Pierre Fermat and Blaise Pascal
Rene Descartes, Pierre Fermat and Blaise Pascal Descartes, Fermat and Pascal: a philosopher, an amateur and a calculator Rene Descartes 1596 - 1650 Pierre de Fermat 17th August 1601 or 1607 – 12th January 1665 Blaise Pascal, 1623 - 1662 Descartes René Descartes was a French philosopher whose work, La géométrie, includes his application of algebra to geometry from which we now have Cartesian geometry. His work had a great influence on both mathematicians and philosophers. Descartes Descartes was educated at the Jesuit college of La Flèche in Anjou. • The Society of Jesus (Latin: Societas Iesu, S.J. and S.I. or SJ, SI ) is a Catholic religious order of whose members are called Jesuits, He entered the college at the age of eight years, just a few months after the opening of the college in January 1604. He studied there until 1612, studying classics, logic and traditional Aristotelian philosophy. He also learnt mathematics from the books of Clavius. Descartes While in the school his health was poor and he was granted permission to remain in bed until 11 o'clock in the morning, a custom he maintained until the year of his death. In bed he came up with idea now called Cartesian geometry. Clavius 1538 - 1612 Christopher Clavius was a German Jesuit astronomer who helped Pope Gregory XIII to introduce what is now called the Gregorian calendar. Descartes School had made Descartes understand how little he knew, the only subject which was satisfactory in his eyes was mathematics. This idea became the foundation for his way of thinking, and was to form the basis for all his works. The above statement was echoed by Einstein in the 20th century. Descartes Descartes spent a while in Paris, apparently keeping very much to himself, then he studied at the University of Poitiers. He received a law degree from Poitiers in 1616 then enlisted in the military school at Breda. In 1618 he started studying mathematics and mechanics under the Dutch scientist Isaac Beeckman, and began to seek a unified science of nature. Wrote on the theory of vortices Desartes After two years in Holland he travelled through Europe. In 1619 he joined the Bavarian army. After two years in Holland he travelled through Europe. Descartes From 1620 to 1628 Descartes travelled through Europe, spending time in Bohemia (1620), Hungary (1621), Germany, Holland and France (1622-23). He 1623 he spent time in Paris where he made contact with Mersenne, an important contact which kept him in touch with the scientific world for many years. Descartes From Paris he travelled to Italy where he spent some time in Venice, then he returned to France again (1625). Mersenne Marin Mersenne was a French monk who is best known for his role as a clearing house for correspondence between eminent philosophers and scientists and for his work in number theory. Similar to Bourbaki • Nicholas Bourbaki is the collective pseudonym under which a group of (mainly French) 20thcentury mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, Mersenne prime In mathematics, a Mersenne number is a positive integer that is one less than a power of two Some definitions of Mersenne numbers require that the exponent n be prime. 2 1 n Mersenne prime A Mersenne prime is a Mersenne number that is prime. As of September 2008, only 46 Mersenne primes are known; the largest known prime number is: (243,112,609 − 1) is a Mersenne prime, and in modern times, the largest known prime has almost always been a Mersenne prime. Descartes By 1628 Descartes was tired of the continual travelling and decided to settle down. He gave much thought to choosing a country suited to his nature and chose Holland. It was a good decision which he did not seem to regret over the next twenty years. Descartes Soon after he settled in Holland Descartes began work on his first major treatise on physics, Le Monde, ou Traité de la Lumière. This work was near completion when news that Galileo was condemned to house arrest reached him. He, perhaps wisely, decided not to risk publication and the work was published, only in part, after his death. He explained later his change of direction saying:- Descartes ... in order to express my judgment more freely, without being called upon to assent to, or to refute the opinions of the learned, I resolved to leave all this world to them and to speak solely of what would happen in a new world, if God were now to create ... and allow her to act in accordance with the laws He had established. Galileo, portrait by Justus Sustermans painted in 1636 Galileo Galilei Galileo Galilei was an Italian scientist who formulated the basic law of falling bodies, which he verified by careful measurements. He constructed a telescope with which he studied lunar craters, and discovered four moons revolving around Jupiter and espoused the Copernican cause. Nicolaus Copernicus 1473 - 1543 Copernicus Copernicus was a Polish astronomer and mathematician who was a proponent of the view of an Earth in daily motion about its axis and in yearly motion around a stationary sun. • Helio-Centric universe This theory profoundly altered later workers' view of the universe, but was rejected by the Catholic church. Galileo Galilei Galileo Galilei: considered the father of experimental physics along with Ernest Rutherford. Galileo Galilei's parents were Vincenzo Galilei and Guilia Ammannati. Vincenzo, who was born in Florence in 1520, was a teacher of music and a fine lute player. After studying music in Venice he carried out experiments on strings to support his musical theories. Galileo Galilei Guilia, who was born in Pescia, married Vincenzo in 1563 and they made their home in the countryside near Pisa. Galileo was their first child and spent his early years with his family in Pisa. Galilean transformation The Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian Physics. The equations below, although apparently obvious, break down at speeds that approach the speed of light due to physics described by Einstein’s theory of relativity. Galileo formulated these concepts in his description of uniform motion . Galileo The topic was motivated by Galileo’s description of the motion of a ball rolling down a ramp, by which he measured the numerical value for the acceleration due to gravity, g at the surface of the earth. The descriptions below are another mathematical notation for this concept. Translation (one dimension) In essence, the Galilean transformations embody the intuitive notion of addition and subtraction of velocities. The assumption that time can be treated as absolute is at heart of the Galilean transformations. • Relativity insists that the speed of light is constant and thus time is different for different observers. This assumption is abandoned in the Lorentz transformations Hendrik Antoon Lorentz 1853 - 1928 Lorentz Lorentz is best known for his work on electromagnetic radiation and the FitzGerald-Lorentz contraction. He developed the mathematical theory of the electron. Galileo These relativistic transformations are deemed applicable to all velocities, whilst the Galilean transformation can be regarded as a lowvelocity approximation to the Lorentz transformation. The notation below describes the relationship of two coordinate systems (x′ and x) in constant relative motion (velocity v) in the xdirection according to the Galilean transformation: Translation (one dimension) x ' x vt y' y z' z t't Lorenz transformations Lorenz transformations The spacetime coordinates of an event, as measured by each observer in their inertial reference frame (in standard configuration) are shown in the speech bubbles. Top: frame F′ moves at velocity v along the xaxis of frame F. Bottom: frame F moves at velocity −v along the x′-axis of frame F Translation (one dimension) Note that the last equation (Galileo) expresses the assumption of a universal time independent of the relative motion of different observers. Galileo In 1572, when Galileo was eight years old, his family returned to Florence, his father's home town. However, Galileo remained in Pisa and lived for two years with Muzio Tedaldi who was related to Galileo's mother by marriage. When he reached the age of ten, Galileo left Pisa to join his family in Florence and there he was tutored by Jacopo Borghini. Galileo Once he was old enough to be educated in a monastery, his parents sent him to the Camaldolese Monastery at Vallombrosa which is situated on a magnificent forested hillside 33 km southeast of Florence. Galileo The Order combined the solitary life of the hermit with the strict life of the monk and soon the young Galileo found this life an attractive one. He became a novice, intending to join the Order, but this did not please his father who had already decided that his eldest son should become a medical doctor. Galileo Vincenzo had Galileo returned from Vallombrosa to Florence and gave up the idea of joining the Camaldolese order. He did continue his schooling in Florence, however, in a school run by the Camaldolese monks. In 1581 Vincenzo sent Galileo back to Pisa to live again with Muzio Tedaldi and now to enrol for a medical degree at the University of Pisa. Galileo Although the idea of a medical career never seems to have appealed to Galileo, his father's wish was a fairly natural one since there had been a distinguished physician in his family in the previous century. Galileo Galileo never seems to have taken medical studies seriously, attending courses on his real interests which were in mathematics and natural philosophy (physics). His mathematics teacher at Pisa was Filippo Fantoni, who held the chair of mathematics. Galileo returned to Florence for the summer vacations and there continued to study mathematics. Galileo In the year 1582-83 Ostilio Ricci, who was the mathematician of the Tuscan Court and a former pupil of Tartaglia, taught a course on Euclid’s Elements at the University of Pisa which Galileo attended. However Galileo, still reluctant to study medicine, invited Ricci (also in Florence where the Tuscan court spent the summer and autumn) to his home to meet his father. Nicolo Fontana Tartaglia Nicolo Fontana Tartaglia 1500 - 1557 Tartaglia was an Italian mathematician who was famed for his algebraic solution of cubic equations which was eventually published in Cardan's Ars Magna. He is also known as the stammerer. François Viète, 1540 - 1603 François Viète François Viète was a French amateur mathematician and astronomer who introduced the first systematic algebraic notation in his book • In artem analyticam isagoge . He was also involved in deciphering codes. • Alan Turin Ricci and Galileo’s father Ricci tried to persuade Vincenzo to allow his son to study mathematics since this was where his interests lay. Ricci flow is of paramount importance in the solution to the Poincare Conjecture. The Poincaré Conjecture Explained The Poincaré Conjecture is first and only of the Clay Millennium problems to be solved, (2005) It was proved by Grigori Perelman who subsequently turned down the $1 million prize money, left mathematics, and moved in with his mother in Russia. Here is the statement of the conjecture from wikipedia: The Poincare conjecture Every simply connected, closed 3manifold is homeomorphic to the 3sphere. Topology This is a statement about topological spaces. Let’s define each of the terms in the conjecture: Simply connected space – This means the space has no “holes.” A football is simply connected, but a donut is not. Technically we can say ( X ) 0 to be explained further on. 1 1 Closed space – The space is finite and has no boundaries. A sphere (more technically a 2-sphere or S 2 ) is closed, but the plane ( R 2 ) is not because it is infinite. A disk is also not because even though it is finite, it has a boundary. manifold – At every small neighbourhood on the space, it approximates Euclidean space. A standard sphere is called a 2-sphere because it is actually a 2-manifold. Its surface resembles the 2d plane if you zoom into it so that the curvature approaches 0. Continuing this logic the 1-sphere is a circle. A 3-sphere is very difficult to visualize because it has a 3d surface and exists in 4d space. homeomorphic – If one space is homeomorphic to another, it means you can continuously deform the one space into the other. The 2-sphere and a football are homeomorphic. The 2-sphere and a donut are not; no matter how much you deform a sphere, you can’t get that pesky hole in the donut, and vice-versa. Morphing Galileo Certainly Vincenzo did not like the idea (of his son studying mathematics) and resisted strongly but eventually he gave way a little and Galileo was able to study the works of Euclid and Archimedes from the Italian translations which Tartaglia had made. Of course he was still officially enrolled as a medical student at Pisa but eventually, by 1585, he gave up this course and left without completing his degree. Galileo Galileo began teaching mathematics, first privately in Florence and then during 1585-86 at Siena where he held a public appointment. During the summer of 1586 he taught at Vallombrosa, and in this year he wrote his first scientific book The little balance [La Balancitta] which described Archimedes method of finding the specific gravities (that is the relative densities) of substances using a balance. • Essentially looking at the use of plumblines Galileo In the following year he travelled to Rome to visit Clavius who was professor of mathematics at the Jesuit Collegio Romano there. Galileo A topic which was very popular with the Jesuit mathematicians at this time was centres of gravity and Galileo brought with him some results which he had discovered on this topic. Despite making a very favourable impression on Clavius, Galileo failed to gain an appointment to teach mathematics at the University of Bologna. Galileo After leaving Rome Galileo remained in contact with Clavius by correspondence and Guidobaldo del Monte who was also a regular correspondent. Certainly the theorems which Galileo had proved on the centres of gravity of solids, and left in Rome, were discussed in this correspondence. It is also likely that Galileo received lecture notes from courses which had been given at the Collegio Romano, for he made copies of such material which still survive today. Galileo The correspondence began around 1588 and continued for many years. Also in 1588 Galileo received a prestigious invitation to lecture on the dimensions and location of hell in Dante's Inferno at the Academy in Florence. Galileo Fantoni left the chair of mathematics at the University of Pisa in 1589 and Galileo was appointed to fill the post (although this was only a nominal position to provide financial support for Galileo). Not only did he receive strong recommendations from Clavius, but he also had acquired an excellent reputation through his lectures at the Florence Academy in the previous year. The young mathematician had rapidly acquired the reputation that was necessary to gain such a position, but there were still higher positions at which he might aim. Galileo Galileo spent three years holding this post at the University of Pisa and during this time he wrote • De Motu a series of essays on the theory of motion which he never published. It is likely that he never published this material because he was less than satisfied with it, and this is fair for despite containing some important steps forward, it also contained some incorrect ideas. Galileo Perhaps the most important new ideas which De Motu contains is that one can test theories by conducting experiments. • The beginnings of the scientific method that had escaped the Greeks due to Aristotle. In particular the work contains his important idea that one could test theories about falling bodies using an inclined plane to slow down the rate of descent. Galileo In 1591 Vincenzo Galilei, Galileo's father, died and since Galileo was the eldest son he had to provide financial support for the rest of the family and in particular have the necessary financial means to provide dowries for his two younger sisters. Being professor of mathematics at Pisa was not well paid, so Galileo looked for a more lucrative post. Galileo With strong recommendations from del Monte, Galileo was appointed professor of mathematics at the University of Padua (the university of the Republic of Venice) in 1592 at a salary of three times what he had received at Pisa. On 7 December 1592 he gave his inaugural lecture and began a period of eighteen years at the university, years which he later described as the happiest of his life. Galileo At Padua his duties were mainly to teach Euclid’s geometry and standard (geocentric) astronomy to medical students, who would need to know some astronomy in order to make use of astrology in their medical practice. Galileo However, Galileo argued against Aristotle’s view of astronomy and natural philosophy in three public lectures he gave in connection with the appearance of a New Star (now known as ‘Kepler’s supernova') in 1604. The belief at this time was that of Aristotle, namely that all changes in the heavens had to occur in the lunar region close to the Earth, the realm of the fixed stars being permanent. Johann Kepler 1571 - 1630 Johannes Kepler Johannes Kepler was a German mathematician and astronomer who discovered that the Earth and planets travel about the sun in elliptical orbits. He gave three fundamental laws of planetary motion. He also did important work in optics and geometry. Kepler’s laws The first law says: "The orbit of every planet is an ellipse with the sun at one of the foci." The second law: "A line joining a planet and the sun sweeps out equal areas during equal intervals of time.“ The third law : "The squares of the orbital periods of planets are directly proportional to the cubes of the axes of the orbits." First law Second law Galileo Galileo used parallax arguments to prove that the New Star could not be close to the Earth. In a personal letter written to Kepler in 1598, Galileo had stated that he was a Copernican (believer in the theories of Copernicus). However, no public sign of this belief was to appear until many years later. Galileo At Padua, Galileo began a long term relationship with Maria Gamba, who was from Venice, but they did not marry perhaps because Galileo felt his financial situation was not good enough. In 1600 their first child Virginia was born, followed by a second daughter Livia in the following year. In 1606 their son Vincenzo was born. Galileo We mentioned above an error in Galileo's theory of motion as he set it out in De Motu around 1590. He was quite mistaken in his belief that the force acting on a body was the relative difference between its specific gravity and that of the substance through which it moved. • Ether Galileo wrote to his friend Paolo Sarpi, a fine mathematician who was consultor to the Venetian government, in 1604 and it is clear from his letter that by this time he had realised his mistake. Galileo In fact he had returned to work on the theory of motion in 1602 and over the following two years, through his study of inclined planes and the pendulum, he had formulated the correct law of falling bodies and had worked out that a projectile follows a parabolic path. However, these famous results would not be published for another 35 years. Galileo In May 1609, Galileo received a letter from Paolo Sarpi telling him about a spyglass that a Dutchman had shown in Venice. Galileo wrote in the Starry Messenger (Sidereus Nuncius) in April 1610:- Galileo About ten months ago a report reached my ears that a certain Fleming had constructed a spyglass by means of which visible objects, though very distant from the eye of the observer, were distinctly seen as if nearby. Of this truly remarkable effect several experiences were related, to which some persons believed while other denied them. Galileo A few days later the report was confirmed by a letter I received from a Frenchman in Paris, Jacques Badovere, which caused me to apply myself wholeheartedly to investigate means by which I might arrive at the invention of a similar instrument. This I did soon afterwards, my basis being the doctrine of refraction. Galileo From these reports, and using his own technical skills as a mathematician and as a craftsman, Galileo began to make a series of telescopes whose optical performance was much better than that of the Dutch instrument. His first telescope was made from available lenses and gave a magnification of about four. Galileo To improve on this Galileo learned how to grind and polish his own lenses and by August 1609 he had an instrument with a magnification of around eight or nine. Galileo immediately saw the commercial and military applications of his telescope (which he called a perspicillum) for ships at sea. Galileo He kept Sarpi informed of his progress and Sarpi arranged a demonstration for the Venetian Senate. They were very impressed and, in return for a large increase in his salary, Galileo gave the sole rights for the manufacture of telescopes to the Venetian Senate. It seems a particularly good move on his part since he must have known that such rights were meaningless, particularly since he always acknowledged that the telescope was not his invention Descartes In Holland Descartes had a number of scientific friends as well as continued contact with Mersenne. His friendship with Beeckman continued and he also had contact with Huygens. Christiaan Huygens 1629 - 1695 Christiaan Huygens 1629 - 1695 Christiaan Huygens was a Dutch mathematician who patented the first pendulum clock, which greatly increased the accuracy of time measurement. He laid the foundations of mechanics and also worked on astronomy and probability. • Proponent of the wave theory He was a contempory of Isaac Newton Isaac Newton Born: 4 Jan 1643 in Woolsthorpe, Lincolnshire, England Died: 31 March 1727 in London, England Isaac Newton Descartes Descartes was pressed by his friends to publish his ideas and, although he was adamant in not publishing Le Monde, he wrote a treatise on science under the title • Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences. Three appendices to this work were • • • La Dioptrique, Les Météores, La Géométrie. The treatise was published at Leiden in 1637 and Descartes wrote to Mersenne saying:- Descartes I have tried in my "Dioptrique" and my "Météores" to show that my Méthode is better than the vulgar, and in my "Géométrie" to have demonstrated it. Descartes The work describes what Descartes considers is a more satisfactory means of acquiring knowledge than that presented by Aristotle’s logic. Only mathematics, Descartes feels, is certain, so all must be based on mathematics. Descartes La Dioptrique is a work on optics and, although Descartes does not cite previous scientists for the ideas he puts forward, in fact there is little new. However his approach through experiment was an important contribution. Descartes Les Météores is a work on meteorology and is important in being the first work which attempts to put the study of weather on a scientific basis. However many of Descartes' claims are not only wrong but could have easily been seen to be wrong if he had done some easy experiments. Descartes For example Roger Bacon had demonstrated the error in the commonly held belief that water which has been boiled freezes more quickly. However Descartes claims:- ... and we see by experience that water which has been kept on a fire for some time freezes more quickly than otherwise, the reason being that those of its parts which can be most easily folded and bent are driven off during the heating, leaving only those which are rigid. Roger Bacon Roger Bacon, (c. 1214–1294), also known as Doctor Mirabilis (Latin: "wonderful teacher"), was an English philosopher and Franciscan friar who placed considerable emphasis on empiricism. • In philosophy, empiricism is a theory of knowledge which proports that knowledge arises from experience. Descartes Despite its many faults, the subject of meteorology was set on course after publication of Les Météores particularly through the work of Boyle Hooke Halley. Robert Boyle, 1627 - 1691 Robert Boyle Robert Boyle, 1627 - 1691 Robert Boyle was an Irish-born scientist who was a founding fellow of the Royal Society. His work in chemistry was aimed at establishing it as a mathematical science based on a mechanistic theory of matter. Robert Hooke, 1635 - 1703 Robert Hooke was an English scientist who made contributions to many different fields including : mathematics, optics, mechanics, architecture and astronomy. He had a famous quarrel with Newton. Edmond Halley, 1656 - 1742 Edmond Halley was an English astronomer who calculated the orbit of the comet now called Halley's comet. He was a supporter of Newton. Descartes La Géométrie is by far the most important part of Descartes’ work. Descartes He makes the first step towards a theory of invariants. Algebra makes it possible to recognise the typical problems in geometry and to bring together problems which in geometrical dress would not appear to be related at all. Algebra imports into geometry the most natural principles of division and the most natural hierarchy of method. Not only can questions of solvability and geometrical possibility be decided elegantly, quickly and fully from the parallel algebra, without it they cannot be decided at all. Descartes Descartes' Meditations on First Philosophy, was published in 1641, designed for the philosopher and for the theologian. It consists of six meditations, Of the Things that we may doubt Of the Nature of the Human Mind Of God: that He exists Of Truth and Error Of the Essence of Material Things Of the Existence of Material Things The Real Distinction between the Mind and the Body of Man. Descartes The most comprehensive of Descartes' works, • Principia Philosophiae was published in Amsterdam in 1644. In four parts, The Principles of Human Knowledge, The Principles of Material Things, Of the Visible World and The Earth, it attempts to put the whole universe on a mathematical foundation reducing the study to one of mechanics. • Not the resemblance to the title of Newton’s great publication Descartes This is an important point of view and was to point the way forward. Descartes did not believe in action at a distance. • Newton’s gravitation employs this. Therefore, given this, there could be no vacuum around the Earth otherwise there was no way that forces could be transferred. In many ways Descartes's theory, where forces work through contact, is more satisfactory than the mysterious effect of gravity acting at a distance. Descartes However Descartes' mechanics leaves much to be desired. He assumes that the universe is filled with matter which, due to some initial motion, has settled down into a system of vortices which carry the sun, the stars, the planets and comets in their paths. Despite the problems with the vortex theory it was championed in France for nearly one hundred years even after Newton showed it was impossible as a dynamical system. As Brewster, one of Newton’s 19th century biographers, puts it:- Newton-Descartes Thus entrenched as the Cartesian system was ... it was not to be wondered at that the pure and sublime doctrines of the Principia were distrustfully received ... The uninstructed mind could not readily admit the idea that the great masses of the planets were suspended in empty space, and retained their orbits by an invisible influence... Descartes's Pleasing as Descartes's theory was even the supporters of his natural philosophy, such as the Cambridge metaphysical theologian Henry More, found objections. Certainly More admired Descartes, writing:- Descartes's I should look upon Des-Cartes as a man most truly inspired in the knowledge of Nature, than any that have professed themselves so these sixteen hundred years... Descartes However between 1648 and 1649 they exchanged a number of letters in which More made some telling objections. Descartes however in his replies making no concessions to More’s points. Descartes In 1644, the year his Meditations were published, Descartes visited France. He returned again in 1647, when he met Pascal and argued with him that a vacuum could not exist, and then again in 1648. Descartes In 1649 Queen Christina of Sweden persuaded Descartes to go to Stockholm. However the Queen wanted to draw tangents at 5 a.m. and Descartes broke the habit of his lifetime of getting up at 11:00am. After only a few months in the cold northern climate, walking to the palace for 5 o'clock every morning, he died of pneumonia. Fermat [In the margin of his copy of Diophantus' Arithmetica, Fermat wrote] To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it. And perhaps, posterity will thank me for having shown it that the ancients did not know everything. Quoted in D M Burton, Elementary Number Theory (Boston 1976). Diophantus of Alexandria Born: about 200 BCE Died: about 284 BCE Diophantus, often known as the 'father of algebra', is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers. However, essentially nothing is known of his life and there has been much debate regarding the date at which he lived. Fermat Whenever two unknown magnitudes appear in a final equation, we have a locus, the extremity of one of the unknown magnitudes describing a straight line or a curve. Introduction to Plane and Solid Loci Fermat Born: 17 Aug 1601 in Beaumont-de-Lomagne, France Died: 12 Jan 1665 in Castres, France Fermat was a lawyer and government official most remembered for his work in number theory, in particular for Fermat's Last Theorem. • He used to pose problems for the mathematics community to solve and the last one to be solved is the so called Fermat’s Last Theorem We will discuss this theorem later in the course. Fermat Pierre Fermat's father was a wealthy leather merchant and second consul of Beaumont- deLomagne. Pierre had a brother and two sisters and was almost certainly brought up in the town of his birth. Although there is little evidence concerning his school education it must have been at the local Franciscan monastery. He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. Fermat In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius’s Plane loci to one of the mathematicians there. Apollonius of Perga about 262 BC about 190 BC Apollonius of Perga Apollonius was a Greek mathematician known as 'The Great Geometer'. His works had a very great influence on the development of mathematics and his famous book Conics introduced the terms parabola ellipse hyperbola. Fermat 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. Elementary calculus Jean Beaugrand about 1590 - 1640 Jean Beaugrand was, it is believed, the son of Jean Beaugrand who was an author of the works La paecilographie (1602) and Escritures (1604) and the calligraphy teacher to Louis XIII who was king of France from 1610 to 1643. Very little is known about the life of Jean Beaugrand, the subject of this biography, and what we do know has been pieced together from references to him in the correspondence of Descartes, Fermat and Mersenne. Aristotle Aristotle (384-322BCE) Born at Stagira in Northern Greece. Aristotle was the most notable product of the educational program devised by Plato; he spent twenty years of his life studying at the Academy. When Plato died, Aristotle returned to his native Macedonia, where he is supposed to have participated in the education of Philip's son, Alexander (the Great) Aristotle He came back to Athens with Alexander's approval in 335 and established his own school at the Lyceum, spending most of the rest of his life engaged there in research, teaching, and writing. His students acquired the name "peripatetics" from the master's habit of strolling about as he taught. Aristotle Although the surviving works of Aristotle probably represent only a fragment of the whole, they include his investigations of an amazing range of subjects, from logic philosophy ethics, physics biology psychology politics rhetoric. Aristotle Aristotle appears to have thought through his views as he wrote, returning to significant issues at different stages of his own development. The result is less a consistent system of thought than a complex record of Aristotle's thinking about many significant issues. Mersenne Marin Mersenne was born into a working class family in the small town of Oizé in the province of Maine on 8 September 1588 and was baptised on the same day. From an early age he showed signs of devotion and eagerness to study. So, despite their financial situation, Marin's parents sent him to the Collège du Mans where he took grammar classes. Later, at the age of sixteen, Mersenne asked to go to the newly established Jesuit School in La Flèche which had been set up as a model school for the benefit of all children regardless of their parents' financial situation. Mersenne It turns out that Descartes, who was eight years younger than Mersenne, was enrolled at the same school although they are not thought to have become friends until much later. Mersenne Mersenne's father wanted his son to have a career in the Church. Mersenne, however, was devoted to study, which he loved, and, showing that he was ready for responsibilities of the world, had decided to further his education in Paris. He left for Paris staying en route at a convent of the Minims. This experience so inspired Mersenne that he agreed to join their Order if one day he decided to lead a monastic life. Mersenne After reaching Paris he studied at the Collège Royale du France, continuing there his education in philosophy and also attending classes in theology at the Sorbonne where he also obtained the degree of Magister Atrium in Philosophy. He finished his studies in 1611 and, having had a privileged education, realised that he was now ready for the calm and studious life of a monastery. Jean Beaugrand It is said that he was a pupil of Viete but since Viete died in 1603 this must have been at a very early stage in Beaugrand's education. Viète Viète Born: 1540 in Fontenay-le-Comte, Poitou (now Vendée), France Died: 13 Dec 1603 in Paris, France François Viète's father was Étienne Viète, a lawyer in Fontenay-le-Comte in western France about 50 km east of the coastal town of La Rochelle. François' mother was Marguerite Dupont. He attended school in Fontenay-le-Comte and then moved to Poitiers, about 80 km east of Fontenay-leComte, where he was educated at the University of Poitiers. Viète Given the occupation of his father, it is not surprising that Viète studied law at university. After graduating with a law degree in 1560, Viète entered the legal profession but he only continued on this path for four years before deciding to change his career. Viète In 1564 Viète took a position in the service of Antoinette d'Aubeterre. He was employed to supervise the education of Antoinette's daughter Catherine, who would later become Catherine of Parthenay (Parthenay is about half-way between Fontenay-le-Comte and Poitiers). Catherine's father died in 1566 and Antoinette d'Aubeterre moved with her daughter to La Rochelle. Viète moved to La Rochelle with his employer and her daughter. Viète Viète introduced the first systematic algebraic notation in his book In artem analyticam isagoge published at Tours in 1591. The title of the work may seem puzzling, for it means "Introduction to the analytic art" which hardly makes it sound like an algebra book. However, Viète did not find Arabic mathematics to his liking and based his work on the Italian mathematicians such as Cardan, and the work of ancient Greek mathematicians. Viète One would have to say, however, that had Viète had a better understanding of Arabic mathematics he might have discovered that many of the ideas he produced were already known to earlier Arabic mathematicians. Cardan Born: 24 Sept 1501 in Pavia, Duchy of Milan (now Italy) Died: 21 Sept 1576 in Rome (now Italy) Girolamo or Hieronimo Cardano's name was Hieronymus Cardanus in Latin and he is sometimes known by the English version of his name Jerome Cardan. Girolamo Cardano 1501 - 1576 Girolamo Cardan or Cardano was an Italian doctor and mathematician who is famed for his work Ars Magna which was the first Latin treatise devoted solely to algebra. In it he gave the methods of solution of the cubic and quartic equations which he had learnt from Tartaglia. Fermat From Bordeaux Fermat went to Orléans where he studied law at the University. He received a degree in civil law and he purchased the offices of councillor at the parliament in Toulouse. So by 1631 Fermat was a lawyer and government official in Toulouse and because of the office he now held he became entitled to change his name from Pierre Fermat to Pierre de Fermat. For the remainder of his life he lived in Toulouse but as well as working there he also worked in his home town of Beaumont-de-Lomagne and a nearby town of Castres. Fermat From his appointment on 14 May 1631 Fermat worked in the lower chamber of the parliament but on 16 January 1638 he was appointed to a higher chamber, then in 1652 he was promoted to the highest level at the criminal court. Fermat Still further promotions seem to indicate a fairly meteoric rise through the profession but promotion was done mostly on seniority and the plague struck the region in the early 1650s meaning that many of the older men died. Fermat himself was struck down by the plague and in 1653 his death was wrongly reported, then corrected:- Fermat I informed you earlier of the death of Fermat. He is alive, and we no longer fear for his health, even though we had counted him among the dead a short time ago. The following report, made to Colbert the leading figure in France at the time, has a ring of truth:Fermat, a man of great erudition, has contact with men of learning everywhere. But he is rather preoccupied, he does not report cases well and is confused. Fermat Of course Fermat was preoccupied with mathematics. He kept his mathematical friendship with Beugrand after he moved to Toulouse but there he gained a new mathematical friend in Carcavi. Fermat met Carcavi in a professional capacity since both were councillors in Toulouse but they both shared a love of mathematics and Fermat told Carcavi about his mathematical discoveries. Fermat In 1636 Carcavi went to Paris as royal librarian and made contact with Mersenne and his group. Mersenne's interest was aroused by Carcavi's descriptions of Fermat's discoveries on falling bodies, and he wrote to Fermat. Fermat replied on 26 April 1636 and, in addition to telling Mersenne about errors which he believed that Galileo had made in his description of free fall, he also told Mersenne about his work on spirals and his restoration of Apollonius's Plane loci. Fermat His work on spirals had been motivated by considering the path of free falling bodies and he had used methods generalised from Archimedes' work On spirals to compute areas under the spirals. In addition Fermat wrote:- Fermat I have also found many sorts of analyses for diverse problems, numerical as well as geometrical, for the solution of which Vietes’ analysis could not have sufficed. I will share all of this with you whenever you wish and do so without any ambition, from which I am more exempt and more distant than any man in the world. Fermat It is somewhat ironical that this initial contact with Fermat and the scientific community came through his study of free fall since Fermat had little interest in physical applications of mathematics. Even with his results on free fall he was much more interested in proving geometrical theorems than in their relation to the real world. Fermat This first letter did however contain two problems on maxima which Fermat asked Mersenne to pass on to the Paris mathematicians and this was to be the typical style of Fermat's letters, he would challenge others to find results which he had already obtained. Fermat Roberval and Mersenne found that Fermat's problems in this first, and subsequent, letters were extremely difficult and usually not soluble using current techniques. They asked him to divulge his methods and Fermat sent Method for determining Maxima and Minima and Tangents to Curved Lines, his restored text of Apollonius’s Plane loci and his algebraic approach to geometry Introduction to Plane and Solid Loci to the Paris mathematicians. Fermat His reputation as one of the leading mathematicians in the world came quickly but attempts to get his work published failed mainly because Fermat never really wanted to put his work into a polished form. However some of his methods were published, for example Herigone added a supplement containing Fermat's methods of maxima and minima to his major work Cursus mathematicus. The widening correspondence between Fermat and other mathematicians did not find universal praise. Frenicle de Bessy became annoyed at Fermat's problems which to him were impossible. Fermat He wrote angrily to Fermat but although Fermat gave more details in his reply, Frenicle de Bessy felt that Fermat was almost teasing him. Fermat However Fermat soon became engaged in a controversy with a more major mathematician than Frenicle de Bessy Having been sent a copy of Descartes' La Dioptrique by Beaugrand, Fermat paid it little attention since he was in the middle of a correspondence with Roberval and Etienne Pascal over methods of integration and using them to find centres of gravity. Mersenne asked him to give an opinion on La Dioptrique which Fermat did, describing it as groping about in the shadows. Fermat He claimed that Descartes had not correctly deduced his law of refraction since it was inherent in his assumptions. To say that Descartes was not pleased is an understatement. Descartes soon found reason to feel even more angry since he viewed Fermat's work on maxima, minima and tangents as reducing the importance of his own work La Géométrie which Descartes was most proud of and which he sought to show that his Discours de la méthode alone could give. Fermat Descartes attacked Fermat's method of maxima, minima and tangents. Roberval and E. Pascal became involved in the argument and eventually so did Desargues who Descartes asked to act as a referee. Fermat proved correct and eventually Descartes admitted this writing:- Girard Desargues, 1591 - 1661 Girard Desargues was a French mathematician who was a founder of projective geometry. His work centred on the theory of conic sections and perspective. Example from projective geometry Projective Geometry Projective geometry is a non-metrical form of geometry. Projective geometry grew out of the principles of perspective art established during the Renaissance period, and was first systematically developed by Desargues in the 17th century, although it did not achieve prominence as a field of mathematics until the early 19th century through the work of Poncelet and others. Jean Victor Poncelet, 1788 1867 Poncelet was one of the founders of modern projective geometry. His development of the pole and polar lines associated with conics led to the principle of duality. Fermat ... seeing the last method that you use for finding tangents to curved lines, I can reply to it in no other way than to say that it is very good and that, if you had explained it in this manner at the outset, I would have not contradicted it at all. Did this end the matter and increase Fermat's standing? Not at all since Descartes tried to damage Fermat's reputation. Fermat For example, although he wrote to Fermat praising his work on determining the tangent to a cycloid (which is indeed correct), Descartes wrote to Mersenne claiming that it was incorrect and saying that Fermat was inadequate as a mathematician and a thinker. Descartes was important and respected and thus was able to severely damage Fermat's reputation. Fermat The period from 1643 to 1654 was one when Fermat was out of touch with his scientific colleagues in Paris. There are a number of reasons for this. Firstly pressure of work kept him from devoting so much time to mathematics. Secondly the Fronde, a civil war in France, took place and from 1648 Toulouse was greatly affected. Fermat Finally there was the plague of 1651 which must have had great consequences both on life in Toulouse and of course its near fatal consequences on Fermat himself. However it was during this time that Fermat worked on number theory. Fermat Fermat is best remembered for this work in number theory, in particular for Fermat’s last Theorem. This theorem states that Fermat’s last theorem x y z n2 n n n Fermat has no non-zero integer solutions for x, y and z when n > 2. Fermat wrote, in the margin of Bachet’s translation of Diophantus’s Arithmetica I have discovered a truly remarkable proof which this margin is too small to contain. Fermat These marginal notes only became known after Fermat's son Samuel published an edition of Bachet’s translation of Diophantus’s Arithmetica with his father's notes in 1670. It is now believed that Fermat's proof was wrong although it is impossible to be completely certain. The truth of Fermat's assertion was proved in June 1993 by the British mathematician Andrew Wiles, but Wiles withdrew the claim to have a proof when problems emerged later in 1993. Fermat In November 1994 Wiles again claimed to have a correct proof which has now been accepted. Unsuccessful attempts to prove the theorem over a 300 year period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries. Fermat's correspondence with the Paris mathematicians restarted in 1654 when Blaise Pascal, E Pascal's son, wrote to him to ask for confirmation about his ideas on probability. Blaise Pascal knew of Fermat through his father, who had died three years before, and was well aware of Fermat's outstanding mathematical abilities. Fermat Their short correspondence set up the theory of probability and from this they are now regarded as joint founders of the subject. Fermat however, feeling his isolation and still wanting to adopt his old style of challenging mathematicians, tried to change the topic from probability to number theory. Pascal was not interested but Fermat, not realising this, wrote to Carcavi saying:- Fermat am delighted to have had opinions conforming to those of M Pascal, for I have infinite esteem for his genius... the two of you may undertake that publication, of which I consent to your being the masters, you may clarify or supplement whatever seems too concise and relieve me of a burden that my duties prevent me from taking on. Fermat However Pascal was certainly not going to edit Fermat's work and after this flash of desire to have his work published Fermat again gave up the idea. He went further than ever with his challenge problems however:Two mathematical problems posed as insoluble to French, English, Dutch and all mathematicians of Europe by Monsieur de Fermat, Councillor of the King in the Parliament of Toulouse. Fermat His problems did not prompt too much interest as most mathematicians seemed to think that number theory was not an important topic. The second of the two problems, namely to find all solutions of Nx2 + 1 = y2 for N not a square, was however solved by Wallis and Brouncker and they developed continued fractions in their solution. Brouncker produced rational solutions which led to arguments. De Bessy was perhaps the only mathematician at that time who was really interested in number theory but he did not have sufficient mathematical talents to allow him to make a significant contribution. Fermat Fermat posed further problems, namely that the sum of two cubes cannot be a cube (a special case of Fermat's Last Theorem which may indicate that by this time Fermat realised that his proof of the general result was incorrect), that there are exactly two integer solutions of x2 + 4 = y3 and that the equation x2 + 2 = y3 has only one integer solution. He posed problems directly to the English. Everyone failed to see that Fermat had been hoping his specific problems would lead them to discover, as he had done, deeper theoretical results. Fermat Around this time one of Descartes' students was collecting his correspondence for publication and he turned to Fermat for help with the Fermat Descartes correspondence. This led Fermat to look again at the arguments he had used 20 years before and he looked again at his objections to Descartes' optics. In particular he had been unhappy with Descartes ' description of refraction of light and he now settled on a principle which did in fact yield the sine law of refraction that Snell and Descartes had proposed. Fermat However Fermat had now deduced it from a fundamental property that he proposed, namely that light always follows the shortest possible path. Fermat's principle, now one of the most basic properties of optics, did not find favor with mathematicians at the time Fermat In 1656 Fermat had started a correspondence with Huygens. This grew out of Huygens interest in probability and the correspondence was soon manipulated by Fermat onto topics of number theory. This topic did not interest Huygens but Fermat tried hard and in New Account of Discoveries in the Science of Numbers sent to Huygens via Carcavi in 1659, he revealed more of his methods than he had done to others. Fermat Fermat described his method of infinite descent and gave an example on how it could be used to prove that every prime of the form 4k + 1 could be written as the sum of two squares. For suppose some number of the form 4k + 1 could not be written as the sum of two squares. Then there is a smaller number of the form 4k + 1 which cannot be written as the sum of two squares. Continuing the argument will lead to a contradiction. Fermat What Fermat failed to explain in this letter is how the smaller number is constructed from the larger. One assumes that Fermat did know how to make this step but again his failure to disclose the method made mathematicians lose interest. It was not until Euler took up these problems that the missing steps were filled in. Fermat Fermat is described as: Secretive and taciturn, he did not like to talk about himself and was loath to reveal too much about his thinking. ... His thought, however original or novel, operated within a range of possibilities limited by that [1600 - 1650] time and that [France] place. Leonhard Euler, 1707 - 1783 Leonhard Euler was a Swiss mathematician who made enormous contributions to a wide range of mathematics and physics including analytic geometry, trigonometry, geometry, calculus and number theory Fermat Carl B Boyer, writes:Recognition of the significance of Fermat's work in analysis was tardy, in part because he adhered to the system of mathematical symbols devised by Viete, notations that Descartes’ "Géométrie" had rendered largely obsolete. The handicap imposed by the awkward notations operated less severely in Fermat's favorite field of study, the theory of numbers, but here, unfortunately, he found no correspondent to share his enthusiasm. Fermat's last theorem Fermat died in 1665. Today we think of Fermat as a number theorist, in fact as perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was in fact a lawyer and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous article written as an appendix to a colleague's book. Fermat in Toulouse The is a picture of a statue of Fermat and his muse in his home town of Toulouse: Fermat Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat’s letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel written as a marginal note in his father's copy of Diophantus’s Arithmetica. Fermat Fermat almost certainly wrote the marginal note around 1630, when he first studied the Arithmetica. It may well be that Fermat realised that his remarkable proof was wrong, however, since all his other theorems were stated and restated in challenge problems that Fermat sent to other mathematicians. Although the special cases of n = 3 and n = 4 were issued as challenges (and Fermat did know how to prove these) the general theorem was never mentioned again by Fermat. Fermat In fact in all the mathematical work left by Fermat there is only one proof. Fermat proves that the area of a right triangle cannot be a square. Clearly this means that a rational triangle cannot be a rational square. In symbols, there do not exist integers x, y, z with x2 + y2 = z2 such that xy/2 is a square. From this it is easy to deduce the n = 4 case of Fermat's theorem. Fermat It is worth noting that at this stage it remained to prove Fermat's Last Theorem for odd primes n only. For if there were integers x, y, z with xn + yn = zn then if n = pq, (xq)p + (yq)p = (zq)p. Fermat Euler wrote to Goldbach on 4 August 1753 claiming he had a proof of Fermat's Theorem when n = 3. However his proof in Algebra (1770) contains a fallacy and it is far from easy to give an alternative proof of the statement which has the fallacious proof. There is an indirect way of mending the whole proof using arguments which appear in other proofs of Euler so perhaps it is not too unreasonable to attribute the n = 3 case to Euler. Fermat Euler’s mistake is an interesting one, one which was to have a bearing on later developments. He needed to find cubes of the form p2 + 3q2 and Euler shows that, for any a, b if we put p = a3 - 9ab2, q = 3(a2b - b3) then p2 + 3q2 = (a2 + 3b2)3. Fermat This is true but he then tries to show that, if p2 + 3q2 is a cube then an a and b exist such that p and q are as above. His method is imaginative, calculating with numbers of the form a + b√-3. However numbers of this form do not behave in the same way as the integers, which Euler did not seem to appreciate. Fermat The next major step forward was due to Sophie Germain. A special case says that if n and 2n + 1 are primes then xn + yn = zn implies that one of x, y, z is divisible by n. Hence Fermat's Last Theorem splits into two cases. Case 1: None of x, y, z is divisible by n. Case 2: One and only one of x, y, z is divisible by n. Marie-Sophie Germain, 1776 1831 Germain made a major contribution to number theory, acoustics and elasticity. Fermat Sophie Germain proved Case 1 of Fermat's Last Theorem for all n less than 100 and Legendre extended her methods to all numbers less than 197. At this stage Case 2 had not been proved for even n = 5 so it became clear that Case 2 was the one on which to concentrate. Now Case 2 for n = 5 itself splits into two. One of x, y, z is even and one is divisible by 5. Case 2(i) is when the number divisible by 5 is even; Case 2(ii) is when the even number and the one divisible by 5 are distinct. Adrien-Marie Legendre 1752 - 1833 Legendre's major work on elliptic integrals provided basic analytical tools for mathematical physics. He gave a simple proof that π is irrational as well as the first proof that π2 is irrational. Fermat Case 2(i) was proved by Dirichlet and presented to the Paris Academie des Sciences in July 1825. Legendre was able to prove Case 2(ii) and the complete proof for n = 5 was published in September 1825. In fact Dirichlet was able to complete his own proof of the n = 5 case with an argument for Case 2(ii) which was an extension of his own argument for Case 2(i). Johann Peter Gustav Lejeune Dirichlet, 1805 - 1859 Dirichlet proved in 1837 that in any arithmetic progression with first term coprime to the difference (i.e. no factors in common) there are infinitely many primes. Fermat In 1832 Dirichlet published a proof of Fermat's Last Theorem for n = 14. Of course he had been attempting to prove the n = 7 case but had proved a weaker result. The n = 7 case was finally solved by Lame in 1839. It showed why Dirichlet had so much difficulty, for although Dirichlet’s n = 14 proof used similar (but computationally much harder) arguments to the earlier cases, Lame had to introduce some completely new methods. Fermat Lame’s proof is exceedingly hard and makes it look as though progress with Fermat's Last Theorem to larger n would be almost impossible without some radically new thinking. The year 1847 is of major significance in the study of Fermat's Last Theorem. On 1 March of that year Lame announced to the Paris Academie that he had proved Fermat's Last Theorem. He sketched a proof which involved factorizing xn + yn = zn into linear factors over the complex numbers. Lame acknowledged that the idea was suggested to him by Liouville. Joseph Liouville 1809 - 1882 Liouville is best known for his work on the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers. Fermat However Liouville addressed the meeting after Lame and suggested that the problem of this approach was that uniqueness of factorisation into primes was needed for these complex numbers and he doubted if it were true. Cauchy supported Lame but, in rather typical fashion, pointed out that he had reported to the October 1847 meeting of the Academie an idea which he believed might prove Fermat's Last Theorem. Augustin Louis Cauchy 1789 - 1857 Cauchy pioneered the study of analysis, both real and complex, and the theory of permutation groups. He also researched in convergence and divergence of infinite series, differential equations, determinants, probability and mathematical physics. Fermat Much work was done in the following weeks in attempting to prove the uniqueness of factorization. Wantzel claimed to have proved it on 15 March but his argument It is true for n = 2, n = 3 and n = 4 and one easily sees that the same argument applies for n > 4 Fermat Wantzel is correct about n = 2 (ordinary integers), n = 3 (the argument Euler got wrong) and n = 4 (which was proved by Gauss). On 24 May Liouville read a letter to the Academie which settled the arguments. The letter was from Kummer, enclosing an offprint of a 1844 paper which proved that uniqueness of factorization failed but could be 'recovered' by the introduction of ideal complex numbers which he had done in 1846. Ernst Eduard Kummer 1810 - 1893 Kummer's main achievement was the extension of results about the integers to other integral domains by introducing the concept of an ideal Related to ring theory Fermat Kummer had used his new theory to find conditions under which a prime is regular and had proved Fermat's Last Theorem for regular primes. Kummer also said in his letter that he believed 37 failed his conditions Fermat By September 1847 Kummer sent to Dirichlet and the Berlin Academy a paper proving that a prime p is regular (and so Fermat's Last Theorem is true for that prime) if p does not divide the numerators of any of the Bernoulli numbers B2 , B4 , ..., Bp-3 . The Bernoulli number Bi is defined by Fermat x/(ex - 1) = Bi xi /i! Kummer shows that all primes up to 37 are regular but 37 is not regular as 37 divides the numerator of B32 . Fermat The only primes less than 100 which are not regular are 37, 59 and 67. More powerful techniques were used to prove Fermat's Last Theorem for these numbers. This work was done and continued to larger numbers by Kummer, Mirimanoff, Wieferich, Furtwängler, Vandiver and others. Although it was expected that the number of regular primes would be infinite even this defied proof. In 1915 Jensen proved that the number of irregular primes is infinite. Fermat Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved. It has the dubious distinction of being the theorem with the largest number of published false proofs. For example over 1000 false proofs were published between 1908 and 1912. The only positive progress seemed to be computing results which merely showed that any counter-example would be very large. Using techniques based on Kummer's work, Fermat's Last Theorem was proved true, with the help of computers, for n up to 4,000,000 by 1993. Fermat In 1983 a major contribution was made by gerd Faltings who proved that for every n > 2 there are at most a finite number of coprime integers x, y, z with xn + yn = zn. This was a major step but a proof that the finite number was 0 in all cases did not seem likely to follow by extending Faltings ' arguments. The final chapter in the story began in 1955, although at this stage the work was not thought of as connected with Fermat's Last Theorem. Yutaka Taniyama asked some questions about elliptic curves, i.e. curves of the form y2 = x3 + ax + b for constants a and b. Further work by Weil and Shimura produced a conjecture, now known as the Shimura-Taniyama-Weil Conjecture. Fermat In 1986 the connection was made between the Shimura-Taniyama-Weil Conjecture and Fermat's Last Theorem by Frey at Saarbrücken showing that Fermat's Last Theorem was far from being some unimportant curiosity in number theory but was in fact related to fundamental properties of space. Fermat Further work by other mathematicians showed that a counterexample to Fermat's Last Theorem would provide a counter example to the Shimura-Taniyama-Weil Conjecture. The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge, England the first on Monday 21 June, the second on Tuesday 22 June. In the final lecture on Wednesday 23 June 1993 at around 10.30 in the morning Wiles announced his proof of Fermat’s Last Theorem as a corollary to his main results. Fermat Having written the theorem on the blackboard he said I think I will stop here and sat down. In fact Wiles had proved the ShimuraTaniyama-Weil Conjecture for a class of examples, including those necessary to prove Fermat’s Last Theorem. Fermat This, however, is not the end of the story. On 4 December 1993 Andrew Wiles made a statement in view of the speculation. He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However one problem remains and Wiles essentially withdrew his claim to have a proof. He states The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semisquare case (of the symmetric square representation associated to a modular form) is not yet complete as it stands. I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures. Fermat In March 1994 Faltings, writing in Scientific American, said If it were easy, he would have solved it by now. Strictly speaking, it was not a proof when it was announced. Weil also in Scientific American, wrote I believe he has had some good ideas in trying to construct the proof but the proof is not there. To some extent, proving Fermat's Theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest. Fermat In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work. They tried a new approach with a similar lack of success. In August 1994 Wiles addressed the International Congress of Mathematicians but was no nearer to solving the difficulties. Taylor suggested a last attempt to extend Flach's method in the way necessary and Wiles, although convinced it would not work, agreed mainly to enable him to convince Taylor that it could never work. Wiles worked on it for about two weeks, then suddenly inspiration struck. Fermat In a flash I saw that the thing that stopped it [the extension of Flach's method] working was something that would make another method I had tried previously work. On 6 October Wiles sent the new proof to three colleagues including Faltings. All liked the new proof which was essentially simpler than the earlier one. Faltings sent a simplification of part of the proof. No proof of the complexity of this can easily be guaranteed to be correct, so a very small doubt will remain for some time. However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat's Last Theorem Blaise Pascal Pascal Blaise Pascal was the third of E.Pascal's children and his only son. Blaise's mother died when he was only three years old. In 1632 the Pascal family, Étienne and his four children, left Clermont and settled in Paris. Blaise Pascal's father had unorthodox educational views and decided to teach his son himself. E.Pascal decided that Blaise was not to study mathematics before the age of 15 and all mathematics texts were removed from their house. Blaise however, his curiosity raised by this, started to work on geometry himself at the age of 12. He discovered that the sum of the angles of a triangle are two right angles and, when his father found out, he relented and allowed Blaise a copy of Euclid. Pascal In December 1639 the Pascal family left Paris to live in Rouen where Étienne had been appointed as a tax collector for Upper Normandy. Shortly after settling in Rouen, Blaise had his first work, Essay on Conic sections published in February 1640. Pascal invented the first digital calculator to help his father with his work collecting taxes. He worked on it for three years between 1642 and 1645. The device, called the Pascaline, resembled a mechanical calculator of the 1940s. This, almost certainly, makes Pascal the second person to invent a mechanical calculator. Pascal There were problems faced by Pascal in the design of the calculator which were due to the design of the French currency at that time. There were 20 sols in a livre and 12 deniers in a sol. The system remained in France until 1799 but in Britain a system with similar multiples lasted until 1971. Pascal had to solve much harder technical problems to work with this division of the livre into 240 than he would have had if the division had been 100. However production of the machines started in 1642 but, as Adamson writes, By 1652 fifty prototypes had been produced, but few machines were sold, and manufacture of Pascal's arithmetical calculator ceased in that year. Pascal Events of 1646 were very significant for the young Pascal. In that year his father injured his leg and had to recuperate in his house. He was looked after by two young brothers from a religious movement just outside Rouen. They had a profound effect on the young Pascal and he became deeply religious. From about this time Pascal began a series of experiments on atmospheric pressure. By 1647 he had proved to his satisfaction that a vacuum existed. Descartes visited Pascal on 23 September. His visit only lasted two days and the two argued about the vacuum which Descartes did not believe in. Descartes wrote, rather cruelly, in a letter to Huygens after this visit that Pascal ...has too much vacuum in his head. Pascal Through the period of this correspondence Pascal was unwell. In one of the letters to Fermat written in July 1654 he writes ... though I am still bedridden, I must tell you that yesterday evening I was given your letter. However, despite his health problems, he worked intensely on scientific and mathematical questions until October 1654. Sometime around then he nearly lost his life in an accident. The horses pulling his carriage bolted and the carriage was left hanging over a bridge above the river Seine. Although he was rescued without any physical injury, it does appear that he was much affected psychologically. Not long after he underwent another religious experience, on 23 November 1654, and he pledged his life to Christianity.