The discovery of the algebraic solution of cubic equations in the 16th century is connected with three Italian mathematicians: Scipione dal Ferro (1465-1526), Nicolo Tartaglia (1499/1500-1557), and Hieronimo Cardano (1501-1576). (Pronunciation: Sci- is pronounced like she, -gl- in Tartaglia -ly-.)

The Bolognese university professor Scipione dal Ferro found the solution of the cubic equations without the quadratic term (modern) x³ + px = q, and x³ = px + q according to one source about 1505 and according to another around 1515, but he did not publish it. He guarded it in his closest circle of family, colleagues and disciples because such secret knowledge was extremely valuable as an intellectual weapon of attack and defense at a time when the reappointment of a university teacher and his pay depended on how he fared in scholarly competitions in which two contestants set real mathematical problems to each other. (There is a third cubic equation without the quadratic term: x³ + q = px. But mostly it was not treated because it has a negative solution, and negative quantities were rejected – called false or fictitious – at that time because one could not represent them geometrically, for example, as the side of a square or the edge of a cube.)

In January 1535 the reckoning master Tartaglia was challenged by a pupil of dal Ferro, the Venetian arithmetician Antoniomaria Fior, to a mathematical contest. Each one put 30 problems to the other. Fior’s problems were all cubic equations of the type (modern) x³ + px = q. Fortunately, Tartaglia found the solution of this equation eight days before the deadline, on February 12, 1535, and of the type x³ = px + q one day later. He won the contest easily.

Nicolo Tartaglia (1500-1557)

Cardano, who had heard about this feat, pressed Tartaglia to divulge to him the formulas and promised to keep them secret. Therefore, Tartaglia disclosed to Cardano the solutions on March 25, 1539, in Cardano's house in Milan. But Cardano broke his promise (according to Tartaglia, even an oath), and published in 1545 the book Ars magna (Latin: The great art, meaning the algebra, as opposed to the ars minor, arithmetic), in which solutions to all 13 types of cubic equations, including those with a quadratic term, and even the solution of biquadratic equations found by Cardano’s disciple Lodovico Ferrari (1522-1565), were published. Cardano stated in the book that he had obtained the formulas from Tartaglia. Moreover, he had seen them in 1542 in a copybook of the late original discoverer, Scipione dal Ferro, when he and Ferrari visited his son-in-law Annibale della Nave in Bologna.

If you look into many books and reference works on the history of mathematics, you will find the assertion that the real name of Tartaglia was Fontana. [Editor's note: See, for example, Burton, p. 291; Cooke, p. 307; Eves, p. 272; O'Connor and Robertson; and Suzuki, p. 313.]  But this author is willing to make a bet: If someone can show me a passage in Tartaglia’s works where he calls himself Fontana, or a document where he is called so, I will pay him $1,000 U.S. I am sure that nobody will get this money because I have seen all his works in the Austrian National Library in Vienna and in Italian libraries, and I own facsimile editions of two of his works and even an original 1550 edition of his first book, La Nova Scientia. You will find not only in all titles of his works the author Nicolo Tartaglia (before 1550, Tartalea) but also in descriptions of conversations he had and published correspondences the name Nicolo Tartaglia or Tartalea. In every description and correspondence he was always called Nicolo Tartalea or Tartaglia, and he himself signed only Nicolo Tartalea or Tartaglia. (The spelling of his given name, Niccolò, found in many papers and books, even Italian ones, is wrong!)

How did the unfortunately ineradicable myth that Tartaglia in reality was called Fontana come about?

He did not know a family name of his father – given name: Michele – who died when Nicolo was 6 years old. On February 19 or 20, 1512, when French troops sacked his home town Brescia (pronunciation: bresha) and conducted a terrible massacre, a soldier split with a sword the palate and the teeth of the 12-year-old boy. The result was that he stuttered for some time, and, therefore, the other boys called him "stutterer," in Italian Tartalea or Tartaglia. In remembrance, he adopted this as his family name, and published all his works under this name. In documents in Italian archives, only Tartalea or Tartaglia is written.

In his last will of December 10, 1557 (he died three days later), which is preserved in the Governmental Archive in Venice, Tartaglia appointed his brother his universal heir, and this brother had the name Zuampiero (John-Peter) Fontana. Therefore, some historians concluded that he also must have had the same surname. [Editor's note: Others noted his brother’s name but did not assume he had the same name; see, for example, D. E. Smith, Vol. I, p. 297, and Arnaldo Masotti, p. 258.] But that is wrong. Probably the brother adopted the name Fontana as an adult in the same way as Nicolo adopted the name Tartaglia. If someone immigrates into the U.S. and adopts there a new name, that does not mean at all that also his brothers and sisters take the same name. There are many cases where immigrant brothers and sisters have different family names.

The notary public Rocho de Benedetti had the duty to write the proper name of the decedent. But although he must have noticed that his brother had the family name Fontana, he called the testator Nicolo Tartalia or Tartalea, and not Fontana. The notary also did not call his father Michele Fontana but in Italian Michiel di Bressa, and in Latin Michaelis de Brixia. The words di and de mean from, and the town Brescia is called Bressa in the Venetian dialect and Brixia in Latin. At that time, it was not unusual to have no family name. Think of the Italian painter, sculptor, architect, engineer, and scientist Leonardo da Vinci (1452-1519), who also had no family name; da Vinci means from Vinci near Empoli, about 20 miles from Florence.

In a Latin document in the Governmental Archive in Verona from 1529, our mathematician is called Nicolaus brixiensis magister Abbachi 30, that is, Nicolo from Brescia, Master of the Abbacus, 30 years. Abacus at that time did not mean a calculating device with beads on wires, but rather the art of reckoning, arithmetic. It was used in the same sense – calculating with the Hindu-Arabic numerals – in the famous work of Leonardo of Pisa or Fibonacci, Liber abaci (Book of the abacus) of 1202. It shows that in 1529, eight years before he published his first book, La Nova Scientia, Nicolo did not yet have a family name.

I appeal to all mathematicians: Give Tartaglia his name back! Do not call him Fontana any more! It should be a matter of course in the literature to use only that name of a scientist or artist he used himself in all his works. I am sure your name is sacrosanct to you. You would protest if your name were garbled. But to change it completely is still worse!

Tartaglia authored all his works with the family name he had given himself. Therefore, do not call him wrongly by the name of his brother, a name he NEVER used in all his lifetime!

Burton, David. The History of Mathematics: An Introduction. Dubuque, Iowa: William C. Brown, 1991.

Cooke, Roger. The History of Mathematics: A Brief Course. New York: John Wiley & Sons, 1997.

Eves, Howard. An Introduction to the History of Mathematics. Philadelphia: Saunders College Publishing, 1990.

Masotti, Arnaldo. “Tartaglia,” Dictionary of Scientific Biography. New York: Charles Scribner’s Sons Inc., 1976, Vol. XIII, pp. 258-262.

O'Connor, J. J. and E. F. Robertson, "Nicolo Tartaglia," MacTutor History of Mathematics Archive, 2005. http://www-history.mcs.st-and.ac.uk/Biographies/Tartaglia.html

Smith, David E. History of Mathematics. 2 vols. New York: Dover Publications, 1950.

Suzuki, Jeff. A History of Mathematics. Upper Saddle River, NJ: Prentice Hall, 2002.

For more on Tartaglia and the solution to cubic equations, see the the Convergence article, How Tartaglia Solved the Cubic Equation, by the same author.