Planetary Latitudes in Babylonian Mathematical Astronomy

For a classification and description of the various categories of text, see Sachs A. , “A classification of the Babylonian astronomical tablets of the Seleucid period”, Journal of cuneiform studies , ii (1948), 271–90; Hunger H. , “Non-mathematical astronomical texts and their relationships”, Ancient astronomy and celestial divination , ed. by Swerdlow N. M. (Cambridge, Mass., 1999), 77–96; and Hunger H. Pingree D. , Astral sciences in Mesopotamia (Leiden, 1999).

Since the Goal Year periods are not perfect, small corrections of a few days and/or a few degrees were needed to obtain accurate predictions. A few texts containing such corrections are known (e.g. Text E Neugebauer O. Sachs A. , “Some atypical astronomical cuneiform texts I”, Journal of cuneiform studies , xxi (1967), 183–217, and LBAT 1515), but Hunger , op. cit. (ref. 1), has shown that the corrections were not always rigorously applied in practice.

See, for example, Aaboe A. , “On period relations in Babylonian astronomy”, Centaurus , x (1965), 213–31; Neugebauer O. , A history of ancient mathematical astronomy (Berlin, 1975; hereafter HAMA ), 380–473; and Swerdlow N. M. , The Babylonian theory of the planets (Princeton, 1998).

The main body of texts are published in Neugebauer O. , Astronomical cuneiform texts (London, 1955; hereafter ACT). To those Systems identified in ACT can now be added a true System A scheme for Venus found in a template text by Hamilton N. T. Aaboe A. , “A Babylonian Venus text computed according to System A: ACT No. 1050”, Archive for history of exact science , liii (1998), 215–21; a System B scheme for Mars identified by Peter Huber and published by Aaboe A. , “On Babylonian planetary theories”, Centaurus , v (1958), 204–77; and a System A scheme for Saturn found in template texts by Aaboe A. Sachs A. , “Some dateless computed lists of longitudes of characteristic planetary phenomena from the Late Babylonian period”, Journal of cuneiform studies , xx (1966), 1–33, and in a fragment of an ephemeris by Steele J. M. , “BM 36948: A Saturn ephemeris calculated using System A from Babylon”, Journal for the history of astronomy , xxxiii (2002), 261–4.

See, for example, Neugebauer Sachs , op. cit. (ref. 2), 208.

Huber P. J. , “Zur täglichen Bewegung des Jupiter nach babylonischen Texten”, Zeitschrift für Assyriologie , lii (1957), 265–303; Neugebauer , HAMA , 412–19.

Neugebauer Sachs , op. cit. (ref. 2).

Hunger Pingree , op. cit. (ref. 1), 205–6.

BM 36680 and BM 40659, which will be published by the author in due course. I wish to thank the Trustees of the British Museum for permission to study these tablets.

Aaboe A. Henderson J. A. , “The Babylonian theory of lunar latitude and eclipses according to System A”, Archive internationales d'histoire des sciences , lxxx (1975), 181–222.

Neugebauer Sachs , op. cit. (ref. 2), 200–5. To the duplicates of this section of Text E listed by Neugebauer Sachs can now be added the unpublished fragment BM 36874.

http://ssd.jpl.nasa.gov/horizons.html. Note that the coordinates have not been corrected for refraction. Refraction is only significant when the planet is near the horizon, and incorporating it into this study would require assumptions about the time of observation, etc., that are hard to justify.

Huber P. J. , “Ueber den Nullpunkt der babylonischen Ekliptik”, Centaurus , v (1958), 192–208.

The difference between a uniform correction of 7° and a true precessional correction which varies with date over the period covered by these illustrations is negligible.

Visibility phases were taken from Roughton's N. A. computations using Schoch's arcus visionis. See Roughton N. A. , “A study of normal star almanacs and observational texts from Babylon”, Under one sky: Astronomy and mathematics in the ancient Near East , ed. by Steele J. M. Imhausen A. (Münster, 2002), 367–78. The accuracy of Schoch's method is not crucial for my purposes. I wish to thank Professor Roughton for making available his database of planetary phenomena.

Almagest , XIII, 1, transl. by Toomer G. J. , Ptolemy's Almagest (London, 1984), 597. See Pedersen O. , A survey of the Almagest (Odense, 1974), 356–7; Neugebauer , HAMA , 206ff; and Swerdlow N. M. , “Ptolemy's theories of the latitude of the planets in the Almagest, Handy tables, and Planetary hypotheses” (forthcoming).

However, Mathieu Ossendrijver (personal communication) has shown that for all of the planets — including Venus and Mercury — the latitudes of the individual Greek Letter phenomena do fall into thin zodiacally fixed bands. Thus one could develop zodiacally fixed schemes for each individual Greek Letter phenomena of each planet (with the possible exception of Venus where the individual Greek Letter phenomena cluster into groups at five longitudes leaving much of the zodiac unfilled due to the 8-year cycle).

Neugebauer Sachs , op. cit. (ref. 2), 208–10.

See Neugebauer Sachs , op. cit. (ref. 2), 209–10, and Hunger Pingree , op. cit. (ref. 1), 206.

See Oelsner J. , “Von Iqišâ und einigen anderen spätgeborenen Babyloniern”, Studi sol vicino oriente antico dedicati alla memoria di Luigi Cagni , ed. by Graziani S. (Napoli, 2000), 797–814, for a complete list and a reconstructed family tree.

In their commentary Neugebauer Sachs interpret, as I do, these phrases as meaning that the planet is at highest and lowest latitude respectively, but their translations “it goes up” and “it goes down” seem more appropriate for when the latitude is increasing and decreasing. Hunger Pingree , op. cit. (ref. 1), 205–6 also point out that Neugebauer and Sachs's translation “it goes up/down” means that the planet is at its maximum northern/southern latitude.

Neugebauer Sachs , op. cit. (ref. 2), 209.

See the figure on p. 209 of Neugebauer Sachs , op. cit. (ref. 2).

See Neugebauer , ACT , 250.

This was pointed out to me by Peter Huber at the Regensburg workshop.

Hunger Pingree , op. cit. (ref. 1), 205.

Neugebauer Sachs , op. cit. (ref. 2), 200–5.

Aaboe Henderson , op. cit. (ref. 10), 196–7.

Neugebauer Sachs , op. cit. (ref. 2), 209.

Neugebauer , HAMA , 554.

Hunger Pingree , op. cit. (ref. 1), 205.

This is only true if the beginning of the zodiacal signs is meant in lines 3 and 4. If the middle were meant, as Neugebauer Sachs assumed, the whole scheme would be shifted 15° to the right; the scheme would still not be a bad reflection of nature, although the ascending branch of the function would lie just outside Jupiter's band of latitude.

Hunger Pingree , op. cit. (ref. 1), 206, remark that they should be ±3°, and then incorrectly continue that this would require an increase and decrease of 12 fingers for every 30° of longitude. The correct figure required for extremes of ±3° is 18 fingers increase or decrease for every 30° of longitude.

The translation is essentially that of Neugebauer , ACT , 404–5 of ACT 813, but I have incorporated readings from the duplicates.

Computed data were again taken from Roughton's N. A. database. I have applied the same longitude correction to convert to the Babylonian zodiac as described above.

See Neugebauer , ACT , 405 and 424.

It is not necessary that the initial date-longitude pair is for the earliest entry on the tablet since the System A and B schemes allow one to compute forwards as well as backwards.

See, for example, Rochberg-Halton F. , “Between observation and theory in Babylonian astronomical texts”, Journal of Near Eastern studies , 1 (1991), 107–20, and Brown D. , “The cuneiform conception of celestial space and time”, Cambridge archaeological journal , x (2000), 103–22.

Sachs A. J. Hunger H. , Astronomical diaries and related texts from Babylonia (Vienna, 1988–).

For example in BM 76738+76816 published by Walker C. , “Babylonian observations of Saturn during the reign of Kandalanu”, Ancient astronomy and celestial divination , ed. by Swerdlow N. M. (Cambridge, Mass., 1999), 61–76 (this text is also discussed by Brown , op. cit. (ref. 38), 112); BM 36823 by Sachs Hunger , op. cit. (ref. 39), v, no. 54; and W 23293/13 by von Weiher E. , Spätbabylonische Texte aus dem Planquadrat U18, Teil V (Mainz am Rheim, 1998), no. 268.

For example, Neugebauer , ACT , 39 and Brown , op. cit. (ref. 38), 112.

Powell M. A. , “Masse und Gewichte”, Reallexikon der Assyriologie und Vorderasiatischen Archaeologie , vii (1987–90), 457–517.

Powell , op. cit. (ref. 42), 470; Friberg J. , “On the stucture of cuneiform metrological table texts from the −1st millennium”, Die Rolle der Astronomie in den Kulturen Mesopotamiens , ed. by Galter H. D. (Graz, 1993), 383–405.

Kugler F. X. , Sternkunde und Sterndienst in Babel , Teil II, Heft 2 (Münster in Westfalen, 1924), 547–50.

This was pointed out to me by my former student, Mark Stringer , who investigated all distances between planets and Normal Stars in the Astronomical Diaries. It is also clear from table 5 in Grasshoff G. , “Normal star observations in Late Babylonian astronomical diaries”, Ancient astronomy and celestial divination , ed. by Swerdlow N. M. (Cambridge, Mass., 1999), 97–147, although Grasshoff fails to draw this conclusion. See also the comments by Sachs Hunger , op. cit. (ref. 39), i, 22–23.

BM 36609+ will be edited by Roughton N. A. Steele J. M. Walker C. B. F. .

Neugebauer , ACT , 74–75.

Neugebauer O. , “Studies in ancient astronomy VII: Magnitudes of lunar eclipses in Babylonian mathematical astronomy”, Isis , xxxvi (1945), 10–15.

Thureau-Dangin F. , “Notes assyriologiques: Le še, measure linéaire”, Revue d'assyriologie , xxiii (1926), 33–34. The source of this text is not given in either this article or, to my knowledge, in any of the subsequent references to it.

Friberg , op. cit. (ref. 43), 391 reads 5? for the number of barleycorn in a finger, without giving any indication of whether this reading comes from collation of the tablet. Thureau-Dangin's copy indicates that the sign is damaged, but looks more like 6 than 5. Neugebauer , op. cit. (ref. 48), 13 and Powell , op. cit. (ref. 42), 458 accept the reading 6 without question, and so I follow them.

Powell , op. cit. (ref. 42), 477–9; Neugebauer , op. cit. (ref. 48), 13.

For example by Powell , op. cit. (ref. 42), 463; Neugebauer , ACT , 39 and Brown , op. cit. (ref. 38), 118 note 15.

Powell , op. cit. (ref. 42), 482–3; Friberg , op. cit. (ref. 43).

Neugebauer Sachs , op. cit. (ref. 2), 204.

Neugebauer Sachs , op. cit. (ref. 2), 205.