History of Mathematics
Ancient Mathematics
Some thoughts on dating:
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History does not consist mainly of dates. It is
important to have a sense of how events fit
together. However the “landmarks” should not
overshadow the “landscape.”
The AD/BC convention proclaims the central
importance of an event that which is actually only
believed to be central by a small proportion of
mankind.
Ancient Ages
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Old Stone Age: 5,000,000 BC 10,000 BC
Middle Stone Age: 10,000 BC 7,000 BC
New Stone Age: 7,000 BC  3,500 BC
Bronze Age: 3500 BC  1400 BC
Some Early Dates: Stone Age
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2,400,000 BC
1,500,000 BC
750,000 BC
Hominoids in Africa manufacture crude stone tools
Homo Erectus (stone tools)
Ancient hearths found in caves near Marseilles indicate
that Homo Erectus uses fires
350,000 BC
Homo Sapiens moved to caves
100,000 BC
Earliest known ornament-an amulet made from
mammoth’s tooth by a Neanderthal. (Hungrary)
79,000 BC
Lamps fueled with animal fat, uses grass for wick.
30,000 BC
Tally Sticks; Beads, Bracelets and Pendants worn. Fire
ceramics found in Czechoslovakia (not used yet in making bowls)
25,000 BC
Artifacts with primitive geometrical designs; Venus
figurines; Music (cave paintings, footprints of dancers, carved bonesinstruments?)
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20,000 BC
Bow & Arrow (Spain), Boomerangs, Sewing
Needle (France), Tailored Clothes (Russia), Spear-thrower
13,000 BC
Calendars, Maps (Ukraine), Ropes
10,000 BC
Dogs (Mesopotamia), Goats & Sheep (Iran,
Afghanistan)
9,000 BC
Mayan astronomical inscriptions
8,000 BC
Potatoes & beans (Peru), Rice (Indo China),
Pumpkin (Latin America), Wheat & Barley (Israel)
7,000 BC
Clay tokens record number of animals,
measure grain in Mesopotamia
6,000 BC
Corn (Mexico)
5,000 BC
Earliest cities in Mesopotamia (carbon dating),
Llama & Alpacas (Peru)
4,241 BC
First Recorded Date (Mesopotamia)
Bronze Age: 3500 BC 1400 BC
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3500 BC
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3300 BC
2600 BC
3000 BC
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2700 BC
2400 BC
Bronze (Egyptians & Babylonians), Potters
Wheel, Papyrus (Egyptians), Wheeled
Vehicles, Wine (Turkestan), Beer
(Mesopotamia), Cattle (Thailand)
Egypt united
Great Pyramid at Giza
Eclipse Predicted; 365 Day Calendar,
Sailing ships (Egypt)
Silk worms in China
Chinese introduce method of taking
observations of the sky, still in use today
The Ancient World
Ishango Bone
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Every culture in the world creates some mathematics, though
it may be on a basic level. The most ancient evidence of
mathematical activity comes from a wolf’s bone on which 55
notches are carved, grouped in sets of 5. It was found in
Czech Republic and believed to be about 35,000 years old.
The Ishango bone (from a baboon) dates to around 18,000
BC, and is particularly intriguing since the notches are
grouped into sets of 11, 13, 17, and 19, suggesting an interest
in prime numbers.
Civilization
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(from civilis, “city-dweller” in Latin)
At this time, mankind obtained food by hunting animals and
gathering plants. About 10,000 years ago an unknown group of
people invented agriculture. This forced people to establish
permanent settlements so the plants could be cared for until
harvest: the first cities.
Since cities were originally farming communities, they were
established in regions that provided the 2 fundamental needs of
agriculture: fertile soil and a reliable water supply.
River flood-plains provided both, and most ancient civilizations
developed around them.
Ancient Egypt
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Civilization developed around the Nile river, whose annual
flooding deposited silt that fertilized the fields of Egyptian
farmers.
As early as 4000 BC, they may have noticed that it took 365
days from one Nile to the next.
The Egyptian calendar divided that year into 12 months of 360
days, with 5 extra days celebrated as the birthdays of the main
gods of the Egyptian pantheon (holidays or “holy days”).
Egypt is divided into two main parts, Lower Egypt and Upper
Egypt.
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Lower Egypt consists of the
marshy areas where the Nile
empties into the
Mediterranean, known as the
Delta (because of it’s
resemblance to the Greek
letter, pointing southwards);
the term would later be
applied to any river’s outlet
into a sea or lake.
According to tradition, Upper
and Lower Egypt were united
by Narmer (Menes in Greek)
around 3100 BC.
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The kings of Egypt were known as pharaohs (“Great House”).
They were revered as gods though this did not spare them from
being criticized, plotted against, and deposed.
The best known feature of Egyptian civilization are the pyramids.
Great Pyramid of Giza, finished around 2500 BC is the oldest and
is the only one of the Seven Wonders of the World still standing!
By 2700 BC, a form of writing had been invented. Since many
examples were found adorning the walls of Egyptian temples, it
was erroneously believed that the writings were religious in nature.
Hence this form of writing became known as hieroglyphic (“sacred
writing” in Greek).
By 2600 BC, a cursive form of hieroglyphics, called hieratic, was
developed ; it was suitable for writing on soft materials such as
papyrus, cloth, and leather.
Egyptian Mathematics
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Mathematics is oldest and most
continuous pursued of exact
sciences.
Aristotle in Metaphysics said
math began with Egyptian
priests .
Egypt mathematics was mostly
utilitarian but developed into
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Algebra: from techniques of
calculation
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Theoretical Geometry: land
measurement
All civilizations have developed special
symbols for numbers. The use of a
stroke to represent “one” is universal.
Napoleon’s invasion of Egypt
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Disastrous military campaign in 1798.
38000 soldiers sailed from Toulon in 328 ships to try to seize Egypt,
but 12 month campaign against English Admiral Nelson stopped
them.
Napoleon had 167 scholars to make comprehensive inquiry into life of
Egypt in ancient and modern times (including Fourier, Monge).
Two particular areas of study resulted in significant and long-lasting
discoveries or achievements. The savants traveled with the army and
the geography was charted and a map of Egypt was drawn
(completed in 1806) that remained classified until the end of
Napoleon's reign.
The great monuments were examined and the science of Egyptology
was founded.
One of the most important discoveries in that field was the Rosetta
Stone.
Mathematics on Expedition
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Monge (1746-1818) is considered the father of differential
geometry because of his work Application de l'analyse à la
géométrie where he introduced the concept of lines of
curvature of a surface in 3-space.
Fourier (1768-1830) studied the mathematical theory of heat
conduction- established the partial differential equation
governing heat diffusion and solved it by using infinite series
of trigonometric functions.
Napoleon
Monge
Fourier
Napoleon
Description de l'Egypte, first appeared in
1809; work of 160 scholars on Napolean’s
expedition.
Etat Moderne,Plate 87,
Views of Qait Bey Fortress and the Diamant Rock, drawn
c.1798, published in the Panckoucke edition of 1821.
Rosetta
Stone
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The Rosetta Stone is a block of basalt with engravings made on its
polished surface. It was named after the village where it was found in
1799, Rashid (known as Rosetta to Europeans) located a few miles
from the sea in the western delta of the Nile.
It measures 3'9" (114 cm) in height, 2'4-1/2" (72cm) in width and 11"
(28cm) in thickness. It weighs just under a ton (762kg). It is somewhat
damaged, missing a large part of the upper left-hand corner, and a
smaller part of its lower right corner.
The chiseled inscriptions are in two languages, Greek and Egyptian,
but three scripts. The first of the Egyptian scripts is Hieroglyphs, used
3,000 years ago at the time of the First Dynasty. The second was later
determined to be Demotic, a cursive language that evolved from
Hieroglyphs and dating from 643 B.C.
Napoleon order rubbings for Europeans and 4 casts for Oxford,
Cambridge, Edinburgh, Dublin.
Hieroglyphic
Hieratic
Ancient
Greek
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Jean-Francois Champollion, 1790-1832, born in
Lot, last of 7 children, raised in humble
circumstances; Met Fourier at 11; Egyptology
became his life.
He spoke several languages by age of 16; by
age of 20, he could was fluent in Latin, Greek,
Hebrew, Amharic, Sanskrit, Avestan, Pahlavi,
Arabic, Syriac, Chaldean, Persian and Ge'ez in
addition to his native French .
At age 17 became faculty at University of
Grenoble;
Father of Modern Egyptology. Finished
complete reading of upper panel of Rosetta
Stone in 1822.
Grave of Champollion (Paris)
The names of Cleopatra
and Ptolemy were the
first words deciphered by
Champollion.
Champollion's notes of
his study of the cartouche
of Cleopatra, inscribed on
an obelisk found at Philae
by Belzoni.
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By analyzing the texts of the Rosetta Stone
and comparing them with those on the
obelisk of Philae, Champollion had the
brilliant intuition that the names of the
pharaohs in cartouches were in hieroglyphs
with a phonetic value.
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It was therefore possible to establish an
equivalence between hieroglyphic and
alphabetic signs.
The Rhind Papyrus
Rhind (Ahmes) Papyrus: 2000-1800 BC
• The papyrus was bought by Rhind in Luxor
in 1858 and willed to the British Museum.
• Hieratic script from 1650BC by scribe Ahmes
• 18ft long and 13in high (center missing)
• US Egyptologist Edwin Smith uncovered
missing section and sent it in 1906 to NY
Historical Society and then in 1922 to British
Museum.
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The Rhind Papyrus
The text contains eighty-four problems
concerned with numerical operations,
practical problem-solving, and
geometrical shapes.
It claims to be a ``thorough study of all
things, insight into all that exists,
knowledge of all obscure secrets." In
fact, it is somewhat less.
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It is not a theoretical treatise, but a list of practical problems
encountered in administrative and building works.
It is a collection of exercises, substantially rhetorical in form,
designed primarily for students of mathematics. Included
are exercises in
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fractions
notation
arithmetic
algebra
geometry
mensuration
The practical mathematical tools for construction.
Early Egyptian Multiplication
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Egyptian mathematics was recorded and taught by means of
problems that were intended as examples to be imitated.
Unclear what the Egyptian mathematicians developed their
science beyond what was needed for everyday work (unlike
the Greeks). It seems remarkably uniform throughout its
long history. It was at all stages, built around the operation
of addition – so primitive. What do you think that the
Egyptians were preoccupied with?
Basic operation was adding and doubling.
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Multiply: doubling one number and then add the appropriate duplications to form the
product.
Division: divisor repeatedly doubled to give dividend.
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