Plimpton 322

The Plimpton 322 tablet.

Of the approximately half million Babylonian clay tablets excavated since the beginning of the 19th century, several thousand are of a mathematical nature. Probably the most notable of these examples of Babylonian mathematics is the tablet called Plimpton 322, referring to the fact that 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.

Although the table has been interpreted by leading mathematicians as a listing of Pythagorean triples, more recently published theories give it a different function. 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.

Provenance and dating

Plimpton 322 is a partly broken clay tablet, approximately 13 cm wide, 9 cm tall, and 2 cm thick. New York publisher George A. 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 BCE, 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 can be dated to the period 1822–1784 BCE. 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 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	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.

Interpretation

In each row, the number in the second column can be interpreted as the shortest side s of a right triangle, and the number in the third column can be interpreted as the hypotenuse d of the triangle. The number in the first column is either the fraction $\scriptstyle {\tfrac {s^{2}}{l^{2}}}$ or $\scriptstyle {\tfrac {d^{2}}{l^{2}}}$, where l denotes the longest side of the same right triangle. Scholars still differ, however, on how these numbers were generated.

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, then $\scriptstyle (p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})$ form a Pythagorean triple, and all Pythagorean triples can be formed in this way. For instance, line 11 can be generated by this formula with p = 1 and q = 1/2. 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) discussed a possible trigonometric explanation: the values of the first column can be interpreted as the squared cosine 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. However, Robson argues on linguistic grounds that this theory is "conceptually anachronistic": it depends on too many other ideas not present in the record of Babylonian mathematics from that time.

Robson (2002) rejects earlier explanations of the tablet and claims that historical, cultural and linguistic evidence all reveal the tablet to be more likely "a list of regular reciprocal pairs." 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, $\scriptstyle x-{\tfrac {1}{x}}=c$, 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 likely 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 likely have been used by a teacher as a problem set to assign to students.

Reconciliation

Just in case the solution algorithm for the quadratic equation might seem divorced from Pythagorean triples, observe that if s, l and d are in Neugebauer's interpretation, so ss + ll = dd, we have ll = dd - ss = (d+s)(d-s), while 2s = (d+s)-(d-s). Hence, indeed, we have "solved" $\scriptstyle x-{\tfrac {1}{x}}=c$, where now c = 2s/l, x = (d+s)/l and x = (d-s)/l. By the same token, we have also solved $\scriptstyle x+{\tfrac {1}{x}}=k$, where k = 2d/l. What the algorithm does then is help identify s, l and d given c, with an an analogous algorithm to identify them in terms of k. Thus, the Pythagorean rule linking s, l and d also facilitates the solution of other problems, notably quadratic equations.

A natural setting for this discussion is a square grid, which, together with overlaid, offset grids, provides a mechanism for solving quadratics. In particular, by looking at the diagonal of rectangles in overlaid grids in two ways we can find right triangles all of whose sides can be computed as integral grid line segments in the underlying square grid. In effect, this is the property that right triangles with commensurable sides are characterised as having half angles with rational tangents. Thus, we can be acquainted with the Pythagorean triples independently of both the Pythagorean rule and number theory, so long as we know to fold sides of right triangles onto one another.

References

• Bruins, Evert M. (1949), "On Plimpton 322, Pythagorean numbers in Babylonian mathematics", Koninklijke Nederlandse Akademie van Wetenschappen Proceedings, 52: 629–632.
• Buck, R. Creighton (1980), "Sherlock Holmes in Babylon", American Mathematical Monthly, 87 (5), Mathematical Association of America: 335–345, doi:10.2307/2321200.
• Bruins, Evert M. (1951), "Pythagorean triads in Babylonian mathematics: The errors on Plimpton 322", Sumer, 11: 117–121.
• Conway, John H.; Guy, Richard K. (1996), The Book of Numbers, Copernicus, pp. 172–176, ISBN 0-387-97993-X.
• Neugebauer, Otto (1969) [1957], The Exact Sciences in Antiquity (2 ed.), Dover Publications, ISBN 978-048622332-2
• Neugebauer, O.; Sachs, A. J. (1945), Mathematical Cuneiform Texts, American Oriental Series, vol. 29, New Haven: American Oriental Society and the American Schools of Oriental Research.
• Robson, Eleanor (2001), "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322", Historia Math., 28 (3): 167–206, doi:10.1006/hmat.2001.2317, MR1849797.
• Robson, Eleanor (2002), "Words and pictures: new light on Plimpton 322" (PDF), American Mathematical Monthly, 109 (2), Mathematical Association of America: 105–120, doi:10.2307/2695324, JSTOR 2695324, MR1903149.

External links

• Casselman, Bill (2003), The Babylonian tablet Plimpton 322, University of British Columbia.