The Greek astronomer, astrologer, and geographer Claudius Ptolemy (ca. 100-ca. 170) established the system of mathematical astronomy that remained standard in Christian and Moslem countries until the 16th century.

Ptolemy is known to have made astronomical observations at Alexandria in Egypt between 127 and 141, and he probably lived on into the reign of Marcus Aurelius (161-180). Beyond the fact that his *On the Faculty of Judgment* indicates his adherence to Stoic doctrine, nothing more of his biography is available.

## The Almagest

The earliest and most influential of Ptolemy's major writings is the *Almagest.* In 13 books it establishes the kinematic models (purely mathematical and nonphysical) used to explain solar, lunar, and planetary motion and determines the parameters which quantify these models and permit the computation of longitudes and latitudes; of the times, durations, and magnitudes of lunar and solar eclipses; and of the times of heliacal risings and settings. Ptolemy also provides a catalog of 1, 022 fixed stars, giving for each its longitude and latitude according to an ecliptic coordinate system.

Ptolemy's is a geocentric system, though the earth is the actual center only of the sphere of the fixed stars and of the "crank mechanism" of the moon; the orbits of all the other planets are slightly eccentric. Ptolemy thus hypothesizes a mathematical system which cannot be made to agree with the rules of Aristotelian physics, which require that the center of the earth be the center of all celestial circular motions.

In solar astronomy Ptolemy accepts and confirms the eccentric model and its parameters established by Hipparchus. For the moon Ptolemy made enormous improvements in Hipparchus's model, though he was unable to surmount all the difficulties of lunar motion evident even to ancient astronomers. Ptolemy discerned two more inequalities and proposed a complicated model to account for them. The effect of the Ptolemaic lunar model is to draw the moon close enough to the earth at quadratures to produce what should be a visible increase in apparent diameter; the increase, however, was not visible. The Ptolemaic models for the planets generally account for the two inequalities in planetary motion and are represented by combinations of circular motions: eccentrics and epicycles. Such a combination of eccentric and epicyclic models represents Ptolemy's principal original contribution in the Almagest.

## Canobic Inscription

This brief text was inscribed on a stele erected at Canobus near Alexandria in Egypt in 146 or 147. It contains the parameters of Ptolemy's solar, lunar, and planetary models as given in the *Almagest* but modified in some instances. There is also a section on the harmony of the spheres. The epoch of the *Canobic Inscription* is the first year of Augustus, or 30 B.C.

## Planetary Hypotheses

In the two books of *Planetary Hypotheses,* an important cosmological work, Ptolemy "corrects" some of the parameters of the *Almagest* and suggests an improved model to explain planetary latitude. In the section of the first book preserved only in Arabic, he proposes absolute dimensions for the celestial spheres (maximum and minimum distances of the planets, their apparent and actual diameters, and their volumes). The second book, preserved only in Arabic, describes a physical actualization of the mathematical models of the planets in the *Almagest.* Here the conflict with Aristotelian physics becomes unavoidable (Ptolemy uses Aristotelian terminology but makes no attempt to reconcile his views of the causes of the inequalities of planetary motion with Aristotle's), and it was in attempting to remove the discrepancies that the "School of Maragha" and also Ibn al-Shatir in the 13th and 14th centuries devised new planetary models that largely anticipate Copernicus's.

## The Phases

This work originally contained two books, but only the second has survived. It is a calendar of the *parapegma* type, giving for each day of the Egyptian year the time of heliacal rising or setting of certain fixed stars. The views of Eudoxus, Hipparchus, Philip of Opus, Callippus, Euctemon, and others regarding the meteorological phenomena associated with these risings and settings are quoted. This makes the *Phases* useful to the historian of early Greek astronomy, though it is certainly the least important of Ptolemy's astronomical works.

## The Apotelesmatica

Consisting of four books, the *Apotelesmatica* is Ptolemy's contribution to astrological theory. He attempts in the first book to place astrology on a sound scientific basis. Astrology for Ptolemy is less exact than astronomy is, as the former deals with objects influenced by many other factors besides the positions of the planets at a particular point in time, whereas the latter describes the unswerving motions of the eternal stars themselves. In the second book, general astrology affecting whole states, societies, and regions is described; this general astrology is largely derived from Mesopotamian astral omina. The final two books are
devoted to genethlialogy, the science of predicting the events in the life of a native from the horoscope cast for the moment of his birth. The Apotelesmatica was long the main handbook for astrologers.

## The Geography

In the eight books of the *Geography,* Ptolemy sets forth mathematical solutions to the problems of representing the spherical surface of the earth on a plane surface (a map), but the work is largely devoted to a list of localities with their coordinates. This list is arranged by regions, with the river and mountain systems and the ethnography of each region also usually described. He begins at the West in book 2 (his prime meridian ran through the "Fortunate Islands, " apparently the Canaries) and proceeds eastward to India, the Malay Peninsula, and China in book 7.

Despite his brilliant mathematical theory of map making, Ptolemy had not the requisite material to construct the accurate picture of the world that he desired. Aside from the fact that, following Marinus in this as in much else, he underestimated the size of the earth, concluding that the distance from the Canaries to China is about 180° instead of about 130°, he was seriously hampered by the lack of all the gnomon observations that are necessary to establish the latitudes of the places he lists. For longitudes he could not utilize astronomical observations because no systematic exploitation of this method of determining longitudinal differences had been organized. He was compelled to rely on travelers' estimates of distances, which varied widely in their reliability and were most uncertain guides. His efforts, however, provided western Europe, Byzantium, and Islam with their most detailed conception of the inhabited world.

## Harmonics and Optics

These, the last two works in the surviving corpus of Ptolemy's writings, investigate two other fields included in antiquity in the general field of mathematics. The *Harmonics* in three books became one of the standard works on the mathematical theory of music in late antiquity and throughout the Byzantine period. The *Optics* in five books discussed the geometry of vision, especially mirror reflection and refraction. The *Optics* survives only in a Latin translation prepared by Eugenius, Admiral of Sicily, toward the end of the 12th century, from an Arabic version in which the first book and the end of the fifth were lost. The doubts surrounding its authenticity as a work of Ptolemy seem to have been overcome by recent scholarship.

## His Influence

Ptolemy's brilliance as a mathematician, his exactitude, and his masterful presentation seemed to his successors to have exhausted the possibilities of mathematical astronomy and geography. To a large extent they were right. Without better instrumentation only minor adjustments in the Ptolemaic parameters or models could be made. The major "improvements" in the models—those of the School of Maragha—are designed primarily to satisfy philosophy, not astronomy; the lunar theory was the only exception. Most of the deviations from Ptolemaic methods in medieval astronomy are due to the admixture of non-Greek material and the continued use of pre-Ptolemaic elements. The *Geography* was never seriously challenged before the 15th century.

The authority of the astronomical and geographical works carries over to the astrological treatise and, to a lesser extent, to the *Harmonics* and *Optics.* The *Apotelesmatica* was always recognized as one of the works most clearly defending the scientific basis of astrology in general, and of genethlialogy in particular. But Neoplatonism as developed by the pagans of Harran provided a more extended theory of the relationship of the celestial spheres to the sublunar world, and this theory was popularized in Islam in the 9th century. The *Harmonics* ceased to be popular as Greek music ceased to follow the classical modes, and the *Optics* was rendered obsolete by Moslem scientists. Ptolemy's fame and influence, then, rest primarily on the *Almagest,* his most original work, justly subtitled *The Greatest.*

## Further Reading on Claudius Ptolemy

There is no comprehensive study of Ptolemy's life and works. Most of the scholarly discussion of Ptolemy is contained in critical editions of the Greek texts (there is still no critical edition of the *Geography*) and in numerous scholarly periodicals. One fairly complete bibliography is William H. Stahl, *Ptolemy's Geography: A Select Bibliography* (1953). For general background see H. F. Tozer, *A History of Ancient Geography* (1897; 2d ed. 1955); Percy Sykes, *A History of Exploration* (1934; 3d ed. 1950); James Oliver Thomson, *History of Ancient Geography* (1948); and C. Van Paassen, The *Classical Tradition of Geography* (1957).

## Additional Biography Sources

Newton, Robert R., *The crime of Claudius Ptolemy,* Baltimore: Johns Hopkins University Press, 1977.