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Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
Lesson 5 gambling, betrayal & murder   algebra
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Lesson 5 gambling, betrayal & murder algebra

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  • 1. Mathematics of the Italian Renaissance MAT 112-16 Summer 2011 Prof. Douglas Furman – Mathematics Dept. SUNY Ulster – International Programs Gambling, Betrayal & Murder: Cardano & The Italian Algebraists
  • 2. The Italian algebraists of the Renaissance were able to accomplish what had eluded humanity for over three millenia!
  • 3.  
  • 4. Let’s go back 3000 years before the Renaissance! BM 13901
  • 5. Babylonian Sexagesimal Numerals
  • 6. QUIZ!
    • How do you write 212 (decimal) in Babylonian sexagesimal?
    • 212 10 = 3,32 60 =
  • 7. Let’s Look at a Babylonian Quadratic Equation
    • A rectangular plot of land is 20 yards longer than it is wide and has an area of 800 square yards. What are the dimensions of the plot?
    • Here is a modern algebraic solution:
    • let W = width of rectangle L = W + 20
    • let L = length of rectangle L · W = 800
    • (W + 20)·W = 800
    • W 2 + 20W – 800 = 0
    • (W + 40)(W – 20) = 0
    • W = - 40 (not possible) or W = 20 (Solution)
    • So, the width is 20 yards and the length is 40 yards.
  • 8. How did the Babylonians do it?
    • L = W + 20
    • L · W = 800
    • (W + 20)·W = 800
    • Demonstrarte
  • 9. What did the Greeks contribute to Algebra? Geometric/mechanical solutions to certain particular cubic equations
  • 10. Directrix??
  • 11. Along comes the Islamic Age
    • c. 800, House of Wisdom in Baghdad
    • Abu Ja'far Muhammad
    • ibn Musa al-Khwarizmi
    • (Father of Abdullah, Muhammad,
    • son of Moses, native of Khwārizm)
      • Al-Kitab al-mukhtaṣar fi
      • hisab al-gabr w’al-muqabala
      • ( The Compendious Book on
      • Calculation by Completion and Balancing )
  • 12.
      • al-gabr translates as “completion”, we think of it as moving a negative from one side of the equation to the other and making it positive
      • al-muqabala translates as “balancing”, we think of it as
      • combining like positive terms on either side of the equation
      • by subtracting the smaller from the larger
  • 13.
    • In Al-Khwarizmi’s book on Algebra he gives algorithms for 5 forms of quadratic equations:
  • 14. The Arabs solve the cubic equation by geometrical means.
    • Umar ibn Ibrahim Al-Nisaburi al-Khayyami
      • Known to the west as Omar Khayyam (c. 1044 – c. 1123)
      • Know to the west more famously as a Persian poet.
      • He wrote The Rubaiyat translated by Edward Fitzgerald in 1859.
  • 15. The Arabs solve the cubic equation by geometrical means.
    • Umar ibn Ibrahim Al-Nisaburi al-Khayyami
      • Known to the west as Omar Khayyam (c. 1044 – c. 1123)
      • Know to the west more famously as a Persian poet.
      • He wrote The Rubaiyat translated by Edward Fitzgerald in 1859.
      • Khayyam’s mathematical works first published in the West in 1851.
      • Not available in English until 1931.
      • He solved 13 different cases of the cubic equation
      • Through ingenious geometric reasoning Khayyam is able to find the solutions to the various cubic equations as intersections of two conic sections (hyperbolas, parabolas, & circles)
      • But these are not numerical solutions they can only provide geometric solutions.
  • 16. Europe Slowly Awakens Mathematically
    • Leonardo Pisano
    • (1170-1250), Fibonacci
      • Introduces the Hindu-Arabic numerals (0 – 9)
      • in his Liber abaci (1202)
      • Finds approximate solution to x 3 + 2 x 2 +10 x = 20 Flos (1225)
        • x = 1.3688081075 (correct to 9 d.p.)
  • 17. Luca Pacioli (1445-1517)
  • 18. Luca Pacioli
    • Professor of Mathematics at University of Perugia
      • Founded 1308
    • Mentored by Francesca & Alberti
    • 1494 Summa de arithmetica, geometria, proportioni et proportionalita (The Collected Knowledge of Arithmetic, Geometry, Proportion and Proportionality)
      • “ Father of Accounting”
      • Claims there is no general solution to the cubic
    • 1496 Invited to Ludovico Sforza’s Court in Milan as court mathematician
      • Befriends Leonardo da Vinci
    • 1499 French Armies of Louis XII entered Milan
      • Luca & Leonardo flee together to Mantua, Venice & then Florence
      • Pacioli taught mathematics at Univ. of Bologna 1501-1502
    • 1509 Divina Proportione
      • Illustrated by Leonardo da Vinci
  • 19. Luca Pacioli
  • 20. Luca Pacioli
  • 21. Scipione del Ferro (1465-1526)
    • 1496-1526 Lectured at University of Bologna
    • First to solve a particular type of cubic equation
      • x 3 + m x = n (depressed cubic)
      • Kept solution a secret
    • Del Ferro, on his deathbed, shares his secret with his student Antonio Maria Fior
  • 22. Niccolo Fontana “Tartaglia” ( 1500 – 1557)
    • 1512 French army sacks Brescia, 46,000 Brescians killed.
    • Niccolo suffers a saber wound to his jaw & palate.
    • Goes to Padua to study mathematics
    • Gradually earns a reputation as a mathematician by winning many public debates
  • 23. Fior challenges Tartaglia to a Debate
    • 1535 Fior challenges Tartaglia to a debate of 30 problems each. The winner receives a banquet for each correct solution.
    • Tartaglia had previously discovered the solution to
      • x 3 + m x 2 = n
    • Tartaglia poses a variety of problems, while Fior poses all depressed cubics.
    • 8 days prior to the debate Tartaglia figures out Fior’s depressed cubic
    • Tartaglia answers all 30 questions & wins the contest.
  • 24. Girolamo Cardano (1501-1576)
  • 25. Girolamo Cardano (1501-1576)
    • Cardano hears of Tartaglia’s victory and that Tartaglia has solved a cubic equation. So Cardano sends a messenger to ask if Tartaglia will share his method.
    • Who is Cardano?
  • 26. Girolamo Cardano (1501-1576)
    • “ the most bizarre character in the whole history of mathematics”
    • – William Dunham, Journey Through Genius
    • Illegitimate son of Fazio Cardano, a lawyer/ mathematician.
    • Fazio lectured at Univ. of Pavia and helped Da Vinci with geometry.
  • 27. Girolamo Cardano (1501-1576)
    • In The Book of My Life he writes
      • “ Although various abortive medicines … were tried in vain … I was normally born on the 24 th day of September in the year 1500”
      • his mother was in labor for 3 days and he was born “almost dead”
      • “ was revived in a bath of warm wine, which might have been fatal to any other child.”
  • 28. Girolamo Cardano (1501-1576)
    • Attends Univ. of Pavia for Medicine
    • War breaks out he goes to Univ of Padua
      • Campaigns to be rector of the students, though he is not well liked.
      • “ This I recognize as unique and outstanding amongst my faults - the habit, which I persist in, of preferring to say above all things what I know to be displeasing to the ears of my hearers. I am aware of this, yet I keep it up willfully, in no way ignorant of how many enemies it makes for me.”
          • The Book of My Life
      • 1525 Doctorate in Medicine
  • 29. Girolamo Cardano (1501-1576)
    • Refused admission to the College of Physicians in Milan (1525).
    • Struggling medical practice in a small village outside of Padua.
    • 1531-1532 Marries, moves outside Milan, once again the is rejected by the College of Physicians.
    • Resorts to gambling.
      • Pawning wife’s jewelry, eventually ending up in the poorhouse.
  • 30. Girolamo Cardano (1501-1576)
    • “ I was inordinately addicted to the chess-board and the dicing table…I gambled at both for many years; and not only every year, but – I say with shame- every day.”
          • The Book of My Life
    • He once slashed a man across the face who he thought had cheated him in cards.
    • He eventually writes Liber de Ludo Aleae (Book on Games of Chance), published posthumously in 1663.
      • “ ...in times of great anxiety and grief, it is considered to be not only allowable, but even beneficial.”
      • In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice.”
  • 31. Girolamo Cardano (1501-1576)
    • Eventually gets his father’s old lecturing job at the Piatti Foundation, Milan
    • Treats patients in spare time and his reputation grows.
    • 1536 publishes a book criticizing the local doctors in Milan.
    • But Cardano is eventually accepted by the college of physicians
    • Becomes so famous he is called on to treat the Pope and the Archbishop in Scotland.
  • 32. Girolamo Cardano (1501-1576)
    • "Count no man happy until he be dead“
        • Solon, Athenian Statesman, c. 6 th century BCE
      • Wife dies at age 31
      • Oldest son (Giambattista) marries a woman “utterly without dowry or recommendation.”
        • She boasts that none of their 3 children are fathered by him
        • In despair Giambattista serves her a poisoned meal.
        • He is convicted of murder and executed (1560)
      • 1570 Cardano is charged with heresy for casting a horoscope of Jesus.
      • Eventually get’s released from prison and receives a pension from the Pope and lives out his life quietly.
  • 33. Tartaglia and Cardano
  • 34. Tartaglia and Cardano
    • Recall in 1539 Tartaglia had refused Cardano’s request for the solution to the cubic.
    • Cardano mentions that he has been discussing Tartaglia’s ingenuity with the Governor of Milan
    • Tartaglia leaves Venice to visit Cardano in Milan…
    • Eventually Tartaglia shares his solution in the form of a poem and Cardano swears to keep it secret.
      • “ I swear to you, by God's holy Gospels, and as a true man of honour, not only never to publish your discoveries, if you teach me them, but I also promise you, and I pledge my faith as a true Christian, to note them down in code, so that after my death no one will be able to understand them.”
  • 35. Lodovico Ferrari (1522 – 1565)
    • Raised by his uncle after his father’s death
    • His cousin, Luke, runs away to Milan and took a job as Cardano’s servant.
    • Luke eventually returns home without notifying Cardano
    • Cardano complains to Luke’s father, who sends Lodovico (14 years old) in Luke’s place…
  • 36. Cardano & Ferrari
    • With Tartaglia’s solution to a depressed cubic, Cardano & Ferrari work for six years and discover methods to solve all the other cases of cubic equations, thus, in essence, they have discovered a general solution!
    • But the other cases hinge in the depressed cubic
    • x 3 + m x = n.
    • Cardano wants to publish his historic “discoveries” but is prevented by his oath to Tartaglia!
  • 37. Hannibal della Nave (Del Ferro’s Son-in-Law)
    • 1543 Cardano & Ferrari travel from Milan to Bologna to visit Hannibal della Nave.
    • Upon Del Ferro’s death Della Nave inherited his Father-in-Law’s notebooks, which included his solution to the depressed cubic!
    • Cardano now feels he is free of his oath.
  • 38. Artis Magnae, Sive de Regulis Algebraicis Liber Unus (1954) ( Book number one about The Great Art, or The Rules of Algebra )
  • 39.  
  • 40.  
  • 41.  
  • 42.  
  • 43.  
  • 44. Epilogue…
  • 45.  
  • 46. Epilogue…
    • Lodovico Ferrari (Cardano’s “student”) solved the general 4 th degree equation! (c. 1540)
      • Cardan published solutions to 20 cases of the quartic equation in Ars Magna
    • So who solved the general 5 th degree equation?
      • Nobody!
    • 1824 Niels Henrik Abel (1802-1829) proved that the general quintic equation (and higher) is not solvable by radicals
    • 1832 Evariste Galois (1811-1832) completes the theory of which equations are solvable by radicals.

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