1 Introduction
Our first knowledge
of mankindís use of mathematics comes from the
Egyptians and Babylonians. Both civilizations developed mathematics that was similar in scope but different in
particulars. There can be no denying the fact that
the totality of their mathematics
was profoundly elementary2, but their astronomy of later
times did achieve a level comparable to the Greeks.
2 Basic Facts
The Babylonian
civilization has its roots dating to 4000BCE with the Sumerians
in Mesopotamia. Yet
little is known about
the Sumerians. Sumer was first
settled between 4500 and 4000 BC by a non-Semitic people who did not
speak the Sumerian language.
These people now are called
Ubaidians, for the village Al-Ubaid, where their remains were first
uncovered. Even less
is known about
their mathematics. Of the
little that is
known, the Sumerians
of the Mesopotamian
valley built homes
and temples and
decorated them with
artistic pottery and mosaics in geometric patterns. The Ubaidians were the first civilizing
force in
the region. They
drained marshes for
agriculture, developed trade
and established industries including weaving, leatherwork, metalwork,
masonry, and pottery.
The people called
Sumerians, whose language
prevailed in the territory, probably came from around Anatolia, probably
arriving in Sumer about 3300 BC. For a brief chronological outline of Mesopotamia see
http://www.gatewaystobabylon.com/introduction/briefchonology.htm.
See also
http://www.wsu.edu:8080/òdee/MESO/TIMELINE.HTM
for more detailed information. The early Sumerians did have writing for
numbers as shown below. Owing to the
scarcity of resources, the Sumerians adapted the ubiquitous clay in the region developing a writing that
required the use of a stylus to carve
into a soft clay tablet. It predated
the
cuneiform (wedge)
pattern of writing that the Sumerians had developed during the fourth millennium. It probably antedates the Egyptian
hieroglyphic may have been the earliest form of written communication. The Babylonians, and
other cultures including the
Assyrians, and Hittites, inherited Sumerian law and
literature and importantly their style of writing. Here we focus on the later period of the
Mesopotamian civilization which engulfed the Sumerian civilization. The Mesopotamian civilizations are often
called Babylonian, though this is not correct.
Actually, Babylon was
not the first great
city, though the whole civilization is
called Babylonian. Babylon, even
during its existence, was not always The first reference to the Babylon site of
a temple occurs in about 2200 BCE. The name means ìgate of God.î It
became an independent city-state
in 1894 BCE and Babylonia was the surrounding area. Its location is about 56 miles south of modern
Baghdad. Babylonian Mathematic the
center of Mesopotamian culture. The region, at least that between the two
rivers, the Tigris and the Euphrates, is also called Chaldea. The dates
of the Mesopotamian
civilizations date from
2000-600 BCE. Somewhat
earlier we see the unification of
local principates by powerful
leaders ó not unlike that in China. One
of the most powerful was Sargon the Great (ca. 2276-2221 BC). Under his rule the
region was forged into an empire called
the dynasty of Akkad and the Akkadian language began to replace Sumerian. Vast public works, such as irrigation canals and embankment fortifications,
were completed about this time. These were needed because of
the nature of the geography combined
with the need to feed a large population.
Because the Trigris and Euphrates would flood in heavy rains and the
clay soil was not very absorptive, such constructions were necessary if a large
civilization was to flourish.
Later
in about 2218
BCE tribesmen from
the eastern hills,
the Gutians, overthrew Akkadian rule
giving rise to the 3rd Dynasty of Ur. They
ruled much of
Mesopotamia. However, this
dynasty was soon to
perish by the
influx of Elamites
from the north,
which eventually destroyed the
city of Ur in about 2000 BC. These tribes took command of all the ancient
cities and mixed with the local people.
No city gained overall control until Hammurabi of Babylon (reigned about
1792-1750 BCE) united the country for a few years toward the end of his reign.
The
Babylonian ìtextsî come
to us in
the form of
clay tablets, usually about the
size of a hand. They were inscribed in
cuneiform, a wedge-shaped writing owing its appearance to the stylus that was
used to make it.
Two types of
mathematical tablets are
generally found, table-texts and
problem texts . Table-texts are just
that, tables of values for some purpose, such as multiplication tables, weights
and measures tables, reciprocal tables, and the like. Many of the table texts are clearly ìschool
textsî, written by apprentice scribes.
The second class of tablets are concerned with the solutions or methods
of solution to algebraic or geometrical problems. Some tables contain up to two hundred
problems, of gradual increasing difficulty.
No doubt, the role of the teacher was significant.
Babylon fell to Cyrus of Persia in 538 BC,
but the city was spared.
The great empire
was finished. However, another period of
Babylonian mathematical history
occurred in about
300BCE, when the Seleucids, successors of Alexander the
Great came into command.
The 300 year period has
furnished a great
number of astronomical
records which are remarkably
mathematical ó comparable
to Ptolemyís Almagest. Mathematical texts though are rare
from this period. This points to the
acuity and survival of the mathematical texts from the old-Babylonian period
(about 1800 to 1600 BCE), and it is the old period we will focus on.
The use of cuneiform script formed a
strong bond. Laws, tax accounts, stories, school lessons, personal letters were
impressed on soft clay tablets and then were baked in the hot sun or in
ovens. From one region, the site of
ancient Nippur, there have been recovered some 50,000 tablets. Many university libraries have
large collections of
cuneiform tablets. The largest
collections from the
Nippur excavations, for
ex- ample, are to
be found at
Philadelphia, Jena, and
Istanbul. All total, at
least 500,000 tablets
have been recovered
to date. Even
still, it is estimated that the vast bulk of existing
tablets is still buried in the ruins of ancient cities.
Deciphering cuneiform
succeeded the Egyptian
hieroglyphic. Indeed, just as for hieroglyphics, the key to deciphering
was a trilingual inscription found by
a British office,
Henry Rawlinson (1810-1895), stationed as an advisor to the
Shah. In 516 BCE Darius the Great, who
reigned in 522-486 BCE, caused a lasting monument to his rule to be engraved in
bas relief on a 100 150 foot surface on a rock cliff, the
ìMountain of the Godsî at Behistun at
the foot of the Zagros Mountains in the Kermanshah region of modern Iran along
the road between modern Hamadan (Iran)
and Baghdad, near
the town of
Bisotun. In antiquity, the name of the village was Bagast‚na, which
means ëplace where the gods dwellí.
Like the Rosetta stone, it was inscribed
in three languages ó Old Persian, Elamite, and Akkadian (Babylonian). However, all three were then unknown.
Only because Old Persian
has only 43 signs
and had been the subject of
serious investigation since the beginning of the cen- tury was the deciphering
possible. Progress was very slow. Rawlinson was able to correctly assign
correct values to 246 characters, and more- over, he
discovered that the
same sign could
stand for different
consonantal sounds, depending
on the vowel
that followed. (polyphony ) It has
only been in the 20th century that substantial
publications have appeared. Rawlinson published the completed translation
and grammar in 1846-1851. He
was eventually knighted
and served in
parliament (1858, 1865-68). For more details on this inscription, see
the article by Jona Lendering
at http://www.livius.org/be-bm/behistun/behistun01.html. A translation is included.
3 Babylonian Numbers
In mathematics, the
Babylonians (Sumerians) were somewhat more advanced than the Egyptians. ï
Their mathematical notation was positional but sexagesimal.
According to some
sources, the actual
events described in the monument
took place between 522 and 520 BCE. also spelled
Bistoun.
·
They used no zero.
·
More general fractions, though not
all fractions, were admitted.
·
They could extract square roots.
They could solve linear systems.
·
They worked with Pythagorean
triples.
·
They solved cubic equations with
the help of tables.
·
They studied circular measurement.
·
Their geometry was sometimes
incorrect.
For enumeration the Babylonians used
symbols for 1, 10, 60, 600, 3,600,
36,000, and 216,000,
similar to the
earlier period. Below are four of the symbols. They did arithmetic in base 60, sexagesimal.
The story is a little more complicated. A few shortcuts or abbreviation were allowed,
many originating in the
Seleucid period.
There is no clear
reason why the Babylonians selected the sexagesimal system. It was possibly
selected in the interest of metrology, this according to Theon of Alexandria, a
commentator of the fourth century A.D.:
i.e. the values 2,3,5,10,12,15,20, and 30 all divide 60. Remnants still exist today with time and
angular measurement. However, a number
of theories have been posited for the Babylonians choosing the base of 60. For example7
1.
The number
of days, 360,
in a year
gave rise to the
subdivision of the circle into
360 degrees, and that the chord of one sixth of a circle is equal to the radius gave rise to a
natural division of the circle into six equal parts. This in turn made 60 a natural unit of
counting. (Moritz Cantor, 1880)
2.
The Babylonians
used a 12
hour clock, with
60 minute hours. That
is, two of
our minutes is
one minute for
the Babylonians. (Lehmann-Haupt, 1889) Moreover, the
(Mesopotamian) zodiac was divided
into twelve equal sectors of 30 degrees each.
3.
The base 60 provided a convenient
way to express fractions from a variety of systems as may be needed in
conversion of weights and measures.
In the Egyptian
system, we have seen
the values 1/1, 1/2, 2/3, 1, 2, .
. . , 10. Combining we see the factor of 6 needed in
the denominator of fractions. This with
the base 10 gives 60 as the base of the new system. (Neugebauer, 1927)
4.
The number 60 is the product of
the number of planets (5 known at the time) by the number of months in the
year, 12. (D. J. Boorstin, Recall, the
very early use of the sexagesimal system in China. There may well be a connection. See Georges
Ifrah, The Universal History of Numbers, Wiley, New York, 2000. (1986)
5. The combination of the duodecimal system (base 12) and the base10
system leads naturally to a base 60 system.
Moreover, duodecmal systems have their remnants even today where we
count some commodities such as
eggs by the dozen.
The English system
of fluid measurement has numerous base twelve values. As we
see in the charts
below, the base
twelve (base 3, 6?)
and base
two graduations are mixed.
Similar values exist in the ancient Roman, Sumerian, and Assyrian measurements.
|
|
teaspoon
|
tablespoon
|
Fluid ounce
|
|
1 teaspoon
|
1
|
1/3
|
1/6
|
|
1 teablespoon
|
3
|
1
|
½
|
|
Fluid ounce
|
6
|
2
|
1
|
|
1 gill
|
24
|
8
|
4
|
|
1 cup
|
48
|
16
|
8
|
|
1 pint
|
96
|
32
|
16
|
|
1 quart
|
192
|
64
|
32
|
|
1 gallon
|
768
|
256
|
128
|
|
1 firkin
|
6912
|
2304
|
1152
|
Note that missing in the first column of
the liquid/dry measurement table is the
important cooking measure 1/4 cup, which equals 12 teaspoons.
6.
The explanations above have the
common factor of attempting to give
a plausibility argument
based on some
particular aspect of their society. Having witnessed various systems evolve in
modern times, we are tempted to conjecture that a certain arbitrariness maybe
at work. To create or impose a number
system and make it apply to an entire
civilization must have
been the work
of a political system of great power and
centralization. (We need only consider the
failed American attempt to go metric beginning in 1971. See http://lamar.colostate.edu/
hillger/dates.htm) The decision to adapt the base may have been may been made
by a ruler with little more than
the advice merchants
or generals with
some vested need. Alternatively, with
the consolidation of
power in Sumeria,
there may have been competing systems of measurement. Perhaps, the base
60 was chosen as a compromise.
4 Babylon Algebra
In Greek mathematics there is a clear
distinction between the geometric and algebraic. Overwhelmingly, the Greeks assumed a
geometric position wherever possible.
Only in the later work of Diophantus do we see algebraic methods of
significance. On the other hand, the
Babylonians assumed just as definitely, an algebraic viewpoint. They allowed opera- tions that were forbidden
in Greek mathematics and even later until th
th 16 century of our own
era. For example, they would freely
multiply areas and lengths, demonstrating that the units were of less
importance. Their methods of
designating unknowns, however,
does invoke units. First, mathematical expression was
strictly rhetorical, symbolism would not come for another two millenia with
Diophantus, and then not sig- nificantly until Vieta in the 16th century.
For example, the designation of the unknown was length. The designation of the square of the un-
known was area.
In solving linear
systems of two
dimensions, the unknowns were
length and breadth, and length, breadth, and width for three dimensions.
5 Pythagorean Triples.
As we have seen
there is solid evidence that the ancient Chinese were aware of the Pythagorean
theorem, even though they may not have had anything near to a proof. The Babylonians, too, had such an awareness.
Indeed, the evidence here is very much stronger, for an entire tablet of
Pythagoreantriples has been discovered. The events surrounding them reads much
like a modern detective story, with the sleuth being archaeologist Otto
Neugebauer. We begin in about 1945
with the Plimpton 322 tablet, which is now the Babylonian collection at Yale
University, and dates from
about 1700 BCE.
It appears to
have the left
section broken away. Indeed, the
presence of glue
on the broken
edge indi- cates that
it was broken
after excavation. What the tablet
contains is fifteen rows of
numbers, numbered from 1 to 15. Below we
list a few of them in decimal form. The first column is descending numerically.
The deciphering of what they mean is due mainly to Otto Neugebauer in about
1945.
6 Babylonian Geometry
Circular Measurement.
We find that
the Babylonians used
π = 3 for practical
computation. But, in 1936 at Susa
(captured by Alexander the Great in 331 BCE), a number of tablets with
significant geometric results were unearthed.
One tablet compares the areas and the squares of the sides of the
regular polygons of three to seven sides.
For example, there is the approximation
Volumes. There are two forms for the volume of a
frustum given
The second is correct, the first is not.
There are many
geometric problems in the cuneiform
texts. For example, the Babylonians were aware that
·
The altitude of an isosceles
triangle bisects the base.
·
An angle inscribed in a semicircle
is a right angle. (Thales)
Source: http://www.math.tamu.edu/~dallen/masters/egypt_babylon/babylon.pdf
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