Niccolò Tartaglia (1499 - 1557)

His real surname was Fontana, but he was always known as ‘Tartaglia’. This pseudonym comes from the Italian verb “tartagliare”, meaning “to stammer” and was the result of a tragic incident in his youth.

When he was just 12, French troops attacked his hometown of Brescia close to the Italian Alps. They rampaged through the town and during the ensuing mayhem, Niccolò was badly injured by a sabre which cut his jaw and palate.

Nursed back to health by his mother he recovered, but had scars and a speech impediment for the rest of his life.

His life was never easy. Legend has that he had to leave the local school when he had learnt up to the letter ‘k’ in the alphabet as his family ran out of money. He carried on teaching himself and moved to Venice where he survived as a teacher of mathematics.

Eventually, his years of struggle looked like they might pay off. He published a book (the first to apply mathematics to the science of artillery fire), engaged in many debates and began to develop a name for himself.

Fior, another Venetian mathematician challenged him to a contest - each would submit 30 questions to the other. Fior had a secret weapon for the competition. He had learnt from his teacher, del Ferro, the unpublished and secret solution to solving cubic equations.

During the contest, however, Tartaglia solved the given cubic equation using a method superior to that of Fior. This though was to cause him more troubles, for he then came to the attention of the acclaimed mathematician, Cardan.

Cardan invited Tartaglia to Milan. Tartaglia went at once, hoping to gain influence with Cardan’s sponsor the powerful governor of the Holy Roman Emperor’s army there. In this he was unsuccesful but after much persuasion and cajoling, and a pledge of confidentiality, he gave Cardan his formula for solving the cubic equation.

Tartaglia returned to Venice and Cardan then discovered that del Ferro had actually been the first to solve the cubic equation - even though the solution was inferior to that of Tartaglia.

He went on to publish an extensive book on the subject, including del Ferro and Tartaglia’s methods. While Cardan was scrupulous about giving due credit, Tartaglia was incensed and never forgave Cardan for breaking his word.

Tartaglia’s mathematics

In order to understand Tartaglia’s solution, we review the solution of other equations that had already been solved. The equation of degree 1 - a so-called linear equation - is,

Degree 1 \qquad{ax+b=0}

This is easily solved, and its solution is,


The equation of degree 2 - the so-called quadratic equation - is,

Degree 2 \qquad{ax^2}+bx+c=0

This has the solution,


Degree 3 \qquad{ax^3}+ax+b=0

This cubic equation is ‘missing’ a term in x^2; it was common knowledge at the time that if this simplified equation could be solved, then so could the more complicated equation.

This has the solution,

x=\quad\root 3 \of{-{b\over{2}}+{\sqrt{{a^3\over{27}}+{b^2\over{4}}}}}+\quad\root 3 \of{-{b\over{2}}-{\sqrt{{a^3\over{27}}+{b^2\over{4}}}}}

a solution involving cube roots as well as square roots.

Mathematicians were quick to find a way to solve the next equation - the equation of degree 4, the quartic equation. Try as they may however, they could not solve the equation of degree 5, the quintic equation. There is a good reason for this - it does not have an algebraic solution, involving roots, like the equations above. It would not be until the nineteenth century and Galois and Abel that this would be fully understood and accounted for.

Alongside this fundamental work in equations, Tartaglia write a popular arithmetic text and was the first Italian translator and publisher of Euclid’s Elements; he also wrote and published an edition of Archimedes’ works.