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date: 01 February 2023

astronomical tablesfree

astronomical tablesfree

  • Alexander Jones

Summary

During the Hellenistic and Roman periods, tables were extensively employed in Graeco-Roman astronomy to present structured, quantitative astronomical data for reference, calculation, and display of patterns of data. Media for tables included papyrus, in roll or codex format, wooden boards, and occasionally inscriptions. Aside from their didactic function in writings on theoretical astronomy such as Ptolemy’s Almagest, the chief practical applications of astronomical tables were in astrology. Tables for calculating celestial positions and phenomena of the heavenly bodies represented two distinct traditions: an originally Babylonian tradition based on arithmetical operations and a Greek tradition, best known from Ptolemy’s works, based on trigonometry relating to geometrical theories for the motions of the Sun, Moon, and planets. Both traditions made use of sexagesimal place-value notation. Additionally, almanacs and calendrically structured ephemerides presented celestial positions calculated over long spans of dates as a convenient tool for horoscopy and the astrological evaluation of days.

Subjects

  • Science, Technology, and Medicine

To a much greater extent than other Graeco-Roman scientific traditions, astronomy made extensive use of tables as a mode of presenting data. From the late Hellenistic period to the end of antiquity, a great variety of tables circulated in manuscript form (typically papyrus rolls and codices, or wooden boards) as independent units, in sets, or embedded in treatises, and astronomical inscriptions on stone could also employ tabular format (e.g., IG 12.1.913 = SEG 56.953, c. 100 bce from Keskintos, Rhodes).

The layout of an astronomical table was almost always a rectangular array of data in which each column represented a specific kind of quantity or category of information. The most basic type was a structured list of some kind of entity together with associated data, commonly quantitative. For example, the star catalogue in Books 7–8 of Ptolemy’s Almagest is a tabular list of more than a thousand stars. In each row, the first column contains the name of each star, and subsequent columns give the positional coordinates (celestial longitude expressed as zodiacal sign, degrees, and fractions of a degree, celestial latitude expressed as direction north or south of the ecliptic, degrees, and fractions) and magnitude expressed as a number between 1 and 6. As one reads down the catalogue, the stars are grouped according to their constellations, with a broad north-to-south trend. Ptolemy’s catalogue served a twofold purpose: to enable one to look up the position and brightness of a specific star and to provide the locations and sizes of the marks portraying all the stars on a celestial globe.

The diffusion and adaptation of late Babylonian mathematical astronomy in the Mediterranean world during the last two centuries bce brought with it a more tightly organized type of table in which the rows represented a series of computations of successive occurrences of an astronomical phenomenon or quantity such as the first visibility of Jupiter before sunrise, the Moon’s opposition to the Sun at full Moon, or Mercury’s celestial longitude at daily intervals. Most columns of these tables contained numerals with their fractional parts in sexagesimal (base-60) place-value notation (including a special symbol representing zero), and the number in each cell was computed according to a rule associated with the column in question and involving arithmetical operations performed on the number immediately above or on numbers in other columns of the same row or both. Such arithmetical tables were primarily instruments for obtaining data for astrological interpretation (see astrology). They are known to us almost exclusively from papyri from Roman Egypt, and by the end of the 4th century ce they seem no longer to have been in use.

What superseded the Babylonian-style approach was the one associated par excellence with Ptolemy: tables based on theories that decomposed the movements of the heavenly bodies into combinations of uniform circular motions such as circular orbits eccentric to the Earth and epicycles (motions on circles whose centres themselves revolve around the Earth). To calculate the apparent position of a heavenly body at a given time according to such a theory, one required two kinds of table: a table of mean motions and an anomaly table. A table of mean motions was a specialized variety of multiplication table, to convert elapsed time since an epoch date into a proportional angular motion. An anomaly table gave the results of trigonometrical calculations (see trigonometry) for converting these angles into a direction from the Earth, which was always treated as the stationary centre of the system. In the Almagest, his systematic deductive treatise on the heavenly bodies, Ptolemy intersperses these tables among the empirical and analytical chapters that provide their foundation. In this context they are not merely tools of practical computation but also didactic means of displaying patterns of data, comparable to modern graphs. In the Handy Tables, Ptolemy extracted, revised, and augmented the tables of the Almagest as an independent corpus for astrological applications, and the large number of extant papyrus fragments of the Handy Tables and of instruction manuals for them testify to his success. Although Ptolemy’s were the most widely circulated tables representing the geometrical approach, other mean motion and anomaly tables coexisted with and indeed preceded them, as we know from such papyri as PFouad 267A. Certain “Aeon-Tables” mentioned by Vettius Valens (Anthologiae 6.2) and castigated by Ptolemy (Alm. 9.2) were probably of this kind.

To determine the astronomical data for a horoscope or other astrological prognostication by means of arithmetically based or trigonometrically based tables required facility with arithmetic in the sexagesimal notation—in the Graeco-Roman context employed only in astronomy—and could be laborious. In the papyrus record, not surprisingly, these kinds of tables are outnumbered by almanacs listing precomputed positions of heavenly bodies for long stretches of dates at intervals ranging from five days to a month, or all the dates when a planet passed from one zodiacal sign into another, so that one could easily look up which sign each planet occupied on a chosen date, though without obtaining the further precision of degrees and minutes. Likewise, the great majority of personal horoscopes preserved on papyri only specify the zodiacal sign where each body was on the birthdate.

Related to the almanacs were tables known already in antiquity as “ephemerides.” In an ephemeris, the celestial longitudes of just the Moon or of the Sun, Moon, and all five planets known in antiquity were tabulated at daily intervals in a calendrical framework structured on either the Roman or the Egyptian calendar. The primary application of ephemerides was not horoscopy but the evaluation of the current day as auspicious or inauspicious for various activities, an assessment that depended chiefly on the rapidly changing angles between the Moon and the other heavenly bodies. Between the earliest known papyrus ephemeris, pertaining to 24 bce (POxy. astr. 4175) and the latest ones dating from the 5th century ce, ephemerides became progressively more elaborate, and in the 5th-century examples there is a column explicitly evaluating the quality of the day. Whereas other astronomical tables constituted the toolbox of the professional astrologer, the ephemeris had a broader, lay market, as we learn from Pliny the Elder (HN 29.9) and Juvenal (6,572–6,576), and the genre continued into the Islamic Middle Ages and early modern Europe.

Primary Texts

  • Fournet, Jean-Luc, and Anne Tihon, eds. Conformément aux observations d’Hipparque: Le Papyrus astronomique Fouad 267A. Publications de l’Institut Orientaliste de Louvain 67. Louvain-la-Neuve: Peeters, 2014.
  • Jones, Alexander. Astronomical Papyri from Oxyrhynchus. Memoirs of the American Philosophical Society 233. Philadelphia: American Philosophical Society, 1999.
  • Ptolemy. Claudii Ptolemaei Opera quae exstant omnia. Vol. 1: Syntaxis mathematica. ed. Johan Ludvig Heiberg. Leipzig, Germany: Teubner, 1898–1903.
  • Ptolemy. Ptolemy’s Almagest. Translated by Gerald J. Toomer. London: Duckworth, 1984.
  • Ptolemy. Πτολεμαίου Πρόχειροι Κανόνες‎. Les Tables faciles de Ptolémée. Ptolemy’s Handy Tables. ed. Anne Tihon and Raymond Mercier. Vols. 1a and 1b. Louvain-la-Neuve, Belgium: Université catholique de Louvain, 2011.

Bibliography

  • Jones, Alexander. “A Classification of Astronomical Tables on Papyrus.” In Ancient Astronomy and Celestial Divination. Edited by Noel M. Swerdlow, 299–340. Cambridge, MA: MIT Press, 1999.
  • Jones, Alexander. “The Keskintos Astronomical Inscription: Text and Interpretations.” SCIAMVS 7 (2006): 3–41.
  • Neugebauer, Otto. A History of Ancient Mathematical Astronomy. Berlin: Springer, 1975.
  • Sidoli, Nathan. “Mathematical Tables in Ptolemy’s Almagest.” Historia Mathematica 41 (2014): 13–37.