3 Renaissance mathematics Flashcards
RENAISSANCE MATHEMATICS
REBIRTH IN LITERATURE AND ART
Revival in mathematics slightly later
GREAT CUBIC CONTROVERSY
Time : 16th Century
Place : Northern Italy
solve cubic equations algebraically by formula (NOT BY DIAGRAM)
ALGEBRAIC SOLUTIONS
Dramatis Personae
Scipione del Ferro (1465 - 1526) Antonio Maria Fior (c. 1506) Niccolo Tartaglia (1500 - 1557) Girolamo Cardano (1501 - 1576) Ludovico Ferrari (1522 - 1565) Rafael Bombelli (1526 - 1572)
NUMBER SYSTEM FOR CUBICS
- integers used
- irrationals used
- couldnt cope with negative numbers
- no complex numbers but they became “accepted”
- RHETORICAL ALGREBRA- written in words
Prologue: Sixteenth century
Prologue: Sixteenth century
University positions temporary
Public problem-solving contests
Scipione del Ferro
In around 1515 Scipione del Ferro (1465 - 1526), a Mathematics professor at the University of Bologna, solved cubic equations of the form:
x^3 + ax=b
‘cube and things equal numbers’
His solution was never published due to the common practice of withholding new mathematical methods; Universities had no tenure and so problem solving contests were often held in sixteenth century Italy, with the winners likely to maintain their positions. Before Del Ferro died in 1526 he managed to tell his son-in-law della Nave and student Fiore.
Fiore replaced del Ferro at the University, however, he is described as a mediocre mathematician.
He became increasingly boastful about being able to solve cubics and challenged Tartaglia in 1535.
Niccolò Tartaglia (1500 - 1557),
Niccolò Tartaglia (1500 - 1557), ‘the stammerer’, was particularly interested in ballistics and published the first italian translations of Euclid and Archimedes. He was also able to solve cubics such as:
x^2+bx^2=d
‘cube and squares equal numbers’
Tartaglia independently solved del Ferro’s type in time for the competition. Thus, he was able to solve all 30 of the del Ferro type problems. Unfortunately for Fiore, the problems set by Tartaglia proved much too challenging and Tartaglia won.
Ten days before contest, Tartaglia found algebraic solutions to cubic equations of del Ferro type
22 February 1535 contest in Venice: Tartaglia winner
All Fior’s challenges reduced to del Ferro type cubics
Tartaglia initially refused to reveal his algebraic solutions to
cubics because was translating the Elements into Italian and hoped to publish his own research on cubics for the whole world to see.
Girolamo Cardano
Girolamo Cardano (1501 - 1576) published an important book on algebra, the Ars Magna.
Eventually, tempted by the promise of meeting the governor of Milan as a prospective patron, Tartaglia visited Cardan in Milan
He contacted Tartaglia with the hopes of learning his method. He was able to convince him by promising to keep it a secret and introduce. Tartaglia to the governor of Milan, as he hoped to gain a job at the Milanese court. Tartaglia provided his solution in the form of a poem but began to question his decision. Cardan made the effort to continue the friendship however he rebuffed him.
Cardan traveled to Bologna in 1543 and learned that del Ferro was the first to solve this type. He used this to justify publishing his Ars Magna but still credits Tartaglia, del Ferro and student Ferrari (who was able to solve quartic equations).
Aware of complex but ignored
Poem told to Cardan by Tartaglia in 1539
“When the cube and things together
Are equal to some discrete number,
Find two other numbers differing in this one.
Then you will keep this as a habit
That their product should always be equal
Exactly to the cube of a third of the things.
The remainder then as a general rule
Of their cube roots subtracted
Will be equal to your principal thing”
Poem told to Cardan by Tartaglia in 1539
25 March 1539: Cardan’s will prevailed. He swore an oath never to make public Tartaglia’s secret, revealed to him in twenty-five lines of rhyming verse.
BOOKS IN CUBIC CONTROVERSY
Tartaglia felt betrayed and included solutions on cubics along with the problems from 1535 in Quesiti et Inventioni the following year.
1546 Tartaglia publishes Quesiti et inventioni diverse
In 1548, a competition was held between Ferrari and Tartaglia, however, Tartaglia fled in the night.
Rafael Bombelli
In 1572 Rafael Bombelli published parts of Algebra, giving rules for calculating with complex numbers.
There is a tree 12 braccia high,….
Example Tree 12 units high is broken in two. Height of tree
remaining is cube root of length cut away. What is height of
tree remaining? [ x3
+ x = 12, x is height remaining]
17th problem set by Fiore to Tartaglia in Venice, 1535.
Tartaglia won
CUBIC TIMELINE
1515 ‘del Ferro type’ solved
1526 del Ferro dies
1535 Competition between della Nave & Tartaglia in Venice
1539 Meeting between Cardan & Tartaglia in Milan
1543 Cardan travels to Bologna
1545 Ars Magna published
1546 Tartaglia publishes Quesiti et inventioni diverse
1548 Competition between Ferrari & Tartaglia
1572 Bombelli publishes parts of Algebra
NOT IN EXAM
using verses to solve
x^3 + cx =d
Find u and v st u-v=d and uv= (c/3)^3
Then x= cuberoot(u) - cuberoot(v)
These things I found, and not with sluggish steps,
In the year one thousand five hundred, four and thirty.
With foundations strong and sturdy
In the city girdled by the sea.
Tartaglia 1539
CITY GIRDLED BY THE SEA
VENICE comp 1534
MILAN told
ACT III: Scene I - 1536
CARDAN
Cardan hires 14 year old Ludovico Ferrari
Relationship between them rapidly changed:
master ~ servant, teacher ~ pupil, colleague ~ colleague
Discovered algebraic solutions to cubics & quartics
Wished to publish. Stymied by Cardan’s oath to Tartaglia
Visited Bologna in 1543 to inspect del Ferro’s papers
del Ferro first to find algebraic solutions to cubics
