 So, a recap of our story so far. Mathematicians sought algebraic solutions to the cubic equations, solutions that only used the basic operations of arithmetic. Scipione del Faro discovered a solution to some types of cubic equations, and taught his student Antonia Maria Fiori. Fiori challenged newcomer Tartaglia to a mathematical dual, and Tartaglia won by figuring out how to solve xq plus px equal to q type equations. And after his stunning victory, Tartaglia might have settled into comfortable obscurity, but for another character. Chirolamo Cardano, who lived in the 16th century, studied medicine at the University of Pavia. Now, at the time, astrology was an important part of medicine. This was because of a belief that the position of the planets might influence patient's health, and so a doctor had to be able to cast horoscopes. In case it's not obvious, the word influence in Italian is influenza. Now, because doctors had to be able to cast horoscopes, this also meant that Cardano had to learn mathematics. Which was a good thing because he graduated from Pavia in 1525 and was denied a license to teach medicine because he was illegitimate. And so to support himself, Cardano eventually took up teaching mathematics in Milan and practicing medicine on the side, unofficially. In 1539, Giurane Detonini d'Acoi visited Cardano. Now, d'Acoi was from Brescia, and boasted about the triumph of his fellow Brescian Tartaglia. Cardano was intrigued and visited Tartaglia to find more about solving the cubic. After much flattery, Tartaglia told Cardano how to solve the cubic, but made him swear not to publish a secret until after Tartaglia himself did. Cardano agreed. Now, we enter a new character. Around 1540, Cardano hired Ludovico Ferrari as a valet. Basically, a man-servant, he would help Cardano get dressed, iron his shirts, do things like that. He realized Ferrari's potential and taught him mathematics as well. Cardano and Ferrari began to investigate other types of cubic equations and eventually found algebraic solutions to all of them, except for the nonsense one. Remember, you can't put together a bunch of objects and get nothing. And this by itself was pretty remarkable. And then Ferrari discovered a way to solve quartic equations. This put Cardano in a difficult position. He'd sworn not to reveal Tartaglia's solution, but he and Ferrari had extended it, covering all cubics, and Ferrari had done something completely unprecedented. If he waited until Tartaglia published, someone else might discover a method of solving quartic, and Ferrari would lose his chance to become the most famous mathematician in Europe. What to do? What to do? So part of what makes this story appealing to a history of mathematics is that Cardano actually did a little history of mathematics. So Cardano thought about things. He knew that Tartaglia won his reputation by defeating Fiore. But Fiore must have known how to solve the cubic, and he might have learned it from his teacher, Delfero. So in 1542, Cardano and Ferrari went to Bologna to investigate. Now by then Delfero had died, but his position and his papers were in the hands of his nephew, Annibale Dallenape. And Dallenape let Cardano and Ferrari look through Delfero's unpublished manuscripts, and they found that Delfero had, in fact, solved the Cosa and Cube equations. And this gave Cardano a loophole. Either Cardano had claimed the result as his own, so the oath of secrecy had been obtained under false pretenses, or Cardano could keep Tartaglia's method secret by publishing Delfero's method. In either case, he could publish a solution to the cubic and quartic equations. And so in 1545, Cardano published Art Magna, the great art, which gave solutions to the cubic and quartic equations. Now it's also important to note that Cardano gave proper academic credit and cited his sources. And so he said, about 30 years ago, Schipione Delfero of Bologna discovered the solution, and passed it to Antonio Maria Fiori of Venice, whose contest with Nicola Tartaglia of Russia caused him to discover it. That didn't help. Tartaglia was furious and challenged Cardano to an academic duel. Cardano declined, but Ferrari accepted. And on August 10th, 1548, in the Church of Santa Maria del Giardino de Midori o Cervante in Milan, before a packed crowd the two fought it out. This was actually a major event attended by many notables, including the Mayor of Milan. We don't know who won, but we do know Tartaglia returned to Russia, and Ferrari got offered the job of tutoring Emperor Charles V's son, which was probably about the most prestigious position an academic could obtain. Ferrari decided to stay in Milan and took a job with the Gonzaga family instead. Now bad luck seems to have followed everybody involved in solving these equations. So what happened? Tartaglia died in 1557, aged 58, never having obtained the lucrative position he sought. Ferrari died in 1565, aged 43. According to rumors, he was poisoned by his sister in an inheritance dispute. Cardano lived until 1576, a few days short of his 75th birthday, but in his later years, Cardano's eldest son was put to death for poisoning his, the son's, wife. Another of Cardano's sons sold substantial amounts of money to pay off gambling debts, and Cardano himself was arrested by the Inquisition. We don't know why. And as we'll see a little bit later, the bad luck continued to plague those who were involved with solving cubic and quartic equations.