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Measuring the Gardens of Eden
A map of the area around Gasur, near Kirkuk in northern Iraq, drawn up
some time in the Sargonic period (2200 BCE). The central area, below
the Rahium river, is described as irrigated gardens belonging to Arala.
• Mesopotamia, Ancient Greek Μεσοποταµία, "between rivers” (Euphrates and Tigris)
• The Sumerian civilization appeared before 3500 BCE - cradle of civilization
• Cities, irrigation systems, a legal system, administration, and even a postal service.
The first writing script developed. Sumerians and Akkadians (including Assyrians
and Babylonians) dominated Mesopotamia until the fall of Babylon in 539 BCE,
when it was conquered by the Achaemenid (Old Persian) Empire.
First Work of Literature
•
The Epic of Gilgamesh originated with Sumerian poems
dating from the Third Dynasty of Ur (around 2100 BCE).
•
The 11th tablet describes the meeting of Gilgamesh with
Utnapishtim. Like Noah in the Hebrew Bible,
Utnapishtim had been forewarned of a plan by the gods
to send a great flood. He built a boat and loaded it with
all his possessions, his kith and kin, domesticated and
wild animals and skilled craftsmen of every kind.
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First Code of Laws
• King Hammurabi established the Old
Babylonian state. Mathematical
culture flourished in this period.
• The Code of Hammurabi (around 1750
BCE) contains 282 laws.
• According to tradition, Hammurabi
received the code from Shamash, the
patron Sun god of Babylon
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First Bureaucracy
• The highly centralized Sumerian and Accadian
states required large bureaucracies.
• “They recorded aspects of the quantitative
measurement of land, livestock, and labour, not
only to account for what had already been
acquired or produced but also to make shortterm predictions of costs and yields” - E. Robson
• Pedagogical curricula started to develop to train
scribes more effectively
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Cuneiform
• The cuneiform culture in Mesopotamia was coextensive with the
mathematical culture.
• Cuneiform was incised on virtually indestructible clay tablets.
• One of the earliest known systems of writing
• The original Sumerian script (4th millennium BCE) was adapted for
writing in Akkadian and other languages just like the Roman
alphabet is now used for many languages, not just Latin. Sumerian
was the first language in the world to be written down. It became
completely extinct around 1 BCE.
• By the 2nd century CE all knowledge of how to read cuneiform was
lost until its decipherment in the 19th century. About half a million
cuneiform tablets have been excavated. Only about 1/10 of them
have been read in modern time as there are only a few hundred
qualified cuneiformists in the world.
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In 1835, Henry Rawlinson, a British East India Company army officer,
visited the Behistun Inscriptions in Persia. Carved in the reign of King
Darius of Persia (522–486 BCE), they consisted of identical texts in the
three official languages of the empire: Old Persian, Babylonian, and
Elamite. The Behistun inscription was to the decipherment of cuneiform
what the Rosetta Stone was to the decipherment of Egyptian hieroglyphs.7
YBC 7302
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What’s on the tablet?
!
!
!
!
!
• YBC 7302 is from the Old Babylonian period (1700BCE)
• circular form indicates an exercise by a trainee scribe
• a vertical wedge = 1 and a corner wedge = 10
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Interpretation
• This tablet is an exercise in computing the
area of the circle (which is rather faint)
• 3 is the circumference.
• We usually visualize the circle as the area
generated by a rotating radius.
In ancient Mesopotamia, the circle was
visualized as the shape contained within a
circumference, no radius is drawn.
!
•
!
c2 c2
A=
≈
4π 12
• So 9 is just 32 (calculation on the side)
• But why is the answer 45 ???
• And where is division by 12 ???
10
Sexagesimal System
• A place-value number system with base 60 was introduced
around 2100 BCE
!
!
• Used now to measure angles, geographic coordinates, and time.
• For example, a GPS-style latitude N 51° 28' 38” of Greenwich is
sexagesimal for a decimal fraction
• 51+28/60+38/3600=51.4772222..
• Babylonians didn’t use a “sexagesimal point” or zero
!
may mean 45, but also 45/60 (=3/4), 45*60; etc.
•
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• Babylonian sexagesimal system was the first ever positional
(or place-value) system. In these systems the value of the digit
depends on its place. For example, in the decimal system,
one “2” in 22 means 2 and another means 20.
may mean 45, but
• Likewise, in the Babylonian system
also 45/60 (=3/4), 45*60; etc. To complicate matters (for us),
Babylonians didn’t use a “sexagesimal point” or zero.
• The first systems were sign-value systems. For example,
familiar Roman numerals is a sign-value system. MMXV=?
• Or Egyptian numerals
!
!
= ?
•
!
• What are the advantages of sign-value systems?
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• Sign-value systems were used in Mesopotamia as well.
Often, different symbols would be used for different
commodities, measures, or in different trades (similar
to imperial units).
• The sexagesimal system was used only for
calculations. When a result was to be inserted into a
contract or an account, Babylonian scribes would use
one of the sign-value systems.
• Are there analogues in the modern world when a
numerical system is used for calculations but not
revealed to the end-user?
13
What’s so nice about 60?
• 60 has a lot of factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
For example, one hour can be divided evenly into sections of
30, 20, 15, 12, 10, etc. minutes.
• Instead of dividing by a number, Babylonians would multiply by
its reciprocal (and shift an invisible sexagesimal point)
!
•
number
3
5
12
8
18
27
!reciprocal
!
20
12
5
7 30
3 20
2 13 20
• What would be an analogous calculation in the decimal system?
14
• Now we can solve the mystery of YBC 7302
• Instead of dividing 9 by 12, Babylonians would
multiply by a reciprocal: 9*5=45
• 45 is in fact not 45 but 45/60=3/4 !!
15
Some tablets contain land area
measurements, e.g. this field plan
(YBC 3900) drawn up in 2045
BCE. The reverse gives the total
area, minus the 'claimed land’. It
states that the land, belonging to
the temple of goddess Ninurra in
Umma, was measured 'at the
command of the king Amar-Suen.
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YBC 7289
• This tablet approximates the diagonal of a square. For this you
need a square root of 2
!1;24,51,10 = 1+ 24 + 51 + 10 = 30547 ≈ 1.41421296…
!
60 60 2 60 3 21600
• What’s the number in the upper-left corner and on the bottom line?
• 30 and 42;25,35 (which is 30 times √2).
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• 1.41421296… is very accurate. The right number is
1.41421356…
• This is more accurate than anything useful in practice why are the scribes so precise?
• In solving abstruse puzzles about measured space, the
true scribe demonstrated his or her technical capability
and moral fitness for upholding justice and maintaining
social and political stability - Robson
• Why couldn’t they compute π more accurately than 3?
• In my opinion, the main reason is that they knew a
recursive algorithm for computing √a with arbitrary
precision: start with any x and iterate x->(x+a/x)/2.
The first algorithm for computing π with arbitrary
precision was discovered only by Archimedes.
Plimpton 322
The clay tablet # 322 in the Plimpton Collection at Columbia University.
Written about 1800BCE, this tablet is a table with 4 columns and 15 rows.
It looks possible that other columns on the left were broken off.
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• Let’s translate it into decimal notation. With the first column,
we’ll make a guess that the number is between 1 and 2.
!
!
!
!
!
!
!
!
!
!
!
!
1.9834028
1.9491586
1.9188021
1.8862479
1.8150077
1.7851929
1.7199837
1.6845877
1.6426694
1.5861226
1.5625
1.4894168
1.4500174
1.4302388
1.3871605
119
3367
4601
12709
65
319
2291
799
481
4961
45
1679
161
1771
28
• What are these numbers?
169
4825
6649
18541
97
481
3541
1249
769
8161
75
2929
289
3229
53
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
20
Pythagorean triples theory
Pythagorean triples are triples of integers (s,l,d) which satisfy equation s2+l2=d2.
They give all possible integer sides of a right triangle. For example, (3,4,5) or (5,12,13).
(d/l)
1.9834028
1.9491586
1.9188021
1.8862479
1.8150077
1.7851929
1.7199837
1.6845877
1.6426694
1.5861226
1.5625
1.4894168
1.4500174
1.4302388
1.3871605
s
119
3367
4601
12709
65
319
2291
799
481
4961
45
1679
161
1771
28
d
169
4825
6649
18541
97
481
3541
1249
769
8161
75
2929
289
3229
53
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
l
120
3456
4800
13500
72
360
2700
960
600
6480
60
2400
240
2700
45
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“Brilliant mathematician” theory
Nowadays, we know that Pythagorean triples can be obtained as follows.
Take any coprime integers p>q>0, which are not both odd.
Then s=p2-q2, l=2pq, d=p2+q2 give all Pythagorean triples, and each
triple only once. It’s tempting to say that the scribe was a brilliant
mathematician who used this formula to make a list of Pythagorean triples.
p
q
(d/l)
12
5
64
s
d
#
l
1.9834028 119
169
1
120
27
1.9491586 3367
4825
2
3456
75
32
1.9188021 4601
6649
3
4800
125
54
1.8862479 12709
18541
4
13500
9
4
1.8150077 65
97
5
72
20
9
1.7851929 319
481
6
360
54
25
1.7199837 2291
3541
7
2700
32
15
1.6845877 799
1249
8
960
25
12
1.6426694 481
769
9
600
81
40
1.5861226 4961
8161
10
6480
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Reciprocal pairs theory
According to this theory, the table was produced based on the table of
reciprocals (x and 1/x) with x in the decreasing order. These tables were
very common because they were used to divide numbers. Starting with
x and 1/x from the table of reciprocals, the scribe could compute
s’=(x-1/x)/2, l’=1, d’=(x+1/x)/2 (a Pythagorean triple of rational numbers!),
and scale them up or down by common factors until they become integers.
x
1/x
(d/l)
s
d
#
l
2 24
25
1.9834028
119
169
1
120
2 22 13 20 25 18 45
1.9491586
3367
4825
2
3456
2 20 37 40 25 36
1.9188021
4601
6649
3
4800
2 18 53 20 25 55 12
1.8862479
12709
18541
4
13500
2 15
26 40
1.8150077
65
97
5
72
2 13 20
27
1.7851929
319
481
6
360
2 09 36
27 46 40
1.7199837
2291
3541
7
2700
2 08
28 07 30
1.6845877
799
1249
8
960
2 05
28 48
1.6426694
481
769
9
600
2 01 30
29 37 46 40 1.5861226
4961
8161
10
6480
23
• Babylonian scribes were routinely taught to solve
problems that nowadays we solve by setting a
quadratic equation.
• Our modern abstraction of a “quadratic equation”
appeared (much later) to teach students a standard
way to solve problems like this.
• One of these typical problems was (see, for example
YBC 6967): “A number exceeds its reciprocal by r.
Find the number and its reciprocal.”
• Babylonians would solve this geometrically.
One can speculate that problems of this sort have
geometric origins (designing plots of land).
24
Babylonian algorithm for finding a number and its
reciprocal which differ by a given number r
25
Suppose you are a teacher and you want to design a problem like that. You
can start with a known number and its reciprocal and find relevant geometric
attributes: the short side, the long side, and the area of the little square.
These numbers are in the rows of the Plimpton 322
(after rescaling by the common denominator)
26
• Another evidence is in the heading of the mystery column:
The holding square of the diagonal from which 1 is torn
out so that the short side comes up.
• So who wrote the tablet and why?
• We can speculate that he was a teacher preparing a test for
his or her students containing many problems of the sort
we discussed. He worked backwards: he started with a
known table of reciprocals and produced many numerical
problems to give to students.
• This is similar to how your calculus professor comes up
with problems to give you on the midterm.
• But a big question remains: did he even noticed all these
Pythagorean triples?
27
Further Reading
• E. Robson, Words and Pictures: New Light on
Plimpton 322 (linked from the website)
• E. Robson, Mathematics in Ancient Iraq
• D. Struik, A Concise History of Mathematics
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