Quotations by Thales of Miletus

					                  HISTORY NOTEBOOK
                                  MME 518

                                                Dannette B. Daniels

                          Thales of Miletus

         Born: about 624 BC in Miletus, Asia Minor (now Turkey)
         Died: about 547 BC in Miletus, Asia Minor (now Turkey)

Little is known of Thales. He was born about 624 BC in Miletus, Asia Minor (now
Turkey) and died about 546 BC in Miletos, Turkey

Some impression and highlights of his life and work follow:

      Thales of Miletus was the first known Greek philosopher, scientist and
       mathematician. Some consider him to be the teacher of of Pythagoras, though it
       may be only that he advised Pythagoras to travel to Egypt and Chaldea.
      From Eudemus of Rhodes (fl ca. 320 B.C) we know that he studied in Egypt and
       brought these teachings to Greece. He is unanimously ascribed the introduction of
       mathematical and astronomical sciences into Greece.
      He is unanimously regarded as having been unusally clever--by general
       agreement the first of the Seven Wise Men, a pupil of the Egyptians and the
      None of his writing survives; this makes it is difficult to determine his philosophy
       and to be certain about his mathematical discoveries.
     There is, of course, the story of his successful speculation in oil presses -- as
      testament to his practical business acumen.
     It is reported that he predicted an eclipse of the Sun on May 28, 585 BC, startling
      all of Ionia.
     He is credited with five theorems of elementary geometry.

                  Eratosthenes of Cyrene

Born: 276 BC in Cyrene, North Africa (now Shahhat, Libya)
Died: 194 BC in Alexandria, Egypt


  The man who first measured the world, the
  Greek astronomer Eratosthenes (c. 276-196
  B.C.), lived in Alexandria during the 3rd
  century B.C. He noticed that on the first day
  of summer in Syene (now Aswan), Egypt,
  the Sun appeared directly overhead at noon.
  At the same time in Alexandria, however,
  the Sun appeared slightly south (about 7
  degrees) of the zenith. Knowing the distance
  between Syene and Alexandria and assuming
  that the Sun’s rays were parallel when they
  struck the curved Earth, he calculated the
  size of our planet using simple geometry.
 His result, about 25,000 miles for the
 circumference, proved remarkably accurate.

      Eratosthenes wasn’t the only Greek who
 tried to measure the Earth. About a century
 later, Posidonius copied this feat, using the
 star Canopus as his light source and the
 cities of Rhodes and Alexandria as his
 baseline. Although his technique was sound,
 he had the wrong value for the distance
 between Rhodes and Alexandria, so his
 circumference came out too small. Ptolemy
 recorded this smaller figure in his geography
 treatise, where it was seized upon by
 Renaissance explorers looking for a quicker
 way to the Indies. Had Ptolemy used
 Eratosthenes’ larger figure instead,
 Columbus might never have sailed west.

                    Euclid of Alexandria

                         Born: about 325 BC
                Died: about 265 BC in Alexandria, Egypt

Euclid (ca. 325-ca. 270 BC)

                                   Greek geometer who wrote the Elements , the
                                   world's most definitive text on geometry. The book
                                   synthesized earlier knowledge about geometry, and
                                   was used for centuries in western Europe as a
                                   geometry textbook. The text began with definitions,
                                   postulates ("Euclid's postulates "), and common
                                   opinions, then proceeded to obtain results by
                                   rigorous geometric proof. Euclid also proved what is
                                   generally known as Euclid's second theorem: the
                                   number of primes is infinite. The beautiful proof
Euclid gave of this theorem is still a gem and is generally acknowledged to be one of
  the "classic" proofs of all times in terms of its conciseness and clarity. In the Elements
    , Euclid used the method of exhaustion and reductio ad absurdum. He also discussed
  the so-called Euclidean algorithm for finding the greatest common divisor of two
  numbers, and is credited with the well-known proof of the Pythagorean theorem.

  Neither the year nor place of his birth have been established, nor the circumstances of
  his death, although he is known to have lived and worked in Alexandria for much of
  his life. In addition, no bust which can be verified to be his likeness is known (Tietze
  1965, p. 8).

Euclid of Alexandria is the most prominent mathematician of antiquity best known for
his treatise on mathematics The Elements. The long lasting nature of The Elements must
make Euclid the leading mathematics teacher of all time. However little is known of
Euclid's life except that he taught at Alexandria in Egypt. Proclus, the last major Greek
philosopher, who lived around 450 AD wrote (see [1] or [9] or many other sources):-

                      Pythagoras of Samos

                      Born: about 569 BC in Samos, Ionia
                              Died: about 475 BC

Pythagoras of Samos is often described as the first pure mathematician. He is an
extremely important figure in the development of mathematics yet we know relatively
little about his mathematical achievements. Unlike many later Greek mathematicians,
where at least we have some of the books which they wrote, we have nothing of
Pythagoras's writings. The society which he led, half religious and half scientific,
followed a code of secrecy which certainly means that today Pythagoras is a mysterious

Of Pythagoras's actual work nothing is known. His school practised secrecy and
communalism making it hard to distinguish between the work of Pythagoras and that of
his followers. Certainly his school made outstanding contributions to mathematics, and it
is possible to be fairly certain about some of Pythagoras's mathematical contributions.
First we should be clear in what sense Pythagoras and the mathematikoi were studying
mathematics. They were not acting as a mathematics research group does in a modern
university or other institution. There were no 'open problems' for them to solve, and they
were not in any sense interested in trying to formulate or solve mathematical problems.

Rather Pythagoras was interested in the principles of mathematics, the concept of
number, the concept of a triangle or other mathematical figure and the abstract idea of a

Heath [7] gives a list of theorems attributed to Pythagoras, or rather more generally to the

(i) The sum of the angles of a triangle is equal to two right angles. Also the Pythagoreans
knew the generalisation which states that a polygon with n sides has sum of interior
angles 2n - 4 right angles and sum of exterior angles equal to four right angles.

(ii) The theorem of Pythagoras - for a right angled triangle the square on the hypotenuse
is equal to the sum of the squares on the other two sides. We should note here that to
Pythagoras the square on the hypotenuse would certainly not be thought of as a number
multiplied by itself, but rather as a geometrical square constructed on the side. To say that
the sum of two squares is equal to a third square meant that the two squares could be cut
up and reassembled to form a square identical to the third square.

(iii) Constructing figures of a given area and geometrical algebra. For example they
solved equations such as a (a - x) = x2 by geometrical means.

(iv) The discovery of irrationals. This is certainly attributed to the Pythagoreans but it
does seem unlikely to have been due to Pythagoras himself. This went against
Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio
of two whole numbers. However, because of his belief that all things are numbers it
would be a natural task to try to prove that the hypotenuse of an isosceles right angled
triangle had a length corresponding to a number.

(v) The five regular solids. It is thought that Pythagoras himself knew how to construct
the first three but it is unlikely that he would have known how to construct the other two.

(vi) In astronomy Pythagoras taught that the Earth was a sphere at the centre of the
Universe. He also recognised that the orbit of the Moon was inclined to the equator of the
Earth and he was one of the first to realise that Venus as an evening star was the same
planet as Venus as a morning star
                        Apollonius of Perga

 Born: about 262 BC in Perga, Pamphylia, Greek Ionia (now Murtina,
                          Antalya, Turkey)
              Died: about 190 BC in Alexandria, Egypt

The mathematician Apollonius was born in Perga, Pamphylia which today is known as
Murtina, or Murtana and is now in Antalya, Turkey. Perga was a centre of culture at this
time and it was the place of worship of Queen Artemis, a nature goddess. When he was a
young man Apollonius went to Alexandria where he studied under the followers of
Euclid and later he taught there. Apollonius visited Pergamum where a university and
library similar to Alexandria had been built. Pergamum, today the town of Bergama in
the province of Izmir in Turkey, was an ancient Greek city in Mysia. It was situated 25
km from the Aegean Sea on a hill on the northern side of the wide valley of the Caicus
River (called the Bakir river today).

While Apollonius was at Pergamum he met Eudemus of Pergamum (not to be confused
with Eudemus of Rhodes who wrote the History of Geometry) and also Attalus, who
many think must be King Attalus I of Pergamum. In the preface to the second edition of
Conics Apollonius addressed Eudemus (see [4] or [5]):-

If you are in good health and things are in other respects as you wish, it is well; with me
too things are moderately well. During the time I spent with you at Pergamum I observed
your eagerness to become aquatinted with my work in conics.

The only other pieces of information about Apollonius's life is to be found in the prefaces
of various books of Conics. We learn that he had a son, also called Apollonius, and in
fact his son took the second edition of book two of Conics from Alexandria to Eudemus
in Pergamum. We also learn from the preface to this book that Apollonius introduced the
geometer Philonides to Eudemus while they were at Ephesus.

We are in a somewhat better state of knowledge concerning the books which Apollonius
wrote. Conics was written in eight books but only the first four have survived in Greek.
In Arabic, however, the first seven of the eight books of Conics survive.
Apollonius of Perga was known as 'The Great Geometer'. Little is known of his life but
his works have had a very great influence on the development of mathematics, in
particular his famous book Conics introduced terms which are familiar to us today such
as parabola, ellipse and hyperbola.

                             René Descartes

  Born: 31 March 1596 in La Haye (now Descartes),Touraine, France
              Died: 11 Feb 1650 in Stockholm, Sweden

René Descartes was a philosopher whose work, La géométrie, includes his application of
algebra to geometry from which we now have Cartesian geometry.

We may consider Descartes as the first of the modern school of mathematics. René
Descartes was born near Tours on March 31, 1596, and died at Stockholm on February
11, 1650; thus he was a contemporary of Galileo and Desargues. His father, who, as the
name implies, was of good family, was accustomed to spend half the year at Rennes
when the local parliament, in which he held a commission as councillor, was in session,
and the rest of the time on his family estate of Les Cartes at La Haye. René, the second of
a family of two sons and one daughter, was sent at the age of eight years to the Jesuit
School at La Flêche, and of the admirable discipline and education there given he speaks
most highly. On account of his delicate health he was permitted to lie in bed till late in
the mornings; this was a custom which he always followed, and when he visited Pascal in
1647 he told him that the only way to do good work in mathematics and to preserve his
health was never to allow anyone to make him get up in the morning before he felt
inclined to do so.

Pierre de Fermat
           Born: 17 Aug 1601 in Beaumont-de-Lomagne, France
                   Died: 12 Jan 1665 in Castres, France

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.

While Descartes was laying the foundations of analytical geometry, the same subject was
occupying the attention of another and not less distinguished Frenchman. This was
Fermat. Pierre de Fermat, who was born near Montauban in 1601, and died at Castres on
January 12, 1665, was the son of a leather-merchant; he was educated at home; in 1631
he obtained the post of councillor for the local parliament at Toulouse, and he discharged
the duties of the office with scrupulous accuracy and fidelity. There, devoting most of his
leisure to mathematics, he spent the remainder of his life - a life which, but for a
somewhat acrimonious dispute with Descartes on the validity of certain analysis used by
the latter, was unruffled by any event which calls for special notice. The dispute was
chiefly due to the obscurity of Descartes, but the tact and courtesy of Fermat brought it to
a friendly conclusion. Fermat was a good scholar, and amused himself by conjecturally
restoring the work of Apollonius on plane loci.

Except a few isolated papers, Fermat published nothing in his lifetime, and gave no
systematic exposition of his methods. Some of the most striking of his results were found
after his death on loose sheets of paper or written in the margins of works which he had
read and annotated, and are unaccompanied by any proof. It is thus somewhat difficult to
estimate the dates and originality of his work. He was constitutionally modest and
retiring, and does not seem to have intended his papers to be published. It is probable that
he revised his notes as occasion required, and that his published works represent the final
form of his researches, and therefore cannot be dated much earlier than 1660. I shall
consider separately (i) his investigations in the theory of numbers; (ii) his use in geometry
of analysis and of infinitesimals; and (iii) his method for treating questions of probability
Kepler's laws of planetary motion
      Kepler's first law (1609): The orbit of a planet about a star is an ellipse with the
       star at one focus.

      Kepler's second law (1609): A line joining a planet and its star sweeps out equal
       areas during equal intervals of time.

      Kepler's third law (1618): The square of the sidereal period of an orbiting
       planet is directly proportional to the cube of the orbit's semimajor axis.

   In 1611, Kepler    proposed that close packing (either cubic or hexagonal close

   packing, both of which have maximum densities of                           ) is the
   densest possible sphere packing, and this assertion is known as the Kepler conjecture.
   Finding the densest (not necessarily periodic) packing of spheres is known as the
   Kepler problem

John Harrison
 (March 1693 - March 24, 1776) was an English clock designer, who developed and built
the world's first successful maritime clock, one whose accuracy was great enough to
allow the determination of longitude over long distances.

Harrison was born at Foulby in Yorkshire, the eldest son of a carpenter. A carpenter by
initial trade, Harrison built and repaired clocks in his spare time. Legend has it that he
was given a watch when he was six to amuse him while in bed with smallpox, spending
hours listening to it and studying its moving parts. Scholars today, however, consider this
unlikely to be true, as clocks and watches of all kinds were rare and expensive at the
time, and Harrison came from a family of fairly modest means.

He was a man of many skills and used these to improve on the way clocks were built. For
example, he developed the gridiron pendulum, consisting of alternating brass and steel
rods assembled so that the different expansion and contraction rates cancelled each other
out. Another example of his inventive genius was the grasshopper escapement -- a control
device for the step-by-step release of a clock's driving power. Being almost frictionless, it
required no oiling.

In 1728 Harrison packed up full scale models of his inventions and drawings for a
proposed marine clock to compete for the Longitude Prize and headed for London
seeking financial assistance. He met with Edmond Halley, the Astronomer Royal, and
presented his ideas. Halley sent him to meet George Graham, the country's foremost
horologist (clockmaker). He must have been impressed with Harrison, for Graham
personally loaned him money and told him to build a model of his marine clock.

It took Harrison seven years to build Harrison Number One or H1. He presented it to
members of the Royal Society who spoke on its behalf to the Board of Longitude. The
board was so skeptical of any such design, after 14 years of failures, that they demanded
a sea trial. Harrison boarded a small ship to Lisbon and back, and on their return the
captain and navigator both praised the design. The navigator noted that his own
calculations estimated they were 90 miles offshore on their return to Britain, but the H1
put them just offshore right when the shoreline appeared.

This was not the transatlantic voyage demanded by the Board of Longitude, but the


              Johann Carl Friedrich Gauss

      Born: 30 April 1777 in Brunswick, Duchy of Brunswick (now
       Died: 23 Feb 1855 in Göttingen, Hanover (now Germany)

At the age of seven, Carl Friedrich Gauss started elementary school, and his potential
was noticed almost immediately. His teacher, Büttner, and his assistant, Martin Bartels,
were amazed when Gauss summed the integers from 1 to 100 instantly by spotting that
the sum was 50 pairs of numbers each pair summing to 101.

From the outside, Gauss' life was very simple. Having brought up in an austere childhood
in a poor and uneducated family he showed extraordinary precocity. He received a
stipend from the duke of Brunswick starting at the age of 14 which allowed him to devote
his time to his studies for 16 years. Before his 25th birthday, he was already famous for
his work in mathematics and astronomy. When he became 30 he went to Göttingen to
become director of the observatory. He rarely left the city except on scientific business.
From there, he worked for 47 years until his death at almost 78. In contrast to his external
simplicity, Gauss' personal life was tragic and complicated. Due to the French
Revolution, Napoleonic period and the democratic revolutions in Germany, he suffered
from political turmoil and financial insecurity. He found no fellow mathematical
collaborators and worked alone for most of his life. An unsympathetic father, the early
death of his first wife, the poor health of his second wife, and terrible relations with his
sons denied him a family sanctuary until late in life.

Even with all of these troubles, Gauss kept an amazingly rich scientific activity. An early
passion for numbers and calculations extended first to the theory of numbers, to algebra,
analysis, geometry, probability, and the theory of errors. At the same time, he carried on
intensive empirical and theoretical research in many branches of science, including
observational astronomy, celestial mechanics, surveying, geodesy, capillarity,
geomagnetism, electromagnetism, mechanism optics, actuarial science. His publications,
abundant correspondence, notes, and manuscripts show him to have been one of the
greatest scientific virtuosos of all time

He published his second book, Theoria motus corporum coelestium in sectionibus conicis
Solem ambientium, in 1809, a major two volume treatise on the motion of celestial
bodies. In the first volume he discussed differential equations, conic sections and elliptic
orbits, while in the second volume, the main part of the work, he showed how to estimate
and then to refine the estimation of a planet's orbit. Gauss's contributions to theoretical
astronomy stopped after 1817, although he went on making observations until the age of

      Georg Friedrich Bernhard Riemann

        Born: 17 Sept 1826 in Breselenz, Hanover (now Germany)
                   Died: 20 July 1866 in Selasca, Italy

Bernhard Riemann's father, Friedrich Bernhard Riemann, was a Lutheran minister.
Friedrich Riemann married Charlotte Ebell when he was in his middle age. Bernhard was
the second of their six children, two boys and four girls. Friedrich Riemann acted as
teacher to his children and he taught Bernhard until he was ten years old. At this time a
teacher from a local school named Schulz assisted in Bernhard's education
First published by Riemann (1859), the Riemann hypothesis states that the nontrivial
Riemann zeta function zeros, i.e., the values of s other than -2, -4, -6, ... such that

         (where       is the Riemann zeta function) all lie on the "critical line"

                (where       denotes the real part of s). While it was long believed that
Riemann's hypothesis was the result of deep intuition on the part of Riemann, an
examination of his papers by C. L. Siegel showed that Riemann had made detailed

numerical calculations of small zeros of the Riemann zeta function         to several
decimal digits (Granville 2002; Borwein and Borwein 2003, p. 68).

A more general statement known as the generalized Riemann hypothesis conjectures that
neither the Riemann zeta function nor any Dirichlet L-series has a zero with real part
larger than 1/2

Nikolai Ivanovich Lobachewsky (1793-1856) "On
the Principles of Geometry" (1829)
The first published account of hyperbolic geometry, in Russian. Lobachewsky developed
his ideas from an analytical (trigonometric) viewpoint
Greek science is the classic example, par excellence, of a normal science, whose
paradigms, according to Kuhn (TK), have lasted thousands of years. The geometry of
Euclid (330 BC) reigned triumphant and undisputed until the introduction of non-
Euclidean geometries by Gauss (1777-1855), Lobachewsky (1793-1856), Boylai (1802-
60), and Riemann (1826-66). The logic of Aristotle (382-322 BC) is even more resistant
to change because it is immersed in the languages which reflect the Western mode of

Immanuel Kant (1724-1804), philosopher and Professor of Logic at Koenigsberg, was
fully convinced that "Aristotle did not omit any essential aspect of knowledge; it only
remains for us to become more precise, methodical, and orderly."

The research of the Polish thinker J. Lukasiewicz was a sharp departure from the
Aristotelian interpretation of logic. Lukasiewicz, a leading member of the Warsaw school
of logic, published his paper "0 logice trojwartoscioweJ " ("On Trivalent Logic") in 1920.
This publication, the point of departure for non-Aristotelian systems of logic, was not
translated into Spanish until 1975 (JL1).

According to J. Ferrater Mora (JFM), there is some evidence that William of Occam
(1298-1349) had already suggested the use of three truth-values. Ferrater Mora also
indicates that around 1910, the "Russian mathematician N. N. Vasilev of the University
of Kazan, published several articles in which he put forward and developed a three-
valued logic. Vasilev's fundamental idea consisted in transposing to Logic the rules
followed by Lobachewsky in founding his non-Euclidean geometry. Lobachewsky, who
had been a Professor at the same University, developed his geometry by eliminating the
parallel postulate. Likewise, Vasilev developed his trivalent logic, which he called "non-
Aristotelian logic", by eliminating the law of excluded middle. However, the most
important and influential contemporary publications on polyvalent logic have been
published by Jan Lukasiewicz, Emi1 L. Post, and Alfred Tarski."

In 1930, Lukasiewicz published his paper "Philosophische Bemerkungen zu
mehrwertigen Systemen des Aussagenkalkuels" (Philosophical Observations on
Polyvalent Systems of Propositional Logic). In this paper the author explains his ideas in
great detail, from the point of view of both logic and philosophy. He analyzes the
consequences of modal statements which, within the limited framework of bivalent logic,
"go against all our intuitions." He also clearly demonstrated the incompatibilities of
theorems regarding modal propositions in bivalent propositional calculus

                             Albert Einstein

           Born: 14 March 1879 in Ulm, Württemberg, Germany
            Died: 18 April 1955 in Princeton, New Jersey, USA

Around 1886 Albert Einstein began his school career in Munich. As well as his violin
lessons, which he had from age six to age thirteen, he also had religious education at
home where he was taught Judaism. Two years later he entered the Luitpold Gymnasium
and after this his religious education was given at school. He studied mathematics, in
particular the calculus, beginning around 1891.
     Albert Einstein – Biography
      Albert Einstein was born at Ulm, in Württemberg, Germany, on March 14, 1879. Six weeks
      later the family moved to Munich and he began his schooling there at the Luitpold Gymnasium.
      Later, they moved to Italy and Albert continued his education at Aarau, Switzerland and in
      1896 he entered the Swiss Federal Polytechnic School in Zurich to be trained as a teacher in
      physics and mathematics. In 1901, the year he gained his diploma, he acquired Swiss
citizenship and, as he was unable to find a teaching post, he accepted a position as technical
assistant in the Swiss Patent Office. In 1905 he obtained his doctor's degree.

During his stay at the Patent Office, and in his spare time, he produced much of his remarkable
work and in 1908 he was appointed Privatdozent in Berne. In 1909 he became Professor
Extraordinary at Zurich, in 1911 Professor of Theoretical Physics at Prague, returning to Zurich in
the following year to fill a similar post. In 1914 he was appointed Director of the Kaiser Wilhelm
Physical Institute and Professor in the University of Berlin. He became a German citizen in 1914 and
remained in Berlin until 1933 when he renounced his citizenship for political reasons and emigrated
to America to take the position of Professor of Theoretical Physics at Princeton*. He became a
United States citizen in 1940 and retired from his post in 1945.

After World War II, Einstein was a leading figure in the World Government Movement, he was
offered the Presidency of the State of Israel, which he declined, and he collaborated with Dr. Chaim
Weizmann in establishing the Hebrew University of Jerusalem.

Einstein always appeared to have a clear view of the problems of physics and the determination to
solve them. He had a strategy of his own and was able to visualize the main stages on the way to
his goal. He regarded his major achievements as mere stepping-stones for the next advance.

At the start of his scientific work, Einstein realized the inadequacies of Newtonian mechanics and his
special theory of relativity stemmed from an attempt to reconcile the laws of mechanics with the
laws of the electromagnetic field. He dealt with classical problems of statistical mechanics and
problems in which they were merged with quantum theory: this led to an explanation of the
Brownian movement of molecules. He investigated the thermal properties of light with a low
radiation density and his observations laid the foundation of the photon theory of light.

In his early days in Berlin, Einstein postulated that the correct interpretation of the special theory of
relativity must also furnish a theory of gravitation and in 1916 he published his paper on the
general theory of relativity. During this time he also contributed to the problems of the theory of
radiation and statistical mechanics.

In the 1920's, Einstein embarked on the construction of unified field theories, although he continued
to work on the probabilistic interpretation of quantum theory, and he persevered with this work in
America. He contributed to statistical mechanics by his development of the quantum theory of a
monatomic gas and he has also accomplished valuable work in connection with atomic transition
probabilities and relativistic cosmology.

After his retirement he continued to work towards the unification of the basic concepts of physics,
taking the opposite approach, geometrisation, to the majority of physicists.

Einstein's researches are, of course, well chronicled and his more important works include Special
Theory of Relativity (1905), Relativity (English translations, 1920 and 1950), General Theory of
Relativity (1916), Investigations on Theory of Brownian Movement (1926), and The Evolution of
Physics (1938). Among his non-scientific works, About Zionism (1930), Why War? (1933), My
Philosophy (1934), and Out of My Later Years (1950) are perhaps the most important.

Albert Einstein received honorary doctorate degrees in science, medicine and philosophy from many
European and American universities. During the 1920's he lectured in Europe, America and the Far
East and he was awarded Fellowships or Memberships of all the leading scientific academies
throughout the world. He gained numerous awards in recognition of his work, including the Copley
Medal of the Royal Society of London in 1925, and the Franklin Medal of the Franklin Institute in
Einstein's gifts inevitably resulted in his dwelling much in intellectual solitude and, for relaxation,
music played an important part in his life. He married Mileva Maric in 1903 and they had a daughter
and two sons; their marriage was dissolved in 1919 and in the same year he married his cousin,
Elsa Löwenthal, who died in 1936. He died on April 18, 1955 at Princeton, New Jersey.

                         Gaston Maurice Julia

                   Born: 3 Feb 1893 in Sidi Bel Abbès, Algeria
                      Died: 19 March 1978 in Paris, France

When only 25 when Gaston Julia published his 199 page masterpiece Mémoire sur
l'iteration des fonctions rationelles which made him famous in the mathematics centres
of his days.

As a soldier in the First World War, Julia had been severely wounded in an attack on the
French front designed to celebrate the Kaiser's birthday. Many on both sides were
wounded including Julia who lost his nose and had to wear a leather strap across his face
for the rest of his life. Between several painful operations he carried on his mathematical
researches in hospital.

Later he became a distinguished professor at the École Polytechnique in Paris.

In 1918 Julia published a beautiful paper Mémoire sur l'itération des fonctions
rationnelles, Journal de Math. Pure et Appl. 8 (1918), 47-245, concerning the iteration of
a rational function f. Julia gave a precise description of the set J(f) of those z in C for
which the nth iterate fn(z) stays bounded as n tends to infinity. It received the Grand Prix
de l'Académie des Sciences.

Seminars were organised in Berlin in 1925 to study his work and participants included
Brauer, Hopf and Reidemeister. H Cremer produced an essay on his work which included
the first visualisation of a Julia set.

Although he was famous in the 1920s, his work was essentially forgotten until B
Mandelbrot brought it back to prominence in the 1970s through his fundamental
computer experiments.
                        Benoit Mandelbrot

                    Born: 20 Nov 1924 in Warsaw, Poland

Benoit Mandelbrot was largely responsible for the present interest in fractal geometry.
He showed how fractals can occur in many different places in both mathematics and
elsewhere in nature.

Mandelbrot was born in Poland in 1924 into a family with a very academic tradition. His
father, however, made his living buying and selling clothes while his mother was a
doctor. As a young boy, Mandelbrot was introduced to mathematics by his two uncles.

Mandelbrot's family emigrated to France in 1936 and his uncle Szolem Mandelbrojt, who
was Professor of Mathematics at the Collège de France and the successor of Hadamard in
this post, took responsibility for his education. In fact the influence of Szolem
Mandelbrojt was both positive and negative since he was a great admirer of Hardy and
Hardy's philosophy of mathematics. This brought a reaction from Mandelbrot against
pure mathematics, although as Mandelbrot himself says, he now understands how
Hardy's deep felt pacifism made him fear that applied mathematics, in the wrong hands,
might be used for evil in time of war.

Mandelbrot attended the Lycée Rolin in Paris up to the start of World War II, when his
family moved to Tulle in central France. This was a time of extraordinary difficulty for
Mandelbrot who feared for his life on many occasions. In [3] the effect of these years on
his education was emphasised:-

The war, the constant threat of poverty and the need to survive kept him away from
school and college and despite what he recognises as "marvellous" secondary school
teachers he was largely self taught.

Mandelbrot now attributed much of his success to this unconventional education. It
allowed him to think in ways that might be hard for someone who, through a
conventional education, is strongly encouraged to think in standard ways. It also allowed
him to develop a highly geometrical approach to mathematics, and his remarkable
geometric intuition and vision began to give him unique insights into mathematical
Mandelbrot returned to France in 1955 and worked at the Centre National de la
Recherche Scientific. He married Aliette Kagan during this period back in France and
Geneva, but he did not stay there too long before returning to the United States. Clark
gave the reasons for his unhappiness with the style of mathematics in France at this time

Still deeply concerned with the more exotic forms of statistical mechanics and
mathematical linguistics and full of non standard creative ideas he found the huge
dominance of the French foundational school of Bourbaki not to his scientific tastes and
in 1958 he left for the United States permanently and began his long standing and most
fruitful collaboration with IBM as an IBM Fellow at their world renowned laboratories
in Yorktown Heights in New York State.

IBM presented Mandelbrot with an environment which allowed him to explore a wide
variety of different ideas. He has spoken of how this freedom at IBM to choose the
directions that he wanted to take in his research presented him with an opportunity which
no university post could have given him. After retiring from IBM, he found similar
opportunities at Yale University, where he is presently Sterling Professor of
Mathematical Sciences.

In 1945 Mandelbrot's uncle had introduced him to Julia's important 1918 paper claiming
that it was a masterpiece and a potential source of interesting problems, but Mandelbrot
did not like it. Indeed he reacted rather badly against suggestions posed by his uncle sice
he felt that his whole attitude to mathematics was so different from that of his uncle.
Instead Mandelbrot chose his own very different course which, however, brought him
back to Julia's paper in the 1970s after a path through many different sciences which
some characterise as highly individualistic or nomadic. In fact the decision by
Mandelbrot to make contributions to many different branches of science was a very
deliberate one taken at a young age. It is remarkable how he was able to fulfil this
ambition with such remarkable success in so many areas.

With the aid of computer graphics, Mandelbrot who then worked at IBM's Watson
Research Center, was able to show how Julia's work is a source of some of the most
beautiful fractals known today. To do this he had to develop not only new mathematical
ideas, but also he had to develop some of the first computer programs to print graphics.

You can see the Mandelbrot set

His work was first put elaborated in his book Les objets fractals, forn, hasard et
dimension (1975) and more fully in The fractal geometry of nature in 1982.


Mandelbrot has received numerous honours and prizes in recognition of his remarkable
achievements. For example, in 1985 Mandelbrot was awarded the Barnard Medal for
Meritorious Service to Science. The following year he received the Franklin Medal. In
1987 he was honoured with the Alexander von Humboldt Prize, receiving the Steinmetz
Medal in 1988 and many more awards including the Légion d'Honneur in 1989, the
Nevada Medal in 1991, the Wolf prize for physics in 1993 and the 2003 Japan Prize for
Science and Technology.

                     Lewis Fry Richardson

Born: 11 Oct 1881 in Newcastle upon Tyne, Northumberland, England
           Died: 30 Sept 1953 in Kilmun, Argyll, Scotland

Lewis Fry Richardson was born into a Quaker family. His mother was Catherine Fry
who came from a family of corn merchants, and his father was David Richardson who
came from a family of tanners, and he had gone himself into the family business. David
and Catherine Richardson had seven children, and Lewis was the youngest of this large
family. He attended Newcastle Preparatory School where his favourite subject was the
study of Euclid..

It was Richardson who was the first to apply mathematics, in particular the method of
finite differences, to predicting the weather in Weather Prediction by Numerical Process
(1922). He first developed his method of finite differences in order to solve differential
equations which arose in his work for the National Peat Industries concerning the flow of
water in peat. Having developed these methods by which he was able to obtain highly
accurate solutions, it was a natural step to apply the same methods to solve the problems
of the dynamics of the atmosphere which he encountered in his work for the
Meteorological Office. In this important treatise he used data from work by Vilhelm
Bjerknes published in Dynamical meteorology and hydrography and constructed, in his
own words:-

... a scheme of weather prediction which resembles the process by which the Nautical
Almanac is produced in so far as it is founded upon the differential equations and not
upon the partial recurrence of phenomena in their ensemble.

Making observations from weather stations would provide data which defined the initial
conditions, then the equations could be solved with these initial conditions and a
prediction of the weather could be made. It was a remarkable piece of work but in a sense
it was ahead of its time since the time taken for the necessary hand calculations in a pre-
computer age took so long that, even with many people working to solve the equations,
the solution would be found far too late to be useful to predict the weather. He calculated
himself that it would need 60,000 people involved in the calculations in order to have the
prediction of tomorrow's weather before the weather actually arrived. Despite this,
Richardson's work laid the foundations for present day weather forecasting.

In addition to his 1922 book, Richardson published about 30 papers on the mathematics
of the weather and in these he made contributions to the calculus and to the theory of
diffusion, in particular eddy-diffusion in the atmosphere. The 'Richardson number', a
fundamental quantity involving gradients of temperature and wind velocity is named after
him. His achievements were recognised by election to the Royal Society in 1926.

Another application of mathematics by Richardson was in his study of the causes of war
and he published the results of his analysis in a number of major books: Generalized
Foreign Politics (1939), Arms and Insecurity (1949), and Statistics of Deadly Quarrels
(1950). Again Richardson made novel applications of mathematics. Previously it had
been assumed that war was a rational national policy, to be used in the interests of a
nation. However the way that Richardson modelled the causes of war was quite different,
giving systems of differential equations which governed the interactions between
countries caused by such things as attitudes and moods. Here were quantities which were
little to do with individual leaders yet, he claimed, were major factors. Psychology of a
whole population was what was relevant, an underlying factor which emerged when
attitudes of individuals were averaged. As he wrote:-

The equations are merely a description of what people would do if they did not stop and

His first paper on this topic The mathematical psychology of war was written in 1919 and
privately printed at that time. It was not widely published until 1935. In fact before this
Richardson had returned to university study and obtained a B.Sc. in psychology as an
external University College, London, student in 1929.

He set up equations governing arms build-up by nations, taking into account factors such
as the expense of an arms race, grievances between states, ambitions of states, etc.
Choosing different values for the various parameters in the equation he then tried to
investigate when situations were stable and when they were unstable. It is clear that in all
this work he had no false illusions regarding its value for contributing to the prevention
of wars, yet he was clearly motivated by his strong hatred of war. Unlike some who
believe that war is part of the normal behaviour of nations, he clearly treated war as an
affliction from which the human race was suffering.

After he retired from Paisley College of Technology in 1940 Richardson began another
major piece of work related to wars. He gathered data on all "deadly quarrels" that had
taken place since the end of the Napoleonic Wars. He developed a magnitude scale for
such quarrels defined to be the logarithm of the number who were killed. He then
analysed a large number of factors associated with such "deadly quarrels" looking for
relations between them. Is there a relation between the frequency or wars and their
magnitude? Is there a relation between frequency and a common language for the two
sides? Is there a relation between frequency and a common religion? Is there one between
frequency and common frontiers?

He looked for factors which would reduce the frequency of wars. Do sporting fixtures
between nations reduce the frequency of war? Does strong military force on each side
reduce the chance of war? Does a common hatred of a third country reduce the chance of
war between two countries? It is a fascinating approach to try to understand war, yet none
of the factors Richardson investigated seemed statistically significant. The most
promising factor to prevent war between two countries appeared to be strong
international trade between them.











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