An early system A-type scheme for Saturn from Babylon

In this paper we publish three fragments of a cuneiform tablet that, when complete, contained the dates and zodiacal positions of Saturn’s synodic phenomena for roughly 60 years. The text is unique in containing comparisons of computed data with observations. Through an analysis of the preserved data we propose that the dates and positions were computed by an otherwise unknown two-zone System A-type scheme and show that the computed data in the tablet can be dated to the fourth century BC. This early date and the comparisons with observations suggest that the text was produced during the period of active development of the planetary systems.


Introduction
Two types of schemes are used for computing the dates and positions in the zodiac of the synodic phenomena of the planets in Babylonian mathematical astronomy: System A, in which the synodic arc and synodic time between consecutive phenomena of the same kind depends upon the planet's position in the zodiac and varies according to a step function, and System B, in which the synodic arc and synodic time are given by linear zigzag functions. Both kinds of schemes are known for Saturn (Ossendrijver 2012: 106-109). System A is a two-zone step function with synodic arcs of 11;43,7,30°a nd 14;3,45°and zone boundaries at 10°Leo and 30°Aquarius. A variant, System A , Communicated by Alexander Jones. which is only attested in one template text, uses the same synodic arcs but shifts the zone boundaries to 20°Leo and 10°Aquarius. Three System B-type schemes are known: Systems B, B , and B . The zigzag functions used in these three schemes have the same difference of 0;12°, but slightly different values for the maximum and minimum.
In Steele (2010), one of us published BM 42878 and BM 45807, two fragments containing lists of the synodic phenomena of Saturn computed according to a previously unattested System A-like scheme. Steele showed that the longitudes were computed using a two-zone scheme with synodic arcs equal to 11;20°and 13;50°. Unfortunately, too little data was preserved on BM 42878 and BM 45807 to allow the boundaries between the fast and slow zones to be identified or for the texts to be dated.
As discussed already by Steele (2010), the scheme does not strictly follow the mathematical rules of System A. In particular, the ratio of the two synodic arcs (13;50: 11;30 = 83:68) is a non-terminating sexagesimal fraction. As a consequence, the ratio between the subdivisions of the synodic arc in the two zones are not precisely equal either to each other or to that of the total synodic arc. More importantly, from a computational perspective, because the ratio of the two synodic arcs is not a terminating sexagesimal fraction, the normal procedure for computing positions which cross zone boundaries cannot be used precisely. In practice, therefore, some approximations must have been made when crossing the zone boundaries, which stands in contrast to the strict mathematical precision and consistency usually found in Babylonian mathematical astronomy. Note also that the subdivision of the synodic arc during Saturn's retrograde motion is symmetrical around acronychal rising. Most schemes for the subdivision of the synodic arc of an outer planet have a shorter time interval and smaller distance between morning station and acronychal rising than between acronychal rising and evening station, in line with what one would expect (Swerdlow 1999;Hollywood and Steele 2004). In assuming a symmetrical division, this Saturn scheme conflates acronychal rising with opposition.
In 2018, Steele identified two further fragments containing the same material. BM 45726, which joins BM 45807, and BM 46004. All three fragments, BM 42878, BM 45726+45807, and BM 46004, are almost certainly part of the same tablet. The new fragments allow for a full, if still provisional, reconstruction of the computational scheme. They also allow the text to be dated to the fourth century BC, right around the time when the System A and System B methods for computing planetary phenomena were actively being developed. But most importantly, some lines in the new fragments contain comparisons between the computed phenomena of Saturn and observations of those same phenomena. Explicit evidence for comparison between computation and observation in order to test computational methods is extremely rare in Babylonian sources, and, indeed, in ancient astronomy generally. We will return to the significance of the comparison of computed and observational data at the end of this paper.
The text presents a list of consecutive phenomena of Saturn in several columns in a prose list (not tabular) form. Each entry contains at a minimum the date and the longitude of Saturn when it exhibits a synodic phenomenon. The date is presented first and is given to a whole number of days. The longitude is preceded by the sign ina ('at') and is given to a precision of 0;10°. For the first and last visibilities, the longitude is followed by an integer number and the sign BE. The earliest entries for Saturn's stations, i.e., those in Obv. I and II, simply give the computed date and position. Beginning in Obv. III, however, a report of the position of Saturn at the station relative to a Normal Star is inserted after the date and before the computed longitude. Entries for Saturn's acronychal rising only give the computed date and longitude.
The names of the signs of the zodiac are given using the short forms (e.g., GÍR instead of GÍR.TAB for Scorpio) which are typical in texts of mathematical astronomy (Steele 2018). The writing ABSIN 0 (= KI) for Virgo instead of ABSIN is worth noting because this form appears frequently in texts from the fourth or early third century BC (e.g., Atypical Text C (BM 36301), Atypical Text H (MNB 1856), ACT 70 (BM 34934), ADART VII 1 (BM 65156), BM 36822+37022, BM 36599+36941, and BM 36737+47912), but only rarely after this, suggesting a fourth century BC date for the tablet. Similarly, UR instead of A for Leo points to a fourth century BC date. The scribe writes the numeral 9 using the three-wedge cursive form.
We edit the three fragments of this tablet below. Column numbers follow our restoration of the complete tablet. Line numbers are given separately for each fragment with, where known, an estimate of the missing number of lines between fragments indicated. The surfaces of all three fragments, especially BM 45726+45807 and the obverse of BM 46004, are badly abraded and much of the text is illegible. We encourage the reader to take very seriously the uncertain nature of the readings of many damaged signs, especially those where we append a superscript question mark ( ? ).

Obv
Rev

Critical apparatus
Obv. I A8 Only the faintest traces of the 10 remain. The damage here is particularly unfortunate because this line when combined with Obv. I A2 provides the only direct evidence for the value of the synodic arc in the fast zone Obv. II B4 -5 The scribe has omitted an entry for Saturn's acronychal rising, skipping straight from western station in B4 to eastern station in B5 Obv. III B2 The wedges for 14 are preserved. There appear to be traces before the first wedge which could be either the remains of another winkelhaken, making the number 24, or they could just be damage Obv. III B3 Following the ina sign there appears to be a small vertical wedge, which is in turn followed by at least one, and perhaps several, winkelhaken. We are tempted to read ina 1,40, but this does not fit in with the positions given in the surrounding lines. We therefore assume that what appears to be a vertical wedge is simply damage and read ina 10 + [x] Obv. III B15 A few traces can be made out before the MÁŠ sign. The traces do not look particularly like the expected 21,10 Obv. III B19 The day number could be either 16 or 26 Obv. III B28 Only the final winkelhaken of the 30 remains Obv. III B29 -C1 From the content, we would expect that line B29 would join C1 , but this is not physically possible. It is possible that B30 and C1 are the same line, but there is not a physical join and the placement of the fragments looks a little tight. Alternatively, C1 may follow B30 , which would be quite possible from the physical fragments. We cannot explain, however, what would have been written in lines B30 and C1 in either case; we would not expect anything in these lines Obv. III C2 The number after ina is probably 18,20 Obv. IV B9 The second sign looks somewhat like an A sign, but there seem to be some additional traces towards the bottom of the sign Obv. IV B15 The number following ina could be either 12 or 13 Obv. IV B20 The tablet breaks off immediately following the two winkelhaken of the 20. Any number between 20 and 29 is possible Rev. III C1 We only see three wedges of the 40, but the spatial arrangement of those wedges is what we would expect for 40 not 30

Date
Several lines begin with what appear to be year numbers: Obv. III B21 and C3 , Obv. IV B9 , B16 , and B17 , and Rev. III C1 and C6. Unfortunately, the reading of the year numbers in Obv. III are all uncertain and, as we will discuss below, the most likely readings do not fit the longitude scheme. Similarly, the year numbers in Obv. III and Rev. III do not fit the dates implied by the longitude scheme. We will return to this problem below. The most secure year numbers seem to be those preserved in Obv. IV. Obv. IV B9 and B16 mention a year 2 and B17 mentions a year 3. Furthermore, B16 indicates that year 2 contained an intercalary month XII. Over the whole of the Late Babylonian period, an intercalary month XII in the second year of a king's reign is attested only for Nabopolassar (624/623 BC), Darius II (422/421 BC), Artaxerxes II (403/402 BC), and Artaxerxes III (357/356 BC). Although Obv. IV is badly damaged, most of the dates are preserved, as well as indications of which entries include reports of observations of positions relative to Normal Stars, which must therefore be the stations. A quick comparison of the dates of the phenomena with computed data for the second and third years of Nabopolassar, Darius II, Artaxerxes II and Artaxerxes III shows that only Artaxerxes III's reign is a possibility.
On the assumption that these lines concern Artaxerxes III, we can then fix the date of the longitude scheme for the whole tablet. It likely covered a 59-year period beginning in about year 16 of Artaxerxes II (389 BC) and ending in about year 1 of Alexander III (330 BC). Comparison of the reconstructed longitude data with modern computation shows excellent agreement (see Sect. 5 below), which confirms our dating.
Given the strength of the agreement between the longitude scheme and modern computation, and the fact that the implied date agrees with the year numbers in Obv. IV, we are confident in the dating of the calculated phenomena. As mentioned above, year numbers in Obv. III and Rev. III do not fit, however. In Obv. III B21 and C3 respectively we have what seem to be the year numbers 28 ? and 30 ? ; according to the longitude scheme, these should correspond to Artaxerxes II years 40 and 42. It seems that the scribe made a 12 year error somewhere in moving back from the preserved year number in Obv. IV to those in Obv. III 4 ; the obvious place where such an error could be introduced is at the reign transition between Artaxerxes II and Artaxerxes III. In Rev. III C1 and C6, we have the year numbers 19 and 20 (unlike the numbers in Obv. III, the reading of these numbers is certain), but the longitude data is for Artaxerxes III years 17 and 18. Thus, between Obv. IV and Rev. III, a 2 year error (in the opposite direction to that between Obv. III and Obv. IV) has occurred. We can offer no explanation for these errors. It seems simply that the scribe was careless in keeping track of the years. While the presence of these errors is of some concern, we remain confident in our dating of the phenomena based upon the reconstructed longitude scheme.
The inclusion of observational data in this tablet means that it must have been compiled after the data of the last observation it contains. This would situate the composition of the tablet in the late fourth century BC. This date is in line with the use of UR and ABSIN 0 to write the names of the zodiacal signs Leo and Virgo and puts the tablet right in the time period where the various systems of mathematical astronomy were being actively developed.

The system A 0 scheme underlying the Saturn ephemeris
The three fragments BM 42878, BM 45726+45807 and BM 46004 are part of a tablet that originally contained a list of longitudes of Saturn at its consecutive synodic phenomena, probably from 389 to 330 BC covering one full 59-year period of Saturn. The longitudes of Saturn appear to have been computed according to a System A model, that we will call System A 0 to distinguish it from the previously known Systems A and A (for a recent summary see Ossendrijver 2012, 106-108).
In the earlier paper based on fragments BM 42878 and BM 45807, Steele (2010) was able to derive the following properties of System A 0 : 1. The variable motion of Saturn is approximated by a step function with two zones: a slow zone where the synodic arc equals 11;20°and a fast zone where the synodic arc equals 13;50°. 2. In both zones the synodic arc is split up in a number of different intervals (pushes 5 ) when Saturn moves from one synodic phase to the next one. These intervals are listed in Table 1.
As we have discussed above a peculiar feature of the Babylonian Saturn ephemeris BM 42878+ is the fact that the longitudes are not presented in the typical way of later synodic tables, i.e., for each synodic phase separately, but instead in the order in which they are observed, from one synodic phase to the next one; this feature suggests that the data was computed from one synodic phase to the next. The intervals in Table 1 in principle allow us to do so, once we know the longitudes of the two transitions between the slow and the fast zones. As pointed out by Steele (2010) the longitudes preserved on fragments BM 42878 and BM 45807 are not sufficient to determine the zone boundaries but, as we will show below, including the two additional fragments BM 45726 and BM 46004, the tablet now contains just enough information to reconstruct the full System A 0 scheme.
Our reconstruction of the longitudes of Saturn preserved on BM 42878+ is shown in Table 2 where we list computed positions of Saturn at 205 successive synodic phases while it moves almost one and a half times through the zodiac from Cancer to Cancer to Sagittarius. Slow-to-fast and fast-to-slow boundary crossings are indicated by dashed lines in Table 2. The lengths of the zones and the longitudes of the zone boundaries in this reconstruction were determined as follows.
We first note that the interval from 6;00°Virgo (line 28 in Table 2) to 21;10°C apricorn (line 85) lies in the slow zone and that the interval from 30;00°Capricorn (line 89) to 18;20°Aquarius (line 95) lies in the fast zone because all preserved Table 2 Reconstruction of the longitudes of Saturn at its synodic phases preserved on BM 42878+ longitudes in these intervals can be reconstructed by applying the pushes listed for the slow or the fast zones in Table 1. This implies that the boundary between the slow and the fast zone must lie somewhere between 21;10°and 30;00°Capricorn.
Longitudes of Saturn when it passes zone boundaries, while moving from one synodic phase to the next one, can be calculated by applying the usual interpolation algorithm for Babylonian System A step functions (see e.g. Ossendrijver 2012, 48): where λ i is the initial longitude of Saturn at synodic phase i, λ i is the interval from synodic phase (i) to (i + 1) in the zone in which λ i is located, λ b is the longitude of the zone boundary crossed, r is the interpolation factor and λ i+1 is the longitude of Saturn at the synodic phase (i + 1) located in the next zone.
To be able to apply this algorithm we must know the value of the interpolation factor r. In the traditional Babylonian System A theory this interpolation factor is equal to the ratio of the step function amplitudes (synodic arcs), usually chosen such that it results in simple ratios of "nice" numbers like 2/3, 3/4, 4/5, 5/6 for computational convenience. 6 In the System A 0 model of Saturn the situation is different and numerically more complicated because the synodic arc is split up in different intervals and the ratios of these intervals are slightly different for each set of intervals and not equal to simple ratios of "nice" numbers. In fact, the numbers in columns (iv) and (v) of Table 1 show that the adopted ratios are virtually identical (within less than 1%) for all intervals and that only three correspond to terminating sexagesimal numbers: 1;13,20 for the fast/slow zone transition and 0;48,53,20 and 0;49,12 for the slow/fast transition. It seems plausible to assume that 0;49,12 and 1;13,20 were the values adopted for the interpolation factors r used to compute the longitudes in our text.
Using these values of the interpolation factors in Eq.
(1) we then find from the preserved longitudes of Saturn in lines 85-89 of Table 2 three values for the longitude of the zone boundary λ b of 23;30°Capricorn (lines 85-86), 23;31°Capricorn (lines 87-88) and 23;30°Capricorn (lines 88 to 89). Since the small difference of 0;01°b etween these values can be attributed to the fact that all preserved longitudes on BM 42878+ are rounded off to an accuracy of 0;10°we find that the transition of the slow to the fast zone occurs at 23;30°Capricorn. We next turn to the transition from the fast to the slow zone. From the data in Table  2 we find that this transition must be located somewhere between 5;10°Leo (line 11 in Table 2) and 6;00°Virgo (line 28). The exact value can be found by numerically experimenting with different values of the fast to slow boundary longitude. This trialand-error approach leads in a few steps to a boundary value of 20;30°Leo, resulting in fast and slow zone lengths of 207°and 153°. We further find that going from the fast to the slow zone the boundary value is crossed once between lines 19 and 20 and three times between lines 160 and 161, lines 162 and 163 and lines 163 to 164 in Table 2. While the boundary values of the slow and the fast zone are not "nice" integer values as they are in the usual Babylonian System A models of the planets, we note that the amount of computational effort required to generate the longitudes of Saturn on BM 6 These ratios are attested in the System A schemes of the outer planets (see e.g. de Jong 2019a, Table 2). Table 3 Babylonian system A parameters for Saturn 42878+ is minimal. Assuming that the tablet originally covered one 59-year period of Saturn the computation of the full run of data involves only about ten boundary crossings.
Using the boundary values of 20;30°Leo and 23;30°Capricorn between the slow and fast zones of the A 0 step function and the interpolation factors 0;49,12 and 1;13,20 discussed above, we computed the tabulated values of the longitudes in the text of BM 42878+ as shown in Table 2. This computation requires minimal arithmetic effort because it involves simple adding and subtraction of intervals and interpolation at only seven boundary crossings. Rounding errors which occur at the boundary crossings go both ways so that they do not accumulate but will statistically average out.
A comparison of the reconstructed longitudes of Saturn with those preserved on BM 42878+ shows overall excellent agreement. The differences in lines 87 and 91 of Table  2 can be attributed to uncertain readings and the discrepancies in lines 200 and 203 are probably due to rounding errors generated in the computation of the longitudes at the transition between the fast and the slow zones in lines 160-164 since they differ by 0;10°from the reconstructed values. All together it appears that the computation of the longitudes of Saturn's phenomena was carried out by a quite competent Babylonian scholar.
In Table 3 we list the parameters characterizing the System A 0 step function together with those of the previously known Systems A and A of Saturn. The parameters of the canonical system A of Saturn are derived from a period relation which has a clear relation to astronomical reality. 7 In 265 years Saturn experiences 256 synodic events while it completes 9 passages of Normal Stars in the sky (orbits around the Sun). This set of parameters results in a mean synodic arc of λ = 9 × 360°/256 = 12;39,22,30°(exactly). If system A 0 is similarly formulated in terms of a period relation the parameters turn out to be unrealistically large (see Table 3): in 4891 years Saturn experiences 4725 synodic events while it completes 166 passages in the sky resulting in a mean synodic arc of λ = 166 × 360°/4725 = 12;38,51,…°. Satisfying a period relation is equivalent to ensuring that the parameters reproduce the correct average synodic arc so that the model does not derail over long time intervals. However, this can also -and even more simply -be done by making sure that the combination of amplitudes and zone lengths of the step function reproduces the mean synodic arc derived from an observed period relation. The mean synodic arc of system A 0 is very close to the value of the canonical system A (see Table 3) so that it is quite possible that the scholar who constructed system A 0 started out with an accurate value of the mean synodic arc, chose two synodic arcs (one for each zone) based on a direct comparison with observations of Saturn near Normal Stars at its stations (see de Jong 2019a) and experimented with different zone lengths to find a combination that reproduced the mean synodic arc. 8 Notice that in system A 0 the slow arc is reduced in size by almost 50°compared to system A (and the fast arc similarly enlarged), and that the symmetry axes of both systems differ by 13°(37°-217°in System A 0 compared to 50°-230°in System A).
System A 0 differs from most other system A models of the planets in the choice of the amplitudes of its step function with the rather awkward ratio of 68/83. While in practice computation with this ratio may be avoided by using the somewhat more user friendly values of 41/50 and 9/11, it is clear that the author of System A 0 gave higher priority to selecting "nice" sexagesimal numbers for the amplitudes of the step function (the synodic arcs) and the pushes than to "nice" numbers for the amplitude ratio which would have simplified the interpolation at the crossing of zone boundaries in the computation of the ephemeris. This is actually a quite sensible policy because, as shown above, the number of zone boundary crossings in computing an ephemeris of Saturn is quite limited.
While we can understand the choices of the numerical values of the amplitudes and of the zone lengths of the System A 0 step function, the reason for choosing non-integer values of 20;30°Leo and 23;30°Capricorn for the zone boundaries is unexpected because in all but one of the presently known System A-type models of the planets the longitudes of the zone boundaries are integer values (Neugebauer 1975, 423). Moreover, since the accuracy of the System A ephemeris of Saturn has been shown to be quite insensitive to shifting the position of the zones by ± 10°in the zodiac (de Jong 2019a, 30-32), the choice of non-integer values for the zone boundaries in the System A 0 step function is puzzling. In an attempt to come up with an explanation we first note that the choice of non-integer values for the zone boundaries in System A 0 does not affect the computational effort of calculating the longitude date on BM 42878+ because the total number of boundary crossings is limited to about ten. We further note that, once a run of longitudes from a System A scheme has been computed, it may be shifted in longitude by any number of (fractional) degrees as long as the computed longitudes and the zone boundaries are shifted by the same amount. With this in mind we suggest that the choice of non-integer boundary values may have been driven by shifting a previously computed run by a small amount to anchor it to a specific observation of Saturn at one of its stations. This procedure is known from several Babylonian planetary tables that appear to be anchored to a specific observation of the planet at one of its stations when it happened to be particularly close to one of the Normal Stars so that its position in the Babylonian zodiac could be accurately determined (de Jong 2019a,b;2021).
System A , known from tablet BM 78080 (Aaboe and Sachs 1966, Text C; for the parameters see Table 3) is closely related to System A because it employs exactly the same values of the amplitudes (synodic arcs) in the slow and the fast zones but uses different zone lengths and boundaries. System A results in a very inaccurate ephemeris of Saturn because the value of the mean synodic arc is about 11 arcminutes too large. Since Saturn experiences roughly one synodic event per year this implies a runaway of the computed longitudes of some 5°every thirty years. Aaboe and Sachs suggest that the scribe may have made a mistake by accidentally taking the beginning of the slow zone at 20°Leo instead of 20°Cancer.
However, it may not be accidental but a deliberate choice to put the value of the beginning of the slow zone in model A at 20°Leo, almost identical to that in model A 0 . In that case we may consider model A as a failed attempt to model the synodic phases of Saturn using parameters which are a mixture of those in the early model A 0 and the final model A. This could be understood if the Babylonian scholar(s) went through a phase (sometime during the fourth century BC) where they were experimenting with different approaches to model Saturn's motion, going from models based on selecting values (with "nice" numbers) for the amplitudes (model A 0 ), to the final system A models in which the emphasis was on choosing numerically convenient amplitude ratios. 9 Such a scenario is consistent with the usage of the logogram ABSIN 0 for Virgo in both BM 42878+ and in BM 78080, a writing habit known to have been en vogue during the fourth century BC.

Reconstruction of tablet BM 42878+
In Sect. 3 we argued that the regnal years preserved in Obv. IV indicated that the data were computed for dates during the fourth century BC and that it may have covered one full 59-year period of Saturn. Having determined the System A 0 parameters underlying the computation of the ephemeris in Sect. 4 we now attempt to reconstruct the layout of the tablet and to place the different fragments in the reconstructed tablet. The results are shown in Tables 4 and 5 where we list dates in the Julian calendar and zodiacal longitudes of successive synodic phases of Saturn distributed over 4 columns on the obverse (from left to right) and over 4 columns on the reverse (from right to left) of tablet BM 42878+. The successive synodic phases of Saturn are indicated by the acronyms: S2 for second station, LA for last appearance in the west, FA for first appearance in the east, S1 for first station and AR for acronychal rising. The longitudes of Saturn at each of its synodic phases are longitudes in the fixed Babylonian zodiac so that they can be directly compared to the longitudes preserved on the three fragments A, B and C of the tablet shown in the shaded areas.
The dates and longitudes in Tables 4 and 5 of Saturn at its first and last appearance and at its stations are taken from the database of synthetic observations of Saturn during the fourth century BC generated by de Jong (2019a). Dates and longitudes for Saturn at its acronychal rising during the fourth century BC were newly computed Table 4 Reconstruction of BM 42878+ and the placement of fragments based on a comparison of the longitudes in the text with synthetic observations of Saturn-Obverse Table 5 Reconstruction of BM 42878+ and the placement of fragments based on a comparison of the longitudes in the text with synthetic observations of Saturn-Reverse based on the assumption that here acronychal rising is equivalent to exact opposition of Saturn with the Sun, consistent with the algorithm employed in the computation of the ephemeris: S1 → AR = AR → S2 (see discussion above in Sect. 1 and Table 1). 10 Our reconstruction of the tablet and the placement of the different fragments in time is based on a comparison of the preserved longitudes in the text with the computed longitudes of Saturn in the synthetic observational database. The exact first entry in column I on the Obverse and the exact last entry in column IV of the Reverse cannot be determined but assuming that the tablet originally covered one full 59-year period of Saturn we expect in total 5 (number of synodic phenomena in one synodic period of Saturn) × 57 (number of similar synodic events in 59 years) = 285 entries distributed over eight columns, or on average about 35.5 entries in eight columns. According to our reconstruction the obverse of the tablet contained 45 lines with 36 to 45 synodic events per column, and the reverse contained 36 lines with 30 to 34 synodic events per column. The number of synodic events tabulated per column varies depending on the number of Normal Star observations included for comparison in each column because these require two to three lines per observation. The reconstruction is further constrained by the requirement that fragment C contains the last lines of columns III and IV on the obverse and the first lines of columns II, III and IV on the reverse side of the tablet. As mentioned above the precise beginning of the tabulated events in Obv. I and the precise end in Rev. IV cannot be determined but in our reconstruction these are based on the assumption that a full 59-year period of Saturn was tabulated so that the tablet covered two runs of Saturn through the zodiac, starting around 0°Aries.
Notice that based on the preserved longitudes our reconstruction shows that fragments B and C should join in column III on the obverse and that the horizontal rulings in column IV of the obverse indeed indicate the separation between successive Babylonian years. As discussed above in Sect. 2, there is no physical join between fragments B and C, indicating that there must be at least one or perhaps two extra and unexpected lines of text here. We can offer no explanation of why this is the case or what that text might have been.
In Tables 4 and 5 we also list values of δλ, the difference between the longitudes of Saturn preserved on the tablet and the longitudes in the synthetic observational database. Inspection of these values shows that the System A 0 ephemeris BM 42878+ is quite accurate over large sections of the zodiac. The largest errors seem to occur in Pisces and Cancer.
The synthetic observational data in Tables 4 and 5 allow us to investigate the accuracy of the System A 0 model of Saturn in more detail. Computing synodic arcs for all observations listed in Tables 4 and 5 and plotting them as a function of initial longitude, we show in Fig. 1 a graphical representation of the results for each synodic phase separately. These graphs can be directly compared with those for System A of Saturn, discussed by de Jong (2019a; see his Fig. 3). 11 There it is also explained why the graphs for all four synodic phenomena are quite similar (primary synodic Fig. 1 Variation of the synodic arc at first appearance, first station, second station and last appearance of Saturn as a function of Babylonian zodiacal longitude. Dots represent values derived from synthetic observations of Saturn in the fourth century BC. Also shown is the Babylonian system A 0 model step function for the synodic arc (thin line marked by squares) and the interpolated model (thick line) phenomena, see also Ossendrijver 2012, 59) and how the System A step functions were constructed based on observations of Saturn at one of its stations with respect to nearby Normal Stars.
A comparison of System A 0 with System A shows that the System A step function of Saturn provides a better fit to the observational data from Aries to Cancer (0°-120°) and from Aquarius to Pisces (300°to 360°) than the early variant System A 0 . This is also reflected in the standard deviations of the longitude differences δλ averaged over one century which for System A 0 amount to ± 1.6°(FA), ± 2.2°(S1), ± 1.8°(S2) and ± 1.8°(LA), compared to the smaller values (better accuracy) for system A in column (i) of Table 4 of de Jong (2019a) where we find ± 1.0°(FA), ± 1.3°(S1), ± 1.1°(S2) and ± 1.0°(LA).

The dates and the BE values
The full dates (day, month, and year) of 25 synodic phenomena are preserved or can be restored from surrounding entries. In Table 6 we compare these dates listed in column (iv) with computed dates of these phenomena in the synthetic observation database Footnote 11 continued rising is put identical to exact opposition with the Sun so that the System A 0 graph is virtually identical to the ones for first and second station. Table 6 Comparison of preserved dates of synodic phases of Saturn in BM 42878+ with synthetic observation dates converted into the Babylonian calendar in column (iii). The number of days difference between the dates given in the text and the synthetic dates in column (v) have been computed ignoring the year numbers and assuming that months have 30 days. Similar to the synthetic longitudes, for achronycal rising we list the date of opposition. We have assumed that the scribe made a one-month error in the date of acronychal rising given at Obv. III B6 , writing month IV instead of month III. Support for this date being an error is provided by the implied interval between this and following synodic phenomena. If we assume that the date is correct, the time interval between this acronychal rising and the following evening station would be 37 days, which is far too short. If we correct the date of the acronychal rising from Month IV to Month III, the time interval becomes 67 days, which is more in line with other values for this interval found in the text. The error can be understood by noting that the time interval that should be added to the date of the previous acronychal rising (or the preceding morning station) to obtain the date of the acronychal rising recorded at Obv. III B6 contains an intercalary second Addaru (Month XII 2 ) which may have been overlooked.
Inspection of Table 6 reveals several things. First, there is general agreement between the preserved months and days of the phenomena with observation. However, as mentioned above, there are serious problems with the year numbers in Obv. III and Rev. III. In Obv. III, the year numbers seem to be 12 years too early; in Rev. III, they are two years too late. We cannot provide a good explanation for this miscounting of the years. Secondly, we see that there are two abrupt changes in the difference between Table 7 Time intervals between the synodic phases of Saturn the dates given in the text and the computed dates: From the earliest preserved dates up to those in Obv. III B7 , the dates given in the text are systematically later than those given by modern computation. However, for the remainder of Obv. III and Obv. IV, the dates switch to being systematically early. The entries on the reverse are systematically late again. One possible explanation for this pattern in the recorded dates is that the scribe has incorrectly intercalated somewhere between these preserved runs of dates. Given the general confusion in the year numbers, omitting/adding an extra intercalary month seems quite plausible.
There are two important questions that need to be addressed: (1) are the dates in the text computed or observed, and (2) what is the meaning of the BE values listed for a number of first and last appearances of Saturn? In an attempt to provide an answer to the first of these questions we show in the upper part of Table 7 values of the time intervals between successive synodic phases of Saturn derived from the preserved dates in the text listed in Table 6. 12 Due to the scarcity of data we have not discriminated between time intervals for Saturn in the slow and the fast zone of the zodiac.
For comparison we also show in the middle section of Table 7 the time intervals between successive synodic phases of Saturn in its fast and slow zone computed from the synthetic observations of Saturn between 389 and 330 BC listed in Tables 4, 5. These time intervals do not include the variation in the observed dates of the first and last appearance of Saturn due to variable atmospheric conditions and the observational uncertainty in the dates of the stations due to the difficulty of exactly determining the date of standstill of the planet. These effects cause an additional spread in the observed dates of first and last appearance of Saturn of about ± 3 days and in the station dates of at least ± 1 week, which should be added to the range of values of the synthetic synodic intervals displayed in the middle section of Table 7.
The data in Table 7 allow two conclusions: (1) the spread in the synodic time interval values in the text are significantly smaller than expected for observed values so that they most probably are computed rather than observed, and (2) the time intervals of FA → S1 and of S2 → LA in the text are on average systematically about 10 days larger than the actual values. This implies that the dates when Saturn reaches its stations are systematically about 10 days late for the morning station (S1) and about 10 days early for the evening station (S2). Apparently, the Babylonian observers determined the date of standstill for the morning station of Saturn as the first day on which it was observed to start moving backwards (1-2 weeks after standstill) and the date of the evening station as the first day on which it was observed to halt its backward motion (1-2 weeks before standstill).
This observational practice is confirmed by a comparison of the dates of synthetic observations of Saturn between 389 and 330 BC, the period covered by the tablet, with records of preserved observations of Saturn in the Diaries. The results are summarized in Table 8. Observed dates of first and last appearances of Saturn differ by up to three days from the expected (synthetic) dates while the observed dates of the morning station S1 of Saturn are about 2 weeks late and those of the evening station (S2) are about 2 weeks early. A similar conclusion about the station dates was reached by Steele and Meszaros (2021) who studied all observations of Saturn at its stationary points preserved in the Astronomical Diaries. 13 There is one observation in Table 8 which unambiguously proves that the dates in BM 42878+ are computed rather than observed: the observation of the evening station of Saturn in 367 BC. The exact standstill of Saturn occurred on August 30 of that year, or day 13 of month V in year 38 of Artaxerxes II according to the Babylonian calendar. In the Diary of that year (No. − 366) the Babylonian observer(s) recorded that Saturn reached its evening station on day 29 of month IV, 14 days earlier, consistent with the Babylonian observing practice discussed above. By a lucky coincidence it so happens that among the roughly 20 preserved dates on BM 42878+ the text gives for this evening station day 17 of month V, 4 days later instead of 14 days earlier than the actual date of standstill. The only reasonable explanation for this discrepancy is that this date and by analogy all dates in the text are computed. As we shall see not only this date but all dates in this part of the text are shifted to later dates by about 2 weeks.
Given that the dates in BM 42878+ are indeed computed the question arises whether the underlying algorithm can be reconstructed from the preserved dates and BE values.
In an attempt to answer this question, we have computed dates of the synodic phases (FA, S1, AR, S2 and LA) of Saturn by applying the standard system A algorithm which prescribes that successive dates of each synodic phase can be found by adding a synodic time interval t = λ + c, where λ is the synodic arc and the parameter c is a constant. For system A 0 of Saturn we have c = 11;27,20 (see Table 3) so that t = 360 + 11;20 + 11;27,20 = 382;47,10 tithis in the slow zone and t = 360 + 13;50 + 11;27,20 = 385;17,10 tithis in the fast zone. As a working hypothesis, we will assume that the BE values correspond to the difference between the date computed by the System A 0 scheme and the observed date of the phenomenon. Since all but one of the preserved BE values and four out of twenty-five preserved dates fall between May 377 BC and January 365 BC when Saturn moved from 6;00°Virgo to 21;10°C apricorn in the slow zone we restrict the computation to this period. Initial dates in 377/376 BC for S2, LA, FA, S2 an AR were chosen in such a way that the preserved dates and the BE values are on average reproduced. The results of the computation are shown in Table 9.
The data in Table 9 cover the period from May 377 BC up to and including January 365 BC with a gap between 373 and 368 BC when no data are preserved on the tablet. Synthetic observation dates of Saturn at its successive synodic phases in the Julian calendar in column (ii) are converted to the Babylonian lunar calendar in column (iii). In column (iv) we list the dates of the synodic phases of Saturn computed according to the Babylonian date algorithm and column (v) shows computed BE numbers, defined as the difference in days between dates computed according to the algorithm and the synthetic observational dates. Column (vi) contains the line numbers of the text and the dates and BE numbers preserved on the tablet. In column (vii) we show the difference in days between the dates in the text and those computed according to the Table 9 Synthetic and computed dates of the synodic phases of Saturn compared to preserved data on BM 42878+ algorithm and in column (viii) the differences between the BE numbers in the text and the computed ones in column (v).
Our reconstruction of the dates and the BE values is still somewhat preliminary because it is based on only four preserved dates and ten BE values. In spite of the preliminary nature of our reconstruction, the data in Table 9 suggest that the Babylonian scholars may have used a more refined algorithm than the straightforward system A algorithm used here because the computed dates and the BE values differ by up to −3/+2 days from the dates preserved in the text (see column (vii)). We know from several ACT texts that the Babylonian scholars occasionally used more refined algorithms to compute dates of the synodic phenomena. A well-known example is ACT 300a where a more complicated algorithm is applied to compute the dates of Mercury at its last appearance (Ossendrijver 2012, 72).
Based on the choice of the initial dates and implicit in the computation of the dates in Table 9 are the time intervals to go from one synodic phase to the next one: LA + 35;47,20 tithis → FA + 114 tithis → S1 + 58 tithis → AR + 62 tithis → S2 + 113 tithis → LA (listed in the bottom section of Table 7). These intervals add up to 382;47,20 tithis, as they should in the slow zone. Notice that these intervals are in reasonable agreement with the ones derived from the preserved dates in the text also displayed Table 7. Also notice the asymmetry in the intervals S1 → AR and AR → S1 which implies that the dates of acronychal rising fall two days before opposition, consistent with Babylonian observational practice (see Hollywood and Steele 2004). 14 The BE values in the text are a few days larger than predicted for FA and a few days smaller for LA (see column (viii) of Table 9). This is to be expected because the predicted values of BE are based on synthetic observations computed for an average atmospheric extinction of 0.27 magnitudes per airmass (de Jong 2019a). Under realistic atmospheric conditions with variations in the atmospheric extinction from day to day first appearance will often be observed a few days earlier than predicted and last appearance a few days later (see de Jong 2012).
Taking all dates in Table 9 together, both of the stations and of the first and last appearances of Saturn, we find that the whole scheme is on average about 14 days late (see column (v)). The reason for this is unclear. One possibility is that the scribe of BM 42878+ anchored his computation of the ephemeris to an observation of Saturn at one of its stations, a procedure known from several later System A ephemerides (see de Jong 2019a,b;2021). Indeed, as shown by the observational data in Table 8, if the author of BM 42878+ used an observation of Saturn at its morning station as initial condition the whole scheme would be about 2 weeks late. On the other hand, it may be just due to some scribal error because continuing the computation of the dates after the period covered in Table 9, using a similar slightly adapted algorithm for the fast zone, we encountered differences between computed and preserved dates in the text running up to 22 days (see above and column (v) of Table 6).
It is worth noticing that the system A modelling of Saturn in BM 42878+, apart from the systematic offset in the dates and in spite of the approximate nature of the date algorithm, is fairly successful. In the previous section we have seen that the computed longitudes are on average accurate to within 1°-2°, while the present analysis shows that the computed dates in the text are on average off by only 2-3 days (column (vii)) and that the BE values are on average correct to within ± 2 days (column (vii)). However, in later sections of the text the scribe appears to have made gross errors in the computation of the dates and in the comparison with observational data (see above and Sect. 7 below).
We conclude that: (1) in tablet BM 42878+ the dates of Saturn at its five synodic phases were computed, that (2) the computation was probably anchored to an initial observation of Saturn at is morning station (S1), and that (3) at the first and last appearances of Saturn the computed dates were compared to observed dates.

Normal Star observations
The text BM 42878+ is unique in illustrating that during the early phase in the development of Babylonian planetary theory the results of the computations were compared to observational data to check on the quality of the System A 0 model of Saturn and on the accuracy of the predicted longitudes and dates. In the previous section we have shown that computed dates of Saturn at its first and last appearances were compared to observed dates and that the differences were listed as BE numbers in the text. In this section we discuss the Normal Star observations which are occasionally included in the text in lines which contain computed values of the date and of the longitude of Saturn at its evening or morning station.
We have identified 14 preserved entries of computed dates and longitudes of Saturn at either one of its two stations where Normal Star observations are, or possibly were, included in the text for comparison. The relevant data are collected in Table 10 Sachs and Hunger (1988, 16-19) and Jones (2004).  16 On 13 March 340 BC Saturn was 11.0°behind (and not in front of, sic!) θ Oph, and it was 1.0°in front of the star μ Sag and 4.0°in front of the star λ Sag. These latter two stars have been suggested as candidates for Normal Stars used in an early text with observations of Saturn dating from 647 to 634 BC by Hunger (1999) andde Jong (2002). Either one of these two stars might be referred to in the second half of the observational report quoted in these lines although the text has again "behind" rather than the actual "in front of". 13. 13. Rev. III, lines C7-C8. Second (evening) station of Saturn on 21 July 341 BC.
Three lines of text with reference to an observation of Saturn with respect to The Tip of Pabilsag's Arrow (θ Oph). This observational record may again have been erroneously inserted here since it better fits an observation of Saturn one year later when Saturn reached its first station on 2 August 340 BC. On this date Saturn was 4.4°behind θ Oph rather than 6.7°in front of θ Oph. However, both dates are possible given the lack of detail in the observational record. 14. 14. Rev. IV, lines C3-C5. First (morning) station of Saturn on 30 April 336 BC.
Two lines of text with reference to an observation of Saturn with respect to a star in the Goat-fish. On that date Saturn was 3.8°in front of The Front Star of the Goat-fish (γ Cap).
On the basis of our analysis of the Normal Star observations inserted in the text of BM 42878+ we may conclude that these observations will have assisted the author of the text in verifying that the positions of Saturn at its stations computed according to his System A 0 model were overall in agreement with the positions of Saturn derived from Normal Star observations. This was to be expected because we have seen in Sect. 5 above that the accuracy of the longitudes of Saturn computed according to the System A 0 model is of order of a few degrees. This accuracy is also reflected in the δλ-values displayed for the stations of Saturn in column (ix) of Table 10, as far as they are preserved.
In this section we have seen that the author of BM 42878+ apparently compared the computed longitudes of Saturn at its stations with observations of Saturn at its stations with respect to nearby Normal Stars. This early Saturn text is unique in showing a Babylonian astronomer at work in the construction and verification of the System A 0 model of Saturn.

Conclusion
BM 42878+ contains the positions and dates of the synodic phenomena for Saturn computed according to a previously unknown two-zone System A-type scheme which we name System A 0 . The computed values correspond to dates from roughly 390 to 330 BC and the data were likely computed around the end of this period, i.e., in the late fourth century BC. This date places BM 42878+ around the time when the various System A and System B planetary schemes seem to have been actively in development. The scheme on BM 42878+ differs from other System A-type schemes in that it apparently prioritizes ease of calculation of one synodic phenomenon to the next by means of nice sexagesimal values for the subdivisions of the synodic arc and the synodic arcs themselves rather than obeying the normal System A rule of a nice value for the ratio of (the subdivisions of) the synodic arc in the two zones.
A similar approach is encountered in Text M, an early Mercury ephemeris first discussed by Aaboe et al. (1991). This text, which probably dates from around 400 BC, was recently rediscussed by de Jong (2021) who suggested that the choice of "nice" sexagesimal values for the synodic arcs (the amplitudes of a System A step function) in different zones of the zodiac is a typical feature in the early development of Babylonian planetary theory. In the canonical System A planetary theory the values of the synodic arcs are chosen in such a way that their ratios are "nice" sexagesimal terminating fractions. This choice significantly simplifies the numerical computation of planetary ephemerides and apparently became standard procedure in the computation of planetary ephemerides from about 300 BC onwards.
BM 42878+ is of particular interest not only because it attests to this newly identified System A 0 but also -indeed more importantly -because it appears to show evidence of the scribe testing the accuracy of the computed data against observations. This testing was performed in two ways: (i) determining the difference between the computed dates of first and last appearance and those found by observation, with this difference noted in the text as a value followed by the term BE, and (ii) comparing the computed longitudes of Saturn at its stations with observations of the position of Saturn relative to a Normal Star at that station; these positions could be compared by converting between a longitude and a Normal Star position using the known position of the Normal Stars in the zodiac. 17 The scribe demonstrated good judgement in choosing these two types of comparison. The dates of first and last appearance are by definition precise determinations (either the planet is seen or it is not), even if these dates are (from a modern perspective) inherently uncertain because of variable atmospheric conditions. 18 The longitude of a planet at its first or last appearance, on the other hand, is difficult to determine because few if any stars may be visible near the planet due to the sky brightness caused by the sun being only a little below the horizon. 19 By contrast, the dates of planetary stations, especially for Saturn which moves so slowly, are very difficult to determine whereas its position at a station can be measured precisely (and repeatedly on several nights) because the planet will be well above the horizon and moves imperceptibly for many days (Steele and Meszaros 2021). The scribe was clearly fully aware of these issues and chose the most reliable observational data at his disposal to test the computed data. Unfortunately for the scribe of this tablet, however, the value of these comparisons was to some extent vitiated by the errors that he made in computing the dates of the phenomena, especially dates in the year count, which seem to have led to at least one case of the scribe comparing a computed station with the observation of a station in a different year.
We only know of one other possible example of the testing of astronomical computation against observation from Babylonia: the so-called Text S preserved on two tablets, BM 36910+ and BM 34597 (Aaboe and Sachs 1969;Britton 1989). This text contains the values of various lunar functions for the dates of solar eclipse possibilities along with what seem to be the details of those eclipses as predicted by  The computed data in Text S refer to the early fifth century BC but the tablets were probably written during the fourth century. If Text S does indeed include comparisons between solar eclipses computed by mathematical astronomy and solar eclipses computed using goal-year methods, then we would appear to have two texts demonstrating an interest in testing systems of mathematical astronomy from the fourth century BC, a period from which we have considerable other evidence for the development of these systems into their final forms.
BM 42878+ is therefore of considerable interest in providing a rare insight into the process by which Babylonian astronomers tested systems of mathematical astronomy during the process of their development.