Aug 172016
 

pf1

Today is the birthday of Pierre de Fermat, a French lawyer at the Parlement of Toulouse and a mathematician who is given credit for early developments that led to the development of calculus. His year of birth is given variously as 1601 and 1607. He is best known, publicly and in the world of mathematics, for Fermat’s Last Theorem, which he described in a note in the margin of a copy of Diophantus’ Arithmetica. I’ll try not to wear you out with mathematics, but I do want to celebrate the life of a person who tends to be forgotten these days.

Fermat was born in Beaumont-de-Lomagne in southern France. The late 15th-century mansion where Fermat was born is now a museum. His father, Dominique Fermat, was a wealthy leather merchant, and served three one-year terms as one of the four consuls of Beaumont-de-Lomagne. His mother was either Françoise Cazeneuve or Claire de Long. Pierre had one brother and two sisters and was almost certainly brought up in the town of his birth. There is little evidence concerning his school education, but it was probably at the Collège de Navarre in Montauban.

pf5

He began studies at the University of Orléans in1623 and received a bachelor in civil law in 1626, before moving to Bordeaux. In Bordeaux he began his first serious mathematical researches, and he produced important work on maxima and minima which he gave to Étienne d’Espagnet who clearly shared mathematical interests with Fermat. There he became much influenced by the work of François Viète.

In 1630, he bought the office of a councilor at the Parlement de Toulouse, one of the High Courts of Judicature in France, and was sworn in by the Grand Chambre in May 1631. He held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat. Fermat was fluent in six languages, French, Latin, Occitan, classical Greek, Italian, and Spanish, and was praised for his written verse in several languages, as well as his advice regarding the emendation of Greek texts.

pf2

He communicated most of his work in letters to friends, often with little or no proof of his theorems. Secrecy was common in European mathematical circles at the time. This naturally led to priority disputes with contemporaries such as Descartes and Wallis. Fermat’s mathematics derived mainly from  classical Greek treatises combined with new algebraic methods he learned from colleagues.

Fermat’s pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes’ famous La géométrie. This manuscript was published posthumously in 1679 in Varia opera mathematica, as Ad Locos Planos et Solidos Isagoge, (“Introduction to Plane and Solid Loci”). In Methodus ad disquirendam maximam et minimam and in De tangentibus linearum curvarum, Fermat developed a method (adequality) for determining maxima, minima, and tangents to various curves that was equivalent to differential calculus. Fermat was the first person known to have evaluated the integral of general power functions. With his method, he was able to reduce this evaluation to the sum of geometric series. The resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus.

OK, I’ll spare you too much more. Briefly, Fermat was fascinated by number theory, and, in particular with whole numbers. As such you could say that he was a true disciple of Pythagoras whose philosophical school saw whole numbers as mystical as well as being the bedrock of the laws of the cosmos. Although Fermat claimed to have proven all of his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss, doubted several of his claims, especially given the difficulty of some of the problems and the limited mathematical methods available to Fermat. His famous Last Theorem was first discovered by his son in the margin in his father’s copy of an edition of Diophantus, and included the statement that the margin was too small to include the proof. It seems that he had not written to anyone about it. It was finally proven in 1994 by Sir Andrew Wiles, using techniques unavailable to Fermat.

Through their correspondence in 1654, Fermat and Blaise Pascal helped lay the fundamental groundwork for the theory of probability. From this brief but productive collaboration on the problem of points, they are now regarded as joint founders of probability theory. Fermat is credited with carrying out the first ever rigorous probability calculation. In it, he was asked by a professional gambler why if he bet on rolling at least one six in four throws of a die he won in the long term, whereas betting on throwing at least one double-six in 24 throws of two dice resulted in his losing. Fermat proved why this was the case mathematically.

pf7

I should at least give a special nod to Fermat’s Last Theorem, sometimes called Fermat’s Conjecture, especially in older texts, because it was not proven. This shouldn’t baffle too many readers, I hope. It states that no three positive integers (whole numbers) a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 have been known to have infinitely many solutions since antiquity. The case of n =1 is trivial because any number raised to the first power is itself. So you can do this in your head – for example, 11 + 21 = 31 and so forth. If you remember geometry and Pythagoras you’ll also remember classic cases such as 32 + 42 = 52 (9 + 16 = 25). It’s one thing to go through countless examples of numbers raised to the 3rd, 4th, 5th, millionth power etc and show that they don’t work. It’s quite another to prove that none work.

The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof, it was in the Guinness Book of Records as the “most difficult mathematical problem,” one of the reasons being that it has the largest number of unsuccessful proofs.

pf6

Fermat died at Castres, in the department of Tarn. Isaac Newton once said when someone praised him that he saw so far because he stood on the shoulders of giants. Fermat was one of those giants.

pf3

Toulouse was Fermat’s home most of his life, so let’s talk about Toulouse sausage to begin with. Toulouse sausages are legendary and nowadays can be found over most of France and parts of western Europe. I don’t know if they have protected status, but obviously they are best in and around Toulouse. They are made from coarsely chopped fatty pork, smoked bacon, garlic, pepper and red wine. They are sold raw and must be cooked before eating. They are commonly used in cassoulet, which is also native to Toulouse. Cassoulet is a rich, slow-cooked casserole originating in Languedoc, which contains meat (typically pork sausages, goose, duck and sometimes mutton), pork skin (couennes) and white beans (haricots blancs). In Toulouse, sausage, pork, and mutton are the most common meats. The dish is named after its traditional cooking vessel, the cassole, a deep, round, earthenware pot with slanting sides.

pf4

Modern cooks usually use pre-cooked beans, or beans simmered in a broth with vegetables, and pre-cooked meats for simplicity. But this practice is not traditional. According to common folklore, home peasant cooks had one cassole that they used exclusively for cassoulet which they never washed. They simply deglazed it and started again, so that it was imbued with the flavors of cassoulets past. In theory, therefore, today’s cassoulet could be the end product of years, or decades, of continuous use. I’m perfectly in tune with this impulse. My cast-iron skillets and wooden salad bowl went unwashed for years in my old kitchen. No doubt germophobes will protest, and I am not recommending the habit. However, I will note that I am still alive, and my cooking did not make anyone sick. I will also note that I took reasonable precautions to make sure that harmful stuff was not lurking about despite not washing my pots.

Cassoulet

Ingredients

1 lb/½kg dried haricot beans
salt and pepper
4 cups chicken stock
3 packets unflavored gelatin
2 tbsp duck fat (or vegetable oil)
8 oz/250g  salt pork, cut into small cubes
1 lb/½ kg Toulouse sausage (about 2 to 4 links)
1 large onion, finely diced
1 carrot, unpeeled, cut into large sections
2 stalks celery, cut into large sections
1 whole head garlic, peeled and thinly sliced
4 sprigs parsley
2 bay leaves
6 whole cloves

Instructions

Place the dried beans in salted water to cover in a large pot, and soak overnight.

Next day, preheat the oven to 300°F/150°C.

Warm the stock slightly and sprinkle the gelatin over the top. Stir and set aside.

Heat the duck fat  in a Dutch oven over high heat.  Add the salt pork and sauté until browned on all sides. Remove the salt pork with a slotted spoon and set aside.

Add the sausages to the pot and sauté until well-browned on both sides. Add to the cooked salt pork and set aside. Remove all but about 2 tablespoons fat from pot.

Add the onions to the pot and sauté until they are translucent. Drain the beans and add them to the pot along with the carrot, celery, garlic, parsley, bay leaves, cloves, and stock/gelatin mixture. Bring to a simmer over high heat. Reduce to low, cover and cook until the beans are almost tender, about 45 minutes.

Pick out the carrots, celery, parsley, bay leaves, and cloves and discard. Add the sausage and salt pork to the pot and mix everything together.

Transfer the pot to the oven and cook, uncovered, until a thin crust forms on top, about 2 hours, adding more water by pouring it carefully down the side of the pot as necessary to keep beans mostly covered.

Break the crust with a spoon and shake the pot gently to redistribute the beans and meat. Return to the oven and continue cooking, stopping to break and shake the crust every 30 minutes until the 4 ½ hour mark.

Return to the oven and continue cooking undisturbed until the crust is deep brown and thick, about 5 to 6 hours total. Serve immediately.

 Leave a Reply

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>

(required)

(required)

This site uses Akismet to reduce spam. Learn how your comment data is processed.