What did Girolamo Cardano contribute to polynomials?

Answer 1

Gerolamo Cardano or Girolamo Cardano, in English Jerome Cardan, or in Latin Hieronymus Cardanus (September 24, 1501 - September 21, 1576) was a celebrated Italian Renaissance mathematician, physician, astrologer, and gambler.

He was born in Pavia, Italy, the illegitimate child of a mathematically gifted lawyer who was a friend of Leonardo da Vinci. In his autobiography, Cardano claimed that his mother had attempted to abort him. Shortly before his birth, his mother had to move from Milan to Pavia to escape the plague; her three other children died from the disease. In 1520, he entered the University of Pavia and later in Padua studying medicine. His eccentric and confrontational style did not earn him many friends and he had a difficult time finding work after his studies had ended. In 1525, Cardano repeatedly applied to the College of Physicians in Milan, but was not allowed due to his reputation and illegitimate birth.

Eventually, he managed to develop a considerable reputation as a physician and his services were highly valued at the courts. He was the first to describe typhoid fever.

Today, he is best known for his achievements in algebra. He published the solutions to the cubic and quartic equations in his 1545 book Ars magna. The solution to one particular case of the cubic, x^3 + ax = b (in modern notation), was communicated to him by Niccolo Fontana Tartaglia (who later claimed that Cardano had sworn not to reveal it, and engaged Cardano in a decade-long fight), and the quartic was solved by Cardano's student Lodovico Ferrari. Both were acknowledged in the foreword of the book, as well as in several places within its body. In his exposition, he acknowledged the existence of what are now called imaginary numbers, although he did not understand their properties.

Cardano was notoriously short of money and kept himself afloat by being an accomplished gambler and chess player. His book about games of chance, Liber de ludo aleae, written in the 1560s but published only in 1663 after his death, contains the first systematic treatment of probability, as well as a section on effective cheating methods.

Cardano invented several mechanical devices including the combination lock, the gimbal consisting of three concentric rings allowing a supported compass or gyroscope to rotate freely, and the Cardan shaft with universal joints, which allows the transmission of rotary motion at various angles and is used in vehicles to this day. He made several contributions to hydrodynamics and held that perpetual motion is impossible, except in celestial bodies. He published two encyclopedias of natural science which contain a wide variety of inventions, facts, and occult superstitions. He also introduced the Cardan grille, a cryptographic tool, in 1550.

Significantly, in the history of deaf education, he was one of the first to state that deaf people could learn without learning how to speak first.

Cardano's eldest and favorite son was executed in 1560 after he confessed to having poisoned his cuckolding wife. His other son was a gambler who stole money from him. Cardano himself was accused of heresy in 1570 because he had computed and published the horoscope of Jesus in 1554. Apparently, his own son contributed to the prosecution. He was arrested and had to spend several months in prison, was forced to abjure his professorship. He moved to Rome, received a lifetime annuity from Pope Gregory XIII (after first having been rejected by Pope Pius V) and finished his autobiography. He died there on the day he had (supposedly) astrologically predicted earlier (some suspect he may have committed suicide).

Cubic equations were first discovered by Jaina mathematicians in ancient India sometime between 400 BC and 200 CE.

The Persian mathematician Omar Khayyám (1048–1123) constructed solutions of cubic equations by intersecting a conic section with a circle. He showed how this geometric solution could be used to get a numerical answer by consulting trigonometric tables.

In the early 16th century, the Italian mathematician Scipione del Ferro (1465-1526) found a method for solving a class of cubic equations, namely those of the form x3 + mx = n. In fact, all cubic equations can be reduced to this form if we allow m and n to be negative, but negative numbers were not known at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fiore about it.

In 1530, Niccolò Tartaglia (1500-1557) received two problems in cubic equations from Zuanne da Coi and announced that he could solve them. He was soon challenged by Fiore, which led to a famous contest between the two. Each contestant had to put up a certain amount of money and to propose a number of problems for his rival to solve. Whoever solved more problems within 30 days would get all the money.

Tartaglia received questions in the form x3 + mx = n, for which he had worked out a general method. Fiore received questions in the form x3 + mx2 = n, which proved to be too difficult for him to solve, and Tartaglia won the contest.

Later, Tartaglia was persuaded by Gerolamo Cardano (1501-1576) to reveal his secret for solving cubic equations. Tartaglia did so only on the condition that Cardano would never reveal it. A few years later, Cardano learned about Ferro's prior work and broke the promise by publishing Tartaglia's method in his book Ars Magna (1545) with credit given to Tartaglia. This led to another competition between Tartaglia and Cardano, for which the latter did not show up but was represented by his student Lodovico Ferrari (1522-1565). Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and income.

Cardano noticed that Tartaglia's method sometimes required him to extract the square root of a negative number. He even included a calculation with these complex numbers in Ars Magna, but he did not really understand it. Rafael Bombelli studied this issue in detail and is therefore often considered as the discoverer of complex numbers.

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Answer 2

The entire listing of builders is an old boys club. Yes Hal singlehandedly destroyed the Leafs but when you read about his hockey achievements before gaining control of the Leafs he should be rewarded for those alone. Hal was two different people. His public demeanor was famous. He followed no rules and broke all the others. His private life was different. The Leaf players under Ballard were required to visit the Hospitals in Toronto. They had to take part in Variety Village activities and the gardens was open for many charity functions for free. All the school championship hockey games were played at MLG. There was no charge for the ice or the facility. Just think how many twelve year old kids had their chance to play on the same ice as Conacher, Apps, Kennedy, and Keon. Ballard Did that. Busher Jackson was blackballed from the HHOF Ballard fought the Old Boys to get him inducted. Should he be removed? My answer is no. But then I'm biased. Ballard was a friend. He had Loblaw's buddy Stuckless fired from the Garden staff when to explain why would have caused a black mark over the club. He did all he could to help the victims. Quietly and without fanfare. He was a great owner but a lousy business man. Stein had himself inducted. That was changed and now the HHOF has no link to the NHL. If non hockey considerations are removed from the reasons for non inclusion of players. Then Dino should be included but how many players are inducted because of their non hockey activities. Answering Bob's question on Henderson. Paul Henderson should not be in the Hall because it is based on a lifetime of achievement not a single game of excellence. If we are to reward single achievements Ian Turnbull should be in but not bloody likely.

Answer 3

Cardan's solution for Cubic equations.