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Asunto:[TA] Mas Matematicas babilonias
Fecha:Miercoles, 4 de Agosto, 2004  21:41:48 (+0200)
Autor:Ana Maria Vazquez <avhoys>
En respuesta a:Mensaje 5335 (escrito por José Luis Santos)

Se me  olvidaban algunos links.El tema es TAN BONITO, que os ruego me disculpeis el inglés( no tengo tiempo de traducirlo) y el cuneiforme lo dejo para que veais de qué hablamos cuando nos referirmos a la matemática sumeria. Por cierto:Javier Sierra en su programa no me dejó ni nombrar el tema, porque es "demasiado fuerte"...Por "eso" yo defiendo la primacía de Oriente en MUCHAS cosas, entre ellas, las matemáticas.
Un saludo.Dra.Vázquez Hoys

Babylonian Pythagoras - [ Traduzca esta página ]
... The four tablets which interest us here we will call the Yale tablet YBC 7289,
Plimpton 322 (shown below), the Susa tablet, and the Tell Dhibayi tablet. ... HistTopics/Babylonian_Pythagoras.html - 20k -


Math in the Media 0201 - [ Traduzca esta página ]
... Plimpton 322. Plimpton 322, George A. Plimpton Collection, Rare Book and
Manuscript Library, Columbia University. ... Rereading Plimpton 322. ... - 14k -
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Pagina nueva 1
Cuestiones. Tablilla Plimpton 322. ... - 95k -
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Babylonian Mathematics (YBC 7289, Plimpton 322) - [ Traduzca esta página ]
... Plimpton 322. Papyrus was a precious and highly expensive material,
and there is little writing space on a Babylonian clay tablet. ... - 88k -
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plimpton 322 - [ Traduzca esta página ]
Tablette Plimpton 322, Université de Columbia, USA. © Serge MEHL. momo/chronomath/anx3/plimpton.html - 2k -
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Hay 10 páginas más, si alguien quiere saber más.


Part 1: Introduction & bibliography

The clay tablet with the catalog number 322 in the G. A. Plimpton Collection at Columbia University may be the most well known mathematical tablet, certainly the most photographed one, but it deserves even greater renown. It was scribed in the Old Babylonian period between -1900 and -1600 and shows the most advanced mathematics before the development of Greek mathematics.


    O. Neugebauer and A. Sachs. Mathematical Cuneiform Texts. Amer. Oriental Series 29. American Oriental Society, New Haven, 1945

    E. M. Bruins. On Plimpton 322, Pythagorean numbers in Babylonian mathematics. Afdeling Naturkunde, Proc. 52 (149), 629-632.

    Akad. v. Wetenshoppen, Amsterdam.

    Kurt Vogel. Vorgriechische Mathematik. Two volumes. Hermann Schroedel Verlag KG, Hannover, 1959.

    O. Neugebauer. The Exact Sciences in Antiquity. Second edition, 1957, Brown Univ. Press. Dover reprint, 1969.

    R. Creighton Buck. Sherlock Holmes in Babylon. Amer. Math. Monthly 87 (1980), 335-345.

    Jöran Fribert. Methods and traditions of Babylonian Mathematics I: Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations. Historia Math. 8 (1981), 277-318.

  • H Eves, An Introduction to the History of Mathematics, Saunders College Publishing, 1990.
  • R J Gillings, The Australian Journal of Science, 16(1953):34-36.
  • O Neugebauer, The Exact Sciences in Antiquity, Second ed., New York: Harper and Row, 1962. Reprinted by Dover: New York.
  • O Neugebauer and A J Sachs, eds., Mathematical Cuneiform Texts, American Oriental Series, Vol. 29, New Haven: American Oriental Society, 1946.
  • D J de Solla Price, The Babylonian "Pythagorean Triangle" Table, Centarus, 10 (1964) 219-231.


Part 2: Transcriptions of the tablet

Here's a transcription of the Plimpton 322 tablet using modern digits. The numbers on the cuneiform tablet are written sexigesimally (in base 60) with combinations of two symbols, one for tens and one for units. Several unit symbols are placed in particular patterns to form the digits 1 through 9, and several ten symbols are placed in particular patterns to form the five digits 10, 20, 30, 40, and 50. Spaces are left to indicate zeros in the unit's place and zeros in the ten's place. In this transcription a colon is used to separate different sexagesimal digits for ease of reading. This is the same notation we use for time. For instance, 3:10:15 means 10 and 1/4 minutes after 3 o'clock.

	                           width     diagonal

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

There are fifteen rows and four columns on the tablet. It isn't clear whether the first column has all its numbers beginning with 1 or not; unfortunately the tablet is broken off at that point. There may have been more columns in the broken off portion of the tablet. Also, there are a couple of chips off the tablet, but it's clear how to fill in the missing digits. Within each column the numbers aren't lined up quite as nicely as they are here, but the numbers in each column are left justified.

The last column is clearly just the row number. The second and third columns are headed by words that can be interpreted as "width" and "diagonal."

The transcription above already includes four corrections. In the second row, third column, the original number was 3:12:01 rather than 1:20:25. In the ninth row, second column, the original number was 9:01 instead of 8:01. In the 13th row, second column, the original number was 7:12:01 instead of 2:41. And in the 15th row, third column, the original number was 53 rather than 1:46.

Often, this tablet is converted to decimal form, but it isn't clear whether the numbers are fractions or integers. The last column clearly contains integers, the first almost certainly contains fractions, and the second and third might contain fractions but probably contain integers. Here's a translation of the second, third, and fourth columns into decimal form under the assumption that all the numbers are integers

	width  diagonal

119 169 1 3367 4825 2 4601 6649 3 12709 18541 4 65 97 5 319 481 6 2291 3541 7 799 1249 8 481 769 9 4961 8161 10 45 75 11 1679 2929 12 161 289 13 1771 3229 14 56 106 15

Part 3: Interpretation and possible construction

The numbers in the first column are interesting since each is a perfect square and subtracting one from each leaves a perfect square. Consider, for instance, line 11. The number 1:33:45 represents 1 + 33/60 + 45/3600  =  1 +  9/16  = 25/16 which is the square of 5/4. One less than 25/16 is 9/16, the square of 3/4. The second and third entries in this row represent these fractions: 45 represents 45/60 = 3/4 and 1:15 represents 1 + 15/60 = 5/4. (Or perhaps 45 represents 45 and 1:15 represents 75 in which case these two entries are proportional to the fractions.)

For another example, consider row 5. 1:48:54:01:40 represents 1 + 48/60 + 54/3600 + 1/216000 + 40/12960000 =  1 + 4225/5184 = 9409/5184, which is the square of 97/72. And 4225/5184 is the square of 65/72. The second and third entries in row 5 are 1:05 representing 65, and 1:37 representing 97. Nearly always, the second and third entries aren't equal to the square roots, but just proportional to them.

This table is usually considered in relation to Pythagorean triples. In that interpretation, the second column is a, one side of a right triangle or the width of a rectangle, and the third column is c, the hypotenuse of the right triangle or the diagonal of the rectangle, and the other side of the triangle or rectangle, b, doesn't appear on the table. In this interpretation the first column is then (c/b)2  =  1 + (a/b)2.

The Old Babylonians knew the Pythagorean theorem (better called the rule of the right triangle for them since there's no evidence that they had a proof; Gillings calls the term "the Pythagorean theorem" a true mumpsimus), since there are examples of its use in various problems of the period. Along with the headings of the second and third columns, that justifies believing that this table relates to Pythagorean triples and right triangles.

Some historians have noticed that (1) each first column entry is the square of the cosecant of an angle of a right triangle, and (2) the associated angles are roughly one degree apart. So they have suggested that this is a trigonometric table of squares of cosecants for 45° down to 30°. This would be the earliest instance of cosecants by millennia, and the earliest instance of a trigonometric function by over a thousand years (the first trig function being the chord of an angle). It would also be the earliest instance of degree measurement by over a thousand years. In other words, that's a bold claim. A weaker claim is that the table was constructed to make the first column uniformly decreasing.

Anyway, here are the actual associated angles in a table. The angles are given in degrees, but for perversity, their fractions are given decimally. A column for the remaining side is also displayed as well as columns for parameters p and q generating Pythagorean triples where a = p2 - q2, b = 2pq, and c = p2 + q2.

angle A    (c/b)2                   a         c        b 
   p     q     n

44.76 1:59:00:15 1:59 2:49 2 12 5 1

44.25 1:56:56:58:14:50:06:15 56:07 1:20:25 57:36 1:04 27 2

43.79 1:55:07:41:15:33:45 1:16:41 1:50:49 1:20 1:15 32 3

43.27 1:53:10:29:32:52:16 3:31:49 5:09:01 3:45 2:05 54 4

42.08 1:48:54:01:40 1:05 1:37 1:12 9 4 5

41.54 1:47:06:41:40 5:19 8:01 6 20 9 6

40.32 1:43:11:56:28:26:40 38:11 59:01 45 54 25 7

39.77 1:41:33:45:14:03:45 13:19 20:49 16 32 15 8

38.72 1:38:33:36:36 8:01 12:49 10 25 12 9

37.44 1:35:10:02:28:27:24:26 1:22:41 2:16:01 1:48 1:21 40 10

36.87 1:33:45 3 5 4 2 1 11

34.98 1:29:21:54:02:15 27:59 48:49 40 48 25 12

33.86 1:27:00:03:45 2:41 4:49 4 15 8 13

33.26 1:25:48:51:35:06:40 29:31 53:49 45 50 27 14

31.89 1:23:13:46:40 56 1:46 1:30 9 5 15

Note that the every value of p and q is a regular sexagesimal integer, that is, their only prime divisors are 2, 3, and 5. That is necessary to make b regular so that the fraction c/b terminates. The Babylonians didn't like to divide by irregular numbers.

It may be that the Babylonians actually used the parameters p and q to generate this table. There is no direct evidence that they did, but there is some indirect evidence. When you consider all possible regular values of p and q with p less than or equal to 125 (and q < p, of course), tabulate the associated triangles, and sort them by p/q, or what is the same order, by (c/b)2, then there are exactly 16 triangles in the range of the table. That is, only one extra triangle.

angle A    (c/b)2                   a         c        b 
   p     q     n

44.76 1:59:00:15 1:59 2:49 2 12 5 1

44.25 1:56:56:58:14:50:06:15 56:07 1:20:25 57:36 1:04 27 2

43.79 1:55:07:41:15:33:45 1:16:41 1:50:49 1:20 1:15 32 3

43.27 1:53:10:29:32:52:16 3:31:49 5:09:01 3:45 2:05 54 4

42.08 1:48:54:01:40 1:05 1:37 1:12 9 4 5

41.54 1:47:06:41:40 5:19 8:01 6 20 9 6

40.32 1:43:11:56:28:26:40 38:11 59:01 45 54 25 7

39.77 1:41:33:45:14:03:45 13:19 20:49 16 32 15 8

38.72 1:38:33:36:36 8:01 12:49 10 25 12 9

37.44 1:35:10:02:28:27:24:26 1:22:41 2:16:01 1:48 1:21 40 10

36.87 1:33:45 3 5 4 2 1 11

35.78 1:31:09:09:25:42:02:15 3:12:09 5:28:41 4:26:40 2:05 1:04 **

34.98 1:29:21:54:02:15 27:59 48:49 40 48 25 12

33.86 1:27:00:03:45 2:41 4:49 4 15 8 13

33.26 1:25:48:51:35:06:40 29:31 53:49 45 50 27 14

31.89 1:23:13:46:40 56 1:46 1:30 9 5 15

Only one extra line. Perhaps it wasn't used because of the large values of q and b. Perhaps it wasn't copied into the table; there were, after all, four other errors. Under two other hypotheses, it ought to be in the table, too, since it fills in a missing angle and a gap in the (c/b)2 column.

David E. Joyce, 1995. (djoyce@...)
Department of Mathematics and Computer Science
Clark University

The Babylonian tablet Plimpton 322

This mathematical tablet was recovered from an unknown place in the Iraq desert. It can be determined, apparently from its style, that it was written originally sometime around 1800 BCE. It is now located at Columbia University.

[ An image at higher resolution ]

How to read it

The Babylonian number system. The Babylonians `wrote' on wet clay by pressing a stylus or wedge into it in one of a small number of ways. The Babylonian number system, already from a very early era, was remarkably sophisticated. The number 1 was `written' with a single `stroke' and the numbers 2 through 9 were written by combining multiples of a single stroke:

The number 10 was written in a single character and the numbers 20 to 50 were written with multiples of this character:

The numbers up to 59 were obtained by combining these much as we do. For example, 11 was written as . But the system was not actually based on 10, but rather on 60 - and it was a true floating point system so that 1 and 60 (as well as, say, 3600) were indistiguishable. All numbers could be obtained by combining these in order, writing from left to right as we do. Thus seventy was written as . Sometime before 300 BCE, but after Plimpton 322 was written, a special symbol was devised as a zero, but in Plimpton 322 there is potential confusion because of this problem. The conventional way to write floating point sexagesimal numbers is by using comma separators, so that 1,29 is 60+29 = 89 in decimal notation, and 1,1,1 is 3661.

The last two columns. At any rate, we now know almost enough to read the right hand columns of the tablet. The last column (with a few natural interpolations to take into account missing symbols for 5, 6, and 15, ) simply numbers the line of numerical data. In the fourth column is written the word pronounced ki which can be translated loosely here as number, so that the fourth and fifth column together (number 1, number 2, etc.) just enumerate the different lines of data. Interpolations are in green.

For the rest of the columns, we show in order the original column, the sexagesimal numbers written in conventional notation, and then the decimal equivalents. Error corrections are in red. Both interpolations and corrections are explained later on. All the zeroes are interpolated, also, since the Babylonians at this time did not have them.

Column two. The heading of the second column includes the word for width.

Column three. The heading of the third column includes the word for diagonal.

Column one. The heading of the first column has not been translated. We shall see in a moment the reasons for placing the `decimal point'.

What it means

Following perhaps the hints given by the words diagonal and width, Neugebauer and Sachs discovered that if w is the entry in the second column and d is the entry in the third column, then in all but a few cases d2-w2 turned out to be a perfect integer square l2. In other words, assuming the exceptions to be caused by error, this table contains part of a list of Pythagorean triples, that is to say integers w, l, d with

w2 + l2 = d2

which form the sides of right triangles. Here is the resulting table of calculations, in modern notation (with discrepancies in square brackets):

A bit later, we shall confirm this theory by explaining the errors.

As for the first column, it contains the values of d2 / l2 . For example, in the first row d = 169, l = 120, and d2/l2 = 1.9834 ... = 1,59,0,15.

How the table was produced

How are the Pythagorean triples in this table distinguished? If (A, B, C) is Pythagorean triple, then we can write it as (ma, mb, nc) where (a, b, c) is a primitive Pythagorean triple - one in which the numbers are relatively prime in pairs. Primitive Pythagorean triples are parametrized by pairs of intgers (p, q) satisfying these conditions:

  • p and q are both positive;
  • p is greater than q;
  • One of them is odd, the other even;
  • p and q are relatively prime.

The pair (p, q) gives rise to the triple (p2-q2, 2pq, p2+q2). The pair (p, q) can be easily recovered from (a, b, c) by the formulas p2 = (a + c)/2, q = b/2p.

Here are the values of p, q, and m for the triples in the table:

The ratio c/b is equal to (p2 + q2)/2pq = (1/2)(p/q + q/p). Therefore this ratio, the square of which appears in the first column of the tablet, will have a finite expression in base 60 if 1/p and 1/q do. The Babylonians almost certainly understood the difference between finite sexagesimal expansions and repeating ones, and in particular we have found tables of reciprocals 1/p for many values of p where the expansion is finite. Such numbers p are called regular by Neugebauer. It is not likely to be a coincidence that the values of both p and q associated to the rows of the tablet are regular, and in fact that in all but one case the expansions of 1/p and 1/q appear in the tables of reciprocals that have been found. It seems plausible, therefore, that the Babylonians knew how to generate primitive Pythagorean triples.

We know something, therefore, about how the tablet was constructed, but we do not know exactly why it was constructed. The ordering of the rows according to the size of the first column suggests that it might have been used in an early form of trigonometry. Perhaps it was constructed from Pythagorean triples just to make arithmetic easier.

However incomplete our present knowledge of Babylonian mathematics may be, so much is establshed beyond any doubt: we are dealing with a level of mathematical development which can in many aspects be compared with the mathematics, say, of the early Renaissance. (O. Neugebauer in The exact sciences in antiquity)

Accounting for the errors

As confirmation of both the interpretation of the table and this conjecture regarding p and q, the four apparent errors can be reasonably explained:

  • The number [9, 1] in row 9 should be [8, 1] - a simple copying error.
  • [7,12,1] in row 13 is the square of [2,41], which would be the correct value - a mistake particularly easy to make since the squares also appear in the conjectured calculation.
  • The correct value to replace [53] in row 15 is [1,46], which is twice the erroneous value.
  • As for the fourth error in row 2, where [3,12,1] occurs instead of [1,20,25], there have been a couple of solutions proposed. None are entirely convincing. The possibility proposed by Gillings suggests strongly that those who made up the table had values of p and q at hand.


  • O. Neugebauer and A. Sachs, Mathematical cuneiform texts, American Oriental Society, 1945. This, as far as I know, is where the tablet was first analyzed.
  • O. Neugebauer, The exact sciences in antiquity, Dover, 1969. This is a reprint of the 1957 second edition. Chapter 2 spends a great deal of effort on the tablet.
  • R. J. Gillings, Australian Journal of Science 16 (1953), pages 54-56. Mentiuoned in Neugebauer's book.
  • R. Creighton Buck, Sherlock Holmes in Babylon, the Mathematical Monthly, 1964.
  • The Akkadian home page. The Internet has a large collection of material on cuneiform and Bablyonian tablets, although not much concerned with mathematics, and no recent photographs of the tablet Plimpton 322.

Written by Bill Casselman.

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