Misplaced Pages

#433566

70-477: Fermat was born in 1607 in Beaumont-de-Lomagne , France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony , where 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 Claire de Long. Pierre had one brother and two sisters and was almost certainly brought up in

140-516: A x 1 + Δ x f ( t ) d t − ∫ a x 1 f ( t ) d t = ∫ x 1 x 1 + Δ x f ( t ) d t , {\displaystyle {\begin{aligned}F(x_{1}+\Delta x)-F(x_{1})&=\int _{a}^{x_{1}+\Delta x}f(t)\,dt-\int _{a}^{x_{1}}f(t)\,dt\\&=\int _{x_{1}}^{x_{1}+\Delta x}f(t)\,dt,\end{aligned}}}

210-423: A x f ( t ) d t . {\displaystyle F(x)=\int _{a}^{x}f(t)\,dt.} Then F is uniformly continuous on [ a , b ] and differentiable on the open interval ( a , b ) , and F ′ ( x ) = f ( x ) {\displaystyle F'(x)=f(x)} for all x in ( a , b ) so F is an antiderivative of f . The fundamental theorem

280-401: A x f ( t ) d t = G ( x ) − G ( a ) {\textstyle F(x)=\int _{a}^{x}f(t)\,dt=G(x)-G(a)} . Then F has the same derivative as G , and therefore F ′ = f . This argument only works, however, if we already know that f has an antiderivative, and the only way we know that all continuous functions have antiderivatives

350-1442: A ) = F ( x n ) − F ( x 0 ) . {\displaystyle F(b)-F(a)=F(x_{n})-F(x_{0}).} Now, we add each F ( x i ) along with its additive inverse, so that the resulting quantity is equal: F ( b ) − F ( a ) = F ( x n ) + [ − F ( x n − 1 ) + F ( x n − 1 ) ] + ⋯ + [ − F ( x 1 ) + F ( x 1 ) ] − F ( x 0 ) = [ F ( x n ) − F ( x n − 1 ) ] + [ F ( x n − 1 ) − F ( x n − 2 ) ] + ⋯ + [ F ( x 2 ) − F ( x 1 ) ] + [ F ( x 1 ) − F ( x 0 ) ] . {\displaystyle {\begin{aligned}F(b)-F(a)&=F(x_{n})+[-F(x_{n-1})+F(x_{n-1})]+\cdots +[-F(x_{1})+F(x_{1})]-F(x_{0})\\&=[F(x_{n})-F(x_{n-1})]+[F(x_{n-1})-F(x_{n-2})]+\cdots +[F(x_{2})-F(x_{1})]+[F(x_{1})-F(x_{0})].\end{aligned}}} The above quantity can be written as

420-428: A continuous function f , an antiderivative or indefinite integral F can be obtained as the integral of f over an interval with a variable upper bound. Conversely, the second part of the theorem, the second fundamental theorem of calculus , states that the integral of a function f over a fixed interval is equal to the change of any antiderivative F between the ends of the interval. This greatly simplifies

490-615: A ) is dependent on ‖ Δ x i ‖ {\displaystyle \|\Delta x_{i}\|} , so the limit on the left side remains F ( b ) − F ( a ) . F ( b ) − F ( a ) = lim ‖ Δ x i ‖ → 0 ∑ i = 1 n [ f ( c i ) ( Δ x i ) ] . {\displaystyle F(b)-F(a)=\lim _{\|\Delta x_{i}\|\to 0}\sum _{i=1}^{n}[f(c_{i})(\Delta x_{i})].} The expression on

560-406: A ) . In other words, G ( x ) = F ( x ) − F ( a ) , and so ∫ a b f ( x ) d x = G ( b ) = F ( b ) − F ( a ) . {\displaystyle \int _{a}^{b}f(x)\,dx=G(b)=F(b)-F(a).} This is a limit proof by Riemann sums . To begin, we recall the mean value theorem . Stated briefly, if F

630-581: A , b ] , and let f admit an antiderivative F on ( a , b ) such that F is continuous on [ a , b ] . Begin with the quantity F ( b ) − F ( a ) . Let there be numbers x 0 , ..., x n such that a = x 0 < x 1 < x 2 < ⋯ < x n − 1 < x n = b . {\displaystyle a=x_{0}<x_{1}<x_{2}<\cdots <x_{n-1}<x_{n}=b.} It follows that F ( b ) − F (

700-403: A continuous function on [ a , b ] {\displaystyle [a,b]} which is an antiderivative of f {\displaystyle f} in ( a , b ) {\displaystyle (a,b)} : F ′ ( x ) = f ( x ) . {\displaystyle F'(x)=f(x).} If f {\displaystyle f}

770-421: A corresponding "area function" x ↦ A ( x ) {\displaystyle x\mapsto A(x)} such that A ( x ) is the area beneath the curve between 0 and x . The area A ( x ) may not be easily computable, but it is assumed to be well defined. The area under the curve between x and x + h could be computed by finding the area between 0 and x + h , then subtracting

SECTION 10

#1732764971434

840-643: A factorization method— Fermat's factorization method —and popularized the proof by infinite descent , which he used to prove Fermat's right triangle theorem which includes as a corollary Fermat's Last Theorem for the case n = 4. Fermat developed the two-square theorem , and the polygonal number theorem , which states that each number is a sum of three triangular numbers , four square numbers , five pentagonal numbers , and so on. Although Fermat claimed to have proven all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss , doubted several of his claims, especially given

910-634: A hobby than a profession. Nevertheless, he made important contributions to analytical geometry , probability, number theory and calculus. Secrecy was common in European mathematical circles at the time. This naturally led to priority disputes with contemporaries such as Descartes and Wallis . Anders Hald writes that, "The basis of Fermat's mathematics was the classical Greek treatises combined with Vieta's new algebraic methods." Fermat's pioneering work in analytic geometry ( Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum )

980-415: A mathematician of rare power. He was an independent inventor of analytic geometry , he contributed to the early development of calculus, he did research on the weight of the earth, and he worked on light refraction and optics. In the course of what turned out to be an extended correspondence with Blaise Pascal , he made a significant contribution to the theory of probability. But Fermat's crowning achievement

1050-413: A method ( adequality ) for determining maxima, minima, and tangents to various curves that was equivalent to differential calculus . In these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature . Fermat was the first person known to have evaluated the integral of general power functions. With his method, he

1120-402: A perfect equality when h approaches 0: f ( x ) = lim h → 0 A ( x + h ) − A ( x ) h   = def   A ′ ( x ) . {\displaystyle f(x)=\lim _{h\to 0}{\frac {A(x+h)-A(x)}{h}}\ {\stackrel {\text{def}}{=}}\ A'(x).} That is,

1190-433: A velocity as the function, you can integrate it from the starting time up to any given time to obtain a distance function whose derivative is that velocity. (To obtain your highway-marker position, you would need to add your starting position to this integral and to take into account whether your travel was in the direction of increasing or decreasing mile markers.) There are two parts to the theorem. The first part deals with

1260-430: Is Riemann integrable on [ a , b ] {\displaystyle [a,b]} then ∫ a b f ( x ) d x = F ( b ) − F ( a ) . {\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).} The second part is somewhat stronger than the corollary because it does not assume that f {\displaystyle f}

1330-587: Is a commune in the Tarn-et-Garonne department in the Occitanie region in southern France. The river Gimone runs through the town. Beaumont-de-Lomagne, bastide , was founded in 1276 following the act of coregency between the abbey of Grandselve and King Philip III of France – the King was represented by his seneschal for the former County of Toulouse , Eustache de Beaumarchais . In 1278

1400-449: Is a theorem that links the concept of differentiating a function (calculating its slopes , or rate of change at each point in time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). Roughly speaking, the two operations can be thought of as inverses of each other. The first part of the theorem, the first fundamental theorem of calculus , states that for

1470-421: Is an antiderivative of f {\displaystyle f} in [ a , b ] {\displaystyle [a,b]} , then ∫ a b f ( t ) d t = F ( b ) − F ( a ) . {\displaystyle \int _{a}^{b}f(t)\,dt=F(b)-F(a).} The corollary assumes continuity on the whole interval. This result

SECTION 20

#1732764971434

1540-486: Is approximate the curve with n rectangles. Now, as the size of the partitions get smaller and n increases, resulting in more partitions to cover the space, we get closer and closer to the actual area of the curve. By taking the limit of the expression as the norm of the partitions approaches zero, we arrive at the Riemann integral . We know that this limit exists because f was assumed to be integrable. That is, we take

1610-483: Is by the first part of the Fundamental Theorem. For example, if f ( x ) = e , then f has an antiderivative, namely G ( x ) = ∫ 0 x f ( t ) d t {\displaystyle G(x)=\int _{0}^{x}f(t)\,dt} and there is no simpler expression for this function. It is therefore important not to interpret the second part of the theorem as

1680-413: Is continuous on the closed interval [ a , b ] and differentiable on the open interval ( a , b ) , then there exists some c in ( a , b ) such that F ′ ( c ) ( b − a ) = F ( b ) − F ( a ) . {\displaystyle F'(c)(b-a)=F(b)-F(a).} Let f be (Riemann) integrable on the interval [

1750-476: Is continuous. For a given function f , define the function F ( x ) as F ( x ) = ∫ a x f ( t ) d t . {\displaystyle F(x)=\int _{a}^{x}f(t)\,dt.} For any two numbers x 1 and x 1 + Δ x in [ a , b ] , we have F ( x 1 + Δ x ) − F ( x 1 ) = ∫

1820-492: Is continuous. When an antiderivative F {\displaystyle F} of f {\displaystyle f} exists, then there are infinitely many antiderivatives for f {\displaystyle f} , obtained by adding an arbitrary constant to F {\displaystyle F} . Also, by the first part of the theorem, antiderivatives of f {\displaystyle f} always exist when f {\displaystyle f}

1890-582: Is named after him: the Lycée Pierre-de-Fermat . French sculptor Théophile Barrau made a marble statue named Hommage à Pierre Fermat as a tribute to Fermat, now at the Capitole de Toulouse . Together with René Descartes , Fermat was one of the two leading mathematicians of the first half of the 17th century. According to Peter L. Bernstein , in his 1996 book Against the Gods , Fermat "was

1960-409: Is often employed to compute the definite integral of a function f {\displaystyle f} for which an antiderivative F {\displaystyle F} is known. Specifically, if f {\displaystyle f} is a real-valued continuous function on [ a , b ] {\displaystyle [a,b]} and F {\displaystyle F}

2030-450: Is strengthened slightly in the following part of the theorem. This part is sometimes referred to as the second fundamental theorem of calculus or the Newton–Leibniz theorem . Let f {\displaystyle f} be a real-valued function on a closed interval [ a , b ] {\displaystyle [a,b]} and F {\displaystyle F}

2100-464: The mean value theorem implies that F − G is a constant function , that is, there is a number c such that G ( x ) = F ( x ) +  c for all x in [ a , b ] . Letting x = a , we have F ( a ) + c = G ( a ) = ∫ a a f ( t ) d t = 0 , {\displaystyle F(a)+c=G(a)=\int _{a}^{a}f(t)\,dt=0,} which means c = − F (

2170-545: The plague which killed 500 inhabitants. By the sixteenth century, Beaumont, a catholic town, was surrounded by three protestant towns: Montauban , Mas-Grenier and Mauvezin . In 1577, Henri III sold Beaumont to Henri III of Navarre (future Henri IV ), leader of the Protestants and whose troops came to massacre a hundred Beaumontois. In December 1580, 600 mercenaries of Montauban demobilized and took Beaumont. They remained for two months, and caused much damage to

Pierre de Fermat - Misplaced Pages Continue

2240-637: The English, made it his episcopal seat until 1432. The market hall, in the centre of the town square, was designed for the markets that took place every Saturday. The fourteenth century marked the beginning of the Hundred Years' War . Taken by the English in 1345, Beaumont was recaptured in 1350 but continued to be plundered by " Great Companies " and experienced civil war due to the opposition of two military chiefs: Count of Foix and John I, Count of Armagnac . The century ended with an epidemic of

2310-834: The above into ( 1' ), we get F ( b ) − F ( a ) = ∑ i = 1 n [ F ′ ( c i ) ( x i − x i − 1 ) ] . {\displaystyle F(b)-F(a)=\sum _{i=1}^{n}[F'(c_{i})(x_{i}-x_{i-1})].} The assumption implies F ′ ( c i ) = f ( c i ) . {\displaystyle F'(c_{i})=f(c_{i}).} Also, x i − x i − 1 {\displaystyle x_{i}-x_{i-1}} can be expressed as Δ x {\displaystyle \Delta x} of partition i {\displaystyle i} . We are describing

2380-776: The area between 0 and x . In other words, the area of this "strip" would be A ( x + h ) − A ( x ) . There is another way to estimate the area of this same strip. As shown in the accompanying figure, h is multiplied by f ( x ) to find the area of a rectangle that is approximately the same size as this strip. So: A ( x + h ) − A ( x ) ≈ f ( x ) ⋅ h {\displaystyle A(x+h)-A(x)\approx f(x)\cdot h} Dividing by h on both sides, we get: A ( x + h ) − A ( x ) h   ≈ f ( x ) {\displaystyle {\frac {A(x+h)-A(x)}{h}}\ \approx f(x)} This estimate becomes

2450-430: The area of a rectangle, with the width times the height, and we are adding the areas together. Each rectangle, by virtue of the mean value theorem , describes an approximation of the curve section it is drawn over. Also Δ x i {\displaystyle \Delta x_{i}} need not be the same for all values of i , or in other words that the width of the rectangles can differ. What we have to do

2520-575: The calculation of a definite integral provided an antiderivative can be found by symbolic integration , thus avoiding numerical integration . The fundamental theorem of calculus relates differentiation and integration, showing that these two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that these two operations were related. Ancient Greek mathematicians knew how to compute area via infinitesimals , an operation that we would now call integration. The origins of differentiation likewise predate

2590-441: The conjecture and the proof of the fundamental theorem of calculus, calculus as a unified theory of integration and differentiation is started. The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, was by James Gregory (1638–1675). Isaac Barrow (1630–1677) proved a more generalized version of the theorem, while his student Isaac Newton (1642–1727) completed

2660-420: The cultivation of garlic . It retains much of its history through its old buildings: the church, its fortress – whose imposing mass dominates the town – the large market with its distinctive roof, as well as approximately fifteen private mansions, the majority of which date from the seventeenth to nineteenth centuries. List of mayors: Fundamental theorem of calculus The fundamental theorem of calculus

2730-446: The definition of the derivative, the continuity of f , and the squeeze theorem . Suppose F is an antiderivative of f , with f continuous on [ a , b ] . Let G ( x ) = ∫ a x f ( t ) d t . {\displaystyle G(x)=\int _{a}^{x}f(t)\,dt.} By the first part of the theorem, we know G is also an antiderivative of f . Since F ′ − G ′ = 0

2800-445: The derivative of an antiderivative , while the second part deals with the relationship between antiderivatives and definite integrals . This part is sometimes referred to as the first fundamental theorem of calculus . Let f be a continuous real-valued function defined on a closed interval [ a , b ] . Let F be the function defined, for all x in [ a , b ] , by F ( x ) = ∫

2870-409: The derivative of the area function A ( x ) exists and is equal to the original function f ( x ) , so the area function is an antiderivative of the original function. Thus, the derivative of the integral of a function (the area) is the original function, so that derivative and integral are inverse operations which reverse each other. This is the essence of the Fundamental Theorem. Intuitively,

Pierre de Fermat - Misplaced Pages Continue

2940-417: The development of the surrounding mathematical theory. Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today. The first fundamental theorem may be interpreted as follows. Given a continuous function y = f ( x ) {\displaystyle y=f(x)} whose graph is plotted as a curve, one defines

3010-581: The difficulty of some of the problems and the limited mathematical methods available to Fermat. His 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 Marin Mersenne about it. It was first proven in 1994, by Sir Andrew Wiles , using techniques unavailable to Fermat. Through their correspondence in 1654, Fermat and Blaise Pascal helped lay

3080-445: The first part. Similarly, it almost looks like the first part of the theorem follows directly from the second. That is, suppose G is an antiderivative of f . Then by the second theorem, G ( x ) − G ( a ) = ∫ a x f ( t ) d t {\textstyle G(x)-G(a)=\int _{a}^{x}f(t)\,dt} . Now, suppose F ( x ) = ∫

3150-807: The following sum: The function F is differentiable on the interval ( a , b ) and continuous on the closed interval [ a , b ] ; therefore, it is also differentiable on each interval ( x i −1 , x i ) and continuous on each interval [ x i −1 , x i ] . According to the mean value theorem (above), for each i there exists a c i {\displaystyle c_{i}} in ( x i −1 , x i ) such that F ( x i ) − F ( x i − 1 ) = F ′ ( c i ) ( x i − x i − 1 ) . {\displaystyle F(x_{i})-F(x_{i-1})=F'(c_{i})(x_{i}-x_{i-1}).} Substituting

3220-549: The foundation 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 showed mathematically why this

3290-560: The fundamental theorem of calculus by hundreds of years; for example, in the fourteenth century the notions of continuity of functions and motion were studied by the Oxford Calculators and other scholars. The historical relevance of the fundamental theorem of calculus is not the ability to calculate these operations, but the realization that the two seemingly distinct operations (calculation of geometric areas, and calculation of gradients) are actually closely related. From

3360-426: The fundamental theorem states that integration and differentiation are inverse operations which reverse each other. The second fundamental theorem says that the sum of infinitesimal changes in a quantity (the integral of the derivative of the quantity) adds up to the net change in the quantity. To visualize this, imagine traveling in a car and wanting to know the distance traveled (the net change in position along

3430-440: The highway). You can see the velocity on the speedometer but cannot look out to see your location. Each second, you can find how far the car has traveled using distance = speed × time , that is, multiplying the current speed (in kilometers or miles per hour) by the time interval (1 second = 1 3600 {\displaystyle {\tfrac {1}{3600}}} hour). By summing up all these small steps, you can approximate

3500-496: The king. The incident ended without conflict, but Beaumont, ruined, had to pay a large fine; another plague epidemic also occurred during this event. In 1702, the town had only 2,400 inhabitants but during this period of peace, it undertook various works and became prosperous again. In 1777, the ramparts were destroyed. After sending a delegate to the Estates General , Beaumont created a revolutionary club, but from 1790

3570-1170: The latter equality resulting from the basic properties of integrals and the additivity of areas. According to the mean value theorem for integration , there exists a real number c ∈ [ x 1 , x 1 + Δ x ] {\displaystyle c\in [x_{1},x_{1}+\Delta x]} such that ∫ x 1 x 1 + Δ x f ( t ) d t = f ( c ) ⋅ Δ x . {\displaystyle \int _{x_{1}}^{x_{1}+\Delta x}f(t)\,dt=f(c)\cdot \Delta x.} It follows that F ( x 1 + Δ x ) − F ( x 1 ) = f ( c ) ⋅ Δ x , {\displaystyle F(x_{1}+\Delta x)-F(x_{1})=f(c)\cdot \Delta x,} and thus that F ( x 1 + Δ x ) − F ( x 1 ) Δ x = f ( c ) . {\displaystyle {\frac {F(x_{1}+\Delta x)-F(x_{1})}{\Delta x}}=f(c).} Taking

SECTION 50

#1732764971434

3640-881: The limit as Δ x → 0 , {\displaystyle \Delta x\to 0,} and keeping in mind that c ∈ [ x 1 , x 1 + Δ x ] , {\displaystyle c\in [x_{1},x_{1}+\Delta x],} one gets lim Δ x → 0 F ( x 1 + Δ x ) − F ( x 1 ) Δ x = lim Δ x → 0 f ( c ) , {\displaystyle \lim _{\Delta x\to 0}{\frac {F(x_{1}+\Delta x)-F(x_{1})}{\Delta x}}=\lim _{\Delta x\to 0}f(c),} that is, F ′ ( x 1 ) = f ( x 1 ) , {\displaystyle F'(x_{1})=f(x_{1}),} according to

3710-773: The limit as the largest of the partitions approaches zero in size, so that all other partitions are smaller and the number of partitions approaches infinity. So, we take the limit on both sides of ( 2' ). This gives us lim ‖ Δ x i ‖ → 0 F ( b ) − F ( a ) = lim ‖ Δ x i ‖ → 0 ∑ i = 1 n [ f ( c i ) ( Δ x i ) ] . {\displaystyle \lim _{\|\Delta x_{i}\|\to 0}F(b)-F(a)=\lim _{\|\Delta x_{i}\|\to 0}\sum _{i=1}^{n}[f(c_{i})(\Delta x_{i})].} Neither F ( b ) nor F (

3780-401: The modern theory of such curves. It naturally falls into two parts; the first one ... may conveniently be termed a method of ascent, in contrast with the descent which is rightly regarded as Fermat's own." Regarding Fermat's use of ascent, Weil continued: "The novelty consisted in the vastly extended use which Fermat made of it, giving him at least a partial equivalent of what we would obtain by

3850-517: The path of shortest time " now known as the principle of least time . For this, Fermat is recognized as a key figure in the historical development of the fundamental principle of least action in physics. The terms Fermat's principle and Fermat functional were named in recognition of this role. Pierre de Fermat died on January 12, 1665, at Castres , in the present-day department of Tarn . The oldest and most prestigious high school in Toulouse

3920-406: The right side of the equation defines the integral over f from a to b . Therefore, we obtain F ( b ) − F ( a ) = ∫ a b f ( x ) d x , {\displaystyle F(b)-F(a)=\int _{a}^{b}f(x)\,dx,} which completes the proof. As discussed above, a slightly weaker version of the second part follows from

3990-399: The summing up corresponds to integration . Thus, the integral of the velocity function (the derivative of position) computes how far the car has traveled (the net change in position). The first fundamental theorem says that the value of any function is the rate of change (the derivative) of its integral from a fixed starting point up to any chosen end point. Continuing the above example using

4060-400: The systematic use of the group theoretical properties of the rational points on a standard cubic." With his gift for number relations and his ability to find proofs for many of his theorems, Fermat essentially created the modern theory of numbers. Beaumont-de-Lomagne Beaumont-de-Lomagne ( French pronunciation: [bomɔ̃ də lɔmaɲ] ; Languedocien : Bèumont de Lomanha )

4130-699: The total distance traveled, in spite of not looking outside the car: distance traveled = ∑ ( velocity at each time ) × ( time interval ) = ∑ v t × Δ t . {\displaystyle {\text{distance traveled}}=\sum \left({\begin{array}{c}{\text{velocity at}}\\{\text{each time}}\end{array}}\right)\times \left({\begin{array}{c}{\text{time}}\\{\text{interval}}\end{array}}\right)=\sum v_{t}\times \Delta t.} As Δ t {\displaystyle \Delta t} becomes infinitesimally small,

4200-473: The town became part of the Haute-Garonne department and became isolated, to the advantage of Grenade , its neighbour and rival. Grenade became the chief town of district. In 1808, new department divisions were brought in by Napoleon and Beaumont began to be within the Tarn-et-Garonne region. Though the importance of large fairs has decreased, Beaumont remains an important agricultural market due to

4270-705: The town of his birth. He attended the University of Orléans from 1623 and received a bachelor in civil law in 1626, before moving to Bordeaux . In Bordeaux, he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apollonius 's De Locis Planis to one of the mathematicians there. Certainly, in Bordeaux he was in contact with Beaugrand and during this time 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

SECTION 60

#1732764971434

4340-532: The town was granted a very liberal charter of laws, by the standards of the period, defining the rights and duties of its inhabitants. In 1280, work commenced on a large church; its flat apse shows the influence of Cîteaux . The bell-tower , was made in the fifteenth century and resembles that of Saint-Sernin in Toulouse. Construction finished around 1430 and the Bishop of Montauban , driven out of his city by

4410-475: The town. When peace returned, many Beaumontois adopted the policy of religious tolerance as advocated by Henri IV. The eminent mathematician Pierre de Fermat , famous for Fermat's Last Theorem , was born in Beaumont in either 1601 or 1607. There is a statue and museum to him in the town. In the seventeenth century, Louis XIII besieged several cities in the south-west including Beaumont; the "Chateau de Roi"

4480-965: 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. On 1 June 1631, Fermat married Louise de Long, a fourth cousin of his mother Claire de Fermat (née de Long). The Fermats had eight children, five of whom survived to adulthood: Clément-Samuel, Jean, Claire, Catherine, and Louise. Fluent in six languages ( French , Latin , Occitan , classical Greek , Italian and Spanish ), Fermat

4550-445: 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 . In number theory, Fermat studied Pell's equation , perfect numbers , amicable numbers and what would later become Fermat numbers . It was while researching perfect numbers that he discovered Fermat's little theorem . He invented

4620-488: Was circulated in manuscript form in 1636 (based on results achieved in 1629), predating the publication of Descartes' La géométrie (1637), which exploited the work. 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 et de tangentibus linearum curvarum , Fermat developed

4690-509: Was destroyed by royal decree . In 1639 Louis sold Beaumont to the Prince of Condé. Under Louis XIV , Beaumont was still under the jurisdiction of viscount Armand de Bourbon, prince de Conti , one of the nobility involved in the Fronde , Beaumont was therefore part of the rebellion and this caused considerable losses to the town. There was an occupation in 1651 by Conti troops, rebelling against

4760-404: Was in the theory of numbers." Regarding Fermat's work in analysis, Isaac Newton wrote that his own early ideas about calculus came directly from "Fermat's way of drawing tangents." Of Fermat's number theoretic work, the 20th-century mathematician André Weil wrote that: "what we possess of his methods for dealing with curves of genus 1 is remarkably coherent; it is still the foundation for

4830-412: Was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often with little or no proof of his theorems. In some of these letters to his friends, he explored many of the fundamental ideas of calculus before Newton or Leibniz . Fermat was a trained lawyer making mathematics more of

4900-411: Was the case. The first variational principle in physics was articulated by Euclid in his Catoptrica . It says that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection . Hero of Alexandria later showed that this path gave the shortest length and the least time. Fermat refined and generalized this to "light travels between two given points along

#433566