Pierre Fermat and Blaise Pascal

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Pierre Fermat and Blaise Pascal
Rene Descartes, Pierre
Fermat and Blaise Pascal
Descartes, Fermat and
Pascal: a philosopher, an
amateur and a calculator
Rene Descartes
1596 - 1650
Pierre de Fermat 17th August 1601 or
1607 – 12th January 1665
Blaise Pascal, 1623 - 1662
René Descartes was a French
philosopher whose work, La géométrie,
includes his application of algebra to
geometry from which we now have
Cartesian geometry.
His work had a great influence on both
mathematicians and philosophers.
Descartes 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
While in the school his health was poor
and he was granted permission to
remain in bed until 11 o'clock in the
morning, a custom he maintained until
the year of his death.
In bed he came up with idea now called
Cartesian geometry.
Clavius 1538 - 1612
Christopher Clavius was a German
Jesuit astronomer who helped Pope
Gregory XIII to introduce what is now
called the Gregorian calendar.
School had made Descartes understand
how little he knew, the only subject
which was satisfactory in his eyes was
This idea became the foundation for his
way of thinking, and was to form the
basis for all his works.
The above statement was echoed by
Einstein in the 20th century.
Descartes 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
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.
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.
From Paris he travelled to Italy where he
spent some time in Venice, then he
returned to France again (1625).
Marin Mersenne was a French monk who is best
known for his role as a clearing house for
correspondence between eminent philosophers and
scientists and for his work in number theory.
Similar to Bourbaki
• Nicholas Bourbaki is the collective pseudonym
under which a group of (mainly French) 20thcentury mathematicians wrote a series of books
presenting an exposition of modern advanced
Mersenne prime
In mathematics, a
Mersenne number
is a positive integer
that is one less than
a power of two
Some definitions of
Mersenne numbers
require that the
exponent n be prime.
2 1
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.
By 1628 Descartes was tired of the
continual travelling and decided to settle
He gave much thought to choosing a
country suited to his nature and chose
It was a good decision which he did not
seem to regret over the next twenty
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
... 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
portrait by Justus Sustermans
painted in 1636
Galileo Galilei
Galileo Galilei was an Italian scientist
who formulated the basic law of falling
bodies, which he verified by careful
He constructed a telescope with which
he studied lunar craters, and discovered
four moons revolving around Jupiter and
espoused the Copernican cause.
Nicolaus Copernicus
1473 - 1543
Copernicus was a Polish astronomer
and mathematician who was a
proponent of the view of an Earth in daily
motion about its axis and in yearly
motion around a stationary sun.
• Helio-Centric universe
This theory profoundly altered later
workers' view of the universe, but was
rejected by the Catholic church.
Galileo Galilei
Galileo Galilei: considered the father of
experimental physics along with Ernest
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
Galileo Galilei
Guilia, who was born in Pescia, married
Vincenzo in 1563 and they made their
home in the countryside near Pisa.
Galileo was their first child and spent his
early years with his family in Pisa.
Galilean transformation
The Galilean transformation is used to transform
between the coordinates of two reference frames
which differ only by constant relative motion within
the constructs of Newtonian Physics.
The equations below, although apparently obvious,
break down at speeds that approach the speed of
light due to physics described by Einstein’s theory of
Galileo formulated these concepts in his description
of uniform motion .
The topic was motivated by Galileo’s
description of the motion of a ball rolling
down a ramp, by which he measured the
numerical value for the acceleration due to
gravity, g at the surface of the earth.
The descriptions below are another
mathematical notation for this concept.
Translation (one dimension)
In essence, the Galilean transformations embody the
intuitive notion of addition and subtraction of
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
Hendrik Antoon Lorentz
1853 - 1928
Lorentz is best known for his work on
electromagnetic radiation and the
FitzGerald-Lorentz contraction.
He developed the mathematical theory
of the electron.
These relativistic transformations are deemed
applicable to all velocities, whilst the Galilean
transformation can be regarded as a lowvelocity approximation to the Lorentz
The notation below describes the relationship
of two coordinate systems (x′ and x) in
constant relative motion (velocity v) in the xdirection according to the Galilean
Translation (one dimension)
x '  x  vt
y'  y
z' z
Lorenz transformations
Lorenz transformations
The spacetime
coordinates of an event,
as measured by each
observer in their inertial
reference frame (in
standard configuration)
are shown in the speech
Top: frame F′ moves at
velocity v along the xaxis of frame F.
Bottom: frame F moves
at velocity −v along the
x′-axis of frame F
Translation (one dimension)
Note that the last equation (Galileo)
expresses the assumption of a universal
time independent of the relative motion
of different observers.
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.
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.
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
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.
Although the idea of a medical career never
seems to have appealed to Galileo, his
father's wish was a fairly natural one since
there had been a distinguished physician in
his family in the previous century.
Galileo 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
In the year 1582-83 Ostilio Ricci, who was the
mathematician of the Tuscan Court and a former
pupil of Tartaglia, taught a course on Euclid’s
Elements at the University of Pisa which Galileo
However Galileo, still reluctant to study medicine,
invited Ricci (also in Florence where the Tuscan
court spent the summer and autumn) to his home to
meet his father.
Nicolo Fontana Tartaglia
Nicolo Fontana Tartaglia
1500 - 1557
Tartaglia was an Italian mathematician
who was famed for his algebraic solution
of cubic equations which was eventually
published in Cardan's Ars Magna.
He is also known as the stammerer.
François Viète, 1540 - 1603
François Viète
François Viète was a French amateur
mathematician and astronomer who
introduced the first systematic algebraic
notation in his book
In artem analyticam isagoge .
He was also involved in deciphering
• Alan Turin
Ricci and Galileo’s father
Ricci tried to persuade Vincenzo to allow his
son to study mathematics since this was
where his interests lay.
Ricci flow is of paramount importance in the
solution to the Poincare Conjecture.
The Poincaré
Conjecture Explained
The Poincaré Conjecture is first and only
of the Clay Millennium problems to be
solved, (2005)
It was proved by Grigori Perelman who
subsequently turned down the $1 million
prize money, left mathematics, and
moved in with his mother in
Russia. Here is the statement of the
conjecture from wikipedia:
The Poincare conjecture
Every simply connected, closed 3manifold is homeomorphic to the 3sphere.
This is a statement about topological
spaces. Let’s define each of the terms in
the conjecture:
Simply connected space
– This means the space has no
“holes.” A football is simply connected,
but a donut is not. Technically we can
say  ( X )  0 to be explained further on.
Closed space
– The space is finite and has no
boundaries. A sphere (more technically
a 2-sphere or S 2 ) is closed, but the plane
( R 2 ) is not because it is infinite.
A disk is also not because even though it
is finite, it has a boundary.
– 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
– 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
The 2-sphere and a donut are not; no
matter how much you deform a sphere,
you can’t get that pesky hole in the
donut, and vice-versa.
Certainly Vincenzo did not like the idea (of
his son studying mathematics) and resisted
strongly but eventually he gave way a little
and Galileo was able to study the works of
Euclid and Archimedes from the Italian
translations which Tartaglia had made.
Of course he was still officially enrolled as a
medical student at Pisa but eventually, by
1585, he gave up this course and left without
completing his degree.
Galileo 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
Essentially looking at the use of plumblines
In the following year he travelled to Rome to
visit Clavius who was professor of
mathematics at the Jesuit Collegio Romano
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.
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.
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
Fantoni left the chair of mathematics at the
University of Pisa in 1589 and Galileo was appointed
to fill the post (although this was only a nominal
position to provide financial support for Galileo).
Not only did he receive strong recommendations
from Clavius, but he also had acquired an excellent
reputation through his lectures at the Florence
Academy in the previous year.
The young mathematician had rapidly acquired the
reputation that was necessary to gain such a
position, but there were still higher positions at which
he might aim.
Galileo 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.
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.
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.
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.
At Padua his duties were mainly to
teach Euclid’s geometry and
standard (geocentric) astronomy to
medical students, who would need
to know some astronomy in order
to make use of astrology in their
medical practice.
However, Galileo argued against Aristotle’s view
of astronomy and natural philosophy in three
public lectures he gave in connection with the
appearance of a New Star (now known as
‘Kepler’s supernova') in 1604.
The belief at this time was that of Aristotle,
namely that all changes in the heavens had to
occur in the lunar region close to the Earth, the
realm of the fixed stars being permanent.
Johann Kepler 1571 - 1630
Johannes Kepler
Johannes Kepler was a German
mathematician and astronomer who
discovered that the Earth and planets
travel about the sun in elliptical orbits.
He gave three fundamental laws of
planetary motion.
He also did important work in optics and
Kepler’s laws
The first law says: "The orbit of every planet is
an ellipse with the sun at one of the foci."
The second law: "A line joining a planet and
the sun sweeps out equal areas during equal
intervals of time.“
The third law : "The squares of the orbital
periods of planets are directly proportional to
the cubes of the axes of the orbits."
First law
Second law
Galileo 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
However, no public sign of this belief was
to appear until many years later.
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.
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.
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.
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.
In May 1609, Galileo received a letter
from Paolo Sarpi telling him about a
spyglass that a Dutchman had shown in
Galileo wrote in the Starry Messenger
(Sidereus Nuncius) in April 1610:-
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.
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.
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
His first telescope was made from available
lenses and gave a magnification of about
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.
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
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
In Holland Descartes had a number of
scientific friends as well as continued
contact with Mersenne.
His friendship with Beeckman continued
and he also had contact with Huygens.
Christiaan Huygens
1629 - 1695
Christiaan Huygens
1629 - 1695
Christiaan Huygens was a Dutch
mathematician who patented the first
pendulum clock, which greatly increased
the accuracy of time measurement.
He laid the foundations of mechanics
and also worked on astronomy and
• Proponent of the wave theory
He was a contempory of Isaac Newton
Isaac Newton
Born: 4 Jan 1643 in Woolsthorpe,
Lincolnshire, England
Died: 31 March 1727 in London, England
Isaac Newton
Descartes 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:-
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.
The work describes what Descartes
considers is a more satisfactory means
of acquiring knowledge than that
presented by Aristotle’s logic.
Only mathematics, Descartes feels, is
certain, so all must be based on
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
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
For example Roger Bacon had demonstrated the
error in the commonly held belief that water which
has been boiled freezes more quickly.
However Descartes claims:-
... and we see by experience that water which
has been kept on a fire for some time freezes
more quickly than otherwise, the reason being
that those of its parts which can be most easily
folded and bent are driven off during the
heating, leaving only those which are rigid.
Roger Bacon
Roger Bacon, (c. 1214–1294), also
known as Doctor Mirabilis (Latin:
"wonderful teacher"), was an English
philosopher and Franciscan friar who
placed considerable emphasis on
• In philosophy, empiricism is a theory of
knowledge which proports that knowledge
arises from experience.
Despite its many faults, the subject of
meteorology was set on course after
publication of Les Météores particularly
through the work of
Robert Boyle, 1627 - 1691
Robert Boyle
Robert Boyle, 1627 - 1691
Robert Boyle was an Irish-born scientist
who was a founding fellow of the Royal
His work in chemistry was aimed at
establishing it as a mathematical science
based on a mechanistic theory of matter.
Robert Hooke, 1635 - 1703
Robert Hooke was an English
scientist who made contributions to
many different fields
including :
mathematics, optics,
mechanics, architecture and
He had a famous quarrel with
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.
La Géométrie is by far the most
important part of Descartes’ work.
He makes the first step towards a theory of invariants.
Algebra makes it possible to recognise the typical problems in
geometry and to bring together problems which in geometrical
dress would not appear to be related at all.
Algebra imports into geometry the most natural principles of
division and the most natural hierarchy of method.
Not only can questions of solvability and geometrical possibility
be decided elegantly, quickly and fully from the parallel algebra,
without it they cannot be decided at all.
Descartes' 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.
The most comprehensive of Descartes' works,
• Principia Philosophiae
was published in Amsterdam in 1644.
In four parts, The Principles of Human Knowledge, The
Principles of Material Things, Of the Visible World and The
Earth, it attempts to put the whole universe on a mathematical
foundation reducing the study to one of mechanics.
• Not the resemblance to the title of Newton’s great
This is an important point of view and was to point
the way forward.
Descartes did not believe in action at a distance.
Newton’s gravitation employs this.
Therefore, given this, there could be no vacuum
around the Earth otherwise there was no way that
forces could be transferred.
In many ways Descartes's theory, where forces work
through contact, is more satisfactory than the
mysterious effect of gravity acting at a distance.
However Descartes' mechanics leaves much to be
He assumes that the universe is filled with matter
which, due to some initial motion, has settled down
into a system of vortices which carry the sun, the
stars, the planets and comets in their paths.
Despite the problems with the vortex theory it was
championed in France for nearly one hundred years
even after Newton showed it was impossible as a
dynamical system.
As Brewster, one of Newton’s 19th century
biographers, puts it:-
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...
Pleasing as Descartes's theory was even
the supporters of his natural philosophy,
such as the Cambridge metaphysical
theologian Henry More, found
Certainly More admired Descartes,
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
However between 1648 and 1649 they
exchanged a number of letters in which
More made some telling objections.
Descartes however in his replies making
no concessions to More’s points.
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.
In 1649 Queen Christina of Sweden persuaded
Descartes to go to Stockholm.
However the Queen wanted to draw tangents
at 5 a.m. and Descartes broke the habit of his
lifetime of getting up at 11:00am.
After only a few months in the cold northern
climate, walking to the palace for 5 o'clock
every morning, he died of pneumonia.
[In the margin of his copy of Diophantus' Arithmetica,
Fermat wrote]
To divide a cube into two other cubes, a fourth power or
in general any power whatever into two powers of the
same denomination above the second is impossible, and
I have assuredly found an admirable proof of this, but the
margin is too narrow to contain it.
And perhaps, posterity will thank me for having shown it
that the ancients did not know everything.
Quoted in D M Burton, Elementary Number Theory
(Boston 1976).
Diophantus of Alexandria
Born: about 200 BCE
Died: about 284 BCE
Diophantus, often known as the 'father of
algebra', is best known for his Arithmetica, a
work on the solution of algebraic equations and
on the theory of numbers.
However, essentially nothing is known of his
life and there has been much debate regarding
the date at which he lived.
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
Born: 17 Aug 1601 in Beaumont-de-Lomagne,
Died: 12 Jan 1665 in Castres, France
Fermat was a lawyer and government official most
remembered for his work in number theory, in
particular for Fermat's Last Theorem.
He used to pose problems for the mathematics community
to solve and the last one to be solved is the so called
Fermat’s Last Theorem
We will discuss this theorem later in the course.
Pierre Fermat's father was a wealthy leather
merchant and second consul of Beaumont- deLomagne.
Pierre had a brother and two sisters and was almost
certainly brought up in the town of his birth.
Although there is little evidence concerning his
school education it must have been at the local
Franciscan monastery.
He attended the University of Toulouse before
moving to Bordeaux in the second half of the 1620s.
In Bordeaux he began his first serious
mathematical researches and in 1629 he
gave a copy of his restoration of
Apollonius’s Plane loci to one of the
mathematicians there.
Apollonius of Perga
about 262 BC about 190 BC
Apollonius of Perga
Apollonius was a Greek mathematician known as
'The Great Geometer'.
His works had a very great influence on the
development of mathematics and his famous book
Conics introduced the terms
Certainly in Bordeaux he was in contact
with Beaugrand and during this time he
produced important work on maxima and
minima which he gave to Étienne
d'Espagnet who clearly shared
mathematical interests with Fermat.
Elementary calculus
Jean Beaugrand
about 1590 - 1640
Jean Beaugrand was, it is believed, the son of
Jean Beaugrand who was an author of the
works La paecilographie (1602) and Escritures
(1604) and the calligraphy teacher to Louis XIII
who was king of France from 1610 to 1643.
Very little is known about the life of Jean
Beaugrand, the subject of this biography, and
what we do know has been pieced together
from references to him in the correspondence
of Descartes, Fermat and Mersenne.
Aristotle (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
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)
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.
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
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.
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
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.
It turns out that Descartes, who was
eight years younger than Mersenne, was
enrolled at the same school although
they are not thought to have become
friends until much later.
Mersenne'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.
After reaching Paris he studied at the Collège
Royale du France, continuing there his
education in philosophy and also attending
classes in theology at the Sorbonne where
he also obtained the degree of Magister
Atrium in Philosophy.
He finished his studies in 1611 and, having
had a privileged education, realised that he
was now ready for the calm and studious life
of a monastery.
Jean Beaugrand
It is said that he was a pupil of Viete but
since Viete died in 1603 this must have
been at a very early stage in
Beaugrand's education.
Born: 1540 in Fontenay-le-Comte, Poitou (now
Vendée), France
Died: 13 Dec 1603 in Paris, France
François Viète's father was Étienne Viète, a lawyer
in Fontenay-le-Comte in western France about 50
km east of the coastal town of La Rochelle. François'
mother was Marguerite Dupont.
He attended school in Fontenay-le-Comte and then
moved to Poitiers, about 80 km east of Fontenay-leComte, where he was educated at the University of
Given the occupation of his father, it is
not surprising that Viète studied law at
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
In 1564 Viète took a position in the service of
Antoinette d'Aubeterre.
He was employed to supervise the education
of Antoinette's daughter Catherine, who
would later become Catherine of Parthenay
(Parthenay is about half-way between
Fontenay-le-Comte and Poitiers).
Catherine's father died in 1566 and
Antoinette d'Aubeterre moved with her
daughter to La Rochelle. Viète moved to La
Rochelle with his employer and her daughter.
Viète 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.
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.
Born: 24 Sept 1501 in Pavia, Duchy of
Milan (now Italy)
Died: 21 Sept 1576 in Rome (now
Girolamo or Hieronimo Cardano's
name was Hieronymus Cardanus in
Latin and he is sometimes known by the
English version of his name Jerome
Girolamo Cardano
1501 - 1576
Girolamo Cardan or Cardano was an
Italian doctor and mathematician who is
famed for his work Ars Magna which was
the first Latin treatise devoted solely to
In it he gave the methods of solution of
the cubic and quartic equations which he
had learnt from Tartaglia.
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
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.
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.
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:-
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.
Of course Fermat was preoccupied with
He kept his mathematical friendship with
Beugrand after he moved to Toulouse but
there he gained a new mathematical friend in
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.
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.
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
In addition Fermat wrote:-
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
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.
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.
Roberval and Mersenne found that Fermat's
problems in this first, and subsequent, letters
were extremely difficult and usually not soluble
using current techniques.
They asked him to divulge his methods and
Fermat sent Method for determining Maxima
and Minima and Tangents to Curved Lines, his
restored text of Apollonius’s Plane loci and his
algebraic approach to geometry Introduction to
Plane and Solid Loci to the Paris
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.
He wrote angrily to Fermat but although
Fermat gave more details in his reply,
Frenicle de Bessy felt that Fermat was
almost teasing him.
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
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
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.
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
Girard Desargues, 1591 - 1661
Girard Desargues was a French
mathematician who was a founder of
projective geometry.
His work centred on the theory of
conic sections and perspective.
Example from projective
Projective Geometry
Projective geometry is a non-metrical form of
Projective geometry grew out of the principles
of perspective art established during the
Renaissance period, and was first
systematically developed by Desargues in the
17th century, although it did not achieve
prominence as a field of mathematics until the
early 19th century through the work of
Poncelet and others.
Jean Victor Poncelet, 1788 1867
Poncelet was one of the
founders of modern projective
His development of the pole and
polar lines associated with
conics led to the principle of
... 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.
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
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.
Finally there was the plague of 1651
which must have had great
consequences both on life in Toulouse
and of course its near fatal
consequences on Fermat himself.
However it was during this time that
Fermat worked on number theory.
Fermat is best remembered for this work
in number theory, in particular for
Fermat’s last Theorem.
This theorem states that
Fermat’s last theorem
x y z
has no non-zero integer solutions for x, y
and z when n > 2.
Fermat wrote, in the margin of Bachet’s
translation of Diophantus’s Arithmetica
I have discovered a truly remarkable
proof which this margin is too small to
These marginal notes only became known after
Fermat's son Samuel published an edition of
Bachet’s translation of Diophantus’s Arithmetica with
his father's notes in 1670.
It is now believed that Fermat's proof was wrong
although it is impossible to be completely certain.
The truth of Fermat's assertion was proved in June
1993 by the British mathematician Andrew Wiles, but
Wiles withdrew the claim to have a proof when
problems emerged later in 1993.
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.
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:-
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.
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.
His problems did not prompt too much interest as most
mathematicians seemed to think that number theory was
not an important topic.
The second of the two problems, namely to find all
solutions of Nx2 + 1 = y2 for N not a square, was however
solved by Wallis and Brouncker and they developed
continued fractions in their solution. Brouncker produced
rational solutions which led to arguments.
De Bessy was perhaps the only mathematician at that
time who was really interested in number theory but he
did not have sufficient mathematical talents to allow him
to make a significant contribution.
Fermat 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
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.
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.
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
In 1656 Fermat had started a correspondence with
This grew out of Huygens interest in probability and
the correspondence was soon manipulated by
Fermat onto topics of number theory.
This topic did not interest Huygens but Fermat tried
hard and in New Account of Discoveries in the
Science of Numbers sent to Huygens via Carcavi in
1659, he revealed more of his methods than he had
done to others.
Fermat 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
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
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
It was not until Euler took up these problems
that the missing steps were filled in.
Fermat is described as:
Secretive and taciturn, he did not like to
talk about himself and was loath to
reveal too much about his thinking. ...
His thought, however original or novel,
operated within a range of possibilities
limited by that [1600 - 1650] time and
that [France] place.
Leonhard Euler, 1707 - 1783
Leonhard Euler was a Swiss
mathematician who made
enormous contributions to a wide
range of mathematics and physics
including analytic geometry,
trigonometry, geometry, calculus
and number theory
Carl B Boyer, writes:Recognition of the significance of Fermat's work in
analysis was tardy, in part because he adhered to the
system of mathematical symbols devised by Viete,
notations that Descartes’ "Géométrie" had rendered
largely obsolete.
The handicap imposed by the awkward notations
operated less severely in Fermat's favorite field of study,
the theory of numbers, but here, unfortunately, he found
no correspondent to share his enthusiasm.
Fermat's last theorem
Fermat died in 1665.
Today we think of Fermat as a number theorist, in
fact as perhaps the most famous number theorist
who ever lived.
It is therefore surprising to find that Fermat was in
fact a lawyer and only an amateur mathematician.
Also surprising is the fact that he published only one
mathematical paper in his life, and that was an
anonymous article written as an appendix to a
colleague's book.
Fermat in Toulouse
The is a picture of a statue of Fermat
and his muse in his home town of
Because Fermat refused to publish his work, his friends
feared that it would soon be forgotten unless something
was done about it.
His son, Samuel undertook the task of collecting Fermat’s
letters and other mathematical papers, comments written
in books, etc. with the object of publishing his father's
mathematical ideas.
In this way the famous 'Last theorem' came to be
It was found by Samuel written as a marginal note in his
father's copy of Diophantus’s Arithmetica.
Fermat almost certainly wrote the marginal note
around 1630, when he first studied the Arithmetica.
It may well be that Fermat realised that his
remarkable proof was wrong, however, since all his
other theorems were stated and restated in
challenge problems that Fermat sent to other
Although the special cases of n = 3 and n = 4 were
issued as challenges (and Fermat did know how to
prove these) the general theorem was never
mentioned again by Fermat.
In fact in all the mathematical work left by
Fermat there is only one proof. Fermat proves
that the area of a right triangle cannot be a
square. Clearly this means that a rational
triangle cannot be a rational square. In
symbols, there do not exist integers x, y, z with
x2 + y2 = z2 such that xy/2 is a square.
From this it is easy to deduce the n = 4 case of
Fermat's theorem.
It is worth noting that at this stage it
remained to prove Fermat's Last
Theorem for odd primes n only.
For if there were integers x, y, z with
xn + yn = zn then if n = pq,
(xq)p + (yq)p = (zq)p.
Euler wrote to Goldbach on 4 August 1753 claiming
he had a proof of Fermat's Theorem when n = 3.
However his proof in Algebra (1770) contains a
fallacy and it is far from easy to give an alternative
proof of the statement which has the fallacious proof.
There is an indirect way of mending the whole proof
using arguments which appear in other proofs of
Euler so perhaps it is not too unreasonable to
attribute the n = 3 case to Euler.
Euler’s mistake is an interesting one, one
which was to have a bearing on later
developments. He needed to find cubes of
the form
p2 + 3q2
and Euler shows that, for any a, b if we put
p = a3 - 9ab2, q = 3(a2b - b3) then
p2 + 3q2 = (a2 + 3b2)3.
This is true but he then tries to show
that, if p2 + 3q2 is a cube then an a and b
exist such that p and q are as above.
His method is imaginative, calculating
with numbers of the form a + b√-3.
However numbers of this form do not
behave in the same way as the integers,
which Euler did not seem to appreciate.
The next major step forward was due to
Sophie Germain.
A special case says that if n and 2n + 1 are
primes then xn + yn = zn implies that one of x,
y, z is divisible by n. Hence Fermat's Last
Theorem splits into two cases.
Case 1: None of x, y, z is divisible by n.
Case 2: One and only one of x, y, z is divisible
by n.
Marie-Sophie Germain, 1776 1831
Germain made a major contribution to
number theory, acoustics and elasticity.
Sophie Germain proved Case 1 of Fermat's Last
Theorem for all n less than 100 and Legendre
extended her methods to all numbers less than 197.
At this stage Case 2 had not been proved for even n
= 5 so it became clear that Case 2 was the one on
which to concentrate. Now Case 2 for n = 5 itself
splits into two. One of x, y, z is even and one is
divisible by 5. Case 2(i) is when the number divisible
by 5 is even; Case 2(ii) is when the even number
and the one divisible by 5 are distinct.
Adrien-Marie Legendre
1752 - 1833
Legendre's major work on elliptic
integrals provided basic analytical tools
for mathematical physics. He gave a
simple proof that π is irrational as well
as the first proof that π2 is irrational.
Case 2(i) was proved by Dirichlet and
presented to the Paris Academie des
Sciences in July 1825.
Legendre was able to prove Case 2(ii) and
the complete proof for n = 5 was published in
September 1825.
In fact Dirichlet was able to complete his own
proof of the n = 5 case with an argument for
Case 2(ii) which was an extension of his own
argument for Case 2(i).
Johann Peter Gustav Lejeune
Dirichlet, 1805 - 1859
Dirichlet proved in 1837 that in any
arithmetic progression with first term
coprime to the difference (i.e. no
factors in common) there are
infinitely many primes.
In 1832 Dirichlet published a proof of Fermat's
Last Theorem for n = 14.
Of course he had been attempting to prove the
n = 7 case but had proved a weaker result.
The n = 7 case was finally solved by Lame in
1839. It showed why Dirichlet had so much
difficulty, for although Dirichlet’s n = 14 proof
used similar (but computationally much harder)
arguments to the earlier cases, Lame had to
introduce some completely new methods.
Lame’s proof is exceedingly hard and makes it look as
though progress with Fermat's Last Theorem to larger n
would be almost impossible without some radically new
The year 1847 is of major significance in the study of Fermat's
Last Theorem. On 1 March of that year Lame announced to the
Paris Academie that he had proved Fermat's Last Theorem.
He sketched a proof which involved factorizing xn + yn = zn into
linear factors over the complex numbers.
Lame acknowledged that the idea was suggested to him by
Joseph Liouville
1809 - 1882
Liouville is best known for his work on
the existence of a transcendental number
in 1844 when he constructed an infinite
class of such numbers.
However Liouville addressed the meeting after Lame
and suggested that the problem of this approach
was that uniqueness of factorisation into primes was
needed for these complex numbers and he doubted
if it were true.
Cauchy supported Lame but, in rather typical
fashion, pointed out that he had reported to the
October 1847 meeting of the Academie an idea
which he believed might prove Fermat's Last
Augustin Louis Cauchy
1789 - 1857
Cauchy pioneered the study of analysis,
both real and complex, and the theory
of permutation groups.
He also researched in convergence and
divergence of infinite series,
differential equations, determinants,
probability and mathematical physics.
Much work was done in the following
weeks in attempting to prove the
uniqueness of factorization.
Wantzel claimed to have proved it on
15 March but his argument
It is true for n = 2, n = 3 and n = 4 and
one easily sees that the same
argument applies for n > 4
Wantzel is correct about n = 2 (ordinary
integers), n = 3 (the argument Euler got wrong)
and n = 4 (which was proved by Gauss).
On 24 May Liouville read a letter to the
Academie which settled the arguments.
The letter was from Kummer, enclosing an offprint of a 1844 paper which proved that
uniqueness of factorization failed but could be
'recovered' by the introduction of ideal complex
numbers which he had done in 1846.
Ernst Eduard Kummer
1810 - 1893
Kummer's main achievement was the extension
of results about the integers to other integral
domains by introducing the concept of an ideal
Related to ring theory
Kummer had used his new theory to find
conditions under which a prime is regular
and had proved Fermat's Last Theorem
for regular primes.
Kummer also said in his letter that he
believed 37 failed his conditions
By September 1847 Kummer sent to
Dirichlet and the Berlin Academy a paper
proving that a prime p is regular (and so
Fermat's Last Theorem is true for that
prime) if p does not divide the
numerators of any of the Bernoulli
numbers B2 , B4 , ..., Bp-3 . The Bernoulli
number Bi is defined by
x/(ex - 1) =  Bi xi /i!
Kummer shows that all primes up to 37
are regular but 37 is not regular as 37
divides the numerator of B32 .
The only primes less than 100 which are not
regular are 37, 59 and 67. More powerful
techniques were used to prove Fermat's Last
Theorem for these numbers. This work was
done and continued to larger numbers by
Kummer, Mirimanoff, Wieferich, Furtwängler,
Vandiver and others.
Although it was expected that the number of
regular primes would be infinite even this defied
proof. In 1915 Jensen proved that the number of
irregular primes is infinite.
Despite large prizes being offered for a solution, Fermat's
Last Theorem remained unsolved.
It has the dubious distinction of being the theorem with the
largest number of published false proofs.
For example over 1000 false proofs were published between
1908 and 1912.
The only positive progress seemed to be computing results
which merely showed that any counter-example would be
very large. Using techniques based on Kummer's work,
Fermat's Last Theorem was proved true, with the help of
computers, for n up to 4,000,000 by 1993.
In 1983 a major contribution was made by gerd Faltings who
proved that for every n > 2 there are at most a finite number
of coprime integers x, y, z with xn + yn = zn.
This was a major step but a proof that the finite number was
0 in all cases did not seem likely to follow by extending
Faltings ' arguments.
The final chapter in the story began in 1955, although at this
stage the work was not thought of as connected with
Fermat's Last Theorem.
Yutaka Taniyama asked some questions about elliptic
curves, i.e. curves of the form y2 = x3 + ax + b for constants
a and b. Further work by Weil and Shimura produced a
conjecture, now known as the Shimura-Taniyama-Weil
In 1986 the connection was made
between the Shimura-Taniyama-Weil
Conjecture and Fermat's Last Theorem by
Frey at Saarbrücken showing that
Fermat's Last Theorem was far from being
some unimportant curiosity in number
theory but was in fact related to
fundamental properties of space.
Further work by other mathematicians showed that a counterexample to Fermat's Last Theorem would provide a counter example to the Shimura-Taniyama-Weil Conjecture.
The proof of Fermat's Last Theorem was completed in 1993 by
Andrew Wiles, a British mathematician working at Princeton in
the USA.
Wiles gave a series of three lectures at the Isaac Newton
Institute in Cambridge, England the first on Monday 21 June,
the second on Tuesday 22 June. In the final lecture on
Wednesday 23 June 1993 at around 10.30 in the morning Wiles
announced his proof of Fermat’s Last Theorem as a corollary to
his main results.
Having written the theorem on the
blackboard he said I think I will stop here and
sat down.
In fact Wiles had proved the ShimuraTaniyama-Weil Conjecture for a class of
examples, including those necessary to
prove Fermat’s Last Theorem.
This, however, is not the end of the story. On 4 December 1993
Andrew Wiles made a statement in view of the speculation. He
said that during the reviewing process a number of problems
had emerged, most of which had been resolved. However one
problem remains and Wiles essentially withdrew his claim to
have a proof. He states
The key reduction of (most cases of) the Taniyama-Shimura
conjecture to the calculation of the Selmer group is correct.
However the final calculation of a precise upper bound for the
Selmer group in the semisquare case (of the symmetric square
representation associated to a modular form) is not yet
complete as it stands. I believe that I will be able to finish this in
the near future using the ideas explained in my Cambridge
In March 1994 Faltings, writing in Scientific American,
If it were easy, he would have solved it by now. Strictly
speaking, it was not a proof when it was announced.
Weil also in Scientific American, wrote
I believe he has had some good ideas in trying to
construct the proof but the proof is not there. To some
extent, proving Fermat's Theorem is like climbing
Everest. If a man wants to climb Everest and falls short of
it by 100 yards, he has not climbed Everest.
In fact, from the beginning of 1994, Wiles began to collaborate
with Richard Taylor in an attempt to fill the holes in the proof.
However they decided that one of the key steps in the proof,
using methods due to Flach, could not be made to work. They
tried a new approach with a similar lack of success. In August
1994 Wiles addressed the International Congress of
Mathematicians but was no nearer to solving the difficulties.
Taylor suggested a last attempt to extend Flach's method in the
way necessary and Wiles, although convinced it would not
work, agreed mainly to enable him to convince Taylor that it
could never work. Wiles worked on it for about two weeks, then
suddenly inspiration struck.
In a flash I saw that the thing that stopped it [the extension of
Flach's method] working was something that would make
another method I had tried previously work.
On 6 October Wiles sent the new proof to three colleagues
including Faltings. All liked the new proof which was
essentially simpler than the earlier one. Faltings sent a
simplification of part of the proof.
No proof of the complexity of this can easily be guaranteed
to be correct, so a very small doubt will remain for some
time. However when Taylor lectured at the British
Mathematical Colloquium in Edinburgh in April 1995 he gave
the impression that no real doubts remained over Fermat's
Last Theorem
Blaise Pascal
Blaise Pascal was the third of E.Pascal's children and
his only son. Blaise's mother died when he was only three
years old. In 1632 the Pascal family, Étienne and his four
children, left Clermont and settled in Paris. Blaise
Pascal's father had unorthodox educational views and
decided to teach his son himself.
E.Pascal decided that Blaise was not to study
mathematics before the age of 15 and all mathematics
texts were removed from their house.
Blaise however, his curiosity raised by this, started to
work on geometry himself at the age of 12. He discovered
that the sum of the angles of a triangle are two right
angles and, when his father found out, he relented and
allowed Blaise a copy of Euclid.
In December 1639 the Pascal family left Paris to live in
Rouen where Étienne had been appointed as a tax
collector for Upper Normandy. Shortly after settling in
Rouen, Blaise had his first work, Essay on Conic sections
published in February 1640.
Pascal invented the first digital calculator to help his
father with his work collecting taxes. He worked on it for
three years between 1642 and 1645. The device, called
the Pascaline, resembled a mechanical calculator of the
1940s. This, almost certainly, makes Pascal the second
person to invent a mechanical calculator.
There were problems faced by Pascal in the design of the
calculator which were due to the design of the French
currency at that time. There were 20 sols in a livre and 12
deniers in a sol. The system remained in France until
1799 but in Britain a system with similar multiples lasted
until 1971. Pascal had to solve much harder technical
problems to work with this division of the livre into 240
than he would have had if the division had been 100.
However production of the machines started in 1642 but,
as Adamson writes,
By 1652 fifty prototypes had been produced, but few
machines were sold, and manufacture of Pascal's
arithmetical calculator ceased in that year.
Events of 1646 were very significant for the young Pascal. In
that year his father injured his leg and had to recuperate in his
house. He was looked after by two young brothers from a
religious movement just outside Rouen. They had a profound
effect on the young Pascal and he became deeply religious.
From about this time Pascal began a series of experiments on
atmospheric pressure. By 1647 he had proved to his
satisfaction that a vacuum existed.
Descartes visited Pascal on 23 September. His visit only lasted
two days and the two argued about the vacuum which
Descartes did not believe in. Descartes wrote, rather cruelly, in
a letter to Huygens after this visit that Pascal
...has too much vacuum in his head.
Through the period of this correspondence Pascal was unwell.
In one of the letters to Fermat written in July 1654 he writes
... though I am still bedridden, I must tell you that yesterday
evening I was given your letter.
However, despite his health problems, he worked intensely on
scientific and mathematical questions until October 1654.
Sometime around then he nearly lost his life in an accident. The
horses pulling his carriage bolted and the carriage was left
hanging over a bridge above the river Seine. Although he was
rescued without any physical injury, it does appear that he was
much affected psychologically. Not long after he underwent
another religious experience, on 23 November 1654, and he
pledged his life to Christianity.
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