 In the 16th century, France tried to solve the problem of having Catholics and Protestants in the same country by massacring the Protestants. This did not go over well, and in fact this led to the War of the Three Henrys, between Henry III, the King, a Catholic, Henry of Guise, a high-ranking nobleman, also Catholic, and Henry of Navarre, a high-ranking nobleman, but Protestant. For reasons that are, well, ludicrous, the two Catholic candidates, Henry III and Henry of Guise, were killed by Catholic fanatics. Like I said, the reasons are rather ludicrous. This left Henry of Navarre, a Protestant, the King of France. Spain, also Catholic, didn't like that, so they invaded. And let's now talk mathematics. Francois Vietta was trained as a lawyer and became an advisor to Henry of Navarre. During the Spanish invasion, some communications fell into Henry's hands. They were encrypted, so he gave them to Vietta, who decoded them and gained information about Spanish troop movements. This played at least some role in Henry's eventual victory, and as a matter of political expediency, Henry of Navarre, once Protestant, declared himself Catholic. But he did guarantee the right of the Protestants to worship as they pleased. This was the so-called Edict of Nantes. More importantly, the cryptography that Vietta did may have caused Vietta to revolutionize mathematics. Vietta began the practice of using letters to represent the unknown, the vowels A, E, I, O, U, and Y. While this is true, it wasn't revolutionary. Remember Deophantus used the terminal sigma, and while Cardano didn't use letters, he and the other Italian algebras used abbreviations like cos and quant to indicate the unknown value, and other authors used various other symbols. Vietta's real contributions were much more significant. He allowed consonants to represent definite but unspecified coefficients, B, G, D, C, and so on. And this allows us to write a general equation. We'll talk about why that's so very important in a moment, but where did Vietta get this idea? One theory is that he got the idea from his work decrypting Spanish messages. Now a defect of the Spanish code is that they didn't encrypt numbers. For example, a message might say something like, send 10,000 x and 500 y. And Vietta used the magnitude of these numbers to give him some insight into what x and y might represent. In this case, since there's a lot of x, you could suspect it represented something you typically have a lot of, for example, infantry. On the other hand, since there's a lot less y, they might be something a lot more rare, for example, cavalry. And perhaps if Vietta got to thinking something along the following, if the Spanish used letters to represent both the number and the quantity, the codes would have been much harder to break. So why is it so very important that Vietta started using letters to represent definite but unspecified values? Well, let's consider two solutions to the quadratic equation. Cardano solves a number of quadratic equations, then gives a rule. So for square and roots equal number, Cardano directs the reader to add the number to the square of half the number of roots. Take the square root of the whole and then subtract half the number of roots. And this is an algorithm. It tells the reader exactly how they solve the quadratic equation by giving them a sequence of steps to follow. But now for Vietta, square and roots equal number is the equation a squared plus 2 dA equal to c, where he used the two for reasons that will become obvious below. So if we follow the same steps as Cardano, we want to add the number to the square of half the number of roots. Well, we're going to add the number c and half the number of roots squared b squared. Then take the square root, then subtract half the number of roots. And here's the important difference. The expression incorporates all the steps into a single formula. In other words, Vietta was the first to write down the quadratic formula. And here's one of those early appearances. But wait, there's more. We can go further. Consider an equation where the numbers are written out. For example, 3x plus 5 equals 12. If you ask someone to solve for 3, they're not going to see the point. You already know what 3 is, so why would you want to solve for it? But if we've replaced the numbers with parameters, px plus q equals r, now we can solve for p, x, or q. And more importantly, this allows us to consider the relationship between the solutions and the coefficients of an equation. So a typical example of this analysis occurs in his first treatise on recognition of equations. In this treatise, Vietta examines the relationship between the solutions of an equation and its parameters. For example, consider the equation ba minus a squared equals c. If e is a different solution, what is the relationship between a and e? So Vietta proceeds as follows. If a and e are solutions, then it must be true that be minus e squared equals c, and also ba minus a squared equals c. Since these are both equal to c, we can set them equal to each other, and we can solve for b. And so this says that if a and e are two solutions to ba minus a squared equals c, then b, the coefficient of a, is e plus a. But we can go further. Since b is equal to e plus a, then from ba minus a squared equals c, we can replace, and that tells us something about c is equal to ea. And there's several important consequences. One of the consequences is that this means that if we know one solution, e or a, we can find the other, because we know that the sum of the two solutions is b and their product is c. Moreover, if this means that there can only be two solutions, because as soon as you have one, you have the other solution.