COMMENTS AND EXAMPLES
The following provides a brief introduction and a few examples for
those unfamiliar with the sexagesimal system and the Babylonian
astronomical and mathematical cuneiform texts. Additional information
may be obtained from Otto Neugebauer's The Exact Sciences in
Antiquity (Barnes & Noble, New York, 1993), Bartel Van Der
Waerden's Science Awakening
II (Oxford University Press, New York, 1974), Neugebauer's Mathematical
Cuneiform Texts (with A. Sachs: Sources in the History of
Mathematics and the Physical Sciences 5, Springer-Verlag, Berlin,
1983).
The notation applied here (semi-colons separate values above zero, commas successive sexagesimal places ) generally follows that used by Otto Neugebauer in his monumental three-volume work Astronomical Cuneiform Texts (ACT, Lund Humphreys, London, 1955). It appears necessary to point out, however, that although this source provides a fund of details and insights concerning Babylonian astronomy, it is unfortunately largely unreadable at first acquaintance because of unknown terms and unusual methodology.
Most readers will already be acquainted with base-60 notation, if only from its use in modern time-keeping (hours, minutes, seconds, and elapsed time, etc.) and will in all likelihood also be familiar with the square root of two. It is safe to suggest, however, that few could express theSQUARE ROOT of TWO HOURS in HOURS, MINUTES, and SECONDS. The answer is approximately one hour and twenty-five minutes (e.g.,1;25 hours squared = 2;00,25 or two hours, zero minutes, and twenty-five seconds). A better value that is both convenient and reasonably accurate, however, would be one hour, twenty-four minutes, fifty-one seconds and ten sixtieths of a second, i.e., 1;24,51,10 (1.414212962,...). This was in fact the constant applied by Babylonian mathematicians in the Old Babylonian Period [1900 BC - 1650 BC] to obtain the diagonal of a square of side a, i.e., a (1;24,51,10) or what we today would understand to be the use of a "Pythagorean" square root obtained from: a ^{2}+ a ^{2} = 2a ^{2} (cuneiform text YBC 7289). The actual square root of 2 is: 1;24,51,10, 07,46,06,04,44... Here the sexagesimal divisions continue past the "seconds" position, which is generally where modern base-10 and Babylonian base-60 notations diverge with the former inconsistently switching to tenths and hundredths of a second, etc.
A system with a base of 60 might appear cumbersome, but it has distinct advantages, as most modern users have come to recognize, especially with respect to the number of integer divisors and associated reciprocals, i.e., 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 (base-10 has only the related pair 2 and 5). Convenient divisors are not limited to integers alone, but although the equivalents of recurring decimal fractions such as 0.333* and 0.666* are simply 0;20 and 0;40, awkward recurring sexagesimal numbers also exist, e.g., the fractions: 1/7 = 0;08,34,17,08,34,17,08,34,.. and 5760/177 = 32;32,32,32,32,.., the former perhaps influencing the choice of 3 (or 25/8) rather than the fraction 22/7 for PI. On a more practical note, the accurate (but still rounded?) Babylonian mean synodic month of 29;31,50,8,20 days (29.530594...days; modern value: 29.5305903...days) provided a fundamental unit of time and motion in Seleucid Era astronomy which could be replaced by an equivalent interval of 30 "mean lunar days" (tithis) to simplify computation. From a modern viewpoint the duration of one "tithi" would be obtained from the mindless division of 29;31,50,8,20 days by 30. However, instead of starting with "29" and carrying out successive divisions in the modern manner, all that is necessary is to start at the other end, multiply by 2, and shift the sexagesimal place accordingly to obtain the required result of 0;59,03,40,16,40 tithi per day. This is a very trivial example; for a vastly more complex application see Friberg's "algorithm for the factorization of a regular sexagesimal number n (or reduction of a semi-regular number)" in his paper "On the Big 6-Place Table of Reciprocals and Squares from Seleucid Babylon and Uruk, and their Old Babylonian and Sumerian Predecessors," (SUMER, Vol.42, 1986) 81-87).
In the sexagesimal system, the extended use of reciprocals effectively reduces division and multiplication to a single operation, i.e., multiplication by numbers which are either larger or smaller than unity. This concept was maximized by Babylonian multiplication tables which extended from 1 through 19 with further entries for 20, 30, 40, and 50 to cover the entire range between 0 and 60. Babylonian mathematical tables were not, however, confined to simple values, as atypical multiplication table U91 (Istanbul) attests (Aaboe, Journal of Cuneiform Studies, Vol. 22, 1969, pp. 88-91). Although unrecognized, U91 also appears to include the tabulated slope for the "trapezoid" mentioned in two Seleucid Era astronomical procedure texts for Jupiter (ACT 813 Section 5, and ACT 817 Section 4) and quite possibly slopes for similar trapezoids associated with the four remaining planets.
Returning to the substitution of
30 tithi for the mean synodic month of 29;31,50,08,20 days, the use of
the conveniently rounded year of 12;22,08 mean synodic months permitted
Babylonian astronomers to replace the awkward unit of daily motion of
1;00,52,36,22,57,24,.. days per degree (i.e., 12;22,08 x 29;31,50,08,20
days divided by 360 degrees) by a much more convenient constant,
namely: 12;22,08 x 30 ^{r} / 360 degrees, which
for
tithi per degree (^{r/0}) reduces further to
12;22,08 / 12, or as explained above, simply 12;22,08 x
"5" = 1;01,50,40^{r/}^{0}.
Applied to the motion of
Jupiter this
method nevertheless produced a mean synodic period of 13;30,27,46,40
mean
synodic months from ACT 812, Section 2 and 13;30,27,46 (stated
in
Section 1 but oddly unrecognized by Neugebauer). Once so obtained the
period
could finally be converted to days by multiplying by 29;31,50,08,20
(the
result is: 398;53,27,10,09,24,.. the decimal equivalent is
398.8908803559.., which compares well with modern estimates). The same
method was employed to
compute the times for the varying synodic velocities; it goes without
saying
that rounded parameters, convenient shortcuts, and simplified
methodology suggest more than a passing acquaintance with the phenomena
and problems under
consideration.
More significantly (and also a
latter-day puzzle), few if any modern
commentators appear to have recognized the most obvious feature of the
above, i.e., that it is the direct sidereal motion per unit time
of Jupiter that is being computed. In detail, the Babylonian
mean synodic arc of Jupiter of 33;8,45 degrees is accomplished in
13:30,27,46,40 mean synodic months of 29;31,50,8,20
days. Thus the mean daily motion will be 0;4,59,8,31,3,28,53,..
degrees
and hence the time required for Jupiter to complete 360 degrees, i.e., one
mean sidereal period of revolution will be
4332;23,28,0,21,45,30,..days. Divided by the best estimate for the
Babylonian year of 365.25646981187.. days (obtained from the integral
relationship between the Babylonian mean synodic and mean sidereal
months) the result is 11.861230577 years, indisputably the mean
sidereal period of Jupiter.
It is also a slightly more
accurate value
(11;51,40,25,48 years) than the period of 11.86111* years (11;51,40)
obtained
from the Babylonian integer period relationship for 427 years, which
corresponds to
36 mean sidereal periods, 391 mean synodic periods and a rounded mean
synodic
arc of 33;8,45, degrees. As for the interval of 13:30,27,46 months in
ACT 812 Section 1--a value that Neugebauer found to be "completely
dark"--
this is most simply, sensibly and reasonably understood in essentially
heliocentric terms,
i.e.,
the time expressed in mean synodic months for Earth to move
one
complete sidereal
revolution of 360 degrees plus the additional 33;8,45 degrees for the
mean synodic arc of Jupiter. In other
words,
the mean lap time (i.e., mean synodic period) of Jupiter from a modern
heliocentric
viewpoint.
For further details on the heliocentric nature of Babylonian planetary
theory
see the link below.
Even 1500 years or so
before the Seleucid Era Babylonian mathematicians were already
utilizing tables
of squares and cubes, and they were also capable of extracting cube
roots, although not without peculiarities, as Sachs (Journal of
Cuneiform Studies, Vol. 6, 1952, pp.151-156) and Muroi (Centaurus,
Vol. 31, 1989, pp.181-188) have noted. For more on this last issue and
the attendant implications, see "On the Babylonian Method
of extracting
Root Squares" by Vilma A.S. Sant'Anna and Adonai S. Saint'Anna.
It is uncertain how the Babylonians obtained their approximation for the square root of two, but it has been suggested that a Babylonian predecessor of Newton's iterative method may have been employed, albeit predating the latter by some 3000 years. Although unattested, it may also be relevant to note here that the iterative approach can easily be expanded beyond square and cube roots since both are special cases (for n = 2 and n = 3 respectively) of the general iterative formula for the Nth root of x, i.e.,
Nth Root: 1/n{(n-1)Estimate + x/(Estimate^{ n - 1})}
1. Take one half of the difference 1, the result is 0;30 [ Hold the result in your hand ]and secondly in decimal notation:
2. Take the half-difference and square it, the result is 0;15
3. Take the 0;15 and add it to the area 1, the result is 1;15
4. Take the square root of 1;15, the result is 1;7,4,55,20,29,39,6,54
5. Add the half 0;30 (from step1) to the square root, the result is 1;37,4,55,20,29,39,6,54
6. What value when multiplied by 1;37,4,55,20,29,39,6,54 gives 1 (the area)?
7. 1;37,4,55,20,29,39,6,54 multiplied by 0;37,4,55,20,29,39,6,54 gives 1
8. 1;37,4,55,20,29,39,6,54 is the Length, 0;37,4,55,20,29,39,6,54is the Width.
1. Take one half of the difference 1, the result is 0.5 [ carry the result ]
2. Take the half-difference and square it, the square is 0.25
3. Take the 0.25 and add it to the area 1, the sum is 1.25
4. Take the square root of 1.25, which is 1.118033989
5. Add the 0.5 (from step1) to the last square root to obtain 1.618033989
6. What value when multiplied by 1.618033989 results in an area of 1?
7. 1.618033989 multiplied by 0.618033989 gives 1
8. 1.618033989 is the Length, 0.618033989 is the Width.
The Babylonians utilized "pythagorean" sets well before the time of Pythagoras, applied algorithms to solve a variety of linear equations (including cubics and quadratics), possessed a knowledge of logarithms and also carried out mathematical operations of still unknown significance during the earliest period (see: J. Friberg, "Methods and Traditions of Babylonian Mathematics II: An Old Babylonian Catalogue Text with Equations for Squares and Circles," Journal of Cuneiform Studies, Vol. 33, 1981, pp. 57-64). In some respects certain Babylonian mathematical tables from the later Seleucid Era are quite mystifying, especially those given to more than twelve sexagesimal places (see: John P. Britton, "A Table of 4th Powers and Related Texts from Seleucid Babylon," Journal of Cuneiform Studies Vol. 43-45, 1991-93, pp.71-87). To put the latter in meaningful perspective, working to even the ninth sexagesimal place corresponds to an accuracy of better than to 10^{-16}.
Babylonian computational methodology may be considered merely "arithmetical" by some, but this is surely a vast over-simplification and there are in addition enormous time-scales involved in its possible refinement and development. Moreover, it is not known what percentage of Babylonian methodology and understanding is represented by the extant material. Consequently, what is known with respect to the astronomical cuneiform texts of the Seleucid Era should, perhaps, be judged mainly on its practical merit, namely the simple yet successful description of complex celestial phenomena associated with the mean, varying, and apparent motions of the five known planets, Sun and Moon.
Lastly, for more on the
sophisticated nature of Babylonian methodology see:Aspects of
Abstraction in Mesopotamian Mathematics by Matheus da Rocha
Grasselli, Instituto de Fisca da Universidade de Sao Paulo, Brasil.1996.
Bibliography
of Mesopotamian Mathematics
Babylonian Planetary Theory and the Heliocentric Concept.
Last Updated on April 2, 2009. Open University Links added February 3, 2013