Fresh Fish 8/FreshFishVol8-CD1.bin/new/misc/math/makers/makers

HyperBook | 1994-05-16 | 648.3 KB | 2,146 lines

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"Title"
The Makers of Mathematics
Part 1
"Riemann.Note"
Georg F. B. Riemann
(1826 - 1866)
"LaGrange.Note"
Joseph Louis Lagrange
(1736 - 1813)
"MacLaurin.Note"
Colin Maclaurin
(1704 - 1746)
"Gauss.Note"
Johann Friederich Carl
Gauss
(1777 - 1855)
"Agnesi.Note"
Maria Agnesi
(1718 - 1799)
"Newton.Note"
Sir Isaac Newton
(1642 - 1727)
Note8
Preface
"Newton"
    RAM DISK:
Newton
"Gauss"
    RAM DISK:
Gauss
    "Riemann"
    Carmen#1:
riemann1
"MacLaurin"
    RAM DISK:
    Maclaurin
"LaGrange"
    Carmen#1:
    Lagrange1
Note9
How to
Picture6
    Carmen#1:
Archimedes.new
"Archimedes.Note"
Archimedes
(287 BC - 212 BC)
"Agnesi"
    Carmen#1:
Agnesi1
Note11
Table of Contents
"CoverPage"
Note1
Copyright Notice
Copyright
 1993 by Carmen Q. Artino. All rights reserved. No
part of this
Hyperbook
 may be reproduced in any form or by any
means nor may this book be disassembled in any manner without the
expressed written consent of the author.
Note2
Table of Contents
"Copyright"
Note1
Preface
Note2
Cover Page
Note3
This Table of Contents
presents a list of the
Makers
 in this book. Use
the arrows on the bottom
right hand side to scroll
the list. The items in the
list are active so that
clicking on them will take
you to a brief account of
the life of the person and
his or her contribution to
mathematics. The
 Basic
Facts
 button presents the
birth and death dates of
Maker
Note4
Basic Facts
Note5
Table of Contents
Note6
The Makers
"Contents"
Maria Agnesi
Archimedes
Isaac Barrow
Jakob Bernoulli
Johann Bernoulli
Augustine-Louis Cauchy
Bonaventura Cavalieri
 Descartes
Leonhard Euler
Pierre de Fermat
Joseph Fourier
Johann Carl Gauss
David Hilbert
Hypatia
Sonia Kowalewski
Note7
Bibliography
Note8
How to use
Note9
Copyright
"Table of Contents"
Group1
Button1
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"Agnesi"
    Carmen#1:
Agnesi1
Note1
Maria Agnesi
Note1
The first great female mathematician in western
culture was most likely
Maria Gaetana Agnesi
. She
was born in Milan, Italy in 1718 and was a very
precocious child. By age 4 she was studying
foreign languages and by age 9 she had mastered
Latin, Greek, Hebrew, French, German, and Spanish
besides her native Italian. Her father, a
mathematics professor at the University of
Bologna, had her tutored by some of the most
distinguished professors in Europe; he also
exploited her brilliance by establishing
afternoon get-togethers at which learned men
would sit in a circle and listen to her lecture
on some topic, usually in Latin. She would, of
course, respond to questions in the native tongue
of the questioner. When she was 10, she published
her first book which advocated the education of
Note2
Table of Contents
Note3
girls and women. Her second book was a
collection of 190 of the lectures she
gave those afternoons; it was published
Group2
Note1
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"Agnesi"
Note1
when she was 21. By age 20, though, she began retreating into the world of
mathematics in rebellion against being put on display by her father. Her
work in mathematics was quite imaginative and influential. The fact that she
was the eldest of 21 children and hence expected to help raise them lead
Note2
her to write the book,
Instituzioni analitiche ad us
della gioventu italiana
This was the first text on
mathematics for teenagers and young adults and was
written in Italian rather than Latin to make it more
accessible to this audience. The book was a massive
two-volume set of some 1070 pages and was a
masterfully written integrated treatment of algebra,
trigonometry, the conic sections, and curve sketching.
The main portion of the text delt with the newly
emerging differential and integral calculus. The book,
published in 1748 when Maria was 30, immediately won
world-wide acclaim and was translated into English. It
Note3
Table of Contents
Note4
is in this book that we find a discussion of the curve that, through an
error in translation, became known as the "
Witch of Agnesi
" in English.
Note5
By age 40, she lost interest in mathematics
and turned to charitable work. She was 80
years old when she died.
Picture1
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"Witch"
In parametric form, the "witch" is
 x =  2a tan(t)
 y =  2a cos
(t) for -
/2 < t <
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Note1
Drawing1
"Agnesi, II"
Note1
Table of Contents
Note2
It is probable that no mathematical historian would
doubt that
Archimedes
 was the greatest mathematician
in antiquity and one of the three or four greatest
mathematicians of all time! He was born in Syracuse,
Sicily circa 287 B.C. Most of what is known about him
comes to us from the Greek biographer
 Plutarch
devoted a few pages to Archimedes in his account of
the Roman general,
Marcellus
Much of Archimedes work is so modern in spirit that,
except for the notation, it is practically
indistinguishable from the mathematical work of the
seventeenth century.
 Among his many achievements,
Archimedes developed the general method of exhaustion
Note3
introduced about 200 years earlier by the Greek mathematician
Eudoxus
. Using
this method, Archimedes was able to find areas bounded by parabolas and find
volumes of cylinders, paraboloids, and segments of spheres. In addition, he
Note4
used this method to obtain a very good
approximation of
. He found it to lie
between 3
 and 3
. This was rather
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Note1
Archimedes
Button1
"Plutarch"
Plutarch, A.D. 46?-120?
A Greek scholar,
moralist, and biographer. Wrote
Lives.
Button2
"Marcellus"
Marcus Claudius Marcellus, 268?-208
B.C., A Roman general in the Second
Punic War.
Button3
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Note1
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Button4
    "Eudoxus"
c. 408 - 355 B.C. The Greatest of
the Greek classical mathematicians;
second only to Archimedes. Founded
a school at Cyzicus in Asia Minor.
"Archimedes"
Note1
Table of Contents
Note2
remarkable considering the greek numbering system.
Using the method of exhaustion, Archimedes came very
close to discovering the calculus two thousand years
before
Newton
. In 1906, a manuscript was discovered in a
library in Constantinople. This manuscript, copied by a
tenth-century scribe, contains several works of
Archimedes. In particular, it contains part of a letter
he wrote to his friend
 Eratosthenes
 in which he describes
Note3
how he used the method and reveals how he thought. He says, "
I do believe
that men of my time and the future, and through this method, might find
still other theorems which have not yet come to my mind.
Archimedes other contributions include the creation of the discipline of
hydrostatics which he used to find equilibrium positions of various floating
bodies ("Eureka, Eureka!"). He set down the fundamental principles of
mechanics which he used to compute centers of gravity of various objects.
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Note4
Of his discovery of the mathematical
laws of the lever, he is said to
declared,
Button1
"Eratosthenes"
Eratosthenes, 276?-195? B.C.
Greek astronomer and geographer.
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"Archimedes, II"
Note1
Table of Contents
Note2
Give me a place to stand and I will move the earth
Archimedes engineering genius came to fore during the second Punic War when
Syracuse was constantly being attacked by Marcellus. Archimedes' military
inventions held off the Roman invaders for more than three years. The Roman
army was eventually victorious and Archimedes was killed by a Roman soldier
against the specific orders of Marcellus; he was 75 years old at the time
of his death.
Archimedes believed that his greatest discovery was the following theorem:
If a right circular cylinder is circumscribed about a sphere, then the area
of the sphere is two-thirds the area of the cylinder and the volume of the
sphere is two-thirds the volume of the cylinder
. He desired to have this
theorem engraved on his tombstone. Some three hundred years after
Archimedes death, the Roman statesman
Cicero
 hunted for his grave and
actually found the tombstone with the theorem inscribed upon it.
Button1
"Cicero"
Marcus Tullius Cicero 106-43 B.C.
Roman orator, author, and
statesman.
Button2
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"Archimedes, III"
Note1
Table of Contents
Note2
Isaac Barrow
 was born in England in 1630 and so
predated
Isaac Newton
 by 12 years. He was a
precocious child but unlike many of the other
akers
, he was quite rebellious driving both his
parents and his teachers to distraction. It is
said that his father would pray to God that if any
of his children had to die (infant mortality was
quite high at the time) that Isaac could be most
easily spared.
Barrow survived and went on to study at Trinity
College where he later stayed on as a scholar. At
the rather young age of 33, he was made first
Lucasian professor of mathematics at Cambridge.
The appointment was due to his royalist leanings
and the restoration of
 Charles II
to the throne of
England. This was an endowed chair with almost no
Note3
duties. He left this position after six
years. Rumor had it that he resigned so
that Newton might fill the professorship;
it was also rumored that
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Note1
Isaac Barrow
Button1
"CharlesII"
Charles II, 1630-1685. King of England
from 1660 to 1685.
"right"
Note1
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"Barrow"
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Note1
Newton was his pupil and that Barrow recognized
Newton's superior genius. However, there is no
evidence to support either rumor although Barrow and
Newton did interact on several occasions and he may
have stimulated Newton's interest in the calculus.
Although his work was not totally original, Barrow
published some of the basic concepts of the calculus
including a general method for finding tangent lines
(derivatives) and a crude version of the fundamental
theorem of calculus. But he had no theory of limits
Note2
and never realized the inverse relationship between differentiation and
integration.
After his resignation at Cambridge, Barrow had hoped for another
appointment but to a different position. He actually considered himself to
be a theologian rather than a mathematician. Within a year, he was
appointed royal chaplain in London.
Note3
Table of Contents
Note4
Barrow died at age 47 in 1677; his death was
said to have been caused by a drug overdose.
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Note1
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"Barrow, II"
Note1
Table of Contents
Note2
Johann and Jakob
Bernoulli
 were two of a
dozen mathematicians of
that name that stretched
over several generations;
they were members of one
of the most amazing
families of scientists
and mathematicians all of
whom were outstanding.
The two brothers shown
here are the most famous.
Jakob is the older of the
two having been born in
1654 and was self-taught
in mathematics. Johann
was born in 1667.
Note3
When he started working in mathematics, Jakob
knew nothing about the work of
Newton
Leibniz
. He eventually became aware of Newton's
work but because Leibniz published
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Jakob Bernoulli
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Johann Bernoulli
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"Bernoulli"
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Note2
so little, Jakob duplicated much of Leibniz' results. In
fact, both Jakob and Johann corresponded for many years
with Leibniz about calculus and made many discoveries
with which we are familiar.
Johann was first to introduce integration by partial
fractions. He discovered a relation between the
logarithmic and trigonometric functions that lead to the
notion that there are only two basic types of elementary
functions: polynomial, rational, and algebraic functions
on the one hand and the elementary transcendental
functions on the other. Jakob was the first to publish
the polar coordinate system even though Newton had the
basic idea somewhat earlier. He also studied the
catenary or hyperbolic cosine curve.
The brothers did have a mutual passion for critizing
each others work which often erupted in nasty confront-
Note3
ations. Leibniz
attempted to
mediate some of
Group1
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Note1
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"Bernoulli, II"
Note1
the disputes but because Jakob resented Leibniz' superior intellect, he
accused him of siding with Johann. Thus Leibniz often became entangled in the
disputes even though it was not his intention to do so.
Note2
Table of Contents
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Note3
The brothers often worked on common problems; for example, they
both studied arc length, curvature, and points of inflection.
Johann, interested in fame, often used unethical means to make
himself appear as the originator of an idea so that it is often
difficult to determine which of the two actually deserves credit
for a result.
Note4
After Johann and Jakob, the most famous Bernoulli was probably
Daniel
 (1700-1782) who was Johann's son. Like his father and
uncle, he also did work in calculus but his most impressive work
was in physics where he established a basic law in fluid dynamics
now known as Bernoulli's Law. He also did award winning work in
vibrating strings and the kinetic theory of gases.
Group1
Note1
Drawing1
"Bernoulli, III"
Note1
integrals that formed the basis for modern
complex function theory. He was also the first
to define the integral as a limit of sums and
not as the inverse of differentiation
Note2
Augustine-Louis Cauchy
 was born in 1789 during the
years of the French revolution. His early
education was obtained from his father who was a
barrister and well-versed in the classics. Cauchy
entered college, at the L'Ecole Polytechnique, to
study engineering. The reasons that he eventually
took up the study of mathematics vary. Some say it
was due to ill health while others say it was due
to the influence of
Pierre Simon Laplace
Joseph Lagrange
. It seems that Laplace was
Cauchy's neighbor and that Laplace introduced him
to Lagrange.
Whatever the reasons, we can be thankful that he
did. His contributions to mathematics were
brilliant and the quantity was overwhelming. Some
of his major contributions include a treatise on
Note3
Table of Contents
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Augustine-Louis Cauchy
"Cauchy"
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Note1
He was the first to give a rigorous definition of the limit concept and
formally define the notion of a continous function. Cauchy was also the first
to define the derivative as a limit of difference quotients as it is done
today. In 1816, he wrote a paper on the propagation of waves in liquids which
won a prize from the French Academy; he also wrote a definitive work which
formed the basis of modern elasticity theory. In all, Cauchy wrote some 700
papers filling about 26 modern volumes. It is said that more theorems and
concepts are named after Cauchy than any other mathematician.
Note2
Cauchy became professor at the L'Ecole Polytechnique
where he had studied. One cannot get a clear picture
of his teaching because some writers praised it while
others claimed he rambled on and on. One report, from
a student, claims that once he spent an entire lecture
extracting the square root of 17 to ten decimal
places!
His personal life was just as unclear. It is known
that he was devoutly Catholic and there were reports
Note3
Table of Contents
Note4
that he once
tried to
convert
Gauss
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"Cauchy, II"
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Table of Contents
Note2
he returned to France but refused to accept a position at a university until
the government dropped its requirement that Cauchy sign an oath of loyalty.
In 1857, Cauchy died rather abruptly. It is said that he was talking to the
Archbishop of Paris and commented that, "
Men pass away, but their deeds
abide.
" With those words, Cauchy dropped dead on the spot.
Note3
to Catholicism. The Norwegian mathematician
 Neils Henrik Abel
described Cauchy as, "mad, infinitely Catholic, and bigoted".
On the other hand, Cauchy sponsored charitable work for unwed
mothers and criminals.
His loyalty to the Bourbon throne was unparalleled. In 1830
he left his wife and children to enter exile with
Charles X
he was given the title of Baron for his loyalty. Eventually,
Button1
"Abel"
Neils Henrik Abel, 1802-1829. Brilliant
Norwegian mathematician who died at a
young age.
Button2
"CharlesX"
Charles X, 1757-1836. King of
France, 1824-1830.
Group2
Note1
Drawing1
"Cauchy, III"
Note1
Table of Contents
Note2
Bonaventura Cavalieri
 was one of the men of
the future that
Archimedes
 spoke of. He was
born in 1598 and was a pupil of
Galileo
 and a
contemporary of
Kepler
. Cavalieri was urged by
Galileo to look into the problems of the
calculus dealing with its mathematical rigor.
He developed the thoughts of Galileo and
others of his time on the subject of
"indivisibles" into a geometrical method that
was a precursor of integration and an
extension of the work of Archimedes. He
published a book in 1635, containing his work
on the subject, entitled "
Geometria
Indivisibilibus Continuorum Nova quadam
Ratione Promota
" or Geometry Advanced thus far
by an Unknown Method, Indivisibles of
Continua. In this work, Cavalieri regards an
area as made up of an indefinite number of
Button1
    "Galileo"
Galileo Galilei, (1564 - 1642).
Brilliant Italian scientist and
mathematician.
Button2
"Kepler"
Johannes Kepler (1571 - 1630).
Dutch scientist and astronomer.
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Note1
Bonaventura
Cavalieri
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"Cavalieri"
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Note1
Table of Contents
Note2
equidistant parallel line segments and a volume as composed of an indefinite
number of parallel plane areas. It is these elements that Cavalieri referred
to as the indivisibles of area and volume, respectively. He never elaborated
on what indefinite meant but he did realize that indefinite meant really large
in number! He also stated that, for example, if the parallel line segements
making up a planar region were slid into other configurations, the total area
would remain the same; the same is true for the volume indivisibles. This
became known as Cavalieri's principle. This principle is
 illustrated in a
Note3
theorem proved in high school solid geometry books known
 Cavalieri's Theorem.
 Using his principle, Cavalieri
proved many other results; for example, he was able to
prove that the volume of a right circular cone is
volume of the circumscribed cylinder. Another of his
results is what we now refer to as the power rule for
indefinite integrals
Button1
"Cavalieri's Thm"
If two solids have equal altitudes and if sections made
by planes parallel to the bases at equal distances from
them always have a given ratio, then the volumes of these
two solids have this given ratio to each other.
Note5
Cavalieri's contemporaries criticized his indivisibles; he
answered them but had no justifying proofs. At times he
claimed that the principle was just a pragmatic device to
avoid the method of exhaustion. Despite the criticisms,
Note6
many mathematicians of the time, including Pascal
and Fermat, used the principle intensively.
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"Cavalieri, II"
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 Descartes
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Table of Contents
Note2
Rene Descartes
is sometimes considered to be
the grandfather of modern philosophy and only
incidentally a mathematician. He was born in
Touraine, France on March 31, 1596 into a
moderately wealthy family. His early school
years were spent at a Jesuit school in Anjou.
At age 16 he left there and at age 20
graduated from the University of Poitiers
with a degree in law. He then spent some time
in Paris were he met
 Father Marin Mersenne
He studied mathematics with Mersenne for
about a year but became restless. He then
spent about nine years as a soldier in
several armies most notably in the army of
Prince Maurice of Orange
. Throughout this
period, however, he continued to study
mathematics. Descartes began to take the
study of mathematics seriously at a time when
Note3
he was in the Netherlands and noticed a
mathematical problem posted on a billboard.
That he was able to solve the problem
Button1
"Mersenne"
Mathematician of the 16th and 17th
centuries. Primarily interested in
Number Theory. Mersenne primes are
named after him.
Button2
    "Maurice"
Maurice of Nassau, 1567 - 1625.
Prince of Orange. Dutch General and
son of William the Silent.
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"Descartes"
Note1
Table of Contents
Note2
convinced him that he did indeed have
ability in mathematics. In addition,
he became more and more interested in
intellectual matters and in 1628 he
moved to Holland to take advantage of
Note3
it, Descartes developed the basic
ideas of what we now call Analytic
Geometry. It is interesting to note
that this was Descartes' only book
on mathematics although he did write
Note4
the freer and quieter
intellectual life there.
Descartes' most famous work
Discourse on the Method
of Rightly Conducting the
Reason and Seeking Truth in
the Sciences
and was a
philosophical work of major
proportions.
 With this work
Descartes includes three
 included three app
Note5
appendices. Their intent was to give
illustrations of the
 Method.
 The first
two dealt with optics and meteorlogy.
The third appendix, however, was
called
La geometrie,
The Geometry
Note6
numerous letters in which
he communicated many math-
ematical ideas.
The Geometry
, coordinate
axes were implied but not
used and most readers
believe that the axes were
oblique; i.e., meeting at
an angle that is not right,
rather than perpendicular.
Picture1
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Descartes2
Note7
The ideas of slope, distance, and
angle did not appear and Descartes
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"Descartes, II"
Note1
did not even plot a curve from an equation! The only equation considered in
detail was the general second degree equation in two unknowns. Descartes gave
conditions under which the equation represented an ellipse, parabola, or
hyperbola. Many details were omitted so the whole work was quite difficult to
read; in fact, it was often difficult to see the relationship between the main
philosophical work and the appendix.
The method of establishing rules came to Descartes in a dream he had on
November 10, 1619 -- it was the method of mathematics. The proofs of
mathematics were based on its axioms which to Descartes were unimpeachable.
Mathematics provided for him the method of achieving certainties; he saw
clearly that the method of mathematics transcended its subject matter. To
quote him, "
It is a more powerful instrument of knowledge than any other that
has been bequeathed to us by human agency, as being the source of all others
In 1649, Descartes was invited to become the private tutor of
 Queen Christina
of Sweden. Tempted by the glamour of royalty, he was lured away from the
Netherlands. Descartes was somewhat sickly throughout his life and the cold,
Note2
Table of Contents
Note3
damp conditions in Sweden did not agree with
him. In 1650 he contracted pneumonia and died.
He was 54 years old.
Button1
"Christina"
Christina of Sweden, 1626 - 1689
Queen from 1632 to 1654
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"Descartes, III"
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Table of Contents
Note2
Perhaps the most prolific mathematician of all
time was
Leonhard Euler
 not only in
mathematics but in other areas as well. He was
the central figure in 18th century mathematics
and is on everyone's list of the top ten
mathematicians of all time.
Euler was born in Basel, Switzerland in 1707.
His father was a Calvinist minister and had
wanted Leonhard to study theology and enter
the ministry also. Euler entered the
university at Basel and by age 16 had
completed his bachelors degree. While there,
he had the good fortune to be tutored in
mathematics by
Johann Bernoulli
. While Euler's
father urged him to study theology, the lure
of mathematics proved to be too great and
Euler decided to pursue the subject seriously.
Note3
By age 18, Euler began publishing and at age
19, he won a prize from the French Academy
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Leonhard Euler
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The "Eu" here is pronounced OIL-er
and not YOU-ler as in Euclid
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of Sciences for research on the masting of ships. Through the influence of
Nicholas
 Daniel Bernoulli
, Euler obtained a position at the St. Petersburg
Academy in Russia in 1733. He spent several years there under the autocratic
government of the czar and did an amazing amount of research. In 1741,
Frederick the Great
 of Prussia asked him to join the Berlin Academy
Note3
of Sciences. Euler spent 25 years in Berlin but never
severed his connection with the St. Petersburg
Academy. He sent hundreds of papers to St. Petersburg
and advised the Academy there on many of its affairs.
In 1766
 Catherine the Great
 asked Euler to return to
Russia. Several years before, Euler's eyesight grew
weak and he was concerned that the harsh climate in
Russia would only make matters worse. His worse fears
were realized when shortly after he returned to Russia
he became totally blind. Nevertheless, his
mathematical and scientific output did not abate.
Euler had a phenomenal memory. He knew by heart all
Note4
the formulas of trigonometry and
analysis and the first 100 primes
together with their powers up to the
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"NicholasB."
1695-1726. There appears to have been two
Bernoullis with the name Nicholas. The one
named here was the son of Johann and the
brother of Daniel.
"FrederickII"
Also known as Frederick II, 1712 - 1786.
He was King of Prussia from 1740 to
1786.
"CatherineII"
Also known as Catherine II, 1729 -
1796. She was empress of Russia from
1762 to 1796.
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"Euler,II"
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sixth. He also knew several hundred poems including the entire
Aeneid.
ability to perform calculations in his head was nothing short of spectacular.
It is said that he solved problems in his head on lunar motion that baffled
Newton
As mentioned, Euler's productivity was phenomenal. His major mathematical
fields were analysis, calculus, differential equations, differential and
analytic geometry, number theory, and the calculus of variations. His
contributions in physics are much too numerous to list here. Suffice it to say
that he published in analytic mechanics, optics, the theory of heat, and
acoustics. In addition, he was also interested in chemistry, geography, and
cartography.
In mathematics, Euler wrote texts and papers on algebra, mathematical
analysis, differential geometry, and a two-volume work,
Introductio in
Analysin Infinitorum
which was a landmark text. The amount and quantity of his
work was tremendous. During most of his life he was publishing at the rate of
about 800 pages a year. The quality of his work is exemplified by the fact the
he won so many prizes for it that they became a regular addition to his
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"Aeneid"
Latin epic poem written by
Virgil
which recounts the adventures and
travels of
Aeneas
Note4
income! What is even more amazing is
that about four hundred of his research
papers were written after he was
totally
"Euler, III"
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was totally blind. His collected works, when finished, will fill about 80
large quarto volumes. No mathematician of the time could handle the resources
of algebra, geometry, and analysis so deftly as he; he was a skilled
technician and very inventive in his methodology. His name appears in all
branches of science and mathematics. There are Euler polynomials, Euler
constants, Euler equations, Euler lines, and Euler integrals.
 It has been
said that, "Euler wrote mathematics as effortlessly as most men breath".
Note2
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Note3
One might expect that Euler was so prolific at the
expense of other interests. Nothing could be
further from the truth. Euler was married and had
13(!) children. He was very attentive to his
family's needs. He instructed both his children
and his grandchildren. He even created scientific
games for them to play and spent evenings reading
the bible to them.
Euler died on September 7, 1783 at the age of 76.
He was surrounded by universal respect and could,
at the end of his life, count most of the math-
Note4
ematicians of Europe
as his pupils.
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Pierre de Fermat
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Note2
Pierre de Fermat
 was not a mathematician, at least not
professionally. He was born into a family of leather
merchants in 1601. He studied law in the city of
Toulouse and spent some time there as an advisor to its
parliament.
Fermat was an amateur mathematician in the truest sense
of the term: he was a lover of mathematics. Though he
could only devote his spare time to the subject, he
contributed first rate results. His major areas of
interest were number theory and what is now known as
calculus; he was a co-discoverer, with
Descartes
coordinate geometry and was one of the first to work on
the development of probability theory. He also worke
Note3
on problems in science.
 Since Fermat was reluctant to publish, most of what
is known about his mathematics comes to us from letters he wrote to his
mathematical acquaintences. Fermat was the first to produce new results in
the theory of numbers since
Diophantus
 and in all likelyhood is most known
Note4
for the result which carries the name,
Fermat's Last Theorem
" and states that there
are no solutions in positive integers of the
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"Diophantus"
Diophantus of Alexandria, c. A.D. 250.
Greek arithmetician. Diophantine
equations are named after him.
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"Fermat"
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equation x
 when n > 2. In the margin of his copy of the latin
translation of Diophantus'
Arithmetica
he wrote, "
... it is impossible to
separate a cube into [as the sum of] two cubes, two biquadrates (fourth
powers) into two biquadrates, or generally any power except a square into
two powers of the same exponent. I have discovered a truly marvelous proof
of this, which however the margin is not large enough to contain
." Fermat's
proof, if indeed it existed, was never found and thousands of
mathematician-hours have been spent trying to resolve this conjecture. In
fact, it has been said that more new mathematics has been discovered trying
to prove Fermat's claim than any other outstanding mathematical problem!
Note3
Besides number theory, Fermat made many contributions
to calculus. Although he lacked the limit notion, he
developed a method to differentiate polynomials, he
devised methods for finding the tangent lines to curves
in form y = f(x) and for finding maxima and minima. He
was able to determine volumes and centers of gravity
but never made the connection that integration and
differentiation are inverse processes.
 He thus lost out
Note4
on being acknowledged as a
creator of the calculus.
Fermat died in 1665.
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Joseph Fourier
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Note2
Born to a tailor in 1768,
Joseph Fourier
 did very
well as a student of mathematics. However, his
interests were in the military and his heart was
set on becoming an army officer. Because of his
family's status he was denied a commission and as
a result, he turned to the priesthood. Fourier
was a supporter of the French revolution and as
such was rewarded with an appointment to the
L'Ecole Polytechnique. He later resigned the
position to follow
 Napoleon
 on a military
campaign to Egypt. He was then appointed Governor
of Lower Egypt in 1798. In 1801, the British took
Egypt and Fourier returned to France. When he was
offered a professorship at the military school he
had attended, he readily accepted and devoted the
rest of his life to mathematics and science.
Like many of the mathematicians of the late 18th
Note3
and early 19th centuries, Fourier studied the
flow of heat in metal plates and rods. In 1807
he submitted a paper on heat conduction to the
French Academy of Science in Paris and
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"Napoleon"
Napoleon Bonaparte, 1769 - 1821. French
military leader and conquerer; emperor of
France from 1804 to 1815 as Napoleon I.
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although he received encouragement for the ideas he presented, the
paper
rejected because of its lack of rigor. The subject of heat conduction was
made the subject of a prize to be awarded in 1812. Fourier submitted a
revised paper in 1811 which won the prize but was still rejected for
publication because of its lack of mathematical rigor. Undaunted but somewhat
resentful he continued to work on heat conduction. In 1822, he published a
classic in mathematics,
Theorie Analytique de la chaleur
The Analytic Theory
of Heat. In 1824, he became secretary of the Academy and was able to have his
1811 paper published without change.
Note3
Many of Fourier's contributions to mathematics grew out
of his study of heat conduction. He discovered that
temperature must satisfy a partial differential equation
now known as the heat equation. He also discovered that
certain functions could be expressed as infinite sums of
sines and cosines. Such sums are now called
Fourier
Series
 in his honor.
Later in his life, Fourier developed the peculier notion
that high heat was the natural condition of the
 human
Note4
body and took to wearing heavy cloths at all
times. He died, it is said, "
thoroughly cooked
in 1830.
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"Quote"
It was said by Howard Eves in his
book,
An introduction to the History
of Mathematics.
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"Paper"
It is interesting to note that the
paper was reviewed by Laplace,
Lagrange, and Legendre.
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Johann Carl Gauss
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His full name is
Johann Friederich Carl Gauss
and he was born the son of a
mason
 in 1777. Most
mathematical historians would have difficulty
denying the fact that Gauss was one of the three
or four greatest mathematicians of all time and
among the top scientists as well. His
precociousness in mathematics began to show
itself when at age three he corrected a
computational error his father had made in his
payroll. It is said that he worked out the
rudiments of arithmetic before he could talk. A
teacher at his elementary school was struck by
Gauss' intelligence and eventually brought him
to the attention of
 Duke Carl Wilhelm
. The Duke
sent Gauss to a higher school and subsequently
to the University of G
ttingen. It was 1795 and
Gauss was 17 years old. The following year he
worked out the method of least squares and one
year later, at age 19, Gauss solved a 2000 year
Note3
old problem by demonstrating that a seventeen -
sided polygon is constructible using the usual
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"GaussName"
In his later life, he signed his
name,
Carl Freiderich Gauss.
"Ferdinand"
Carl Wilhelm Ferdinand, Duke of
Brunswick
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"FatherJob"
Gauss' father was also a
gardener and canal tender
"Gauss"
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Euclidean tools.
When Gauss turned twenty
he began a mathematical
diary which was to
become one of the most
important documents in
mathematical history. In
it he recorded many of
his mathematical and
scientific discoveries
that were never formally
published. The very
first entry records his
discovery concerning the
17-sided polygon. When
the diary became known
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to the mathematical community it was
determined that not all of his
discoveries were noted; however, any
one of them would have established
Gauss as a first rate researcher in a
Note4
he gave the first proof of what is
now known as the fundamental theorem
of algebra. He believed this result
Note5
variety of different
mathematical fields.
But what contributions
did Gauss make to
mathematics? The list is
so long that it would be
impossible to reproduce
them all. However, here
are some of them in no
particular order of
importance. We have
already mentioned the
method of least squares
and the constructibility
of the 17-sided polygon.
For his doctoral thesis,
"Gauss, II"
Note1
friend, Wolfgang Bolyai.
One of Gauss' finest mathematical masterpieces was (is!) the
Disquisitiones
Arithmeticae
 (Researches into Arithmetic)
and deals primarily with what we now
know as the theory of numbers. It is arranged into several sections; the first
three develop the theory of congruences. The fourth develops the theory of
quadratic residues and contains the first proof of
 the golden theorem
gem of arithmetic
; the result known as the
law of quadratic reciprocity
. This
result, which is too intricate to discuss here, also commanded Gauss'
attention throughout his lifetime. He gave six proofs of this result.  In  the
Note2
to be so important that over his lifetime he gave four
independent proofs. The last of these was made when Gauss
was seventy years old! He made a systematic study of the
distribution of errors in computations that lead to what is
now known as the
normal
Gaussian
 distribution. He
established, in general, the double periodicity of certain
elliptic functions. He developed the major details of
non-Euclidean geometry which were later independently
developed by Johann Bolyai, the son of Gauss' very good
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Note4
fifth section we are presented with the
theory of binary quadratic forms and
the sixth section contains various
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"Gauss, III"
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applications of the theory presented in the first five sections to special
cases. The seventh section is the capstone of the work. It contains a
discussion of the equation
 where
 is an integer. It brings together
arithmetic, algebra, and geometry into a perfect pattern. This equation is
the algebraic formulation of the geometric problem to construct a regular
polygon of
 sides.
Gauss by no means limited his great intellect to mathematics. His other
interests in science included physics and astronomy. He calculated the orbit
of the asteroid Ceres much to the amazement of the scientific community and
to the astonishment of certain philosophers of the time. He also invented the
heliotrope, a device for tracking the Sun; the magnetometer, a device for
measuring the intensity and direction of a magnetic force, and an electric
telegraph.
As mentioned earlier, there are so many contributions that Gauss made to
mathematics that it is impossible to list them all. Suffice it to say that he
has easily and justly earned the title, Prince of Mathematicians.
In his later years, Gauss suffered from an enlarged heart  and  shortness  of
Note3
breath. He eventually developed dropsy, an
abnormal accumulation of fluids in the tissues.
He died peacefully on February 23, 1855.
"Gauss, IV"
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David Hilbert
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David Hilbert
 was born in 1862 in K
nigsburg,
Germany and was to become one of the great
mathematicians of the early 20th century. His
mathematical education was obtained at the
University of K
nigsburg where he studied with some
of the great mathematical minds of the 19th
century. While there, Hilbert made many friends,
among them was
Herman Minkowski
. His first major
work in mathematics was stimulated by his
friendship with Minkowski and was in the relatively
new field of invariant theory. He worked in this
area of mathematics until about 1892. During the
period 1892-1895 Hilbert was promoted from
privatdozent
 to associate professor, got married,
and was promoted again to full professor. Then in
1895, he moved to G
ttingen and was made successor
to the famous
Heinrich Weber
Hilbert tended to concentrate his mathematical work
Note3
on one subject at a time. After completing his
work on invariant theory he became interested,
with his friend Minkowski, in a theory dealing
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Note4
Herman Minkowski, 1864-1909.
Developer of Geometric Number
Theory
Button2
Note5
A professor who got paid
only if students signed up
for his or her courses.
Button3
Note6
Heinrich Weber, 1842-1913.
Professor of mathematics at
nigsburg and G
ttingen.
    "Hilbert"
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with numbers. Specifically, he became
interested the theory of algebraic
numbers. His studies in the area lead to
the now classic work,
"The Theory of
Algebraic Number Fields
" commonly known
Note2
as the
"Zahlbericht
. In it, he
expressed a point of view on
mathematics that was to become
typical of his later work and
influence in mathematics:
abstraction, arithmetization,
and logical development of
concepts. These ideas were
prevelant in Hilbert's later
development of Geometry.
Note3
Hilbert wanted to resolve many of the
logical inadequacies and concealed
assumptions that were repleat in
Euclid's
Elements
In a small but rather
celebrated volume entitled,
 Grundlagen
der Geometrie
, (Foundations of Geometry)
Note4
Hilbert began with the notions of
point, line, and plane and stressed
that these notions were to be
undefined
 objects. In this way, he
was able to axiomatize geometry in
Note5
a way in which Euclid could
not. To stress the need for
abstraction, he once said of
the basic notions,
 "One must
at all times be able to
replace points, lines, and
planes by tables, chairs,
and beermugs"
In 1900 the International
Congress  of  Mathematicians
Note6
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was held in Paris and Hilbert gave
a talk entiled, "Mathematical
Problems". In it, he listed twenty-
"Hilbert, II"
Note1
three unsolved problems whose solutions he felt would play a major role in the
development of twentieth-century mathematics. These problems have since become
the focus of much of the mathematical work of this century and many
mathematicians have made their reputations by solving one or another of
Hilbert's problems.
Hilbert worked in many other areas in mathematics including integral
equations, differential equations, number theory, and mathematical physics.
Hilbert died in 1943.
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"Hilbert, III"
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Hypatia
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Hypatia
 is usually considered to be the
first of the great women mathematicians;
she is certainly the first mentioned in
most histories of mathematics. She was
probably born in Alexandria in 370 A.D.
and lived most of her life there much of
which is clouded in legend. Her father was
the Greek mathematician and astronomer,
Theon. He guided every aspect of her
education, physical as well as mental. He
devised a regimen of rigorous exercises
for the improvement of her body and taught
her the philosphy and mathematics of the
Greek civilization. It is said that she
possessed uncommon physical beauty and
extraordinary mathematical talent. As a
lecturer at the School of Alexandria, she
was immensely popular. Students came from
Europe, Asia, Africa and other parts of
the world to hear her lecture on the works
Diophantus
 and
Appollonius
Note2
See the entry under
Fermat
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"right"
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Note4
Appollonius of Perga, founder of
Mathematical Astronomy. Wrote a great
work on the Conic Sections c. 262 B.C.
to c. 190 B.C.
    "Hypatia"
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Very little is known for certain about Hypatia's work in mathematics because
no copies of anything she wrote exist today. It is believed that she was
inspired by Diophantus and as such she was thought to be an arithmetician. She
also wrote a book,
On the Conics of Appollonius
and commentaries on the works
of Ptolemy and Diophantus. Much of what is known about her comes to us from
letters she wrote to her students. Some of them contain instructions for
building scientific instruments. It is therefore probable that she lectured on
simple mechanics as well as mathematics and philosophy.
Hypatia was an ardent follower and devotee of pagan learning and so was in a
dangerous position in the Alexandria of her time. The new Christianity was
developing and spreading throughout the Roman empire and most Christians were
Note3
hostile to the pagan ideas. In 415 A.D., she incurred
the wrath of a fanatical Christian mob probably
inspired by Cyril, the patriarch of Alexandria who came
to power in 412 A.D. At the hands of this mob, she
suffered a rather gruesome death. The dramatic impact
of her death has caused many historians to mark that
year as the end of the era of ancient mathematics.
"Hypatia, II"
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Sonia Kowalewski
Note2
She was born
Sonia Corvin-Kroukowsky
in Russia on January 15, 1850
was the middle child in a family of
minor nobility having an older
sister and younger brother. Despite
several obstacles to her education,
she was destined to become the
leading woman mathematician of the
nineteenth century.
Kowalewski began her study of
mathematics at the age of fifteen
receiving her first lessons in
differential equations. By the time
she was seventeen, she had become
acutely aware of the very
restrictive and limited educational
opportunities open to women in
Russia. She also knew that higher
education was  more open to women in
Button1
Note3
Her last name is also spelled,
Krukovsky
 and we sometimes see her
first name written
Sonja
Sonya
. Her
married name is sometimes spelled,
Kovalevskaya
Kovalevsky
"Sonia"
Note1
in Europe but being an unmarried young women, travel abroad was out of the
question. Kowalewski then took the only option open to her: she married for
convenience. Her husband was a promising young geologist,
Vladimir Kowalewski
After a brief stay in Russia, the couple traveled to Germany where Sonia
matriculated at the University of Heidelberg. While there, she studied with
Leo K
nigsberger
, a former student of
 Karl Weierstrass
, one of the great
mathematicians of the nineteenth century. K
nigsberger eventually brought
Kowalewski to the attention of his former teacher who was professor of
mathematics at the University of Berlin. The University there was not,
however, as willing to admit women as was Heidelberg so Weierstrass taught
Kowalewski at her home during the following four years. The two became
lifelong friends even after her regular tutoring sessions with Weierstrass
ended.
At the end of this period, Kowalewski produced three outstanding research
papers. One of them, on partial differential equations, was a remarkable
contribution to the field. The three papers qualified her for a doctorate,
absentia
, in mathematics from the University of G
ttingen. The papers were so
well done that the University waived the oral exam when she petitioned that
her German was inadequate.
Note2
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During the ensuing years, Kowalewski
had a daughter and returned to Russia
but the political situation  became un-
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Note4
Little is known about him. He was born
in 1837 and was a student of
Weierstrass
Note5
After they arrived in Heidelberg, Vladimer left
after a few months to study elsewhere and
eventually earned a doctorate. After Sonia's father
died, she turned to her husband for consolation and
their marriage was finally consumated about eight
years after their marriage. Foufie, Sonia's
daughter was born in October of 1878.
Button2
"Sonia, II"
Note1
On February 10, 1891, Sonia Kowalewski died of
influenza at the age of forty-one. She is buried
in Stockholm.
Note2
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Note3
bearable for her so she returned to Berlin leaving her husband and his shady
business associates. Sometime later, in 1883, she received news that her
husband had committed suicide. She became quite despondent and for four days
she shut herself up refusing food and drink. On the fifth day she lost
consciousness and on the sixth day she had recovered enough to ask for pencil
and paper. She then proceeded to cover the paper with mathematics. By Fall,
she was well enough to attend the scientific congress at Odessa.
A former student of Weierstrass,
 Gosta Mittag-Leffler
, was to form the
mathematics department at the newly created University of Sweden. He had
heard of Kowalewski's work and was eager to attract the first great woman
mathematician. She accepted the position with great delight and by the Fall
of 1884 she was lecturing at the University and in 1889 she was appointed
professor for life.
Weierstass and Kowalewski remained friends throughout their lives
corresponding regularly mostly about mathematics but often times about other
matters as well. One of Weierstrass' great joys in his later life was the
recognition that Sonia finally received. On Christmas eve in 1888, the French
Academy of Sciences honored her in person with its Bordin Prize for her
paper,
On the rotation of a solid body about a fixed point
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Swedish mathematician, 1846 -
1927. Made important contributions
to complex analysis and founded
the journal,
Acta Mathematica
"Sonia, III"
"Intro"
inspiration for this Hyperbook was derived in part from my desire to
have some mathematical history available to students at "the touch of a
button". It is my belief that mathematics learned in the presence of its
history only enhances its richness and beauty. Brief biographies of some of
the great and near great mathematicians seemed to be a natural way to
achieve this. The Amiga 3000 computer and the
Hyperbook
 authoring system
made the task easy and fun to do.
There
 are more than likely several omissions, most notably, Galois and
Abel and others. On the other hand, there are several
Makers
 whose
 inclusion
some would question. I make no excuse for these
 perhaps
the contents will
change in a
future revision.
Note2
Cover Page
Note3
Table of Contents
Note4
Preface
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Note5
 biographies given here are
clearly not meant to be all
inclusive; further information on
    "Preface"
Note1
any of the
Makers
in this
Hyperbook
 may be obtained from the sources given in
the bibliography. Those sources consist of commonly known books on the
history of mathematics and mathematicians and are available in most college
libraries; they are meant to be a starting point and the list is far from
complete.
There
 are several people whose help I wish to acknowledge in the creation
of this
Hyperbook
; without their help, this project could not have been
completed. Thanks are due to
Marcia J. Felix
 of John Wiley & Sons for
providing me with some of the pictures and to
Cliff Mac Adams
 and
Brian
Ginett
 for creating the digitized images. Thanks are also due to my
colleague,
 Mary Ann McLoughlin
at the College of Saint Rose for providing me
with some of the source material on the great women mathematicians that
appear here.
Carmen Q. Artino
Department of Mathematics
The College of Saint Rose
Albany, NY 12203
May, 1993
Note2
Table of Contents
Note3
Cover Page
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"Preface, II"
Note1
This
HyperBook
 contains brief accounts of the lives of some famous
mathematicians. To use it, simply note that when the mouse pointer moves
over certain parts of the screen such as a button, note or other object,
it may change shape. This means that a "hot spot" has been encountered.
Click the left mouse button over such a spot to activate it. For example,
each picture on the cover page is hot. Try them and see. Also note that
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Bibliography
Note2
Bell, Eric T.,
Men of Mathematics
, New York, Simon and Schuster, 1937.
Boyer, Carl B. and Uta C. Merzbach,
A History of Mathematics
, Second Edition,
New York, John Wiley & Sons, Inc., 1991.
Kline, Morris,
Mathematical Thought from Ancient to Modern Times
, New York,
Oxford University Press, 1972.
Perl, Teri,
Math Equals. Biographies of Women Mathematicians
, Reading, MA,
Addison-Wesley Publishing Co., 1978.
"The Faces of Calculus", A poster published by John Wiley & Sons, Inc., 1991.
Note3
Table of Contents
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Cover Page
"Bibliography"
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Basic Facts
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Note3
Maria Gaetana Agnesi
1718 - 1799
Archimedes
c. 287 B.C. - 212 B.C.
Isaac Barrow
1630 - 1677
Jakob Bernoulli
1654 - 1705
Johann Bernoulli
1667 - 1748
Augustine Louis Cauchy
1789 - 1857
Bonaventura Cavalieri
1598 - 1647
Rene Descartes
1596 - 1650
Note4
Leonhard Euler
1707 - 1783
Pierre de Fermat
1601 - 1665
Joseph Fourier
1768 - 1830
Johann Carl Gauss
1777 - 1855
David Hilbert
1862 - 1943
Hypatia
370 - 415
Sonia Kowalewski
1850 - 1891
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"BasicFacts"
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Note8
Preface
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Cover Page
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Table of Contents
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More. . .
"Contents"
Archimedes
Maria Agnesi
Isaac Barrow
Jakob Bernoulli
Johann Bernoulli
Augustine-Louis Cauchy
Bonaventura Cavalieri
Gabriel Cramer
 Descartes
Leonhard Euler
Pierre de Fermat
Joseph Fourier
Johann Carl Gauss
Josiah Willard Gibbs
David Hilbert
Joseph Lagrange
Pierre Simon Laplace
Gottfried Leibniz
Marquis de L'Hopital
Colin Maclaurin
Sir Isaac Newton
Georg Bernard Riemann
George G. Stokes
Brook Taylor
John Von Neumann
John Wallis
Karl Weierstrass
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Bibliography
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