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8 W 19408, 76: Introduction

“Some five thousand clay tablets, reused as building rubble in the central temple precinct of the city of Uruk, constitute the world’s oldest assemblage of written records, dating from sometime in the last quarter of the fourth millennium BCE.”[1]

Among those thousands were seven tablets[2] that are the oldest known mathematical exercises pertaining to field areas.[3] Among these seven, W 19408, 76 is exceptional that it has a mathematical problem on each side. This tablet is now kept at the German Archaeological Institute, Berlin, Germany.

A greyscale photo of the front and back of the tablet W 19408, 76. The tablet contains symbols from System S. The tablet has a large chunk missing from the lower right corner.
Figure 9. The front and back of W 19408, 76.

Figure 9 shows how badly damaged and incomplete W 19408, 76 is. Using earlier photographs of a more complete version of W 19408, 76[4] experts have been able to recreate the contents of W 19408, 76 as pictured in Figure 10.[5]

A rendering of both sides W 19408, 76. Each side is divided into 4 cells each cell contains numbers from System S and either a horizontal or a vertical stroke.
Figure 10. A simplified rendering of the front and back of W 19408, 76.

Each side of the tablet contains four cells. These four cells contain numerals from System S and either a vertical or horizontal stroke. Assyriologists agree that these four numbers refer to lengths of a quadrilateral in ninda, where opposite sides are grouped based on the orientation of the strokes. Notice on the front of W 19408, 76 that the numbers labeled by the horizontal stroke are the same number 2 gèš’u ninda. A quadrilateral with two equal sides could potentially be a trapezoid.

A trapezoid that is nearly square on a grid.
Figure 11. The trapezoid corresponding to the front side of W 19408, 76.

Activity 7. Refer to Figure 10 and to Figure 11

  1. Convert the dimensions of the trapezoid listed on the front side of W 19408, 76 into modern base 10.
  2. Use the Pythagorean theorem to find the distance between the parallel sides of the trapezoid in Figure 11.
  3. Use the modern area formula for a trapezoid to find this trapezoid’s area.
  4. Convert this area into the standard Sumerian unit for area gána. Remember that 1 iku gána is the same as 100 square ninda. Leave your answer in base 10.

 

Media Attributions

  1. Eleanor Robson, "Mesopotamian Mathematics," in The Mathematics of Egypt, Mesopotamia, China, India, and Islam  ed. Victor J. Katz (2007), 73. ↵
  2. Jöran Friberg, "Round and Almost Round Numbers in Proto-Literate Metro-Mathematical Field Texts." Archiv für Orientforschung 44/45 (1997/1998), 8. ↵
  3. Hans J. Nissen, Peter Damerow, and Robert K. Englund. Archaic Bookkeeping: Writing and Techniques of Economic Administration in the Ancient near East (1993), 55. ↵
  4. A large fragment of the tablet has been lost! Nissen et al. (1993), 55. ↵
  5. Rendering based on the one in Friberg (1997/1998), 10. ↵

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

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