Rene Descartes, Pierre Fermat and Blaise Pascal

1. Rene Descartes, Pierre Fermat and Blaise Pascal Descartes, Fermat and Pascal: a philosopher, an amateur and a calculator

2. Rene Descartes 1596 - 1650

3. Pierre de Fermat 17th August 1601 or 1607 – 12th January 1665

4. Blaise Pascal, 1623 - 1662

5. 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.

6. 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.

7. 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.

8. Clavius 1538 - 1612 • Christopher Clavius was a German Jesuit astronomer who helped Pope Gregory XIII to introduce what is now called the Gregorian calendar.

9. 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.

10. 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

11. 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.

12. 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.

13. Descartes • From Paris he travelled to Italy where he spent some time in Venice, then he returned to France again (1625).

14. 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) 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics,

15. 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.

16. 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.

17. 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.

18. 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:-

19. 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.

20. Galileo, portrait by Justus Sustermans painted in 1636

21. 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.

22. Nicolaus Copernicus1473 - 1543

23. 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.

24. 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.

25. 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.

26. 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.

27. 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.

28. 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

29. Hendrik Antoon Lorentz1853 - 1928

30. Lorentz • Lorentz is best known for his work on electromagnetic radiation and the FitzGerald-Lorentz contraction. • He developed the mathematical theory of the electron.

31. Galileo • These relativistic transformations are deemed applicable to all velocities, whilst the Galilean transformation can be regarded as a low-velocity 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 x-direction according to the Galilean transformation:

32. Translation (one dimension)

33. Lorenz transformations

34. 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 x-axis of frame F.Bottom: frame F moves at velocity −v along the x′-axis of frame F

35. Translation (one dimension) • Note that the last equation (Galileo) expresses the assumption of a universal time independent of the relative motion of different observers.

36. 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.

37. 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.

38. 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.

39. 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.

40. 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.

41. 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.

42. 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.

43. Nicolo Fontana Tartaglia

44. Nicolo Fontana Tartaglia1500 - 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.

45. François Viète, 1540 - 1603

46. 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

47. 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.

48. 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:

49. The Poincare conjecture • Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

50. Topology • This is a statement about topological spaces.  Let’s define each of the terms in the conjecture: