IM 67118, also known as Db2-146, is an Old Babylonian clay tablet in the collection of the Iraq Museum that contains the solution to a problem in plane geometry concerning a rectangle with given area and diagonal. In the last part of the text, the solution is proved correct using the Pythagorean theorem. The steps of the solution are believed to represent cut-and-paste geometry operations involving a diagram from which, it has been suggested, ancient Mesopotamians might, at an earlier time, have derived the Pythagorean theorem.
Description The tablet was excavated in 1962 at Tell edh-Dhiba'i, an Old Babylonian settlement near modern Baghdad that was once part of the kingdom of Eshnunna, and was published by Taha Baqir in the same year. It dates to approximately 1770 BCE (according to the middle chronology), during the reign of Ibal-pi-el II, who ruled Eshnunna at the same time that Hammurabi ruled Babylon. The tablet measures 11.5 cm × 6.8 cm × 3.3 cm (4+1⁄2 in × 2+3⁄4 in × 1+1⁄4 in). Its language is Akkadian, written in cuneiform script. There are 19 lines of text on the tablet's obverse and six on its reverse. The reverse also contains a diagram consisting of the rectangle of the problem and one of its diagonals. Along that diagonal is written its length in sexagesimal notation; the area of the rectangle is written in the triangular region below the diagonal.
Problem and its solution In modern mathematical language, the problem posed on the tablet is the following: a rectangle has area A = 0.75 and diagonal c = 1.25. What are the lengths a and b of the sides of the rectangle? The solution can be understood as proceeding in two stages: in stage 1, the quantity c 2 − 2 A {\displaystyle {\sqrt {c^{2}-2A}}} is computed to be 0.25. In stage 2, the well-attested Old Babylonian method of completing the square is used to solve what is effectively the system of equations b − a = 0.25, ab = 0.75. Geometrically this is the problem of computing the lengths of the sides of a rectangle whose area A and side-length difference b−a are known, which was a recurring problem in Old Babylonian mathematics. In this case it is found that b = 1 and a = 0.75. The solution method suggests that whoever devised the solution was using the property c2 − 2A = c2 − 2ab = (b − a)2. It must be emphasized, however, that the modern notation for equations and the practice of representing parameters and unknowns by letters were unheard of in ancient times. It is now widely accepted as a result of Jens Høyrup's extensive analysis of the vocabulary of Old Babylonian mathematics, that underlying the procedures in texts such as IM 67118 was a set of standard cut-and-paste geometric operations, not a symbolic algebra.
From the vocabulary of the solution Høyrup concludes that c2, the square of the diagonal, is to be understood as a geometric square, from which an area equal to 2A is to be "cut off", that is, removed, leaving a square with side b − a. Høyrup suggests that the square on the diagonal was possibly formed by making four copies of the rectangle, each rotated by 90°, and that the area 2A was the area of the four right triangles contained in the square on the diagonal. The remainder is the small square in the center of the figure. The geometric procedure for computing the lengths of the sides of a rectangle of given area A and side-length difference b − a was to transform the rectangle into a gnomon of area A by cutting off a rectangular piece of dimensions a×½(b − a) and pasting this piece onto the side of the rectangle. The gnomon was then completed to a square by adding a smaller square of side ½(b − a) to it. In this problem, the side of the completed square is computed to be A + 1 4 ( b − a ) 2 = 0.75 + 0.015625 = 0.875 {\displaystyle {\sqrt {A+{\tfrac {1}{4}}(b-a)^{2}}}={\sqrt {0.75+0.015625}}=0.875} . The quantity ½(b − a)=0.125 is then added to the horizontal side of the square and subtracted from the vertical side. The resulting line segments are the sides of the desired rectangle. One difficulty in reconstructing Old Babylonian geometric diagrams is that known tablets never include diagrams in solutions—even in geometric solutions where explicit constructions are described in text—although diagrams are often included in formulations of problems. Høyrup argues that the cut-and-paste geometry would have been performed in some medium other than clay, perhaps in sand or on a "dust abacus", at least in the early stages of a scribe's training before mental facility with geometric calculation had been developed. Friberg does describe some tablets containing drawings of "figures within figures", including MS 2192, in which the band separating two concentric equilateral triangles is divided into three trapezoids. He writes, "The idea of computing the area of a triangular band as the area of a chain of trapezoids is a variation on the idea of computing the area of a square band as the area of a chain of four rectangles. This is a simple idea, and it is likely that it was known by Old Babylonian mathematicians, although no cuneiform mathematical text has yet been found where this idea enters in an explicit way." He argues that this idea is implicit in the text of IM 67118. He also invites a comparison with the diagram of YBC 7329, in which two concentric squares are shown. The band separating the squares is not subdivided into four rectangles on this tablet, but the numerical value of the area of one of the rectangles area does appear next to the figure.
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