Transcript Tartaglia

Tartaglia vs
Cardano
History of a great fight
Tartaglia
1500-1557
Niccolò Fontana Tartaglia
was a mathematician, an
engineer, and a
bookkeeper from the
Republic of Venice.
During the War of League
of Cambrai, a French
soldier sliced Niccolò's jaw
and palate with a saber.
This made it impossible for
Niccolò to speak normally,
prompting the nickname
"Tartaglia"
("stammerer").
His great work for science
He published many books, for example the first Italian translations of
Archimedes and Euclid, and a famous compilation of maths.
Tartaglia was the first to apply mathematics to the investigation of the paths
of cannonballs; his work was later validated by Galileo's studies on falling
bodies.
Tartaglia is also known for having given an
expression (Tartaglia's formula) for the volume
of tetrahedon (incl. any irregular tetrahedra) as
the determinant of the distance values
measured pairwise between its four corners
He invented the Tartaglia’s triangle , a method to
obtain binomial coefficents
Tartaglia's greates legacy to mathematical history, though,
occurred when he won the 1535 Bologna University
mathematics competition by demonstrating a general algebric
formula fto solve cubic equations (equations with terms
including x3), something which was thought as something
impossible, requiring as it does an understanding of the square
roots of negative numbers. In the competition, he beat Scipione
del Ferro (or at least del Ferro's assistant, Fior), who had
coincidentally produced his own partial solution to the cubic
equation problem not long before. Although del Ferro's solution
perhaps predated Tartaglia’s, it was much more limited, and
Tartaglia is usually credited with the first general solution. In the
highly competitive and cut-throat environment of 16th Century
Italy, Tartaglia even encoded his solution in the form of a poem
in an attempt to make it more difficult for other mathematicians
to steal it.
ONE MOMENT!!!
Are you sure you know what is a cubic
function???
In mathematics this is the cubic function’s form:
F(x)= x³+bx²+cx+d
Where a cannot be 0; so it is a polynomial of degree three.
Setting ƒ(x) = 0 and assuming a ≠ 0 produces a cubic equation of
the form:
ax³+bx²+cx+d=0
The coefficients a, b,c, d are generally real numbers, though most
of the theory is also valid if they belong to a field of characteristic
other than 2 or 3. Solving a cubic equation amounts to finding the
roots of a cubic function
Cardano
1501-1576
Gerolamo Cardano was an Italian
Renaissance mathematician, physician,
astrologer and gambler.
He was born in Pavia as illegitimate child.
Shortly before his birth, his mother had to
move from Milan to Pavia to escape the
Plague; her three other children died from
the disease.
In 1520, he entered the University of Pavia
and later in Padua studied medicine.
His eccentric and confrontational style did
not earn him many friends and he had a
difficult time finding work after his studies
ended.
In 1525, Cardano repeatedly applied to the
College of Physicians in Milan, but was not
admitted owing to his combative reputation
and illegitimate birth.
What about his job?
His gambling led him to formulate elementary rules in
probability, making him one of the founders of the field.
Today, he is best known for his achievements in algebra.
Cardano was the first mathematician to make systematic use
of numbers less than zero
The solution to one particular case of the cubic equation 0=
ax^3+bx^2+c was communicated to him by Tartaglia and
the quartic was solved by Cardano’s student: Lodovico
Ferrari.
In 1545 Cardano published his, and Ferrari’s, solution in his
book "Ars Magna". Ferrari, on seeing Tartaglia's cuubic
solution, had realized that he could use a similar method to
solve quartic equations (equations with terms including x4).
In this work, Tartaglia, Cardano and Ferrari between them
demonstrated the first uses of what are now known as
complex numbers, combinations of real and imaginary
numbers of the type a + bi, where i is the imaginary unit √1. It fell to another Bologna resident, Rafael Bombelli, to
explain, at the end of the 1560's, exactly what imaginary
numbers really were and how they could be used.
Del Ferro
1465-1526
the third in the dispute..
Scipione del Ferro was an Italian
mathematician who first discovered a method
to solve the depressed cubic equation.
his job...
Mathematicians from del Ferro's time knew that the general cubic
equation could be simplified to one of two cases called the
depressed cubic equation, for positive numbers p,q,x
x3+px=q
x3=px+q
Then with the appropriate substitution of parameters, one derive a
solution to the depressed cubic:
What did they argue about?
Around 1542 Cardano hired a young man
named Ferrari as a servant. He quickly
realised that Ferrari was very gifted and
Cardano became his teacher. They worked
on maths together, and during one of their
sessions Cardano revealed Tartaglia’s
tecnique to the young guy.
Working together they expanded upon
Tartaglia’s initial work, and made a
number of new discoveries.
They knew, because of Cardano’s promise to
Tartaglia, they could not make their work
public without revealing Tartaglia’s work.
Knowing that Tartaglia had first used his
method to win a competition against a
mathematician by the name of Scipione
Del Ferro some 30 years earlier, they
decided to research the archieves.
In 1543 they discovered in the writing of Del Ferro the same solution that
Tartaglia had given Cardano. Since the tecnique appared in Del Ferro’s
paper, Cardano no longer felt obligated to keep his oath.
In 1545 Cardano published his algebra book, the “Ars Magna” (in which he
stated that it was Del Ferro who was the first to solve the cubic equation
and that the solution he gave was Del Ferro’s method).
Even thought Cardano gave appropriate credit, Tartaglia was very upset when
the book came out accusing Cardano of being a thief, a scoundrel, and of
breaking a sacred oath.
Tartaglia continued his attack for many years. Ferrari took up the defence of
Cardano by strongly responding to Tartaglia’s letters and challenging him
to public debate.
How is ended the discussion?
Mathematical historians now credit both with
the paternity of the formula to solve cubic
equations, referring to it as:
“Cardano-Tartaglia Formula”.
Filippo Romanengo and Stefano Amoretti
(who really wanted to come to France!!)
Liceo-Ginnasio Classico Cristoforo Colombo,
Genova,
ITALY