Ptolemy's Astronomical Works (other than the Almagest)

Canobic Inscription

The text known as the Canobic Inscription is a transcription made in late antiquity of a lost public inscription that Ptolemy erected in A.D. 149/150 at Canopus in Egypt. The inscription records the numerical parameters of Ptolemy's models for the motions of the sun, moon, and planets. Most of the data in the Canobic Inscription agree with the Almagest, but there are a few significant differences which are now recognized as proving that the Almagest was completed later than the inscription. (see also here)

Translation of the Canobic Inscription.

Sources and further reading:

Edition of the text: Claudii Ptolemaei opera quae exstant omnia. II. Opera astronomica minora, ed. J. L. Heiberg. Leipzig, 1907. [I know of no modern translation of the inscription other than the draft linked above.]

N. T. Hamilton, N. M. Swerdlow, and G. J. Toomer, "The Canobic Inscription: Ptolemy's Earliest Work." In From Ancient Omens to Statistical Mechanics: Essays on the Exact Sciences Presented to Asger Aaboe, ed. J. L. Berggren and B. R. Goldstein. Copenhagen, 1987. 55-73.

N. M. Swerdlow, "Ptolemy's Harmonics and the 'Tones of the Universe' in the Canobic Inscription." In Studies in the History of the Exact Sciences in Honour of David Pingree, ed. C. Burnett, J. P. Hogendijk, K. Plofker, and M. Yano (Islamic Philosophy Theology and Science 54) Leiden, 2004. 137-180.

Phaseis

The Byzantine encyclopedia known as the Suda reports that Ptolemy's works included "On phases and weather-changes of fixed stars, two books." The work that has come down to us under the title Phaseis aplanon asteron kai synagoge episemasion = "phases of fixed stars and collection of weather-changes" is not divided into books, and it is generally presumed that this was originally Book 2 of Ptolemy's work, Book 1 being lost. The introductory section of our Phaseis does indeed refer to a more mathematically technical work in which Ptolemy discussed the calculation of visibility phenomena of fixed stars, but the way in which Ptolemy speaks of it implies that this was a separate treatise rather than an immediately preceding section of the Phaseis. A passage in a work of the ninth century scientist Thabit ibn Qurra ascribes to "Ptolemy's book on the phenomena of the fixed stars" an algorithm for calculating the conditions for stellar visibility that is not found in our Phaseis, so that this is presumably a reference to the lost technical treatise.

According to Ptolemy's summary at the beginning of the Phaseis, the lost treatise was a theoretical treatment of the conditions of visibility of the fixed stars, providing the means for determining where the sun must be on the ecliptic for a star to make its first annual appearance before sunrise, its last appearance after sunset, its rising at sunset, or its setting at sunrise. In Almagest 8.4-6 Ptolemy deals in a comparatively cursory manner with the conditions of stellar visibility, excusing the omission of a more detailed treatment of such phenomena and their supposed connection with weather on the grounds that the Almagest was limited to what Ptolemy thought of as exact science valid for all time whereas the dates of stellar phases shift over time because of precession. He says nothing, however, about having written elsewhere more fully on these topics. Presumably, then, both the technical treatise on stellar phases and the Phaseis were written later than the Almagest. This is confirmed by the fact that Ptolemy had refined his theory of the conditions of visibility of a heavenly body at the horizon close to sunset or sunrise by the time he came to write the specialized treatise and its perergon, the Phaseis.

The core of the Phaseis is a parapegma or weather-calendar. Parapegmata were a traditional part of Greek astronomy, going back in some form to the fifth century B.C., although the oldest extant examples date from the third century B.C. and after. A parapegma was a list of dates of more or less regular weather changes, first appearances and last appearances of stars or constellations, and solar events such as solstices, organized according to the solar year.

Ptolemy's parapegma gives the dates of such events according to the Alexandrian (or "reformed Egyptian") calendar, which, like the Julian calendar, had a fixed cycle of three years of 365 days followed by a year of 366 days, so that at least in the short term a particular Alexandrian calendar date would remain fixed relative to the natural seasons. The dates of the astronomical phenomena in the parapegma are all computed, not directly observed, and Ptolemy provides the dates for a range of different terrestrial latitudes. The weather phenomena, on the other hand, are a digest of "observations" (presumably generalizations based at some remove on local observation) made by various authorities of the past, including Demokritos, Meton, Euktemon, Philippos, Eudoxos, Callippos, Conon, Dositheos, Metrodoros, Hipparchos, (Julius?) Caesar, and "the Egyptians."

Ptolemy believed that there was a causal relationship between the astronomical phenomena and the changes in weather, but that the correlation of these events was not perfectly regular or predictable because other factors (especially the physical influences of other heavenly bodies) come into play. Hence for him weather prediction was a special division of astrology.

Translation of the Phaseis.

Sources and further reading:

Edition of the text: Claudii Ptolemaei opera quae exstant omnia. II. Opera astronomica minora, ed. J. L. Heiberg. Leipzig, 1907. [I know of no modern translation of the complete Phaseis other than the draft version linked above.]

G. Grasshoff, "The Babylonian Tradition of Celestial Phenomena and Ptolemy's Fixed Star Calendar." In Die Rolle der Astronomie in den Kulturen Mesopotamiens, ed. H. D. Galter (Grazer morgenlaendische Studien 3) Graz, 1993. 95-134.

D. Lehoux, Parapegmata. (Ph.D. Thesis, University of Toronto, 2000).

D. Lehoux, Astronomy, Weather, and Calendars in the Ancient World. (forthcoming with Cambridge University Press)

R. Morelon, "Fragment arabe du premier livre du Phaseis de Ptolemee." Journal for the History of Arabic Science 5, 1981. 3-22.

O. Neugebauer, A History of Ancient Mathematical Astronomy. 3 vols. Berlin, 1975. [especially v. 2, 926-931]

Taub, L. Ancient Meteorology. London, 2003.

G. J. Toomer, trans., Ptolemy's Almagest. London, 1984. [especially 407-417]

G. J. Toomer, article "Ptolemy." Dictionary of Scientific Biography 11, 186-206.

Handy Tables

The Handy Tables (Procheiroi Kanones) are a revised and expanded version of the astronomical tables in the Almagest. Ptolemy designed this set of tables for practical use (especially among astrologers); for this reason, although he wrote a short introduction giving instructions for using the tables, he said nothing about the theory underlying them, even though in a few instances he had modified the theory since writing the Almagest. Most of Ptolemy's revisions to the tables are designed to make them more convenient. Among the additions to the tables are a chronological table (known as the Kanon Basileon or "Ptolemy's regnal canon") and a geographical table (the list of "noteworthy cities", poleis episemoi) extracted from his Geography.

The tables of the Handy Tables that reflect changes in Ptolemy's astronomical theories are those pertaining to the latitudinal motion of the planets (i.e. the tilt of their orbits relative to the plane of the ecliptic) and the conditions of visibility of the planets.

The Handy Tables achieved a more widespread distribution in antiquity than the Almagest. Fragmentary copies of several papyrus copies of the Handy Tables have been found in Egypt. Interestingly, most of these copies are codices (pages bound in quires) rather than continuous rolls, the normal vehicle for texts in Ptolemy's time. Ptolemy himself may have designed the tables with the codex format in mind, because of its greater convenience for ready reference. In late antiquity many commentaries were written on the Handy Tables. These were mostly limited to explanations of how to use the tables, but an exception is the so-called Great Commentary by Theon of Alexandria (late fourth century A.D.), which attempts to explain the relationship between the Almagest and the Handy Tables.

Modern scholars have often supposed that the version of the Handy Tables surviving in medieval manuscripts was a revision by Theon of Alexandria (or other writers after Ptolemy's time). This hypothesis is now discredited, and it is now generally accepted that what we have is substantially Ptolemy's work, although with certain tables added.

Sources and further reading:

Edition of the tables: N. Halma, Commentaire de Th��on d���Alexandrie sur le livre III de l���Almageste de Ptolem��e; Tables manuelles des mouvemens des astres. Paris, 1822. Tables manuelles astronomiques de Ptolem��e et de Th��on. II and III. Paris, 1823-1825. [This is still the only printed edition of the Handy Tables, and not a very satisfactory one at that. Halma provides a French translation.]

W. D. Stahlman, The Astronomical Tables of Codex Vaticanus graecus 1291. Doctoral dissertation, Brown University, 1960. [Reliable transcription and commentary on most of the tables of one of the oldest medieval copies of the Handy Tables]

Edition of the text of Ptolemy's introduction: Claudii Ptolemaei opera quae exstant omnia. II. Opera astronomica minora, ed. J. L. Heiberg. Leipzig, 1907. [No modern translation exists]

B. L. van der Waerden, "Bemerkungen zu den 'Handlichen Tafeln' des Ptolemaios." Sitzungsber. Bayer. Akad. Wiss., Math. Naturwiss. Kl., 1953, 261-72.

O. Neugebauer, A History of Ancient Mathematical Astronomy. 3 vols. Berlin, 1975. [especially v. 2, 969-1028 and v. 1, 256-261]

A. Aaboe, "On the Tables of Planetary Visibility in the Almagest and the Handy Tables." Danske Vidensk. Selskab, Hist.-filos. Medd. 37.8 (1960).

N. M. Swerdlow and O. Neugebauer, Mathematical Astronomy in Copernicus' De Revolutionibus. 2 vols. New York, 1984. [especially v. 1, 483-486, on the latitude tables]

A. Tihon, "Les Tables Faciles de Ptol��m��e dans les manuscrits en onciale (ixe���xe si��cles)." Revue d'histoire des textes 22 (1992) 47���87.

A. Tihon, "Th��on d���Alexandrie et les Tables Faciles de Ptol��m��e." Archives Internationales d���Histoire des Sciences 35 (1985) 106-123.

A. Tihon, Le "Petit Commentaire" de Th��on d���Alexandrie aux Tables Faciles de Ptol��m��e: Livre I. (Studi e Testi 282) Vatican, 1978. [Theon's so-called Little Commentary is a clearer set of practical instructions for the tables than Ptolemy's own. Tihon provides an edition and French translation.]

J. Mogenet and A. Tihon, Le "Grand Commentaire" de Th��on d���Alexandrie aux Tables Faciles de Ptol��m��e. (Studi e Testi 315, 340, 390) Vatican, 1985���1999. [Edition and French translation of the Great Commentary]

A. Jones, Astronomical Papyri from Oxyrhynchus. (Memoirs of the American Philosophical Society 233) Philadelphia, 1999. [Contains editions and translations of several papyrus fragments of the tables and of early commentaries]

Planetary Hypotheses

The Planetary Hypotheses (Hypotheseis ton planomenon) represents, so far as we know, Ptolemy's last word on the models for the motions of the heavenly bodies. The work comprised two books, of which the first part of Book 1 survives in Greek, while a medieval Arabic translation of the entire work also exists. The principal object of the Planetary Hypotheses is to set out a physical interpretation in three dimensions of the idealized circles that compose the astronomical models of the Almagest. Ptolemy seems to have in mind both a description of the real physical nature of the mechanisms of planetary motion, and a prescription of how one might construct an orrery-like tangible model of these mechanisms. Unfortunately, in the large parts of the text for which we depend on the Arabic translation, it is not always clear whether Ptolemy is speaking of the reality or the imitation; but it is at least clear that he believed in the existence of invisible aetherial spheres in the heavens.

One of the most interesting features of the Planetary Hypotheses, which appears in a part of the work that was only rediscovered recently, is that Ptolemy was the true originator of the medieval cosmological system of tightly-packed nested spheres traditionally called the "Ptolemaic System." When he wrote the Almagest Ptolemy could see no grounds on which even the relative order, let alone the absolute distances, of the remaining planets could be established with certainty, given that they exhibit no observable parallax. Nevertheless he tentatively approves the opinion of "the more ancient astronomers" that the sequence from the earth outwards is: moon, Mercury, Venus, sun, Mars, Jupiter, Saturn, fixed stars. By the time that he came to write the Planetary Hypotheses, however, Ptolemy had made a discovery, based on the models deduced in the Almagest, that promised to settle the question even of the absolute distances for good. Following a traditional physical interpretation of kinematic models, each of Ptolemy's models could be described as a system of nested spheres, in such a way that the whole model is bounded by two spheres concentric with the earth. The ratio of radii of the inner and outer spheres is determined for each model by Ptolemy���s parameters. Moreover, the actual dimensions of the solar and lunar models are known. Ptolemy now discovered that, with his parameters, the models for Mercury and Venus would fit almost perfectly between the outermost sphere of the moon and the innermost sphere of the sun. In the Planetary Hypotheses he therefore assumes that the entire cosmos is so arranged that the planetary models proceed outward from the moon's with no vacant space.

Translation of Book 1 part 1 of the Planetary Hypotheses [based on Heiberg's Greek text]

Sources and further reading:

Edition of the Greek text (of the first part of Book 1) and German translation by L. Nix of the Arabic version (of Book 1 part one and Book 2 only): Claudii Ptolemaei opera quae exstant omnia. II. Opera astronomica minora, ed. J. L. Heiberg. Leipzig, 1907.

B. R. Goldstein, The Arabic Version of Ptolemy���s Planetary Hypotheses. (Transactions of the American Philosophical Society 57.4) Philadelphia, 1967. [Includes translation of the "missing" part of Book 1 and a reproduction of one of the Arabic manuscripts of the entire work with collations of a second manuscript and of the Hebrew translation of the Arabic version.]

R. Morelon, La version arabe du Livre des Hypoth��ses de Ptol��m��e, ��dition et traduction de la premire partie, M��langes de l'Institut dominicain d'��tudes orientales 21 (1993), pp. 7-85.

A. Murschel, "The Structure and Function of Ptolemy���s Physical Hypotheses of Planetary Motion." Journal for the History of Astronomy 26 (1995) 33���61.

Analemma

The Analemma is a monograph on the mathematical and practical determination of certain angles useful in the construction of sundials. ("Analemma" is the technical term for a method of converting problems in spherical geometry into planar constructions by rotating components of the three-dimensional "diagram" into a single plane of reference.) The only substantially complete text that we have of this book is in a Latin translation made in or soon after 1269 by William of Moerbeke from a Greek manuscript that vanished soon afterwards. (This version is missing all but the first of the numerical tables that, so far as we know, concluded the Analemma.) Parts of the original Greek text survive as the older writing in a palimpsest manuscript now in the Biblioteca Ambrosiana, Milan. It is possible that further fragments lie concealed on some of the up-to-now unread pages of the palimpsest.

Sources and further reading:

Edition of the Latin translation and Greek fragments: Claudii Ptolemaei opera quae exstant omnia. II. Opera astronomica minora, ed. J. L. Heiberg. Leipzig, 1907.

D. R. Edwards, Ptolemy's Peri Analemmatos - An Annotated Transcription of Moerbeke's Latin Translation and of the Surviving Greek Fragments with an English Version and Commentary. Dissertation, Brown University, 1984.

O. Neugebauer, A History of Ancient Mathematical Astronomy. 3 vols. Berlin, 1975. [especially v. 2, 839-856]

Planisphaerium

The Planisphaerium is another mathematical monograph dealing with a specialized astronomical problem, in this case how to construct a diagram in a plane representing the celestial sphere, in such a way that circles on the sphere (e.g. the equator and circles parallel to it) are represented by circles in the plane. Ptolemy's construction amounts in fact to producing a stereographic projection, that is, a projection through one of the sphere's poles upon a plane parallel to the equator; it is, however, open to dispute whether Ptolemy was aware of this fact. Ptolemy has in mind the construction of an instrument, which has usually been understood to be a form of the plane astrolabe familar from medieval astronomy, though his description more closely fits a star chart.

Sources and further reading:

Edition of the Latin translation: Claudii Ptolemaei opera quae exstant omnia. II. Opera astronomica minora, ed. J. L. Heiberg. Leipzig, 1907.

C. Anagnostakis, The Arabic Version of Ptolemy's Planisphaerium. Dissertation, Yale University, 1984.

O. Neugebauer, A History of Ancient Mathematical Astronomy. 3 vols. Berlin, 1975. [especially v. 2, 857-868]

J. L. Berggren, "Ptolemy's Maps of Earth and the Heavens: A New Interpretation." Archive for History of Exact Sciences 43, 1991, 133-144.

Return to Index page for Ptolemy and his Geography