Niccolò Fontana Tartaglia

Italian mathematician

• Born: c. 1500
• Birthplace: Brescia, Republic of Venice (now in Italy)
• Died: December 13, 1557
• Place of death: Venice, Republic of Venice (now in Italy)

Tartaglia helped create the field of algebra, once considered a branch of geometry, by using the wisdom of ancient Greek mathematics in unforeseen ways. His discovery of the cubic formula in algebra was one of the first distinctive advances in theoretical mathematics beyond the accomplishments of the Greeks.

Early Life

Niccolò Fontana Tartaglia (nee-kohl-LAW fohn-TAH-nah tahr-TAHL-yah) was the son of a postal carrier named Michele, who died in 1506. It is not known what his family’s name was, although a brother of Tartaglia used the name “Fontana.” The name “Tartaglia,” which is connected with the Italian verb for stammering, was assumed after Niccolò was attacked by a French soldier during France’s military activities in Brescia in 1512; only the assiduous attention of Tartaglia’s mother kept him alive. Tartaglia was the name he used for all his publications.

Early efforts at education were frustrated by the family’s inability to cover the costs, so Tartaglia was almost entirely self-taught. When he went to Verona in 1516, he was already capable of teaching calculation. He was never able to keep his family in the best of financial conditions, compared with his contemporaries with similar familial backgrounds. Nevertheless, he persevered in his own studies, which included the text of Euclid’s Stoicheia (c. 300 b.c.e.; Elements). It has been suggested that part of Tartaglia’s originality stemmed from his not having followed the more traditional pattern of education available to his wealthier fellow citizens.

In 1534, Tartaglia moved to Venice to take up a professorship in mathematics and to give public lessons, and he remained there for the rest of his life.

Life’s Work

When Tartaglia was in Venice, he also took part in public disputations. In mathematics, such disputations involved challenges to solve problems of various degrees of difficulty and provided a kind of excitement more familiar in sporting competitions. In Italy, one of the most challenging disputations was the solving of equations. An equation in which the variable occurs to the second power is called a “quadratic equation,” and methods for solving those had been known to several ancient civilizations. There had been no general technique for solving equations in which the variable occurred to a third power (called “cubic”). The reason, in part, was that the influence of Euclid’s geometry encouraged those tackling equations to think in geometric terms. In addition, there was no system for translating a problem into the kind of shorthand that an algebraic equation represents.

Tartaglia was not the first to approach the problem of solving cubic equations. Ancient mathematicians tried to extract cube roots, and so even the Greeks considered the issue of whether they could handle such calculations with Euclidean methods.

In medieval Islamic mathematics, trigonometry afforded a new set of techniques for looking at algebraic equations. It was not clear how much Islamic material was brought back to Europe, along with the text of Euclid used by mathematicians in the Near East. The mathematician Scipione del Ferro had come up with a method for solving cubic equations in which there was no quadratic term.

In the highly competitive atmosphere of intellectual competitions, it was worth keeping methods secret, and del Ferro passed his along to a student named Antonio Fiore. Fiore challenged Tartaglia to a mathematical duel, and Tartaglia’s reputation depended on the result of the competition. Tartaglia, however, had managed to go beyond the technique of del Ferro and could solve cubic equations even when they included a quadratic term. As a result, Tartaglia emerged victorious, the “star” among mathematicians of the time.

Tartaglia also was reluctant to let the details of his method enter the public realm. He had discussed them with mathematician Gerolamo Cardano, but Tartaglia swore him to secrecy. Cardano subsequently claimed to have found a way to solve a quartic equation (solving with a fourth power of the variable) that used but went beyond the cubic equation of Tartaglia and so felt that he was released from his vow of secrecy. The controversy that resulted when Cardano published Tartaglia’s method, even though he gave Tartaglia credit by name, is an indication of the extent to which intellectual property was an important issue in the Italy of the Renaissance. It also shows that Tartaglia had no shortage of strong language to use when he was indulging in acrimonious debate. Tartaglia was not always able to make the same impression in public confrontations as he had done with Fiore, and that probably contributed to his relative isolation later in life. He died with relatively little to show for his success in solving cubic equations.

In addition to his work in algebra, Tartaglia produced in 1543 the first printed translation of the text of Euclid into any modern language. The influence of Euclid had remained strong in mathematics throughout the medieval period, and Tartaglia argued that some of the foundations of Euclidean mathematics could be jettisoned without having to abandon the results gathered by Euclid. As an example, Euclid did not include the number one among his list of numbers and, instead, assigned it a special place as the “root” of all numbers. Tartaglia argued that mathematicians could make more sense out of their calculations by including one as a number. The inclusion of one as a number helped to cast off some of the restrictions that the Euclidean formulation of mathematics had imposed more than eighteen centuries before.

Of greater contemporary interest was Tartaglia’s work on gunnery. Just as he had not been inhibited by Euclidean precedent from striking out in new directions, so he did not let the dominance of Aristotle in the study of motion get in his way. Trying to describe the trajectory of cannon shells was an important matter for the many military forces involved in sieges of walled cities during Tartaglia’s lifetime. Tartaglia studied the work of Archimedes, the third century b.c.e. Greek mathematician who had done much after Euclid to make applied mathematics a respectable field. His work on the parabolic shape of the trajectories of projectiles was not entirely theoretical, but it gave subsequent generations of students of ballistics a sense of freedom from the kinds of explanation that Aristotle had proposed.

Significance

Tartaglia’s accomplishments in the field of mathematics helped to create the field of algebra as an autonomous discipline rather than as a branch of geometry. His willingness to deviate from the text of Euclid on matters that were not primarily geometrical enabled subsequent mathematicians to look at algebraic questions in their own right.

Contemporary mathematician François Viète established a system for equations that would have been impossible without the work of Tartaglia. The introduction of analytic geometry by René Descartes, which enabled both geometers and algebraists to benefit from each other’s work, would have been senseless if algebra had not been made to stand on its own as a method.

It is difficult to know what Tartaglia could have accomplished with a notation better suited to describe algebraic manipulations, but those who did create that notation were inspired by Tartaglia’s accomplishments.

Bibliography

Bergamini, David. Mathematics. New York: Time, 1967. A lively narration of the public confrontation, with illustrations intended to capture the flavor of the period.

Dauben, Joseph W., and Christoph J. Scriba, eds. Writing the History of Mathematics:Its Historical Development. Basel, Switzerland: Birkhauser, 2002. Scholarly assessment of some of the legends associated with Tartaglia and his contemporaries, arranged by region.

Drake, Stillman, and I. E. Drabkin, eds. Mechanics in Sixteenth-Century Italy. Madison: University of Wisconsin Press, 1969. Still the best collection of material on the innovations that Tartaglia and his students introduced in transforming the study of projectiles from a theoretical and philosophical discipline to a practical and mathematical one.

Dunham, William. Journey Through Genius. New York: John Wiley and Sons, 1990. Looking at Tartaglia primarily through the lens of Cardano, a more-thorough presentation of the algebraic details than the presentation of strictly biographical accounts.

Mahoney, Michael Sean. The Mathematical Career of Pierre de Fermat. 2d ed. Princeton, N.J.: Princeton University Press, 1998. Simply the best account of the influence of Tartaglia on the subsequent development of algebra, not just for specific achievements but also for his transforming algebra into a separate discipline.

Mankiewicz, Richard. The Story of Mathematics. Princeton, N.J.: Princeton University Press, 2000. Further details about the later years of Tartaglia’s career after Cardano had given him fame but taken away his proprietary interest in the cubic formula.

Stillwell, John. Mathematics and Its History. New York: Springer, 2002. Presents the familiar biographical narrative but indicates the algebraic details of Tartaglia’s advances.