Rene Descartes, Pierre Fermat and Blaise Pascal

- Descartes, Fermat and Pascal a philosopher, an

amateur and a calculator

Rene Descartes 1596 - 1650

Pierre de Fermat 17th August 1601 or 1607 12th

January 1665

Blaise Pascal, 1623 - 1662

Descartes

- René Descartes was a French philosopher whose

work, La géométrie, includes his application of

algebra to geometry from which we now have

Cartesian geometry. - His work had a great influence on both

mathematicians and philosophers.

Descartes

- Descartes was educated at the Jesuit college of

La Flèche in Anjou. - The Society of Jesus (Latin Societas Iesu, S.J.

and S.I. or SJ, SI ) is a Catholic religious

order of whose members are called Jesuits, - He entered the college at the age of eight years,

just a few months after the opening of the

college in January 1604. - He studied there until 1612, studying classics,

logic and traditional Aristotelian philosophy. - He also learnt mathematics from the books of

Clavius.

Descartes

- While in the school his health was poor and he

was granted permission to remain in bed until 11

o'clock in the morning, a custom he maintained

until the year of his death. - In bed he came up with idea now called Cartesian

geometry.

Clavius 1538 - 1612

- Christopher Clavius was a German Jesuit

astronomer who helped Pope Gregory XIII to

introduce what is now called the Gregorian

calendar.

Descartes

- School had made Descartes understand how little

he knew, the only subject which was satisfactory

in his eyes was mathematics. - This idea became the foundation for his way of

thinking, and was to form the basis for all his

works. - The above statement was echoed by Einstein in the

20th century.

Descartes

- Descartes spent a while in Paris, apparently

keeping very much to himself, then he studied at

the University of Poitiers. - He received a law degree from Poitiers in 1616

then enlisted in the military school at Breda. - In 1618 he started studying mathematics and

mechanics under the Dutch scientist Isaac

Beeckman, and began to seek a unified science of

nature. Wrote on the theory of vortices

Desartes

- After two years in Holland he travelled through

Europe. - In 1619 he joined the Bavarian army.
- After two years in Holland he travelled through

Europe.

Descartes

- From 1620 to 1628 Descartes travelled through

Europe, spending time in Bohemia (1620), Hungary

(1621), Germany, Holland and France (1622-23). - He 1623 he spent time in Paris where he made

contact with Mersenne, an important contact which

kept him in touch with the scientific world for

many years.

Descartes

- From Paris he travelled to Italy where he spent

some time in Venice, then he returned to France

again (1625).

Mersenne

- Marin Mersenne was a French monk who is best

known for his role as a clearing house for

correspondence between eminent philosophers and

scientists and for his work in number theory. - Similar to Bourbaki
- Nicholas Bourbaki is the collective pseudonym

under which a group of (mainly French)

20th-century mathematicians wrote a series of

books presenting an exposition of modern advanced

mathematics,

Mersenne prime

- In mathematics, a Mersenne number is a positive

integer that is one less than a power of two - Some definitions of Mersenne numbers require that

the exponent n be prime.

Mersenne prime

- A Mersenne prime is a Mersenne number that is

prime. - As of September 2008, only 46 Mersenne primes are

known the largest known prime number is

(243,112,609 - 1) is a Mersenne prime, and in

modern times, the largest known prime has almost

always been a Mersenne prime.

Descartes

- By 1628 Descartes was tired of the continual

travelling and decided to settle down. - He gave much thought to choosing a country suited

to his nature and chose Holland. - It was a good decision which he did not seem to

regret over the next twenty years.

Descartes

- Soon after he settled in Holland Descartes began

work on his first major treatise on physics, Le

Monde, ou Traité de la Lumière. - This work was near completion when news that

Galileo was condemned to house arrest reached

him. - He, perhaps wisely, decided not to risk

publication and the work was published, only in

part, after his death. - He explained later his change of direction

saying-

Descartes

- ... in order to express my judgment more freely,

without being called upon to assent to, or to

refute the opinions of the learned, I resolved to

leave all this world to them and to speak solely

of what would happen in a new world, if God were

now to create ... and allow her to act in

accordance with the laws He had established.

Galileo, portrait by Justus Sustermans painted

in 1636

Galileo Galilei

- Galileo Galilei was an Italian scientist who

formulated the basic law of falling bodies, which

he verified by careful measurements. - He constructed a telescope with which he studied

lunar craters, and discovered four moons

revolving around Jupiter and espoused the

Copernican cause.

Nicolaus Copernicus1473 - 1543

Copernicus

- Copernicus was a Polish astronomer and

mathematician who was a proponent of the view of

an Earth in daily motion about its axis and in

yearly motion around a stationary sun. - Helio-Centric universe
- This theory profoundly altered later workers'

view of the universe, but was rejected by the

Catholic church.

Galileo Galilei

- Galileo Galilei considered the father of

experimental physics along with Ernest

Rutherford. - Galileo Galilei's parents were Vincenzo Galilei

and Guilia Ammannati. - Vincenzo, who was born in Florence in 1520, was a

teacher of music and a fine lute player. - After studying music in Venice he carried out

experiments on strings to support his musical

theories.

Galileo Galilei

- Guilia, who was born in Pescia, married Vincenzo

in 1563 and they made their home in the

countryside near Pisa. - Galileo was their first child and spent his early

years with his family in Pisa.

Galilean transformation

- The Galilean transformation is used to transform

between the coordinates of two reference frames

which differ only by constant relative motion

within the constructs of Newtonian Physics. - The equations below, although apparently obvious,

break down at speeds that approach the speed of

light due to physics described by Einsteins

theory of relativity. - Galileo formulated these concepts in his

description of uniform motion .

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.

Translation (one dimension)

- In essence, the Galilean transformations embody

the intuitive notion of addition and subtraction

of velocities. - The assumption that time can be treated as

absolute is at heart of the Galilean

transformations. - Relativity insists that the speed of light is

constant and thus time is different for different

observers. - This assumption is abandoned in the Lorentz

transformations

Hendrik Antoon Lorentz1853 - 1928

Lorentz

- Lorentz is best known for his work on

electromagnetic radiation and the

FitzGerald-Lorentz contraction. - He developed the mathematical theory of the

electron.

Galileo

- These relativistic transformations are deemed

applicable to all velocities, whilst the Galilean

transformation can be regarded as a 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

Translation (one dimension)

Lorenz transformations

Lorenz transformations

The spacetime coordinates of an event, as

measured by each observer in their inertial

reference frame (in standard configuration) are

shown in the speech bubbles.Top frame F' moves

at velocity v along the x-axis of frame

F.Bottom frame F moves at velocity -v along the

x'-axis of frame F

Translation (one dimension)

- Note that the last equation (Galileo) expresses

the assumption of a universal time independent of

the relative motion of different observers.

Galileo

- In 1572, when Galileo was eight years old, his

family returned to Florence, his father's home

town. - However, Galileo remained in Pisa and lived for

two years with Muzio Tedaldi who was related to

Galileo's mother by marriage. - When he reached the age of ten, Galileo left Pisa

to join his family in Florence and there he was

tutored by Jacopo Borghini.

Galileo

- Once he was old enough to be educated in a

monastery, his parents sent him to the

Camaldolese Monastery at Vallombrosa which is

situated on a magnificent forested hillside 33 km

southeast of Florence.

Galileo

- The Order combined the solitary life of the

hermit with the strict life of the monk and soon

the young Galileo found this life an attractive

one. - He became a novice, intending to join the Order,

but this did not please his father who had

already decided that his eldest son should become

a medical doctor.

Galileo

- Vincenzo had Galileo returned from Vallombrosa to

Florence and gave up the idea of joining the

Camaldolese order. - He did continue his schooling in Florence,

however, in a school run by the Camaldolese

monks. - In 1581 Vincenzo sent Galileo back to Pisa to

live again with Muzio Tedaldi and now to enrol

for a medical degree at the University of Pisa.

Galileo

- Although the idea of a medical career never seems

to have appealed to Galileo, his father's wish

was a fairly natural one since there had been a

distinguished physician in his family in the

previous century.

Galileo

- Galileo never seems to have taken medical studies

seriously, attending courses on his real

interests which were in mathematics and natural

philosophy (physics). - His mathematics teacher at Pisa was Filippo

Fantoni, who held the chair of mathematics. - Galileo returned to Florence for the summer

vacations and there continued to study

mathematics.

Galileo

- In the year 1582-83 Ostilio Ricci, who was the

mathematician of the Tuscan Court and a former

pupil of Tartaglia, taught a course on 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.

Nicolo Fontana Tartaglia

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.

François Viète, 1540 - 1603

François Viète

- François Viète was a French amateur mathematician

and astronomer who introduced the first

systematic algebraic notation in his book - In artem analyticam isagoge .
- He was also involved in deciphering codes.
- Alan Turin

Ricci and 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.

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 3-manifold is

homeomorphic to the 3-sphere.

Topology

- This is a statement about topological spaces.

Lets 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 to be

explained further on.

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.

manifold

- At every small neighbourhood on the space, it

approximates Euclidean space. - A standard sphere is called a 2-sphere because it

is actually a 2-manifold. Its surface resembles

the 2d plane if you zoom into it so that the

curvature approaches 0. - Continuing this logic the 1-sphere is a circle.

A 3-sphere is very difficult to visualize because

it has a 3d surface and exists in 4d space.

homeomorphic

- If one space is homeomorphic to another, it

means you can continuously deform the one space

into the other. - The 2-sphere and a football are homeomorphic.
- The 2-sphere and a donut are not no matter how

much you deform a sphere, you cant get that

pesky hole in the donut, and vice-versa.

Morphing

Galileo

- Certainly Vincenzo did not like the idea (of his

son studying mathematics) and resisted strongly

but eventually he gave way a little and Galileo

was able to study the works of Euclid and

Archimedes from the Italian translations which

Tartaglia had made. - Of course he was still officially enrolled as a

medical student at Pisa but eventually, by 1585,

he gave up this course and left without

completing his degree.

Galileo

- Galileo began teaching mathematics, first

privately in Florence and then during 1585-86 at

Siena where he held a public appointment. - During the summer of 1586 he taught at

Vallombrosa, and in this year he wrote his first

scientific book - The little balance La Balancitta which

described Archimedes method of finding the

specific gravities (that is the relative

densities) of substances using a balance. - Essentially looking at the use of plumblines

Galileo

- In the following year he travelled to Rome to

visit Clavius who was professor of mathematics at

the Jesuit Collegio Romano there.

Galileo

- A topic which was very popular with the Jesuit

mathematicians at this time was centres of

gravity and Galileo brought with him some results

which he had discovered on this topic. - Despite making a very favourable impression on

Clavius, Galileo failed to gain an appointment to

teach mathematics at the University of Bologna.

Galileo

- After leaving Rome Galileo remained in contact

with Clavius by correspondence and Guidobaldo del

Monte who was also a regular correspondent. - Certainly the theorems which Galileo had proved

on the centres of gravity of solids, and left in

Rome, were discussed in this correspondence. - It is also likely that Galileo received lecture

notes from courses which had been given at the

Collegio Romano, for he made copies of such

material which still survive today.

Galileo

- The correspondence began around 1588 and

continued for many years. - Also in 1588 Galileo received a prestigious

invitation to lecture on the dimensions and

location of hell in Dante's Inferno at the

Academy in Florence.

Galileo

- Fantoni left the chair of mathematics at the

University of Pisa in 1589 and Galileo was

appointed to fill the post (although this was

only a nominal position to provide financial

support for Galileo). - Not only did he receive strong recommendations

from Clavius, but he also had acquired an

excellent reputation through his lectures at the

Florence Academy in the previous year. - The young mathematician had rapidly acquired the

reputation that was necessary to gain such a

position, but there were still higher positions

at which he might aim.

Galileo

- Galileo spent three years holding this post at

the University of Pisa and during this time he

wrote - De Motu
- a series of essays on the theory of motion which

he never published. - It is likely that he never published this

material because he was less than satisfied with

it, and this is fair for despite containing some

important steps forward, it also contained some

incorrect ideas.

Galileo

- Perhaps the most important new ideas which De

Motu contains is that one can test theories by

conducting experiments. - The beginnings of the scientific method that had

escaped the Greeks due to Aristotle. - In particular the work contains his important

idea that one could test theories about falling

bodies using an inclined plane to slow down the

rate of descent.

Galileo

- In 1591 Vincenzo Galilei, Galileo's father, died

and since Galileo was the eldest son he had to

provide financial support for the rest of the

family and in particular have the necessary

financial means to provide dowries for his two

younger sisters. - Being professor of mathematics at Pisa was not

well paid, so Galileo looked for a more lucrative

post.

Galileo

- With strong recommendations from del Monte,

Galileo was appointed professor of mathematics at

the University of Padua (the university of the

Republic of Venice) in 1592 at a salary of three

times what he had received at Pisa. - On 7 December 1592 he gave his inaugural lecture

and began a period of eighteen years at the

university, years which he later described as the

happiest of his life.

Galileo

- At Padua his duties were mainly to teach 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.

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.

Johann Kepler 1571 - 1630

Johannes Kepler

- Johannes Kepler was a German mathematician and

astronomer who discovered that the Earth and

planets travel about the sun in elliptical

orbits. - He gave three fundamental laws of planetary

motion. - He also did important work in optics and

geometry.

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."

First law

Second law

Galileo

- Galileo used parallax arguments to prove that the

New Star could not be close to the Earth. - In a personal letter written to Kepler in 1598,

Galileo had stated that he was a Copernican

(believer in the theories of Copernicus). - However, no public sign of this belief was to

appear until many years later.

Galileo

- At Padua, Galileo began a long term relationship

with Maria Gamba, who was from Venice, but they

did not marry perhaps because Galileo felt his

financial situation was not good enough. - In 1600 their first child Virginia was born,

followed by a second daughter Livia in the

following year. - In 1606 their son Vincenzo was born.

Galileo

- We mentioned above an error in Galileo's theory

of motion as he set it out in De Motu around

1590. - He was quite mistaken in his belief that the

force acting on a body was the relative

difference between its specific gravity and that

of the substance through which it moved. - Ether
- Galileo wrote to his friend Paolo Sarpi, a fine

mathematician who was consultor to the Venetian

government, in 1604 and it is clear from his

letter that by this time he had realised his

mistake.

Galileo

- In fact he had returned to work on the theory of

motion in 1602 and over the following two years,

through his study of inclined planes and the

pendulum, he had formulated the correct law of

falling bodies and had worked out that a

projectile follows a parabolic path. - However, these famous results would not be

published for another 35 years.

Galileo

- In May 1609, Galileo received a letter from Paolo

Sarpi telling him about a spyglass that a

Dutchman had shown in Venice. - Galileo wrote in the Starry Messenger (Sidereus

Nuncius) in April 1610-

Galileo

- About ten months ago a report reached my ears

that a certain Fleming had constructed a spyglass

by means of which visible objects, though very

distant from the eye of the observer, were

distinctly seen as if nearby. - Of this truly remarkable effect several

experiences were related, to which some persons

believed while other denied them.

Galileo

- A few days later the report was confirmed by a

letter I received from a Frenchman in Paris,

Jacques Badovere, which caused me to apply myself

wholeheartedly to investigate means by which I

might arrive at the invention of a similar

instrument. - This I did soon afterwards, my basis being the

doctrine of refraction.

Galileo

- From these reports, and using his own technical

skills as a mathematician and as a craftsman,

Galileo began to make a series of telescopes

whose optical performance was much better than

that of the Dutch instrument. - His first telescope was made from available

lenses and gave a magnification of about four.

Galileo

- To improve on this Galileo learned how to grind

and polish his own lenses and by August 1609 he

had an instrument with a magnification of around

eight or nine. - Galileo immediately saw the commercial and

military applications of his telescope (which he

called a perspicillum) for ships at sea.

Galileo

- He kept Sarpi informed of his progress and Sarpi

arranged a demonstration for the Venetian Senate.

- They were very impressed and, in return for a

large increase in his salary, Galileo gave the

sole rights for the manufacture of telescopes to

the Venetian Senate. - It seems a particularly good move on his part

since he must have known that such rights were

meaningless, particularly since he always

acknowledged that the telescope was not his

invention

Descartes

- In Holland Descartes had a number of scientific

friends as well as continued contact with

Mersenne. - His friendship with Beeckman continued and he

also had contact with Huygens.

Christiaan Huygens1629 - 1695

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

Isaac Newton

- Born 4 Jan 1643 in Woolsthorpe, Lincolnshire,

EnglandDied 31 March 1727 in London, England

Isaac Newton

Descartes

- Descartes was pressed by his friends to publish

his ideas and, although he was adamant in not

publishing Le Monde, he wrote a treatise on

science under the title - Discours de la méthode pour bien conduire sa

raison et chercher la vérité dans les sciences. - Three appendices to this work were
- La Dioptrique,
- Les Météores,
- La Géométrie.
- The treatise was published at Leiden in 1637 and

Descartes wrote to Mersenne saying-

Descartes

- I have tried in my "Dioptrique" and my "Météores"

to show that my Méthode is better than the

vulgar, and in my "Géométrie" to have

demonstrated it.

Descartes

- The work describes what Descartes considers is a

more satisfactory means of acquiring knowledge

than that presented by Aristotles logic. - Only mathematics, Descartes feels, is certain, so

all must be based on mathematics.

Descartes

- La Dioptrique is a work on optics and, although

Descartes does not cite previous scientists for

the ideas he puts forward, in fact there is

little new. - However his approach through experiment was an

important contribution.

Descartes

- Les Météores is a work on meteorology and is

important in being the first work which attempts

to put the study of weather on a scientific

basis. - However many of Descartes' claims are not only

wrong but could have easily been seen to be wrong

if he had done some easy experiments.

Descartes

- For example Roger Bacon had demonstrated the

error in the commonly held belief that water

which has been boiled freezes more quickly. - However Descartes claims-
- ... and we see by experience that water which has

been kept on a fire for some time freezes more

quickly than otherwise, the reason being that

those of its parts which can be most easily

folded and bent are driven off during the

heating, leaving only those which are rigid.

Roger Bacon

- Roger Bacon, (c. 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.

Descartes

- Despite its many faults, the subject of

meteorology was set on course after publication

of Les Météores particularly through the work of - Boyle
- Hooke
- Halley.

Robert Boyle, 1627 - 1691

Robert Boyle

- Robert Boyle, 1627 - 1691
- Robert Boyle was an Irish-born scientist who was

a founding fellow of the Royal Society. - His work in chemistry was aimed at establishing

it as a mathematical science based on a

mechanistic theory of matter.

Robert Hooke, 1635 - 1703

Robert Hooke was an English scientist who made

contributions to many different fields including

mathematics, optics, mechanics, architecture

and astronomy. He had a famous quarrel with

Newton.

Edmond Halley, 1656 - 1742

Edmond Halley was an English astronomer who

calculated the orbit of the comet now called

Halley's comet. He was a supporter of Newton.

Descartes

- La Géométrie is by far the most important part of

Descartes work.

Descartes

- He makes the first step towards a theory of

invariants. - Algebra makes it possible to recognise the

typical problems in geometry and to bring

together problems which in geometrical dress

would not appear to be related at all. - Algebra imports into geometry the most natural

principles of division and the most natural

hierarchy of method. - Not only can questions of solvability and

geometrical possibility be decided elegantly,

quickly and fully from the parallel algebra,

without it they cannot be decided at all.

Descartes

- Descartes' Meditations on First Philosophy, was

published in 1641, designed for the philosopher

and for the theologian. - It consists of six meditations,
- Of the Things that we may doubt
- Of the Nature of the Human Mind
- Of God that He exists
- Of Truth and Error
- Of the Essence of Material Things
- Of the Existence of Material Things
- The Real Distinction between the Mind and the

Body of Man.

Descartes

- The most comprehensive of Descartes' works,
- Principia Philosophiae
- was published in Amsterdam in 1644.
- In four parts, The Principles of Human Knowledge,

The Principles of Material Things, Of the Visible

World and The Earth, it attempts to put the whole

universe on a mathematical foundation reducing

the study to one of mechanics. - Not the resemblance to the title of Newtons

great publication

Descartes

- This is an important point of view and was to

point the way forward. - Descartes did not believe in action at a

distance. - 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.

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-

Newton-Descartes

- Thus entrenched as the Cartesian system was ...

it was not to be wondered at that the pure and

sublime doctrines of the Principia were

distrustfully received ... The uninstructed mind

could not readily admit the idea that the great

masses of the planets were suspended in empty

space, and retained their orbits by an invisible

influence...

Descartes's

- Pleasing as Descartes's theory was even the

supporters of his natural philosophy, such as the

Cambridge metaphysical theologian Henry More,

found objections. - Certainly More admired Descartes, writing-

Descartes's

- I should look upon Des-Cartes as a man most truly

inspired in the knowledge of Nature, than any

that have professed themselves so these sixteen

hundred years...

Descartes

- However between 1648 and 1649 they exchanged a

number of letters in which More made some telling

objections. - Descartes however in his replies making no

concessions to Mores points.

Descartes

- In 1644, the year his Meditations were published,

Descartes visited France. - He returned again in 1647, when he met Pascal and

argued with him that a vacuum could not exist,

and then again in 1648.

Descartes

- In 1649 Queen Christina of Sweden persuaded

Descartes to go to Stockholm. - However the Queen wanted to draw tangents at 5

a.m. and Descartes broke the habit of his

lifetime of getting up at 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.

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).

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.

Fermat

- Whenever two unknown magnitudes appear in a final

equation, we have a locus, the extremity of one

of the unknown magnitudes describing a straight

line or a curve.Introduction to Plane and Solid

Loci

Fermat

- Born 17 Aug 1601 in Beaumont-de-Lomagne, France
- Died 12 Jan 1665 in Castres, France
- Fermat was a lawyer and government official most

remembered for his work in number theory, in

particular for Fermat's Last Theorem. - He used to pose problems for the mathematics

community to solve and the last one to be solved

is the so called Fermats Last Theorem - We will discuss this theorem later in the course.

Fermat

- Pierre Fermat's father was a wealthy leather

merchant and second consul of Beaumont- 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.

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.

Apollonius of Perga

- about 262 BC - about 190 BC

Apollonius of Perga

- Apollonius was a Greek mathematician known as

'The Great Geometer'. - His works had a very great influence on the

development of mathematics and his famous book

Conics introduced the terms - parabola
- ellipse
- hyperbola.

Fermat

- Certainly in Bordeaux he was in contact with

Beaugrand and during this time he produced

important work on maxima and minima which he gave

to Étienne d'Espagnet who clearly shared

mathematical interests with Fermat. - Elementary calculus

Jean Beaugrand

- about 1590 - 1640
- Jean Beaugrand was, it is believed, the son of

Jean Beaugrand who was an author of the works La

paecilographie (1602) and Escritures (1604) and

the calligraphy teacher to Louis XIII who was

king of France from 1610 to 1643. - Very little is known about the life of Jean

Beaugrand, the subject of this biography, and

what we do know has been pieced together from

references to him in the correspondence of

Descartes, Fermat and Mersenne.

Aristotle

Aristotle (384-322BCE)

- Born at Stagira in Northern Greece.
- Aristotle was the most notable product of the

educational program devised by Plato he spent

twenty years of his life studying at the Academy.

- When Plato died, Aristotle returned to his native

Macedonia, where he is supposed to have

participated in the education of Philip's son,

Alexander (the Great)

Aristotle

- He came back to Athens with Alexander's approval

in 335 and established his own school at the

Lyceum, spending most of the rest of his life

engaged there in research, teaching, and writing.

- His students acquired the name "peripatetics"

from the master's habit of strolling about as he

taught.

Aristotle

- Although the surviving works of Aristotle

probably represent only a fragment of the whole,

they include his investigations of an amazing

range of subjects, from - logic
- philosophy
- ethics,
- physics
- biology
- psychology
- politics
- rhetoric.

Aristotle

- Aristotle appears to have thought through his

views as he wrote, returning to significant

issues at different stages of his own

development. - The result is less a consistent system of thought

than a complex record of Aristotle's thinking

about many significant issues.

Mersenne

- Marin Mersenne was born into a working class

family in the small town of Oizé in the province

of Maine on 8 September 1588 and was baptised on

the same day. - From an early age he showed signs of devotion and

eagerness to study. - So, despite their financial situation, Marin's

parents sent him to the Collège du Mans where he

took grammar classes. - Later, at the age of sixteen, Mersenne asked to

go to the newly established Jesuit School in La

Flèche which had been set up as a model school

for the benefit of all children regardless of

their parents' financial situation.

Mersenne

- It turns out that Descartes, who was eight years

younger than Mersenne, was enrolled at the same

school although they are not thought to have

become friends until much later.

Mersenne

- Mersenne's father wanted his son to have a career

in the Church. - Mersenne, however, was devoted to study, which he

loved, and, showing that he was ready for

responsibilities of the world, had decided to

further his education in Paris. - He left for Paris staying en route at a convent

of the Minims. - This experience so inspired Mersenne that he

agreed to join their Order if one day he decided

to lead a monastic life.

Mersenne

- After reaching Paris he studied at the Collège

Royale du France, continuing there his education

in philosophy and also attending classes in

theology at the Sorbonne where he also obtained

the degree of Magister Atrium in Philosophy. - He finished his studies in 1611 and, having had a

privileged education, realised that he was now

ready for the calm and studious life of a

monastery.

Jean Beaugrand

- It is said that he was a pupil of Viete but since

Viete died in 1603 this must have been at a very

early stage in Beaugrand's education.

Viète

Viète

- Born 1540 in Fontenay-le-Comte, Poitou (now

Vendée), 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.

Viète

- Given the occupation of his father, it is not

surprising that Viète studied law at university. - After graduating with a law degree in 1560, Viète

entered the legal profession but he only

continued on this path for four years before

deciding to change his career.

Viète

- In 1564 Viète took a position in the service of

Antoinette d'Aubeterre. - He was employed to supervise the education of

Antoinette's daughter Catherine, who would later

become Catherine of Parthenay (Parthenay is about

half-way between Fontenay-le-Comte and Poitiers).

- Catherine's father died in 1566 and Antoinette

d'Aubeterre moved with her daughter to La

Rochelle. Viète moved to La Rochelle with his

employer and her daughter.

Viète

- Viète introduced the first systematic algebraic

notation in his book In artem analyticam isagoge

published at Tours in 1591. The title of the work

may seem puzzling, for it means "Introduction to

the analytic art" which hardly makes it sound

like an algebra book. - However, Viète did not find Arabic mathematics to

his liking and based his work on the Italian

mathematicians such as Cardan, and the work of

ancient Greek mathematicians.

Viète

- One would have to say, however, that had Viète

had a better understanding of Arabic mathematics

he might have discovered that many of the ideas

he produced were already known to earlier Arabic

mathematicians.

Cardan

- Born 24 Sept 1501 in Pavia, Duchy of Milan (now

Italy)Died 21 Sept 1576 in Rome (now Italy) - Girolamo or Hieronimo Cardano's name was

Hieronymus Cardanus in Latin and he is sometimes

known by the English version of his name Jerome

Cardan.

Girolamo 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.

Fermat

- From Bordeaux Fermat went to Orléans where he

studied law at the University. - He received a degree in civil law and he

purchased the offices of councillor at the

parliament in Toulouse. - So by 1631 Fermat was a lawyer and government

official in Toulouse and because of the office he

now held he became entitled to change his name

from Pierre Fermat to Pierre de Fermat. - For the remainder of his life he lived in

Toulouse but as well as working there he also

worked in his home town of Beaumont-de-Lomagne

and a nearby town of Castres.

Fermat

- From his appointment on 14 May 1631 Fermat worked

in the lower chamber of the parliament but on 16

January 1638 he was appointed to a higher

chamber, then in 1652 he was promoted to the

highest level at the criminal court.

Fermat

- Still further promotions seem to indicate a

fairly meteoric rise through the profession but

promotion was done mostly on seniority and the

plague struck the region in the early 1650s

meaning that many of the older men died. - Fermat himself was struck down by the plague and

in 1653 his death was wrongly reported, then

corrected-

Fermat

- I informed you earlier of the death of Fermat. He

is alive, and we no longer fear for his health,

even though we had counted him among the dead a

short time ago. - The following report, made to Colbert the leading

figure in France at the time, has a ring of

truth- - Fermat, a man of great erudition, has contact

with men of learning everywhere. But he is rather

preoccupied, he does not report cases well and is

confused.

Fermat

- Of course Fermat was preoccupied with

mathematics. - He kept his mathematical friendship with Beugrand

after he moved to Toulouse but there he gained a

new mathematical friend in Carcavi. - Fermat met Carcavi in a professional capacity

since both were councillors in Toulouse but they

both shared a love of mathematics and Fermat told

Carcavi about his mathematical discoveries.

Fermat

- In 1636 Carcavi went to Paris as royal librarian

and made contact with Mersenne and his group.

Mersenne's interest was aroused by Carcavi's

descriptions of Fermat's discoveries on falling

bodies, and he wrote to Fermat. - Fermat replied on 26 April 1636 and, in addition

to telling Mersenne about errors which he

believed that Galileo had made in his description

of free fall, he also told Mersenne about his

work on spirals and his restoration of

Apollonius's Plane loci.

Fermat

- His work on spirals had been motivated by

considering the path of free falling bodies and

he had used methods generalised from Archimedes'

work On spirals to compute areas under the

spirals. - In addition Fermat wrote-

Fermat

- I have also found many sorts of analyses for

diverse problems, numerical as well as

geometrical, for the solution of which Vietes

analysis could not have sufficed. - I will share all of this with you whenever you

wish and do so without any ambition, from which I

am more exempt and more distant than any man in

the world.

Fermat

- It is somewhat ironical that this initial contact

with Fermat and the scientific community came

through his study of free fall since Fermat had

little interest in physical applications of

mathematics. - Even with his results on free fall he was much

more interested in proving geometrical theorems

than in their relation to the real world.

Fermat

- This first letter did however contain two

problems on maxima which Fermat asked Mersenne to

pass on to the Paris mathematicians and this was

to be the typical style of Fermat's letters, he

would challenge others to find results which he

had already obtained.

Fermat

- Roberval and Mersenne found that Fermat's

problems in this first, and subsequent, letters

were extremely difficult and usually not soluble

using current techniques. - They asked him to divulge his methods and Fermat

sent Method for determining Maxima and Minima and

Tangents to Curved Lines, his restored text of

Apolloniuss Plane loci and his algebraic

approach to geometry Introduction to Plane and

Solid Loci to the Paris mathematicians.

Fermat

- His reputation as one of the leading

mathematicians in the world came quickly but

attempts to get his work published failed mainly

because Fermat never really wanted to put his

work into a polished form. - However some of his methods were published, for

example Herigone added a supplement containing

Fermat's methods of maxima and minima to his

major work Cursus mathematicus. - The widening correspondence between Fermat and

other mathematicians did not find universal

praise. Frenicle de Bessy became annoyed at

Fermat's problems which to him were impossible.

Fermat

- He wrote angrily to Fermat but although Fermat

gave more details in his reply, Frenicle de Bessy

felt that Fermat was almost teasing him.

Fermat

- However Fermat soon became engaged in a

controversy with a more major mathematician than

Frenicle de Bessy - Having been sent a copy of Descartes' La

Dioptrique by Beaugrand, Fermat paid it little

attention since he was in the middle of a

correspondence with Roberval and Etienne Pascal

over methods of integration and using them to

find centres of gravity. - Mersenne asked him to give an opinion on La

Dioptrique which Fermat did, describing it as

groping about in the shadows.

Fermat

- He claimed that Descartes had not correctly

deduced his law of refraction since it was

inherent in his assumptions. - To say that Descartes was not pleased is an

understatement. - Descartes soon found reason to feel even more

angry since he viewed Fermat's work on maxima,

minima and tangents as reducing the importance of

his own work La Géométrie which Descartes was

most proud of and which he sought to show that

his Discours de la méthode alone could give.

Fermat

- Descartes attacked Fermat's method of maxima,

minima and tangents. Roberval and E. Pascal

became involved in the argument and eventually so

did Desargues who Descartes asked to act as a

referee. Fermat proved correct and eventually

Descartes admitted this writing-

Girard Desargues, 1591 - 1661

Girard Desargues was a French mathematician who

was a founder of projective geometry. His work

centred on the theory of conic sections and

perspective.

Example from projective geometry

Projective Geometry

- Projective geometry is a non-metrical form of

geometry. - Projective geometry grew out of the principles of

perspective art established during the

Renaissance period, and was first systematically

developed by Desargues in the 17th century,

although it did not achieve prominence as a field

of mathematics until the early 19th century

through the work of Poncelet and others.

Jean Victor Poncelet, 1788 - 1867

Poncelet was one of the founders of modern

projective geometry. His development of the pole

and polar lines associated with conics led to

the principle of duality.

Fermat

- ... seeing the last method that you use for

finding tangents to curved lines, I can reply to

it in no other way than to say that it is very

good and that, if you had explained it in this

manner at the outset, I would have not

contradicted it at all. - Did this end the matter and increase Fermat's

standing? - Not at all since Descartes tried to damage

Fermat's reputation.

Fermat

- For example, although he wrote to Fermat praising

his work on determining the tangent to a cycloid

(which is indeed correct), Descartes wrote to

Mersenne claiming that it was incorrect and

saying that Fermat was inadequate as a

mathematician and a thinker. - Descartes was important and respected and thus

was able to severely damage Fermat's reputation.

Fermat

- The period from 1643 to 1654 was one when Fermat

was out of touch with his scientific colleagues

in Paris. - There are a number of reasons for this. Firstly

pressure of work kept him from devoting so much

time to mathematics. - Secondly the Fronde, a civil war in France, took

place and from 1648 Toulouse was greatly

affected.

Fermat

- Finally there was the plague of 1651 which must

have had great consequences both on life in

Toulouse and of course its near fatal

consequences on Fermat himself. - However it was during this time that Fermat

worked on number theory.

Fermat

- Fermat is best remembered for this work in number

theory, in particular for Fermats last Theorem. - This theorem states that

Fermats last theorem

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.

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.

Fermat

- In November 1994 Wiles again claimed to have a

correct proof which has now been accepted. - Unsuccessful attempts to prove the theorem over a

300 year period led to the discovery of

commutative ring theory and a wealth of other

mathematical discoveries. - Fermat's correspondence with the Paris

mathematicians restarted in 1654 when Blaise

Pascal, E Pascal's son, wrote to him to ask for

confirmation about his ideas on probability. - Blaise Pascal knew of Fermat through his father,

who had died three years before, and was well

aware of Fermat's outstanding mathematical

abilities.

Fermat

- Their short correspondence set up the theory of

probability and from this they are now regarded

as joint founders of the subject. - Fermat however, feeling his isolation and still

wanting to adopt his old style of challenging

mathematicians, tried to change the topic from

probability to number theory. - Pascal was not interested but Fermat, not

realising this, wrote to Carcavi saying-

Fermat

- am delighted to have had opinions conforming to

those of M Pascal, for I have infinite esteem for

his genius... the two of you may undertake that

publication, of which I consent to your being the

masters, you may clarify or supplement whatever

seems too concise and relieve me of a burden that

my duties prevent me from taking on.

Fermat

- However Pascal was certainly not going to edit

Fermat's work and after this flash of desire to

have his work published Fermat again gave up the

idea. - He went further than ever with his challenge

problems however- - Two mathematical problems posed as insoluble to

French, English, Dutch and all mathematicians of

Europe by Monsieur de Fermat, Councillor of the

King in the Parliament of Toulouse.

Fermat

- His problems did not prompt too much interest as

most mathematicians seemed to think that number

theory was not an important topic. - The second of the two problems, namely to find

all solutions of Nx2 1 y2 for N not a square,

was however solved by Wallis and Brouncker and

they developed continued fractions in their

solution. Brouncker produced rational solutions

which led to arguments. - De Bessy was perhaps the only mathematician at

that time who was really interested in number

theory but he did not have sufficient

mathematical talents to allow him to make a

significant contribution.

Fermat

- Fermat posed further problems, namely that the

sum of two cubes cannot be a cube (a special case

of Fermat's Last Theorem which may indicate that

by this time Fermat realised that his proof of

the general result was incorrect), that there are

exactly two integer solutions of x2 4 y3 and

that the equation x2 2 y3 has only one

integer solution. - He posed problems directly to the English.
- Everyone failed to see that Fermat had been

hoping his specific problems would lead them to

discover, as he had done, deeper theoretical

results.

Fermat

- Around this time one of Descartes' students was

collecting his correspondence for publication and

he turned to Fermat for help with the Fermat -

Descartes correspondence. - This led Fermat to look again at the arguments he

had used 20 years before and he looked again at

his objections to Descartes' optics. In

particular he had been unhappy with Descartes '

description of refraction of light and he now

settled on a principle which did in fact yield

the sine law of refraction that Snell and

Descartes had proposed.

Fermat

- However Fermat had now deduced it from a

fundamental property that he proposed, namely

that light always follows the shortest possible

path. - Fermat's principle, now one of the most basic

properties of optics, did not find favor with

mathematicians at the time

Fermat

- In 1656 Fermat had started a correspondence with

Huygens. - This grew out of Huygens interest in probability

and the correspondence was soon manipulated by

Fermat onto topics of number theory. - This topic did not interest Huygens but Fermat

tried hard and in New Account of Discoveries in

the Science of Numbers sent to Huygens via

Carcavi in 1659, he revealed more of his methods

than he had done to others.

Fermat

- Fermat described his method of infinite descent

and gave an example on how it could be used to

prove that every prime of the form 4k 1 could

be written as the sum of two squares. - For suppose some number of the form 4k 1 could

not be written as the sum of two squares. Then

there is a smaller number of the form 4k 1

which cannot be written as the sum of two

squares. Continuing the argument will lead to a

contradiction.

Fermat

- What Fermat failed to explain in this letter is

how the smaller number is constructed from the

larger. - One assumes that Fermat did know how to make this

step but again his failure to disclose the method

made mathematicians lose interest. - It was not until Euler took up these problems

that the missing steps were filled in.

Fermat

- Fermat is described as
- Secretive and taciturn, he did not like to talk

about himself and was loath to reveal too much

about his thinking. ... His thought, however

original or novel, operated within a range of

possibilities limited by that 1600 - 1650 time

and that France place.

Leonhard Euler, 1707 - 1783

Leonhard Euler was a Swiss mathematician who

made enormous contributions to a wide range of

mathematics and physics including analytic

geometry, trigonometry, geometry, calculus and

number theory

Fermat

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