Apr 152018
 

Today is the birthday (1707) of Leonhard Euler, a Swiss-born mathematician, physicist, astronomer, logician and engineer, who was unquestionably the most prolific, and one of the most influential, mathematicians in the West of all time. His written works fill around 80 quarto volumes. He, like so many other great mathematicians of the past, is not a household name these days, although you may know what an Euler diagram is, or you may know that the mathematical constant e is also known as Euler’s number, because he was the first to prove that e is irrational (“e” stands for “Euler”). I am going to spare you a diatribe on mathematics, working on the assumption that most people’s eyes glaze over when I stray too far from 2 + 2 = 4. This fact of life is a great pity in my ever-humble opinion. Mathematics and mathematical logic are useful intellectual tools. They are not the only tools in the toolbox, nor necessarily the most useful, but deep thinking is difficult without them. Care to build a shed without a hammer? It can be done, but is easier with one. I’ll delve into Euler’s life and influence mostly, and just give you a taste of what his mathematics can (and cannot) do.

Euler was born in Basel in Switzerland to Paul Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastor’s daughter. He had two younger sisters: Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family. Johann Bernoulli was then regarded as Europe’s foremost mathematician, and would eventually be the most important influence on young Leonhard.

Euler’s formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged 13, he enrolled at the University of Basel, and in 1723 (aged 16), he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his pupil’s incredible aptitude for mathematics. At that time Euler’s main studies included theology, Greek, and Hebrew at his father’s urging in order to become a pastor, but Bernoulli convinced his father that Euler was destined to become a great mathematician.

In 1726, Euler completed a dissertation on the propagation of sound, titled De Sono. At that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition; the problem that year was to find the best way to place the masts on a ship. Pierre Bouguer, who became known as “the father of naval architecture,” won and Euler took second place. Euler later won this annual prize 12 times.

Around this time Johann Bernoulli’s two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. On 31st July 1726, Nicolaus died of appendicitis after spending less than a year in Russia, and when Daniel assumed his brother’s position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler accepted the offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.

Euler arrived in Saint Petersburg on 17th May 1727. He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he often worked in close collaboration. Euler mastered Russian and settled into life in Saint Petersburg. He also took on an additional job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy in order to lessen the faculty’s teaching burden, and the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions.

The Academy’s patron, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler’s arrival. The Russian nobility then gained power upon the ascension of the 12-year-old Peter II. The nobility was suspicious of the academy’s foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues. Conditions improved slightly after the death of Peter II, and Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731. Two years later, Daniel Bernoulli, who was fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department.

Concerned about the continuing turmoil in Russia, Euler left St. Petersburg  in June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. He lived for 25 years in Berlin, where he wrote over 380 articles. In Berlin, he published the two works for which he would become most renowned: the Introductio in analysin infinitorum, a text on functions, published in 1748, and the Institutiones calculi differentialis, on differential calculus, published in 1755.

Euler was asked to tutor Friederike Charlotte of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Frederick’s niece. Euler wrote over 200 letters to her in the early 1760s, which were later compiled into a best-selling volume: Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess. This work contained Euler’s exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler’s personality and religious beliefs. This book became more widely read than any of his mathematical works and was published across Europe and in the United States. The popularity of the “Letters” testifies to Euler’s ability to communicate scientific matters effectively to a lay audience.

Despite Euler’s immense contribution to the Academy’s prestige, he eventually incurred the wrath of Frederick and ended up having to leave Berlin. The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs, in many ways the polar opposite of Voltaire, who enjoyed a high place of prestige at Frederick’s court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire’s wit. Frederick also expressed disappointment with Euler’s practical engineering abilities:

I wanted to have a water jet in my garden: Euler calculated the force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry!

Euler’s eyesight worsened throughout his mathematical career. In 1738, three years after nearly dying from a fever, he became almost blind in his right eye, but Euler preferred to blame the painstaking work on cartography he performed for the St. Petersburg Academy for his condition. Euler’s vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as “Cyclops”. Euler later developed a cataract in his left eye, which was discovered in 1766. Just a few weeks after its discovery, he was rendered almost totally blind. However, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and exceptional memory. Upon losing the sight in both eyes, Euler remarked, “Now I will have fewer distractions.” Euler could repeat Virgil’s Aeneid from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last. With the aid of his scribes, Euler’s productivity on many areas of study actually increased. He produced, on average, one mathematical paper every week in the year 1775. The Eulers bore a double name, Euler-Schölpi, the latter of which derives from schelb and schief, signifying squint-eyed, cross-eyed, or crooked. This suggests that the Eulers may have had a genetic disposition to eye problems.

In 1760, with the Seven Years’ War raging, Euler’s farm in Charlottenburg was ransacked by advancing Russian troops. Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for the damage caused to Euler’s estate, later Empress Elizabeth of Russia added a further payment of 4000 rubles – an exorbitant amount at the time. The political situation in Russia stabilized after Catherine the Great’s accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were steep – a 3000 ruble annual salary, a pension for his wife, and the promise of high-ranking appointments for his sons. All of these requests were granted. He spent the rest of his life in Russia. However, his second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife Katharina after 40 years of marriage. Three years after his wife’s death, Euler married her half-sister, Salome Abigail Gsell (1723–1794). This marriage lasted until his death.

In St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet Uranus and its orbit with a fellow academician Anders Johan Lexell, when he collapsed from a brain hemorrhage. He died a few hours later. French mathematician and philosopher Marquis de Condorcet, wrote: “il cessa de calculer et de vivre” (he ceased to calculate and to live). Euler was buried next to Katharina at the Smolensk Lutheran Cemetery on Goloday Island. In 1785, the Russian Academy of Sciences put a marble bust of Leonhard Euler on a pedestal next to the Director’s seat and, in 1837, placed a headstone on Euler’s grave. To commemorate the 250th anniversary of Euler’s birth, the headstone was moved in 1956, together with his remains, to the 18th-century necropolis at the Alexander Nevsky Monastery.

Here I am going to touch on Euler’s contributions to mathematics and related fields, so, if your eyes glaze over at this stuff, skip to the recipe. This section is not really technical (just a tiny bit). Euler worked in almost all areas of mathematics, such as geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. He is the only mathematician to have two numbers named after him: the Euler number, e, approximately equal to 2.71828, and the Euler–Mascheroni constant γ (gamma) sometimes referred to as just “Euler’s constant,” approximately equal to 0.57721.

Euler introduced and popularized several notational conventions, that are now commonplace, through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (not originally “e” for “Euler’s number”), the Greek letter Σ for summations, and the letter i to denote the square root of -1. The use of the Greek letter π to denote the ratio of a circle’s circumference to its diameter was also popularized by Euler, although it originated with Welsh mathematician William Jones.

The development of infinitesimal calculus was at the forefront of 18th-century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Because of their influence, studying calculus became a major focus of Euler’s work. Newton and Leibniz got the ball rolling by showing that if you tolerated the concept of infinity (in mathematics) a giant new world opened up that had not been known in the West before. You have to grasp – maybe against your intuition, or common sense – that as you get closer and closer and ever closer to infinity with a series of numbers that are getting smaller and smaller and yet smaller, a simple answer (almost magically) pops out when you get all the way to infinity (known as the limit). Getting the answer is almost a leap of faith although mathematicians won’t admit this. The classic “explanation” is to take the number 1.999999999999999999999999999 with the 9s extending all the way to infinity. As the list of 9s gets longer and longer, the number gets closer and closer to 2. So, at the limit 1 followed by infinite 9s is the same as 2. There’s your leap of faith. It is not just a tiny bit smaller than 2, it is exactly equal to 2.

Brilliant mathematicians like Euler appear to be able, not only to grasp mathematical concepts intuitively, but also to see patterns between seemingly disparate mathematical expressions. The ratio pi, for example, concerning the diameter and circumference of a circle, shows up all over the place in expressions that do not seem to have anything to do with circles. It is almost mystical. Mathematicians like Euler are not worried by this oddity; they see much deeper into the structure of mathematics than ordinary mortals, in ways that seem obvious to them, but are opaque to the rest of us. For example, he derived the formula known as Euler’s identity:

e i π + 1 = 0

Richard Feynman called it the “most remarkable formula in all mathematics” because it pulls together fundamental, but rather quirky, constants of mathematics in one neat bundle combining the operations of addition, multiplication, exponentiation, and equality.

Euler also pioneered the use of analytic methods to solve number theory problems. Euler’s interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. A lot of Euler’s early work on number theory was based on the works of Pierre de Fermat. Euler developed some of Fermat’s ideas and disproved some of his conjectures. He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. A perfect number is a number that is the sum of all of its positive divisors (excepting itself). So, for example, 6 is a perfect number because its divisors are 1, 2, and 3, and 1 + 2 + 3 = 6.

In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg. The city of Königsberg in Prussia was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. Euler proved it is not possible. This solution is considered to be the first theorem of graph theory, specifically of planar graph theory. Euler also discovered the formula V − E + F = 2 relating the number of vertices (V), edges (E) and faces (F) of a convex polyhedron, and hence of a planar graph.

One of Euler’s more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.

Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering. Aside from successfully applying his analytic tools to problems in classical mechanics, Euler also applied these techniques to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the parallax of the sun. His calculations also contributed to the development of accurate longitude tables.

In addition, Euler made important contributions in optics. He disagreed with Newton’s corpuscular theory of light in the Opticks, which was then the prevailing theory. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christiaan Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light.

Euler is also credited with using closed curves to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagrams which are sometimes confused with Venn diagrams. Here is a series of images that might help explain the difference.

An Euler diagram is a diagrammatic means of representing sets and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict sets. Each Euler curve divides the plane into two regions or “zones”: the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. In Venn diagrams every closed curve must intersect every other curve, but in Euler diagrams they do not.

Much of what is known of Euler’s religious beliefs can be deduced from his Letters to a German Princess and an earlier work, Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister (Defense of the Divine Revelation against the Objections of the Freethinkers). These works show that Euler was a devout Christian who believed the Bible to be inspired; the Rettung was primarily an argument for the divine inspiration of scripture.

The dish known as French Meat was developed in St Petersburg in Euler’s time – a time when the Russian aristocracy wanted to appear more cosmopolitan to the outside world. The dish is unknown in France, of course, but it has remained popular in parts of Europe.  Worth a try, I’d say. I like it, and it is simple to make. The order of layers in the dish may vary depending on your preferences. The bottom layer can be onions to create a stratum between the meat and the baking tray, or potatoes, which will result in the dish saturated with pork fat. Some people make French Meat without potatoes. In this case, the pork chunks should be larger. Some don’t use mayonnaise, but the cheese-mayonnaise layer should always be on top, creating an aromatic gratin cheese crust while the dish is in the oven.

French Meat

Ingredients

500 gm/ 1lb moderately fat pork, cut in small chunks
600 gm/ 1 ¼ lb potatoes, peeled and sliced
4 large onions, peeled and sliced
300 gm/ 10 ½ oz melting cheese, grated
200 grams/ 7 oz (approx.) mayonnaise
salt, pepper to taste

Instructions

Pre-heat the oven to 200˚C/400˚F.

Grease a casserole and spread the pork in a layer on the bottom. Cover the pork evenly with a layer of onion slices. Put a layer of thin slices of potato on top of the onions. Season with salt and pepper to taste. Top with a layer of grated cheese smothered in mayonnaise using a tablespoon or a cooking brush.

Bake the dish  for about 30 minutes. The dish is ready when the top layer of cheese is golden and bubbly. Remove the casserole from the oven and let it cool for 10 minutes before serving in blocks or slices.

Aug 172016
 

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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.

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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.

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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.

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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.

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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.

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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.

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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.