Mean Motions in Ptolemy’s Planetary Hypotheses

  • Dennis DukeEmail author


In the Planetary Hypotheses, Ptolemy summarizes the planetary models that he discusses in great detail in the Almagest, but he changes the mean motions to account for more prolonged comparison of observations. He gives the mean motions in two different forms: first, in terms of ‘simple, unmixed’ periods and next, in terms of ‘particular, complex’ periods, which are approximations to linear combinations of the simple periods. As a consequence, all of the epoch values for the Moon and the planets are different at era Philip. This is in part a consequence of the changes in the mean motions and in part due to changes in Ptolemy’s time in the anomaly, but not the longitude or latitude, of the Moon, the mean longitude of Saturn and Jupiter, but not Mars, and the anomaly of Venus and Mercury, the former a large change, the latter a small one. The pattern of parameter changes we see suggests that the analyses that yielded the Planetary Hypotheses parameters were not the elegant trio analyses of the Almagest but some sort of serial determinations of the parameters based on sequences of independent observations.


Northern Limit Continue Fraction Expansion Outer Planet Period Relation Emend 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Duke D.W. (2005) Ptolemy’s treatment of the outer planets. Archive for History of Exact Sciences 59: 169–187CrossRefMathSciNetGoogle Scholar
  2. Duke D.W. (2008) Mean motions and longitudes in Indian astronomy. Archive for History of Exact Sciences 62: 489–509zbMATHCrossRefMathSciNetGoogle Scholar
  3. Evans J. (1998). The theory and practice of ancient astronomy. Oxford University Press, New York, pp 362–368Google Scholar
  4. Fowler D.H. (1987) The Mathematics of Plato’s academy. Oxford University Press, New YorkzbMATHGoogle Scholar
  5. Goldstein B.R. (2002) On the Babylonian discovery of the periods of lunar motion. Journal for the History of Astronomy 33: 1–13Google Scholar
  6. Hamilton, N.T., N.M. Swerdlow and G.J. Toomer. 1987. The Canobic inscription: Ptolemy’s earliest work. In From ancient Omens to statistical mechanics, eds. J.L. Berggren and B.R. Goldstein, 55–73. Copenhagen.Google Scholar
  7. Heiberg, J.L. 1907. Claudii Ptolemaei opera quae exstant omnia. Volumen II. Opera astronomica minora, 69–145. Leipzig. There is an English translation of the Greek part of Book 1 by A. Jones (unpublished, 2004) and it is the source of all English excerpts that appear in this paper.Google Scholar
  8. Jones A., Duke D. (2005) Ptolemy’s planetary mean motions revisited. Centaurus 47: 226–235zbMATHCrossRefMathSciNetGoogle Scholar
  9. Neugebauer O. (1975) A history of ancient mathematical astronomy. Springer–Verlag, New York, pp 781–833zbMATHGoogle Scholar
  10. Stahlman, W.D. 1960. The Astronomical tables of Codex Vaticanus Graecus 1291. Ph.D. dissertation, Brown University.Google Scholar
  11. Toomer G.J. (1984) Ptolemy’s Almagest. Duckworth, LondonzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of PhysicsFlorida State UniversityTallahasseeUSA

Personalised recommendations