9 W 19408, 76: Analysis
While we were able to use our modern mathematical knowledge to compute the area of the trapezoid listed on the front of W 19408, 76 in Chapter 8, it remains for us to understand how a Sumerian trainee scribe would be meant to compute this area six thousand years ago. Luckily there are examples of tablets dating to later time periods with similar problems and intended solutions.[1] One technique seems to be employed across vast time periods throughout ancient Mesopotamia.
This technique is often called by modern mathematicians “the surveyors method.” If a quadrilateral has two parallel faces of length [latex]F_1[/latex] and [latex]F_2[/latex] and two parallel sides of length [latex]L_1[/latex] and [latex]L_2[/latex], then the surveyors method gives us the following approximation of its area [latex]A[/latex].
[latex]A =\dfrac{F_1+F_2}{2}\times \dfrac{S_1+S_2}{2}.[/latex]
In other words, its area is approximately equal to the area of the rectangle whose dimensions are the average of corresponding dimensions of our quadrilateral.
In order to apply the surveyors method to the trapezoid on the front of W 19408, 76, we begin by listing the dimensions found on Figure 10.
\(F_1=\) 1 géš’u 5 géš 3 u ninda
\(F_2=\) 1 géš’u 4 géš 3 u ninda
\(S_1=\) 2 géš’u ninda
\(S_2=\) 2 géš’u ninda
We then take the average of the pair of parallel sides:
\( \dfrac{F_1+F_2}{2} = \) 1 géš’u 5 géš ninda
\( \dfrac{S_1+S_2}{2} = \) 2 géš’u ninda
In order to multiply these averages together, we note that 1 géš’u 5 géš ninda is 1½ géš’u. So the product of these averages will be 3 square géš’u ninda. Luckily VAT 12593 tells us that a square géš’u ninda is 3 šár 2 bùr’u gána. Thus the area of this trapezoid will be 9 šár 6 bùr’u or exactly 1 šár’u gána.
Activity 8.
- Convert 1 šár’u gána into a modern base ten number of iku gána using Table 3. Remember that 1 šár’u gána is not the same as 1 šár’u.
- In Activity 7 we found that the area of the trapezoid on the front of W 19408, 76 was approximately 10796.62 iku gána.
- What is the error of the approximation given by the surveyor’s method in this example? Remember that the error of an approximation is the difference between the approximation and the real value as a percentage of the real value.
We conclude that the surveyor’s method is very good when applied to the trapezoid listed on the front of W 19408, 76.
- Jöran Friberg, "Round and Almost Round Numbers in Proto-Literate Metro-Mathematical Field Texts." Archiv für Orientforschung 44/45 (1997/1998), 11. ↵