## A historical interlude Pierre de Fermat 16011665

Born near Montauban, France, in 1601, Pierre de Fermat was a lawyer and a government official. In 1648, he rose to the position of chief spokesman for the parliament, and president of the Chambre de l'Edit, which had jurisdiction over lawsuits between Huguenots and Catholics. Despite his nominally high office, he seems not to have been particularly devoted to matters of law and government; indeed, a confidential report by the intendant, sent to Jean Baptiste Colbert* in 1664, was quite critical of his performance.

For Fermat, mathematics was a hobby. One can surmise that his 'day job' bored him and that he devoted more of his time to mathematics than was considered proper by his superiors. He published very little of his work and most of his results were found after his death, written on loose sheets of paper or on margins of textbooks. He had a mischievous streak and enjoyed teasing other mathematicians by stating results and theorems without revealing the proof. His son Samuel found what is his now-famous Last Theorem, scribbled as a marginal note in his father's copy of Diophantus'Arithmetica.

Fermat's Last Theorem states that the equation xn + yn = zn

* Chief Financial Minister of Louis XIV.

has no non-zero integer solutions for x, y and z when n > 2. Fermat wrote: "I have discovered a truly remarkable proof which this margin is too narrow to contain."

In 1908, the mathematician Paul Wolfskehl of Darmstadt, Germany, bequeathed a sum of 100,000 marks to the Academy of Sciences in Gottingen. It was to be awarded to the first person to publish a complete proof of Fermat's theorem for all values of n. Nobody succeeded in finding a proof until 1997, when Andrew Wiles, an English mathematician, following 11 years' work, published what has been generally accepted as a complete proof. It showed that Fermat was correct in his result although, given the complexity of the ultimate theorem, Fermat's own 'remarkable proof' may well have been wrong.

Fermat was also interested in the mathematics of maxima and minima, and developed methods for finding tangents to curved lines and surfaces. It is this work which led to his principle of least time. His mathematical methods formed the central issue of a debate with Descartes in the spring of 1638. Descartes, who had an aggressive temperament and a sharp tongue, considered Fermat a rival and disapproved of his mathematical reasoning. As a result there was little cooperation between the two men, who were arguably the greatest mathematicians of the time. The two made a formal peace when finally Descartes admitted the error of his criticisms of Fermat's methods, but even then there seems to have been little love lost between them.

It is remarkable that Fermat was able to come to his conclusions not only geometrically, but also analytically, using methods similar to calculus, the invention of which is normally attributed to Isaac Newton (1642-1727) and Gottfried Leibnitz (1646-1716) about 50 years later.

The principle of least time implies a fixed and therefore finite speed of light. In those days this speed was totally unknown. Opinion in Fermat's era still followed the belief of Aristotle that the speed of light was infinite until the first measurement of c was made in 1677 by Olaf Römer (see Chapter 1).

Fermat was not sure whether or not light is transmitted instantaneously, and even less sure that it had a finite speed which differed from medium to medium. Nevertheless, he undertook a mathematical derivation of the physical law of refraction on the basis of the postulates that:

(1) 'the speed of light varies as the rarity of the medium through which it passes' and

(2) 'nature operates by the simplest and most expeditious ways and means'.

To Fermat's surprise (expressed in a letter to Clerselier in 1662), his mathematical analysis led to the experimental law now known as Snell's law of refraction.

Appendix 2.1 The Parabolic mirror

The parabolic mirror — a perfect illustration of the law of least time

A definition of a parabola is 'the locus of all points equidistant from a point called the focus, and a straight line called the directrix'. This is exactly what is needed to construct a mirror with no spherical aberration. The surface of such a mirror has the shape of a paraboloid of revolution.

In the case of the parabolic mirror it is easier to apply the law of least time directly. Consider a point source of light at an infinite distance from the mirror. Imagine that at a given instant a number of photons leave the source, and spread out in all directions. At any given time later they will occupy a growing spherical surface, which at infinite distance from the source will become a plane of photons travelling parallel to one another.

property of a parabola: FA' = AA'', FB' = BB', FC' = CC'

Figure 2.8 Parabolic mirror.

directrix A''I

property of a parabola: FA' = AA'', FB' = BB', FC' = CC'

### Figure 2.8 Parabolic mirror.

Consider three such photons arriving in a parallel beam ABC, as illustrated in Figure 2.8. What is the shortest and therefore quickest route to F, assuming that they are going to be reflected by the mirror?

The geometrical property of the parabola gives FA' = A'A", FB' = B'B" and FC' = C'C"

Since the planes ABC and A"B"C" are parallel, AF = BF = CF.

This means that the travel times for all light rays to F from the plane ABC (and therefore also from the 'point at infinity') are equal.

To show that the equal paths are also the shortest paths:

The paths taken by the three rays in Figure 2.8 are exactly equal. They are also the shortest paths.

It is easy to see that, for example, there is no shorter path from A to F via reflection at the mirror than AA'F.

Try for instance AB'F:

AA'F = AA'', BB'F = BB'' = AA'' but AB' > BB' ^ AB'F > AA'F The path via B' is longer.

We can conclude that:

Photons which cross the plane ABC together arrive at the focus together.

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