babylonian planetary theory and the heliocentric concept

27/02/11 18:05Babylonian Planetary Theory and the Heliocentric Concept

Pagina 1 van 24http://www.spirasolaris.ca/sbb2c.html

BABYLONIAN PLANETARY THEORY AND THE HELIOCENTRICCONCEPT

I. PERIOD RELATIONS, PARAMETERS, AND PLANETARY THEORIES

OVERVIEW Although the numerical methods and parameters found in the Babylonian AstronomicalProcedure texts and Ephemerides of the Seleucid Era [310 B.C. - 75 A.D.] have been describedin some detail, notably by Neugebauer (1955),1 (1975), 2 Van der Waerden (1974) 3 and others, itis far from certain whether the extant material represents the state of Babylonian astronomy perse, or merely scattered remnants of a larger corpus of knowledge. Moreover, what has beenreclaimed can hardly be considered sequential or self-explanatory, while base-60 notation,unusual terminology, little-known phenomena and proclamations concerning the lack of a fictivemodel for Babylonian astronomy all prevent easy assimilation of the details. Then again, theinformation that has come down to us is itself scattered and fragmentary, ranging from earlier"Omens" through the detailed astronomical cuneiform texts of the Seleucid Era. In the lattercontext results exist in the form of "ephemerides" and at least part of the methodology isdescribed (albeit in condensed form) in a number of related "procedure" texts, though none areparticularly simple in the first place nor necessarily complete in the second. The recovery of theBabylonian astronomical cuneiform texts is also relatively recent (late 19th century onwards) asis the current understanding of their contents--an understanding unfortunately complicated bythe necessary inclusion of Babylonian methodology within long-established time frames and theperceived flow of Greek astronomical thought. But while admiring Babylonian methodology mostcommentators nevertheless appear unwilling to grant that the Babylonians really understoodwhat they were doing, or that they ever proceeded to determine a fictive planetary model. It istrue (as far as is known) that the Babylonians did not use trignometrical methods in theirastronomy, but then again neither did they confine their treatment of the planets to meancircular motion, or use auxiliary devices to reinforce unsupportable geocentric premises either.But there may be more sides to this issue in any event. We can hardly claim to know what exactprerogatives produced the known remnants of Babylonian astronomy from the Seleucid Era, andmoreover, as evidenced by mathematical cuneiform texts of the Old Babylonian Period [1900-1650 BCE] Babylonian mathematics had already reached a remarkably high level some 1500year earlier. Unfortunately there are no astronomical texts from the earlier period comparable tothose of the Seleucid Era, but there is little doubt that a sufficiently high level of mathematicswas already in place. Thus there are considerable gaps--gaps during which time who knowswhat manner of investigations were carried out and what conclusions were reached, held,discarded and also perhaps passed down. Unfortunately, short of the recovery of further texts itis likely that we will never know, but nevertheless we can at least examine Babylonianastronomy in terms of its own distinctive successes in accounting for regular variations in bothluni-solar and planetary motion.



Fig. 1. The Seleucid Era and Later Planetary Theories

With the above in mind one of the main purposes of the present paper is to simplify matterswherever possible, and like it or not, the simplest way to achieve this gaol is to treat Babylonianmethodology from a fictive heliocentric viewpoint. And indeed why not? This is surely a perfectlyreasonable premise for the period in question. After all, it includes the time of Aristarchus ofSamos, and moreover, it is also generally accepted that single, modified period relations ofBabylonian origin provide the underlying bases for the later and diverse planetary planetarymodels proposed by Ptolemy, Al-Bitruji and Copernicus. But while the latter all employeduniform circular motion and auxiliary devices of one sort or another to account for variations inplanetary velocity, the Babylonians for their part appear to have determined fundamental periodrelationships subsequently applied in numerical schemes concerned with fixed, mean andvarying orbital motion. This said, it is not the intention here to give a detailed comparativeanalysis of the later applications, but rather point out that there are distinct and criticaldifferences between the latter and the more extensive corpus of Babylonian planetary periodrelations. There no doubt remain champions of the status quo who will dutifully insist thatBabylonian methodology represents mere counting and that their approach to planetary motionwas entirely non-fictive. However, it is safe to suggest that none could explain how theBabylonians were able to differentiate the sidereal, anomalistic and draconic months from thesynodic month by simple counting. Why stress luni-solar parameters when discussingBabylonian planetary theory and the heliocentric concept? Simply stated, the two areinextricably linked--not merely because of the fundamental units of time applied in Babylonianastronomy (days, tithi, mean synodic months and years) but also the inescapable fact that onlythe synodic month is directly observable. All the rest must be deduced, and this could hardly hadbeen accomplished without applied conceptual reasoning and understanding--an understandingthat would reasonably include planetary motion, though it need not have developed in preciselythis way. In one sense the very simplicity of Babylonian methodology is itself misleading; buttheir concern with characteristic synodic phenomena leads naturally enough to an awareness of



varying planetary motion, along with an understanding of the apparent retrogradations,stationary points and dates of appearance and disappearance, etc. And after gatheringextensive sets of planetary period relations and generating various schemes to account forsuccessive synodic phenomena and variations in velocity, it seems highly unlikely that theBabylonians managed to do so without developing a fictive model of any kind.

PERIOD RELATIONS Apart from the terminology and notation, the extant planetary material of the Seleucid Erasupplies reasonably straightforward procedures that are both instructive and informative. Forexample, for the three superior planets the Babylonians appear to have determined finalplanetary period relationships based on pairs of integer periods (called here T1 and T2) close tothe mean sidereal period (or multiples thereof) with corresponding numbers of synodic periodsand small, convenient corrections for longitude of opposite sign. In the case of Jupiter,Babylonian period relationships of 12 Years, 71 years, 83 Years, 95 years, 166 years and 261years result in a final integer period relation of 427 years to which correspond 391 meansynodic arcs and 36 sidereal revolutions.4 The inter-relationship between the initial pair and thefinal 427-year period for Jupiter is shown in Table 1, the essence of a letter to the Editor of ISIS5 published 26 years ago in 1977 in an unsuccessful bid to not only "close the circle" withrespect to Babylonian orbits, but also to raise the issue of the heliocentric nature of Babylonianastronomy itself.

Table.1 Babylonian Period Relations and the 427-year Period for Jupiter

Here the Babylonian 12-year and 71-year period relations are clearly intermediate in nature withpositive and negative corrections for longitude that permit the generation of all the rest en routeto the disarmingly simple 427-year final period (for the Babylonian application and use of thisresult see below). In this context the Babylonian period relations for 12 and 71 years (T1 and T2respectively) represent an initial set, with neither one capable of generating particularly accurateresults. Here the 71-year period corresponds to slightly less than 6 sidereal periods and 65mean synodic arcs, i.e., six sidereal revolutions less a small correction; for the latter theBabylonians used two sets for both the fast and slow segments of the orbit, i.e.,

71 years corresponds to 6 x 360 - 5;00 degrees, 65 mean synodic periods/meansynodic arcs 71 years corresponds to 6 x 360 - 4;50 degrees, 65 mean synodic periods/meansynodic arcs



The historical significance of the above is seen in its similarity to the single period relationshipfor Jupiter utilized in Ptolemy's geocentric planetary model, i.e.,

71 Years - 4;54 days = 6 x 360 - 4;50 degrees = 65 Cycles (Anomaly) 7

Moreover, although applied to the heliocentric concept, a similar variant was used in turn byNicholas Copernicus, i.e.,

(71 Years - 5;54,13 days = 6 x 360 - 5;42,32 degrees = 65 Rotations in parallax 8 )

while a further close variant was applied by Al-Bitruji. Without going into greater detail, for Marsthe Babylonian initial and final periods with attendant small corrections for longitude of oppositesign (six sets) appear to have been: T1 = 47 years, T2 = 79 years for a final integer relationshipof: 284 years, 133 mean synodic periods and 151 sidereal periods.For Saturn the corresponding periods appear to have been T1 = 29 years, T2 = 59 years,leading in turn to a final integer period relationship of 265 Years, 256 mean synodic periods and9 sidereal revolutions. Nevertheless, instead of the 284-year and 265-year final periods, all three later planetarytheories used modified variants of the 79-year and 59-year T2 periods alone for Mars andSaturn respectively while further Babylonian periods appear to have been applied for the twoinferior planets. Thus there exist three quite different fictive planetary models--two geocentricand one heliocentric--based on similar, fractional and quite possibly misunderstood periodrelations of undoubted Babylonian origin.

At first acquaintance one might think that the 71-year period relation used by Ptolemyrepresents a refined and superior approach, but detailed examination reveals that this is notnecessarily the case. In fact, instead of contriving to fit an inflexible geocentric framework anduniform circular motion to varying orbital motion and sequential synodic phenomena, theBabylonians delineated the latter with remarkable clarity and simplicity. Simple, yes, but hardlymindless or non-fictive. The planets do indeed move with varying velocity and the varioussynodic motions (i.e., relative motion as observed from Earth) do indeed exhibit apparentstationary points, apparent retrograde motions, along with first and last appearances in the eastand the west as Earth moves around the Sun. The operative word here, of course, is "apparent"since to "save the phenomena" the successful accounting of the apparent motions of theplanets is of paramount importance. Thus the method applied by Ptolemy wins hands down?Hardly; for none of Ptolemy's impressive looking planetary period relations produce the claimedmotions in either longitude or anomaly, as Robert R. Newton pointed out in 1977 in The Crime ofClaudius Ptolemy 9 and other related works.10, 11 Given to the sixth sexagesimal place and long touted as the epitome of accuracy, Ptolemy'sdaily planetary velocities in fact diverge from the stated values at or beyond the fourthsexagesimal place. Small errors? Perhaps, but hardly insignificant. If the phenomena cannot besaved by the data, then Ptolemy's geocentric model cannot be upheld either, never mind itsfundamentally incorrect nature, its contrived use of uniform circular motion and cumbersomeauxiliary devices. Nor is it a question of the precise value of the year used by Ptolemy in hisperiod relations either. Simply stated, there is no one single value for the year that willsimultaneously correct the deviations in the cited velocities. Why the difference between the Babylonian and later applications? A complex question, nodoubt, but one can suggest a number of factors that may have played their various roles, notleast of all the intrusion of religious dogma on the scientific process and the fact that the laterapplications were all inherited, partial data, whereas the Babylonians were themselves theoriginators and the collators of the original material.



Frankly, one might have hoped that Newton's detailed analysis (one of the few original works onthe subject since Al-Bitruji's earlier criticisms12 ) might have helped generate fresh interest inBabylonian methodology, but unfortunately the subject still remains largely overshadowed bythe Ptolemaic system, despite the latter's clear inaccuracies, fundamentally incorrect premisesand dubious heritage. As for the earlier Babylonian approach, even a brief acquaintance withthe parameters and the methodology should serve to raise a number of questions, not least ofall how the notion that Babylonian astronomy lacked of a fictive planetary theory ever arose, letalone how it came to take root.

II. LUNI-SOLAR PARAMETERS AND PRECESSION One of the fundamental units of time and motion applied in Babylonian astronomy appears tohave been the mean synodic month of 29;31,50,8,20 days.13 This accurate Babylonian constant(29.53059413 days; modern estimate: 29.53059027) is applied to mean and varying orbitalvelocities in both planetary and luni-solar contexts. The latter includes further attestedBabylonian months, e.g., the mean sidereal month of 27;19,18 days (27.3216667),14 theanomalistic month of 27;33,16,20 days (27.5545370) 15 and the draconic month of 27;12,43,56days (27.21220370)16 in complex numerical contexts concerned largely with the 223-monthSaros eclipse cycle. But the real significance of the inclusion of both the synodic and siderealmonths in this context lies firstly in their difference, which for the mean values provides themean motion of Earth and an implicit Babylonian sidereal year of 365.25646981 days (modern sidereal year: 365.25635674 days). Although the implicit value for the sidereal year mentioned above is not attested in theBabylonian material, a year of 365;14,44,51 days (365.24579166) is nevertheless mentioned ina lunar procedure text.17 The calendaric year on the other hand was obtained from a 19-year,235 mean synodic month relationship (365.24682220 days) while the conveniently rounded yearof 12;22,8 mean synodic months (365.26063766 days) appears to have been more generallyemployed in astronomical contexts. But apart from Hartner's obliquely asserted Babylonianestimate for annual precession of approximately 45 seconds of arc,18it is not generallyacknowledged that the Babylonians differentiated between the tropical and sidereal years at all.However, the latter Babylonian pair could also represent convenient approximations that aregreater than their modern equivalents by 0.43% and 0.46% respectively. As a consequence,they implicitly maintain the correct relationship between the two types of years with a differencethat yields an excellent value for annual precession of slightly more than 49 seconds of arc. Andindeed why not; with the heliocentric concept deduced and firmly in place all manner of detailscould be investigated and over time amply refined. But like much of the extant material, the various lengths of the "year" in Babylonian astronomydeserve more than casual consideration. The rounded Babylonian estimate for year of 12;22,8mean synodic months is almost certainly one of convenience but it nevertheless has subtleunderpinnings with 360 degrees of motion (solar or terrestrial) corresponding to a mean monthlyprogress of 29;06,19,00,55,37,24,... degrees. This may be compared to the mean monthly arcof 29;6,19,20 degrees 19 utilized in a Babylonian System B variable velocity function thatincreased and decreased between minimum and maximum monthly arcs of m = 28;10,39,40degrees and M = 30;1,59,20 degrees respectively. The division of 360 degrees by the meanvalue (29;06,19,20) results in a year of 12;22,7,51,54,7,4,..months (365.2595295509 days) witha corresponding mean daily velocity of 0;59,8,9,43,22,7,. degrees. Abbreviated at the fifthsexagesimal place (from "22" to "20"), the latter produces a corresponding year of365.2595305603 days (the modern anomalistic year is 365.259641204 days). This dailyparameter may also have been employed in a daily System B velocity function determined by



parameter may also have been employed in a daily System B velocity function determined byAaboe 20 from a fragmentary set of undated solar longitudes which apparently increased anddecreased by 0;00,01,43,42,13,20 (0.0004801097) degrees per day . Although unattested, theresulting extrema for a mean daily velocity (u) of 0;59,8,9,43,20 degrees per day would have hada maximum daily velocity (M) = 1;01,45,59,24,50 degrees and minimum daily velocity (m) of0;56,30,20,01,50 degrees respectively.

III. THE CLIMATE OF THE TIME Just why the notion that Babylonian astronomers possessed no fictive planetary model of theirown persists is unclear, especially since the variable velocity functions used to account for themotions of the Sun (or Earth), Moon and the three known superior planets suggest that theBabylonians were at least half right in their approach to planetary motion. Moreover, the variousBabylonian schemes employed to concurrently describe the uninterrupted synodic arc in termsof forward motion, stationary points and retrogradations prove on further examination to be morethan sufficient to "save the phenomena." Nor should this be any real surprise given the "definiteopinion" of Seleucus (ca. 150 B.C.) on this matter as recorded by Plutarch (On the Face in theMoon's Orb):

Did Plato put the Earth in motion as he did the sun, the moon and the five planets which hecalled 'the instruments of time' on account of their turnings, and was it necessary to conceivethat the Earth ... was not represented as being (merely) held together and at rest but asturning and revolving, as Aristarchus and Seleucus afterwards maintained that it did, theformer of whom stated this as only a hypothesis, the latter as a definite opinion? (emphasisadded)

The attested subdivision of integer multiples of 360 degrees of uninterrupted sidereal motion bythe number of synodic occurrences in the final Babylonian period relationships result in thedetermination of the mean synodic arcs. But such subdivisions can hardly take place withoutrespect to a center if they are to have any meaning whatsoever. As a consequence, it is naturaland necessary to ask whether a common center can be found in Babylonian astronomy, and ifso, where the common center might lie. Furthermore, if a possibly out-of-context fraction of theBabylonian material sufficed to provide the frameworks for the later planetary theories ofPtolemy, Al-Bitruji, and Copernicus, then what would have prevented the Babylonians - theoriginators, observers, and collators of extended sets of periods and related data - fromdeveloping a fictive planetary model of their own? Babylonian methodology clearly involved bothsidereal and synodic motion, and although the latter was subdivided into "characteristicphenomena" rarely applied in modern astronomy the concept is nevertheless useful and logicalin both its execution and its outcome.

Moreover, even the simplest Babylonian treatment of varying synodic velocity involved the division of360 degrees of sidereal motion into "Fast" and "Slow" arcs while Babylonian varying velocity functions(System B) are obviously quite sophisticated, especially in the case of Jupiter, with the line of apsidesand mean velocity defined to one degree. The corresponding maximum, mean, and minimum synodicvelocities for Jupiter were determined to be 38;02 degrees, 33;08,45 degrees and 28;15,30 degreesrespectively, with a rate of change of velocity that was understood to increase (or decrease, depending onlocation) by 1;48 degrees per synodic cycle. Although the available information for Saturn and Mars isless extensive, it is now known that the Babylonians determined System B functions for both. Given therelatively high eccentricity, swift motion and proximity to Earth, this was no mean achievement in thecase of Mars. Although conjectural, it also appears possible that System B was similarly determined forMercury (as deduced from ACT 816, pp. 425-428; for further details see the Methodology below).



IV. BABYLONIAN PERIOD RELATIONS As noted earlier, the Babylonian fundamental period relationships for the three superior planetsappear to have depended on the selection of two integer periods (T1 and T2) close to meansidereal periods (or multiples) for which small, convenient corrections for longitude of oppositesign were determined. The frame of reference for these corrections was provided by some 33"normal" or "Goal-Year" Stars distributed along and around the ecliptic. The details are found inthe "Goal-Year" Texts 21 while intriguing atypical examples that include latitude provide furtherinsights.22

The key period relations for Jupiter are given in Section 1 of ACT 813 (translator: A. Sachs): 23

Compute for the whole zodiac (or: for each sign) according to the day and the velocity. In 12years you add 4;10, in 1,11 years you subtract 5, in 7,7 years the longitude (returns) to itsoriginal longitude

The Babylonians possessed two sets of initial corrections assigned to the fast and the slowsynodic arcs; the second correction in longitude given above (5;00 degrees) concerns theformer; for the slow arc the correction was the 4;50 degrees above in association with the 71-year period relation. The full set of periods for Jupiter are given in ACT 813, Section 20 24

namely intervals of 12, 71, 83, 95, 166, 261 and 427 years (7,7) leading to a final integer periodrelationship to which corresponded 36 sidereal revolutions, 391 synodic periods, and a totalsidereal motion for the 427-year interval of 36 x 360 degrees (3,36,0) with the meansynodic arc of 33;8,45 degrees as explicitly stated in ACT 813, Section 21:

"[7,7, years (corresponds to) 6,31 appearances ] 36 rotations, 3,36,0 motion. 33,8,[4]5 (is the)mean value of the longitudes."

or more simply in decimal notation and general terms:

427 years corresponds to 391 mean synodic appearances, 36 sidereal revolutions, 12,960degrees total sidereal progress, and 33;8,45 degrees (rounded) for the mean synodic arc.

In Neugebauer's terminology (ACT, pp. 282-283), the relationship is expressed as: N Years = IIsynodic "appearances" and Z sidereal "rotations" of 360 degrees, although the use of "rotation"in this context is fundamentally inappropriate since the latter undoubtedly represent siderealrevolutions. Nor can there be any doubt that for the above to have any meaning the siderealrevolutions in question must represent closed orbits, thus the revolutions must necessarily takeplace with respect to a specific centre, as indeed must the mean synodic arc for it to have anymeaning whatsoever. To which may also be added the attested Babylonian awareness to withinone degree of what we today recognize to be the line of apsides, along with the location of theline that corresponded to the mean values. At which point one begins to suspect thatNeugebauer's claim that the Babylonians never possessed a fictive approach to planetarymotion was not only premature, but also likely erroneous.

To continue, the mean synodic arcs for both Jupiter and Mars were apparently rounded at thethird sexagesimal place (in the present case 33;8,45 rounded from: Z x 360 / II =33;8,44,48,29,...degrees). It is generally understood that the number of mean synodic arcs (II)can be obtained from the relation: II = N - Z. The determination of the Final, or "long" babylonianperiod is therefore simply an intermediate step to firstly obtain mean values. The next steps



concern the detemination of the variable velocities and the variable times according toBabylonian System A or System B methodology.

Expressed in tithis and synodic months the synodic times for System B were also derivedaccording to the convoluted method provided in Section 2 of Jupiter text ACT 812 25 involvingthirtieths of the mean synodic month (tithis) and the Babylonian year of 12;22,8 mean synodicmonths (371;4 r ) split into two constants, k1 = 12 months (360 r ) and k2 = 11;4 r. Because thetime required to travel one degree was taken to be 371;4/360 degrees = 1;1,50,40 r/o

(Neugebauer, ACT, p.286 and p.393) the time for the mean synodic arc (u) would be:u(1;1,50,40) plus one year, or as explicitly given in Section 2 of ACT 812 , [u + u(0;1,50,40)+11;4 r +12 months]. This multiplicative process could have been applied each time the synodicarcs changed, but instead the segment [u(0;1,50,40)] was combined with k2 (11;4 r ) to form afundamental constant (k3) which was added to both the mean and the varying synodic arcs with(presumably) acceptable marginal deviations in the results. Dividing by 30 and combining withk1 produces synodic times expressed in mean synodic months, i.e., u = 33;8,45 degrees, k3 =u(0;1,50,40)+11;4 = 12;5,8,8,20 r therefore the mean synodic time for Jupiter is obtained from[{(u+k3)/30}+k1]= 13;30,27,46,16,40 months.

Fragments of Section 1 of ACT 812, however, mention the total number of synodic arcs (391)and the value "13,30,27,46," (a parameter of far-reaching significance that Neugebauer foundto be "completely dark" even though he was only one step from it in Section 2 of the same text;see ACT, pp.392-393). This parameter in fact indicates that the mean synodic time can bederived simply and directly from the fundamental period relationship for Jupiter, i.e., from therelation: N = 427 Years, II = 391 mean synodic arcs, Z = 36 revolutions, resulting in 427 x12;22,8 / 391 = 13;30,27,45,52,.. mean synodic months. Rounded at the third sexagesimalplace this gives the value stated in Section 1 of ACT 812 of 13;30,27,46.

On checking further the method proves to be readily applicable to all the final Babylonianperiod relations, i.e.,

SATURN: 265 Years, 256 Mean Synodic Arcs, 9 Sidereal Revolutions Mean Synodic Arc = 9 x 360 / 256 = 12;39,22,30 Degrees Mean Synodic Time = 265 x 12;22, 8 / 256 = 12;48,13,26,15 Months

JUPITER: 427 Years, 391 Mean Synodic Arcs, 36 Sidereal Revolutions Mean Synodic Arc = 36 x 360 / 391 = 33;08,45 Degrees (rounded) Mean Synodic Time = 427 x 12;22, 8 / 391 = 13;30,27,46 Months (rounded)

MARS: 284 Years, 133 Mean Synodic Arcs, 151 Sidereal Revolutions Mean Synodic Arc = (151 x 360 / 133) - 360 = 48;43,18,30 Degrees (rounded) Mean Synodic Time= 284 x 12;22, 8 / 133 = 26;24,42,20,45 Months (rounded)

VENUS: 1151 Years, 720 Mean Synodic Arcs ( and 1871 sidereal revolutions ) Mean Synodic Arc = 1151 x 360 / 720 = 575:30 Degrees 1151 x 12;22, 8 / 720 = 19;46,22,57,20 Months

MERCURY: 46 Years, 145 Mean Synodic Arcs ( and 191 sidereal revolutions ) Mean Synodic Arc = 46 x 360 / 145 = 114;12,24,49,40 Degrees (rounded) 46 x 12;22, 8 / 145 = 3; 55,26,7,30 Months (rounded)

Section 1 of ACT 812 thus provides a simple, straight-forward method for obtaining the mean



synodic periods expressed in mean synodic months, whereas the alternative method in Section2 provides the basis for both the mean and varying synodic parameters with the inclusion of thesynodic arcs. Both are fundamental (if not primary) methods associated with the Babylonianapproach to mean and varying planetary motion, yet Neugebauer was nevertheless unable tocome to terms with the given constant in the first section, nor (for whatever reason) did he carrythe methodology to its logical and necessary conclusion in the second.Such fundamental deficiencies combined with Neugebauer's "linear" arithmetical approach toclosed orbits, substitution of "rotations" for orbital revolutions, and not least of all, his failure todeduce an obvious System B for Mars from readily available data in procedure texts such asACT 811 suggest that however erudite and qualified Neugebauer may have been, he was notparticularly well-acquainted with the fundamental framework, and thus far from justified in hisassertions that Babylonian astronomers possessed no cinematic approach to planetary motion.Moreover, in spite of the wealth of technical details in his 1955 opus Astronomical CuneiformTexts, it is likely that his non-cinematic, non-model approach sadly rendered the Babylonianmaterial largely unreadable on one hand and hardly worth reading on the other. Thus fewuncommitted astronomers probably ever bothered to read the work, while the majority of thosethat did likely preferred to take Neugebauer's word rather than try to understand convoluteddetails discussed in base-60 without a cinematic model of any kind. Nevertheless, as it nowstands, I would suggest that the cinematic, heliocentric nature of Babylonian astronomy was inreality self-evident ever since the publication of Astronomical Cuneiform Texts, at least foranyone who cared to tackle the material with sufficient industry and an open, inquiring mind.

IVb. BABYLONIAN "CHARACTERISTIC PHENOMENA" The Babylonian use of "characteristic" synodic phenomena appears to have been largelyminimized and generally misunderstood by most modern commentators for reasons that are farfrom clear. It is certainly true that the phenomena in question are not generally treated bymodern astronomers, but even so there are aspects of the methodology that require carefulconsideration--not least of all the twin components provided firstly by the diurnal axial rotation ofEarth about its axis from west to east, and secondly--also from west to east--the annualrevolution of Earth itself. "East" and "west" are therefore loaded terms, but they arenevertheless perfectly understandable in the Babylonian context, especially from the heliocentricviewpoint, as indeed are all the Babylonian synodic phenomena. Take, for example, thefollowing description of the motion of Jupiter with elliptical planetary orbits viewed from abovewith both Jupiter and Earth moving "concentrically" around the Sun from west to east roughly inthe same plane.



Fig. 2. The relative sidereal motions of Earth and Jupiter

According to the procedures outlined in Sections 30 and 31 of ACT 813, a lengthy Babylonianprocedure text for Jupiter, starting with Jupiter positioned at 90 degrees and Earth at 257degrees (say), as faster moving Earth continues to move away from Jupiter there will eventuallybe a "Last Appearance (i.e., last visibility) in the West" for Jupiter when this planet becomesobscured from view, i.e., when Earth moves "behind" the Sun. The next time Jupiter will becomevisible will be the "First Appearance in the East" (after a further 29 days of motion by Earth) asEarth swings around the Sun and Jupiter becomes visible once again as it rises on the easternhorizon on one specific date (i.e., the helical rising). Next, as Earth continues to gain on Jupiter,it will reach a position (the "First Stationary Point") whereafter Jupiter will appear to move"backwards" and then reach opposition when Jupiter, Earth and Sun are in line. Furtherprogress takes Earth to the "second Stationary Point" after which Jupiter's forward motion willapparently resume. Lastly, continuing to move away from Jupiter, Earth will once again reach apoint in the orbit when Jupiter finally disappears from view, i.e., the "Last Appearance in theWest" is reached again, and so on into the next cycle. All of which is perfectly understandable inheliocentric terms and almost meaningless without. It is not certain whether sequential observations of this kind necessarily resulted in theBabylonian determination of the 12 and 71 year periods and ultimately the fundamental periodrelationship for Jupiter of 427 years with its 36 sidereal periods and 391 "First appearances inthe East." But one thing seems clear enough; carrying out continued observations of successivesynodic phenomena around the complete orbit of a planet against the background provided bythe "goal-year" and other stellar reference points would naturally lead to an awareness of the



faster, slower and mean orbital velocities and also where they were located. Thus it is not thatdifficult to envisage how the Babylonian were able to determine varying orbital velocity, therange between extrema, the rate of change and even the location of the line of apsides. Nor is ithard to see that in doing so and also coming to terms with the apparent retrogradation andstationary points, that the Babylonians had no need whatsoever for auxiliary devices. Theirapproach may have been a simple one, but it was the simplicity of Occam's Razor nevertheless,as the following detailed example shows:

Fig. 3. The relative sidereal/synodic motions of Earth and Jupiter for the medium synodicarc.



SYNODIC MOTION Referring to Figure 3, the elliptical orbits of Earth and Jupiter are displayed on a 360 degreesidereal reference frame with Jupiter initially at the 90 degrees at the point that corresponds tothe synodic velocity of 34;30 degrees and Earth initially at 257 degrees. Fixed sidereal velocitiesof one degree per tithi for Earth and a velocity Vk = 34;30/405r = 0;5,6,40 degrees per tithi forJupiter produce the positions for the Babylonian "characteristic" phenomena over one completesynodic cycle for Jupiter and the specific synodic arc in question. The example may perhapsshed some light on the puzzling statement found in ACT 814 (Sect. 2, L9): "for the first station itis high, for the second station it is low" in so much as the synodic velocity that started at 34;30degrees falls to almost 34 degrees by the time the second stationary point is reached. Needlessto say, the above also shows that such phenomena as stationary points and retrograde motionare clearly apparent and it is undoubtedly direct orbital motion that is under considerationthroughout. Thus for mean values, because of the fundamental period relationship for Jupiter,the Mean synodic arc (u) = (Z x 360)/II and Mean synodic time = (N x 12;22,8)/II, unit time perdegree is therefore obtained from:

(N x 12;22,8)/(Zx360) = N/Z(1;1,50,40).r/o

In the case of Jupiter, this parameter (unit time per degree) is: (427/36)(1;1,50,40) r/o =12;13,32,46,40 r/o or 12.0344361337...days per degree, which is unquestionably the siderealmotion of Earth for each degree of Jupiter's sidereal motion. Moreover, Babylonianfundamental period relations for Mars and Saturn also produce corresponding times for themotion of Earth. For example, from the full Babylonian period relationship for Saturn of 265 years, 256 meansynodic arcs and 9 periods of revolution, the mean synodic arc of 9 x 360 / 256 =12;39,22,30degrees Saturn takes 12;48,13,26,15 months and thus the planet moves 0;2,0,30,11,42,...degrees per day. Thus dividing the latter into one sidereal revolution of 360 degrees results in10,754;53,47,35,41,... (10,754. 89655...) days to complete one mean sidereal period. Thefurther division of this total by the number of days in the standard 12;22,8-month Babylonianyear next produces 29;26,40 (29.444* years), the attested Babylonian mean sidereal period forthe planet in question. On the other hand, the Babylonian fundamental period relationships for the two inferior planets(Mercury and Venus) provide only the number of years (N) and the number of synodicoccurrences (II). This would seem to be one of the two the major factors which have hithertomitigated against a fictive understanding of the Babylonian approach to planetary motion; theother is the apparent motion of the Sun in both planetary and luni-solar contexts. Yet these twofactors are necessarily related and the motion of the sun in Babylonian astronomy need be nomore indicative of Babylonian theoretical basis than is our own retention of solar motion forcomputational convenience (i.e., the slow, mean, and fast sun applied to the equation of time,etc). Thus, as Zombeck (1993) explains in a modern astronomical treatise on the motion of themoon:26

It would be natural but impractical to describe the motion of the moon in heliocentriccoordinates. In the method used here to determine the position of the moon we shallconsider that both the Sun and the Moon are in orbit about Earth. The position of the Sunwas calculated in Section 2.1 under this assumption, and we shall use these calculations tocorrect the mean orbital elements of the moon for solar perturbations. (emphases supplied)

With respect to the planets, from a distinctly fictive heliocentric viewpoint, the sidereal motion ofan outer superior planet provides the synodic arc, while the sidereal motion of the inner planet(Earth) supplies the unit of time. In the case of the inferior planets, from the same heliocentric



viewpoint, Earth is now the outer planet, therefore its motion provides both the synodic arc andthe synodic time, which renders the numbers of sidereal periods for Mercury and Venuscompletely superfluous. In other words, the number of years (N) in the period relationships forthe latter pair is also the number of revolutions (Z) of Earth. Even though the Babyloniantreatment of planetary phenomena pertains to synodic rather sidereal velocity, on furtherexamination the approach is nonetheless found to represent direct, forward sidereal motion perunit time. Finally, with Earth in motion, the relations: 12;22, 8 / 360 = 1;1,50,40 r/oand N x 12;22,8 / II apply consistently to the known Babylonian fundamental period relations, as shown withlargely decimal values for simplicity in the following table:

PLANET N Z II T = N/Z SynodicArc

SynodicT1

SynodicT2

Degrees perday

SATURN 265 9 256 29.444444 12.65625 12.80373 378.10183 0;02,00,11,30,42JUPITER 427 36 391 11.861111 33.14578 13.50771 398.89077 0;04,59,08,29,37

MARS 284 151 133 1.8807947 408.72180 26.41176 779.95505 0;31,26,31,01,24VENUS 1151 1871 720 0.6151791 575.50000 19.77304 583.90971 0;59,08,09,04,37Mercury

146 190 144 0.2421053 115.00000 3.95117 116.68048 0;59,08,09,04,37

Mercury2

46 191 145 0.2408377 114.20690 3.92392 115.87579 0;59,08,09,04,37

Mercurya

848 3521 2673 0.2408407 114.20875 3.92399 115.87767 0;59,08,09,04,37

Mercuryb

388 1611 1223 0.2408442 114.21096 3.92406 115.87991 0;59,08,09,04,37

Mercuryb2

480 1993 1513 0.2408430 114.21018 3.92404 115.87912 0;59,08,09,04,37

Mercuryd

217 901 684 0.2408435 114.21053 3.92405 115.87947 0;59,08,09,04,37

Notes: N = The number of years in the final integer period relation. Z = The corresponding number of mean sidereal periods, i.e., sidereal revolutions. II = The corresponding number of mean synodic arcs/mean synodic periods. Synodic T1: Mean synodic time expressed in mean synodic months (decimal). Synodic T2: Mean synodic time expressed in days (decimal). The mean synodic arc for Mars is given in full; the applied value is the excess over 360degrees. The applied value of the mean synodic month is the Babylonian standard value of29;31,50,8,20 days The corresponding sidereal periods for the two inferior planets (in parenthesis) are implicit.For a more accurate set of hypothetical period relations for VENUS see Appendix B.

METHODOLOGY. The periods and velocities in the table are determined from the integer elements of thefundamental relationships (N, Z and II). The mean synodic arcs for Jupiter and Mars wererounded by the Babylonians from 33;8,44,48,29,. to 33;8,45 degrees for the former, and further



reduced for the latter to the excess over one revolution, i.e., 408;43,18,29,46,27,.. minus 360 to48;43,18,30 degrees. The complete synodic arc and synodic period for Mars still provide thecorrect motion for Earth in degrees per day. The standard unit of time in all cases is theBabylonian year of 12;22,8 mean synodic months treated as the time required for Earth tocomplete one sidereal revolution of 360 degrees. This corresponds to the daily motion of Earthof 0;59,8,9,04,36,59,.degrees associated with the period relationships for the two inferior planetsas explained above. The sidereal periods for the latter pair are implicit in the relationships, andalthough not required they may be obtained from the relation ( Z = N + II ), i.e.,:

Number of Sidereal Revolutions (Z) = Number of Years (N) + Number of Synodic Periods (II)

The period relationships for Mercury concern either the general statement (2) "145 phenomenaof the same kind in 46 years," or specific observational phenomena given in ACT (pp.283-288),i.e., from the following:

( a ) "2673 appearances as a morning star" ( First visibility in the east )( b ) "1223 disappearances as a morning star" ( Last visibility in the east )(b2) "1513 appearances as an evening star" ( First visibility in the west )( d ) "684 disappearances as an evening star" ( Last visibility in the west )

The less accurate Mercury (1) relationship from ACT 816 appears to represent a pedagogicalsimplification associated with the determination of a "System B" type variable velocity function.In this case the extremal velocities would be m = 97;00 degrees and M = 133;00 degrees; withthe same value for the difference d, the extremal velocities for the 46-year/145 arc relationshipwould in turn be: 96;04,54,49,4 degrees and 132;19,54,49,40 degrees respectively. It may be remarked that none of the final data for Mercury and Venus necessarily reflectobservational periods per se, any more than do those of the superior planets. In fact it seempossible that the entire corpus of planetary relations based on the T1-T2 pairings could havebeen generated over perhaps a century or less, though this need not be taken as indicative ofthe comparative newness of Babylonian astronomy on one hand or the limits of their inquiries onthe other. They are simply fragments of what have come down to us. How extensive wasBabylonian astronomy? How far back in time did it extend and what else remains? Short ofadditional material these questions may remain unanswered, although there are undoubtedlyintriguing aspects that still defy explanation, especially a strangely ignored mathematicalproblem concerning a trapezoid that occurs in Jupiter procedure texts ACT 813 and ACT 817,along with still unknown corrections and parameters in the luni-solar texts.

SUMMARY Whether one accepts what has been discussed here or not, it should at least be recognized that



complex issues arising from precession, the various types of months, and the definition of the"year" merely represent the luni-solar component of Babylonian astronomy while furtherquestions arise from the limited number and uneven distribution of the planetary texts publishedin ACT and elsewhere. In fact, there would appear to be sufficient gaps and uncorrelatedparameters to suggest that Babylonian astronomy was almost certainly more developed than isnormally assumed. Included in this latter group are unexplained parameters and operations inthe planetary texts and unknown corrections for both the solar velocity 27 and the zodiac 28 in thelunar material. One might also consider the implications of the extensive range of theBabylonian period relations, synodic phenomena in association with varying, direct, andretrograde velocity, closed orbits, lines of apsides, and not least of all, the aforementionedtrapezoid in the astronomical procedure texts for Jupiter.29

Finally, all of the periods and velocities discussed above can be applied to the motion of Earthfrom one consistent heliocentric viewpoint.

Given the undoubted awareness of accurate sidereal periods for the superior planets, implicitsidereal periods for the inferior planets, accurate sidereal, synodic, draconic, and anomalisticmonths, and varying velocity functions for the planets, sun, and moon - all readily understood interms of a cohesive framework - it seems reasonable to conclude that the Babylonians almostcertainly possessed a well-developed, fictive heliocentric planetary model by at least 250 BCE,and quite possibly much earlier.

SUPPLEMENTARY INFORMATION Babylonian Mathematics and Sexagesimal Notion: Comments and a few examples.

REFERENCES1. Neugebauer, O. Astronomical Cuneiform Texts , (Lund Humphreys, 3 Vols, London, 1955).2. Neugebauer, O. A History of Ancient Mathematical Astronomy, (Springer-Verlag, Berlin,1975).3. Van der Waerden, B. Science Awakening II The Birth of astronomy, with contributions byPeter Huber (Oxford University Press, New York, 1975).4. Astronomical Cuneiform Texts, Ed. O. Neugebauer (Lund Humphreys, London, 1955)404.5. Harris, J. Letter to the Editor of ISIS, Vol. 68, No.245, December 1977:626-617.6. Astronomical Cuneiform Texts, Ed. O. Neugebauer (Lund Humphreys, London, 1955)414.7. Manitius, K. Ptolemaus Handbuch Der Astronomie , (B.G. Teubner, Leipzig, 1963)100.8. Duncan, A. On the Revolutions of the Heavenly Spheres, (Barnes and Noble, New York,1976) 235-236.9. Newton, R. The Crime of Claudius Ptolemy , (Johns Hopkins University Press, Baltimoreand London, 1977).10. Newton, R. The Origins of Ptolemy's Astronomical Parameters (Technical Report No. 4,Center for Archaeoastronomy, College Park, Maryland, 1982).11. Newton, R. The Origins of Ptolemy's Astronomical Tables ( Technical Report No. 5,Center for Archaeoastronomy, College Park, Maryland, 1985).12. Goldstein, B. Al-Bitruji: On the Principles of Astronomy, (Yale University Press, NewHaven London, 1971)13. ACT 210, Section 3, Astronomical Cuneiform Texts, (Lund Humphreys, London, 1955)271-273.14. Op. cit., p. 272.

http://www.spirasolaris.ca/sbb1sup1.html



15. Neugebauer, O. A History of Ancient Mathematical Astronomy, (Springer-Verlag, Berlin,1975) 503.16. Op. cit., p.518.17. ACT 210, Section 3, Astronomical Cuneiform Texts, (Lund Humphreys, London, 1955)272.18. Hartner, W. "The Young Avestan Calendar and the Antecedents of Precession," JHA, Vol X(1979) 1-22.19. Neugebauer, O. Astronomical Cuneiform Texts, (Lund Humphreys, London, 1955) 70.20. Aaboe, A. "A Seleucid Table of Daily Solar (?) Positions," Journal of Cuneiform Studies, Vol.18 (1964) 34.21. Sachs, A. "The Goal-Year Texts," Journal of Cuneiform Studies, Vol. 2 (1948).22. Neugebauer, O, and A. Sachs. "Some Atypical Astronomical Cuneiform Texts I," Journal ofCuneiform Studies, Vol. 21 (1967) 183-218.23. ACT 813, Section 1, Astronomical Cuneiform Texts, (Lund Humphreys, London,1955):403-404.24. ACT 813 , Section 20, Astronomical Cuneiform Texts, (Lund Humphreys, London,1955):414-41525. ACT 812, Section 2, Astronomical Cuneiform Texts, (Lund Humphreys, London,1955):393-394.26. Zombeck, M. Astronomical Formulas, Section 2.2, MATHCAD Electronic Books, (MathSoftInc., 1993.).27. ACT 200, Sections 7 and 9, Astronomical cuneiform Texts , (Lund Humphreys, London,1955) 193-195, 198.28. ACT 202, Section 2, Astronomical Cuneiform Texts, (Lund Humphreys, London, 1955)242-244.29. ACT 813, Section 5, Lines 1-4, ACT 817, Section 4, Lines 1-12, Astronomical CuneiformTexts, 405; 430-431.

Copyright © 1997. John N. Harris, M.A.(CMNS). Last updated July 3, 2003.

APPENDIX A: HYPOTHETICAL BABYLONIAN PLANETARY PARAMETERS FORURANUSAPPENDIX B: HYPOTHETICAL PERIOD RELATIONS FOR THE INFERIOR PLANETSMailto: [email protected]

Return to Spira Solaris

APPENDIX AHYPOTHETICAL BABYLONIAN PARAMETERS FOR URANUS

THE NAKED-EYE VISIBILITY OF URANUS

Preliminary Remarks:

1. Discovered fortuitously by William Herschel with the aid of a telescope in 1781, URANUS iswithout question visible to the naked eye1, 2 ,3

2. As Wagner [1991] has pointed out, it is in fact surprising that the planet was not detected inantiquity.4

mailto:[email protected]

http://www.spirasolaris.ca/index.html



3. Babylonian astronomers - long skilled in observing planetary risings and settings etc., wouldhave been prime candidates for the incidental discovery of a faint (but visible) outer planet movingin essentially the same orbital plane as Mars, Jupiter and Saturn. 4. If detected, Uranus could well have been subjected to the same Babylonian procedures adoptedfor the three attested superior planets, leading to the eventual determination of the correspondingplanetary parameters.5. Hypothetical Babylonian parameters for Uranus are provided below for comparison with valuesthat may be encountered in the future.

I. MEAN VALUESAs discussed in detail above, the final Babylonian fundamental period relationships for the three knownsuperior planets appear to have depended on two initial integer periods (T1 and T2) thaty are close to themean sidereal periods (or multiples thereof) for which small, convenient corrections for longitude ofopposite sign were determined; leading in turn to the final integer period when the correctionscompletely cancel out. for example, (using Neugebauer's terminology from ACT, pp.282-283), the relationship for Jupiter wasexpressed as: N Years = II synodic "appearances" and Z sidereal "rotations" of 360 degrees, but in somuch as the mean synodic arcs for both Jupiter and Mars were rounded at the third sexagesimal place,and that of Saturn was exact. (265 Years = 256 synodic appearances and 9 sidereal revolutions; meansynodic arc u = 9 x 360 / 256 = 12;39,22,30 degrees) is would appear that both accurate and roundedvalues were applied. Either way, however,what will be required in the case of Uranus are initally the twoperiods T1 and T2 (with attendant corrections in longitude) that will provide the final integer relationshipTn. Thus, based on a period of revolution of Uranus of approximately 84 years, for example, the initialpairs of periods with the requisite corrections in longitude leading to the final period relation can besuggested:

T1 = 81 Years, = 80 synodic arcs and 1 sidereal revolution of 360 degrees -10;00 degreesT2 = 85 Years, = 84 synodic arcs and 1 sidereal revolution of 360 degrees + 7;30 degrees

leading to a final integer period relationship for Uranus of Fn = 583 years as follows:

T1 = 81 Years, 80 synodic arcs, 1 revolution of 360 degrees - 10;00 Degrees T2 = 85 Years, 84 synodic arcs, 1 revolution of 360 degrees + 7;30 Degrees T3 = 166 Years, 164 synodic arcs, 2 revolutions of 360 degrees - 2;30 degrees (T1 + T2) T4 = 251 Years, 248 synodic arcs, 3 revolutions of 360 degrees + 5;00 degrees (T2 + T3) T5 = 417 Years, 412 synodic arcs, 5 revolutions of 360 degrees + 2.30 degrees (T3 + T4)FN = 583 Years, 576 synodic arcs, 7 revolutions of 360 degrees with 0;00 degreescorrection (T3 + T5)

and, according to standard methodology, the hypothetical mean values for Uranus based on a final periodFn of 583 years would be in turn:

Mean Sidereal Period = N/Z = 583/7 = 83.28571428 Years Mean Synodic Period = N/II = 583/576 = 1.01215277 Years Mean Synodic Period (months) = 583 x 12;22,8 Months / 576 = 12;31,9,8,20 MeanSynodic Months Mean Synodic Arc (u) = N x 360 / II = 7 x 360 / 576 = 4;22,30 degrees

ADDITIONAL OPTIONS



Other possibilities include final period relations of:

a. 249 years (3 sidereal revolutions)b. 420 years (5 sidereal revolutions) c.. 565 years (7 sidereal revolutions)d. 586 years (7 sidereal revolutions)e. 587 years (7 sidereal revolutions)f. 589 years (7 sidereal revolutions)

with intermediate periods, corrections, and mean parameters as follows:

f. N = 589 Years, Z = 7, II = 582, T = 84.14285... Years, u = 4;19,47,37,43,... T1 = 82 Years (360 - 9;10) T2 = 85 Years (360+ 3;40)

e. N = 587 Years, Z = 7, II = 580, T = 83.85714... Years, u = 4;20,41,22,45,... T1 = 81 Years (360 - 12;15) T2 = 85 Years (360+ 4;54)

d. N = 586 Years, Z = 7, II = 579, T = 83.71428... Years, u = 4;21,08,23,37,... T1 = 83 Years (360 - 3;00) T2 = 84 Years (360+ 1;12)

c. N = 565 Years, Z = 7, II = 559, T = 80.71428... Years, u = 4;30,58,03,52,... T1 = 80 Years (360 - 3;10) T2 = 81 Years (360+ 1;16)

b. N = 420 Years, Z = 5, II = 415, T = 84 Years, u = 4;20,14,27,28,...T1 = 81 Years (360 - 12;45) T2 = 86 Years (360 + 8;30)

a. N = 249 Years, Z = 3, II = 246, T = 83 Years, u = 4;23,24,52,40,... T1 = 81 Years (360 - 8;40) T2 = 84 Years (360 +4;20)

NOTES: The selection of the above periods was partly influenced by 589 and 83-year Jupiter period relations inBabylonian "Goal-Year" texts ( the latter period is also the sum of Jupiter T1 = 12 years and Jupiter T2 = 71 years). Thecorrections for the 583-year period are based on information in a lunar text (ACT 210, Section 2) found in a linepreceding the possible mention of the 265-year fundamental period for Saturn. The fragmentary condition of the sectionand the absence of a second correction make this already insecure data doubtful; the resulting 583-year periodnevertheless provides a convenient mean synodic arc of 4;22,30 degrees, which is more in keeping with mean valuesderived by the Babylonians for Mars, Jupiter and Saturn. The less likely data based on a 565-year final period (7 siderealrevolutions; mean synodic arc: 4;30,58,3,52,..) owes its origins to the unexplained occurrence of the number "4 31" foundin an early Babylonian text concerned with "omens" associated with a cryptic reference to a moving "star" in theconstellation of Pisces, i.e., "If the Fish Star approaches the Acre Star..." with the latter considered to be in the adjacentconstellation Pegasus.

II. HYPOTHETICAL SYSTEM A PARAMETERS FOR URANUSBased on modern aphelion and perihelion distances, Babylonian System A synodic arcs for Uranus mightperhaps center around 4;20 degrees for the mean value with 4;2 degrees and 4;40 degrees for "Slow" and"Fast" arcs distributed over 200 and 160 degrees respectively, i.e., as applied in the case of Saturn. Oralternatively, around 4;00 and 5;00 degrees with a corresponding mean synodic arc (u) closer to 4;30



degrees, etc. Finally, for a mean synodic arc of precisley 4;31 degrees the corresponding times forvarious approximations would be:

1. 12;31,26,39,41,20 months (k = 11,12,19,50,40)2. 12;31,26,40 months for k = 11;12,20 r

3. 12;31,26 months for k = 11;12 r.

III. HYPOTHETICAL SYSTEM B VELOCITIES FOR URANUS

BASIS: THE 583-YEAR INTEGER PERIOD RELATIONSHIP FOR URANUS: N = 583 YEARS, II = 576 SYNODIC ARCS, Z = 7 SIDEREAL REVOLUTIONS

P = Number of mean synodic arcs per sidereal revolution = 360/u T = Sidereal Period = P + 1 d = Increase/decrease in velocity (degrees) and time (tithi) per synodic arc = 0;1,10 Amplitude of Synodic Arcs = 1/2Pd = 0;48 (1/4Pd = 0;24) m = Minimum Synodic Arc: ( u - 1/4Pd) = 3;58,30 degrees u = Mean Synodic Arc: [(7 x 360 )/576] = 4;22,30 degrees M = Maximum Synodic Arc (u +1/4Pd) = 4;46,30 degrees

The 583-year period is used here for simplicity. The attested determination of the mean synodic arc (u)from the division of the total sidereal motion by the number of synodic arcs in the final relationshipwould be followed by the derivation of the parameters of a "linear zigzag" function given above andbelow. The difference, d = 0;1,10 is on the high side, but closer to the approximate 9 : 1 ratios of theMars : Jupiter and the Jupiter : Saturn differences. Values for this parameter might range from 0;40 toperhaps 0;1,20. (note: The derivation of the extremal velocities follows the procedure suggested by theremnants of Section 1 of Jupiter procedure text ACT 812 )

IV. HYPOTHETICAL SYSTEM B TIMES FOR URANUS

(a) SYNODIC FACTORS IN TITHIS (Synodic Arc + k3 = Synodic Arc + 11;12,4,10,r

Abbreviated value: +11;12 r )

(m) = 15;10,51,40 r Minimum Synodic Arc (abbreviated value: 15;10,30 ) (u ) = 15;34,34,10 r Mean Synodic Arc (abbreviated value: 15;34,30 ) (M) = 15;58,16,40 r Maximum Synodic Arc (abbreviated value: 15;58,30 )

(b) SYNODIC PERIODS (MONTHS) [ IV (a) Values/30 + 12 Mean Synodic months]

(m) = 12;30,21,43,20 mean synodic months (u ) = 12;31,9,8,20 mean synodic months (369.699569 days) (M) = 12;31,56,33,20 mean synodic months

The synodic times in tithis and mean synodic months were derived according to the method given inSection 2 of Jupiter text ACT 812 (Neugebauer, Astronomical Cuneiform Texts, Lund Humphreys,London 1955:393). The mean synodic time for Uranus is also obtainable from the final integerrelationship and the methodology indicated in Section 1 of the same text, i.e., the mean synodic time isaccordingly:



583 x 12;22,8 / 576 = 12;31,9,8,20 mean synodic months (of 29;31,50,8,20 days).

V. THE SELEUCID ERA The Seleucid Era - a Babylonian astronomical era of unknown significance - begins with Month 0, Year0 in April 310 BC (311 BCE). As it so happened, Uranus was occluded three times by Jupiter around thistime, i.e., on September 23, 312 BCE, January 2, 311 BCE ( Uranus at opposition and nearly at itsbrightest, M = +5.4 ) and April 29, 311 BCE, i.e., April 310 B.C. Those with astronomical software canobserve from the location of Babylon (Iraq: 44 25E, 32 33N) the positions of both planets, theperceptible parallax exhibited by Uranus with respect to Jupiter between the dates given and the planet'slater motion (at its brightest) along the ecliptic through the constellation of Leo.

SUMMARY Firstly, because of the relatively low visual magnitudes of Uranus it is possible that even if sighted, theorbit could not be completely determined. Secondly, although no unambiguous references to an additional planet are apparent in the historicalrecord, there nevertheless remain enigmatic statements and parameters of unknown significance in bothearlier Babylonian material and the astronomical cuneiform texts of the Seleucid Era. Complex issuesarising from "precession", the various types of months, and the definition of the "year" represent merelythe luni-solar component of Babylonian astronomy. Others issues arise from the limited number anduneven distribution of the extant planetary texts. In fact, sufficient gaps and uncorrelated parametersremain to suggest that Babylonian astronomy was quite likely more developed than is normally assumed.Until the matters outlined above and at end of the parent paper are addressed more adequately, it wouldsurely be premature to dismiss the capabilities of Babylonian astronomers, or their possible naked-eyedetection of Uranus, conventional wisdom and the status quo notwithstanding.

APPENDIX BHYPOTHETICAL PERIOD RELATIONS FOR VENUS AND MERCURY

B1: VENUS

PRELIMINARY REMARKSIn general, it may be assumed that shorter Babylonian period relationships will provide less accuratemean values than those obtained from F, the Final (and exact) integer period relationship determinedfrom the initial T1 and T2 periods, e.g.,

F = 284 Years, 133 Mean Synodic Arcs, 151 Orbital Revolutions [MARS]F = 427 Years, 391 Mean Synodic Arcs, 36 Orbital Revolutions [JUPITER]F = 265 Years, 256 Mean Synodic Arcs, 9 Orbital Revolutions [SATURN]F = 583 Years, 576 Mean Synodic Arcs, 7 Orbital Revolutions [URANUS(?)]

However, for Mercury and Venus no corresponding T1 and T2 periods are readily apparent; furthermore,in the case of Venus the 1151-year relationship yields a relatively poor value for the mean synodicperiod. Nor, for that matter, does the corresponding mean synodic arc inspire confidence, being simplyone half of the period itself (i.e., 1151*360/720 = 1151/2 = 575;30 degrees). The latter may well be a



working value, and a convenient one at that, but with a length of 1151 years for the final integer periodone might reasonably have expected more accurate results. Recalling, however, the key period relationsfor Jupiter provided in Section 1 of ACT 813 (see above) 23

"Compute for the whole zodiac (or: for each sign) according to the day and the velocity.In 12 years you add 4;10, in 1,11 years you subtract 5, in 7,7 years the longitude (returns) toits original longitude."(In 12 years you add 4;10 degrees, in 71 years you subtract 5 degrees, in 427 years thelongitude returns to its original longitude)

and the expansion that produced the attested period relation for Jupiter of:

427 years, 391 mean synodic arcs and 36 sidereal revolutions:

Table.1 Babylonian Period Relations and the 427-year Long Period for Jupiter

one could do little more than hope that additional periods for Venus and Mercury might eventually come tolight from newly recovered cuneiform tablets etc., and failing this, other historical sources.

THE 243-YEAR PERIOD IIn the latter category, for example, there is the interval of 243 years mentioned in the following cryptic footnoteby George Burges (1876:171):

. . . . the ratio of 243 to 256 is to that of 35 to 44; especially if we bear in mind what is stated byPlutarch, De Anim., Procreat. ii, p. 1028, B., respecting Lucifer (Venus) being represented by 243, and the Sun by729. (George Burges, The Works of Plato, George Bell and Sons, London, 1876:171) 10

THE 8-YEAR PERIOD Although it may seem an unnecessary elaboration "the ratio of 243 to 256 is to that of 35 to 44 " mayalso be restated as: "the ratio of 243 to 256 is to that of 35 to 28", but apart from the well-knownVenus/Fibonacci relationship of 5 synodic periods, 8 years and 13 orbital revolutions, little in the way ofadditional understanding follows. But this is numerology in any case, is it not? Possibly, but more likely



methodology, and highly condensed methodology at that. But in any event, the parent work itself isfound in "The Treatise of Timaeus the Locrian" in Burge's The Works of Plato (1876) whereas initialreferences to the number "243" occur in a distinctly "Pythagorean" context, i.e., in a footnote to Burge'sown SUPPLEMENTARY NOTE to (yet another extension) "The Notes of Batteaux." However, since weare not so much concerned here with Pythagorean tenets as the determination of fundamental periodrelations for Venus the latter subject is perhaps best left for a more specialized treatment at another time.Nevertheless, it might still be unwise to "criticize without light."

THE 251-YEAR PERIOD; ADDITIONAL COMPLEXITIES So far so good, though merely the 8-year Venus cycle with its 5 corresponding synodic periods and 13corresponding orbital revolutions, and an obscure historical reference to an interval of 243 years. Whatnext? As far as my own efforts were concerned, nothing at all. It was in fact the Internet that supplied theanswer, providing both intermediate periods T1, T2 as well as F, the final"long" integer relationship inone neat, detailed package. The source in question, however, dealt with matters far more difficult thanthe present historical asterisk, namely the complexities that arise from the analysis of the transits ofVenus carried out by Karl-Heinz and Uwe Homan. For present purposes, however, the following periodsdiscussed in detail by the latter ( VENUS TRANSITS AND PRECESSION, May 31, 2004):

A preliminary analysis of the Venus Transit Data has shown that the Earth must go aroundthe Sun 360 degrees in a tropical year, contrary to current lunisolar precession theory. Thefact remains and the evidence suggests that the observed transit cycles reflect a moreaccurate correlation between the periods of 251 tropical years and 408 orbits of Venusaround the Sun, than 243 and 395 respectively. This paper examines what appears to be a pattern of resonance between Venus transit cycles,the mean synodic period and the time interval of the 360-degree tropical year based onEarth's non-precessing axis of rotation relative to the position of the Sun. .... A complete360-degree cycle occurs after 157 mean synodic periods, or exactly 251 tropical years and408 orbits of Venus.( Uwe Homan, The Sirius Research Group, May 31, 2004; emphases suppplied)

plus the 5, 152 and 157 Venus synodic periods and corresponding 13, 395 and 408 orbital periods appliedin an earlier paper (TRANSITS OF VENUS VS NASA'S ASTRONOMICAL DATA, April 21, 2004) provideall that is necessary. In these modern contexts the latter sets are discussed in detail with respect to boththe tropical year and the sidereal year with far-reaching implications; in our present historical context,however, all six periods may simply be used directly after the manner adopted for Jupiter, i.e.,hypothetically:

"In 8 years you add 1;26, In 243 years you subtract 1;26. In 251 years the longitude (returns) to its original longitude."

http://www.siriusresearchgroup.com/articles/Nasa-Venus-Transit.shtml


http://www.siriusresearchgroup.com/




Table 2. Hypothetical Babylonian Period Relations for VenusFinal Period F = 251 years, 157 mean synodic arcs ( 408 orbital revolutions )

In Table 2 the longitude corrections of 1;26 degrees are conveniently truncated from the more accurate valueof 1;26,3,20,47,48,..(1.434262948.. degrees); the positive correction the excess over 360 degrees after 8 years,the negative correction the amount less than 360 degrees after 243 years.These corrections necessarily involvethe annual orbital motions of Earth and Venus, the latter value being 585;10, 45,25,5,58,33,( 585.179282868.degrees from the 251-year relationship, i.e., from 408 x 360 / 251). Finally, though not to be confused with the modern complexities that attend this matter, the mean synodicperiod for Venus (based on the Babylonian year of 12;22,8 mean synodic months) can be otained from thefinal period as before, i.e.,

251 x 12;22,8 / 157 = 19;46,28,4,35,9, (19.774465676...) mean synodic months, or simpler still:19;46,30 months.

B2: MERCURY13-YEAR AND 33-YEAR PERIODSAlthough a similar situation exists for Mercury, i.e., no attested T1 and T2 periods or related FinalPeriod (F), the available material for this planet is nevertheless more extensive. However, remaining withthe better known 46-year period that has come down to us in various planetary theories (e.g., those ofPtolemy, Al-Bitruji, and Copernicus) the methodology applied in the case of Venus -- apart from thereversed polarity of the paired corrections -- remains virtually unchanged, i.e.,

"In 13 years you subtract 7;50, In 33 years you add 7;50. In 46 years the longitude (returns) to its original longitude."

Table 3. Hypothetical Babylonian Period Relations for MercuryFinal Period F = 46 years, 145 mean synodic arcs ( 191 orbital revolutions )

Here again the longitude corrections (7;50 degrees in this instance; to two sexagesimal places perhaps: 7;49,30) are simplified variants of more accurate values obtainable from the 46-year final relationship ( i.e., 7;49,33,54,46,57, ..., 7.826086956 .. degrees) and the combined annual orbital motions of both Earth (360degrees) and Mercury (1494;46,57,23,28,41,44, ... degrees, etc.). In this case, however, the negative correctionis the amount less than 360 degrees for T1 (13 years) and the positive correction the excess for T2 (33 years). Based on the final 46-year integer relation the mean synodic period will accordingly be:

46 x 12;22,8 / 145 = 3;55,26,7,30 months (3;55,26,7,26,53,47..) or more approximately, 3;55,30 months.



REFERENCES [ APPENDICES A and B ]1. Moore, Patrick.Naked Eye Astronomy, W.W. Norton, New York, 1965 2. Webb, Rev. T.W. Celestial Objects for Common Telescopes, Dover, New York, 1962:221.3. Levy, D H. THE SKY - A User's Guide, Cambridge University Press, Cambridge 1991:134. 4. Wagner, Jeffrey K. Introduction to the Solar System, Holt, Rinehart and Winston, Orlando 1991. 5. ACT 813, Section 1, Astronomical Cuneiform Texts, (Lund Humphreys, London, 1955.. 6. Horowitz, W. "Two New Ziqpu-Star Texts and Stellar Circles,"Journal of Cuneiform Studies, Vol 46,1994. 8. Gadd, J. "Omens Expressed as Numbers," Journal of Cuneiform Studies, Vol 21.1967. 9. Van Der Waerden, B. Science Awakening II, Oxford University Press, New York, 1974. 10 Burges, George. The Works of Plato: A new Literal Verson, George Bell and Sons, London, 1876.

Added July 3, 2004.

Mailto: [email protected]

Return to Spira Solaris

mailto:[email protected]

http://www.spirasolaris.ca/index.html

babylonian planetary theory and the heliocentric concept

Documents