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Mathematical Treasure: Old Babylonian Area Calculation

Frank J. Swetz (The Pennsylvania State University)

Among the inscribed clay tablets from Old Babylonia (ca. 1800-1600 BCE in what is now Iraq) in the Yale Babylonian Collection (YBC) are some informative mathematical finds.  YBC 7290, shown above, contains a student scribe’s exercise in which he (scribes were male) recorded the area of a designated trapezoid.  Translating the text from left to right, the base of the trapezoid is given as  \(2\,\,;\,20\)  sexagesimal (base 60) units.  In the current convention used for expressing sexagesimal notation, the semi-colon represents a sexagesimal point and spaces are left between sexagesimal places.  Thus  \(2\,\,;\,20\)  is read as  \(2\) units and \(20\) \(60\)ths; that is, \[2\cdot 1 + 20\cdot{\frac{1}{60}}.\] The sides of the trapezoid are also given as  \(2\,\,;\,20\)  and the remaining base, \(2.\)  The result for the computed area, \(5\,\,;\,03\,\,20,\)  is written within the diagram; we read it as  \(5\) units,  \(3/60\)’s, and  \(20/3600\)’s, or \[5\cdot 1 + 3\cdot{\frac{1}{60}} + 20\cdot{\frac{1}{3600}}.\] It appears that the formula used by ancient land surveyors to approximate a trapezoidal area was employed here; that is, the area is equal to the average of the sides multiplied by the average of the bases.  The viewer may wish to test the accuracy of this approximation against that obtained by using our contemporary formula for the area of a trapezoid.

The reverse side of the tablet YBC 7290 (above) contains a diagram of a trapezoid and is devoid of numerical text.  Perhaps the student scribe was merely practicing drawing the figure or perhaps the diagram he obtained was inadequate for his inscription.  A more extensive discussion of trapezoidal computations from ancient Mesopotamia can be found in Eleanor Robson’s chapter on “Mesopotamian Mathematics” in The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Source Book, Victor Katz, editor, Princeton University Press, 2007, pp. 58-181.

The tablet YBC 11120 contains the diagram of a circle and the computation of its area.  The cuneiform inscription gives the sexagesimal number  \(1\,\,;\,30,\)  the measure of the circumference of the given circle.  The number on the right,  \(2\,\,;\,15,\)  is the circumference squared:  \[{(1\,\,;\,30)}^2 = 2\,\,;\,15.\] In their computation for the area of a circle, the ancient Babylonians employed the circumference as a factor, rather than the radius or diameter as we would.  Their formula for the area of a circle was \[{\rm Area} = (;\,05) \times {({\rm circumference)}^2,}\] where  \(;\,05\) was a standard constant employed in Babylonian computation, which we would write as  \(1/12\)  and interpret as  \(1/4\pi.\)  Old Babylonian scribes also used \(3\) as a value for \(\pi\) elsewhere.  The result of the area computation is \[0;\,11\,\,15\,\,\,{\rm or}\,\,\,\,{\frac{11}{60}} + {\frac{15}{3600}} = {\frac{3}{16}}.\] For further information on Mesopotamian circle computation, consult the reference given above or Eleanor Robson’s article, “Neither Sherlock Holmes nor Babylon: A Reassessment of Plimpton 322,” in Historia Mathematica 28 (2001), pp. 167-206, especially pages 179-183.

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Old Babylonian Area Calculation," Loci (March 2014)