The document traces the history of mathematics from ancient civilizations to the modern era. It discusses how ancient cultures developed numeration systems and arithmetic techniques to solve practical problems. It then covers the major developments in each historical period, including the advances made by Greek mathematicians like Euclid, the transmission of knowledge between cultures during the Islamic Golden Age, and the founding of calculus and other modern branches of mathematics. The history shows how mathematics has continually built upon previous discoveries and adapted to solve new problems over thousands of years.

1. PRESENTATION
RESMI.S
B.Ed, MATHEMATICS
NSS TRAINING COLLEGE,PANDALAM
REG NUM:13304012

2. • Ancient Period
•• Greek Period
• Hindu-Arabic Period
• Period of Transmission
• Early Modern Period
• Modern Period

3. Ancient Period (3000 B.C. to 260 A.D.)
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.

4. 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.

5. 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.

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

7. Greek Period (600 B.C. to 450 A.D.)
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 ffoorr tthheeiirr wwoorrkk..
B. Euclidean Geometry
The first mathematical system based on postulates, theorems and proofs
appears in Euclid's Elements.

8. 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.

9. 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
Proposition 47, the Pythagorean Theorem

10. Archimedes and the Crown
Eureka!

11. 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 wwoorrkkss,, pprriinncciippaallllyy iinn
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

12. 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.

13. 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

14. 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”).
He was also instrumental in promoting
the Hindu-arabic numeration system.

15. Evolution of Hindu-Arabic
Numerals

16. Period of Transmission (1000 AD – 1500 AD)
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.

17. 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.

18. The Abacists and Algorists
Compete
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.

19. 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.

20. 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.

21. 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.

22. Viète and Symbolic Algebra
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

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

24. 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).

25. Newton’s Principia – Kepler’s Laws
“Proved”
Isaac Newton
1642 - 1727
Newton’s Principia Mathematica
(1687) presented, in the style of
Euclid’s Elements, a mathematical
theory for celestial motions due to the
force of gravity. The laws of Kepler
were “proved” in the sense that they
followed logically from a set of basic
postulates.

26. 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.

27. Leibniz’s Calculus
Gottfied Leibniz
1646 - 1716
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.
A diagram from Leibniz's famous
1684 article in the journal Acta
eruditorum.

28. 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, p, i and Σ .
Leonhard Euler
1707 - 1783

29. 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

30. Current Branches of Mathematics
1. Foundations
• Logic Model Theory
• Computability Theory Recursion Theory
• Set Theory
• Category Theory
2. Algebra
• Group Theory
• Ring Theory
(includes elementary algebra)
4. Geometry Topology
• Euclidean Geometry
• Non-Euclidean Geometry
• Absolute Geometry
• Metric Geometry
• Projective Geometry
• Affine Geometry
• Discrete Geometry Graph Theory
• Differential Geometry
• Field Theory
• Module Theory
• Galois Theory
• Number Theory
• Combinatorics
• Algebraic Geometry
3. Mathematical Analysis
• Real Analysis Measure Theory
(includes elementary Calculus)
• Complex Analysis
• Tensor Vector Analysis
• Differential Integral Equations
• Numerical Analysis
• Functional Analysis Theory of Functions
• General Topology
• Algebraic Topology
5. Applied Mathematics
• Probability Theory
• Statistics
• Computer Science
• Mathematical Physics
• Game Theory
• Systems Control Theory