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 Einsteins
  theory of relativity.
• Galileo formulated these concepts in his
  description of uniform motion .

27
Galileo

• The topic was motivated by Galileos 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 Euclids
  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 Galileos 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.
   Lets define each of the terms in the conjecture

51
Simply connected space

• This means the space has no holes.  A
  football is simply connected, but a donut is
  not.  Technically we can say to be
  explained further on.

52
Closed space

• The space is finite and has no boundaries.  A
  sphere  (more technically a 2-sphere or ) is
  closed, but the plane ( ) is not because it
  is infinite. 
• A disk is also not because even though it is
  finite, it has a boundary.

53
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.

54
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 cant get that
  pesky hole in the donut, and vice-versa.

55
Morphing
56
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.

57
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

58
Galileo

• In the following year he travelled to Rome to
  visit Clavius who was professor of mathematics at
  the Jesuit Collegio Romano there.

59
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.

60
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.

61
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.

62
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.

63
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.

64
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.

65
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.

66
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.

67
Galileo

• At Padua his duties were mainly to teach Euclids
  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.

68
Galileo

• However, Galileo argued against Aristotles view
  of astronomy and natural philosophy in three
  public lectures he gave in connection with the
  appearance of a New Star (now known as Keplers
  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.

69
Johann Kepler 1571 - 1630
70
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.

71
Keplers 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."

72
First law
73
Second law
74
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.

75
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.

76
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.

77
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.

78
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-

79
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.

80
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.

81
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.

82
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.

83
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

84
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.

85
Christiaan Huygens1629 - 1695
86
Christiaan Huygens1629 - 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

87
Isaac Newton

• Born 4 Jan 1643 in Woolsthorpe, Lincolnshire,
  EnglandDied 31 March 1727 in London, England

88
Isaac Newton
89
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-

90
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.

91
Descartes

• The work describes what Descartes considers is a
  more satisfactory means of acquiring knowledge
  than that presented by Aristotles logic.
• Only mathematics, Descartes feels, is certain, so
  all must be based on mathematics.

92
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.

93
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.

94
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.

95
Roger Bacon

• Roger Bacon, (c. 12141294), 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.

96
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.

97
Robert Boyle, 1627 - 1691
98
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.

99
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.
100
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.
101
Descartes

• La Géométrie is by far the most important part of
  Descartes work.

102
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.

103
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.

104
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 Newtons
  great publication

105
Descartes

• This is an important point of view and was to
  point the way forward.
• Descartes did not believe in action at a
  distance.
• Newtons 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.

106
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 Newtons 19th century
  biographers, puts it-

107
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...

108
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-

109
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...

110
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 Mores points.

111
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.

112
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 1100am.
• After only a few months in the cold northern
  climate, walking to the palace for 5 o'clock
  every morning, he died of pneumonia.

113
Fermat

• In the margin of his copy of Diophantus'
  Arithmetica, Fermat wroteTo 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).

114
Diophantus of Alexandria

• Born about 200 BCEDied 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.

115
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

116
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 Fermats Last Theorem
• We will discuss this theorem later in the course.

117
Fermat

• Pierre Fermat's father was a wealthy leather
  merchant and second consul of Beaumont- de-
  Lomagne.
• 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.

118
Fermat

• In Bordeaux he began his first serious
  mathematical researches and in 1629 he gave a
  copy of his restoration of Apolloniuss Plane
  loci to one of the mathematicians there.

119
Apollonius of Perga

• about 262 BC - about 190 BC

120
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.

121
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

122
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.

123
Aristotle
124
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)

125
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.

126
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.

127
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.

128
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.

129
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.

130
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.

131
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.

132
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.

133
Viète
134
Viète

• Born 1540 in Fontenay-le-Comte, Poitou (now
  Vendée), FranceDied 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-le-Comte, where he was educated at the
  University of Poitiers.

135
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.

136
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.

137
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.

138
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.

139
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.

140
Girolamo Cardano1501 - 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.

141
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.

142
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.

143
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-

144
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.

145
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.

146
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.

147
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-

148
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.

149
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.

150
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.

151
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
  Apolloniuss Plane loci and his algebraic
  approach to geometry Introduction to Plane and
  Solid Loci to the Paris mathematicians.

152
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.

153
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.

154
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.

155
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.

156
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-

157
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.
158
Example from projective geometry
159
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.

160
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.
161
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.

162
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.

163
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.

164
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.

165
Fermat

• Fermat is best remembered for this work in number
  theory, in particular for Fermats last Theorem.
• This theorem states that

166
Fermats last theorem
167
Fermat

• has no non-zero integer solutions for x, y and z
  when n gt 2.
• Fermat wrote, in the margin of Bachets
  translation of Diophantuss Arithmetica
• I have discovered a truly remarkable proof which
  this margin is too small to contain.

168
Fermat

• These marginal notes only became known after
  Fermat's son Samuel published an edition of
  Bachets translation of Diophantuss 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.

169
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.

170
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-

171
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.

172
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.

173
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.

174
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.

175
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.

176
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

177
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.

178
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.

179
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.

180
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.

181
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
182
Fermat

• Carl B Boyer, writes-
• Recognition of the sig