PTOLEMY (OR CLAUDIUS PTOLEMAEUS)

(b. c. 100 CE; d. c.170 CE),

mathematical sciences, especially astronomy. For the original article on Ptolemy see DSB, vol. 11.

Doubts were raised in the original DSB article about Ptolemy’s authorship of two works attributed to him in the manuscript tradition: the Canobic Inscription, which purports to be a transcription of a list of astronomical parameters that Ptolemy had erected at Canopus in Egypt in 146/147 CE (not 147/148 as stated in DSB), and On the Criterion (or “On the Faculties of Judgment and Command”). Subsequently, N. T. Hamilton brilliantly vindicated the authenticity of the Canobic Inscription and showed that the version of Ptolemy’s parameters that it presents preceded the Almagest (Hamilton, Noel Swerdlow, and Gerald Toomer). Thus the Almagest must have been completed more than five years after the latest dated observation (2 February 141) that it reports.

The demonstration rests on a passage in Book Four of the Almagest in which Ptolemy described a method that he said he had formerly used to derive a correction to the Babylonian period relation for the Moon’s motion in latitude that he had inherited from Hipparchus. Hamilton showed that by applying this repudiated method to two eclipse observations that Ptolemy used for other purposes in the Almagest, one obtains exactly the latitudinal periodicity in the Canobic Inscription. Ptolemy made several other changes in his system between the Canobic Inscription and the Almagest, including the eccentricity of Mercury, the epicycle radius of Saturn, and the latitudinal models and visibility criteria for the planets. It is noteworthy that Ptolemy continued to tinker with the Mercury model and the theories of planetary latitude and visibility in the later Handy Tables and Planetary Hypotheses, whereas in most other matters the Almagest had the last word. The final section of the Canobic Inscription lists the fixed stars, planets, Sun, Moon, and the four terrestrial elements (fire and air, water and earth) in descending order of distance from the center, and associates with each a whole number representing a musical pitch. This scheme, which had more to do with supposed astrological than with astronomical relations among the heavenly bodies, was apparently the one set out in the lost final three chapters of Book III of the Harmonics, a fact that suggests that the Harmonics may well have been an early work in Ptolemy’s career.

The philosophical monograph On the Criterion has been dismissed by some competent scholars as a slight and probably spurious work, but it shares many stylistic habits with Ptolemy’s accredited writings, and most decisively contains vocabulary that occurs practically nowhere else in the surviving corpus of Greek literature. Its basic argument (leaving aside some strikingly materialistic speculations about the interaction of the soul and body in its final section) is that the soul grasps truth about existing things using two of its faculties: sense perception, which communicates perceived attributes of things to the intellect, and reason, which through comparison and generalization yields knowledge of the essence of the things. Sense perception is thus necessary but not sufficient for scientific knowledge. Each faculty is reliable when operating in isolation on its proper objects, but they lose trustworthiness when they combine to make more complex judgments. Nevertheless, empirically grounded and appropriately generalized apprehensions of the reason constitute reliable knowledge. Surprisingly, in the light of Ptolemy’s other works, the Criterion does not discuss this epistemological model in the light of specific scientific problems, nor is there any serious discussion of the perception and analysis of the quantitative relations, which are fundamental to his approaches to astronomy, harmonic theory, and (to a lesser extent) optics. It is tempting to see the work as a juvenile exercise.

Scientific Epistemology in the Harmonics Although its project, the mathematical modeling of the numerous systems of pitches employed by Greco-Roman musicians, may in the early twenty-first century appear parochial and even wrong-headed, the Harmonics deserves a prominent rank in Ptolemy’s oeuvre because of its philosophically assured articulation of approaches to scientific epistemology that he applied with less overt rationalization elsewhere, especially in the Almagest. As in the Criterion, he started by accepting that reason and sense perception—in the present case, hearing—are necessary means of judging truth. But in place of a simplistic one-way relationship between the sense faculties and the intellect, the opening chapters of the Harmonics argue for the necessity of applying recursive strategies in which reason, initially informed by simple and “manifest” facts reported by the senses, guides sense perception by setting up the conditions of observation to minimize or eliminate errors of judgment resulting from the imperfect capacity of the senses to perceive quantities.

Thus without invoking any measurements or precise observations, Ptolemy argued on the basis of a wide range of broad empirical facts that musical pitches are essentially fixed magnitudes, and hence intervals between pitches have to be modeled as ratios, indeed ratios of whole numbers as in the Pythagorean harmonic theory. The theorist’s problem, then, is to determine what ratios correspond to what intervals, taking into account certain universal perceptions. These perceptions are that pitches separated by certain intervals (e.g. the octave, fifth, and fourth) are “concordant,” that is, have a high degree of likeness, and that a wider variety of smaller intervals are “melodic,” that is, heard as appropriate, esthetically acceptable, steps in the scales in which musicians sing or tune their instruments. Recognizing that the senses are par excellence reliable in identifying the phenomenon of concordance, reason devises an experimental apparatus, the single-stringed kanôn, or monochord, in which a tensed string can be divided accurately by a bridge into sections in simple whole number ratios such as 1 : 2, 2 : 3, and 3 : 4. All that is required of the hearing is to confirm that the pitches produced by these divisions are in concordant intervals. This leads to the next stage of modeling, in which rules governing the kinds of ratios producing concordance are devised according to criteria that are partly aprioristic. But the rules must be consistent with the elementary and uncontroversial facts about concordant intervals, and it is in this respect that the Pythagorean harmonic theory was methodologically unsound.

Ptolemy continued in this vein, progressively refining the models in the light of simple empirical judgments that are themselves only meaningful when interpreted according to the constraints that the models impose. A landmark stage along the way is the end of Book I, the point at which Ptolemy worked out a set of rules constituting a model for the acceptable ways in which the tetrachord(interval of a fourth) can be subdivided into melodic intervals. This model finds its empirical confirmation in the opening chapter of Book II. Here Ptolemy described a tour de force experiment in which pairs of tetrachords are tuned in specified ways by a musician, so that the scientist-auditor can logically deduce all the constituent ratios on the basis of hearing whether a particular note of one tetrachord is higher, lower, or indistinguishable from a particular note of the other tetrachord. Through this experiment Ptolemy showed that the set of distinct tetra-chords conforming to his model can be matched roughly one to one to the set of tunings actually employed by musicians. Subsequently, he constructed systems of ratios comprising complete scales of tuning from these building blocks, and again he invited the reader to see a close correspondence between these theoretically predicted scales and the ones in actual use.

In the Almagest Ptolemy applied this same strategy of progressive refinements and confirmations to the problem of establishing models for the motions of the heavenly bodies, though without explaining or defending the epistemological approach. The basic model structure appropriate for each body is justified in terms of simple, minimally quantitative empirical assertions. For example, the fact that the Sun takes a longer time to go from minimum apparent speed to mean speed than from mean to maximum speed determines that the Sun’s model involves either an eccenter or an epicycle such that the Sun revolves on the epicycle in the reverse direction to the epicycle’s revolution around the Earth. The model is then quantified in stages by a mathematical analysis of specific observations that are chosen to isolate elements of the model and render them reliably perceptible.

Almagest Observations and Calculations Much of the Almagest’s impressiveness as a work of scientific writing consists in its presentation of a systematic route of deduction of practically all the elements of the mathematical models of the heavenly bodies, with generalized empirical phenomena, and, above all, specific dated observations appearing to provide the starting points for an ostensibly rigorous mathematical analysis. This logical structure turns out to be largely artificial, in that some of the reports

of observations can be shown to have been tampered with, if not simply fabricated, and some of the calculations based on observations incorporate significant arithmetical inaccuracies, in both cases for the purpose of obtaining a preconceived result with apparent exactitude.

The longest recognized, and still the most irrefutable, instances of falsified observations in the Almagest are the four dates and times of equinoxes and solstices that Ptolemy says that he determined “with the greatest accuracy” by means of measurements of the Sun’s altitude at its meridian crossings (Almagest I 1, 4, and 7), and on which his solar theory ostensibly depends. Ptolemy reported these times with implausible precision to the nearest hour, and they are all about a day too late. And yet they agree within a fraction of an hour with the times one would predict by using Ptolemy’s tropical year to extrapolate from the older observations of Hipparchus and Meton that Ptolemy paired with them to establish his tropical year. Either Ptolemy simply fabricated these observations or, after making observations, he altered the times to obtain more exact correspondence between the empirical evidence and the theory. No one has found a plausible systematic cause for observing both kinds of equinox as well as summer solstices consistently a day late. (Incidentally, the large error in Ptolemy’s times implies that in his time there was no extensive body of accurate solar observations.)

It has become more and more manifest that many others among the observations that Ptolemy claimed to have made are at the least adjusted to eliminate discrepancies between them and the theoretical parameters that Ptolemy derived from them. For example, if one repeats Ptolemy’s iterative method of finding the eccentricities of the equant models for Mars, Jupiter, and Saturn (Book X, 7 and Book XI, 1 and 5) using exact calculations on Ptolemy’s reported mean oppositions, one finds that they converge, within a minuscule fraction of a unit, on the round numbers that Ptolemy adopted (e.g., for Mars the procedure converges on an eccentricity of 12;0,0 sixtieths of the deferent’s radius) (Duke, 2005, pp. 169–187). Again, the symmetries in the alleged observations of greatest elongations of Mercury and Venus from the mean Sun, from which Ptolemy derives the parameters of their models (Book IX 7–9, Book X 1–3), are much too exact to be genuine (Swerdlow, 1989). In one instance Ptolemy went so far as to present the reader, within a space of two pages, with two observations twenty-seven days apart that are both supposed to have Venus at greatest elongation as evening star, and for both observations the stated locations of Venus relative to nearby stars are about a degree further east than the planet really was on these dates. In general Ptolemy’s practice was to determine each model relying on arguments or authorities that are not reported in the Almagest, and thereafter to construct a set of observation reports and a deductive argument from which the model can be proved logically on the hypothesis that these observations are true. It goes without saying that this practice, unhinted at in the text, makes the epistemological status of the Almagest’s demonstrations much less straightforward than they appear on the surface.

When deducing parameters on the basis of observation reports dating from before his time, Ptolemy often made arbitrary inferences from ambiguities in the reports, and discrepancies in the arithmetic of the mathematical analyses seem to be designed to force out a preordained result, facts that imply that he did not tamper with the wording of the reports themselves as he found them in his sources. He seems in fact to have extracted many of these reports from the theoretical works of his more immediate predecessors rather than from ancient collections of unanalyzed observations. It was to be expected that later astronomers would draw in the same way on the Almagest as a source of observations. Thus Ptolemy’s introduction of factitious observation reports in company with genuine ones is open to severe criticism according to the standards of his own scientific practice.

Ptolemy’s works in fields other than astronomy are less open to the kind of analysis that might reveal analogous manipulations of data. He quite openly used contrived calculations to obtain a figure of almost exactly 180° for the longitudinal extent of the known world in Book I of the Geography. The tables of ostensibly observed angles of refraction corresponding to angles of incidence at ten degree intervals in Optics Book V are patently extrapolated from a very small genuine data set, because they exhibit constant second differences that are always precisely a small multiple of half a degree.

Context of Ptolemy’s Astronomy The demonstrable artificiality of many of Ptolemy’s empirical deductions of his astronomical models means that one cannot take it for granted that the structures and parameters of the models were originally arrived at in the same way that Ptolemy proves them in the Almagest. Some elements of the models, in particular as the length of the tropical year and the size and direction of the solar eccentricity, are explicitly ascribed to Hipparchus, whereas Ptolemy said clearly that he himself found a few others by a stated route, among them the Moon’s second anomaly, some of the mean motions of the Moon, and all the mean motions of the planets. There remain many elements for the discovery of which Ptolemy did not take credit unambiguously. Unfortunately the medieval manuscript tradition of Greco-Roman literature preserved no texts at all comparable to Ptolemy's, so that one is dependent on fragmentary and indirect sources for what is known of the background to his work and the extent to which the Almagest is the product of Ptolemy’s own research.

One such source is the fragments of astronomical tables and texts in papyri from Greco-Roman Egypt, which are largely though not entirely an offshoot of the intense interest in astrology in the first four centuries of the common era. From them one learns that the most common means of making astronomical calculations for horoscopes and other astrological applications in Ptolemy’s time were arithmetical methods descending from Babylonian astronomy, not at all resembling Ptolemy’s tables with their geometrical and trigonometrical foundations. But non-Ptolemaic tables relating to geometrical models were also in current use, and it is particularly interesting to see tables reflecting solar models different from Hipparchus's, for example assuming a moving apogee and a latitudinal wobble. Ptolemy’s adoption of Hipparchus’s solar theory was not a simple matter of acquiescing in the only theory available.

The testimony of Indian astronomical texts is even more important, though not always easy to interpret. Extant Sanskrit writings on astronomy dating from the period after about 500 CE manifestly derive much of their contents from a transmission of Greek astronomy into India that must have taken place during the preceding three centuries. Part of this transmission consisted of the same arithmetical astronomy that is so well represented in the papyri, but the predominant part was based on geometrical modeling involving eccenters and epicycles, similar in general terms to Ptolemy’s models but apparently showing no influences from his works. The numerical parameters of the Indian models are almost invariably different from Ptolemy's, and generally inferior in accuracy. Structurally the models reflect an understanding of astronomical phenomena and methods of computation that are not radically different from Ptolemy's. Most remarkably, algorithms for computing planetary positions are found in Indian texts that closely approximate the behavior of the equant model that Ptolemy assumed for all the planets except Mercury, a fact that casts some doubt on the widely held supposition that Ptolemy devised the equant model itself or that he discovered the inadequacy of a straightforward epicycle-and-eccenter model for the planets (Duke, 2005). Indian models for planetary latitude are nearer to the simpler (and more nearly correct) models that Ptolemy adopted in the Planetary Hypotheses than to those of Book Thirteen of the Almagest.

The existence of the Canobic Inscription proves that Ptolemy took a degree of personal credit for the parameters of his celestial system, taken as a whole, and the inscription, like the Almagest, was originally seen by contemporaries who had access to the astronomical literature that subsequently perished. With few exceptions these parameters are the most accurate examples from antiquity, and it is reasonable to maintain that part of Ptolemy’s achievement as an astronomer consisted in the refinement of methods of calibrating geometrical models. His claim to have discovered the Moon’s second anomaly is not contradicted by any evidence known as of 2007, but broadly speaking it would be a mistake to attribute to him every fundamental innovation in astronomical modeling that had been made since Hipparchus. For his own part, Ptolemy seems to have set more value in establishing a tight logical structure connecting theory with observation, and his complaint about his predecessors is not primarily that their hypotheses were incorrect but that they failed to prove them.

SUPPLEMENTARY BIBLIOGRAPHY

WORKS BY PTOLEMY

Berggren, J. Lennart, and Alexander Jones. Ptolemy’s Geography: An Annotated Translation of the Theoretical Chapters. Princeton, NJ: Princeton University Press, 2000.

Hübner, Wolfgang. Claudii Ptolemaei Opera quae exstant omnia, III, 1, A?OTE?E?MATIKA. Leipzig: Teubner, 1998. Supersedes the previous editions of Boll and Boer and of Robbins.

Lejeune, Albert. L’Optique de Claude Ptolémée. Leiden, Germany: Brill, 1989. Includes a French translation of the work.

Liverpool-Manchester Seminar on Ancient Greek Philosophy. “On the Kriterion and the Hegemonikon.” In The Criterion of Truth, edited by P. Huby and G. Neal. Liverpool, U.K.: Liverpool University Press, 1989. Revision of Lammert’s edition of the Criterion with numerous textual improvements and an English translation.

Morelon, Regis. “La Version arabe du Livre des hypothèses de Ptolémée.” Mélanges de l’Institut dominicain d’études orientales 21 (1993): 7–85. Critical edition and French translation of the Arabic text of the whole of Book One of the Planetary Hypotheses.

Solomon, Jon. Ptolemy: Harmonics. Leiden, Germany: Brill, 2002.

Stückelberger, Alfred, Gerd Grasshoff, et al., Klaudios Ptolemaios: Handbuch der Geographie. Basel: Schwabe, 2006. The first edition of the entire work since Nobbe’s inadequate 1843–1845 text.

Toomer, James Gerald, trans. Ptolemy’s Almagest. London: Duckworth, 1984. Reprinted with errata, Princeton, NJ: Princeton University Press, 1998. Annotated translation based on Heiberg’s edition but incorporating numerous textual improvements derived from the Greek and Arabic manuscript traditions.

OTHER SOURCES

Anagnostakis, Christopher. The Arabic Version of Ptolemy’s Planisphaerium. PhD diss., Yale University, 1984.

Aujac, Germaine. Claude Ptolémée astronome, astrologue, géographe. Paris: Comité des travaux historiques et scientifiques (CTHS), 1993.

Barker, Andrew. Greek Musical Writings, 2 vols. Cambridge, U.K.: Cambridge University Press, 1984–1989. Vol. 2 (pp. 270–391) contains an English translation of the Harmonics.

———. Scientific Method in Ptolemy’s Harmonics. Cambridge, U.K.: Cambridge University Press, 2001.

Britton, John P. Models and Precision: The Quality of Ptolemy’s Observations and Parameters. New York: Garland, 1992. A careful and judicious analysis of Ptolemy’s solar and lunar observation reports and their relation to his models.

Duke, Dennis. “Ptolemy’s Treatment of the Outer Planets.” Archive for the History of Exact Sciences 59 (2005): 169–187.

Grasshoff, Gerd. The History of Ptolemy’s Star Catalogue. New York: Springer, 1990. Refutes Vogt’s often-cited statistical argument that the Almagest’s star catalogue is independent of Hipparchus’ positional data as represented by his Commentary on Aratus and Eudoxus.

Hamilton, Norman T., Noel M. Swerdlow, and Gerald 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, edited by J. Lennart Berggren and Bernard R. Goldstein. Copenhagen: University Library, 1987.

Jones, Alexander. Ptolemy’s First Commentator, Transactions of the American Philosophical Society, 80 Part 7. Philadelphia: American Philosophical Society, 1990. A fragment of an early third century CE commentary on the Handy Tables.

———. Astronomical Papyri from Oxyrhynchus, 2 vols. in 1.Philadelphia: American Philosophical Society, Philadelphia, 1999. Papyrological evidence for the early reception of the Handy Tables among astrologers in Roman Egypt.

———. “A Likely Source of an Observation Report in Ptolemy’s Almagest.” Archive for History of Exact Sciences 54 (1999): 255–258.

———. “Ptolemy’s Canobic Inscription and Heliodorus’ Observation Reports.” SCIAMVS: Sources and Commentaries in Exact Sciences 6 (2005): 53–98.

———. “Ptolemy’s Ancient Planetary Observations.” Annals of Science 63 (2006): 255–290.

Kunitzsch, Paul. Claudius Ptolemäus: Der Sternkatalog des Almagest, 3 vols. Wiesbaden, Germany: Harrassowitz, 1986–1991. A fundamental contribution to knowledge of the catalogue’s medieval transmission and influence.

Mogenet, Joseph, and Anne Tihon. Le “Grand Commentaire” de Théon d’Alexandrie aux Tables Faciles de Ptolémée, 3 vols. Vatican: Biblioteca Apostolica Vaticana, Vatican, 1985–1999.

Murschel, Andrea. “The Structure and Function of Ptolemy’s Physical Hypotheses.” Journal for the History of Astronomy 26 (1995): 33–61.

Neugebauer, Otto. A History of Ancient Mathematical Astronomy, 3 vols. Berlin: Springer, 1975. Book 1 is a detailed and penetrating mathematical commentary on the Almagest.

Newton, Robert R. The Crime of Claudius Ptolemy. Baltimore: Johns Hopkins University Press, 1975. A sustained attack on the veracity and originality of the Almagest, vitiated in part by uneven grasp of historical conditions and oversimplified probabilistic arguments, but the first work to show how pervasively the Almagest conceals the origins of its data.

Pedersen, Olaf. A Survey of the Almagest. Odense, Denmark: Odense University Press, 1974.

Simon, Gerard. Le regard, l’être et l’apparence dans l’optique de l’Antiquité. Paris: Seuil, 1988.

Smith, A. Mark. Ptolemy’s Theory of Visual Perception, Transactions of the American Philosophical Society 86, Part 2. Philadelphia: American Philosophical Society, 1986.

———. Ptolemy and the Foundations of Ancient Mathematical Optics, Transactions of the American Philosophical Society, 89 Part 3. Philadelphia: American Philosophical Society, 1999.

Steele, John M. Observations and Predictions of Eclipse Times by Early Astronomers. Dordrecht, Netherlands: Kluwer, 2000.

———. “Ptolemy, Babylon and the Rotation of the Earth.” Astronomy and Geophysics 46 (2005), 5.11–5.15.

Swerdlow, Noel. “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, edited by Charles Burnett, Jan P. Hogendijk, Kim Plofker, et al. Leiden, Germany: Brill, 2004.

———. “Ptolemy’s Theories of the Latitude of the Planets in the Almagest, Handy Tables, and Planetary Hypotheses.” In Wrong for the Right Reasons, edited by Jed Z. Buchwald and Allan Franklin. Dordrecht, Netherlands: Springer, 2005.

Tihon, Anne. Le “Petit Commentaire” de Théon d’Alexandrie aux Tables Faciles de Ptolémée. Vatican: Biblioteca Apostolica Vaticana, Vatican, 1978.

———. “Théon d’Alexandrie et les Tables Faciles de Ptolémée.” Archives Internationales d’Histoire des Sciences 35 (1985): 106–123. Demonstrates that Theon did not reedit the Handy Tables.

Alexander Jones