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1. History of mathematics

3. BIOGRAPHY
Though his exploits in the field of geometry,
science, and physics are widely famous, not
much is known about his personal life, as all
records have been lost. He was so much in
love with geometry and his inventions that
the last words he uttered were "Do not
disturb my circles."
He was killed in the Second Punic War by a
Roman soldier against the wishes of General
Marcellus. Plutarch writes that Archimedes
was contemplating a mathematical diagram
at the time of his death. His tomb was
engraved with the figure of a sphere and
cylinder as per his wish.

4. INVENTIONS
In the field of mathematics, Archimedes produced several
theorems that became widely known throughout the world.
He is credited with producing some of the principles of
calculus long before Newton and Leibniz. He worked out
ways of squaring the circle and computing areas of several
curved regions. His interest in mechanics is credited with
influencing his mathematical reasoning, which he used in
devising new mathematical theorems. He proved that the
surface area and volume of a sphere are two-thirds that of its
circumscribing cylinder.
He is credited with the invention of Archimedes screw or screw
pump, which is a device used to raise the level of water from
a lower area to a higher elevation. He is known for the
formulation of Archimedes' principle, a hydrostatic principle
stating that an object in any liquid is buoyed by force equal
to the weight of fluid it displaces. Legend has it that he
discovered the principle of buoyancy while taking bath and
following the discovery, he ran naked shouting "Eureka,
Eureka," meaning I have found it.

5. Using the method of exhaustion, he was able to
address irrational numbers, such as square roots
and Pi. He showed how to calculate areas and
tangents. His mastery of applied mathematics
reflects from his work on the Archimedes screw.
From his invention of war machines, such as
parabolic mirrors, Archimedes claw and death ray
and complex lever systems, shows that he played
an important role in guarding Syracuse against the
siege laid by Romans. Though he could not save
Syracuse from being captured by General Marcus
Claudius Marcellus and his Roman forces in 212
B.C., his war machines might have delayed the
capture. Archimedes himself was killed when the
city was captured by the Romans.
Undoubtedly, Archimedes was one of the most
brilliant minds of all times. His contributions in the
field of geometry, science, and physics truly
reflect his genius. He wrote many treatises, but
only a few would survive the Middle Ages. Still his
work and fame live on.

7. BACKGROUND
Charles Babbage was born in London, England December 26,
1791. Babbage suffered from many childhood illnesses,
which forced his family to send him to a clergy operated
school for special care.
Babbage had the advantage of a wealthy father that wished to
further his education. A stint at the Academy at Forty Hills
in Middlesex began the process and created the interest in
Mathematics. Babbage showed considerable talent in
Mathematics, but his disdain for the Classics meant that
more schooling and tutoring at home would be required
before Babbage would be ready for entry to Cambridge.
Babbage enjoyed reading many of the major works in
math and showed a solid understanding of what theories
and ideas had validity. As an undergraduate, Babbage
setup a society to critique the works of the French
mathematician, Lacroix, on the subject of differential and
integral calculus. Finding Lacroix's work a masterpiece and
showing the good sense to admit so, Babbage was asked
to setup a Analytical Society that was composed of
Cambridge undergraduates.

8. CONTRIBUTIONS
Written Works:
A Comparative View of the Various Institutions for the
Assurance of Lives (1826)
Table of Logarithms of the Natural Numbers from 1 to 108,
000 (1827)
Reflections on the Decline of Science in England (1830)
On the Economy of Machinery and Manufactures (1832)
Ninth Bridgewater Treatise (1837)
Passages from the Life of a Philosopher (1864)
Famous Quote:
"The whole of the developments and operations of analysis
are now capable of being executed by machinery. ... As
soon as an Analytical Engine exists, it will necessarily
guide the future course of science."
---Excerpt from the Life of a Philosopher

10. René Descartes was born on March 31, 1596 in a village in Touraine,
France, which is now called La Haye-Descartes. His mother died shortly
after he was born (thirteen months). About 1606, René entered the
Jesuit college of La Fleche, in which a relative of his, Father Charlet, a
theologian would watch out for him. Because of his delicate health,
Descartes was allowed to spend mornings in bed, meditating, reading,
and writing -a habit he maintained for most of his life.
He left La Fleche, because he was more confused about knowledge, and
he did not get his thirst for knowledge fulfilled. He then studied at the
University of Poitiers in 1615-16, earning a bachelor's degree and a
licentiate in law there.
At the age of twenty-two he left Paris and join the army of Prince Maurice
of Nassau. In the next year he was transferred to the army of
Maximailian. Duke of Bavaria. But in the night of November 10, 1619,
he had a series of three dreams, that he interpreted as a message from
God tell him to devote his life to the rational quest for certain truth.
After ending his voluntary military service he went back to Paris.

11. The social life in Paris was too distracting, so he moved to Holland in
1628. He lived in Holland until 1649. During this time, he avoided
reading any scholastic texts.
He wrote a couple of works in Holland, but when he heard of the
Inquisition condemning Galileo to death for his thoughts, as well as,
other thinkers, he decided to suppress his works.
In late 1604, Descartes' daughter, though he was never married, and father
died.
Descartes' philosophy became famous during the last decade of his life.
Descartes was later accused of heresy at the University of Leyden and
wrote a letter of self-defense to its trustees in 1647. He feared that he
might be arrested and killed, like Galileo, but that never happened.
In around 1648, Queen Christina of Sweden invited him to come to her
court to instruct her in philosophy. Despite his cautious reluctance,
Descartes accepted her invitation. She sent an admiral with a warship
to carry him to Sweden, and Descartes left for Stockholm in September
of 1649. This was the costliest mistake of his life.

13. German mathematician who is sometimes called the "prince of
mathematics." He was a prodigious child, at the age of three
informing his father of an arithmetical error in a complicated
payroll calculation and stating the correct answer. In school,
when his teacher gave the problem of summing the integers
from 1 to 100 (an arithmetic series ) to his students to keep
them busy, Gauss immediately wrote down the correct
answer 5050 on his slate. At age 19, Gauss demonstrated a
method for constructing a heptadecagon using only
a straightedge and compass which had eluded the Greeks.
(The explicit construction of the heptadecagon was
accomplished around 1800 by Erchinger.) Gauss also showed
that only regular polygons of a certain number of sides
could be in that manner (a heptagon, for example, could
not be constructed.) Gauss proved the fundamental theorem
of algebra, which states that every polynomial has a root
of the form a+bi. In fact, he gave four different proofs, the
first of which appeared in his dissertation. In 1801, he proved
the fundamental theorem of arithmetic, which states that
every natural number can be represented as
the product of primes in only one way. At age 24, Gauss
published one of the most brilliant achievements in
mathematics, Disquisitiones Arithmeticae (1801). In it, Gauss
systematized the study ofnumber theory (properties of
the integers ). Gauss proved that every number is the sum of
at most three triangular numbers and developed
the algebra of congruences.

14. n 1801, Gauss developed the method of least squares fitting, 10 years
before Legendre, but did not publish it. The method enabled him to calculate
the orbit of the asteroid Ceres, which had been discovered by Piazzi from
only three observations. However, after his independent
discovery, Legendre accused Gauss of plagiarism. Gauss published his
monumental treatise on celestial mechanics Theoria Motus in 1806. He became
interested in the compass through surveying and developed the magnetometer
and, with Wilhelm Weber measured the intensity of magnetic forces.
With Weber, he also built the first successful telegraph. Gauss is reported to
have said "There have been only three epoch-making
mathematicians: Archimedes, Newton and Eisenstein" (Boyer 1968, p. 553). Most
historians are puzzled by the inclusion of Eisenstein in the same class as the
other two. There is also a story that in 1807 he was interrupted in the middle of
a problem and told that his wife was dying. He is purported to have said, "Tell
her to wait a moment 'til I'm through" (Asimov 1972, p. 280). Gauss arrived at
important results on the parallel postulate, but failed to publish them. Credit
for the discovery of non-Euclidean geometry therefore went to Janos
Bolyai and Lobachevsky. However, he did publish his seminal work
on differential geometry in Disquisitiones circa superticies
curvas. The Gaussian curvature (or "second" curvature) is named for him. He
also discovered the Cauchy integral theorem

16. G. W. Leibniz was one of the most important thinkers of his time. His
contributions to such diverse fields as philosophy, linguistics, and history are
undeniable. And yet although he became acquainted quite late in his life with the
mathematical achieveme nts of his generation, it will always be his innovations in
this field that put him to the forefront of the enlightened thinkers of his era.
These achievements are especially remarkable considering that Leibniz often
treated the subject as a corollary to his studies in other fields, notably logic,
philosophy, and even law. It was precisely for this reason that Leibniz had so
much success in the field, in that he was unhampered by much of the dogma that
might have hindered its progress. Leibniz viewed the subject through his own
lens, interpreting the mathematical issues differently from his colleagues. Perhaps
it was the distance from which he viewed the field that allowed Leibniz to be
such an innovator in the rapidly changing subject. He gathered and pr ocessed as
much contem-porary mathematics as possible, reassessed it, and through his
innovative system of notation, repackaged it as a superior product. It was
Leibniz's algebraic symbolism that freed the subject from much of its rigid verbal
structure, allowing it to develop at an even faster rate. Leibniz's modern
mathematical notation probably represents his greatest single contribution to
mathematics. G.W. Leibniz is generally considered, along with Isaac Newton, as a
cofounder of the differential and integral Calculus.
INTRODUCTION

17. Mathematicians had developed algebraic methods for finding areas and volumes of a great variety
geometric figures. This marks one of the greatest developments in m athematics since the Greeks beg
using limits to approximate areas and then find the value of p. It was Cavalieri (1598-1647) who firs
introduced the concept of "indivisible magnitudes" in his Geometry of Indivisibles to study areas und
curves of the form:
y = xn (n(1)
At roughly the same time Descartes published his La Giomitrie, in which he showed, somewhat obsc
how to use Viite's algebra to describe curves and obtain an algebraic analysis of geometric problems
319-331] The work of these two mathematici ans would have an especially great influence on the
development of Leibniz's new calculus. [4, X]

19. Newton, Sir Isaac (1642-1727), English
natural philosopher, generally regarded
as the most original and influential
theorist in the history of science. In
addition to his invention of the
infinitesimal calculus and a new theory
of light and color, Newton
transformed the structure of physical
science with his three laws of motion
and the law of universal gravitation. As
the keystone of the scientific
revolution of the 17th century,
Newton's work combined the
contributions of Copernicus, Kepler,
Galileo, Descartes, and others into a
new and powerful synthesis. Three
centuries later the resulting structure -
classical mechanics - continues to be a
useful but no less elegant monument to
his genius.

20. Life & Character - Isaac Newton was born prematurely on Christmas day
1642 (4 January 1643, New Style) in Woolsthorpe, a hamlet near
Grantham in Lincolnshire. The posthumous son of an illiterate yeoman
(also named Isaac), the fatherless infant was small enough at birth to fit
'into a quartpot.' When he was barely three years old Newton's
mother, Hanna (Ayscough), placed her first born with his grandmother
in order to remarry and raise a second family with Barnabas Smith, a
wealthy rector from nearby North Witham. Much has been made of
Newton's posthumous birth, his prolonged separation from his mother,
and his unrivaled hatred of his stepfather. Until Hanna returned to
Woolsthorpe in 1653 after the death of her second husband, Newton
was denied his mother's attention, a possible clue to his complex
character. Newton's childhood was anything but happy, and
throughout his life he verged on emotional collapse, occasionally
falling into violent and vindictive attacks against friend and foe alike.

21. Mathematics - The origin of Newton's interest in mathematics can be traced to his
undergraduate days at Cambridge. Here Newton became acquainted with a number of
contemporary works, including an edition of Descartes Géométrie, John Wallis' Arithmetica
infinitorum, and other works by prominent mathematicians. But between 1664 and his return
to Cambridge after the plague, Newton made fundamental contributions to analytic
geometry, algebra, and calculus. Specifically, he discovered the binomial theorem, new
methods for expansion of infinite series, and his 'direct and inverse method of fluxions.' As the
term implies, fluxional calculus is a method for treating changing or flowing quantities. Hence,
a 'fluxion' represents the rate of change of a 'fluent'--a continuously changing or flowing
quantity, such as distance, area, or length. In essence, fluxions were the first words in a new
language of physics.
Scientific Achievements

22. Pythagoras of Samos was an Ionian
(Greek) philosopher and founder of the
religious movement called
Pythagoreanism. He is often revered as
a great mathematician, mystic and
scientist; however some have
questioned the scope of his
contributions to mathematics or natural
philosophy. We do know that
Pythagoras and his students believed
that everything was related to
mathematics and that numbers were
the ultimate reality and, through
mathematics, everything could be
predicted and measured in rhythmic
patterns or cycles. The Pythagoreans
were musicians as well as
mathematicians. Pythagoras wanted to
improve the music of his day, which he
believed was not harmonious enough
and was too hectic.

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