The Latitude and Epoch for the Origin of the Astronomical Lore of Eudoxus

Aratus Phaenomena , translated by Kidd D. (Cambridge, 1997). Another widely available translation is by Mair; G. R. it appears in the Loeb Classical Library series and is titled Aratus (Cambridge, Mass., 1921, with many subsequent reprints). A case can be made that the earliest surviving presentation of the Greek constellations was by Hesiod (The astronomy, transl. by Evelyn-White H. G. , in the volume of the Loeb Classical Library series entitled Hesiod (Cambridge, Mass., 1914), 66–71). However, Hesiod merely mentions the mythology associated with the Pleiades, the Hyades, the Great Bear, Bootes, and Orion. The absence of any mention of non-prominent star groups has inevitably lead to speculation that the Greeks had few other constellations at this time, although such a conclusion is unwarranted. Unfortunately, for the purposes of this paper, neither the works of Homer nor those of Hesiod contain astronomical lore that can be used to derive dates or latitudes with any useful accuracy.

Aratus enumerates the constellations, and these appear to include two asterisms (the Waters and the Pleiades) that are not regarded as constellations in modern times. In addition, Aratus's description of the modern Corona Australis (as a “ringed circle”) does not clearly elevate this to the status of a constellation. Of all the constellations specified in the Almagest, the only missing group is Equuleus.

MacFarlane R. T. Mills P. S. , Hipparchus' commentaries on the Phaenomena of Aratus and Eudoxus , manuscript, 2003. This translation is the first in English and the first in modern times.

The area of Mesopotamia has a complex succession of empires and dynasties. In view of the wide range of dates discussed in this paper and the significant uncertainties in these dates, it is difficult to ascribe one political name to the originating civilization. However, there is substantial cultural continuity, so it is sensible to give one name to the culture, which I will label as ‘Babylonian’ throughout this paper.

Maunder E. W. , The astronomy of the Bible (London, 1909), 149–61; Maunder E. W. , “The origin of the constellations”, Observatory , xxxvi (1913), 329–34; Crommelin A. C. D. , “The ancient constellation figures”, in Splendour of the heavens , ed. by Phillips T. E. R. Steavenson W. H. (London, 1923), 640–69; Ovenden M. W. , “The origin of the constellations”, Philosophical journal , iii (1966), 1–18; Ovenden M. W. , “The origin of the constellations”, Journal of the British Astronomical Association , lxxi (1961), 91–96; and Roy A. E. , “The origin of the constellations”, Vistas in astronomy , xxvii (1984), 171–97. These papers share a common purpose of trying to date the origin of the constellations by analysing the southern void (i.e., the region of invisible south circumpolar stars) whose centre and radius should give information about the epoch and latitude for the origin. These papers, plus many older and many popular references, all conclude that the constellations originated in the third millennium b.c. It is convenient to refer to these authors collectively by the name of ‘voidists’.

Ovenden , op. cit. (ref. 5, 1966), 10–12.

Roy , op. cit. (ref. 5), 175–80.

Henriksson G. Blomberg M. , “New arguments for the Minoan origin of the stellar positions in Aratos' Phainomena”, Astronomy and cultural diversity , ed. by Esteban C. Belmonte J. A. (La Laguna, 1999), 303–10.

Zhitomirsky S. , “Aratus' ‘Phaenomena’: Dating and analysing its primary source”, Astronomical and astrophysical transactions , xvii (1999), 483–500.

I have consulted the third translation of Aratus by Lombardo S. F. , “Aratus' Phaenomena: An introduction and translation”, Ph.D. thesis at the University of Texas at Austin, 1976. However, the relevant passages are not clear as to their meaning, supporting neither Kidd nor Mair.

Rogers J. H. , “Origins of the ancient constellations”, Journal of the British Astronomical Society , cviii (1998), 9–27 and 79–89.

Schaefer B. E. , “Latitude and epoch for the formation of the southern Greek constellations”, Journal for the history of astronomy , xxxiii (2002), 313–50.

Frank R. M. Bengoa Arregi J. , “Hunting the European sky bears: On the origins of the non-zodiacal constellations”, Astronomy, cosmology and landscape , ed. by Ruggles C. L. N. Prendergast F. Ray T. (Bognor Regis, 2002), 15–43. An alternative construction of this group is to isolate the constellations that might be very ancient (i.e., formed before 3000 b.c.), as perhaps the associated lore might be equally ancient. In particular, the bear constellations likely date back to the time of the last Ice Age (c. 10,000 b.c. or before) as demonstrated by the widespread presence of similar myths for the same stars across Eurasia and North America. The basic collection of myths from many cultures is given by Gibbon W. M. , “Asiatic parallels in North American star lore: Ursa Major” , Journal of American folklore , lxxvii (1964), 236–50, and by Hagar S. , “The celestial bear”, Journal of American folklore , xiii (1900), 92–103. The idea that the Bear is a very ancient constellation is briefly mentioned and supported in the astronomical literature by Gingerich O. , The great Copernicus chase (Cambridge, 1992), 10–11; Krupp E. C. , Beyond the blue horizon (Oxford 1991), 239; Gurshtein A. A. , “On the origin of the zodiacal constellations”, Vistas in astronomy , xxxvi (1993), 171–90; and Schaefer , op. cit. (ref. 12), 334–5.

Meeus J. , Astronomical algorithms (Richmond, Va, 1991), 126–8. The mean obliquity of the ecliptic was 23.756° in 500 b.c. and 23.821° in 1000 b.c.

Let the two points have right ascensions and declinations (for any epoch) of α₁, δ₁ and α₂, δ₂ respectively. The angular distance between these two points, γ, can be found from the formula cos(γ) = sin(δ₁).sin(δ₂) + cos(δ₁).cos(δ₂).cos(α₁ — α₂).

Bevington P. R. Robinson D. K. , Data reduction and error analysis for the physical sciences , 3rd edn (Boston, 2003), see primarily chapters 4, 8, and 11.

The chi-square is defined as Σ{[(D_i — M_i)/σ]²}, where D_i is the ‘i-th’ datum value, M_i is the model prediction for that particular datum value, and σ_i is the one-sigma (68+) uncertainty for the measurement. The summation is over all data items in the considered sample of data.

For a ‘normal’ or Gaussian distribution of errors and a good model, each observed value typically differs from the model prediction by roughly the one-sigma observational uncertainty. (Some points will be close to the predictions while others will be far, some will be larger than the predictions while others will be smaller. Yet the average deviation will be close to the one-sigma error bars.) In this case, the numerator will, on average, equal the denominator; so the chi-square contribution from each point will be roughly 1. When these chi-square contributions are summed over N data points, the expected total chi-square should be close to N. The number of degrees of freedom in this fit is N minus the number of fit parameters, which will generally be 2 (the latitude and epoch). The quantity called the reduced chi-square is the total chi-square divided by the number of degrees of freedom. For a good model (including the error estimates), the reduced chi-square should be roughly unity. Tables for converting chi-square values into probabilities appear in a large number of mathematics and statistics books.

Gaussian or ‘normal’ distributions are ubiquitous throughout natural and artificial systems: The Central Limit Theorem shows that such distributions always result when multiple sources of errors are added together. For the lore of Eudoxus, the actual error distribution, around the best fit model, is close to Gaussian in shape, and this means that the derived error bars are good. Had the error distribution not been Gaussian, then a more general maximum likelihood technique would have been required.

For reasonable models with the X² value near the number of data items, the best fit model will produce best fit parameters for which the chi-square equals some minimum value, X²_min. The one-sigma confidence region, within which the true values reside 68+ of the time, will be delineated by those parameter sets for which the calculated chi-square value is less than X²_min + 1. In general, the N-sigma confidence region is taken to be the range of parameters for which the calculated chi-square value is less than X²_min + N².

My adoption of these typical RMS scatters as measured in the circle centre method means that there will be little meaning when the chi-square minimization method returns reduced chi-square values near unity. If we had some independent means of knowing the real uncertainties, then these could be used in the chi-square calculation and then the best reduced chi-square would be a good measure for deciding whether the model is valid or not.

If σ_meas is the one-sigma error for the datum (i.e., either 4° or 8°) and σ_size is the one-sigma error associated with the size of the star group (see Table 5), then the final total uncertainty for use in calculating the chi-square is (σ_meas² + σ_size²)^0.5.

Ovenden , op. cit. (ref. 5, 1966).

Schaefer , op. cit. (ref. 12).

Here is a listing of Ovenden's certain mistakes from major to minor: (1) First and most important, Ovenden's star chart of the southern skies includes four false stars added so that Triangulum Australe looks like a ‘ringed circle’, with this being the primary reason for shifting the Southern Crown, which is the only reason to shift the centre of the southern void to derive an early epoch. (2) All the voidist analyses assumed that stars are visible in the south down to the perfect horizon, whereas realistic visibility of southern stars proves that the voidists' derived latitudes are systematically wrong by several degrees to the north. (3) The orientations of the constellations are definitively not centred around any pole, instead they are randomly distributed. (4) Ovenden's ‘ring’ of Auriga, Perseus, Hercules, and Bootes is actually centred on a pole of a.d. 300, not on a pole of 2500 b.c. as claimed. (5) Ovenden's idea that rings can indicate a pole is useless since the multitude of constellations always forces a ‘ring’ about any pole. (6) As described later in this section, Ovenden's analysis of Aratus's rise/set pairs is circular reasoning, and his assumed longitudes for the zodiacal constellations can be varied to produce any desired epoch. (7) The uncertainty for Hydra's fitting along the celestial equator is roughly seven millennia, so that this fact, if intentional, has no utility in dating anything. (8) Dicks ( op. cit. (ref. 47), 160) points out that Ovenden's arguments require anachronisms in that the development of concepts such as colures and the various great circles was certainly long after the third millennium b.c. (9) The use of bends in serpentine constellations can always be used to indicate any desired epoch. (10) Ovenden states that Hipparchus's latitude was 31°, whereas the ancient tradition is that he observed from 36° and Rhodes is at 36.4°. (11) Ovenden claims, without any evidence, that a celestial map drawn in a.d. 1801 contains otherwise lost pre-Hipparchan data. (12) Ovenden claims that the head of Draco was on the arctic circle in 2800 b.c. for a latitude of 34°, whereas this date is in error by 2300 years. (13) Ovenden claims that Ara has the negative declination of Arcturus in 2200 b.c., whereas this is true only around the middle of the second millennium b.c. (14) Ovenden claims that Roy A. E. found a date of 2900 ± 500 b.c. from the orientation of the constellations, whereas Roy himself claims ( Roy , op. cit. (ref. 5), 181) that he got 2300 b.c. by this method. (15) All the voidist analyses have ignored refraction of star light.

The declination can be confirmed by direct calculation from the rigorous equations or it can be confirmed from the extensive tabulations for all stars brighter than V = 3 at century intervals that appear in Hawkins G. S. , “Astro-archaeology”, Vistas in astronomy , x (1968), 45–88.

Roy , op. cit. (ref. 5), 171–97.

Zhitomirsky , op. cit. (ref. 9), 483–500.

Many of the plotted lines disagree with Zhitomirsky's stated dates. For example, the first item is quoted as being for the dates from 3400 to 2200 b.c. yet is plotted as 3500 to 2500 b.c. The second item is listed as 3200 to 2600 b.c. yet is plotted as 3300 to 2800 b.c. The typical plotting error is two to three centuries. The three largest plotting errors are 2400 b.c. to a.d. 2000 appearing as 3400 b.c. to a.d. 2000 for the neck of the Bird, a.d. 400 to 2000 appearing as a.d. 1500 to 1900 for Orion's belt, and 6000 b.c. to a.d. 2000 appearing as 4300 b.c. to a.d. 2000 for the bend in the Water-serpent.

If the ancients meant to indicate one side of the specified arc, then they would not have specified the far end of the arc.

For data with a Gaussian distribution (and the distribution of errors for the lore of Eudoxus has closely a Gaussian distribution), only 68+ of the items will be in the indicated range and 32+ will be more than one-sigma from the middle.

Let me specify two simplified examples of bad situations that can arise from an all-or-nothing statistic: (1) Suppose that there are three lore items whose indicated dates and one-sigma error bars are 1000 ± 100, 2000 ± 800, and 3000 ± 1500 b.c. Zhitomirsky might represent these data as having a statistic of unity within the time ranges 1100–900, 2800–1200, and 4500–1500 b.c. and a statistic of zero elsewhere. He would then note that the sum of the statistics is maximized at two for dates in the time range 2800–1500 b.c., where his derived date would be based on the worst input data and happens to ignore the one accurate input datum. Correctly, a formal weighted average gives a date of 1024 ± 99 b.c. (2) In another example of three lore items indicating 1000 ± 100, 2000 ± 900, and 3000 ± 2000 b.c., Zhitomirsky might represent his statistic as unity within the ranges 1100–900, 2900–1100, and 5000–1000 b.c. His statistic would show a broad peak from 2900–1000 b.c., where the breadth of his peak is being dominated by bad data. Correctly, a formal weighted average gives a date of 1017 ± 99 b.c.

Henriksson Blomberg , op. cit. (ref. 8).

This conclusion is problematic as the Minoan civilization did not start as a distinctive culture until substantially later than 2250 b.c. If a nautical culture from around 2250 b.c. at near a latitude of 36° is sought, a better choice would be to name the “Cyclopean” culture prominent throughout the Aegean at the time. Nevertheless, the exact choice of names for the progenitor culture is of little importance for the questions at hand, and indeed, the roots of the classical Minoan civilization must certainly be around the time and place indicated.

It might be possible that some fraction of the lore had a separate origin, say, the rise/set pairs had a later epoch. However, the statistics are consistent with a single origin and there is no evidence for a multiple origin. So any future advocate who desires to separate some fraction of the lore must come up with some independent source of evidence. Until such time, it is safe and required to conclude that the lore has a single origin.

Similarly, Aratus was passing on lore from his source ∼850 years later, while Hipparchus was passing on lore from almost one millennium earlier.

Again, it is possible that some subset, most likely the rise/set pairs, came from the time of Eudoxus as an update, yet there is no independent evidence for this and the statistical distribution of subsets is consistent with a single origin.

Actually, the oldest known star catalogue is that of Shih Shen in China from 360 b.c., and this is still extant. For the dating of Shih Shen's catalogue, see Ueta J. , “Shih Shen's catalogue of stars, the oldest star catalogue in the Orient”, Publications of the Kwasan Observatory , i (1930), 17–48.

Rogers , op. cit. (ref. 11); Hunger H. Pingree D. , Astral sciences in Mesopotamia (Leiden, 1999), 50–90.

Schaefer , op. cit. (ref. 12).

Pingree D. , “Legacies in astronomy and celestial omens”, The legacy of Mesopotamia , ed. by Dalley S. (Oxford, 1998), 125–37.

Penglase C. , Greek myths and Mesopotamia (London, 1994), and Dalley S. (ed.), The legacy of Mesopotamia (Oxford, 1998).

See Hunger Pingree , op. cit. (ref. 39), 67 for a complete list of the MUL.APIN rise/set pairs. Hunger H. Pingree D. , MUL.APIN: An astronomical compendium in cuneiform (Horn, Austria, 1989), give the original text of the MUL.APIN tablets along with translations and commentary.

Hunger Pingree , op. cit. (ref. 43, 1989), 10–12; Reiner E. Pingree D. , Babylonian planetary omens (Malibu, 1975), 1–6.

Schaefer , op. cit. (ref. 12), 318–19.

Schaefer , op. cit. (ref. 12).

When the sun is at summer solstice, the day-to-night length ratio is 5:3. This is only a function of the observer's latitude, provided that minor corrections like refraction are ignored. The relevant equation has been derived by many authors, including Dicks D. R. , Early Greek astronomy to Aristotle (Ithaca, 1970), 19–23; and Evans J. , The history and practice of ancient astronomy (Oxford, 1998), 119–20. The basic equation is tan(LAT) = cos(DUR_night × 7.5°)/tan(23.8°), where LAT is the observer's latitude and DUR_night is the duration of the night in hours. For a 5:3 day/night ratio, DUR_night = 9 hours which yields LAT = 40.95°.

The duration of the night is given as 3/8 of the full day. This number has uncertainty associated with the quantization to eighths such that the value could be in the range (3 ± 0.5)/8 for DUR_night = 9 ± 1.5 hours. This range of durations (7.5 to 10.5 hours) corresponds to latitudes of 51.55° to 23.86°. The uncertainty in the northward direction is not relevant (see Section 4), so the uncertainty in the southward direction is roughly 40.95° — 23.86° = 17.09°. But the one-sigma uncertainty will be substantially smaller than this total acceptable range, being roughly half the size. Certainly, such quantitative evaluations cannot have high accuracy, so I am representing lore item 20 as implying a latitude of 41° ± 8° to all needed accuracy.

For random data following the model, there is roughly a 3+ chance that X² ≥ 43.9 for 28 degrees of freedom.

I calculate the zenith distance of the star group, Z, at the time when the zodiacal constellation is rising. Let the position of each zodiacal constellation be α_z and δ_z for the epoch under consideration, while the position of the star group has coordinates of α and δ for the same epoch. Label the observer's latitude as λ. The zenith distance of the zodiacal constellation at the time of its rising is Z_z, which should be near 90°. The hour angle of the zodiacal constellation at the time of its rising is HA_z, where cos(HA_z) = [cos(Z_z) — sin(λ) × sin(δ_z)]/[cos(λ) × cos(δ_z)]. The hour angle of the star group, HA, will be (α_z — α) — HA_z if setting or HA_z — (α_z — α) if rising. Finally, cos(Z) = sin(λ) × sin(δ) + cos(λ) × cos(δ) × cos(HA).

The conversion from ecliptic to equatorial coordinates is taken from Equations 12.3 and 12.4 in Meeus , op. cit. (ref. 14), 126–8.

The zenith distance of the star group, either Z or Z_z, will equal 90° — EXT, where EXT is the extinction angle for the star group. Faint stars can only be seen fairly high above the horizon while bright stars can be seen at zenith distances close to 90°. The extinction angles for individual stars is discussed in Schaefer , op. cit. (ref. 12), 319–20. For reasonable extinction coefficients, EXT values are 0.7°, 1.5°, 2.0°, 2.8°, 4.2°, 6.6°, and 13.5° for stars of magnitudes −1, 0, 1, 2, 3, 4, and 5 respectively. A group of stars will have a range of EXT corresponding to their individual magnitudes, so there is no unambiguous answer to evaluating EXT. Nevertheless, I have generally taken EXT for the value corresponding to the brightest star in the group.

See the Almagest, Book VII, chap. 1, last paragraph.

Hermann Hunger (at the Institut für Orientalistik in Vienna) has recently sent me an e-mail describing the earliest evidence for the use of a uniform zodiac. He points to the earliest preserved tablets that record positions such as “Aries 25” as being from the third century b.c. In one of the astronomical diaries dated to 384 b.c. (see Neugebauer O. , Astronomical cuneiform texts (London, 1955)), a planet is said to have moved from one zodiacal sign to the next, which in Hunger's opinion is the earliest unambiguous reference to uniform signs. Hunger also points to a claim by Van der Waerden ( van der Waerden B. L. , “History of the zodiac”, Archiv für Orientforschung , xvi (1952/3), 216–30) that the earliest use of uniform signs is in a diary from 453 b.c. However, Hunger notes that the text includes passages that are apparently references to uniform signs and well as passages that appear to reference fixed stars. In any case, it appears that the first known usage of a uniform zodiac is roughly in the century centred on 400 b.c.

Perhaps the original intent was to place it on the equinoctial colure, or perhaps the position of the left arm in the Almagest (i Cep) is greatly misplaced from an earlier picture of Cepheus? This one lore item is the only one in the entire list of Table 5 that deviates by much more than 3-sigma from the best fit.

Schaefer , op. cit. (ref. 12).

In comparing two hypotheses for the same data set, the two hypotheses will generally have a different number of fit parameters. But adding fit parameters will always improve the fit to any data set. For example, a quadratic fit will always yield a smaller chi-square than does a linear fit, while a polynomial with N terms can fit N data points with a zero chi-square. So the real question is not whether the use of additional fit parameters produces a better chi-square, but whether the improvement in chi-square is sufficiently large to justify the additional fit parameters. The standard test for such questions is the F-Test, which is essentially a quantification of Ockham's Razor wherein the simplest answer that explains the data is preferred. The F-Test uses the ‘F’ parameter, which is a ratio of reduced chi-squares. The relevant version appears as Equation 2 in a paper relating to my search for gamma ray spectral lines in Gamma Ray Burst spectra with the BATSE detectors ( Band D. L. , “BATSE gamma-ray burst line search. III. Line detectability”, Astrophysical journal , cdxlvii (1995), 289–301), as well as in Martin B. R. , Statistics for physicists (London, 1971). The F value (ΔX²/Δν)/(X²/ν), where ΔX² is the improvement in chi-square, Δν is the number of extra fit parameters, and X²/ν is the reduced chi-square. A large F value indicates that the additional parameters are significant. The probability of the F value's exceeding the observed value by chance alone is Q(F|Δν, ν) and is tabulated in Table 26.9 of Abramowitz M. Stegun I. A. , Handbook of mathematical functions , (Washington, DC, 1964).

Here, ΔX² is 10.0, Δν is 2, and X²/ν is 302.9/(172 – 4) = 1.80; so that F = (10.0/2)/1.80 = 2.77. Then, the probability of the improvement in chi-square being due simply to the addition of more fit parameters is Q(2.77|2,168), which equals 0.08. This is an unimpressive probability that is consistent with the number of trials exhibited in Table 12. That is, with a dozen subsets of data, we expect one to produce a probability of Q ∼ 0.08 by random chance. This produces a decisive conclusion that there is no significant evidence for the rise/set pairs' having a separate origin from the rest of the lore.

The ΔX² value is 8.5, Δν is still 2, and X²/ν is 304.4/(172 – 4) = 1.81; so that F = (8.5/2)/1.81 = 2.35. With this, the probability of the improvement in chi-square being due to the addition of two fit parameters is Q(2.35|2,168), which equals 0.10. Again, this is an unimpressive probability that is consistent with the number of trials exhibited in Table 12.

Schaefer B. E. , “The latitude of the observer of the Almagest star catalogue”, Journal for the history of astronomy , xxxii (2001), 1–42; Schaefer B. E. , “Lunar crescent visibility”, Quarterly journal of the Royal Astronomical Society , xxxvii (1996), 759–68; Doggett L. Schaefer B. E. , “Lunar crescent visibility”, Icarus , cvii (1994), 388–403; Schaefer B. E. , “Visibility of sunspots”, Astrophysical journal , cdxi (1993), 909–19; Schaefer B. E. Bulder H. J. J. Bourgeois J. , “Lunar occultation visibility”, Icarus , c (1992), 60–72; Schaefer B. E. , “Length of the lunar month”, Archaeoastronomy , no. 17 (1992), S32–42; Kriciunas K. Schaefer B. E. , “A model of the brightness of moonlight”, Publications of the Astronomical Society of the Pacific , ciii (1991), 1033–9; Schaefer B. E. , “Glare and celestial visibility”, Publications of the Astronomical Society of the Pacific , ciii (1991), 645–60; Schaefer B. E. , “Length of the lunar crescent”, Quarterly journal of the Royal Astronomical Society , xxxii (1991), 265–77; Schaefer B. E. , “Sunspot visibility”, Quarterly journal of the Royal Astronomical Society , xxxii (1991), 35–44; Schaefer B. E. Liller W. , “Refraction near the horizon”, Publications of the Astronomical Society of the Pacific , cii (1990), 796–805; Schaefer B. E. , “Telescopic limiting magnitudes”, Publications of the Astronomical Society of the Pacific , cii (1990), 212–29; Schaefer B. E. , “Heliacal rise phenomena”, Archaeoastronomy , no. 11 (1987), S19–33; Schaefer B. E. , “The Perseus Flasher and satellite glints”, Astrophysical journal , cccxx (1987), 398–404; and Schaefer B. E. , “Atmospheric extinction effects on stellar alignments”, Archaeoastronomy , no. 10 (1986), S32–42.

The average visual extinction coefficient from McDonald Observatory for this time of year is 0.17 magnitudes per airmass; Angione R. J. de Voucouleurs G. , “Twenty years of atmospheric extinction at McDonald Observatory”, Publications of the Astronomical Society of the Pacific , xcviii (1986), 1201–7. McDonald Observatory is at an elevation of 2075 metres above sea level and is well known for its very dark skies. The view from the top of McDonald Observatory was only of distant low mountains that were close to 90° from the zenith. The view from the Texas Star Party had low nearby mountains, which raised the apparent horizon by about 5° on average around the entire sky.

My observations of stars could have been improved by (1) using shadows of the poles to avoid problems of glare from the sun, (2) using only one eye to avoid parallax, (3) using a marker for the exact position of my eye, instead of the position of my feet, (4) using an assistant to provide faint illumination of the pole, and (5) staying at the sightline long enough to watch the star pass over the pole end.

A more sophisticated means is to use an armillary sphere to measure the right ascensions of stars by transferring from a daytime equinoctial sun to the daytime visible moon and then later to stars.

Neugebauer O. , “Studies in ancient astronomy. VIII: The water clock in Babylonian astronomy”, Isis , xxxvii, (1947), 37–43; Pogo A. , “Egyptian water clocks”, Isis , xxv (1936), 403–25; Needham J. , Science and civilisation in China , iii (Cambridge, 1959), 313–29; and Hunger Pingree , op. cit. (ref. 43, 1989), 8–12 and 101–8. Impressive modern theoretical and experimental work on the configuration and accuracy of Babylonian clepsydras are presented in back-to-back papers by Michel-Nozières C. , “Second millennium Babylonian water clocks: A physical study”, Centaurus , xl (2000), 180–209, and Fermor J. Steele J. M. , “The design of Babylonian waterclocks: Astronomical and experimental evidence”, Centaurus , xl (2000), 210–22. They note that the long time intervals were measured with no bias, no long versus short nonlinearities, and without apparent seasonal or diurnal temperature effects. The very next paper is by Huber P. , “Babylonian short-time measurements: Lunar sixes”, Centaurus , xl (2000), 223–34, in which he analyses Babylonian data to determine the measurement accuracy. He finds that short time intervals can be measured to a one-sigma accuracy of ∼144 seconds. The work from these researchers is a beautiful example of combining modern practical and theoretical experience with analysis of ancient data.

Schaefer , op. cit. (ref. 60, 2001), 8–12.