 BABYLONIAN MATHEMATICS AND SEXAGESIMAL NOTATION

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;01,50,40r/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)}
To what ends such techniques were routinely applied remains conjectural. Nevertheless, it seems possible that the likely combination of standard Babylonian procedures could have produced most profound results, e.g., the practical solution to the problem of finding the length and the width of a rectangle with an area of 1 and a difference of 1 between the two sides. Such problems in themselves are simple enough, e.g., the area of a rectangular field with sides of unknown length and width is given, along with the difference between the latter dimensions which are to be determined, i.e., in modern notation: xy = C and x-y = d where both C and d are given as 1 and the dimensions of x and y are to be found. The attested Babylonian solution is similar to the modern equivalent of solving a quadratic equation, albeit by algorithmic means. Thus firstly in sexagesimal notation to the 8th sexagesimal place:
1. Take one half of the difference 1, the result is 0;30 [ Hold the result in your hand ]
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.
and secondly in decimal notation:
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.
Thus the length and the width are found to be 1.618033989 and 0.618033989 respectively, i.e., Phi and the reciprocal of Phi. Moreover, with unity in between, one also obtains three consecutive values from the Phi-Series itself. Not that this is to be taken as historical fact by any means, but the precise determination undoubtedly remains feasible using attested methods from the Old Babylonian period [1900-1600 BCE]. For more on the above topic see the recent publication  Lengths, Widths, Surfaces A Portrait of Old Babylonian Algebra and its Kin by Jens Hoyrup.

On a technical note, the above treatment is linguistically imprecise and more verbose than the instructions given line-by-line in mathematical texts of the Old Babylonian Era; the use of the semi-colon to denote the equivalent of the decimal point is also a modern addition for clarity. The square root of 5/4 given above as 1;7,4,55,20,29,39,6,54 is accurate to 14 decimal places but nevertheless readily obtainable from attested Babylonian procedures from the same era, specifically by the fourth iteration of the Babylonian version of "Newton's" method for approximating square roots. Rounding at the sixth and fifth places would also provide useful if less accurate sexagesimal approximations for Phi of: 1;37,4,55,20,29,40 and 1;37,4,55,20,30 respectively.

As for the Babylonian approach itself, from a modern viewpoint one could suggest that it results from an awareness that for problems of this nature the quadratic formula: can be split into two parts, i.e., with a = 1, into b/2 to be added to the other part, which (if the 4 inside the square root is brought outside and cancelled by the divisor 2) also includes b/2 already calculated, as expressed in the above procedure. Thus the Babylonian algorithm is essentially the simplified form: where b is the difference between the length and the width, and c is their product, the area. Which (although still unattested in the present context) suggests in turn that the simplest way to obtain Phi is from the addition of one half to the square root of five-fourths.

This is not to say that this was how the procedure was arrived at by Babylonian mathematicians. But what can be suggested here is that it takes a fair degree of competency whichever way one looks at it, and this proves to be true in other aspects of Babylonian methodology, including the techniques laid out in the Babylonian astronomical cuneiform texts of the much later Seleucid Era [ 310 BCE - 75 CE ].

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.