What Actually Happened Gary Towsley SUNY Geneseo September 18 2020 Around the year 1510 Scipione del Ferro 1465 1526 a professor at the university of Bologna solves the cubic equation of the form ID: 1019781
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1. The Solution of the Cubic Equation in Sixteenth Century ItalyWhat Actually Happened?Gary Towsley, SUNY Geneseo September 18, 2020
2. Around the year 1510, Scipione del Ferro (1465 – 1526), a professor at the university of Bologna solves the cubic equation of the form :
3. Questions1. What does it mean to solve a cubic equation (algebraically)?2. Why is such a solution important?3. Why did the solution happen when it did?
4. What is a Cubic Equation?What does it mean to Solve a Cubic Equation?
5. What is a Cubic Equation?
6. What is a Cubic Equation?
7.
8. Scipione’s Actual SolutionFor the cubic the solution he computed was not 2 but which a calculator will tell you is 2.
9. Quadratic EquationsProblems we recognize as quadratic equations have been solved for at least the last 4,000 years:A number added to its reciprocal is 4. What is the number?The area of a rectangle is 40 and its perimeter is 28. What are its length and width?I have added a square and 4 times its side. The result is 21. What is the side?
10. A number added to its reciprocal is 4. What is the number?The area of a rectangle is 40 and its perimeter is 28. What are its length and width?I have added a square and 4 times its side. The result is 21. What is the side?
11. Modern SolutionA number added to its reciprocal is 4. Find the number.
12. Ancient Mesopotamian Solution4, the sum, is multiplied by itself: result 16From 16 subtract 4: result 12Take the square root of 12: result Add 4 and :resultDivide by 2: result the desired number
13. Modern Explanation
14. Solutions to Quadratics were GeometricVariants on the above methodLiterally Completing the Square
15. Muhammed al-Khwarizmi (780 – 850)Al-Jabr and al-Muqabala Solutions of all quadratic equations (only positive solutions)Geometric MethodsForms of Quadratics: (all coefficients are positive)
16. What About?Divide out the x.No positive roots.
17. What about Cubic Equations?Considered by Mesopotamians but only solved approximatelyConsidered by Archimedes for specific Geometric ProblemsConsidered by Diophantus of Alexandria but only for situations where the solution was already known and the roots were rational numbers
18. Omar Khayyam (1048 – 1131)Classification of all cubics into two types:Those reducible to quadratics – for exampleFourteen Irreducible Cubics: Three term cubics – Four term cubics –
19. Reducible Cubics
20. 14 Irreducible Cubics
21. Omar Khayyam’s SolutionsFor Reducible Cubics he gave al-Khwarizmi’s solutions with Euclidean ProofsFor Irreducible Cubics he showed that solutions existed using Conic SectionsThese solutions did not result in numerical (or algebraic) solutions except for the very first case:
22. Example of Solutions via ConicsLeft hand side is a parabola, right hand side is a hyperbolaSolution is a point of intersection of the conicsHis solutions were much more complicated than the above example appears to be
23. Khayyam’s Solutions to Irreducible CubicsUsed Conics by Apollonius of Perga (250 – 175 BCE)Gave possible number of positive roots in each caseAnalytic or Coordinate Geometry is 6 centuries in the futureOmar Khayyam was not satisfied with his results, he had not found algebraic solutions to any of the cubics except the case
24. Leonardo of Pisa (Fibonacci) (1170 – 1240)As a youth he traveled with his merchant father around the Mediterranean and learned the mathematics of al-Khwarizmi.He wrote Liber Abacci (Book of Computation) 1. Explained Hindu-Arabic Base 10 Numeration 2. Presented algorithms for all the arithmetic operations 3. Presented and solved a great number of problems using these numerals, proportions, and the algebra of al-Khwarizmi
25. Cossist AlgebraBuilding on the work of Fibonacci and others, the cities of Central and Northern Italy set up “Abacus Schools” as part of the standard education system and thus developed a highly numerate workforce.The most advanced topics in this kind of school involved “Cossist Algebra”, the name coming from the word “cosa” or thing which stood for the unknown in an equation. co - cosa (x)ce - censo or square (x2)cu - cubo or cube (x3).
26. Luca Pacioli (1445 – 1517)Franciscan Monk, friend of Leonardo da Vinci and Piero della FrancescaWrote Summa (1494) – compendium of all known mathematics of the timeIn the Summa he stated that many mathematicians had worked on finding algebraic solutions to the irreducible cubics but none had succeeded beyond solutions by trial and error. Perhaps it was an impossible task.
27. Around the year 1510, Scipione del Ferro (1465 – 1526), a professor at the university of Bologna solves the cubic equation of the form :
28. Mathematicians who worked on CubicsScipione del Ferro (1465 – 1526)Nicolo Fontana (Tartaglia) (1499 – 1557) Girolamo Cardano (1501 – 1576)Ludovico Ferrari (1522 – 1565)
29. Ars Magna (The Great Art)Published by Cardano in 1545 (with some help from Ferrari and Tartaglia)Presented algebraic solutions to all cubic equationsOne chapter to each formPresented algebraic solutions to quartic equations (due to Ferrari)Found negative roots to cubics and quarticsFound complex roots – (but they were useless)
30. Cardano Solves a CubicChapter 11 of Ars MagnaOn the Cube and First Power equal to the Number“For example, let GH3 plus 6 times its side GH equal 20”He solves
31. Step 1Cardano gives a Euclidean proof that if A and B are lengths such that 3 times the area of the rectangle AB is 6 and the difference of the volumes of the two cubes A3 – B3 is 20, then the length A – B satisfies the given equation.
32. Step 2Cardano produces what he calls a rule – namely a recipe or an algorithm to find the required lengths A and B. He offers no proof of this rule.The Rule:“Cube one-third the coefficient of x; add to it the square of one-half the constant of the equation; and take the square root of the whole. You will duplicate this, and to one of the two you add one-half the number you have already squared and from the other you subtract one-half the same. Then subtracting the cube root of the second from the cube root of the first the remainder of what is left is x.”
33. Why didn’t Cardano prove Step 2?It involved squaring a volume which clearly does not exist geometrically (clearly at the time).The derivation used Cossist Algebra which Cardano rather believed was outside the realm of proof.
34. Cardano’s Formula for
35. Our Example p = 6, q = 20
36. How did Cardano find his Rule?
37.
38. Why is this important?1. It solved a problem that had been unsolved for many thousands of years.2. The quartic equation was solved almost immediately afterwards using the solution to the cubic.3. The solution gave great impetus to the development of the symbolic algebra we use today. (Bombelli, Vieta, Fermat, Descartes, etc.)4. Coordinate Geometry is in many ways a byproduct of this solution.
39. Why did the Solution first happen around 1510?The solution to the cubic found by del Ferro required two kinds of mathematics:Euclidean Geometry (The Mathematics of the University)Cossist Algebra (The Mathematics of Commerce)They first came together in Italy around this time.
40. What about the Quintic Equation?Another story, another talk.