 In his method of finding maximum and minimum values, 1636, Fermat outlined a general procedure for solving two types of problems. Today, we'd call them the optimization problem and the tangency problem. They are two of the central problems of differential calculus. Fermat illustrated his method with the problem to divide a given line into two parts, so the area of the rectangle formed by the parts is the greatest possible. Fermat's procedure went like this. Let B be the length of the line, partition it into segments of length A and B minus A. Then the area of the rectangle will be A times B minus A, which we can expand. Now consider a second partition into lengths of A plus E and B minus A minus E. The area will be, at this point Fermat does something that has caused a lot of debate among historians of mathematics. He pulls a term from Diathontis and says, adequate the two expressions for the area of the rectangle. Now if we treat this squiggle as an equals, these two expressions can be simplified and we can divide by E to get. And then Fermat says the following, we'll diminish E and so B is equal to 2A, which gives us the solution A equals B over 2. Fermat also solves the tangent problem in the same way. In Fermat's terminology, the problem is to find the sub-tangent to the parabola y squared equals x. This term sub-tangent might not be familiar, but we can view this as the line below the tangent. So Fermat proceeds as follows. Let C E equal A, the sub-tangent, B E, the actual tangent line, C I equal to E, O, a point on the tangent, and C D equal D. Let I O equal y2 and C B equal y1. By the equation of the parabola, we have D is to D minus E as y1 squared is to y2 squared. Now since this line is tangent, O has to be exterior to the parabola, so O I must be greater than y2, and so that says D over D minus E is greater than y1 squared over O I squared. Now by similar triangles, C B is to O I as E C is to E I. Well C B is just y1, so we have y1 is to O I as A is to A minus E, or squaring both sides. And so D is to D minus E is greater than A squared is to A squared minus 2AE plus E squared. Or we can cross-multiply to obtain the inequality. And we have an inequality, but if we adequate the two sides, simplify, then divide by E, then diminish E, which we can simplify to A equals to D, which gives us the length of the sub-tangent. There's a lot of debate over exactly what Fermat meant by adequate and diminish. Let's take a look at Fermat's work again. Now viewed from the perspective of calculus, it appears Fermat obtains the solution by the following reasoning. If x equals A maximizes our objective function, then x equals A plus E will be very nearly equal to the maximum value. And that's one of the meanings of this word adequate in Diathontis. And so the two values can be equated when E is equal to zero. And so it's possible to view Fermat's work as a precursor to calculus. However, this interpretation relies on what Fermat means by adequate. While he takes the term from Diathontis, it's unclear what Fermat really means by it. Now this led to a series of letters back and forth among French mathematicians that it's convenient to refer to as the Descartes-Fermat correspondence. Although Descartes and Fermat only rarely wrote directly to each other. What's important is that the Descartes-Fermat correspondence makes clear that the theory of equations was the basis for the methods of both Fermat and Descartes. In somewhat modern terms, we can describe their approach as follows. For finding extreme values, in general there are two values A and A plus E that give the same value of the objective function. But at a local extreme value, stop me if this sounds familiar, the two values are the same. And so what that says is that we can find the extreme value by requiring our objective function equal to some value. f of x equals m and otherwise have a repeated root. Similarly, we can view a secant line as intersecting the curve in two places. But as we change where the line meets the axis, those two places become one. And the system of equations giving the tangent line and the curve have a double root at the point of tangency. And so again it appears that Fermat is basing his work on the theory of equations. A non-tangent line will cross the curve at two points, but the two points will coincide if the line is tangent. So we can find the tangent line by requiring the curve and the tangent line have a double solution. This leads to a rather remarkable result. This means we can find tangents and solve optimization problems without resorting to limits. Instead, we can rely on the theory of equations. None of this messing around with things going to zero that aren't actually zero. Except there's one problem. Given an equation, what condition corresponds to a double root? In general, we require that our equation will be of the form x minus a squared times something, where our something could be a polynomial of arbitrary degree. And so now we have the problem. How can we identify the required condition?