The document provides a high-level overview of the history and development of mathematics from ancient civilizations to modern times. It discusses how mathematics originated in ancient Mesopotamia, Egypt, Greece, China, and India, and was further developed during the Greek period with people like Euclid and Archimedes. It then discusses how mathematics progressed during the Hindu-Arabic period with the development of Hindu-Arabic numerals and their spread by Arabs. Key developments of algebra, trigonometry, and analytic geometry during the early modern period are also summarized.

2. Lesson 1:
A
Brief History
of
Mathematics
Ancient period
Greek period
Hindu-arabic period
Period of transmission
Early modern period
Modern period

3. The Mathematics that we know in the modern world has its roots in ancient
Mesopotamia, Egypt and Babylonia. Then it was developed in Greece, and
simultaneously in China and in India. This ancient Greek mathematics, along
with some influence of Hindu mathematics spread to the neighboring countries
in the Middle East. It was translated into Arabic and Latin and was adopted by
Western Europe. Western education was spread throughout the world by
colonization and trade. Today’s Mathematics has been enriched by the
contributions of different civilizations and individual mathematicians who
unselfishly passed on their discoveries and knowledge to us. It is therefore
fitting for us to look back and appreciate how Mathematics have developed
and who made these developments possible.
Overview/Introduction:

4. A. Number Systems and Arithmetic
• Development of numeration systems.
• Creation of arithmetic techniques, lookup tables, the abacus and other
calculation tools.
B. Practical Measurement, Geometry and Astronomy
• Measurement units devised to quantify distance, area, volume, and
time.
• Geometric reasoning used to measure distances indirectly.
• Calendars invented to predict seasons, astronomical events.
• Geometrical forms and patterns appear in art and architecture.
Ancient Period (3000 B.C. to 260 A.D.)

5. Practical Mathematics
As ancient civilizations developed, the
need for practical mathematics
increased. They required numeration
systems and arithmetic techniques for
trade, measurement strategies for
construction, and astronomical
calculations to track the seasons and
cosmic cycles.

6. Babylonian Numerals
The Babylonian Tablet Plimpton 322
This mathematical tablet was recovered from an unknown place in the Iraqi
desert. It was written originally sometime around 1800 BC. The tablet
presents a list of Pythagorean triples written in Babylonian numerals. This
numeration system uses only two symbols and a base of sixty.

7. Chinese Mathematics
Diagram from Chiu Chang
Suan Shu, an ancient Chinese
mathematical text from the
Han Dynasty (206 B.C. to A.D.
220).
This book consists of nine
chapters of mathematical
problems. Three involve
surveying and engineering
formulas, three are devoted to
problems of taxation and
bureaucratic administration,
and the remaining three to
specific computational
techniques. Demonstration of the Gou-Gu
(Pythagorean) Theorem

8. Calculating Devices
Chinese Wooden
Abacus
Roman Bronze
“Pocket” Abacus
Babylonian Marble
Counting Board
c. 300 B.C.

9. A. Greek Logic and Philosophy
• Greek philosophers promote logical, rational explanations of
natural phenomena.
• Schools of logic, science and mathematics are established.
• Mathematics is viewed as more than a tool to solve practical
problems; it is seen as a means to understand divine laws.
• Mathematicians achieve fame, are valued for their work.
B. Euclidean Geometry
• The first mathematical system based on postulates, theorems and
proofs appears in Euclid's Elements.
Greek Period (600 B.C. to 450 A.D.)

10. Area of Greek Influence
Archimedes of
Syracuse
Euclid and Ptolemy of
Alexandria
Pythagoras of
Crotona
Apollonius of
Perga
Eratosthenes of
Cyrene

11. Mathematics and Greek Philosophy
Greek philosophers viewed the universe in mathematical terms.
Plato described five elements that form the world and related
them to the five regular polyhedra.

12. Euclid’s Elements
Greek, c. 800 Arabic, c. 1250 Latin, c. 1120
French, c. 1564 English, c. 1570 Chinese, c. 1607
Translations of Euclid’s Elements of Gemetry:
The Pythagorean Theorem

13. The Conic Sections of Apollonius

14. Archimedes and the Crown
Eureka!

15. Archimedes Screw
Archimedes’ screw is a mechanical device used to lift water and such light
materials as grain or sand. To pump water from a river, for example, the
lower end is placed in the river and water rises up the spiral threads of the
screw as it is revolved.

16. Ptolemaic System
Ptolemy described an Earth-
centered solar system in his book
The Almagest.
The system fit well with the
Medieval world view, as shown
by this illustration of Dante.

17. Hindu-Arabian Period (200 B.C. to 1250 A.D. )
A. Development and Spread of Hindu-Arabic Numbers
• A numeration system using base 10, positional notation, the zero symbol and
powerful arithmetic techniques is developed by the Hindus, approx. 150 B.C. to
800 A.D..
• The Hindu numeration system is adopted by the Arabs and spread throughout
their sphere of influence (approx. 700 A.D. to 1250 A.D.).
B. Preservation of Greek Mathematics
• Arab scholars copied and studied Greek mathematical works, principally in
Baghdad.
C. Development of Algebra and Trigonometry
• Arab mathematicians find methods of solution for quadratic, cubic and higher
degree polynomial equations. The English word “algebra” is derived from the
title of an Arabic book describing these methods.
• Hindu trigonometry, especially sine tables, is improved and advanced by Arab
mathematicians

18. Area of Greek Influence

19. Baghdad and the House of Wisdom
About the middle of the ninth
century Bait Al-Hikma, the "House
of Wisdom" was founded in
Baghdad which combined the
functions of a library, academy, and
translation bureau.
Baghdad attracted scholars from
the Islamic world and became a
great center of learning.
Painting of ancient Baghdad

20. The Great Mosque of Cordoba
The Great Mosque, Cordoba
During the Middle Ages
Cordoba was the greatest
center of learning in Europe,
second only to Baghdad in the
Islamic world.

21. Arabic Translation of Apollonius’ Conic Sections

22. Arabic Translation of Ptolemy’s Almagest
Pages from a
13th century
Arabic edition
of Ptolemy’s
Almagest.

23. Islamic Astronomy and Science
Many of the sciences developed from
needs to fulfill the rituals and duties of
Muslim worship. Performing formal
prayers requires that a Muslim faces
Mecca. To find Mecca from any part of the
globe, Muslims invented the compass and
developed the sciences of geography and
geometry.
Prayer and fasting require knowing the
times of each duty. Because these times
are marked by astronomical phenomena,
the science of astronomy underwent a
major development.
Painting of astronomers at work in
the observatory of Istanbul

24. Al-Khwarizmi
Abu Abdullah Muhammad bin Musa al-
Khwarizmi, c. 800 A.D. was a Persian
mathematician, scientist, and author. He
worked in Baghdad and wrote all his
works in Arabic.
He developed the concept of an
algorithm in mathematics. The words
"algorithm" and "algorism" derive
ultimately from his name. His systematic
and logical approach to solving linear and
quadratic equations gave shape to the
discipline of algebra, a word that is
derived from the name of his book on the
subject, Hisab al-jabr wa al-muqabala
(“al-jabr” became “algebra”).

25. Evolution of Hindu-Arabic Numerals

26. A. Discovery of Greek and Hindu-Arab mathematics
• Greek mathematics texts are translated from Arabic into
Latin; Greek ideas about logic, geometrical reasoning,
and a rational view of the world are re-discovered.
• Arab works on algebra and trigonometry are also
translated into Latin and disseminated throughout
Europe.
B. Spread of the Hindu-Arabic numeration system
• Hindu-Arabic numerals slowly spread over Europe
• Pen and paper arithmetic algorithms based on Hindu-
Arabic numerals replace the use the abacus.
Period of Transmission (1000 AD – 1500 AD)

27. Leonardo of Pisa
From Leonardo of Pisa’s famous book Liber Abaci (1202 A.D.):
"These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1.
With these nine figures, and with this sign 0 which in Arabic is
called zephirum, any number can be written, as will be
demonstrated."

28. “Jealousy” Multiplication
Page from an anonymous Italian treatise
on arithmetic, 1478.
16th century Arab copy of an early
work using Indian numerals to show
multiplication. Top example is 3 x 64,
bottom is 543 x 342.

29. This woodblock engraving of a
competition between arithmetic
techniques is from from
Margarita Philosphica by
Gregorius Reich, (Freiburg, 1503).
Lady Arithmetic, standing in the
center, gives her judgment by
smiling on the arithmetician
working with Arabic numerals
and the zero.
The Abacists and Algorists Compete

30. Rediscovery of Greek Geometry
Luca Pacioli (1445 - 1514), a
Franciscan friar and
mathematician, stands at a
table filled with geometrical
tools (slate, chalk, compass,
dodecahedron model, etc.),
illustrating a theorem from
Euclid, while examining a
beautiful glass
rhombicuboctahedron half-
filled with water.

31. Pacioli and Leonardo Da Vinci
Luca Pacioli's 1509 book The Divine Proportion was illustrated by
Leonardo Da Vinci.
Shown here is a drawing of an icosidodecahedron and an "elevated"
form of it. For the elevated forms, each face is augmented with a
pyramid composed of equilateral triangles.

32. Early Modern Period (1450 A.D. – 1800 A.D.)
A. Trigonometry and Logarithms
• Publication of precise trigonometry tables, improvement of surveying methods using
trigonometry, and mathematical analysis of trigonometric relationships. (approx. 1530 –
1600)
• Logarithms introduced by Napier in 1614 as a calculation aid. This advances science in a
manner similar to the introduction of the computer
B. Symbolic Algebra and Analytic Geometry
• Development of symbolic algebra, principally by the French mathematicians Viete and
Descartes
• The cartesian coordinate system and analytic geometry developed by Rene Descartes
and Pierre Fermat (1630 – 1640)
C. Creation of the Calculus
• Calculus co-invented by Isaac Newton and Gottfried Leibniz. Major ideas of the calculus
expanded and refined by others, especially the Bernoulli family and Leonhard Euler.
(approx. 1660 – 1750).
• A powerful tool to solve scientific and engineering problems, it opened the door to a
scientific and mathematical revolution.

33. In his influential treatise In Artem
Analyticam Isagoge (Introduction
to the Analytic Art, published
in1591) Viète demonstrated the
value of symbols. He suggested
using letters as symbols for
quantities, both known and
unknown.
François Viète
1540-1603
Viète and Symbolic Algebra

34. The Conic Sections and Analytic Geometry
General Quadratic Relation
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
Parabola
-x2 + y = 0
Ellipse
4x2 + y2 - 9 = 0
Hyperbola
x2 – y2 – 4 = 0

35. Some Famous Curves
Fermat’s Spiral
r2 = a2 
Archimede’s Spiral
r = a
Trisectrix of Maclaurin
y2(a + x) = x2(3a - x)
Lemniscate of Bernoulli
(x2 + y2)2 = a2(x2 - y2)
Limacon of Pascal
r = b + 2acos()

36. Curves and Calculus: Common Problems
Find the area between curves.
Find the volume and
surface area of a
solid formed by
rotating a curve.
Find the length of a curve.
Find measures of a curve’s shape.

37. Napier’s Logarithms
In his Mirifici Logarithmorum Canonis
descriptio (1614) the Scottish
nobleman John Napier introduced
the concept of logarithms as an aid
to calculation.
John Napier
1550-1617

38. Henry Briggs and the Development of Logarithms
Napier’s concept of a logarithm is not
the one used today. Soon after
Napier’s book was published the
English mathematician Henry Briggs
collaborated with him to develop the
modern base 10 logarithm. Tables of
this logarithm and instructions for
their use were given in Briggs’ book
Arithmetica Logarithmica (1624). A
page from this work is shown on the
left.
Logarithms revolutionized scientific
calculations and effectively “doubled
the life of the astronomer”. (LaPlace)

39. Kepler and the Platonic Solids
Johannes Kepler
1571-1630
Kepler’s first attempt to
describe planetary orbits used a
model of nested regular
polyhedra (Platonic solids).

40. Newton’s Calculus
Newton developed the main
ideas of his calculus in private
as a young man. This research
was closely connected to his
studies in physics. Many years
later he published his results to
establish priority for himself as
inventor the calculus.
Newton’s Analysis Per Quantitatum
Series, Fluxiones, Ac Differentias,
1711, describes his calculus.

41. Leibniz’s Calculus
Leibniz and Newton independently
developed the calculus during the
same time period. Although
Newton’s version of the calculus led
him to his great discoveries, Leibniz’s
concepts and his style of notation
form the basis of modern calculus.
Gottfied Leibniz
1646 - 1716
A diagram from Leibniz's famous
1684 article in the journal Acta
eruditorum.

42. Leonhard Euler
Leonhard Euler was of the generation that
followed Newton and Leibniz. He made
contributions to almost every field of
mathematics and was the most prolific
mathematics writer of all time.
His trilogy, Introductio in analysin infinitorum,
Institutiones calculi differentialis, and Institutiones
calculi integralis made the function a central part
of calculus. Through these works, Euler had a
deep influence on the teaching of
mathematics. It has been said that all calculus
textbooks since 1748 are essentially copies of
Euler or copies of copies of Euler.
Euler’s writing standardized modern mathematics
notation with symbols such as:
f(x), e, , i and  .
Leonhard Euler
1707 - 1783

43. Modern Period (1800 A.D. – Present)
A. Non-Euclidean Geometry
• Gauss, Lobachevsky, Riemann and others develop alternatives to Euclidean geometry in the 19th century.
• The new geometries inspire modern theories of higher dimensional spaces, gravitation, space curvature and nuclear
physics.
B. Set Theory
• Cantor studies infinite sets and defines transfinite numbers
• Set theory used as a theoretical foundation for all of mathematics
C. Statistics and Probability
• Theories of probability and statistics are developed to solve numerous practical applications, such as weather
prediction, polls, medical studies etc.; they are also used as a basis for nuclear physics
D. Computers
• Development of electronic computer hardware and software solves many previously unsolvable problems; opens
new fields of mathematical research
E. Mathematics as a World-Wide Language
• The Hindu-Arabic numeration system and a common set of mathematical symbols are used and understood
throughout the world.
• Mathematics expands into many branches and is created and shared world-wide at an ever-expanding pace; it is
now too large to be mastered by a single mathematician

44. Non-Euclidean Geometry
In the 19th century Gauss, Lobachevsky, Riemann and other
mathematicians explored the possibility of alternative geometries
by modifying the 5th postulate of Euclid’s Elements.
This opened the door to greater abstraction in geometrical
thinking and expanded the ways in which scientists use
mathematics to model physical space.
Bernhard Riemann
1826 - 1866
Nikolai Lobachevsky
1792 - 1856
Carl Gauss
1777 - 1855

45. Pioneers of Statistics
In the early 20th century a
group of English
mathematicians and
scientists developed
statistical techniques that
formed the basis of
contemporary statistics.
William Gosset
1876 - 1937
Francis Galton
1822 - 1911
Karl Pearson
1857 - 1936
Ronald Fisher
1890- 1962

46. Gossett’s Student t Curve
Diagram from the ground breaking 1908 article “Probable
Error of the Mean” by Student (William S. Gossett).

47. ENIAC: First Electronic Computer
In 1946 John W.
Mauchly and J.
Presper Eckert Jr.
built ENIAC at the
University of
Pennsylvania.
It weighed 30
tons, contained
18,000 vacuum
tubes and could
do 100,000
calculations per
second.

48. Von Neumann and the Theory of Computing
John von Neumann with Robert Oppenheimer
in front of the computer built for the Institute
of Advanced Studies in Princeton, early 1950s.
Von Neumann
Architecture

49. Computer Generated Images
Equicontour Surface of a Random Function

50. Computer Generated Images
Evolution of a three dimensional cellular automata.

51. Check point 1. Historical Development
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