Mathematical Treasure: Mesopotamian Accounting Tokens

Archaeological digs in the Mideast have uncovered thousands of small clay objects, dating from as far back as 7500 BCE. These objects, referred to as “tokens,” have specific shapes and markings indicating a designated, but until recently unknown, purpose. This mystery was solved by the art historian Denise Schmandt-Besserat who began researching these items in 1969. The extraordinary results of her research were published in a number of articles and books, including How Writing Came About (University of Texas Press, 1996). 

Her conclusion? The tokens were counters. Their use evolved over thousands of years from simply shaped tokens (see Figure 1) to more complex tokens bearing markings (see Figure 2). Each counter shape represented a specific quantity of a specific commodity. For example, a cone stood for a small measure of grain and a sphere for a large measure of grain. Using different shapes of counters to count different commodities is evidence of concrete counting, meaning that each category of items was counted with special numerations or number words specific to that category. There is a hint of concrete counting in our own society in our preference for phrases such as "a pair of shoes" or "a couple of days" over "two shoes" or "two days." However, we almost always use abstract counting with our abstract numbers "two," "three," "four," ... that can be used to count any item. After 3300 BCE, the tokens were sometimes stored in clay envelopes with their imprints made on the envelope’s surface to make visible the number and shapes of tokens enclosed (see Figure 3). According to Schmandt-Besserat, the transformation of three-dimensional tokens to two-dimensional signs to communicate information was the beginning of writing. Eventually, the tokens were replaced by signs made by their impressions onto solid balls of clay, or tablets (see Figures 4 and 5). The impressed signs evolved to become cuneiform writing.

Figure 1. The Mesopotamian accounting tokens shown above were found at Tepe Gawra, near present day Mosul, Iraq, and date from about 4000 BCE. The cone, spheres, and flat disc were measures of cereals: smallest, larger, and largest. The tetrahedron designated a unit of work, perhaps one man-day, or the amount of work performed by one man in one day. (Image provided courtesy of Denise Schmandt-Besserat and the University of Pennsylvania Museum of Archaeology and Anthropology, University of Pennsylvania, Philadelphia.)

Figure 2. Incised tokens from Tello, ancient Girsu, present day Iraq, circa 3300 BCE. Viewing the top row from right to left, the tokens represented: one length of textile, one jar of oil, uncertain, one measure of wheat. Continuing along the bottom row from left to right: one sheep, one length of rope, one ingot of metal, one garment. (Image provided courtesy of Denise Schmandt-Besserat and Musée du Louvre, Département des Antiquités Orientales, Paris.)

Figure 3. Envelope and contents from Susa, Iran, circa 3300 BCE. Each lenticular disc stood for “a flock” (perhaps 10 animals). The large cone represented a very large measure of grain; the small cones designated small measures of grain. (Image provided courtesy of Denise Schmandt-Besserat and Musée du Louvre, Département des Antiquités Orientales, Paris.)

Figures 4 and 5. Impressed tablets from Susa, Iran, circa 3200 BCE. Each circular impression stood for one large measure of grain; each wedge or conical impression designated a smaller measure of grain. The lighter impressions covering the tablets – and also the envelope in Figure 3 – are those of official seals. The scribe-accountant would roll his cylindrical seal all over the tablet or envelope, covering the entire surface. He would then press the tokens and/or stylus into the tablet or envelope to make the deeper marks. (Image provided courtesy of Denise Schmandt-Besserat and Musée du Louvre, Département des Antiquités Orientales, Paris.)

Figure 6. Impressed and incised tablet from Godin Tepe, Iran, circa 3100 BCE. The circular imprints stood for tens and the wedges for units. The incised figure to the right is a depiction of a jar of oil, and this tablet was a record of, in total, 33 jars of oil. On this tablet, the six impressions represent abstract numbers and the incised figure a specific object. Schmandt-Besserat has argued that such impressions represent the beginning of abstract numeration and such incised pictographs the beginning of writing. (Image provided courtesy of Denise Schmandt-Besserat and the late T. Cuyler Young, Jr., of the Royal Ontario Museum, Toronto, Canada.)

For further information on the history and use of these tokens, the reader is referred to Denise Schmandt-Besserat’s How Writing Came About or one of the other sources listed below. All four of the images shown here are presented courtesy of Prof. Schmandt-Besserat, who supplied the images to Convergence, and with the cooperation of the indicated museums and/or persons. You may use these images in your classroom; all other uses require permission from both Schmandt-Besserat and the museum holding the item depicted.

References

• Schmandt-Besserat, Denise, How Writing Came About, University of Texas Press, Austin, 1996.
• Website of Denise Schmandt-Besserat, University of Texas: http://sites.utexas.edu/dsb/

For a summary, see one or both of the following articles:

• Schmandt-Besserat, Denise, “Oneness, Twoness, Threeness: How Ancient Accountants Invented Numbers,” The Sciences 27:4 (July-August 1987), pp. 44-48. Reprinted in From Five Fingers to Infinity (Frank Swetz, ed.), Open Court, Chicago, 1994, pp. 45-51.
• Peterson, Ivars, “From Counting to Writing,” Science News. Originally posted Feb. 22, 1997; updated March 11, 2006.

For scholars:

• Schmandt-Besserat, Denise, Before Writing: Vol. I, From Counting to Cuneiform; Before Writing: Vol. II, A Catalogue of Near Eastern Tokens, University of Texas Press, Austin, 1992.

Acknowledgment

The author and editor thank Denise Schmandt-Besserat for the images and information used in this article.

Index of Mathematical Treasures

Index of Mathematical Objects