Mesopotamian Slides

Terminology and Geography

Not an ideal term: some mathematics predates the founding of the city of Babylon.
But most of the mathematical tablets have been found in or near Babylon.

"Mesopotamian" (Greek meso + potamos "river").
Modern day Iraq.

Mesopotamian civilization is based on two rivers: Tigris and Eurphrates.

Flooding less predictable than the Nile.
More draining and cooperation needed for farming.
Rivers changed course over time.
More turbulent history than Egypt.
Most buildings: mud brick (not wood or stone).

British Museum educational website on Mesopotamia.

Sketch of Mesopotamian History

Overview

Main Periods

1. The Sumerians and Early Writing

Organized into city states. No one city was most important.
See above map (not all the cities on the left map are Sumerian, some are later).

The language is unique: much different than the languages that came later in the region which are Semitic or Indo-European.

Writing in Sumeria started around 3200 BC.
About the same time, perhaps even earlier, than in Egypt.
So Sumerians may be the very first civilization with writing.
Among the earliest writings are numbers.
Numbers might have predated actual writing.

Early clay tablet from the city state of Uruk.
It has the old-style numbers.

Later stage: From 26th century BC.

Early Sumerian tablet with numbers.

Eventually writing evolved into Cuneiform ("Cue knee a form").
The pictographic nature lost. (Similar to Hieratic from hieroglyphics in Egypt.).

(Left example: Library of Congress, Washington DC)
(Middle examples: Amarna tablets were found in Egypt)

Note: According to Wikipedia the last cuneiform document dates to AD 75.
Note: the word cuneiform is build out of the Latin word cuneus "wedge".

Interesting art:
Ram in thicket from Ur:

Sculpture of a Summerian:

2. The Akkadian / Sumerian Culture

Some Akkadians, like the famous Sargon, ruled an empire over all of southern Mesopotamia (for 56 years ca. 2300 BC).

Akkadian was a Semitic language (related to Hebrew, Arabic, and Aramaic).
It gradually became the dominant language of the region.
Cuneiform was originally designed for Sumerian,
but when the dominant language gradually changed to Akkadian, cuneiform was adapted to Akkadian.
This cuneiform script had over 600 characters. [EB].
Later, the Babylonians and Assyrians also spoke forms of Akkadian,
and Babylonian mathematics was written in the Akkadian language.

Note: Akkadian has not been spoken for thousands of years: it was replaced by Aramaic.

Akkadians adopted the Sumerian culture.
Example: Epic of Gilgamesh, the world's first known Epic, was best preserved in the Akkadian language.
But the story is Sumerian in origin.

The last major Sumerian dynasty (following Sargon's Akkadian dynasty) was called Ur III.
Ur is famous for its ziggurat. Other Sumerian cities had them as well.
Not a pyramid. Not a tomb, but a temple platform.

Note: The word ziggurat ("zig gah rat") is derived from the Akkadian word ziqqurratu. It entered English in 1877.

Some mathematical tablets date from this period (c. 2100 BC).
Most date from the next period.

3. Old Babylonian Period

Babylon became a major city around 1900 BC.
The period from about 1900 BC to 1600 BC is called the Old Babylonian Period.
Note: Babylon remained a major city as late as the time of Alexandar the Great, over 1500 years after its founding.

Famous ruler Hammurabi (c. 1792 - c. 1750 BC)
Sixth King of Old Babylonian Period.
Hammurabi is famous for his empire and his (harsh) Code of Laws.
The inscription of 282 laws was found in Susa (Iran) 200 miles away. Perhaps a war trophy.
Hammurabi ruled much of Mesopotamia (now called Babylonia) with the capital at Babylon.

Temple of Marduk, the god of the city of Babylon, and Marduk himself.

This was the period of our largest collection of Mathematical texts.

Nabu, god of writing and science
His symbols are the clay tablet and the writing stylus.

4. Kassite Period

The horse was sacred to the Kassites; the Kassites might have introduced the horse to Babylonia. [EB]
Not much is known about this period (c. 1600 BC to c. 1100 BC).

5. Assyrians

The Assyrians spoke a dialect of Akkadian (like the Babylonians) and, later, the Aramaic language.

No known mathematics. They are more known for lion hunting than scholarly pursuits.
Here is the Assyrian King Ashurbanipal.
Sargon II was another ruler.

Famous for human headed lions and bulls (protective dieties). Winged bull.

Ninevah destroyed in 612 BC by an alliance (Medes, Chaldeans, etc.).

6. Neo-Babylonians

Some mathematics from this period, and the later periods.

Hanging Gardens, one of the Seven wonders of the ancient world.

Ishtar Gate built by Nebuchadnezzar II (ca. 575).

Persians captured Babylon in 539 BC, putting an end, for the last time, to the Babylonian empire.

Assyriology

Before the modern discoveries, there were fascinating tales by the Ancient Greek author Herodotus.
And much in the Bible.

Assyriology began at the shear cliffs of Behistun (Bisotun), Iran. 300 feet high.
Trilingual (all cuneiform) inscription: Old Persian, Elamite, Akkadian.
It records the deed of the famous Persian king Darius (c. 520 BC).
Henry Rawlinson (1810-1895) an Englishman interested in Persian antiquities first translated the Old Persian part (starting in 1837).
He an others translated the other parts (by 1857).
He and others risked lives to copy the writing from the cliff.

The Ziggurat of Ur was excavated in the 1920s and 1930s.

There are now a half-million tablets, with thousands mathematical.

Babylonian Mathematics

Our knowledge of Babylonian mathematics dates from 1935.
The scholars Otto Neugebauer and F. Thureau-Dangin are largely responsible for our knowledge.

There over a thousand mathematical tablets. About half are mathematical tables, and the rest are mathematical problems.

Numbers

Table Texts

Example: Multiplication by 9, multiplication by 7 and 1000. Table of inverses (or reciprocals).

Problem Texts

Example 1. When the area of a square is added to four times its width, a big unit (in other words 60) results. What square does this?

Solution. Square 2. Add this to the sum. Now you have 1, 4. Find the square root. This is 8. Now remove the 2. The answer is 6.

Example 2. The area is 7;30. The length and width add to 6; 30. Find the rectangle that does this.

Solution. Divide 6; 30 by 2. (This gives the average 3; 15). Square this to get 10; 33, 45. Subtract the area of the rectangle from this square, to get 3; 3, 45. Find the square root to get 1; 45. Add and subtract this from the average: answer 5 and 1; 30.

Example 3. How to see (a+b)² = a² + b² + 2 ab geometrically.

Accurate Square Roots

Pythagorean Theorem

Here is a drawing, with translation, and related (recreated and speculative) quantities

For example, the first row describes a 119, 120, 169 triangle. The first column (partly broken off) is probably 120² / 169² (or 1; 59, 0, 15), the second column is 119 (or 1, 59), and the third is 169 (or 2, 49).

There seems to be an error in the last row, which different translators fix in different ways. According to one source it describes a 28, 45, 53 triple. (The side 45 is not explicitly written. The first column is 53² / 45² which is 1 ; 23, 13, 46, 40 (or 1.38716 ... in decimal). Other sources describe it as a 56, 90, 106 triangle.

Geometry

Volume of boxes (rectangular parallelpipeds).
Truncated pyramid (frustrum), with a different formula than the Egyptians.
Probably prisms and cyllinders given accurate area computations.

Similar triangles, Pythagorean theorem, even Thales' theorem.

Pi

(The text from Susa says that the perimeter of an inscribed hexagon divided by the circumference is 0; 57, 36.
This fraction, in our notation, is 57/60 + 36/60^2 = 3456/3600 = 2^7 3^3 / 2^4 3^2 5^2
which simplifies to 24/25.
This leads to 25/8 for pi.)