Pierre de Fermat and the Invention
of Differential Calculus
1. The life of Fermat
Pierre de Fermat (1601 C.E. – 1665 C.E.) was one of the most remarkable mathematicians who ever lived. He spent his entire adult life as a magistrate or judge in the city of Toulouse, France. His career was marked by prudence, honesty, and fairness (Figure 1)
Fermat is remembered today by a large statue that is in the basement of the Hotel de Ville in Toulouse. The statue depicts Fermat, dressed in formal attire, and seated.
He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius’s Plane loci to one of the mathematicians there.
From Bordeaux, Fermat went to Orleans where he studied law at the University. So by 1631, Fermat was a lawyer and government official in Toulouse and because of the office he now held, he became entitled to change his name from Pierre Fermat to Pierre de Fermat.
Fermat was struck down by the plague and in 1653 his death was wrongly reported. The period from 1643 to 1654 was one when Fermat was out of touch with his scientific colleagues in Paris.
Fermat is best remembered for this work in number theory, in particular for Fermat’s Last Theorem
xn + yn = zn
has no non-zero integer solutions x, y and z when the integer exponent n > 2.
These marginal notes only became known after Fermat’s death, when his son Samuel published an edition of Bachet’s translation of Diophantus’s Arithmetica with his father notes in 1670. It is now believed that Fermat’s “proof” was wrong although it is impossible to be completely certain.
Fermat correspondence with the Paris mathematicians restarted in 1654 when Blaise Pascal, Etienne Pascal’s son, wrote to him to ask for confirmation about his ideas on probability.
2. Fermat’s Method
One of the fundamental ideas of calculus is to calculate the tangent line to a given curve. Figure 2 exhibits the familiar idea of the tangent line to a circle. This is particularly simple situation. In classical geometry texts, we are told that the tangent line to a circle C at a point P of the circle is that line which passes through P and is perpendicular to the radius at P.
For a more general curve, the tangent line is that it passes through P and touches the curve at P.
Fermat’s idea was this: the tangent line has the special feature that it only intersects the curve at one point.
In order to actually implement Fermat’s idea, we shall need the concept of slope. Recall that if we are given a line l in the plane and two points (p1,q1) and (p2,q2) on that line, than the slope of the line is
Figure 3 illustrates the idea of slope. The number m represents the ratio of “rise” over “run” for this line.
Use Fermat’s idea to find the tangent line to the curve y = x2 at the point (2, 4).
3. More Advanced Ideas of Calculus: The Derivative and the Tangent Line
See figure 4. It takes two points to determine the slope of a line, yet we are only given the point (c, f(c)) on the graph. One reasonable interpretation of the slope at (c, f(c)) is that it is the limit of the slopes of secant lines determined by (c, f(c)) and nearby points (c+h, f(c+h)). See the dotted line in figure 4. When we say limit, we mean to consider the behavior of the expression as h tends to 0. Let us calculate this limit:
Now, this last limit is what we shall call the derivative of f at c. we denote the derivative by f’(c). When the limit exists, we say that the function f is differentiable at c.
4. Fermat’s Lemma and Maximum/Minimum Problems
Fermat’s lemma is based on a simple geometric observation about differentiable functions. Examine the graph exhibited in figure 5. The point P is vertically higher than its neighbors. We say that P is a local maximum. We see that the point Q is vertically lower than its neighbors. We say that Q is local minimum.
The tangent line is horizontal in the point P. That means that its slope is zero. Thus the derivative of a function at a point of differentiability where the function assumes a local maximum is 0. With the same case, the tangent line is horizontal in the point Q. That means that its slope is zero. Thus the derivative of a function at a point of differentiability where the function assumes a local minimum is 0. These two displayed rules are the content of Fermat’s lemma.
A point x is a critical point for the function f if f’(x) = 0. If f’(x) > 0 then this says that the approximating quotients (or Newton quotients)
are positive. If f’(x) < 0 then we see that the approximating quotients (or Newton quotients)