[paper that I wrote for a class in undergrad that mostly served as an exercise in using LaTex]

Abstract

This document will describe the history and methods behind how the cubic equation was solved, specifically focusing on the work of specific sixteenth century Italian mathematicians. It will also touch upon the quartic and quintic equations.

1 The Life and Times of Tartaglia

Niccolo Fontana, known as “Tartaglia,” meaning “the stammerer,” was a Venetian mathematician during the early sixteenth century. His many accomplishments included engineering, working as a bookkeeper, and creating the first Italian translations of the works of Archimedes and Euclid, but he is most famous for his involvement in the 1535 Bologna University Mathematics Competition. Some years before, University of Bologna's Chair of Arithmetic and Geometry, Sciopione dal Ferro, solved one case of the cubic equation, a polynomial equation in which the highest variable exponent is three, a case only involving positive numbers. Though it is believed that dal Ferro solved this form around 1515, he kept it a secret until just before his death in 1526, at which point he shared this solution with his student and far inferior mathematician, Antonio Fior. Rumors began to spread that the cubic had been solved, eventually reaching Tartaglia. Tartaglia managed to find his own partial solution to the cubic in this time, leading the overconfident Fior to challenge him to a public math competition. According to the rules, each gave the other thirty problems to solve over the course of forty to fifty days. All of Fior's problems were set to the form x^3 + mx = n which dal Ferro had found, believing that Tartaglia would not be able to figure it out. However, Tartaglia had not only managed to solve all of the problems Fior had presented him with within the course of two hours, but he found a general method for solving all forms of the cubics before the contest was over. This ensured Tartaglia's victory [1].

Figure 1: A graph resulting from a cubic function

2 Enter Cardano

News of Tartaglia's win spread across Italy, eventually reaching Milan-based mathematician and general eccentric character Gerolamo Cardano. Though brilliant in mathematics and medicine, Cardano's illegitimate birth and generally abrasive personality had limited his opportunities, leading to a somewhat inconsistent professional life. Cardano managed to bring himself to prominence in the world of math with the publication of The Practice of Arithmetic and Simple Mensuration, the first of one hundred and thirty-one books he would write in his lifetime [2]. Cardano approached Tartaglia in 1539 with hopes of adding his solutions to the cubic to his next book, Practica Arithmeticae. Tartaglia was reluctant to share his method, having previously coded his method in a cryptic poem so that no one would figure it out, but did so after Cardano made an oath to not publish his ideas. Using this knowledge, Cardano spent the next six years working on the cubic and quartic equations, sometimes with the help of his assistant Lodovico Ferrari [1].

Ferrari was born in Bologna, Italy in 1522, a particularly politically tumultuous time for Northern Italy. Ferrari's father was killed in the army, meaning he had to move in with his uncle, Vincent. It so happened that Vincent's son Luke had run away to Milan for work and ended up briefly as a servant to Cardano. Luke ended up tiring of the work and going back home without telling Cardano. Cardano contacted Vincent to send his servant back, but Vincent thought that his fourteen-year-old nephew might make a better candidate. Upon Ferrari's arrival to Cardano's in November of 1936, Cardano learned that the teenager was literate and made him a secretary rather than a servant. He began to teach him mathematics, at which Ferrari proved to be so talented that he was able to teach by the age of eighteen and, by the age of twenty, he began working as a public lecturer in geometry. During this time, he and Cardano also worked on Tartaglia's cubic solutions in an attempt to solve the quartic as well. Between Tartaglia's findings and the work of another mathematician, Zuanne da Coi, Ferrari eventually found a solution for the quartic equation, polynomials for which the highest variable exponent is four. He wanted his work published, but it would be impossible to do so without revealing the work Tartaglia had produced, which Cardano had previously vowed to keep a secret. The two did further research and decided that because technically dal Ferro had solved a form of the cubic before Tartaglia, they could publish the findings about the cubic and the quartic without it being considered breaking Cardano’s oath. He featured these works in his book Ars Magna in 1545 [3].

Figure 2: A graph resulting from a quartic function

3 Ferrari Reigns Victorious

Needless to say, Tartaglia did not take well to the news of his findings being published, especially by someone who had previously promised not to. The alleged loophole counted for naught, as the portion about the cubic was very much Tartaglia's work. Ferrari responded to Tartaglia's anger by writing him an insulting letter in which he also challenged him to a public debate. Tartaglia saw no point in debating the relatively-unknown Ferrari, but did respond in an attempt to bring Cardano in. Tartaglia and Ferrari exchanged publicly read insulting letters for about a year. The rivalry was brought to its conclusion in 1548. Tartaglia was offered a job as a lecturer in Brescia and needed to participate in a public math competition with Ferrari to prove that he was worthy of the spot. Though less experienced in the ways of public debate than Tartaglia, Ferrari proved himself far more understanding of the concepts at hand, not only his own quartic equation, but the cubic as well. Tartaglia left that night in shame, leaving Ferrari as the winner [4].

4 The Failed Quintic

Though all proven mathematically brilliant, neither Ferrari, Cardano, nor Tartaglia ever even attempted to breach the quintic equation. It is thought that the first mathematician to attempt to actually solve for a specific quintic equation was French mathematician Joseph-Louis Lagrange in the eighteenth century. He tirelessly studied the previously developed methods for the quadratic, cubic, and quartic equations for clues. He focused on how rational numbers changed under permutations, but could not find an expression for powers above four. His successor came in the form of Italian mathematician Paolo Ruffini. Ruffini's work focused on the properties of permutations with the goal of proving that the quintic was not solvable. Though Ruffini did develop proofs, his work was considered vague for its focus on patterns instead of computation and was not well-accepted by mathematicians at the time. In the nineteenth century, Norwegian mathematician Niels Henrik Abel carried on with Ruffini’s work, publishing the completed versions of the proofs, and confirming that quintic functions cannot be solved with a single equation. This work is known to this day as Abel's Impossibility Theorem,or the Abel-Ruffini Theorem [5].

References

[1] Mastin, Luke. 16th Century Mathematicians. The Story of Mathematics, 2010.

http : //www.storyof mathematics.com/16thtartaglia.html

[2] JJ O’Connor and EF Robertson. Quadratic, Cubic, and Quartic Equations University of St. Andrews, February 1996.

http : //www−groups.dcs.st−and.ac.uk/history/HistT opics/Quadraticetcequations.html

[3] JJ OConnor and EF Robertson. Giralamo Cardano. School of Mathematics and Statistics at the University of Saint Andrews, June 1998

http : //www−history.mcs.st−andrews.ac.uk/Biographies/Cardan.html

[4] JJ OConnor and EF Robertson. Lodovico Ferrari. School of Mathematics and Statistics at the University of Saint Andrews, September 2005.

http : //www−groups.dcs.st−and.ac.uk/history/Biographies/F errari.html

[5] Fiona Brunk. Galois' Predecessors. School of Mathematics and Statistics at the University of Saint Andrews, January 2005.

http : //www−history.mcs.st−and.ac.uk/P rojects/Brunk/Chapters/Ch1.html

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