The Ninsianna tablets, a preliminary reconstruction

by

Michael G. Reade

Michael Reade, D.S.C., a confectionary technologist, is also a specialist in marine navigation, having served in the Royal Naval Volunteer Reserve since 1936. His earlier articles in the Review have dealt with astronomical records from Egypt and with the 6th-century Hindu astronomical manual, the Panchasiddhantika.

A detailed analysis is presented of the succession of disturbances of the Earth's motions indicated by the Ninsianna (Ammisaduqa) Venus tablets. The principal single outcome is a small expansion of the Solar System, an approximate 1% expansion of the orbit of the Earth which was initiated about 882 BC and which was complete by about 729 BC. In the course of this expansion, Venus pushed the Earth out towards Mars, causing the Earth to start pushing against Mars. The reality of the one time 360 day year is confirmed but this was a count of Babylonian days per Babylonian year, any change in the absolute spin rate of the Earth being minimal. A secondary outcome of the analysis is a major disruption of the orthodox system of precessional dating, permitting synchronisation of the Phaeton and the Exodus episodes, together with the somewhat unexpected finding that Stonehenge can be added to the world list of ancient monuments which have their origin in these disturbances.

Method and Assumptions

The 'Ninsianna' or 'Venus tablets of Ammisaduqa' have been the subject of much previous study [1]. The present analysis is based on Rose and Vaughan's reading of the tablets [2]. The basic (Babylonian) dates for the heliacal settings and risings of Venus which Rose and Vaughan find recorded on the tablets are summarised in Table 1 [3]. Whilst there are some minor variations in the readings of the tablets quoted by researchers other than Rose and Vaughan, I know of none which significantly affect the conclusions drawn in the present study except in so far as some commentators have sought to amend the tablets by postulating 'mistakes' of substance, especially omitted intercalary months.

The precise dates of conjunction are of fundamental importance for the present analysis and this set derives from application of Van der Waerden's system of calculation of the invisibilities of Venus [4] to the actual day numbers of appearance and disappearance claimed on the tablets. Van der Waerden computes each invisibility in two parts, so many days from heliacal setting to conjunction plus so many days from conjunction to heliacal rising, thus pin-pointing the date of conjunction with precision when the dates of setting and rising are known.

The basic form of the invisibility calculation is set out in Appendix A. It will be seen that a considerable number of variables is involved and, for the purposes of this work, latitude of observer has been held constant at 32.5°N and inclination of the planes of orbit of Earth and Venus constant at 3.365°. Arcus visionis of Venus, obliquity of ecliptic and heliocentric longitude of the rising node of the orbit of Venus have all been treated as unknowns and various combinations of these have been tested for each Event in turn until a self-consistent pattern of variations has emerged. It is perhaps worth commenting that the ultimate arbiter has frequently proved to be the derived spin rate of the Earth, which can be very sensitive to particular choices of variable. The average spin rate of the Earth during the interval between two conjunctions is found by comparing the interval between the two conjunctions (in days) with the change in mean longitude of the Sun (in degrees, but note that it is mean and not actual longitude which is required).

Basic longitudes of conjunction can only be precisely retro-calculated when the epoch of the Event is known and it has been necessary to repeat the whole process for different epochs until ultimately a self-consistent pattern of variations has again emerged. It has also been necessary to apply two types of retro-calculation to find the longitudes - orthodox Julian retro-calculation and special Babylonian or Assyrian retro-calculation - as it has become apparent in the course of the analysis that a minimum of three different calendars is involved (further discussed below under 'Results').

[*!* Image: Figure 1. Observed spin rate of Earth (Babylonian days per Babylonian year, merging into Assyrian days per Assyrian year). Each point on the graph marks the average spin rate between two neighbouring Events; it only represents a stabilised spin rate when three or more Events share a common average (e.g. the actual minimum spin rates reached will be substantially less than the minima shown on the graph). The interval between neighbouring Events is approx. 10 months.

OPPOSITIONS OF MARS 325.25
1 2 3 4 5A 5B 6 7 8A 8B 9 10 11 12 13A 13B 14 15 16A 16B 17
Event Number Spin rate of Earth (days per year)
320 330 340 350 360 370 380 0.4 0.6 0.8
Earth-Mars distance (AU) at oppositions of Mars]

The whole analysis is essentially a series of successive approximations, which carries with it some risk of circularity of argument, but there are so many different pieces of evidence available that a very fair degree of confidence can be felt in the overall picture which results, even if some of the finer detail would be upset if there were actually faults, or 'mistakes', in the raw data. I find neglect of possible changes in apparent latitude of observer to be a potential cause of minor anomalies. Another secondary factor which has had to be neglected is the possibility (even probability) of shifts of the longitude of perihelion of the orbits of Earth and Venus, but there comes a time when the 'noise' arising from factors such as these and the custom of the Babylonians of not recording events more closely than to the nearest whole day limits the number of successive approximations which it is practicable to attempt. There are also uncertainties as to the precise eccentricities of orbits at particular eras; whilst such uncertainties may eventually be resolvable to a greater degree than has been achieved in the present analysis, the revisions are likely to modify only some of the finer detail, not the broad overall picture which emerges (a great surge in the brightness of Venus at inferior conjunctions, coupled with quite sharply defined decelerations and re-accelerations of Earth spin rate - see Figures 1 & 2; these features have been common to every scheme which has been tested, including ones based on quite widely differing assumptions). Finally, there are some potentially unresolvable uncertainties arising from the unknown clarity of the atmosphere at particular Events, and it is these uncertainties which are a principal cause of the 'tolerances' which are sometimes cited below.

[*!* Image: Figure 2. Changes in the Arcus visionis of Venus. The upper line joins inferior conjunctions, the lower one superior conjunctions.
Event number. 1 2 3 4 5A 5B 6 7 8A 8B 9 10 11 12 13A 13B 14 15 16A 16B 17. Arcus visionis of Venus - Degrees. 10 9 8 7 6 5 4 3 2 1 0 -1]

Before turning to actual results, however, it should be noted that a calculated invisibility cannot be longer than the corresponding observed invisibility but that it can be (and usually will be) shorter than the observed invisibility when visibility is impaired (sometimes very substantially shorter, according to references 5 and 6). Moreover, the 'observed invisibilities' quoted in Table 1 for inferior conjunctions need reducing by 0.5 days, whilst those for superior conjunctions need increasing by 0.5 days. This is because 'disappearance' at an inferior conjunction is observed at or near sunset whilst 're-appearance' is observed at or near sunrise. At superior conjunctions the position is reversed, 'disappearance' being recorded at or near sunrise, 're-appearance' at or near sunset. Similarly, if the Day No. of disappearance is cited as 'N', the most satisfactory estimate of 'the moment of disappearance' is N + 0.75 at an inferior conjunction and N + 0.25 at a superior conjunction (but note that these corrections need revision if the Earth should become inverted). Day No. of conjunction has therefore been computed as:

(Day No. of disappearance + 0.75/0.25) + (calculated invisibility between heliacal setting and conjunction) + ½(observed total invisibility +/- 0.5 - calculated total invisibility)

A very similar formula can be devised on a basis of the day of re-appearance instead of the day of disappearance and it gives an identical result for the computed Day No. of conjunction.

The assumption made in these formulae is that any effect due to poor visibility will be the same at rising as it was at setting; this form of computation also enables one to assess the effects of variable visibility (principally by applying a suitably modified value of 'h' in the invisibility calculation).

There are also further assumptions of potential significance, however. Firstly, it is clear enough that 'Venus appeared on day...' means that Venus was seen on that day; it is less certain that 'Venus disappeared on day...' means that it was last seen on that day (as has been tacitly assumed). Should the day of disappearance actually be the first day on which Venus was not seen, all observed invisibilities would need to be increased by 1 day.

Secondly, the present analysis has been based on an assumption that early morning and late evening observations would be booked to the same day. It has been reported elsewhere that Babylonian days began and ended at sunset [7]; if this rule was respected absolutely, post-sunset observations would actually have been booked to the next following day, which would result in modification of the corrections described above (e.g. 'true invisibility' would be half a day longer than the reported invisibility at inferior conjunctions, half a day shorter at superior conjunctions). The tablets themselves, especially Events 1 and 17, actually suggest that it is improbable that a 'post-sunset' rule was applied [8].

The present analysis has not been continued beyond Event 17. Doubts have been expressed as to whether Events 19-21B are continuous with Events 1-17, but it does appear probable that they are. They present special problems of their own, however, and I have preferred to defer their detailed investigation until later.

Results

First to be remarked is the interval between Events 1 and 11. Going by the mid-points of the invisibilities recorded on the tablets, it was 3481.5 - 16.5 = 3465 days. After successive runs through the invisibility analysis (Appendix A) with gradual refinement of the component data (epoch, varying obliquities of the ecliptic, shifts of vernal equinox and values of Arcus visionis), this figure becomes 3485.9-18.1 = 3467.8 days (as in Table 1). Orthodox retro-calculation (Appendix A) indicates that if the Julian calendar had been in force throughout this era the interval would have been 3500.24 days (varying very slightly according to the epoch selected for Event 1). Clearly, the Julian calendar was not applicable at this era.

Similarly, the interval between Events 11 and 17 is 5833-3481.5=2351.5 days by the mid-points, or 5836.2-3485.9 = 2350.3 days after refinement as described above. If the Julian calendar had been in force at this era, retro-calculation indicates an expected interval of 2339.1 days.

Less strikingly different than the finding for the period of Events 1-11, the difference is nevertheless still enough to suggest that yet another calendar will have been applicable during this era. This second calendar must be presumed to have endured until some later date, when there was a final transition to the modern Julian calendar. It is, of course, impossible that there could have been a really sharp transition from one calendar system to another, and some degree of 'slurred' or 'stepwise' transition between the individual calendars must be expected. It is also clear that the second era, especially, can almost certainly be sub-divided into a number of different periods of differing characteristics, but it seems enough for the purposes of this paper to recognise two principal calendars immediately preceding the (modern) Julian era. I propose naming the one prior to Event 11 'the Babylonian calendar' and the one subsequent to Event 11 'the Assyrian calendar'.

It should also be noted that day lengths were not necessarily the same in each of the three eras, but there does not actually appear to have been any very great difference between average day lengths (though there were undoubtedly individual days which were greatly distorted in length).

As a general guide and as a first approximation, the Babylonian era was one of years of approximately 360 days per Babylonian year and the Assyrian era, at least initially, was one of years of 370-375 days per Assyrian year (subject to considerable deviation from time to time, a distinction between 'absolute spin rate' and 'apparent spin rate' being called for, further discussed below under 'Implications and discussion'). 'Apparent spin rate' is defined here as the observed spin rate expressed in Babylonian days per Babylonian year, Assyrian days per Assyrian year, or Julian days per Julian year, as appropriate for the era concerned.

The location of the break between the Babylonian and the Assyrian eras at Event 11 derives primarily from the numerical analysis of the tablet data first presented in Workshop 1986:1 [9] and is further discussed in the latter part of Appendix A. It suffices at this stage to remark that Event 11 must be regarded more as an effective pivot point between the two calendars than as a truly sharp break point between them.

The spin rate of the Earth during the Babylonian era is of fundamental importance for the present analysis. Its derivation involves an assumption that Events 1, 2 and 3 were unaffected by catastrophic events (or extra-terrestrial forces). It can be seen from Figure 7 (in Appendix A) that Events 1 and 3 present no problems for this hypothesis. Event 2 is at first sight less amenable, but it turns out to be much more strongly affected by a shift of the vernal equinox than is either Event 1 or Event 3 and, in practice, there is no real difficulty about matching up the observed and calculated invisibilities for all three Events (though only if one is prepared to accept that the obliquity of the ecliptic and the position of the vernal equinox which are common to all three differ from the commonly expected values indicated by orthodox retro-calculation).*

[* A change in the obliquity of the ecliptic implies a 'roll' of the Earth's axis; a sudden shift of the vernal equinox implies development of a mis-match between time and season of year, a sort of forced intercalation which is discussed in more detail in Appendix B. Dynamical considerations suggest that both are likely to occur simultaneously and that a change in one without a matching change in the other is likely to be a rare occurrence, though not quite inconceivable.]

Also, Event 2 is a superior conjunction (Events 1 and 3 being inferior conjunctions) and it is almost inconceivable that there could have been a disturbance at the time of Event 2 which would not have had a noticeable effect on Event 3 as well. It is also worth noting that Dr Peter Huber [7] classified the data for Events 1, 2, 3 and 4 as 'good' (i.e. compatible with his retro-calculations), whereas he classified a high proportion of the later Event data as 'mediocre' or 'bad' [10].

[*!* Image: Figure 3. Changes in the obliquity of the ecliptic. The more securely fixed points are those which are shared by three or more consecutive Events; there are possible alternative patterns of development between such 'fixed' points.
Event number. Obliquity of ecliptic - Degrees.]

The apparent changes in spin rate of Earth, Arcus visionis of Venus, obliquity of ecliptic and shift of vernal equinox (from its retro-calculated position) are summarised in graph form in Figures 1, 2, 3 and 4. These graphs derive entirely from application of the invisibility calculation (Appendix A) to the tablet data so as to match up calculated and observed invisibilities with the minimum possible in the way of dislocation between neighbouring Events.

It has been assumed that there are no mistakes at all in the published data, but observed invisibilities have been derived from the published ones by application of the corrections described above. Arcus visionis has been treated throughout as a potentially floating variable; changes in the obliquity of the ecliptic and position of the vernal equinox have been co-ordinated, at least to some degree, and changes in the spin rate of the Earth have been kept to the minimum possible. The basic hypothesis is that, in the absence of catastrophic forces, Arcus visionis may vary but all the other parameters (obliquity, vernal equinox and spin rate) will remain unchanged, though they may well be at altogether unforeseen levels. The whole is, of course, essentially a 'scenario': that is, an explanation which fits all the known facts but which is not necessarily the only possible explanation. A whole range of such 'scenarios' is actually possible, but the choice is greatly narrowed by adoption of the above hypothesis (Arcus visionis the only free variable in the absence of catastrophic forces; if this hypothesis is not respected, the outcome is always frequent and very improbable fluctuations of Earth spin rate).

The first point to be remarked about these four graphs is that the general pattern - a great surge in the brightness of Venus at inferior conjunctions, with matching sharp decelerations and re-accelerations of Earth's spin rate at intervals of 5 conjunctions of Venus (= 4 years) - is virtually independent of the epoch selected for Event 1. Though a change in the dating of Event 1 modifies the detail slightly, the general pattern remains essentially the same. The dating of the tablets is at least to some extent dependent on this observation as Figures 1, 2, 3 & 4 are all based on 868 BC for Event 1, though later considerations (below) suggest amendment of this date to 884 BC.

[*!* Image: Figure 4. Movements of the Vernal Equinox. The more securely fixed points are those which are shared by three or more consecutive Events; the dashed parts of the curve are more speculative but they cannot be amended without at least amending the obliquity as well, sometimes also the associated spin rate and Arcus visionis.

Event number]

Dating the tablets

The reason for starting from 868 BC is that the principal confirmation of otherwise rather weak chronological indications comes from matching the trend of observed Earth spin rates with the retro-calculated dates of oppositions of Mars (included on Figure 1). It needs to be observed that there is a near repetition of this pattern at 16 year intervals. Venus phenomena repeat themselves rather closely at 8 (or 16) year intervals, whilst Mars phenomena repeat at about 15 year intervals, though less closely. There is thus at least a near repetition of both Mars and Venus phenomena at 16 year intervals. If the epoch of Event 1 is moved back from 868 BC by 16 years to 884 BC, the close opposition of Mars occurs shortly after Event 16A instead of shortly before Event 15. At any date intermediate between 868 and 884 BC there is a total mismatch of both Venus and Mars phenomena, though 876 BC would be consistent with the Venus phenomena alone.

It seems clear from Figure 1 that interactions with Venus tended to increase the observed spin rate of the Earth whilst interactions with Mars tended to reduce it again, the maximum observed spin rate being 376.1 Assyrian days per Assyrian year, sustained between Events 10 and 13A. It is amply confirmed in what follows that Venus was in fact pushing the Earth out towards Mars and it is thus eminently plausible that the first brush with Mars should have occurred at about Event 15 (as indicated on Figure 1).

It has to be noted, however, that retro-calculation of close oppositions of Mars is dependent on a knowledge of the orbit of Mars. The present work has not identified any sure way of detecting a shift in the longitude of perihelion of an orbit, for instance, but trial calculations have indicated that a change in the longitude of perihelion of the orbit of Mars could have only a negligible effect on the dates of oppositions (a maximum change of about 0.1 days). Similarly, a straightforward contraction of the (modern) orbit of Mars, such as might be expected (below), also has a negligible effect on the datings, though naturally a very substantial effect on Earth-Mars distance at opposition - the distances cited on Figure 1 must obviously be treated with very considerable reserve. The orbits of both Earth and Venus will very probably also have suffered shifts in their longitudes of perihelion, which might either enhance or suppress effects due to Mars alone. Changes in the eccentricities of the orbits of both Earth and Mars do have a rather more disturbing effect on the dates of opposition but it has proved surprisingly difficult to postulate any 'simple' modification of the (modern) orbit of Mars which could offset the comparatively small change in the date of opposition (relative to Event number) caused by a 16 year change of epoch. To this extent, at least, 868 BC scenarios certainly cannot be dismissed out of hand.

The whole question of dating the tablets is complex and distinctly tortuous; unfortunately, also, no single style of computation has as yet been identified which can be guaranteed precise in every respect. By way of introduction, the general orientation of the dating scheme was shaped by two preliminary findings. Firstly, there is the hypothesis that the Panchasiddhantika [11,12] describes a heliacal setting of Sirius 60 days before a superior conjunction of Venus (identified below as Event 4). The invisibilities of Sirius can be computed by essentially the same equations as those of Venus (the Sirius invisibility calculation is actually appreciably the simpler of the two). Preliminary calculation suggested a date of around 880 BC, but the computation is very sensitive to both the Arcus visionis assumed for Sirius and the location of the vernal equinox at the time concerned. For a long time this result was simply disbelieved.

Secondly, in the course of trying to optimise the eccentricity of the orbit of Venus so as to permit crediting Events 1-4 with a common obliquity and position of the vernal equinox, a 'proof' was derived that it was impossible to date Event 1 later than about 860 BC. The optimum eccentricity for the orbit of Venus (for this purpose) was incidentally indicated as 0.047, to be associated with an eccentricity of Earth's orbit of 0.001 or less. When it proved impossible to marry up all four Events [10], only Events 1, 2 and 3, this 'proof' naturally lapsed. Nevertheless, it evidently retains at least some validity, as the present finding (below) is that a date later than 852 BC is impossible.

The anchor point of the present analysis is the epoch of Event 11, the nominal transition point between the Babylonian and Assyrian eras. This epoch can be derived from the data for the Assyrian era (the period between Event 11 and the start of the Julian era) on the basis of:

a). the observed 2350.3 day count from Event 11 to Event 17,
b). the average spin rate of the Earth during the period between Event 11 and the start of the Julian era,
c). the longitudes of conjunction of Events 12 and 14, both of which are particularly critical Events but which nevertheless tolerate mis-reporting of their invisibilities by up to at least 10 days without significant effect on the end result, and
d). various assumptions as to epochs for the start of the Julian era (the Assyrian/Julian transition point) and the occurrence of Event 1.

Table 2 summarises the effect of specifying various epochs (computed as described in Appendices C, D and E). Given an epoch for Event 11, it is then possible to retro-calculate the corresponding conditions at Events 1-3, especially eccentricities of orbits, obliquity of ecliptic and shift of vernal equinox (Appendix E, step 4). This computation is principally based on the observed day count of 3467.8 days between Event 1 and Event 11.

Table 2 is complex and unfortunately also none too precise. The first snag is that the effective average spin rate between Event 11 and the start of the Julian era can only be assessed much too crudely for comfort. For the computation of Table 2, it was taken as the mid-point between 376.1 (above) and 365.256 (the modern value). A smooth decay of spin rate is assumed by this procedure but it is only too clear that the actual decay was anything but smooth, almost certainly occurring stepwise and being associated with various discontinuities (especially 'jumps' in eccentricities of orbits). Changes in the eccentricities during the period under study also introduce uncertainties into the Table 2 computation. A partial correction can be made by 'retro-retro-calculation', that is, computation forward from Event 11 to Event 17 instead of computation backward from the start of the Julian era to Event 11. True precision of computation is not at present possible but there is at least a clear indication (primarily based on the longitudes of Events 12 and 14) that the Table 2 epochs for Event 11 tend to be too low, possibly by about 2 days (i.e. the 'true' epoch will be up to about 2 days later than that claimed in Table 2). Computations showing the effect of such a correction are included in Table 4 (below).

The second snag is that one can hardly state the longitudes of Events 12 and 14 with absolute precision and a mis-assessment of 0.5° in the longitudes of both (both mis-assessed in the same direction) results in a mis-assessment of slightly more than 0.5° in the value of L_s (the mean longitude of the Sun at the epoch of Event 11).

Actually, such an error is improbable, as the slopes of the plots of total invisibility against longitude of conjunction (for Events 12 and 14) fortunately have opposing signs, making it quite difficult to mis-assess the mid-point between the two (which was the parameter actually used for the computation of L_S in Table 2). Nevertheless, an error of plus or minus 0.5° in L_S (from this cause alone) is easily conceivable and can be seen to be significant by comparison with the range of L_S values quoted in Table 2.

The first point to be noted is that all of the epoch combinations cited in Table 2 produce eccentricities of orbits at the time of Events 1-3 which do not differ at all widely from 0.008 for Earth and 0.050 for Venus (slightly lower values for 852 BC scenarios, slightly higher ones for 884 BC and earlier scenarios - see Table 4). I find these eccentricities strongly suggestive of an expansion of the orbit of the Earth, its eccentricity having been 'bumped up' from 0.008 to 0.018 in the course of the disturbances, whilst the eccentricity of the orbit of Venus was 'knocked back' from 0.050 to 0.008 (0.018 and 0.008 being the approximate retro-calculated eccentricities of the orbits of Earth and Venus at the start of the Julian era, around 700 BC). This finding is ultimately confirmed by the computed Sun-planet distances cited above (Tables 3 and 4).

852 BC scenarios call for Earth eccentricities in the 0.002-0.004 class, which are remarkably small. Any later date calls for a negative eccentricity of the Earth's orbit, which is an impossibility. Whilst 852 BC is not impossible for Event 1, it appears distinctly improbable.

The second point to look at is the eccentricities of the two orbits in the Assyrian era, as cited in Table 2. A glance at Figure 1 will show that, given an expanding Solar System, any changes in the eccentricities would be expected to be somewhere around two-thirds complete at the time of Event 11 - that is, about 0.015 for Earth and about 0.022 for Venus. These figures are essentially crude and speculative, but fortunately the computed eccentricities change quite rapidly for small changes in the epochs. One is therefore not dependent on very precise identification of the eccentricities during the Assyrian era. Note, however, that the computation of these eccentricities actually involves an assumption that they remain constant between Events 11 and 17 (or between Event 11 and the start of the Julian era in the case of the 'uncorrected' Table 2). It appears most unlikely that they would have remained constant for any length of time and it should also be noted that the 0.015 eccentricity quoted for Earth is an assumed one. This computation does not evaluate the eccentricities separately - it only computes the eccentricity of the orbit of Venus which goes with any given eccentricity of the orbit of the Earth; if the Earth's eccentricity is stepped up or down, so also is the corresponding eccentricity of the orbit of Venus. The computation itself is set out in Appendix E, step 3.

A final judgement as to what combinations of epochs can be deemed acceptable (or not acceptable) has to be based on a combination of acceptable eccentricities at the time of Event 11 with acceptable values of obliquity of the ecliptic (etc) at the time of Events 1-3. The characteristics of Events 1-3 actually have a greater influence on the final decision than the eccentricities of the orbits at the time of Events 11-17 (those listed in Table 2).

The simplest criterion for judging the acceptability or otherwise of the computed data for Events 1-3 appears to be G. F. Dodwell's finding that the observed obliquity of the ecliptic in 850-900 BC was just about 24.2 degrees [13]. One need not rely on Dodwell's finding, however, as the obliquity of the ecliptic at Events 1-3 correlates strongly with the Arcus visionis value of Venus at Event 3. A glance at Figure 2 will show that it is distinctly improbable that Venus would be materially fainter at Event 3 than at either Event 1 or Event 2. A low value of the obliquity (at Events 1-3) is always associated with a high value of Arcus visionis at Event 3 (i.e. a faint Venus). I actually attach more weight to the Arcus visionis criterion than to Dodwell's criterion, as will be seen below; note also that the present work suggests that there should actually be a discontinuity in the Bass-Dodwell curve at about 880 BC.

Table 3 presents the computed characteristics of Events 1-3 for the 744/884 epoch combination. They are spin rate dependent and are calculated for three (assumed) spin rates of the Earth (359.8, 360.0 and 360.2 days per year). This table does not of itself identify the correct spin rate. For this, I am currently suggesting an application of Schoch's limit, which claims that the difference between the Arcus visionis values for Events 1 and 2 will be 0.30. Schoch's limit is unlikely to be very precise and it is further discussed in Appendix C; a better criterion is almost certainly the shift of the vernal equinox, which should sooner or later be confirmable from some independent source. Meanwhile it is desirable to use some criterion other than the shift of the vernal equinox, and Schoch's limit seems to be the best available.

Table 3 is reproduced graphically in Figures 5A and 5B, on which Schoch's limit has been superimposed, resulting in an indicated spin rate at Events 1-3 of 360.135 Babylonian days per Babylonian year.

744/884 is not the only possible combination of epochs and Table 4 presents a number of alternatives (all computed in the same fashion as 744/884, above).

[*!* Image: Figure 5A. Spin rate of the Earth. Arcus visionis of Venus. EVENT 1 EVENT 2 EVENT 3 EVENT 4]

[*!* Image: Figures 5A and 5B. Detail of the circumstances at Events 1, 2 and 3 for the Epoch combination 744/884.]

[*!* Image: Figure 5B. Spin rate of Earth. Shift of vernal equinox. Obliquity of ecliptic. OBLIQUITY. SHIFT OF V.E.]

The general pattern of changes in the characteristics with changes of epoch is easily followed on this Table, which is divided into three parts. Both of the first two parts derive from the Table 2 epochs for Event 11. The difference between them is the span of years between the two epochs. On the whole, the first (left hand) part is indicated as the preferable one, but there is not a lot in it (it is particularly supported by Events 12 and 14 independently of one another). The third (right hand) part shows the effect of incorporating 'corrections' (plus 1 day and plus 1.5 days) into the Table 2 epochs, as outlined above. It is worth noting that such a correction cannot be applied to 868 BC scenarios without taking the Event 3 Arcus visionis value above about 6.3°, which seems hardly justifiable; similarly, the same corrections cannot be applied to 900 BC scenarios without taking the obliquity of the ecliptic above about 25.5°, which also seems unjustifiable. Though this may seem to indicate 884 BC with some degree of certainty, the precision of the present retro-calculations (further discussed below under 'Implications and discussion') is not such that very much guarantee can be given, and I suggest that the date of Event 1 should be seen as 884 BC plus or minus 16 years.

After making some allowance for the various uncertainties still present in the raw data, I find that the most probable state of affairs at the time of Events 1, 2 and 3 can be summarised as follows:

• Spin rate of Earth (Babylonian days per Babylonian year) = 360.0 plus or minus 0.15 days
• Obliquity of the ecliptic = 24.9 plus or minus 1 degree
• Shift of vernal equinox (from its retro-calculated position) = +15 plus or minus 3.5 degrees [14].

Were other evidence to support a date for the tablets outside the limits of 884 BC plus or minus 16 years sufficiently strongly, it would probably be possible to amend the present interpretation to match:

• by postulating as yet unidentifiable shifts of longitudes of perihelion of the orbits of Earth, Venus and/or Mars,
• by postulating exceptional visibility conditions at one or more Events,
• by postulating shifts of the effective latitude of observer at particular Events, and so on.

The possibility of imprecision in the retro-calculations has also been aired above and is further discussed below. It should also be noted that the computations can be materially affected by quite minor 'mistakes' in the raw data, the precise invisibility attributed to Event 2 being particularly critical (see Appendix D). Also to be observed is that the present system of dating is critically dependent on the longitudes assigned to the conjunctions of Events 12 and 14, and that these will vary slightly if the Arcus visionis values, obliquities of ecliptic or shifts of vernal equinox with which they are associated should be re-assessed. These two Events are so critical in their requirements, however, that there is very little scope for variation. Also to be noted is that a very similar pattern of events could probably be built up on a basis of dates about 480 years distant from 884 BC - in either direction - but these also seem improbable.

Implications and discussion

The original Sun-Earth distance is indicated as 0.9904 AU or thereabout (Table 4). The final Sun-Earth distance is known to be 1.000 AU (its modern value), making an ultimate expansion of the Earth's orbit of 0.97%. In all probability, however, there was an initial 'over-shoot' to rather more than 1.000 AU, subsequently cut back to 1.000 in the course of further clashes with Mars. If the absolute spin rate of the Earth was the same in the Assyrian era as in the subsequent Julian era, the maximum observed spin rate of 376.1 days per years would be associated with an expansion of the Earth's orbit to 1.0197 AU (= 2.95%) but there seems little doubt that this must be an impracticably large 'over-shoot'. The over-shoot can be eliminated if one allows that the absolute spin rate of the Earth was higher in the Assyrian era than in the Julian era, best expressed in the form of a value 'x', such that 'x' is the ratio of the length of the Assyrian day to the length of the Julian day (see Appendix D for further detail). If there was no over-shoot at all, the value of 'x' would have been 365.256/376.1, = 0.971. At present, one can unfortunately only speculate as to the actual degree of 'over-shoot'. An over-shoot to about 1.003 AU (= 1.27%) or 1.005 AU (= 1.47%) has been suggested, but these are essentially only guesses. They represent 'x' values of 0.975 and 0.978 respectively, the implications of which are further discussed below.

The very high observed spin rate between Events 5B and 7, 375.45 days per year, is potentially anomalous in terms of the explanation so far offered. Whilst it could be that there was a temporary increase in the absolute spin rate of the Earth at this time, it seems more probable that the transition from the Babylonian era to the Assyrian era had already started at Event 5B. If so, there would already have been an expansion of the Earth's orbit by the time of Event 6. This would affect the detail of the sequence of changes of Arcus visionis, obliquity and shift of the vernal equinox in the period between Events 5B and 11 but, as explained in Appendix A, it is not easy to substitute for Event 11 as the 'effective pivot point' between the two calendars. Note also that if there was actually a change in the maintained average spin rate between the times of Events 1 and 11, the precision of the retro-calculations specified in Appendix E, step 4, would be impaired, with probable repercussions on the dating arguments set out above.

The approximate mismatch can be assessed from Figure 1; it is not large but it supports the suggestion that the effective pivot point will actually lie rather earlier than Event 11.

It might now be worth attempting a reconstruction based on Event 6 as the effective Babylonian/Assyrian transition point (see Table 1 and Figures 1-4) but numerical analysis, as in C & C Workshop 1986:1 (above), suggests that Event 6 will be rather too early to mark the true pivot point, which actually appears to lie at a point about one-third to one-half of the way from Event 6 to Event 11. The best possible dating of Event 1 calls for retro-calculation from this point back to Events 1-3, but the period between Events 6 and 11 appears to be too disturbed to offer any satisfactory anchor point, which has to be relatable to a reliable observed day number and which also has to be retro-calculable. Fluctuations in the eccentricities between Event 3 and the anchor point have no back effect on Events 1-3 if the average sidereal period of the Earth is assured, as it is when 'the effective pivot point' can be firmly identified and adopted as anchor point.

Attempts to derive a satisfactory epoch for Event 6 have not met with much success to date, however, always being associated with derived values (synodic periods, predicted intervals, etc) which are both improbable and also incompatible with other evidence, so it may be that Event 6 is too closely associated with major disturbances to be of much value for computational purposes.

Its predicted longitude is also consistently 26-28° higher than its observed longitude, irrespective of the epochs assumed (for prediction purposes). The one pointer it does provide, however, is that the start of the Julian era may have been slightly later than indicated above, either 697 BC (in association with 868 BC for Event 1) or 712 BC (in association with 884 BC for Event 1). 712/884 is indicated as preferable to 697/868 on grounds of the associated Assyrian era eccentricities, but this finding is dependent on acceptance of the 26-28° displacement of the longitude of Event 6 outlined above. The same argument incidentally rules 729/900 out as distinctly improbable (its associated Assyrian era eccentricities are appreciably too high, approx 0.0150 + 0.0360). The immediate upshot of this is that 729 BC must be regarded as preferable to 744 BC for the start of the Julian era; whether it can be even further extended to 712 BC seems to be dependent on just where one locates the effective pivot point between the two earlier eras.

There are, in any case, still some incompletely resolved minor uncertainties, and it is also clear that Step 5 of Appendix E is ripe for renewed implementation, to be combined with the most thorough investigation possible of the limits to which the longitudes of Events 12 and 14 can be stretched (Step 2 of Appendix E), but both of these undertakings are extremely laborious and call for almost unlimited computing power. One would also like to have some confirmation of the true value of 'x' before finally embarking on this step. Meanwhile, even if the absolute dating of the tablets cannot be considered all that secure, much of the associated data - especially the relationships between Earth spin rate, shift of vernal equinox and obliquity of ecliptic - are only marginally affected by a small change of epochs, whilst the evidence that the tablets are describing a small expansion of the orbit of the Earth appears to be all but unassailable.

I actually find it abundantly clear that it was the disturbance immediately following Events 4 and 5A which particularly shook the ancients. Whilst the 274 day invisibility of Event 9 may seem more momentous to us, it seems to have been seen by the ancients as simply part of the general chaos following Events 4 and 5A. It should also be noted that no very close approach of Venus to Earth is indicated at any time. The closest passages are currently seen as 0.258 AU at Event 8A and 0.260 AU at Event 5A (comparing with a present day average distance between Earth and Venus at inferior conjunctions of 0.277 AU, with 0.255 AU being the minimum possible). The separation of Earth and Venus at the time of Event 9 appears to have been about 0.280 AU (and even the most unfavourable combination of eccentricities of orbit would scarcely reduce it below 0.275 AU).

Event 9 (the 274 day invisibility at an inferior conjunction) calls for special comment. Whilst one can get calculated invisibilities of up to 60-80 days at inferior conjunction by specifying suitably high values of Arcus visionis and obliquity of ecliptic, they are always associated with longitudes of conjunction which are 100-180 degrees different from the retro-calculated longitude of conjunction of Event 9 (around 325 degrees, in the absence of any shift of the vernal equinox). Since inversion of the Earth automatically shifts retro-calculated longitudes by 180 degrees, it appears probable that the Earth was at least partially inverted at this era (inversion being defined as occurring when the obliquity of the ecliptic exceeds 90 degrees). This was most probably a partial inversion, the longest invisibilities occurring when the obliquity of the ecliptic is between 57.5 degrees and 122.5 degrees (for an observer situated in latitude 32.5°N). Under static (or 'stable') conditions, such an invisibility would have endured for about six months (180 days), just as in the Arctic and Antarctic zones at the present day. The situation was clearly dynamic rather than static, however, with the Earth continuing to roll so as to hold Venus below the horizon at Babylon for a protracted period. One cannot get an invisibility as long as 274 days in the absence of such continued rolling.

Note that Venus would have been above the horizon in some other parts of the world; one thinks especially of the Polynesian claims that a bright star fished islands up out of the sea. Another consequence of obliquities between about 70 and 110 degrees is that both the Sun and Venus could have remained just below or just above the horizon for periods of several days (and nights) at a time, even for a few weeks, giving rise to prolonged spells of twilight, identical in nature with the Arctic and Antarctic twilights of the present day. Such phenomena would have occurred a few weeks or months to either side of Event 9. The inversion was only temporary, as no other Event is associated with such a very high obliquity of the ecliptic (see Figure 3).

The motions of the Earth indicated by the tablets can be replicated on the Earth inversion model I described in SIS Review V [15] but one needs to distinguish two types of spin rate change. Firstly, there is the short term reversible change of absolute spin rate (a deceleration followed by a re-acceleration), such as is consistent with Newtonian dynamics and which arises from temporary stressing of the Earth by an extra-terrestrial force. This is the only type of spin rate change which can be replicated on the model. The short term sudden collapses of spin rate immediately following Events 5A and 13A are both of this type; note also that both of these Events are inferior conjunctions. The rather more protracted collapse of spin rate at and about the time of Event 9 is also of this type; it produces what I would describe on the model as 'a failed inversion' (a near inversion which fails to go to completion, reverting instead to the starting condition; it is always associated with at least a short spell of sustained 'drunken reeling' of the Earth). This more prolonged but less severe collapse of spin rate started immediately following a superior conjunction and it is also clear that the ancients associated the particularly sharp and disabling catastrophes with immediately preceding inferior conjunctions.

Secondly, there is the change in observed spin rate which is not necessarily associated with a change of absolute spin rate. This type of change, abundantly represented on the tablets, arises from a change of orbital parameters, especially semi-major axis of orbit; if the orbit of the Earth is expanding, the count of days per year will automatically increase for no change in the absolute spin rate of the Earth. Only changes of absolute spin rate can be demonstrated on the model and they only persist whilst an extra-terrestrial force is actually present. A change in apparent spin rate can be much more permanent and will persist until extra-terrestrial forces intervene once again; even then there will only be a change in the long term maintained spin rate if the extra-terrestrial forces are strong enough to produce a further change of orbital parameters.

In this connection, note that the length of a year is defined as the time required for the longitude of the Sun (or a star) to change by 360 degrees; it is totally independent of Earth spin rate for the time being. The length of a day, on the other hand, is defined by the absolute spin rate of the Earth (further detail in Appendix D). Special interest therefore attaches to the value ascribed to 'x'. If 'x' = 1.000, there is no change of absolute spin rate between the Assyrian and the Julian eras. If 'x' is less than 1.000, however, the absolute spin rate of the Earth was higher in the Assyrian era than in the Julian. The only mechanism which can produce a sustained change of absolute spin rate, short of direct frictional contact with an extra-terrestrial body, appears to be a change in the electrical charge on the Earth [16]. If 'x' is less than 1.000, therefore, the inference is that the Earth was losing electrical charge to Venus but re-acquiring it from Mars (as is the prima facie indication of the pattern of changes as a whole, though this inference has to be discarded as an illusion if 'x' = 1.000 or more). If 'x' is greater than 1.000, the direction of flow of charge is reversed. Unfortunately, as already indicated, the true value of 'x' cannot at present be determined with any certainty, but a value of 0.975/0.98 is suggested as a first approximation.

The increase in the brightness of Venus at the superior conjunction of Event 16B (see Figure 2) is also potentially anomalous and could be the subject of a correction at some time, presumably when Events 19-21B come to be analysed in full detail. There is at least some prima facie evidence that Venus did become abnormally bright again, however, as Event 19 is listed as a 15 day invisibility. This is almost certainly a 'too short' invisibility as Event 19 is in the same sequence as Event 3 (20 days) and Event 11 (11 days). It was the realisation that the 11 day invisibility of Event 11 was appreciably too short for compatibility with any orthodox sequence of Venus invisibilities which first alerted me to the possibility that Venus might be abnormally bright at this era (Event 11 and thereabout). One would have expected the invisibilities of Events 11 and 19 both to be in the 19-20 day class.

It has already been commented that the two very sharp decelerations of Earth spin rate immediately follow inferior conjunctions of Venus (Events 5A and 13A). These two decelerations are spaced precisely 10 conjunctions of Venus apart (= approximately 8 years), making it a near certainty that Venus was their prime cause. All scenarios tested have produced a rather weaker and less well defined deceleration at Event 9 (conceivably because the half-way point between the two other decelerations is associated with a superior conjunction instead of an inferior one). Comparisons with the Mayan Codices (Dresden and Paris), especially Gregory M. Severin's excellent analysis of the animals which make up the Mayan zodiac [17], could well lead on to further identification of the parts of the sky in which the critical conjunctions occurred.

In passing, it is also interesting to note that the rate of roll of the Earth, other than during the 'tumbling phase' of Event 9, does not appear to have exceeded about one degree per ten days. This is a far cry from the 'inversion in a day' claim commonly made for the much earlier Phaeton episode.

It is probable that Event 4 is the superior conjunction described in the Panchasiddhantika. It is the only more or less regular superior conjunction listed on the tablets which occurs in a similar longitude to that of Sirius. A point of potential interest is that if Event 4 is dated by the Julian calendar, it falls in the month of July (JDN 1405314.75 = 18th July 866 BC if Event 1 occurs in 868 BC, or 22nd July 882 BC if Event 1 is moved back to 884 BC). This suggests that Event 4 will be a potential candidate for the marker of 'the first Olympiad' (see American Nautical Almanac for 1891, p. VII, on which the July dating is cited and which Dr Velikovsky referenced in Worlds in Collision in this connection).

The previous Workshop 1986:1 article [9], cited above, suggested that the (apparent) spin rate acceleration indicated by the Ninsianna tablets would be identical with the (absolute) spin rate acceleration indicated by the Ramesside Star Tables, but a dating of 880 BC or thereabout for the Ramesside tables is not consistent with our present understanding of Egyptian chronology. One way of escape from this dilemma - and it does still seem to me that the Ramesside tables must be describing the same events as the Ninsianna tablets - is to postulate that the Ramesside tables are commemorative monuments erected long after the events themselves (the Senmut ceiling sets a good precedent), but this may not actually be necessary. There are at least three years in the tablet record in which the absolute spin rate of the Earth accelerates very noticeably and it should eventually be possible to identify the relevant one. It may be necessary first to study possible disturbances of the Moon's motions, however, as the Ramesside observations are principally identified by months and all ancient references to months must be assumed to have been based on actual appearances and disappearances of the Moon. The changes between one calendar system and another will certainly have interfered with orthodox retro-calculated dates for appearances and disappearances of the Moon, and the starting point of any analysis would presumably be retro-calculation of the ADN and/or BDN dates at which the longitude of the Sun is zero, thus pin-pointing the dates of the vernal equinox (and thus also the dates of heliacal rising of Sirius or any other marker of the 'start of the year', = the start of Month 1).

The one biblical historical event which appears to tie in firmly with the Ninsianna events is the spectacular demise of Elijah, and it could be that this event can be used to reconcile biblical dates with the ones proposed here. It is also worth noting that certain sudden shifts of the vernal equinox could produce a 'repetition of births' - that is, a second heliacal rising of Sirius, or any other heavenly body, a week or two after the first one instead of after the normal interval of one year (the shifts concerned must be fairly abrupt and they must be in a positive direction, as defined in Appendix B).

Whilst I have no explanation to offer for the association of Ammisaduqa's name with the tablets, I imagine that 'the year of the heavenly throne' could refer to something spectacular in the skies, such as an extended Aurora which could even have blotted out normal observations of the heavenly bodies. These last proposals are essentially speculative, but they need to be kept in mind whilst acceptable explanations of all the detail are still being sought.

Precessional dating

A fall-out from the present study which is of general interest is recognition of the position of the vernal equinox at Event 1. It seems reasonable to suppose that it will have been in this position (subject to regular annual precession) ever since the previous round of major catastrophes - Exodus, Joshua, Sodom and Gomorrah, etc - and that these catastrophes will have occurred around 500 years earlier (say, 1400 BC). The indicated shift of the vernal equinox from its retro-calculated position (above) is +15° (plus or minus 3.5°); it is associated with an obliquity of the ecliptic of 24.9° (plus or minus 1°). At the present day, regular annual precession shifts the vernal equinox at a rate of 1° per 71.6 years. In round figures, therefore, a shift of the vernal equinox of 15° means that precessionally derived dates for events which appear to occur earlier than 882 BC (Event 4) will be 1075 years in error (plus or minus 250 years). For instance, if a part of Stonehenge is precessionally dated to 1820 BC [Lockyer, 18], its 'true' date will be 1820 - 1075 = 745 BC (plus or minus 250 years). This date will, of course, relate to the final form of Stonehenge, which was not necessarily the same as its initial form. Similarly, the Mandelkehr/Phaeton event is variously dated to 2100-2900 BC (by precession) and will thus become synchronous, or near synchronous, with the Exodus event (currently thought to be around 1450 BC). Note also, however, that if a 'true' date comes out earlier than about 1400 BC, it also will probably be fallacious, as it is barely conceivable that the Earth could survive the Exodus catastrophe (if not others as well) without suffering still another shift of the vernal equinox.

Summary

I find there can be little doubt that the Ninsianna tablets are describing a phase in a minor expansion of the Solar System, the phase when Venus was forcing the Earth out towards Mars. The eventual expansion of the Earth's orbit is indicated as a bare 1% though there appears also to have been a temporary over-shoot, cut back to 0.97% by later interactions with Mars.

Because of the general interest in chronology, special attention has been given to dating these events, but the possible precision of dating appears to be limited. Whilst the dating of Event 1 of the tablets to 884 BC (plus or minus 16 years) appears plausible, it cannot yet be considered absolutely firm.

It is clear that both day and year lengths underwent some change (year lengths being affected to a greater extent than day lengths), principally in the form of a transition from an original Babylonian calendar through an Assyrian one to the (modern) Julian one. It is confirmed that there were very close to 360 days per year in the Babylonian era, but these were Babylonian days and Babylonian years, not necessarily associated with any change in the absolute spin rate of the Earth.

There can also be little doubt that there was a significant shift of the vernal equinox (from its retro-calculated position) at the time of Event 1, a shift which would probably have been applicable throughout the previous 500 years (approximately), with far reaching consequences for precessionally determined dates which at first sight appear to come out earlier than 882 BC (Event 4 of the tablets). The error in these dates is indicated as being 1075 years plus or minus 250 years.

A final caveat: much of the detail listed here must still be regarded as essentially tentative. The only way of escape from the more general consequences, however, would seem to lie in the very dubious claim that the tablets are essentially only an early example of science fiction writing, intended to serve astrological purposes. That the observations were used for astrological purposes is as good as certain, but any suggestion that they represent a deliberate mis-statement of observable phenomena is plainly unsustainable.

Acknowledgements

This study would have been impossible without the pioneering works of Dr Immanuel Velikovsky, Professor Lynn E. Rose and Mr Raymond C. Vaughan. The efforts of other contributors to the publications of the Society for Interdisciplinary Studies have also helped it along its way.

Notes and References

Appendix A: The invisibility calculation

This form of calculation of the invisibilities of Venus derives essentially from the computations set out by B. L. van der Waerden in Berichte uber die Verhandlungen der Sachsischen Akademie der Wissenchaften zu Leipzig, Mathematisch-Naturwiss. Klasse, vol. 94 [1942], pp. 23-56. The detail of the calculation has been modified to suit modern calculating machines, however, and it has been extended to embrace a wider range of circumstances than was envisaged by van der Waerden.

A different definition of Arcus visionis has also been adopted. Van der Waerden used Ptolemy's definition of Arcus visionis, namely the difference in altitude of the Sun and Venus when Venus is just visible on the [mathematical] horizon. This is not a satisfactorily realisable situation in practice, however, and the definition adopted here for Arcus visionis is the difference in altitude of the Sun and Venus when Venus is just visible at an altitude of 0.4 x h degrees above the [mathematical] horizon and the Sun is simultaneously at 0.6 x h degrees below this horizon, 'h' being the Arcus visionis of Venus. This is at least approximately the actual situation at last or first visibility of Venus (before or after a conjunction with the Sun). Note also, however, that the visible horizon is normally about 0.5° lower in the sky (TZD = 90.5°) than the mathematical horizon (TZD = 90.0°), due to refraction.

Symbols used:

• l = ecliptic longitude
• l_H = geocentric longitude of the point of intersection of the ecliptic and the horizon
• l_S = geocentric longitude of the Sun at first or last visibility of Venus
• l_O = geocentric longitude of the Sun at conjunction with Venus
• l_V = geocentric longitude of Venus
• l_VH = heliocentric longitude of Venus (at conjunction or elsewhere)
• l_VHN = heliocentric longitude of the rising node of the orbit of Venus
• i = inclination of the plane of the orbit of Venus to the plane of the ecliptic
• E_G = elongation of Venus, geocentric, in the plane of the ecliptic
• E_H = elongation of Venus, heliocentric, in the plane of the ecliptic

(NB: Geocentric elongation of Venus is here defined as the angular separation of the Sun and Venus in the plane of the ecliptic, as viewed from the Earth; heliocentric elongation of Venus is similarly defined as the angular separation of Venus and Earth, in the plane of the ecliptic, as viewed from the Sun, except when the Sun and Venus are near to superior conjunction, when it is defined as 180° minus the angular separation of Venus and the Earth).

• f = latitude of observer (assumed northerly, unless stated otherwise)
• e = obliquity of the ecliptic
• h = Arcus visionis of Venus (see introductory paragraphs)
• b = ecliptic latitude, geocentric, of Venus
• b = ecliptic latitude, heliocentric, of Venus
• d = declination (d_H = declination of the point of intersection of the ecliptic and the horizon,d_V = declination of Venus)
• R = distance from Sun to Earth (modern average = 1.0 AU)
• r = distance from Sun to Venus (modern average = 0.72333 AU)
• H[dot]_E = rate of change of heliocentric longitude of Earth, degrees per day (modern average = 0.98561 degrees per day)
• H[dot]_V = rate of change of heliocentric longitude of Venus, degrees per day (modern average = 1.60213 degrees per day)
• t = number of days interval between a last or a first visibility of Venus and the associated conjunction with the Sun
• q = angle between the ecliptic and the horizon at their point of intersection
• q' = angle between the ecliptic and a small circle which passes through the projected position of Venus on the ecliptic and which is parallel to the horizon

[*!* Image: Figure 6: Geocentric view of the heavens when Venus is at first or last visibility, RISING or SETTING

SETTING (view westward) RISING (view eastward)

increasing longitude. travel of heavenly bodies. ecliptic. equator. horizon.

Notes to accompany Figure 6:

This diagram is only applicable to a northern hemisphere observer,l_H in the first quadrant (0 < l_H < 90°) and Venus with northerly ecliptic latitude (b positive). Corresponding diagrams for other configurations are easily derived. The usual situation at first and last visibilities (not applicable when the Earth is inverted) is as follows:

• Venus near inferior conjunction, first visibility occurs when RISING
• Venus near inferior conjunction, last visibility occurs when SETTING
• Venus near superior conjunction, first visibility occurs when SETTING
• Venus near superior conjunction, last visibility occurs when RISING

• HP is perpendicular to the equator
• VV' is perpendicular to the ecliptic
• QV' is perpendicular to the horizon
• VQ is perpendicular to QV'
• (all of the above are segments of great circles)
• angle 'A' is the angle at the 'H' corner of [DELTA]HPg
• angle 'B' is the angle at the 'H' corner of [DELTA]HPX (neither of the above angles can exceed 90°)
• Q can only exceed 90° in the 2nd and 3rd quadrants of l_H at rising (sunrise), or the 1st and 4th quadrants at setting (sunset). Beware descriptions of Venus as 'rising' when what is really meant is 'first visibility', which can occur either at sunrise or sunset.

]

[*!* Image: Figure 7: The starting point of the analysis (see notes which follow). Longitude of Conjunction (geocentric). Invisibility (Days).
350 250 150 050 310 210 110 010 270 170 070 330 230 130 030 290
80 70 60 50 40 30 20 10

Notes on Figure 7:

This diagram should be seen as consisting of two parts, one superimposed on the other. The base part comprises the two wavy lines, which mark the calculated invisibilities of Venus for any longitude of conjunction (calculation based on h = 6.0, e = 23.8, Q = 32.5, l_VHN = 51.3, i = 3.365, R = 1.0000, H_E = 0.98561, r = 0.7233, H_V = 1.60213). These calculated invisibilities are essentially only a first approximation, however, as the values of R, H_E, r and H_V must in practice all vary slightly with actual longitude of conjunction and actual duration of invisibility (uncorrected on this diagram); superior conjunctions are appreciably more adversely affected by this effect than inferior conjunctions. The upper line joins superior conjunctions, the lower one inferior conjunctions. The superimposed part of the diagram comprises the circled points, which mark the observed invisibility at each (numbered) Event, as listed in Table 1 (i.e. uncorrected), spaced at approximately equal increments of longitude (288° for a first approximation). For this part of the diagram, however, the longitude scale must be seen as slightly elastic, varying a few degrees either way according to Epoch and eccentricities of orbits of Earth and Venus for the time being. It is legitimate to regard the superimposed part of the diagram (the circled points) as being on tracing paper which can be moved about to find the 'best fit' with the underlying calculated invisibilities. Note that observed invisibilities are expected to be slightly longer than calculated ones; they cannot be shorter. If the tablet observations had been 'regular', every circled spot would have located on (or just above) one or other of the two wavy lines (subject to correct specification of Arcus visionis, an increase of 0.1° in Arcus visionis typically increasing a 65-70 day calculated invisibility by 1 day, and subject also to correct choice of Epoch, which controls the actual longitude of conjunction associated with each Event; note also the limitations on precision of calculated invisibility mentioned above). The whole purpose of the mathematical analysis is to identify the circumstances which marry up the two parts of this diagram as closely as possible (together with elimination of the many minor sources of inaccuracy as yet uncorrected on this 'first approximation' presentation).

The convention which is adopted here is that q and q' are less than 90° when the segment of the ecliptic above the horizon passes short of the zenith and greater than 90° when it passes beyond the zenith. When q is 90°, so also is q'; the ecliptic then passes precisely through the zenith. When the ecliptic passes through the poles (it cannot pass through one without passing through the other as well), the present style of computation of invisibilities is inapplicable (it also presents some difficulties when the ecliptic passes close to the poles, as when the Earth is on the point of inversion, inversion being defined as occurring at the moment when the poles pass through the plane of the ecliptic).

On diagrams, V denotes the true position of Venus, V' the projected position of Venus on the ecliptic. H denotes the intersection of the ecliptic with the horizon, S the position of the Sun (on the ecliptic) at first or last visibility of Venus. QV' is a line through V', perpendicular to the horizon. Q is a point on this line which is at the same altitude above the horizon as V (though not with absolute precision when triangle QVV' is treated as a spherical triangle, as below, but the deviation is extremely small in practice).

QV' = arctan(tan b cos q')

A computation of invisibility is always started from a chosen value of l_H. Inputs needed to complete the computation are f, e, h, l_VHN, i, R, r, H_E and H_V.

1). evaluate q from l_H, f and e
2). evaluate l_S from l_H, q and h
3). test q against 90°

[*!* Image:

Test QV' against 0.4h Test QV' against 0.4h
Evaluate E_G Evaluate E_G Evaluate E_G Evaluate E_G
Evaluate q' Evaluate q' Evaluate q' Evaluate q'
Evaluate t Evaluate b
Evaluate l_O from l_S, t and H_E

E_G is evaluated from l_S, l_H, h, b and q' (iterated values of b and q')
q' is evaluated from q, l_S, l_H, h, b and q' (iterated values of b and q')
t is evaluated from E_G, R, r and H[dot]_V - H[dot]_E (via E_H)
b is evaluated from E_G, l_S, t, H[dot]_V - H[dot]_E, l_VHN and i

]

The individual evaluation equations vary slightly according to whether the conjunction concerned is inferior or superior and whether the occasion is a last or a first visibility of Venus. They must naturally be expressed in forms which tolerate unforeseen changes of quadrant, but this presents no real difficulty in practice. At a last visibility, t is the interval in days between last visibility and conjunction; at a first visibility, t is the interval in days between conjunction and first visibility. Total invisibility is the sum of the two values of t for the conjunction concerned.

Notes on the invisibility calculation

The magnitude of QV' determines whether the projected position of Venus on the ecliptic (at a first or a last invisibility) lies above or below the terrestrial horizon. The calculation has to be started with estimated values of b and q', which are subsequently refined by iterative computation. One usually waits for the value of E_G to stabilise (at any chosen level of precision - 3 decimal places for the present work), then reads off the associated values of t and l_O. The iterations normally converge easily but convergence becomes slow at low values of q; in extreme cases, divergence may occur but can be circumvented by suitable pre-selection of the starting value of b. In extreme cases, the computed value of E_G can also over-run its permitted limit (arcsin r/R, modern average = 46.33°), in which case the particular computation must either be abandoned or approached from some other angle (such effects arise from the slight imprecision in the evaluation of QV', mentioned above; they are avoidable, but only at the cost of considerable complication). There are also rare occasions when multiple visibilities occur - described in detail by Rose & Vaughan in their paper, 'Analysis of the Babylonian observations of Venus', Kronos II:2 [1976], pp. 3-26 - and the effect on the computation is that there are then multiple solutions for E_G in a narrow band of values of l_O; the easiest way of dealing with this circumstance appears to be to make a plot of l_O against t through the critical zone, when it becomes obvious what is happening. Note also, however, that even when the calculation is 'perfect' (no approximations or departures from rigour), there are still several potential sources of error when the invisibility is finite (non-zero). Arcus visionis can and does vary slightly with time, for instance, whilst the values of H[dot]_E and H[dot]_V vary appreciably with time and season. There is no difficulty about computing average values of H[dot]_E and H[dot]_V for any particular duration (and season) of invisibility, however, and this is a necessary step in practice for all invisibilities which endure for more than a day or two. There are also potential complications when the spin rate of the Earth, its angle of tilt and/or the position of the vernal equinox are changing continuously or in an unpredictable manner. See also Appendix B for required modifications to longitudes when the vernal equinox is displaced from its normal (or retro-calculated) position.

The division of h in the proportions of 0.4 and 0.6 is not sacrosanct and it is probable that the 'true ratio' varies slightly with Arcus visionis for the time being (see reference 5), but it is seldom difficult to adjust a computing program appropriately (scarcely necessary in practice, in my experience). It is possible, but complicated, to extend the system of computation to accommodate values of e greater than 57.5° (the value implied by the restriction cited in the heading to the above basic scheme of computation) but such computations have not been relied upon in the present study. When e is greater than 57.5°, there will be seasons when the Sun and/or Venus neither rise nor set, as in the Arctic and Antarctic regions at the present day.

Notes on retrocalculation and sundry other computations

Retrocalculations have been made on the basis of the elements cited in the Explanatory supplement to the Astronomical Ephemeris [19]. The principal basis of comparison available to me has been Tuckerman's tables [20] but these only go as far back as 601 BC. My longitudes for the Sun differ from Tuckerman's in a consistent fashion, -0.19° in 300 BC, -0.25/0.26° in 600 BC (Tuckerman's longitudes being larger than mine for any given date and time). These deviations presumably arise from a small difference in the elements used for the calculation. The deviations for longitude of Venus are similar (+0.1° to -0.3° in 600 BC) but they are not consistent, going one way at inferior conjunctions and the other way at superior conjunctions (plus at inferior, minus at superior conjunctions). This is presumably due to inclusion of a correction for perturbations in Tuckerman's calculations, neglected in mine. Mars longitudes follow a similar pattern to those of Venus (+0.45° at opposition, -0.15° at conjunction; see also ref. 22, where the same effect is noted disapprovingly). I find these deviations small enough to be unimportant for the purposes of the present analysis. Doubtless Tuckerman's figures are preferable to mine, but it is essential for the present work to have a system of retro-calculation from first principles in order:

a). to be able to study effects arising from abnormal Sun-planet distances and abnormal eccentricities of orbit,
b). to be able to derive mean longitudes,
c). to extend the calculations further back than 601 BC, and
d). to be able to retro-calculate in the special circumstances of the Assyrian and Babylonian eras (when modified rates of change of mean longitude are applicable). Details of retro-calculation techniques are discussed in refs. 20 & 23.

The retro-calculation programme is principally used to compute sequences of conjunctions of Venus for any chosen era (i.e. sets of day numbers, JDN, ADN or BDN, usually calculated to two decimal places, occasionally three, at which l_V = l_O). In the course of the computation, geocentric and mean longitudes for the Sun at each conjunction are abstracted, occasionally also the mean longitude of Venus. The computation also yields spot values of R, r, H[dot]_E and H[dot]_V for use in the first run through the invisibility analysis; averaged values of H[dot]_E and H[dot]_V, mentioned above, with which also belong averaged values of R and r, can only be derived subsequently. If one is analysing the (Babylonian) era of Events 1-11, for instance, one needs a sequence of 13 conjunctions, starting from an inferior conjunction (which is eventually chosen in the light of experience of successive runs through the invisibility analysis). The difference between the day numbers for Events 1 and 11 then gives the retro-calculated day count for the interval between them. The resultant geocentric longitudes of conjunction are next applied in the invisibility calculation to derive observed day numbers of conjunction, as in column 5 of Table 1 (in practice, one has to apply successive approximations of l_H until the desired l_O results, with separate calculations for 'rising' and 'setting'; see also notes below re continuous up-dating of Arcus visionis values). The difference between the day numbers for Events 1 and 11 then gives the observed day count for the interval between them. The same process must also be applied separately to the Assyrian era (Events 11-17). For almost all types of further computation, the next step is synchronisation of the retro-calculated and the observed day counts, achievable by suitable adjustment of the synodic period of Venus (which results in automatic modification of some of the inputs into the retro-calculation program; this program is actually set up on a Hewlett-Packard HP 41-CV programmable calculator, most of the time set to home in on the day numbers of Venus conjunctions automatically). The ensuing steps are set out in Appendix E, according to the type of further computation which is required.

A necessary task which presents special computational difficulties is to establish whether there are values for the obliquity of the ecliptic and the shift of the vernal equinox which are common to three or more consecutive Events and which will result in a constant spin rate of the Earth for the whole of the period concerned; it can be done, but probably not without the help of rather complex plotting procedures. A simpler variant of the same problem is optimisation of the obliquity of the ecliptic and the shift of the vernal equinox at Events 1-3, solvable as follows:

• Make invisibility calculations for Events 1-3 on a basis of two positions of the vernal equinox and three obliquities of the ecliptic, = 18 invisibility calculations. Arcus visionis values will have to be deliberately varied so as to preserve a specified differential between calculated and observed visibility (separately for each of the 18 computations - the present work has been standardised on 0.85 day difference at Events 2 and 3, 0.65 day difference at Event 1).
• Derive average spin rates of the Earth between Events 1 and 2 and between Events 2 and 3, = 12 spin rates, 4 for each obliquity.
• Plot these spin rates against position (or shift) of the vernal equinox. Mark the intersections of the diagonals of each set of 4 spin rates (one set for each obliquity), thus determining the position of the vernal equinox and the Earth spin rate associated with each obliquity (if the appropriate diagonals are not easily identifiable, repeat the process with new starting values - the starting values should preferably straddle the expected end values for vernal equinox and obliquity).
• Join the three intersections, which should lie on a straight line with proportional spacing of the obliquities along the line. The intersection of this line with the known (or assumed) spin rate of the Earth at Events 1-3 then gives the optimised position of the vernal equinox directly; the associated obliquity of the ecliptic is found from interpolation along the line.
• If necessary, repeat the process with starting values which have been selected so as to optimise precision - e.g. two positions of the vernal equinox, plus 3° and minus 3° from the expected end result and two obliquities of the ecliptic, plus 1° and minus 1° from the expected end result.

Invisibilities commonly need to be calculated to two decimal places (of days) but three decimal places can be needed for determination of eccentricities of orbit (Appendix E, step 4); Arcus visionis values must usually be calculated to three decimal places, occasionally four, to preserve comparability between the individual invisibility calculations (even though such levels of precision are scarcely realistic, given the nature of the raw data).

The prime justification for declaring Event 11 to be a 'pivot point' lies in the numerical analysis of the tablet data presented in Workshop 1986:1, p. 9 [9]. It was suggested there that the true pivot point might lie somewhat earlier, perhaps around Event 8A, but attempts to move it, even part way back towards Event 10, have so far proved unavailing. It also appears improbable that precision would be improved even if it could be shown with certainty that the true pivot point lay somewhat earlier than Event 11. It is ideal to locate base points for computations at well documented inferior conjunctions (e.g. Events 1, 11 and 17). Moreover, when so much reliance has to be placed on averages, the base points must if at all possible be located in periods when extra-terrestrial (catastrophic) forces are not operative. It can be seen from Figures 1-4 that Event 11 does lie in such a period. It actually strikes one very noticeably when analysing the data that Events 10 and 11 present an apparent oasis of stability in an otherwise very troubled period. Though the invisibilities at Events 10 and 11 appear normal, this is actually an illusion as they do not locate properly within the normal 8 year cycle of Venus phenomena. The Babylonians also seem to have seen in them a hope of renewed stability, or they would hardly have inserted an intercalary month in Year 11.

Appendix B: the effect of a displacement of the vernal equinox

The position of the vernal equinox is defined as one of the (two) points of intersection of the ecliptic with the celestial equator. It marks the direction of zero celestial longitude and it is usually also seen as marking the start of the 'natural' year. If a celestial disturbance should cause a more or less abrupt change in the obliquity of the ecliptic, it will usually also cause a change in the position of the vernal equinox. Let g₁ be the normally expected, or retro-calculated, position of the vernal equinox at any particular era; suppose this now to be more or less abruptly shifted by X° to a new position g₂, such that g₁ + X =g₂.

g₁,g₂ and X must all lie in the plane of the ecliptic, which can itself only become modified if the orbit of the Earth is significantly changed.

[*!* Image: Figure 8. increasing longitude. l_VHN]

[*!* Image: Figure 9. heliocentric longitude. geocentric longitudes]

The invisibility calculation yields a computed l_O as the geocentric longitude of the Sun at conjunction with Venus, relative to the position of the vernal equinox for the time being and irrespective of whether the conjunction concerned is an inferior or a superior one. If the vernal equinox has been shifted from its normal (or retro-calculated) position by +X°, l_VHN (the retro-calculated heliocentric longitude of the rising node of the orbit of Venus) must therefore be replaced in the equations (prior to re-evaluation) by l_VHN minus X° (see Figure 8 for clarification). Note that the displacement of the vernal equinox has been expressed in terms of a shift in its heliocentric longitude but that the position is essentially unchanged when geocentric longitudes are substituted for heliocentric ones (see Figure 9 for clarification). Care is necessary when both are involved in a single calculation, however, for the longitude of the Sun can only be expressed geocentrically (geocentric longitude of Sun = heliocentric longitude of Earth + 180°).

As already mentioned, the invisibility calculation yields the geocentric longitude of the Sun relative to the vernal equinox for the time being. A frequent problem is to calculate the invisibility which results when the conjunction occurs in a longitude which is relative to a shifted vernal equinox. In this case, not only must l_VHN be suitably modified prior to calculation, but l_O as well.

The normal annual drift of the vernal equinox (due to the precession of the equinoxes) is X = -0.014° per year and, in the ordinary way, l_VHN increases slowly with time (about 9° per 1000 years); a positive displacement of the vernal equinox (+X°) therefore produces an illusion that the epoch of the observation is earlier than it actually is (the apparent value of l_VHN, and most other 'observed' longitudes, having been reduced).

So long as the orbits of both Earth and Venus remain undisturbed, the value of i is unaffected by either a displacement of the vernal equinox or a change in the obliquity of the ecliptic.

[*!* Image: Figure 10. geo- or helio- centric longitudes. JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC]

A sudden shift of the vernal equinox also has an effect on the calendar, as it interferes with the normal sequence of the seasons. If the shift is a positive one (as defined above), one or more months will be lost, or 'jumped over' (see Figure 10 for clarification). Should the shift be +30° (or thereabout, dependent on actual month length), the calendar can be rectified by repeating the current month (e.g. the 'second Ulul' of the Ninsianna tablets: in the circumstances of Figure 10, one could proclaim a 'second March' so as to bring g₂ level with the new 21st March). If the shift is negative, similar principles apply. The lengthening of February in leap years is the modern form of rectification of the calendar for the present slight negative shift of the vernal equinox. The reason why the calendar correction appears to be in the same direction as for a positive shift of the vernal equinox lies in our adoption of a 365 day calendar year in place of an approximately 365.25 day 'true' year (there being several slightly differing definitions of the length of the 'true' year, dependent on what one means by 'true'), a practice which grossly over-compensates for the present drift of the vernal equinox (1 degree per 71.6 years).

Appendix C: the 360 day year

It will be well known that Dr Velikovsky argued that the addiction of the ancients to 360 day years was due to there having at one time been actual years of 360 days (or thereabout). The numerical analysis of the tablet data presented in Workshop 1986:1 [9] suggested that if there was once such a year, it would have been applicable at the era of Events 1-11 of the tablets. No satisfying confirmation of the validity of the 360 day year appears ever to have been found in ancient astronomical records and it is therefore of prime interest to establish whether the tablets can confirm a 360 day year or not.

If one applies the optimisation procedure for Events 1-3, described in the latter part of Appendix A, one soon discovers that there is only one combination of eccentricities of orbits of Earth and Venus possible for any given spin rate of Earth (epochs of Event 1 and the various calendar transitions also being given, or provisionally assumed; the observed day count from Event 1 to Event 11 is also of critical significance). The method of computing these eccentricities is complex and cumbersome; it is set out in Appendix E, step 4. There does not appear to be any alternative to making assumptions as to the epochs and Table 2 shows the effect of a number of such assumptions. Initial trials were made with a whole number of synodic periods between Event 11 and the Assyrian/Julian transition point - that is, both dates were sited at inferior conjunctions of Venus, which simplifies determination of a first approximation to the day number of Event 11. I find it more probable that the Assyrian/Julian transition point will have been marked by a close opposition of Mars than by an inferior conjunction of Venus, however, and the epochs suggested for the start of the Julian era in Table 2 are all retro-calculated close oppositions of Mars (below). As can be seen from Table 4, all epoch combinations which produce potentially acceptable values for the obliquity of the ecliptic (etc) at Events 1-3 also produce spin rates at the time of Events 1-3 which do not differ at all widely from 360 (Babylonian) days per year.

Re Schoch's limit, which is used in the main text as the criterion by which to identify the actual spin rate of the Earth, modern data on the brightness of Venus shortly before and shortly after conjunctions appears to suggest that the change in brightness of Venus can be rather variable, but published data (B.A.A. yearbooks) for a set of 18 conjunctions during the period 1977-1991 shows an average visual magnitude of -3.76 to either side of inferior conjunctions and -3.56 to either side of superior conjunctions. These figures are essentially only approximate, however, as the interval between conjunction and observation is not standardised in the published data. Whilst I know of no absolute conversion between visual magnitude and Arcus visionis, this does seem to suggest that Schoch's limit may be slightly on the high side. If Schoch's limit should be reduced, the indicated spin rate of the Earth is marginally reduced but, more importantly, the indicated shift of the vernal equinox is increased. It should also be noted that Schoch's original data calls for a difference of 0.3° only on one side of inferior conjunction, with 0.8° on the other; no such distinction appears in modern observations, nor does it seem reasonable (though there are possibilities arising from differences in brightness between morning and evening skies, but the tablets themselves give no indication of any such distinction, as was briefly discussed in the introductory part of the main text). Schoch's limit has incidentally received qualified approval from both van der Waerden [4] and Huber, the latter describing it as 'surprisingly accurate' [7].

It needs to be observed that dates earlier than 729 BC (or equivalent epoch) cannot properly be described by a BC date as the BC calendar system is tied absolutely to the Julian calendar. In the absence of conventions as to how one is to count years which contain an abnormal number of days, it is probably safest to stick to a day number system of dating. (To find the season of the year associated with any particular day number, the simplest method is probably to retro-calculate the day numbers at which the longitude of the Sun is zero - i.e. the day numbers of the vernal equinox. This applies equally whether the calendar concerned is Babylonian, Assyrian or Julian, as the customary BC calendar is itself imperfectly corrected for leap year effects).

The deviation between the Julian and the other calendars varies quite widely with the choice of epochs, and can be read off (for the epoch of Event 11) from the column headed JDN-ADN in Table 2. For the epochs with which we are principally concerned, the deviation of Event 11 does not amount to more than 300-400 days. In the circumstances, I find it reasonable to continue to use the familiar BC dates for everyday purposes, but one must naturally bear in mind that they are no longer a rigorous description of the true date (they are typically about 1 year out by comparison with a true count of years). JDN-ADN incidentally maximises at Event 11; the deviation at Event 1 (JDN-BDN) is approx. 32.5 days less.

The JDN numbers of the close Mars oppositions which are potential markers of the start of the Julian era (by my retro-calculations) are as follows:

Appendix D: sidereal and synodic periods

Notation: S = synodic period, T = sidereal period, N = the count of days per year, n = daily change of longitude, [alpha] = semi-major axis of orbit (expressed in modern A.U.).

Subscripts: E = Earth, V = Venus, m = 'modern', a = 'ancient'

The sidereal period (T) of a planet (or the Earth) is the time, usually expressed in days, which the planet takes to complete a circuit of the Sun (sometimes also termed the sidereal year). The average daily change of (heliocentric) longitude of the planet (no) is thus n = 360/T (or, T = 360/n).

The more commonly cited tropical year of 365.2422 days is derived from the sidereal year by application of a correction for the precession of the equinoxes, but the sidereal year is the better measure of the 'true' spin rate of the Earth. Sidereal period and observed spin rate of Earth are treated as synonymous throughout this work. A caution is needed, however, that neither the sidereal year nor the observed spin rate are satisfactory measures of the absolute spin rate of the Earth as they are commonly expressed in days per year, both of which can vary in absolute length in catastrophic circumstances.

The modern sidereal period of Earth is T_Em = 365.2564 days. The corresponding average daily change of longitude (n) is n_Em = 0.98561 degrees per day.

The modern sidereal period of Venus is T_Vm = 224.701 days. The corresponding average daily change of longitude is n_Vm = 1.60213 degrees per day.

The synodic period (S) of a planet is the time, usually expressed in days, which elapses between one conjunction with the Sun and the next (like types of conjunction, the interval between an inferior and a superior conjunction being approximately half a synodic period). Synodic period is wholly controlled by the difference in the rates of change of longitude (n) for the Earth and the planet concerned. Assuming the spin rate of the Earth to remain constant, the synodic period (S) of an inner planet (such as Venus) is given by S = 360/(n_V - n_E).

The synodic period of modern Venus is thus 360/(n_Vm-n_Em) = 360/(1.60213 - 0.98561) = 360/0.61652 = 583.92 days. Note, however, that daily rates of change of longitude are not actually constant but vary slightly between one part of the year and another, causing the interval between successive conjunctions to differ slightly from the average (by up to several days, dependent on eccentricities of orbits at the time concerned). It is actually the rates of change of longitude which determine the relationship between spin rate and synodic period, but this relationship is more commonly expressed in the form 1/T_V = 1/S + 1/T_E (only applicable in the case of an inner planet, such as Venus). This relationship has been applied in the present work to find the new value of T_V which applies when a synodic period has to be modified in order to synchronise retro-calculated and observed day counts. The corresponding values of [alpha]_E and [alpha]_V are found by application of Kepler's third law to the values (for the time being) of T_E and T_V. This law states that the ratio [alpha]3/T² is constant, which can be evaluated by setting [alpha] = 1.000 and T = 365.2564 Julian days. A count of Assyrian days must be multiplied by 'x' (the ratio of the length of the Assyrian day to the length of the Julian day) to find its equivalent in Julian days before application in this equation. In this connection, it should also be noted that it has been assumed in the present work that Babylonian and Julian days were equal in length; whilst unlikely to be absolutely true, it appears probable that they will have been at least very similar in length [24].

Note that the length of the day is controlled by the absolute spin rate of the Earth and is independent of year length. For all practical purposes, 1 revolution of Earth = 1 day (subject to a consideration that the Earth moves further along its orbit in the course of a day and so has to turn slightly more than 1 revolution to bring the Sun back to the same position relative to any particular spot on the Earth's surface).

If, at some ancient era, the orbit of the Earth were to be contracted from 1.000 AU semi-major axis to '[alpha]' AU ([alpha]< 1.000) but the Earth maintained the same absolute spin rate as today (= no change in day length), its orbital period would become [alpha]^1.5 years (by Kepler's third law) and the number of days per year (N_a) would be N_m x [alpha]^1.5, where N_m = 365.2564 days (= T_m). The ancient sidereal period of Earth (in a contracted Solar System) would thus have been T_Ea = T_Em x a^1.5 days (equal in length to modern days) and the ancient daily rate of change of longitude would have been n_Ea = 360/(T_Em x [alpha]^1.5).

The above calculation assumes no change in the absolute spin rate of the Earth, but there is not actually any guarantee that the absolute spin rate of the Earth would have remained constant in the course of a contraction (or expansion) of orbits. There are actually three configurations of potential interest, as follows:

i). Earth spin rate variable, no contraction or expansion of orbits. This is the situation envisaged in most (if not all) previous attempts to relate Earth spin rate to synodic period,
ii). Earth spin rate constant, contraction or expansion of orbits is permitted,
iii). Any combination of i). and ii).

i). Earth spin rate variable, orbits unchanged:

Orbital period is independent of spin rate. Sidereal period (T) is therefore unchanged when expressed in years; as the number of days per year (N) has changed, however, sidereal period in days will be T_a = T_m x N/365.256. The sidereal period of Venus will be affected in the same proportion as that of the Earth, as the sidereal period of Venus is reckoned in Earth days (ancient or modern, as appropriate for the era concerned). Daily rates of change of longitude will be n_a = (360/T_m) x (365.256/N). The value of n_V - n_E will be (360/T_Vm - 360/T_Em) x 365.256/N. The quantity inside the brackets is the modern synodic period of Venus (S_m); the ancient synodic period is thus S_a = S_m x N/365.256 (subject to some modification according to whether the spin rate change was absolute or apparent; a change in absolute spin rate changes the day length but a change in apparent spin rate does not).

ii). Earth spin rate constant, orbits contracted in ancient era:

A principal problem is that if the orbit of the Earth is contracted, the orbit of Venus will probably also be contracted, but there does not appear to be any sure way of assessing by how much the orbit of Venus will have been contracted.

Consider firstly the effect of specifying that the synodic period of Venus is to remain constant. That is, n_Va - n_Ea = n_Vm - n_Em = 1.60213 - 0.98561 = 0.61652. In the situation described in the main text (epochs 744/884), the ratio [alpha]_Va/[alpha]_Ea is actually 0.7236. This ratio, in an uncontracted Solar System, is 0.7233. That is, the orbit of Venus was proportionately slightly less contracted than that of Earth, giving rise to a slightly closer approach of Venus to Earth than normal.

For the synodic period of Venus to increase, n_V - n_E must reduce. If n_V is reduced, however, T_V and [alpha]_V are both increased, eventually giving rise to an exceptionally close passage of Earth and Venus at inferior conjunction. Actually, all the evidence is that no such close passage occurred.

It is clear that contraction of orbits without change of Earth spin rate (absolute) is both feasible and compatible with the tablet record, but there does not appear to be sufficient evidence available to permit specification of any one set of parameters as preferable to any other.

iii). Combined systems:

Clearly, a reduction of synodic period of the order of 1% (S_a = 99% x S_m), as indicated by the tablets, can result from either mechanism i). or ii)., or from a combination of the two (though the contribution of each may then be very small).

Also to be noted is that all the 'days' mentioned on the tablets will not have been equal in length, but one must assume that the observers of the time would not have had any means of checking day length with precision, so that observed day counts make no distinction between 'long' and 'short' days. One must also suspect that the observers recorded their observations in the form 'x months + y days' because their months were actually observed lunar months rather than day counts (an assumption has been made that a recorded month was equal to 30 recorded days, but it is doubtfully justifiable). This would mean some mis-assessment (on our part) of the longer invisibilities, but the only 'long' invisibility which is at all critical (for the purposes of this paper) is the 68 day invisibility of Event 2. If this were to be proved 'wrong', the computations for Events 1-3 would be significantly affected.

Each 1 day reduction of the 68 day claimed invisibility advances (makes later) the dating of the tablets by about 10 years. A 10 day reduction actually brings the dating quite close to the 776 BC suggested by Dr Velikovsky but I know of no indication that the present 2 month plus 8 day reading of the tablets may be challengeable.

Appendix E: summary of computations

Notation: 'L' = mean longitude, 'dL₁' = change in mean longitude between the conjunctions of Events 1 and 2, 'dL₂' = change in mean longitude between the conjunctions of Events 2 and 3, e_E = eccentricity of Earth's orbit, e_V = eccentricity of the orbit of Venus.

Step 1

Make some preliminary runs through the invisibility calculation for each Event in turn (Events 1-17) to establish the probable sequence of changes of Arcus visionis, obliquity of the ecliptic, shift of vernal equinox, and spin rate of the Earth (as in Figs. 1-4). An epoch for Event 1 (here taken as 868 BC) has to be assumed but the results prove to be very insensitive to the actual date chosen. It will usually also be necessary to start from an assumption that the Julian calendar (JDN) will be applicable throughout the sequence. Though it transpires subsequently that the Julian calendar must be supplanted by a Babylonian one (BDN) and an Assyrian one (ADN), this substitution also does not affect the results very noticeably (JDN = ADN at the Assyrian/Julian transition point, ADN = BDN at Event 11). Successive approximations to the eccentricities of orbit of Earth and Venus, together with estimates of their variation with time (over the period of Events 1-17), are also required but fortunately the consequences of initial mis-assessments are not such as to threaten eventual convergence of the successive approximations as a whole. Adoption of the hypothesis that Arcus visionis will be the only free variable in the absence of catastrophic events (= abnormal extra-terrestrial forces) permits intelligent choice between possible alternative scenarios. The prime aim at this stage is to arrive at the best possible estimate of the 'observed' day number of each conjunction (as in Table 1, preferably with a precision of the nearest 0.1 day, though this degree of precision will not be attainable in the early stages of the operation; the precision so far attained in the present work appears not to be better than the nearest 0.2 days). Note that the Day Nos. of Table 1 are 'local' ones, not to be confused with ADN or BDN, though relatable to them.

Step 2

Study Events 12 and 14 particularly closely. Both are exceptionally critical in their requirements. Derive a plot of calculated total invisibility against (geocentric) longitude of conjunction over the narrow range of longitudes in which there is a sudden drop of calculated invisibility from a maximum (approx. 85°-120° for Event 12, 250°-260° for Event 14, both before correction for any shift of the vernal equinox). This plot is essentially but not absolutely independent of the epoch selected for Event 1; it is unfortunately rather more sensitive to the value ascribed to 'x'. The current findings are 109.1° for Event 12 and 253.5° for Event 14 (at 'x' = 1.000 and relative to an unshifted vernal equinox, though calculated on the basis of the actual quite substantial shifts of the vernal equinox at these Events).

Step 3

Assume dates ('epochs') for the start of the Julian era and for the time of Event 1 (as in Table 2). Derive an approximate date for Event 11 (usually in the form of an ADN number). Assume an average Earth spin rate for the period concerned (either Events 11-17 or Event 11 to the start of the Julian era), also potential eccentricities of the orbits of Earth and Venus. Adjust the synodic period of Venus till the retro-calculated interval between Event 11 and Event 17 is equal to the observed interval (2350.3 days has been used in the present work; necessary associated adjustments are detailed in Appendix D). Retro-calculate the longitudes of Events 11, 12, 14 and 17. Compare the retro-calculated longitudes of Events 12 and 14 with those found in Step 2. If they do not agree, the retro-calculation must be repeated with new values for the eccentricities of the orbits (including re-assessment of the synodic period of Venus which will match up the retro-calculated and observed intervals between Event 11 and Event 17; with experience, one soon learns the degree of correction of eccentricities which will offset any mismatch of the longitudes). The epoch of Event 11 can now be read off (it will be expressed in ADN but ADN = BDN at Event 11).

Step 4

Assume probable eccentricities of orbits at the era of Events 1-3 and using the above retro-calculated ADN date for Event 11 as epoch, retro-calculate the conjunctions of Events 1-3 for various Earth spin rates (e.g. 359.8, 360.0 and 360.2). As in Step 3, the synodic period of Venus must be re-adjusted for each calculation to keep the retro-calculated interval between Events 1 and 11 equal to the observed interval (here taken as 3467.8 days). Apply the retro-calculated longitudes of Events 1-3 in the invisibility calculation, applying also at least two values of the obliquity of the ecliptic and at least two values of the shift of the vernal equinox so as to find the optimised obliquity of the ecliptic and shift of the vernal equinox at the era of Events 1-3, as described in the latter part of Appendix A. Re-calculate the invisibilities for Events 1-3 with these optimised values and derive rates of spin for Events 1-2 and 2-3. If the derived rates of spin (two, one for Events 1-2 and one for Events 2-3) are not both precisely equal to the spin rate for which the particular retro-calculation was made, the originally assumed eccentricities of the orbits were not correct. One way of finding the corrected eccentricities is to determine the L values (dL₁ +dL₂ and dL₁/dL₂, the latter preferably calculated to 5 decimal places) which would be needed to match the calculated invisibilities to the spin rate used in the particular retro-calculation (a very simple operation), then derive the corresponding retro-calculated L values from retro-calculations made for values of e_E and e_V slightly above and slightly below the expected end values. Four sets of retro-calculated L values will result which, when suitably plotted against e_E and e_V, will permit derivation of the e_E and e_V values which will match up with the L values indicated by the invisibility calculation.

This method appears not to be infallible, however, and there are some combinations for which prediction of the required eccentricities proves exceptionally difficult. It is of potential interest that precision of eccentricity prediction frequently minimises quite sharply at the spin rate which subsequent evaluation indicates as the 'correct' one, but the whole subject is one which calls for further study. Computation of eccentricities to four decimal places is always possible, however, and is also necessary if comparability between the various tables presented in the main text is to be preserved.

The calculations, to be of any value, have to be repeated for more than one spin rate of the Earth (preferably a minimum of three); eccentricities differ for each spin rate but so also do Arcus visionis values, obliquity of ecliptic, shift of vernal equinox, etc. Currently, the difference between the Arcus visionis values for Events 1 and 2 is regarded as the prime indicator of the spin rate which is to be accepted as the correct one, but the shift of the vernal equinox is also a potent indicator which it should eventually be possible to tie in with independent evidence from other sources. Note also that Arcus visionis values applied in the invisibility calculations have to be repeatedly updated so as to preserve specified differentials between calculated and observed invisibilities (0.85 days has been adopted as 'standard' for all Events other than Event 1, for which 0.65 days has been allowed, but there must still be some doubt as to whether 0.85 days will always be enough).

Step 5

Finally, repeat Step 1. If the observed Day Nos. of conjunction (as in Table 1) are now significantly different from what they were at the start, it may be necessary to repeat the whole process, starting from the new intervals between Events 1, 11 and 17.

Appendix F: The expansion indicated by the Panchasiddhantika

A set of retrograde periods for the planets can be derived from the Panchasiddhantika [11]. Known formulae [22] can transform these to Sun-planet distances, though only in the form of a multiple of the Sun-Earth distance for the time being (the modern average distance being 1.000 AU). There is a restriction on precision in that the 'known formulae' are over-simplified, treating orbits as circular rather than elliptic.

On the basis of a Sun-Earth distance of 1.000 AU, the Panchasiddhantika Sun-planet distances compare with modern ones as follows:

There is an anomaly in the above table in that the orbit of Mercury contracts to reach its modern value whilst all the others are either substantially unchanged or expand. If Sun-Earth distance at the ancient era was actually 0.884 AU or less, the anomaly disappears and all Sun-planet distances expand to their modern values. It must be considered probable that the retrograde period cited for Mercury (20 days) in reference 11 is wrong, however; it need only be increased to about 23 days to eliminate the Mercury anomaly, and this correction could be made without violation of the text of the Panchasiddhantika, which is not very specific at this point.

One wonders why there is little or no expansion of the orbit of Mars when those of Jupiter and Saturn expand so very noticeably. It could be that the Panchasiddhantika data was compiled when the expansion of the Solar System had proceeded only as far as Mars, but other explanations are conceivable and probably even preferable when Sun-Earth distances smaller than 1.000 AU are a real possibility at some time in the distant past.