Mathematics of the Italian Mathematics of the Italian RenaissanceRenaissance

MAT 112-16 Summer 2011MAT 112-16 Summer 2011

Prof. Douglas Furman – Mathematics Prof. Douglas Furman – Mathematics Dept.Dept.

SUNY Ulster – International ProgramsSUNY Ulster – International Programs

Gambling, Betrayal & Murder:Gambling, Betrayal & Murder:Cardano & The Italian Cardano & The Italian

AlgebraistsAlgebraists

The Italian algebraists of the The Italian algebraists of the Renaissance were able to Renaissance were able to

accomplish what had eluded accomplish what had eluded humanity for over three humanity for over three

millenia!millenia!

0 bax b xa

22 40

2b b acax bx c x

a

3 2 0 ?ax bx cx d x

Let’s go back 3000 years Let’s go back 3000 years beforebefore the the Renaissance!Renaissance!

BM 13901

Babylonian Sexagesimal Babylonian Sexagesimal NumeralsNumerals

QUIZ!QUIZ!►How do you write 212 (decimal) in How do you write 212 (decimal) in

Babylonian sexagesimal?Babylonian sexagesimal?

►21221210 10 = 3,32= 3,3260 60 = =

Let’s Look at a Babylonian Quadratic Let’s Look at a Babylonian Quadratic EquationEquation

► 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 + 20let L = length of rectangle L · W = 800(W + 20)·W = 800W2 + 20W – 800 = 0(W + 40)(W – 20) = 0W = - 40 (not possible) or W = 20 (Solution)So, the width is 20 yards and the length is 40 yards.

How did the Babylonians do How did the Babylonians do it?it?

L = W + 20L · W = 800(W + 20)·W = 800

► Demonstrarte

What did the Greeks contribute What did the Greeks contribute to Algebra?to Algebra?

Geometric/mechanical solutions to certain particular cubic equations

Directrix??Directrix??

Along comes the Islamic AgeAlong comes the Islamic Age► c. 800, House of Wisdom in Baghdadc. 800, House of Wisdom in Baghdad

► Abu Ja'far Muhammad Abu Ja'far Muhammad ibn Musa ibn Musa al-Khwarizmial-Khwarizmi

(Father of Abdullah, Muhammad, (Father of Abdullah, Muhammad, son of Moses, native of Khwārizm)son of Moses, native of Khwārizm)

Al-Kitab al-mukhtaṣar fi Al-Kitab al-mukhtaṣar fi hisab al-gabr w’al-muqabalahisab al-gabr w’al-muqabala

((The Compendious Book on The Compendious Book on Calculation by Completion and Calculation by Completion and BalancingBalancing))

al-gabral-gabr translates as “completion”, we think of it translates as “completion”, we think of it as moving a negative from one side of the as moving a negative from one side of the equation to the other and making it positiveequation to the other and making it positive

al-muqabalaal-muqabala translates as “balancing”, we think of it as translates as “balancing”, we think of it as combining like positive terms on either side of the equation combining like positive terms on either side of the equation by subtracting the smaller from the largerby subtracting the smaller from the larger

3 2 4 25 2 4x xx

5 2 45 2xx

► In Al-Khwarizmi’s book on Algebra he In Al-Khwarizmi’s book on Algebra he gives algorithms for 5 forms of gives algorithms for 5 forms of quadratic equations:quadratic equations:

2

2

2

2

1. Squares equal to roots ( =bx)

2. Squares equal to numbers ( =c)

3. Squares and roots equal to numbers ( +bx=c)

4. Squares and numbers equal to roots ( +c=bx

ax

ax

ax

ax2

)

5. Roots and numbers equal to squares ( )bx c ax

The Arabs solve the cubic The Arabs solve the cubic equation by geometrical means.equation by geometrical means.►Umar ibn Ibrahim Al-Nisaburi Umar ibn Ibrahim Al-Nisaburi al-Khayyamial-Khayyami

Known to the west as Known to the west as Omar Khayyam Omar Khayyam (c. 1044 – c. 1123)(c. 1044 – c. 1123) Know to the west more famously as a Persian poet.Know to the west more famously as a Persian poet. He wrote He wrote The RubaiyatThe Rubaiyat translated by Edward Fitzgerald in 1859. translated by Edward Fitzgerald in 1859.

The Arabs solve the cubic The Arabs solve the cubic equation by geometrical means.equation by geometrical means.►Umar ibn Ibrahim Al-Nisaburi Umar ibn Ibrahim Al-Nisaburi al-Khayyamial-Khayyami

Known to the west as Known to the west as Omar Khayyam Omar Khayyam (c. 1044 – c. 1123)(c. 1044 – c. 1123) Know to the west more famously as a Persian poet.Know to the west more famously as a Persian poet. He wrote He wrote The RubaiyatThe Rubaiyat translated by Edward Fitzgerald in 1859. translated by Edward Fitzgerald in 1859. Khayyam’s mathematical works first published in the West in 1851.Khayyam’s mathematical works first published in the West in 1851. Not available in English until 1931.Not available in English until 1931.

He solved 13 different cases of the cubic equationHe solved 13 different cases of the cubic equation Through ingenious geometric reasoning Khayyam is able to find Through ingenious geometric reasoning Khayyam is able to find

the solutions to the various cubic equations as intersections of two the solutions to the various cubic equations as intersections of two conic sections (hyperbolas, parabolas, & circles)conic sections (hyperbolas, parabolas, & circles)

But these are not numerical solutions they can only provide But these are not numerical solutions they can only provide geometric solutions. geometric solutions.

Europe Slowly Awakens Europe Slowly Awakens MathematicallyMathematically

►Leonardo Pisano Leonardo Pisano (1170-1250), (1170-1250), FibonacciFibonacci Introduces the Hindu-Introduces the Hindu-

Arabic numerals (0 – 9) Arabic numerals (0 – 9) in his in his Liber abaciLiber abaci (1202) (1202)

Finds approximate Finds approximate solution to solution to x x 33 + 2 + 2x x 22 +10+10xx = 20 = 20 FlosFlos (1225) (1225)► x x = 1.3688081075 = 1.3688081075

(correct to 9 d.p.)(correct to 9 d.p.)

Luca Pacioli Luca Pacioli (1445-1517)(1445-1517)

Luca PacioliLuca Pacioli► Professor of Mathematics at University of PerugiaProfessor of Mathematics at University of Perugia

Founded 1308Founded 1308► Mentored by Francesca & AlbertiMentored by Francesca & Alberti► 1494 1494 Summa de arithmetica, geometria, proportioni et Summa de arithmetica, geometria, proportioni et

proportionalitaproportionalita (The Collected Knowledge of Arithmetic, (The Collected Knowledge of Arithmetic, Geometry, Proportion and Proportionality)Geometry, Proportion and Proportionality) ““Father of Accounting”Father of Accounting” Claims there is no general solution to the cubicClaims there is no general solution to the cubic

► 1496 Invited to Ludovico Sforza’s Court in Milan as court 1496 Invited to Ludovico Sforza’s Court in Milan as court mathematicianmathematician Befriends Leonardo da VinciBefriends Leonardo da Vinci

► 1499 French Armies of Louis XII entered Milan1499 French Armies of Louis XII entered Milan Luca & Leonardo flee together to Mantua, Venice & then Luca & Leonardo flee together to Mantua, Venice & then

FlorenceFlorence Pacioli taught mathematics at Univ. of Bologna 1501-1502Pacioli taught mathematics at Univ. of Bologna 1501-1502

► 1509 1509 Divina ProportioneDivina Proportione Illustrated by Leonardo da VinciIllustrated by Leonardo da Vinci

Luca PacioliLuca Pacioli

Luca PacioliLuca Pacioli

Scipione del Ferro Scipione del Ferro (1465-1526)(1465-1526)► 1496-1526 Lectured at University of Bologna1496-1526 Lectured at University of Bologna

► First to solve a particular type of cubic First to solve a particular type of cubic equationequation x x 33 + m+ mxx = n (depressed cubic) = n (depressed cubic) Kept solution a secretKept solution a secret

► Del Ferro, on his deathbed, shares his secret Del Ferro, on his deathbed, shares his secret with his student Antonio Maria Fiorwith his student Antonio Maria Fior

Niccolo Fontana “Tartaglia” Niccolo Fontana “Tartaglia” ((1500 – 1557)1500 – 1557)

► 1512 French army sacks 1512 French army sacks Brescia, 46,000 Brescians Brescia, 46,000 Brescians killed.killed.

►Niccolo suffers a saber Niccolo suffers a saber wound to his jaw & palate.wound to his jaw & palate.

►Goes to Padua to study Goes to Padua to study mathematicsmathematics

►Gradually earns a Gradually earns a reputation as a reputation as a mathematician by winning mathematician by winning many public debatesmany public debates

Fior challenges Tartaglia to a DebateFior challenges Tartaglia to a Debate► 1535 Fior challenges Tartaglia to a debate of 30 1535 Fior challenges Tartaglia to a debate of 30

problems each. The winner receives a banquet for problems each. The winner receives a banquet for each correct solution.each correct solution.

► Tartaglia had previously discovered the solution to Tartaglia had previously discovered the solution to x x 33 + m + mx x 22 = n = n

► Tartaglia poses a variety of problems, while Fior Tartaglia poses a variety of problems, while Fior poses all depressed cubics.poses all depressed cubics.

► 8 days prior to the debate Tartaglia figures out 8 days prior to the debate Tartaglia figures out Fior’s depressed cubicFior’s depressed cubic

► Tartaglia answers all 30 questions & wins the Tartaglia answers all 30 questions & wins the contest.contest.

Girolamo Cardano Girolamo Cardano (1501-1576)(1501-1576)

Girolamo Cardano Girolamo Cardano (1501-1576)(1501-1576)►Cardano hears of Tartaglia’s victory and Cardano hears of Tartaglia’s victory and

that Tartaglia has solved a cubic that Tartaglia has solved a cubic equation. So Cardano sends a messenger equation. So Cardano sends a messenger to ask if Tartaglia will share his method.to ask if Tartaglia will share his method.

►Who is Cardano?Who is Cardano?

Girolamo Cardano Girolamo Cardano (1501-1576)(1501-1576)►““the most bizarre character in the the most bizarre character in the

whole history of mathematics” whole history of mathematics” – – William Dunham, William Dunham, Journey Through GeniusJourney Through Genius

► Illegitimate son of Fazio Cardano, a Illegitimate son of Fazio Cardano, a lawyer/ mathematician. lawyer/ mathematician.

►Fazio lectured at Univ. of Pavia and Fazio lectured at Univ. of Pavia and helped Da Vinci with geometry.helped Da Vinci with geometry.

Girolamo Cardano Girolamo Cardano (1501-1576)(1501-1576)► In In The Book of My LifeThe Book of My Life he writes he writes

““Although various abortive medicines … were Although various abortive medicines … were tried in vain … I was normally born on the tried in vain … I was normally born on the 2424thth day of September in the year 1500” day of September in the year 1500”

his mother was in labor for 3 days and he was his mother was in labor for 3 days and he was born “almost dead” born “almost dead”

““was revived in a bath of warm wine, which was revived in a bath of warm wine, which might have been fatal to any other child.”might have been fatal to any other child.”

Girolamo Cardano Girolamo Cardano (1501-1576)(1501-1576)► Attends Univ. of Pavia for MedicineAttends Univ. of Pavia for Medicine► War breaks out he goes to Univ of PaduaWar breaks out he goes to Univ of Padua

Campaigns to be rector of the students, though he is Campaigns to be rector of the students, though he is not well liked.not well liked.

“ “This I recognize as unique and outstanding amongst This I recognize as unique and outstanding amongst my faults - the habit, which I persist in, of preferring to my faults - the habit, which I persist in, of preferring to say above all things what I know to be displeasing 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 the ears of my hearers. I am aware of this, yet I keep it up willfully, in no way ignorant of how many it up willfully, in no way ignorant of how many enemies it makes for me.”enemies it makes for me.”

- The Book of My LifeThe Book of My Life

1525 Doctorate in Medicine1525 Doctorate in Medicine

Girolamo Cardano Girolamo Cardano (1501-1576)(1501-1576)► Refused admission to the College of Physicians Refused admission to the College of Physicians

in Milan (1525).in Milan (1525).► Struggling medical practice in a small village Struggling medical practice in a small village

outside of Padua.outside of Padua.► 1531-1532 Marries, moves outside Milan, once 1531-1532 Marries, moves outside Milan, once

again the is rejected by the College of again the is rejected by the College of Physicians.Physicians.

► Resorts to gambling. Resorts to gambling. Pawning wife’s jewelry, eventually ending up in the Pawning wife’s jewelry, eventually ending up in the

poorhouse.poorhouse.

Girolamo Cardano Girolamo Cardano (1501-1576)(1501-1576)► “ “ I was inordinately addicted to the chess-board and the I was inordinately addicted to the chess-board and the

dicing table…I gambled at both for many years; and not dicing table…I gambled at both for many years; and not only every year, but – I say with shame- every day.”only every year, but – I say with shame- every day.”

- The Book of My LifeThe Book of My Life

► He once slashed a man across the face who he thought He once slashed a man across the face who he thought had cheated him in cards.had cheated him in cards.

► He eventually writes He eventually writes Liber de Ludo Aleae Liber de Ludo Aleae (Book on (Book on Games of Chance), published posthumously in 1663.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.”

Girolamo Cardano Girolamo Cardano (1501-1576)(1501-1576)► Eventually gets his father’s old lecturing job at Eventually gets his father’s old lecturing job at

the Piatti Foundation, Milanthe Piatti Foundation, Milan► Treats patients in spare time and his reputation Treats patients in spare time and his reputation

grows.grows.► 1536 publishes a book criticizing the local 1536 publishes a book criticizing the local

doctors in Milan.doctors in Milan.► But Cardano is eventually accepted by the But Cardano is eventually accepted by the

college of physicianscollege of physicians► Becomes so famous he is called on to treat the Becomes so famous he is called on to treat the

Pope and the Archbishop in Scotland.Pope and the Archbishop in Scotland.

Girolamo Cardano Girolamo Cardano (1501-1576)(1501-1576)► "Count no man happy until he be dead“"Count no man happy until he be dead“

- Solon, Athenian Statesman, c. 6Solon, Athenian Statesman, c. 6thth century BCE century BCE

Wife dies at age 31Wife dies at age 31 Oldest son (Giambattista) marries a woman “utterly Oldest son (Giambattista) marries a woman “utterly

without dowry or recommendation.”without dowry or recommendation.”►She boasts that none of their 3 children are fathered by himShe boasts that none of their 3 children are fathered by him► In despair Giambattista serves her a poisoned meal.In despair Giambattista serves her a poisoned meal.►He is convicted of murder and executed (1560)He is convicted of murder and executed (1560)

1570 Cardano is charged with heresy for casting a 1570 Cardano is charged with heresy for casting a horoscope of Jesus.horoscope of Jesus.

Eventually get’s released from prison and receives a Eventually get’s released from prison and receives a pension from the Pope and lives out his life quietly.pension from the Pope and lives out his life quietly.

Tartaglia and CardanoTartaglia and Cardano

Tartaglia and CardanoTartaglia and Cardano► Recall in 1539 Tartaglia had refused Cardano’s request Recall in 1539 Tartaglia had refused Cardano’s request

for the solution to the cubic.for the solution to the cubic.► Cardano mentions that he has been discussing Cardano mentions that he has been discussing

Tartaglia’s ingenuity with the Governor of MilanTartaglia’s ingenuity with the Governor of Milan► Tartaglia leaves Venice to visit Cardano in Milan…Tartaglia leaves Venice to visit Cardano in Milan…► Eventually Tartaglia shares his solution in the form of a Eventually Tartaglia shares his solution in the form of a

poem and Cardano swears to keep it secret.poem and Cardano swears to keep it secret. ““I swear to you, by God's holy Gospels, and as a true man of 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 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 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 true Christian, to note them down in code, so that after my death no one will be able to understand them.”death no one will be able to understand them.”

Lodovico Ferrari Lodovico Ferrari (1522 – 1565)(1522 – 1565)►Raised by his uncle after his father’s Raised by his uncle after his father’s

deathdeath►His cousin, Luke, runs away to Milan and His cousin, Luke, runs away to Milan and

took a job as Cardano’s servant.took a job as Cardano’s servant.►Luke eventually returns home without Luke eventually returns home without

notifying Cardanonotifying Cardano►Cardano complains to Luke’s father, who Cardano complains to Luke’s father, who

sends Lodovico (14 years old) in Luke’s sends Lodovico (14 years old) in Luke’s place…place…

Cardano & FerrariCardano & Ferrari►With Tartaglia’s solution to a depressed cubic, With Tartaglia’s solution to a depressed cubic,

Cardano & Ferrari work for six years and discover Cardano & Ferrari work for six years and discover methods to solve all the other cases of cubic methods to solve all the other cases of cubic equations, thus, in essence, they have equations, thus, in essence, they have discovered a general solution!discovered a general solution!

► But the other cases hinge in the depressed cubic But the other cases hinge in the depressed cubic x x 33 + m+ mxx = n. = n.

► Cardano wants to publish his historic Cardano wants to publish his historic “discoveries” but is prevented by his oath to “discoveries” but is prevented by his oath to Tartaglia!Tartaglia!

Hannibal della NaveHannibal della Nave(Del Ferro’s Son-in-Law)(Del Ferro’s Son-in-Law)

►1543 Cardano & Ferrari travel from Milan 1543 Cardano & Ferrari travel from Milan to Bologna to visit Hannibal della Nave.to Bologna to visit Hannibal della Nave.

►Upon Del Ferro’s death Della Nave Upon Del Ferro’s death Della Nave inherited his Father-in-Law’s notebooks, inherited his Father-in-Law’s notebooks, which included his solution to the which included his solution to the depressed cubic!depressed cubic!

►Cardano now feels he is free of his oath.Cardano now feels he is free of his oath.

Artis Magnae, Sive Artis Magnae, Sive de Regulis de Regulis

Algebraicis Liber Algebraicis Liber UnusUnus (1954) (1954)

((Book number one Book number one about The Great about The Great

Art, or The Rules of Art, or The Rules of AlgebraAlgebra))

3 2 0ax bx cx d Let 3bx za

3 2

03 3 3b b ba z b z c z da a a

2 3 23 2 2

2 3 2

2 03 27 3 9 3

b b b b b ba z z z b z z c z da a a a a a

2 3

32 3 2

23 27 3

c b b bc dz za a a a a

3z mz n 2 3

2 3 2

2,3 27 3

c b b bc dm na a a a a

3z mz n

3 2 3 2

3 3

3 2 2 3 2 2m n n m n nz

2 3

2 3 2

2,3 27 3

c b b bc dm na a a a a

3 22 3 2 3 23

2 3 3

3 22 3 2 3 23

2 3 3

3 2 9 27 2 9 279 54 54

3 2 9 27 2 9 279 54 54

ac b b abc a d b abc a dza a a

ac b b abc a d b abc a da a a

3 22 3 2 3 23

3 22 3 2 3 23

1 4 3 2 9 27 2 9 272

1 4 3 2 9 27 2 9 272

3

ac b b abc a d b abc a d

ac b b abc a d b abc a dz

a

3 2 0ax bx cx d

3 22 3 2 3 23

3 22 3 2 3 23

1 4 3 2 9 27 2 9 272

1 4 3 2 9 27 2 9 272

3 3

ac b b abc a d b abc a d

ac b b abc a d b abc a dbxa a

Recall, 3 3b bx z x za a

0 bax b xa

22 40

2b b acax bx c x

a

3 22 3 2 3 23

3 22 3 2 3 23

3 2

1 4 3 2 9 27 2 9 272

1 4 3 2 9 27 2 9 2720

3

b ac b b abc a d b abc a d

ac b b abc a d b abc a dax bx cx d x

a

Epilogue…Epilogue…

0 bax b xa

22 40

2b b acax bx c x

a

3 22 3 2 3 23

3 22 3 2 3 23

3 2

1 4 3 2 9 27 2 9 272

1 4 3 2 9 27 2 9 2720

3

b ac b b abc a d b abc a d

ac b b abc a d b abc a dax bx cx d x

a

4 3 2 0 ?ax bx cx dx e x

Epilogue…Epilogue…► Lodovico Ferrari (Cardano’s “student”) solved Lodovico Ferrari (Cardano’s “student”) solved

the general 4the general 4thth degree equation! (c. 1540) degree equation! (c. 1540) Cardan published solutions to 20 cases of the quartic Cardan published solutions to 20 cases of the quartic

equation in equation in Ars MagnaArs Magna► So who solved the general 5So who solved the general 5thth degree equation? degree equation?

Nobody!Nobody!► 1824 Niels Henrik Abel (1802-1829) proved that 1824 Niels Henrik Abel (1802-1829) proved that

the general quintic equation (and higher) is not the general quintic equation (and higher) is not solvable by radicalssolvable by radicals

► 1832 Evariste Galois (1811-1832) completes the 1832 Evariste Galois (1811-1832) completes the theory of which equations are solvable by theory of which equations are solvable by radicals.radicals.