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2 VAT 12593: Introduction

How do math historians decipher the tablets of ancient Mesopotamia? Thanks to the philologists and assyriologists, the language on the tablets was already well understood. All that was left was to assign quantitative meaning to symbols in a way that gives a coherent structure to each tablet. And similar to how Assyriologists used their knowledge of modern Semitic languages to decipher Akkadian, the math historians were able to use their own understanding of modern mathematics to decipher mathematical tablets.

In this section we will get to experience this process for ourselves. We will do this by analyzing VAT 12593, pictured below in Figure 1. This is the oldest “securely datable” mathematical table in the world.[1] The tablet was produced in Šuruppag, a large Sumerian settlement upstream of Uruk on the Euphrates, during the late Early Dynastic period (c. 2600 BC). We know the origin and date of this tablet thanks to the German Oriental Society who were responsible for the first excavation of the site during 1902-1903. The tablet is now located at the Vorderasiatisches Museum in Berlin, Germany.

A greyscale image of the clay tablet VAT 12593 showing its obverse on the left and its reverse on the right. To the right of the reverse are three small fragments of the tablets face. The tablet is the shape of a square with rounded edges. It has some places where the face of the tablet has been broken off. Over half of the reverse side's face is missing. The front and back are organized into three columns and several rows containing numerals. These numerals are impressed into the clay in the form of circular or bullet-shaped indentations of varying sizes.
Figure 1. The front and back side of VAT 12593 with fragments.

The tablet is roughly 14.5cm tall, 13cm wide, and 4.2cm thick as seen in Figure 2. The tablet is organized into three columns. There are 15 rows that are still partially intact. It is possible that more rows existed on the badly damaged back side of the tablet.

A line drawing of an outline of VAT 12593 next to a line drawing of a human hand. The tablet is roughly square and extends from the base of the palm to the tip of the pinky finger.
Figure 2. An outline of VAT 12593 next to an average human hand.

Despite the bad damage to the tablet, the clearly present patterns of the text allowed assyriologists to fill in the missing pieces with reasonable accuracy as shown in Figure 3.[2]

A simplified digital rendering of the front and back of VAT 12593. Areas of the tablet that were damaged in the actual picture have been restored based on context.
Figure 3. A rough recreation of the front and back of VAT 12593 based on the drawing in Deimel.

VAT 12593 only has five Sumerian words. As mentioned above, the literal meaning of these words were known already by the time math historians began studying tablets like VAT 12593. The following table gives a description of each of the symbols for the Sumerian words present in VAT 12593 along with how it is pronounced and the literal English translation.

Sumerian Description of Symbol Pronunciation English
The cuneiform symbol "sag." It is meant to look like a head and neck looking to the right, but rotated ninety degrees counterclockwise. a head and neck sag face
The cuneiform symbol for "ninda." It is meant to look like a bowl, but it is rotated ninety degrees counterclockwise. a bowl ninda ration
The cuneiform symbol for "du." It is meant to look like an ankle and foot, but it is rotated ninety degrees counterclockwise. a foot du to walk
The cuneiform for "sa." It looks like a diamond with an equal sign inside it. a diamond with equal sign sá equality
The cuneiform for the word "ganá." It is meant to look like a field of grain. a field of grain ganá field
Table 1. The Sumerian words present in VAT 12593.
All of the other symbols on VAT 12593 pictured in Figure 3 are numerals. They represent different quantities that are somehow related to one another. Activity 1  invites you to look for mathematical meaning in VAT 12593 in an effort to decipher the values of these numerals and the quantities they represent.

Activity 1. Refer to Figure 3 to answer the following questions.

  1. Ignoring the Sumerian words, what patterns do you notice in the tablet? What rules do the numerical symbols seem to obey? How is the tablet organized?
  2. Combining your observations about the numerical symbols with the literal meaning of the Sumerian words provided in Table 1, what do you think is the purpose of VAT 12593? What do you think these Sumerian words actually meant in the context of this mathematical tablet?
  3. Make a guess for the values of each of the numerical symbols use in VAT 12593. What types of quantities do you think these numerals represent?

 

 

Media Attributions

  1. Eleanor Robson, "Mesopotamian Mathematics," in The Mathematics of Egypt, Mesopotamia, China, India, and Islam  ed. Victor J. Katz (2007), 74. ↵
  2. Figure 3 is based on a drawing by Anton Deimel, "Schultexte Aus Fara," Wissenschaftliche Veröffentlichungen Der Deutschen Orient-Gesellschaft Vol. 43 (1923), 75. ↵

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Math History Copyright © by Bradley Burdick. All Rights Reserved.

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