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# Lesson 5 gambling, betrayal & murder algebra

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### Lesson 5 gambling, betrayal & murder algebra

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