The Renaissance Of Mathematics

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The Renaissance Of Mathematics

• Europe in the 14th and 15th Centuries

2
Little Ice Age

• Events following the the collapse of Rome
• Series of rainfall and colder climate compared
  to the great flood of Genesis
• Disastrous crop failure led to famine
• Towns lost 10 percent of their inhabitants in 6
  months

3
Black Death

• Cause of the Black Death
• Lasted for several years and returned in
  intervals for 12 15 years
• Great Plague of London in 1665 was the last
  English eruption
• 800 people died each day in Paris
• 10,000 people were buried in a single mass grave
  the first 6 weeks

4
The Smoke of War

• War was also a factor during this time
• Hundred Years War
• Those who survived the plague may not have been
  so lucky in the war
• Populations of towns still decreasing

5
The Cultural Rebirth

• By approximately 1450, the calamities of war,
  plague, and famine had tapered off.
• The invention of printing revolutionized the
  transmission and dissemination of ideas.
• The study of Mathematics shifted from the means
  of survival to the battle of scholars.
• Italian mathematics of the 1500s can be
  summarized in the names of del Ferro,Tartaglia,
  Cardan, Ferrari, and Bombelli.

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Problems Of The Time

• Problem 1
• Find three numbers, the second of which exceeds
  the first by 2, and the third of which exceeds
  the second by 2 also, and whose product is 1000.
• Problem 2
• Find a number whose cube added to three times
  its square makes 5.

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Battle of the Scholars

• The first person to solve cubic equations
  algebraically was Scipione del Ferro.
• Antonio Maria Fior, a pupil of del Ferro, was
  often entrusted with the solution method.
• Fior used this to his advantage by challenging
  others to contests at mathematical
  problem-solving.

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Battle of the Scholars

• The search for the general solution to the
  previously unsolved cubic equation led to the
  dispute between Gerolamo Cardano and Niccolo
  Tartaglia.
• It is impossible to discuss the debate without
  first giving brief biographies of the antagonists
  to show why they acted as they did.

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The Life of Tartaglia

• Niccolo Tartaglia (whose actual family name was
  Fontana) was born in Brescia, Italy, about the
  turn of the sixteenth century.
• Nearly killed as a teenager when the French
  captured his home town in 1512.

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The Life of Tartaglia

• Although his early years were spent in poverty,
  Tartaglia was determined to educate himself.
• Tartaglia was self taught in mathematics, but was
  able to earn his living teaching at Verona and
  Venice.

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Isnt It Ironic

• Tartaglia, a man disfigured by a saber,
  contributed to the ultimate obsolescence of the
  saber by his pioneering work Nova Scientia
  (1537), on the application of mathematics to
  artillery fire. Tartaglias new science was, of
  course, ballistics. Even though the theories were
  completely wrong, he was the first to offer a
  theoretical discussion as against the so-called
  experience of gunners.

12
A Suspicious Character

• Tartaglias unfortunate early experiences may
  have encouraged a suspicious character.
• Self-taught, he was jealous of his prerogatives
  and constantly impelled to try to establish his
  intellectual credentials.
• Either through intent or simple ignorance of the
  literature, he had a habit of claiming other
  peoples discoveries as his own.

13
Fior Challenges Tartaglia

• About 1510, del Ferro found a general solution to
  x3 ax b, but he died before he could publish
  his discovery.
• Fior, trying to gain a reputation by exploiting
  his masters discovery, challenged Tartaglia to a
  public problem-solving contest.

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The Terms of the Challenge

• Each contestant was to propose 30 problems, the
  victor being the one who could solve the greatest
  number within 50 days.
• Shortly before the appointed date, Tartaglia
  devised a scheme for solving cubics that lacked
  the second-degree term.

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Tartaglia Wins

• Tartaglia entered the competition prepared to
  handle two types of cubics, whereas his opponent
  was equipped for but one.
• Within two hours, Tartaglia had reduced all 30
  problems posed to him to particular cases of the
  equation x3 ax b, for which he knew the
  answer.
• Of the problems Tartaglia put to Fior, the latter
  failed to master a single one.

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The Life of Cardano

• Girolamo Cardano or Jerome Cardan was born in
  Milan (now Italy) in Sept. 1501
• Illegitimate child of Fazio Cardano and Chiara
  Micheria
• Cardan attended a university until war broke out
  and forced to leave
• In 1525 he was awarded his doctorate in medicine

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Cardans Mishaps

• Cardan achieved near miraculous cures and
  developments in the medical field
• Unfortunately, his personal life was not as
  accepting as his career
• His eldest son was put to death for poisoning his
  own wife
• Cardan cut the ears off his youngest son for
  attempting the same crime

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The Mishaps Continue

• Cardan was a compulsive gambler
• However, he lost more than he gained
• He was known as one of the first to think about
  probability as a theory

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Cardan Predicts the Future

• He cast the horoscope of Jesus Christ
• Predicts his own date of death
• Suicide held the prediction to truth

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Mathematical Works of Art

• Cardan wrote a book on the Games of Chance
• This book was the first for the theory of
  probability
• Ars Magna (The Great Art) first printed in 1545
  and is classified as a text on algebraic
  equations.

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Fictitious Numbers

• Cardan was first to take notice that negative
  roots were positive numbers
• He also was first to say that a cubic might have
  3 roots.
• His popularity of the solution to the cubic
  equation is further noted as a dispute between
  he and Tartaglia

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Back to the Challenge

• Cardan was writing the Practica arithmeticae
  generalis at the time of the Fior-Tartaglia
  Challenge.
• Because Pacioli had earlier stated that there
  could not be a general solution to the cubic,
  Cardan had ignored this topic.
• Upon hearing that Tartaglia had a solution for x3
  ax b, he tried to find one.

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The Debate

• Cardan failed to find the general solution, so he
  asked Tartaglia for the solution so that he might
  publish it under Tartaglias name.
• Tartaglia refused, stating he would publish the
  solution himself at a later date. This prompted
  Cardan to label Tartaglia as greedy and unwilling
  to help mankind.

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Tartaglia Tells

• Because of these insults, a correspondence
  developed between the two mathematicians.
• Cardan, in the hope of learning the secret,
  invited Tartaglia to visit him.
• After much flattery, Tartaglia revealed his
  method of solution in a cryptic poem, provided
  that Cardan swore an oath that he would never
  reveal the solution.

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Cardan Breaks His Oath

• Rumors began to circulate that Tartaglia was not
  the first to discover of the cubic formula, and
  in 1543 Cardan journeyed to Bologna to try to
  verify these reports.
• After examining the papers of del Ferro, he
  concluded that del Ferro was the one who had made
  the breakthrough.

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Cardan Breaks His Oath

• In 1545, Cardan published the Ars Magna, which
  contained Tartaglias solution of the cubic with
  a statement that del Ferro and Tartaglia had each
  found solutions by independent research.

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Practice Problems

• Section 7.3
• (1, 2, 3, 5-11)
• Section 7.4 Read section (1, 3, 5)

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Rafael Bombelli

• Born in 1526 in Bologna, Italy
• Bombellis hometown is where all the disputes
  have taken place
• He was 9 years old when the dispute between Fior
  and Tartaglia took place
• He was 19 when Cardan published Ars Magna

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Algebra

• It took several years, but Bombelli wrote the
  text Algebra
• However, out of the 272 problems in the book at
  least 143 of them were originally discovered by
  Diophantus

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Imagine That

• Bombelli was the first mathematician that
  accepted imaginary numbers
• He was also the first to write down rules for
  addition, subtraction, multiplication of
  complex numbers

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Rules

• Plus of minus times plus of minus makes minus
  -n . -n -n
• Plus of minus times minus of minus makes plus
  -n . - -n n
• Minus of minus times plus of minus makes plus -
  -n . -n n
• Minus of minus times minus of minus makes minus
  - -n . - -n -n

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Quartic Equation

• After the cubic had been solved, it was only
  natural that mathematicians should attack the
  quartic (fourth-degree) equation.
• Cardan, unsuccessful at solving the quartic
  equation, turned it over to his disciple Ludovico
  Ferrari (1522-1565).
• Ferrari, using the rules for the solving the
  cubic, eventually succeeded where his master had
  failed.

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The Life of Ferrari

• Ferrari, the son of poor parents, was taken into
  Cardans household as a servant boy at the age of
  14.
• Ferrari was exceptionally gifted, and Cardan
  undertook to instruct him in Latin, Greek, and
  mathematics.
• Ferrari joined the fray surrounding the solution
  of the cubic by swearing that he had been present
  at the fateful meeting between Cardan and
  Tartaglia.

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Ferrari Challenges Tartaglia

• When Tartaglia had seen Ars magna, he publicly
  denounced Cardano for breaking an oath sworn on
  the gospels, and he ridiculed Cardanos
  mathematical ability.
• Cardano disdained to refute the slur, but Ferrari
  attacked Tartaglia.
• Ferrari challenged Tartaglia to a public debate
  on mathematics and all related subjects.

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Tartaglia Answers Back

• Tartaglia answered with further insults and
  refused the debate.
• After a further exchange of insults, each
  proposed thirty-one questions which were
  exchanged, answered, and returned.
• No decision was reached because each tore the
  others answers to shreds.

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The Debate

• Tartaglia accepted a debate to be held in Milan,
  Cardanos stronghold.
• On August 10th, 1548, Tartaglia and Ferrari met
  in combat, Cardan having left town.

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The Debate

• Very little is know about the debate.
• Tartaglia left after the first day, claiming to
  have won, although it seems Ferrari won by
  default.
• An indication of Ferraris triumph is that
  Tartaglia lost his teaching post in Brescia, and
  Ferrari was invited to lecture in Venice.

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Colorful Episode Ends

• Tartaglia died in 1557 without publishing his
  solution to the cubic.
• Ferrari became a professor of mathematics at
  Bologna in 1565 and died the same year, having
  been poisoned with white arsenicby his own
  sister, as rumor had it.
• With Cardans death in 1576, one of the most
  interesting and colorful episodes in the history
  of mathematics ended.

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A mathematical problem should be difficult in
order to entice us, yet not completely
inaccessible, lest it mock at our efforts.