Apart from his outstanding role in number theory, Fermat was also involved in a series of works on other mathematical areas worth mentioning. But at this point it is necessary to understand that for Fermat, mathematics was always just a form of leisure, which meant he never had any ambition in extending his fame among fellow mathematicians. To the contrary, he tried to avoid the work of publication and merely stated his discoveries in letters to some ot the best mathematicians in France. His only published treatise was anonymously published. Without Mersenne, who was in constant correspondence with Fermat and who passed his papers on to other interested mathematicians, Fermat might even have been forgotten.
Another very important aspect is also Fermat’s love for posing problems, for which he already worked out the proofs, to other mathematicians, so as to challenging them and also to lead them into his favoured fields of study. Unfortunately this method annoyed a few of his contemporaries and his ‘unsolvable’ problems were considered both as big challenges and as a source of frustration.
As explained in Chapter II, the reputation of each mathematician depended a great deal on his attitude towards his field of study, and Fermat’s attitude clearly did not make him popular among his colleagues. In light of this, it does not seem so unnatural that Fermat was only immortalized for his note about his Last Theorem, and did not have great popularity among his contemporaries. However, Fermat deserved to be famous for his major contributions in many fields of mathematics and physics, as we shall now see.
Fermat was considered to be one of the founders of analytic geometry. Early in 1637 he sent his Introduction to Plane and Solid Loci to Mersenne, who also received from René Descartes the Discours de la Methode at the same time. Both Fermat and Descartes reached the identification of equations and geometric loci. In the Géomètrie, one of the three books of the Discours, Descartes combined algebra and geometry, using one to illuminate the other and vice versa. This is considered to be the birth of analytic geometry. Apart from the content, the systematical and strictly conforming way in which Descartes used language and formulae is also considered to be the birth of modern mathematical writing. It was only fair that these facts were widely recognized and praised almost immediately after the publication. Fermat on the other hand used algebra to make the same discoveries. But his work was far more difficult to understand due to his complicated and old notations. Shortly after this, he sent his method of maxima and minima to Mersenne, of which the basic idea was the observation that
the difference between a curve and its tangent has an extremum in the tangential point.
Fermat used this observation to determine the tangents to a curve.
Descartes’ used a different approach: he used the idea that
a circle of variable centre on one of the axes has two intersections with the curve in the tangential point.
Both solutions were weak in that they were only applicable to polynomial equations. A few years later Roberval published another approach and from then on these sources gave rise to significant developments for the calculus of tangents.
Fermat wrote about his discovery of maxima and minima:
The method never fails, and it can be extended to a great number of very beautiful questions: through it we have found the centre of gravity of figures limited by curved and straight lines, of solids and of many other things which I will treat separately, if we will have the time.
Together with Blaise Pascal, with whom he was in correspondence but whom he never managed to meet, Fermat founded probability theory as a result of their works on gaming and winning chances. Pascal wrote to Fermat, asking for the solution of the socalled Problem of Points:
Consider two players, A and B. A needs to points to win and B needs 3 points. What's the chance for A to win over B?
Fermat’s answer was a very simple and yet very systematized one:
There are at most 4 games left until either A gets 2 points in which case B cannot get the 3 points to win or B gets the 3 points in which case A cannot win. Therefore there are the following possibilities for the game: (here ‘A’ states that A wins and ‘B’ states that B wins)
| BAAA | BBAA | ABAB | BABB |
| ABAA | BABA | AABB | ABBB |
| AABA | BAAB | BBBA | AAAA |
| AAAB | ABBA | BBAB | BBBB |
Then one just needs to count the winning cases for each player and gets for A the winning chance 11/16 and for B the chance 5/16.
Pascal generalized this problem to: A needs m points to win, whereas B needs n points to win. His solution is that A’s chance of winning is
sum of first m terms in the (m+n)th row of Pascal's Triangle
sum of entire (m+n)th row
and B’s chance of winning is the complement of this.
Another problem they solved together is called the Dice Problem which asks how many times one must throw a pair of dice before one expects a double six.
Fermat found out some basic integral and differential calculus before Newton was even born. As was his style, he did not give any proofs for his statements.
He generalised the parabola and hyperbola:
Parabola: y/a = (x/b)2 to (y/a)n = (x/b)m
Hyperbola: y/a = b/x to (y/a)n = (b/x)m.
In the course of examining y/a = (x/b)p, Fermat computed the sum of rp from
And, in finding his method for maxima and minima, he effectively helped form the idea of derivatives in differential calculus. He devised an algebraic procedure called adequality to determine maxima and minima on a curve, and considered the idea of a whole family of curves together. His use of algebraic methods for determining geometric curves was of great help for the later development of derivative and integral through Newton.
Fermat found the law of reflection and the law of refraction, both about how light behaves at the boundary between two media with different refraction. In 1650 he discovered that both reflection and refraction are the consequence of the same principle.
If we want the light to bounce off a mirror on its way from A to B, then Fermat's principle states that of all the possible paths the light could take with respect to. the boundary conditions, light will take the path which requires the shortest time.
In this diagram light travels from A to B. If we assume that light takes the red path, then its length is
L = (x2 + d2)1/2 + ((D – x)2 + d2)1/2.
Because of the constant speed of the light the shortest path requires the shortest time. To find it we want to find an extremum of the function L and therefore want to differentiate L with respect to x and set it equal to zero.
After simplifying the term we find
d2D2 = 2Dx, or x = D/2.
The path that takes the shortest time is the one for which x = D/2. Fermat’s principle yields the law of reflection.
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