3 VAT 12593: Analysis

We can deduce the mathematical content of VAT 12593 by looking at the last two rows on the back side as seen in Figure 3, recreated below. The presence of the symbol sá (equality), tells us that the two numerals on the left are being combined in some way to get the numerals on the right. The only operation that makes sense in this context is multiplication, since [latex]1\times 1=1[/latex] and [latex]2\times 2 = 4[/latex]. We conclude that VAT 12593 is a multiplication table.

This interpretation that VAT 12593 as a multiplication table is reinforced by the Sumerian words present. Recall the Sumerian words present in Figure 3 are described in Table 1.

The words sag (face) and ganá (field) may seem unrelated, but a Sumerian scribe would have associated the two words together with a third: uš (side).^[1] These three words go together because a rectangular field has a face and a side. For ancient Mesopotamians, the side was always longer than the face.^[2] In situations where the two dimensions are equal, it is reasonable to only use one of the two words. In this case the tablet only lists the length of the sag (face). We conclude that the first two columns give the lengths of the sides of a square field and the third column gives us the area of that field. This conforms with the interpretation as a multiplication table.

The only other words that have not been contextualized are ninda (ration) and du (to walk). It would be understood by a Sumerian reader that the symbols for ninda and du formed a single word. However, what put these words together to form ninda-du, it does not mean “ration walk” like other compound words would. Instead, the word du is functioning as a determinative. A single Sumerian word like ninda can have several different meanings like many English words can. However, unlike English, written Sumerian employed helper words, called determinatives, which clue the reader to which meaning is being employed. That way the reader knows that this ninda refers to a unit of length that modern Assyriologists call a “rod,” which measures approximately 6 meters.^[3] To indicate that du is a determinative and is not pronounced, modern Assyriologists transliterate this word as ninda^du. We conclude that the first two columns of VAT 12593 give us how long the face and side of each field measured in rods.

With a little bit of logic and linguistics we were able to deduce the mathematical context of VAT 12593. However you may have noticed while completing Activity 1 that the numerical symbols in the first two columns are apparently different or inconsistent with those in the third column. Recall that, in our interpretation, the third column records an area while the first two columns record lengths. We are forced to conclude that Sumerian had different number systems for different quantities. Activity 2 invites you to decipher the values of these symbols relative to one another given all that we have learned about VAT 12593.