that Anaxagoras studied but, unfortunately, very little remains in the records to allow us to know of definite results which he may have proved. While in prison he tried to solve the problem of squaring the circle, that is constructing with ruler and compasses a square with area equal to that of a given circle. This is the first record of this problem being studied and this problem, and other similar problems, were to play a major role in the development of Greek mathematics.

One other intriguing piece of information comes from the writing of Vitruvius, a Roman architect, engineer, and author who lived in the first century BC. He records information about the painting of stage scenes for the plays which were performed in Athens and says that Anaxagoras wrote a treatise on how to paint scenes so that some objects appeared to be in the foreground while other appeared in the background. This fascinating comment must mean that Anaxagoras wrote a treatise on perspective, but sadly no such work survives.

Anaxagoras was saved from prison by Pericles but had to leave Athens. He returned to Ionia where he founded a school at Lampsacus. This Greek city on the Asiatic shore of the Hellespont was the place for the worship of Priapus, a god of procreation and fertility. Anaxagoras died there and the anniversary of his death became a holiday for schoolchildren.

The best that we can hope to learn of Anaxagoras's personality is from the story that when once asked what as the point of being born he replied [4]:-

The investigation of sun. moon, and heaven.

Even if this story is fictitious, it is likely to be based on the way that Anaxagoras lived his life and so tells us something of the personality of this remarkable scientist who gave a description of the creation of the solar system that took 2000 years to improve upon.

Panini

2008-01-23T00:51:00.000-08:00

Born: about 520 BC in Shalatula (near Attock), now Pakistan
Died: about 460 BC in India
Panini was born in Shalatula, a town near to Attock on the Indus river in present day Pakistan. The dates given for Panini are pure guesses. Experts give dates in the 4th, 5th, 6th and 7th century BC and there is also no agreement among historians about the extent of the work which he undertook. What is in little doubt is that, given the period in which he worked, he is one of the most innovative people in the whole development of knowledge. We will say a little more below about how historians have gone about trying to pinpoint the date when Panini lived.

Panini was a Sanskrit grammarian who gave a comprehensive and scientific theory of phonetics, phonology, and morphology. Sanskrit was the classical literary language of the Indian Hindus and Panini is considered the founder of the language and literature. It is interesting to note that the word "Sanskrit" means "complete" or "perfect" and it was thought of as the divine language, or language of the gods.

A treatise called Astadhyayi (or Astaka ) is Panini's major work. It consists of eight chapters, each subdivided into quarter chapters. In this work Panini distinguishes between the language of sacred texts and the usual language of communication. Panini gives formal production rules and definitions to describe Sanskrit grammar. Starting with about 1700 basic elements like nouns, verbs, vowels, consonants he put them into classes. The construction of sentences, compound nouns etc. is explained as ordered rules operating on underlying structures in a manner similar to modern theory. In many ways Panini's constructions are similar to the way that a mathematical function is defined today. Joseph writes in [2]:-

[Sanskrit's] potential for scientific use was greatly enhanced as a result of the thorough systemisation of its grammar by Panini. ... On the basis of just under 4000 sutras [rules expressed as aphorisms], he built virtually the whole structure of the Sanskrit language, whose general 'shape' hardly changed for the next two thousand years. ... An indirect consequence of Panini's efforts to increase the linguistic facility of Sanskrit soon became apparent in the character of scientific and mathematical literature. This may be brought out by comparing the grammar of Sanskrit with the geometry of Euclid - a particularly apposite comparison since, whereas mathematics grew out of philosophy in ancient Greece, it was ... partly an outcome of linguistic developments in India.

Joseph goes on to make a convincing argument for the algebraic nature of Indian mathematics arising as a consequence of the structure of the Sanskrit language. In particular he suggests that algebraic reasoning, the Indian way of representing numbers by words, and ultimately the development of modern number systems in India, are linked through the structure of language.

Panini should be thought of as the forerunner of the modern formal language theory used to specify computer languages. The Backus Normal Form was discovered independently by John Backus in 1959, but Panini's notation is equivalent in its power to that of Backus and has many similar properties. It is remarkable to think that concepts which are fundamental to today's theoretical computer science should have their origin with an Indian genius around 2500 years ago.

At the beginning of this article we mentioned that certain concepts had been attributed to Panini by certain historians which others dispute. One such theory was put forward by B Indraji in 1876. He claimed that the Brahmi numerals developed out of using letters or syllables as numerals. Then he put the finishing touches to the theory by suggesting that Panini in the eighth century BC (earlier than most historians place Panini) was the first to come up with the idea of using letters of the alphabet to represent numbers.

There are a number of pieces of evidence to support Indraji's theory that the Brahmi numerals developed from letters or syllables. However it is not totally convincing since, to quote one example, the symbols for 1, 2 and 3 clearly do not come from letters but from one, two and three lines respectively. Even if one accepts the link between the numerals and the letters, making Panini the originator of this idea would seem to have no more behind it than knowing that Panini was one of the most innovative geniuses that world has known so it is not unreasonable to believe that he might have made this step too.

There are other works which are closely associated with the Astadhyayi which some historians attribute to Panini, others attribute to authors before Panini, others attribute to authors after Panini. This is an area where there are many theories but few, if any, hard facts.

We also promised to return to a discussion of Panini's dates. There has been no lack of work on this topic so the fact that there are theories which span several hundreds of years is not the result of lack of effort, rather an indication of the difficulty of the topic. The usual way to date such texts would be to examine which authors are referred to and which authors refer to the work. One can use this technique and see who Panini mentions.

There are ten scholars mentioned by Panini and we must assume from the context that these ten have all contributed to the study of Sanskrit grammar. This in itself, of course, indicates that Panini was not a solitary genius but, like Newton, had "stood on the shoulders of giants". Panini must have lived later than these ten but this is absolutely no help in providing dates since we have absolutely no knowledge of when any of these ten lived.

What other internal evidence is there to use? Well of course Panini uses many phrases to illustrate his grammar any these have been examined meticulously to see if anything is contained there to indicate a date. To give an example of what we mean: if we were to pick up a text which contained as an example "I take the train to work every day" we would know that it had to have been written after railways became common. Let us illustrate with two actual examples from the Astadhyayi which have been the subject of much study. The first is an attempt to see whether there is evidence of Greek influence. Would it be possible to find evidence which would mean that the text had to have been written after the conquests of Alexander the Great? There is a little evidence of Greek influence, but there was Greek influence on this north east part of the Indian subcontinent before the time of Alexander. Nothing conclusive has been identified.

Another angle is to examine a reference Panini makes to nuns. Some argue that these must be Buddhist nuns and therefore the work must have been written after Buddha. A nice argument but there is a counter argument which says that there were Jaina nuns before the time of Buddha and Panini's reference could equally well be to them. Again the evidence is inconclusive.

There are references by others to Panini. However it would appear that the Panini to whom most refer is a poet and although some argue that these are the same person, most historians agree that the linguist and the poet are two different people. Again this is inconclusive evidence.

Let us end with an evaluation of Panini's contribution by Cardona in [1]:-

Panini's grammar has been evaluated from various points of view. After all these different evaluations, I think that the grammar merits asserting ... that it is one of the greatest monuments of human intelligence.

Pythagoras of Samos

2008-01-23T00:49:00.000-08:00

Born: about 569 BC in Samos, Ionia
Died: about 475 BC
Pythagoras of Samos is often described as the first pure mathematician. He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements. Unlike many later Greek mathematicians, where at least we have some of the books which they wrote, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure.

We do have details of Pythagoras's life from early biographies which use important original sources yet are written by authors who attribute divine powers to him, and whose aim was to present him as a god-like figure. What we present below is an attempt to collect together the most reliable sources to reconstruct an account of Pythagoras's life. There is fairly good agreement on the main events of his life but most of the dates are disputed with different scholars giving dates which differ by 20 years. Some historians treat all this information as merely legends but, even if the reader treats it in this way, being such an early record it is of historical importance.

Pythagoras's father was Mnesarchus ([12] and [13]), while his mother was Pythais [8] and she was a native of Samos. Mnesarchus was a merchant who came from Tyre, and there is a story ([12] and [13]) that he brought corn to Samos at a time of famine and was granted citizenship of Samos as a mark of gratitude. As a child Pythagoras spent his early years in Samos but travelled widely with his father. There are accounts of Mnesarchus returning to Tyre with Pythagoras and that he was taught there by the Chaldaeans and the learned men of Syria. It seems that he also visited Italy with his father.

Little is known of Pythagoras's childhood. All accounts of his physical appearance are likely to be fictitious except the description of a striking birthmark which Pythagoras had on his thigh. It is probable that he had two brothers although some sources say that he had three. Certainly he was well educated, learning to play the lyre, learning poetry and to recite Homer. There were, among his teachers, three philosophers who were to influence Pythagoras while he was a young man. One of the most important was Pherekydes who many describe as the teacher of Pythagoras.

The other two philosophers who were to influence Pythagoras, and to introduce him to mathematical ideas, were Thales and his pupil Anaximander who both lived on Miletus. In [8] it is said that Pythagoras visited Thales in Miletus when he was between 18 and 20 years old. By this time Thales was an old man and, although he created a strong impression on Pythagoras, he probably did not teach him a great deal. However he did contribute to Pythagoras's interest in mathematics and astronomy, and advised him to travel to Egypt to learn more of these subjects. Thales's pupil, Anaximander, lectured on Miletus and Pythagoras attended these lectures. Anaximander certainly was interested in geometry and cosmology and many of his ideas would influence Pythagoras's own views.

In about 535 BC Pythagoras went to Egypt. This happened a few years after the tyrant Polycrates seized control of the city of Samos. There is some evidence to suggest that Pythagoras and Polycrates were friendly at first and it is claimed [5] that Pythagoras went to Egypt with a letter of introduction written by Polycrates. In fact Polycrates had an alliance with Egypt and there were therefore strong links between Samos and Egypt at this time. The accounts of Pythagoras's time in Egypt suggest that he visited many of the temples and took part in many discussions with the priests. According to Porphyry ([12] and [13]) Pythagoras was refused admission to all the temples except the one at Diospolis where he was accepted into the priesthood after completing the rites necessary for admission.

It is not difficult to relate many of Pythagoras's beliefs, ones he would later impose on the society that he set up in Italy, to the customs that he came across in Egypt. For example the secrecy of the Egyptian priests, their refusal to eat beans, their refusal to wear even cloths made from animal skins, and their striving for purity were all customs that Pythagoras would later adopt. Porphyry in [12] and [13] says that Pythagoras learnt geometry from the Egyptians but it is likely that he was already acquainted with geometry, certainly after teachings from Thales and Anaximander.

In 525 BC Cambyses II, the king of Persia, invaded Egypt. Polycrates abandoned his alliance with Egypt and sent 40 ships to join the Persian fleet against the Egyptians. After Cambyses had won the Battle of Pelusium in the Nile Delta and had captured Heliopolis and Memphis, Egyptian resistance collapsed. Pythagoras was taken prisoner and taken to Babylon. Iamblichus writes that Pythagoras (see [8]):-

... was transported by the followers of Cambyses as a prisoner of war. Whilst he was there he gladly associated with the Magoi ... and was instructed in their sacred rites and learnt about a very mystical worship of the gods. He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians...

In about 520 BC Pythagoras left Babylon and returned to Samos. Polycrates had been killed in about 522 BC and Cambyses died in the summer of 522 BC, either by committing suicide or as the result of an accident. The deaths of these rulers may have been a factor in Pythagoras's return to Samos but it is nowhere explained how Pythagoras obtained his freedom. Darius of Persia had taken control of Samos after Polycrates' death and he would have controlled the island on Pythagoras's return. This conflicts with the accounts of Porphyry and Diogenes Laertius who state that Polycrates was still in control of Samos when Pythagoras returned there.

Pythagoras made a journey to Crete shortly after his return to Samos to study the system of laws there. Back in Samos he founded a school which was called the semicircle. Iamblichus [8] writes in the third century AD that:-

... he formed a school in the city [of Samos], the 'semicircle' of Pythagoras, which is known by that name even today, in which the Samians hold political meetings. They do this because they think one should discuss questions about goodness, justice and expediency in this place which was founded by the man who made all these subjects his business. Outside the city he made a cave the private site of his own philosophical teaching, spending most of the night and daytime there and doing research into the uses of mathematics...

Pythagoras left Samos and went to southern Italy in about 518 BC (some say much earlier). Iamblichus [8] gives some reasons for him leaving. First he comments on the Samian response to his teaching methods:-

... he tried to use his symbolic method of teaching which was similar in all respects to the lessons he had learnt in Egypt. The Samians were not very keen on this method and treated him in a rude and improper manner.

This was, according to Iamblichus, used in part as an excuse for Pythagoras to leave Samos:-

... Pythagoras was dragged into all sorts of diplomatic missions by his fellow citizens and forced to participate in public affairs. ... He knew that all the philosophers before him had ended their days on foreign soil so he decided to escape all political responsibility, alleging as his excuse, according to some sources, the contempt the Samians had for his teaching method.

Pythagoras founded a philosophical and religious school in Croton (now Crotone, on the east of the heel of southern Italy) that had many followers. Pythagoras was the head of the society with an inner circle of followers known as mathematikoi. The mathematikoi lived permanently with the Society, had no personal possessions and were vegetarians. They were taught by Pythagoras himself and obeyed strict rules. The beliefs that Pythagoras held were [2]:-

(1) that at its deepest level, reality is mathematical in nature,
(2) that philosophy can be used for spiritual purification,
(3) that the soul can rise to union with the divine,
(4) that certain symbols have a mystical significance, and
(5) that all brothers of the order should observe strict loyalty and secrecy.

Both men and women were permitted to become members of the Society, in fact several later women Pythagoreans became famous philosophers. The outer circle of the Society were known as the akousmatics and they lived in their own houses, only coming to the Society during the day. They were allowed their own possessions and were not required to be vegetarians.

Of Pythagoras's actual work nothing is known. His school practised secrecy and communalism making it hard to distinguish between the work of Pythagoras and that of his followers. Certainly his school made outstanding contributions to mathematics, and it is possible to be fairly certain about some of Pythagoras's mathematical contributions. First we should be clear in what sense Pythagoras and the mathematikoi were studying mathematics. They were not acting as a mathematics research group does in a modern university or other institution. There were no 'open problems' for them to solve, and they were not in any sense interested in trying to formulate or solve mathematical problems.

Rather Pythagoras was interested in the principles of mathematics, the concept of number, the concept of a triangle or other mathematical figure and the abstract idea of a proof. As Brumbaugh writes in [3]:-

It is hard for us today, familiar as we are with pure mathematical abstraction and with the mental act of generalisation, to appreciate the originality of this Pythagorean contribution.

In fact today we have become so mathematically sophisticated that we fail even to recognise 2 as an abstract quantity. There is a remarkable step from 2 ships + 2 ships = 4 ships, to the abstract result 2 + 2 = 4, which applies not only to ships but to pens, people, houses etc. There is another step to see that the abstract notion of 2 is itself a thing, in some sense every bit as real as a ship or a house.

Pythagoras believed that all relations could be reduced to number relations. As Aristotle wrote:-

The Pythagorean ... having been brought up in the study of mathematics, thought that things are numbers ... and that the whole cosmos is a scale and a number.

This generalisation stemmed from Pythagoras's observations in music, mathematics and astronomy. Pythagoras noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments. In fact Pythagoras made remarkable contributions to the mathematical theory of music. He was a fine musician, playing the lyre, and he used music as a means to help those who were ill.

Pythagoras studied properties of numbers which would be familiar to mathematicians today, such as even and odd numbers, triangular numbers, perfect numbers etc. However to Pythagoras numbers had personalities which we hardly recognise as mathematics today [3]:-

Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. This feeling modern mathematics has deliberately eliminated, but we still find overtones of it in fiction and poetry. Ten was the very best number: it contained in itself the first four integers - one, two, three, and four [1 + 2 + 3 + 4 = 10] - and these written in dot notation formed a perfect triangle.

Of course today we particularly remember Pythagoras for his famous geometry theorem. Although the theorem, now known as Pythagoras's theorem, was known to the Babylonians 1000 years earlier he may have been the first to prove it. Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see [7]):-

After [Thales, etc.] Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrational and the construction of the cosmic figures.

Again Proclus, writing of geometry, said:-

I emulate the Pythagoreans who even had a conventional phrase to express what I mean "a figure and a platform, not a figure and a sixpence", by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among the sensible objects and so become subservient to the common needs of this mortal life.

Heath [7] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans.

(i) The sum of the angles of a triangle is equal to two right angles. Also the Pythagoreans knew the generalisation which states that a polygon with n sides has sum of interior angles 2n - 4 right angles and sum of exterior angles equal to four right angles.

(ii) The theorem of Pythagoras - for a right angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. We should note here that to Pythagoras the square on the hypotenuse would certainly not be thought of as a number multiplied by itself, but rather as a geometrical square constructed on the side. To say that the sum of two squares is equal to a third square meant that the two squares could be cut up and reassembled to form a square identical to the third square.

(iii) Constructing figures of a given area and geometrical algebra. For example they solved equations such as a (a - x) = x2 by geometrical means.

(iv) The discovery of irrationals. This is certainly attributed to the Pythagoreans but it does seem unlikely to have been due to Pythagoras himself. This went against Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio of two whole numbers. However, because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number.

(v) The five regular solids. It is thought that Pythagoras himself knew how to construct the first three but it is unlikely that he would have known how to construct the other two.

(vi) In astronomy Pythagoras taught that the Earth was a sphere at the centre of the Universe. He also recognised that the orbit of the Moon was inclined to the equator of the Earth and he was one of the first to realise that Venus as an evening star was the same planet as Venus as a morning star.

Primarily, however, Pythagoras was a philosopher. In addition to his beliefs about numbers, geometry and astronomy described above, he held [2]:-

... the following philosophical and ethical teachings: ... the dependence of the dynamics of world structure on the interaction of contraries, or pairs of opposites; the viewing of the soul as a self-moving number experiencing a form of metempsychosis, or successive reincarnation in different species until its eventual purification (particularly through the intellectual life of the ethically rigorous Pythagoreans); and the understanding ...that all existing objects were fundamentally composed of form and not of material substance. Further Pythagorean doctrine ... identified the brain as the locus of the soul; and prescribed certain secret cultic practices.

In [3] their practical ethics are also described:-

In their ethical practices, the Pythagorean were famous for their mutual friendship, unselfishness, and honesty.

Pythagoras's Society at Croton was not unaffected by political events despite his desire to stay out of politics. Pythagoras went to Delos in 513 BC to nurse his old teacher Pherekydes who was dying. He remained there for a few months until the death of his friend and teacher and then returned to Croton. In 510 BC Croton attacked and defeated its neighbour Sybaris and there is certainly some suggestions that Pythagoras became involved in the dispute. Then in around 508 BC the Pythagorean Society at Croton was attacked by Cylon, a noble from Croton itself. Pythagoras escaped to Metapontium and the most authors say he died there, some claiming that he committed suicide because of the attack on his Society. Iamblichus in [8] quotes one version of events:-

Cylon, a Crotoniate and leading citizen by birth, fame and riches, but otherwise a difficult, violent, disturbing and tyrannically disposed man, eagerly desired to participate in the Pythagorean way of life. He approached Pythagoras, then an old man, but was rejected because of the character defects just described. When this happened Cylon and his friends vowed to make a strong attack on Pythagoras and his followers. Thus a powerfully aggressive zeal activated Cylon and his followers to persecute the Pythagoreans to the very last man. Because of this Pythagoras left for Metapontium and there is said to have ended his days.

This seems accepted by most but Iamblichus himself does not accept this version and argues that the attack by Cylon was a minor affair and that Pythagoras returned to Croton. Certainly the Pythagorean Society thrived for many years after this and spread from Croton to many other Italian cities. Gorman [6] argues that this is a strong reason to believe that Pythagoras returned to Croton and quotes other evidence such as the widely reported age of Pythagoras as around 100 at the time of his death and the fact that many sources say that Pythagoras taught Empedokles to claim that he must have lived well after 480 BC.

The evidence is unclear as to when and where the death of Pythagoras occurred. Certainly the Pythagorean Society expanded rapidly after 500 BC, became political in nature and also spilt into a number of factions. In 460 BC the Society [2]:-

... was violently suppressed. Its meeting houses were everywhere sacked and burned; mention is made in particular of "the house of Milo" in Croton, where 50 or 60 Pythagoreans were surprised and slain. Those who survived took refuge at Thebes and other places.

Apastamba

2008-01-23T00:47:00.000-08:00

To write a biography of Apastamba is essentially impossible since nothing is known of him except that he was the author of a Sulbasutra which is certainly later than the Sulbasutra of Baudhayana. It would also be fair to say that Apastamba's Sulbasutra is the most interesting from a mathematical point of view. We do not know Apastamba's dates accurately enough to even guess at a life span for him, which is why we have given the same approximate birth year as death year.

Apastamba was neither a mathematician in the sense that we would understand it today, nor a scribe who simply copied manuscripts like Ahmes. He would certainly have been a man of very considerable learning but probably not interested in mathematics for its own sake, merely interested in using it for religious purposes. Undoubtedly he wrote the Sulbasutra to provide rules for religious rites and to improve and expand on the rules which had been given by his predecessors. Apastamba would have been a Vedic priest instructing the people in the ways of conducting the religious rites he describes.

The mathematics given in the Sulbasutras is there to enable the accurate construction of altars needed for sacrifices. It is clear from the writing that Apastamba, as well as being a priest and a teacher of religious practices, would have been a skilled craftsman. He must have been himself skilled in the practical use of the mathematics he described as a craftsman who himself constructed sacrificial altars of the highest quality.

The Sulbasutras are discussed in detail in the article Indian Sulbasutras. Below we give one or two details of Apastamba's Sulbasutra. This work is an expanded version of that of Baudhayana. Apastamba's work consisted of six chapters while the earlier work by Baudhayana contained only three.

The general linear equation was solved in the Apastamba's Sulbasutra. He also gives a remarkably accurate value for √2 namely

1 + 1/3 + 1/(34) - 1/(3434).

which gives an answer correct to five decimal places. A possible way that Apastamba might have reached this remarkable result is described in the article Indian Sulbasutras.

As well as the problem of squaring the circle, Apastamba considers the problem of dividing a segment into 7 equal parts. The article [3] looks in detail at a reconstruction of Apastamba's version of these two problems

Manava

2008-01-23T00:45:00.000-08:00

Born: about 750 BC in India
Died: about 750 BC in India
Manava was the author of one of the Sulbasutras. The Manava Sulbasutra is not the oldest (the one by Baudhayana is older) nor is it one of the most important, there being at least three Sulbasutras which are considered more important. We do not know Manava's dates accurately enough to even guess at a life span for him, which is why we have given the same approximate birth year as death year. Historians disagree on 750 BC, and some would put this Sulbasutra later by one hundred or more years.

Manava would have not have been a mathematician in the sense that we would understand it today. Nor was he a scribe who simply copied manuscripts like Ahmes. He would certainly have been a man of very considerable learning but probably not interested in mathematics for its own sake, merely interested in using it for religious purposes. Undoubtedly he wrote the Sulbasutra to provide rules for religious rites and it would appear an almost certainty that Manava himself would be a Vedic priest.

The mathematics given in the Sulbasutras is there to enable accurate construction of altars needed for sacrifices. It is clear from the writing that Manava, as well as being a priest, must have been a skilled craftsman.

Manava's Sulbasutra, like all the Sulbasutras, contained approximate constructions of circles from rectangles, and squares from circles, which can be thought of as giving approximate values of π. There appear therefore different values of π throughout the Sulbasutra, essentially every construction involving circles leads to a different such approximation. The paper is concerned with an interpretation of verses 11.14 and 11.15 of Manava's work which give π = 25/8 = 3.125.

Baudhayana

2008-01-23T00:44:00.000-08:00

Born: about 800 BC in India
Died: about 800 BC in India
To write a biography of Baudhayana is essentially impossible since nothing is known of him except that he was the author of one of the earliest Sulbasutras. We do not know his dates accurately enough to even guess at a life span for him, which is why we have given the same approximate birth year as death year.

He was neither a mathematician in the sense that we would understand it today, nor a scribe who simply copied manuscripts like Ahmes. He would certainly have been a man of very considerable learning but probably not interested in mathematics for its own sake, merely interested in using it for religious purposes. Undoubtedly he wrote the Sulbasutra to provide rules for religious rites and it would appear an almost certainty that Baudhayana himself would be a Vedic priest.

The mathematics given in the Sulbasutras is there to enable the accurate construction of altars needed for sacrifices. It is clear from the writing that Baudhayana, as well as being a priest, must have been a skilled craftsman. He must have been himself skilled in the practical use of the mathematics he described as a craftsman who himself constructed sacrificial altars of the highest quality.

The Sulbasutras are discussed in detail in the article Indian Sulbasutras. Below we give one or two details of Baudhayana's Sulbasutra, which contained three chapters, which is the oldest which we possess and, it would be fair to say, one of the two most important.

The Sulbasutra of Baudhayana contains geometric solutions (but not algebraic ones) of a linear equation in a single unknown. Quadratic equations of the forms ax2 = c and ax2 + bx = c appear.

Several values of π occur in Baudhayana's Sulbasutra since when giving different constructions Baudhayana uses different approximations for constructing circular shapes. Constructions are given which are equivalent to taking π equal to 676/225 (where 676/225 = 3.004), 900/289 (where 900/289 = 3.114) and to 1156/361 (where 1156/361 = 3.202). None of these is particularly accurate but, in the context of constructing altars they would not lead to noticeable errors.

An interesting, and quite accurate, approximate value for √2 is given in Chapter 1 verse 61 of Baudhayana's Sulbasutra. The Sanskrit text gives in words what we would write in symbols as

√2 = 1 + 1/3 + 1/(34) - 1/(3434)= 577/408

which is, to nine places, 1.414215686. This gives √2 correct to five decimal places. This is surprising since, as we mentioned above, great mathematical accuracy did not seem necessary for the building work described. If the approximation was given as

√2 = 1 + 1/3 + 1/(34)

then the error is of the order of 0.002 which is still more accurate than any of the values of π. Why then did Baudhayana feel that he had to go for a better approximation?

Ahmes

2008-01-23T00:41:00.000-08:00

Born: about 1680 BC in Egypt
Died: about 1620 BC in Egypt

Ahmes is the scribe who wrote the Rhind Papyrus (named after the Scottish Egyptologist Alexander Henry Rhind who went to Thebes for health reasons, became interested in excavating and purchased the papyrus in Egypt in 1858).

Ahmes claims not to be the author of the work, being, he claims, only a scribe. He says that the material comes from an earlier work of about 2000 BC.

The papyrus is our chief source of information on Egyptian mathematics. The Recto contains division of 2 by the odd numbers 3 to 101 in unit fractions and the numbers 1 to 9 by 10. The Verso has 87 problems on the four operations, solution of equations, progressions, volumes of granaries, the two-thirds rule etc.

The Rhind Papyrus, which came to the British Museum in 1863, is sometimes called the 'Ahmes papyrus' in honour of Ahmes. Nothing is known of Ahmes other than his own comments in the papyrus.