Plimpton 322 is a Babylonian clay tablet, believed to have been written around 1800 BC, that contains a mathematical table written in cuneiform script. Each row of the table relates to a Pythagorean triple, that is, a triple of integers that satisfies the Pythagorean theorem, , the rule that equates the sum of the squares of the legs of a right triangle to the square of the hypotenuse. The era in which Plimpton 322 was written was roughly 13 to 15 centuries prior to the era in which the major Greek discoveries in geometry were made.
At the time that Otto Neugebauer and Abraham Sachs first realized the mathematical significance of the tablet in the 1940s, a few Old Babylonian tablets making use of the Pythagorean rule were already known., p. 396. In addition to providing further evidence that Mesopotamian scribes knew and used the rule, Plimpton 322 strongly suggested that they had a systematic method for generating Pythagorean triples as some of the triples are very large and unlikely to have been discovered by ad hoc methods. Row 4 of the table, for example, relates to the triple (12709,13500,18541).
The table exclusively lists triples in which the longer leg, , (which is not given on the tablet) is a regular number, that is a number whose prime factors are 2, 3, or 5. As a consequence, the ratios and of the other two sides to the long leg have exact, terminating representations in the Mesopotamians' sexagesimal (base-60) number system. The first column most likely contains the square of the latter ratio, , and is in descending order, starting with a number close to 2, the value for the isosceles right triangle with angles , , , and ending with the ratio for a triangle with angles roughly , , . The Babylonians, however, are believed not to have made use of the concept of measured angle. Columns 2 and 3 are most commonly interpreted as containing the short side and hypotenuse. Due to some errors in the table and damage to the tablet, variant interpretations, still related to right triangles, are possible.
The purpose of Plimpton 322 is not known. Neugebauer and Sachs saw Plimpton 322 as a study of solutions to the Pythagorean equation in whole numbers, and suggested a number-theoretic motivation. They proposed that the table was compiled by means of a rule similar to the one used by Euclid in Elements., p. 40., pp. 36–40. Many later scholars have favored a different proposal, in which a number , greater than 1, with regular numerator and denominator, is used to form the quantity . This quantity has a finite sexagesimal representation and has the key property that if it is squared and 1 subtracted, the result has a rational square root also with a finite sexagesimal representation. This square root, in fact, equals . The result is that
is a rational Pythagorean triple, from which an integer Pythagorean triple can be obtained by rescaling. The column headings on the tablet, as well as the existence of tablets YBC 6967, MS 3052, and MS 3971 that contain related calculations, provide support for this proposal.
Most current scholars consider a number-theoretic motivation to be anachronistic, given what is known of Babylonian mathematics as a whole. The proposal that Plimpton 322 is a trigonometric table is ruled out for similar reasons, given that the Babylonians appear not to have had the concept of angle measure. Various proposals have been made, including that the tablet had some practical purpose in architecture or surveying, that it was geometrical investigation motivated by mathematical interest, or that it was compilation of parameters to enable a teacher to set problems for students. With regard to the latter proposal, Creighton Buck, reporting on never-published work of D. L. Voils, raises the possibility that the tablet may have only an incidental relation to right triangles, its primary purpose being to help set problems relating to reciprocal pairs, akin to modern day quadratic-equation problems. Other scholars, such as Jöran Friberg and Eleanor Robson, who also favor the teacher's aid interpretation, state that the intended problems probably did relate to right triangles., p. 202, "Further, if we believe that Plimpton 322 was intended to be a list of parameters to aid the setting of school mathematics problems (and the typological evidence suggests that it was), the question 'how was the tablet calculated?' does not have to have the same answer as the question 'what problems does the tablet set?' The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems.", p. 448. Friberg asserts that Plimpton 322 could be used to set problems like the ones in Problem 3 of the tablet MS 3971, in which the diagonal of a rectangle of length 1 is calculated from a reciprocal pair, and then the width is calculated using the Pythagorean rule.
Provenance and dating Plimpton 322 is partly broken, approximately wide, tall, and thick. New York publisher George Arthur Plimpton purchased the tablet from an archaeological dealer, Edgar J. Banks, in about 1922, and bequeathed it with the rest of his collection to Columbia University in the mid-1930s. According to Banks, the tablet came from Senkereh, a site in southern Iraq corresponding to the ancient city of Larsa., p. 109.
The tablet is believed to have been written around 1800 BC (using the middle chronology),When comparing dates given by different sources, note that many of Wikipedia's articles on the ancient world use the short chronology, while much of the history of mathematics literature uses the middle chronology. An exception is , which uses the long chronology. based in part on the style of handwriting used for its cuneiform script: writes that this handwriting "is typical of documents from southern Iraq of 4000–3500 years ago." More specifically, based on formatting similarities with other tablets from Larsa that have explicit dates written on them, Plimpton 322 might well be from the period 1822–1784 BC., p. 111. Robson points out that Plimpton 322 was written in the same format as other administrative, rather than mathematical, documents of the period., p. 110.
Content The main content of Plimpton 322 is a table of numbers, with four columns and fifteen rows, in Babylonian sexagesimal notation. The fourth column is just a row number, in order from 1 to 15. The second and third columns are completely visible in the surviving tablet. However, the edge of the first column has been broken off, and there are two consistent extrapolations for what the missing digits could be; these interpretations differ only in whether or not each number starts with an additional digit equal to 1.
With the differing extrapolations shown in parentheses, damaged portions of the first and fourth columns whose content is surmised shown in italics, and six presumed errors shown in boldface along with the generally proposed corrections in square brackets underneath, these numbers are
takiltum of the diagonal from which 1 is tornout so that thewidth comes up ÍB.SI8 of the width ÍB.SI8 of the diagonal its line (1) 59 00 15 1 59 2 49 1st (1) 56 56 58 14 56 15 (1) 56 56 58 14 [50 06] 15 56 07 3 12 01 [1 20 25] 2nd (1) 55 07 41 15 33 45 1 16 41 1 50 49 3rd (1) 53 10 29 32 52 16 3 31 49 5 09 01 4th (1) 48 54 01 40 1 05 1 37 5th (1) 47 06 41 40 5 19 8 01 6th (1) 43 11 56 28 26 40 38 11 59 01 7th (1) 41 33 59 03 45 (1) 41 33 [45 14] 03 45 13 19 20 49 8th (1) 38 33 36 36 9 01 01 12 49 9th (1) 35 10 02 28 27 24 26 40 1 22 41 2 16 01 10th (1) 33 45 45 1 15 11th (1) 29 21 54 02 15 27 59 48 49 12th (1) 27 00 03 45 7 12 01 [2 41] 4 49 13th (1) 25 48 51 35 06 40 29 31 53 49 14th (1) 23 13 46 40 56 56 (alt.) 53 [1 46] 53 (alt.) 15th
Two possible alternatives for the correction in Row 15 are shown: either 53 in the third column should be replaced with twice its value, 1 46, or 56 in the second column should be replaced with half its value, 28.
It is possible that additional columns were present in the broken-off part of the tablet to the left of these columns. Babylonian sexagesimal notation did not specify the power of 60 multiplying each number, which makes the interpretation of these numbers ambiguous. The numbers in the second and third columns are generally taken to be integers. The numbers in the first column can only be understood as fractions, and their values all lie between 1 and 2 (assuming the initial 1 is present—they lie between 0 and 1 if it is absent).
These fractions are exact, not truncations or rounded off approximations. The decimal translation of the tablet under these assumptions is shown below. Most of the exact sexagesimal fractions in the first column do not have terminating decimal expansions and have been rounded to seven decimal places.
or Short Side Diagonal Row # (1).9834028 119 169 1 (1).9491586 3,367 4,825 2 (1).9188021 4,601 6,649 3 (1).8862479 12,709 18,541 4 (1).8150077 65 97 5 (1).7851929 319 481 6 (1).7199837 2,291 3,541 7 (1).6927094 799 1,249 8 (1).6426694 481 769 9 (1).5861226 4,961 8,161 10 (1).5625 45* 75* 11 (1).4894168 1,679 2,929 12 (1).4500174 161 289 13 (1).4302388 1,771 3,229 14 (1).3871605 56* 106* 15
*As before, an alternative possible correction to Row 15 has 28 in the second column and 53 in the third column. The entries in the second and third columns of Row 11, unlike those of all other rows except possibly Row 15, contain a common factor. It is possible that 45 and 1 15 are to be understood as 3/4 and 5/4, which is consistent with the standard (0.75,1,1.25) scaling of the familiar (3,4,5) right triangle in Babylonian mathematics.
In each row, the number in the second column can be interpreted as the shorter side of a right triangle, and the number in the third column can be interpreted as the hypotenuse of the triangle. In all cases, the longer side is also an integer, making and two elements of a Pythagorean triple. The number in the first column is either the fraction (if the "1" is not included) or (if the "1" is included). In every case, the long side is a regular number, that is, an integer divisor of a power of 60 or, equivalently, a product of powers of 2, 3, and 5. It is for this reason that the numbers in the first column are exact, as dividing an integer by a regular number produces a terminating sexagesimal number. For instance, line 1 of the table can be interpreted as describing a triangle with short side 119 and hypotenuse 169, implying long side , which is a regular number (23·3·5). The number in Column 1 is either (169/120)2 or (119/120)2.
Column headings Each column has a heading, written in the Akkadian language. Some words are Sumerian logograms, which would have been understood by readers as standing for Akkadian words. These include ÍB.SI8, for Akkadian ("square"), MU.BI.IM, for Akkadian ("its line"), and SAG, for Akkadian ("width"). Each number in the fourth column is preceded by the Sumerogram KI, which, according to , "gives them the character of ordinal numbers." In the sexagesimal table above, italicized words and parts of words represent portions of the text that are unreadable due to damage to the tablet or illegibility, and that have been reconstructed by modern scholars. The terms ÍB.SI8 and takiltum have been left untranslated as there is ongoing debate about their precise meaning.