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

Cardano1501-1576

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 Ferro1465-1526

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