Asunto:  [TA] Mas Matematicas babilonias  Fecha:  Miercoles, 4 de Agosto, 2004 21:41:48 (+0200)  Autor:  Ana Maria Vazquez <avhoys @.....es>

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. ... wwwgap.dcs.stand.ac.uk/~history/
HistTopics/Babylonian_Pythagoras.html  20k 
http://
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.
... www.ams.org/newinmath/022001media.html  14k  En caché  Páginas similares
Pagina nueva 1 Cuestiones. Tablilla Plimpton 322. ... www.personal.us.es/cmaza/mesopotamia/plimpton.htm  95k  En caché  Páginas similares
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. ... www.seshat.ch/home/babylon.htm  88k  En caché  Páginas similares
plimpton 322  [ Traduzca esta página ] Tablette Plimpton
322, Université de Columbia, USA. © Serge MEHL. www.sciencesenligne.com/
momo/chronomath/anx3/plimpton.html  2k  En caché  Páginas similares
Hay 10 páginas más, si alguien quiere saber más.
aleph0.clarku.edu/~djoyce/mathhist/plimpnote.html
PLIMPTON 322
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.
BIBLIOGRAFĎA
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), 629632.
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),
335345.
Jöran Fribert. Methods and traditions of Babylonian Mathematics I: Plimpton
322, Pythagorean triples, and the Babylonian triangle parameter equations.
Historia Math. 8 (1981), 277318.
 H Eves, An Introduction to the History of Mathematics, Saunders College
Publishing, 1990.
 R J Gillings, The Australian Journal of Science, 16(1953):3436.
 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) 219231.
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 = p^{2}  q^{2}, b = 2pq, and
c = p^{2} + q^{2}.
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 d^{2}w^{2} turned out
to be a perfect integer square l^{2}. 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
w^{2} + l^{2} = d^{2}
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 d^{2} / l^{2} . For example,
in the first row d = 169, l = 120, and d^{2}/l^{2} = 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 (p^{2}q^{2}, 2pq, p^{2}+q^{2}). The
pair (p, q) can be easily recovered from (a, b, c) by the formulas p^{2} = (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 (p^{2} + q^{2})/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.
References
 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 5456.
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.
http://www.math.ubc.ca/~cass/courses/m44603/pl322/pl322.html
