# A cuneiform tablet in Pergamon Museum

This map is dated inthe 21st century BC and can be seen in the Pergamon Museum of Berlin. As you can see, there is a plane of some lands and the cuneiform figures tell us the dimensions of it. Of course, this is a small piece of clay rouded by the great Mesopotamian treasures that you can admire in this wonderful museum but… can we forget focussing our attention in this cuneiform tablet? I don’t think so!

**Location**: Pergamon Museum in Berlin (map)

# Mesopotamian Mathematics in the BM

This is the famous Royal Game of Ur (2600-2300 BC). This wooden game board was in at least six graves in the Royal Cemetery so it’s an early example of a game that was played all over the ancient Near East for about 3.000 years.

The game is a race for two players using dice with seven identical pieces each. All playing squares are decorated, but on later boards only the five ‘rosette’ squares are marked. […] Pieces are ‘at war’ along the central path but turn off to their own side to exit.

Playing pieces were discs of shell or lapis lazuli. The tetrahedrical dice of the game are also exhibited.

Apart of the Royal Game of Ur, the only exhibited objects which are related with Mespotamian mathematics are the Archaic and Cuneiform tablets. For example, look at this tablet containing a five day ration list (Jemdet Nasr, 3000-2900 BC):

Each line contains rations for one day and the sign for ‘day’ and numbers 1, 2, 3, 4 and 5 are easily identificable (at the beginning of the line!).

This Gypsum tablet with Archaic numbers (Uruk, 3300 BC) has 3 units (round impressions) and 3 ‘tens’ (elongated impressions).

This tablet above contains the daily barley beer ration for the workers (3300-3100 BC). Here there are also identificable all the marks representing units and tens and it’s the same in the next tablet containig food rations (3300-3100 BC):

Finally, there is another tablet from the Late Uruk Period (3300-3100 BC):

However, mathematical tables are not only clay tablets with figures and numbers. For example, the next tablet contains a set of problems relating to the calculation of volume, together with the solutions.

You can see the details of the tablet in the next two pictures:

There is also a tablet recording observations of the planet Venus from c.1700 BC:

Astronomical tablets were so common in Mesopotamia and here we have a representation of the heavens in eight segments which include drawings of the constellations.

The next piece of cuneiform tablet contains a star chart which was found in Ashurbanipal’s library:

According the British Museum’s web…

## Ashurbanipal, whose name (Ashur-bani-apli) means, ‘the god Ashur is the creator of the heir’, came to the Assyrian throne in 668 BC. He continued to live in the Southwest Palace of his grandfather, Sennacherib, in Nineveh, which he decorated with wall reliefs depicting his military activity in Elam. He also had a new residence built at Nineveh, known today as the North Palace. The famous lion hunt reliefs, some of which are now in The British Museum, formed part of the new palace’s decorative scheme.

Throughout his reign, Ashurbanipal had military problems, mainly at the borders of the empire. He also continued his father’s policy of attacking Egypt. Campaigns in 667 and 664 BC led to the defeat of the Egyptian Twenty-fifth Dynasty and the appointment of a pro-Assyrian ruler in the Nile Delta. Assyria also attacked Elam, possibly in 658-57 BC, following the receipt of insulting letters from the Elamite king. In 652 BC Shamash-shum-ukin, Ashurbanipal’s brother, and ruler of Babylonia, revolted against Assyria with the support of the Elamites. The Assyrian army invaded Elam and Babylonia. Babylon was captured in 648 BC and the following year the Elamite city of Susa was destroyed. There is little surviving evidence that can help us to reconstruct the last years of Ashurbanipal’s reign. Ashurbanipal boasted of his ability to read the cuneiform script, and was responsible for the collection and copying of a major library of contemporary literary and religious texts

**Location**: British Museum in London (map)

# The Ashmolean Museum in Oxford (III)

There also are some Mesopotamian astronomical and mathematcal tablets in the Ashmolean Museum. For example, these two tablets are two proto-cuneiform clay tablets from an administrative building. They contain receipts of objects and grain, accounts and possibly rations and it’s possible to distinguish the units, the tens and the sixties:

Next clay tablet records date palms, orchards and gardeners in Akkadian cuneiform (2350-2150 BC):

Perhaps, the next clay tablet is the most interesting mathematical one because of its diagram. It’s a school tablet from 1900-1600 BC with a mathematical exercise showing a triangle with the incorrect calculation of the area of a field:s

In Eleanor Robson’s Mathematical cuneiform tablets in the Ashmolean Museum, Oxford, we find an explanation about this tablet:

Type IV tablet with upper right portion missing and reverse blank where preserved. Geometrical diagram of a triangle, showing the two lengths and an erroneous value for the area. Found in Trench C-10, 1 metre from surface level, 2 metres from plain level, with two other Type IV tablets bearing elementary exercises […].

The correct answer is 3;45 · 1;52,30 · 0;30 = 3;30,56,15

The error appears to have arisen through misplacing the sexagesimal place of one part of an intermediate calculation […]

There also is a clay prism with table of linear measures and squares roots (1950-1700 BC) from Southern Iraq:

Finally, I took a photography of the clay tablet with astronomical observations copied by a scribe in the early 8th century BC from Iraqian Kish. It gives the dates of the rising and settings of Venus in the reign of Ammizaduqa, king of Babylon in the 17th century BC:

**Location**: The Ashmolean Museum in Oxford (map)

# Archaic Mesopotamian numbers

Finally, I must talk about these two cuneiform tablets where we can see Mesopotamian figures. There are two ancient Sumerian administrative documents (offerings to a temple and a palace) from the Vorderasiastisches Museum of Berlin. They are dated in c. 2300 BC and we can distinguish the Archaic Mesopotamian numerals in them. In the first one (on the left), we can see some Mesopotamian numerals in the upper left corner:

The Archaic Mesopotamian System of Numeration consisted in the following symbols:

From left to right, the values of the respectively symbols are 1, 10, 60, 600, 3.600 and 36.000. So we can see in the cuneiform tablet the numbers 3 x 600 = 1.800, 1 x 10 = 10 and 5 x 1 = 5. The Archaic numeration wasn’t a positional system so each number was constructed from the iteration of the different ciphers. However, the Sumerians adopted a system in which they could subtract ciphers in spite of adding them. For example, in our tablet, the number 1 x 10 + 5 x 1 = 15 is located inside an angle. This angle represents the operation “minus” so the number which we read in the tablet is 1.800- 15 = 1.785.

In the second tablet we can see the number 11 in the lower left corner.

**Location**: CaixaForum (map)

# Two more planes

I have found two more interesting pictures checking my photos of the Caixaforum exhibition. The first one is the plane of a house (c. 2000 BC) from the Vorderasiatisches Museum of Berlin:

The second one is the plane of a sanctuary or a private house (c. 200 BC) from the Musée du Louvre:

**Location**: CaixaForum (map)

# Before the deluge

From November 30, 2012 to February 24, 2013, we can enjoy this wonderful exhibition in the Caixaforum of Barcelona:

Another 2,500 years would go by before the first dolmens and menhirs were built in Europe, and Egypt was not yet a uni_ed state ruled by the Pharaohs. But, in what is now southern Iraq, a people had built a great city with 40,000 inhabitants. Perhaps the first city in history, this was the capital of a kind of “empire”, with colonies as far-flung as southern Turkey. Its name was Uruk.

The first monumental architecture; the first territorial planning; the first writing in history, perhaps even predating Egypt; the first accounting. All this came about in Uruk in around 3500 BC.

The exhibition showed about 400 pieces from the most important museums of the World and it was possible to contact a lost civilization and all its characteristics:

It seems they spoke Sumerian, a tongue with no links to any other known language, past or present. After the fall of this great state in around 2900 BC, a number of independent city-states sprang up along the southern banks of the Tigris and Euphrates rivers and in the marshlands of the delta. Five hundred years later, these cities were united, firstly, under the Akkadian Empire, whose capital, Akkad, may have occupie what is now Baghdad. Short-lived, Akkad was replaced by a second empire, that of Ur III, whose capital was the city of Ur. Replacing Akkadian, Sumerian once more became the language of this empire.

I enjoyed it a lot. Furthermore, the mathematical objects were also exhibited and I could see different mathematical cuneiform tablets. First of all, I could find Bonaventura Ubach’s suitcase:

Ubach (1879-1960) was a Catalan priest who was an orientalist interested for the Bible. He traveled to the lands of the former Mesopotamia and wrote *Dietari d’un viatge per les regions d’Iraq* (1922-1923).

The first exhibited cuneiform tablet is about a contract of sale of lands:

One

sharof house and orchard/Shamash-nâsir’s house/From Shamash-nâsir/Tâbîya/buys/The full price/A silver shekel and a half/ will be paid/ He won’t claim in the future/ In the name of [¿?]

Different planes of fields were also exhibited:

A round tablet with measurements of the Third Dynasty of Ur (2100-2000 BC) from the Musée royaux d’Art et d’Histoire of Brussels:

A map of an arable bounded land (c. XXth. century BC) from the Musée du Louvre:

Undoubtedly, it was a really wonderful exhibition!

**Location**: CaixaForum (map)

# Pythagoras’ Theorem in Ancient Mesopotamia

One of the more clear extant proofs of the knowledge of Pythagoras’ theorem by the ancient Mesopotamians is the clay tablet YBC 7289 (Yale Babylonian Collection). We can see in the tablet a square in which the two diagonals are drawn and three numbers are the clues to understand this representation. First of all, next to one of the four sides we can read the number 30 which is the measure of the side. The other two numbers are written in the middle of the square: 42;25,35 and 1;24,51,10 (in sexagesimal notation). The first one is the corresponding measure of the diagonal and it’s equal to the product 30 · 1;24,51,10 so it’s not difficult to deduce that 1;24,51,10 is the value of the corresponding square root of 2. This value is a very accurate approximation of the real value because we can compute:

1,24,51,10^{2} = 1,59,59,59,38,1,40

If we want to see the details of the tablet we can read page 27 of the book by A.Aaboe entitled *Episodes from the Early History of Mathematics,* Washington, D.C.: MAA, 1998 (originally published in 1964) which is possible to find very easy in internet:

**Location**: University of Yale (Yale Babylonian Collection)