Plimpton 322

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The Plimpton 322 clay tablet, with numbers written in cuneiform script.

Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This tablet, believed to have been written about 1800 BC, has a table of four columns and 15 rows of numbers in the cuneiform script of the period.

This table lists two of the three numbers in what are now called Pythagorean triples, i.e., integers a, b, and c satisfying a² + b² = c². From a modern perspective, a method for constructing such triples is a significant early achievement, known long before the Greek and Indian mathematicians discovered solutions to this problem. At the same time, one should recall the tablet's author was a scribe, in his day, a professional mathematician; it has been suggested that one of his goals may have been to produce, find, and fix school problems.

There has been significant scholarly debate on the nature and purpose of the tablet. For readable popular treatments of this tablet see Robson (2002) or, more briefly, Conway & Guy (1996). Robson (2001) is a more detailed and technical discussion of the interpretation of the tablet's numbers, with an extensive bibliography.

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Provenance and dating

Plimpton 322 is partly broken, approximately 13 cm wide, 9 cm tall, and 2 cm 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.

The tablet is believed to have been written about 1800 BC, based in part on the style of handwriting used for its cuneiform script: Robson (2002) 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. Robson points out that Plimpton 322 was written in the same format as other administrative, rather than mathematical, documents of the period.

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, these numbers with six errors corrected are

(1) 59 00 15	1 59	2 49	1
(1) 56 56 58 14 50 06 15	56 07	1 20 25	2
(1) 55 07 41 15 33 45	1 16 41	1 50 49	3
(1) 53 10 29 32 52 16	3 31 49	5 09 01	4
(1) 48 54 01 40	1 05	1 37	5
(1) 47 06 41 40	5 19	8 01	6
(1) 43 11 56 28 26 40	38 11	59 01	7
(1) 41 33 45 14 03 45	13 19	20 49	8
(1) 38 33 36 36	8 01	12 49	9
(1) 35 10 02 28 27 24 26 40	1 22 41	2 16 01	10
(1) 33 45	45	1 15	11
(1) 29 21 54 02 15	27 59	48 49	12
(1) 27 00 03 45	2 41	4 49	13
(1) 25 48 51 35 06 40	29 31	53 49	14
(1) 23 13 46 40	56	1 46	15

It is possible that additional columns were present in the broken-off part of the tablet to the left of these columns. Conversion of these numbers from sexagesimal to decimal raises additional ambiguities, as the Babylonian sexagesimal notation did not specify the power of the initial digit of each number. The sixty sexagesimal entries are exact, no truncations or rounding off.

Interpretations

In each row, the number in the second column can be interpreted as the shortest side ${\displaystyle s}$ of a right triangle, and the number in the third column can be interpreted as the hypotenuse ${\displaystyle d}$of the triangle. The number in the first column is either the fraction ${\displaystyle \scriptstyle {\tfrac {s^{2}}{l^{2}}}}$ (if the "1" is not included) or ${\displaystyle \scriptstyle {\tfrac {d^{2}}{l^{2}}}\,=\,1+{\tfrac {s^{2}}{l^{2}}}}$ (if the "1" is included), where ${\displaystyle l}$ denotes the longer side of the same right triangle. Scholars still differ, however, on how these numbers were generated. Below is the decimal translation of the tablet.

${\displaystyle d^{2}/l^{2}}$ or ${\displaystyle s^{2}/l^{2}}$	Short Side ${\displaystyle s}$	Diagonal ${\displaystyle d}$	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

Otto E. Neugebauer (1957) argued for a number-theoretic interpretation, pointing out that this table provides a list of (pairs of numbers from) Pythagorean triples. For instance, line 11 of the table can be interpreted as describing a triangle with short side 3/4 and hypotenuse 5/4, forming the side:hypotenuse ratio of the familiar (3,4,5) right triangle. If p and q are two coprime numbers, one odd and one even, then ${\displaystyle \scriptstyle (p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})}$ form a Pythagorean triple, and all Pythagorean triples can be formed in this way or as multiples of a triple formed in this way. For instance, line 11 can be generated by this formula with p = 2 and q = 1. As Neugebauer argues, each line of the tablet can be generated by a pair (p,q) that are both regular numbers, integer divisors of a power of 60. This property of p and q being regular leads to a denominator that is regular, and therefore to a finite sexagesimal representation for the fraction in the first column. Neugebauer's explanation is the one followed e.g. by Conway & Guy (1996). However, as Eleanor Robson (2002) points out, Neugebauer's theory fails to explain how the values of p and q were chosen: there are 92 pairs of coprime regular numbers up to 60, and only 15 entries in the table. In addition, it does not explain why the table entries are in the order they are listed in, nor what the numbers in the first column were used for.

Buck (1980) proposed a possible trigonometric explanation: the values of the first column can be interpreted as the squared secant or tangent (depending on the missing digit) of the angle opposite the short side of the right triangle described by each row, and the rows are sorted by these angles in roughly one-degree increments. In other words, if you take the number in the first column, discounting the (1), and derive its square root, and then divide this into the number in column two, the result will be the length of the long side of the triangle. Consequently, the square root of the number (minus the one) in the first column is what we would today call the tangent of the angle opposite the short side. If the (1) is included, the square root of that number is the secant.

In contraposition with these earlier explanations of the tablet, Robson (2002) claims that historical, cultural and linguistic evidence all reveal the tablet to be more likely "a list of regular reciprocal pairs." Robson argues on linguistic grounds that Buck's trigonometric theory is "conceptually anachronistic": it depends on too many other ideas not present in the record of Babylonian mathematics from that time. In 2003, the MAA awarded Robson with the Lester R. Ford Award for her work, stating it is "unlikely that the author of Plimpton 322 was either a professional or amateur mathematician. More likely he seems to have been a teacher and Plimpton 322 a set of exercises." Robson takes an approach that in modern terms would be characterized as algebraic, though she describes it in concrete geometric terms and argues that the Babylonians would also have interpreted this approach geometrically.

Robson bases her interpretation on another tablet, YBC 6967, from roughly the same time and place. This tablet describes a method for solving what we would nowadays describe as quadratic equations of the form, ${\displaystyle \scriptstyle x-{\tfrac {1}{x}}=c}$, by steps (described in geometric terms) in which the solver calculates a sequence of intermediate values v₁ = c/2, v₂ = v₁², v₃ = 1 + v₂, and v₄ = v₃^1/2, from which one can calculate x = v₄ + v₁ and 1/x = v₄ - v₁.

Robson argues that the columns of Plimpton 322 can be interpreted as the following values, for regular number values of x and 1/x in numerical order:

v₃ in the first column,

v₁ = (x - 1/x)/2 in the second column, and

v₄ = (x + 1/x)/2 in the third column.

In this interpretation, x and 1/x would have appeared on the tablet in the broken-off portion to the left of the first column. For instance, row 11 of Plimpton 322 can be generated in this way for x = 2. Thus, the tablet can be interpreted as giving a sequence of worked-out exercises of the type solved by the method from tablet YBC 6967, and reveals mathematical methods typical of scribal schools of the time, and that it is written in a document format used by administrators in that period. Therefore, Robson argues that the author was probably a scribe, a bureaucrat in Larsa. The repetitive mathematical set-up of the tablet, and of similar tablets such as BM 80209, would have been useful in allowing a teacher to set problems in the same format as each other but with different data. In short, Robson suggests that the tablet would probably have been used by a teacher as a problem set to assign to students.

So what does any of this have anything to do with the occult? Simple mathematics being one of the 7 sciences encompasses what we know as the occult.