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Rene Descartes, Pierre Fermat and Blaise Pascal

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Rene Descartes, Pierre Fermat and Blaise Pascal Descartes, Fermat and Pascal: a philosopher, an amateur and a calculator Fermat Sophie Germain proved Case 1 of Fermat ... – PowerPoint PPT presentation

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Title: 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
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