 The Chinese solved cubic equations through a process of successive approximations. They could find as many digits of the root of a polynomial equation as they wanted. But we do want to ask, does this count as solving the equation? We also saw that Islamic mathematicians solved cubic equations by identifying conic sections whose intersection points corresponded to the solution of geometric problems. But again, we might ask, does this count as solving the equation? And the question comes down to what we mean by a solution. For example, consider the equation x squared equals 3. We might give the solution x equals the square root of 3. But what would Chinese and Islamic mathematicians say about this solution? Well, in China, you might get a response like, well, that's nice, an expression of the solution, but what's the actual value? Meanwhile, an Islamic mathematician might say, OK, then draw me a line with length square root 3. And so here's the important idea. x equals square root 3 is only a solution because we agree that it's a solution, and not because it is in any way better than what Chinese or Islamic mathematicians would have found. Now the reason for that goes back to the Middle Ages, and roughly speaking, since the Middle Ages, European mathematicians increasingly focused on what we would call algebraic solutions. We say that the solution is algebraic if it can be expressed using the basic operations of arithmetic, addition, subtraction, multiplication, division, and roots, applied a finite number of times. And so this expression, which only involves roots, addition, subtraction, is an algebraic solution. On the other hand, if we write something like this, it's not algebraic, because the ellipses here indicates that we're going to perform an infinite number of additional operations. And this is why the Chinese solutions are not considered algebraic solutions. And we might go a little bit further. x equals arctangent of 3 or x equals log 5 are not algebraic solutions. They don't use the basic operations of arithmetic applied a finite number of times. Now since the distinguishing feature of an algebraic solution is the nth root, we often say that algebraic solutions are solutions in radicals. So part of what makes the story of the cubic very interesting is that bad luck seems to have followed everyone who tried to solve it. If we wish to engage in a little melodrama, there is a curse of the cubic. Our story begins in Renaissance Italy, and so in fair Bologna do we set our scene. By 1500 the Italian peninsula became the mathematical center of Europe and the University of Bologna the intellectual center of Italy. Luca Paccioli was among the faculty and Paccioli attempted to find an algebraic solution to the cubic but failed. Consequently in his Suma, a compendium of mathematical, geometrical and artistic knowledge, he declared that the cubic was impossible to solve. We might regard Paccioli's comment as a way to salv his own ego because Scipione del Ferro also taught at the University of Bologna around the same time and he disagreed with his colleague. And around 1501 he found a solution to what was then called the Cosa and Cube equation. Now an Italian Cosa is big and it's a way that the Italian algebras refer to the unknown. So these are equations we'd write as x cubed plus px equals q, where p and q are positive constants. And here's our Cosa, some number of things, and cube. Now an important factor in the story is the nature of academic jobs in the 16th century. In mathematics your reputation was based on two things. Your ability to solve problems, that's kind of obvious, but your ability to pose unsolvable problems that you could still solve. This meant that if you could do something no one else could do, you were assured of continual employment. And whatever you did, you didn't make your method public. Now eventually del Ferro retired and taught his method to one of his favorite students, Antonio Maria Fiori, who lived around 1555 from Venice. And so now a little bit of history. In 1512 French forces invaded Italy as a result of a series of dynastic wars that are kind of complicated to explain and might make for a good series of novels, maybe set in some sort of fantasy setting with dragons. Where was I? Oh yeah, the French invasion went through Russia. Because the inhabitants of Russia fought back a little too effectively, the French retaliated and massacred thousands of inhabitants of the city. One of the survivors was a 13-year-old boy named Nicolo Fontana. However a French knight, the Paragon of Chivalry, slashed Fontana across the face with a sword which left the boy with a permanent speech impediment. And subsequently Fontana became known as Tartalia, the stutterer. In 1534 Tartalia moved to Venice. At the time Venice had a reputation as a town with wild raucous parties. And not much has changed. Now remember Fiori? Fiori had established a reputation as a mathematician while Tartalia was a boy from the backwoods. For reasons that are unclear, Fiori, the established and well-known mathematician, challenged Tartalia, the nobody from nowhere, to a duel. An academic duel. Each would pose 30 questions, and the loser would pay for 30 banquets. Tartalia posed a variety of problems, but Fiori only posed one type of problem. He posed 30 closer and cube equations. When Tartalia saw the questions he panicked. He had no idea how to solve the closer and cube equations. At least not at first. Of February 12-13, 1535, a few days before the answers were due, he had a sudden inspiration. And with that inspiration he solved all 30 of Fiori's problems. And established his reputation as one of the preeminent mathematicians of northeastern Italy. And for now we'll leave Tartalia after his stunning victory and go west to the Duchy of Milan. And here we'll pick up part two of the story of the Curse of the Cubic.