# Ptolemy (or Claudius Ptolemaeus)

# PTOLEMY (OR CLAUDIUS PTOLEMAEUS)

(*b. ca.*a.d. 100; *d. ca.*a.d. 170)

*mathematical sciences, especially astronomy.*

Our meager knowledge of Ptolemy’s life is based mostly on deductions from his surviving works, supplemented by some dubious information from authors of late antiquity and Byzantine times. The best evidence for his dates is the series of his observations reported in his major astronomical work, the *Almagest:* these are all from the reigns of the Roman emperors Hadrian and Antoninus, the earliest 26 March 127 and the latest 2 February 141.^{1} Since he wrote several major works after the *Almagest*, this evidence fits well with the statement of a scholiast attached to works of Ptolemy in several late manuscripts—that he flourished under Hadrian and lived until the regin of Marcus Aurelius (161-180).^{2} The only other explicit date is that of the “Canobic Inscription”: this is found in manuscripts of Ptolemy’s astronomical works and purports to be a copy of an inscription dedicated by Ptolemy to the “Savior God” at Canopus, a town at the western mouth of the Nile, in the tenth year of Antoninus (A.F. 147-147).^{3}. It consists mostly of lists of astronomical parameters determined by Ptolemy; although most of its contents are extracted from the *Almagest* and other genuine works of Ptolemy, I doubt its authenticity. A statement by the sixth-century philosophical commentator Olyumpiodorus that Ptolemy practiced astronomy for forty years in “the so-called wings at Canopus,” and hence set up there the inscription commemorating his astronomical discoveries,^{4} is probably a fictional elaboration on the “Canobic Inscription,” in fact the only place mentioned in any of Ptolemy’s observations is Alexandria, and there is no reason to suppose that he ever lived anywhere else. The statement by Theodore Meliteniotes that he was born in Ptolemais Hermiou (in Upper Egypt) could be correct, ^{5} but it is late (ca. 1360) and unsupported. The belief that he came from Pelusium is a Renaissance misinterpretation of the title “Phelud(i)ensis” attached to his name
in medieval Latin texts, which in turn comes from a corruption of the Arabic “qalūdī,” a misunderstanding of *Κλαύδιоς*.^{6} His name “Ptolemaeus” indicates that he was an inhabitant of Egypt, descended from Greek or hellenized forebears, while “Claudius” shows that he possessed Roman citizenship, probably as a result of a grant to an ancestor by the emperor Claudius or Nero.

It is possible to deduce something about the order of composition of Ptolemy’s surviving works from internal evidence. The *Almagest* is certainly the earliest of the major works: it is mentioned in the introductions to the *Tetrabiblos, Handy Tables*, and *Planetary Hypotheses*, and in book **VIII** , 2, of the *Geography:* a passage of the *Almagest* looks forward to the publication of the *Geography.*^{7} One can occasionally trace development: in the *Handy Tables* many tables are presented in a form more convenient for practical use than the corresponding sections of the *Almagest*, and some parameters are slightly changed. *The Planetary Hypotheses* exhibits considerably more change in parameters, and introduces a notable improvement in the theory of planetary latitudes and an entirely new system for calculating the absolute sizes and distances of the planets. The prime meridian of the *Geography* is not Alexandria, as is promised in the *Almagest*, but a meridian through the “Blessed Isles” (the Canaries) at the extreme west of the ancient known world (which had the advantage that all longitudes were counted in the same direction). The phenomenon of the apparent enlargement of heavenly bodies when they are close to the horizon is explained in the *Almagest* as due to physical causes (the dampness of the atmosphere of the earth),^{8} whereas in the *Optics* Ptolemy gives a purely psychological explanation.^{9} Presumably in the interval between the composition of the two works he had discovered that there is no measurable enlargement. Similarly the Optics discusses the problem of astronomical refraction,^{10} which is never considered in the *Almagest* despite its possible effect on observation. It is hardly possible, however, to trace Ptolemy’s scientific development, except in his astronomical work; and even there the later works contribute only minor modifications to the masterly synthesis of the *Almagest.*

We know nothing of Ptolemy’s teachers or associates, although it is a plausible conjecture that the Theon who is said in the *Almagest* to have “given” Ptolemy observations of planets from between 127 and 132 was his teacher.^{11} Many of the works are addressed to an otherwise unknown Syrus. It would be hasty, but not absurd, to conclude from the fact that the *Geography* and the *Harmonica* are not addressed to Syrus that they are later than all the works so addressed (that is, all the other extant works except for the dubious *περὶ κριτηρίυ* and possibly the *Optics* and the *Phaseis* the relevant sections of which are missing). Living in Alexandria must have been a great advantage to Ptolemy in his work (and perhaps his education). Although much declined from its former greatness as a center of learning, the city still maintained a scholarly tradition and must at the least have provided him with essential reference material from its libraries.

Ptolemy’s chief work in astronomy, and the book on which his later reputation mainly rests, is the *Almagest*, in thirteen books. The Greek title is *μαθηματικὴ σύνταξις* which means “mathematical [that is, astronomical] compilation.” In later antiquity it came to be known informally as *ή μεγάλη σύνταξις* or *ἠ μεγίστη σύνταξις* (“The great [or greatest] compilation”), perhaps in contrast with a collection of earlier Greek works on elementary astronomy called *о̒ μικρо̀ς ἀςτρоνоμоύμενоς* (“the small astronomical collection”).^{12} The translators into Arabic transformed η̒ *μεγιστη* into “al-majisti,” and this became “almagesti” or “almagestum” in the medieval Latin translations. It is a manual covering the whole of mathematical astronomy as the ancients conceived it. Ptolemy assumes in the reader nothing beyond a knowledge of Euclidean geometry and an understanding of common astronomical terms; starting from first principles, he guides him through the prerequisite cosmological and mathematical apparatus to an exposition of the theory of the motion of those heavenly bodies which the ancients knew (sun, moon, Mercury, Venus, Mars, Jupiter, Saturn, and the fixed stars, the latter being considered to lie on a single sphere concentric with the earth) and of various phenomena associated with them, such as eclipses. For each body in turn Ptolemy describes the type of phenomena that have to be accounted for, proposes an appropriate geometric model, derives the numerical parameters from selected observations, and finally constructs tables enabling one to determine the motion or phenomenon in question for a given date.

In order to appreciate Ptolemy’s achievement in the *Almagest*, we ought to know how far Greek astronomy had advanced before his time. Unfortunately the most significant works of his predecessors have not survived, and the earlier history has to be reconstructed almost entirely from secondary sources (chiefly the *Almagest*itself). Much remains uncertain, and the following sketch of that history is merely provisional.

The first serious attempt by a Greek to describe the motions of the heavenly bodies by a mathematical model was the system of “homocentric spheres”
of Eudoxus (early fourth century b.c.). Although mathematically ingenious, this model was ill-suited to represent even the crude data on which it was based; and the approach proved abortive (it would have vanished from history had it not been adopted by Aristotle).^{13} Equally insignificant for the development of astronomy were works on “spherics” that treat phenomena such as the risings and settings of stars in terms of spherical geometry (these appear from the fourth century b.c. on). The heliocentric theory of Aristarchus of Samos (early third century b.c.), perhaps developing ideas of Heraclides Ponticus (*ca*. 360 b.c.), was purely descriptive and also without consequence. After Eudoxus, however, the epicyclic and eccentric models of planetary motion were developed; and the equivalence of the two was proved by Apollonius of Perga (*ca*. 200 b.c.), if not earlier. Apollonius made an elegant application of these models to the problem of determining the stationary points of a planet.^{14}

Meanwhile astronomical observations were being made in the Greek world from the late fifth century b.c. The earliest were mostly of the dates of solstices, but by the early third century Aristyllus and Timo-charis in Alexandria were attempting to determine the positions of fixed stars and observing occultations. These observations, however, were few and unsystematic; no firmly based theory was possible until records of the observers in Babylon and other places in Mesopotamia, reaching back to the eighth century, became available to Greeks. It seems likely that period relations derived by the Babylonian astronomers from these observations were known as early as Eudoxus, but the first Greek who certainly used the observations themselves was Hipparchus: and it is no accident that Greek astronomy was established as a quantitative science with his work. Hipparchus (active from ca. 150 to 127 b.c.) used Babylonian eclipse records and his own systematic observations to construct an epicyclic theory of the sun and moon that produced reasonably accurate predictions of their positions. Hence he was able to predict eclipses. He measured the lunar parallax and evolved the first practical method for determining the distances of sun and moon. By comparing his own observations of the position of the star Spica with those of Timocharis 160 years earlier, he discovered the precession of the equinoxes. He employed plane trigonometry and stereographic projection. In the latter techniques and in the observational instruments he used he may have been a pioneer, but we know too little of his predecessors to be sure. Ptolemy expressly informs us that Hipparchus did not construct a theory of the five planets but contented himself with showing that existing theories did not satisfy the observations.^{15} Between Hipparchus and Ptolemy the only advance was the work of Menelaus (*ca*. a.d. 100) on spherical trigonometry.

Greek astronomy, then, as Ptolemy found it, had evolved a geometric kinematic model of solar and lunar motion that successfully represented the phenomena, at least as far as the calculation of eclipses was concerned, but had produced only unsatisfactory planetary models. It had developed both plane and spherical trigonometry and had adopted the Babylonian sexagesimal place-value system not only for the expression of angles but also (although not systematically) for calculation. As a mathematical science it was already sophisticated. From the point of view of physics it was not a science at all: such physical theories as were enunciated were mere speculation. But there was available a fairly large body of astronomical observations, of which the most important, both for completeness and for the length of time it covered, was the series of eclipses observed in Mesopotamia. Ptolemy made no radical changes in the system he took over; but by intelligent use of available observations and ingenious modification of the basic principle of all existing kinematic models (uniform circular motion), he extended that system to include the five planets and significantly improved the lunar model.

Books I and II of the *Almagest* are devoted to preliminaries. Ptolemy begins by stating and attempting to justify his overall world picture, which is that generally (although not universally) accepted from Aristotle on: around a central, stationary, spherical earth the sphere of the fixed stars (situated at a distance so great that the earth’s diameter is negligible in comparison) revolves from east to west, making one revolution per day and carrying with it the spheres of sun, moon, and planets; the latter have another, slower motion in the opposite sense, in or near a plane (the ecliptic) inclined to the plane of the first motion. He then develops the trigonometry that will be used throughout the work. The basic function is the chord (which we denote Crd) subtended by the angle at the center of a circle of radius 60. This is related to the modern sine function by

The values of some chords (for instance, Crd 60°) are immediately obtainable by elementary geometry. By geometrically developing formulas for Crd (α+β), Crd (α-β), Crd 1/2;α, where Crd α and Crd β are known, and then finding Crd 1° by an approximation procedure, Ptolemy produces a table of chords, at intervals of 1/2° and to three sexagesimal places, which
serves for all trigonometric calculations. There was probably nothing new in his procedure (Hipparchus had constructed a similar table).^{16}

Since the sine function is sufficient for the solution of all plane triangles, the chord function too was sufficient, although the lack of anything corresponding to a tangent table often made the solution laborious, involving the extraction of square roots. For trigonometry on the surface of the sphere Ptolemy uses a figure that we call a Menelaus configuration. It is depicted in Figure 1, where all the arcs *AB, BE*, and

so on are segments of a great circle less than 90°, Ptolemy proves (following Menelaus)that

It can be shown that the four basic formulas used in modern trigonometry for the solution of right spherical triangles are directly derivable from (1) and (2). Hence all problems soluble by means of the former can also be solved by the use of Menelaus configurations. Thus Ptolemaic trigonometry, although sometimes cumbersome, is completely adequate.

This trigonometry is applied in books I and II to various phenomena connected with the annual variation in solar declination. The sole numerical parameter used is the inclination of the ecliptic (e). This is also the amount of the greatest declination of the sun from the equator, and that is the basis of the two simple instruments for measuring E which Ptolemy describes (I, 12). He reports that he measured 2E as between 47;40° and 47;45°. Since the estimation of Eratosthenes and Hipparchus, that 2E is 11/83 of a circle, also falls between these limits, Ptolemy too adopts the latter, taking E as 23; 51, 20°. His failure to find a more accurate result, and hence to discover the slow decrease in the inclination, is explained by the crudity of his instruments. He can now construct a table of the declination of the sun as a function of its longitude, which is a prerequisite for solving problems concerning rising times. The rising time of an arc of the ecliptic is the time taken by that arc to cross the horizon at a given terrestrial latitude. Most of book II is devoted to calculating tables of rising times for various latitudes. Such tables are useful astronomically, for instance, for computing the length of daylight for a given date and latitude (important in ancient astronomy, since the time of day or night was reckoned in “civil hours,” one civil hour being 1/12 of the varying length of day or night); but the space devoted to the topic is disproportionate to its use in the *Almagest.* It is essential in astrology, however (for example, in casting horoscopes); and this is one of the places where astrological requirements may have influenced the *Almagest* discussion (although they are never explicitly mentioned).

Book III treats the solar theory. By comparing his own observations of the dates of equinoxes with those of Hipparchus, and his observation of a solstice with one made by Meton and Euctemon in 432 b.c., Ptolemy confirms Hipparchus’ estimate of the length of the tropical year as 365 1/4—1/300 days. This estimate is notoriously too long (the last fraction should be about 1/128), and the error was to have multiple consequences for Ptolemaic astronomy. It is derivable from the date only because Ptolemy made an error of about one day in the time of each of his observations. This is a gross error even by ancient standards and is the strongest ground of those modern commentators (such as Delambre) who maintain that Ptolemy slavishly capied Hipparchus, to the point of forging observations to obtain agreement with Hipparchus’ results. The conclusion is implausible, but it is likely that in this case Ptolemy was influenced by his knowledge of Hipparchus’ value to select such of his own observations as best agreed with it. For Hipparahus’ solar and lunar theory represented the known facts (that is, eclipse records) very well, and Ptolemy would be reluctant to tamper with those elements of it that would seriously affect the circumstances of eclipses. Using the above year length, Ptolemy constructs a table of the mean motion of the sun that is the pattern for all other mean motion tables: the basis is the Egyptian calendar, in which the year has an unvarying length of 365 days (twelve thirty-day months plus five epagomental days). The motion is tabulated to six sexagesimal places, for hours, days months, years and eighteen-year periods.

The main problem in dealing with all planets is to account for their “anomaly” (variation in velocity). In the case of the sun this variation is apprent from the fact that the seasons are of unequal length—for instance, in Ptolemy’s day the time from spring equinox to autumn equinox. Ptolemy proposes a general model for representing anomalistic motion. The “eccetric’ version is depicted in Figure 2, where

*O* represents the central earth. The body *P* moves with uniform angular velocity on a circle whose center *M* is distant from *O* by an amount e (the eccenricity). It is clear that if the motion of *P* appears uniform from *M*, it will appear nonuniform from *O*, being slowest at its greatest distance, the apogee *A*, and fastest at its least distance, the perigee *Z*. The angle *K* which *P* has traveled from the apogee is derivable from its mean motion *K̄* by the formula

*K* = *K̄* ± δ.

δ is called by Ptolemy the *πρоσθαφ;αίρεσις* and by us, following medieval usage, the “equation.” The same motion can also be represented by an epicyclic model. In Figure 3 an epicycle, center *C*, moves with uniform

angular velocityon the circle (known as the deferent) about *O*, while the body *P* moves uniformly about *C* in the opposite sense. Provided that the angular velocities of body and epicycle are equal and the radius of the epicycle the eccentricity, the two models are completely equivalent, as can be seen from Figure 3, where *α¯* = *K¯* and *r* = *e*. Such is the case of Ptolemy’s solar model. But the angular velocity of *P* may be different from that of *C*, as in the lunar theory (this is equivalent to a rotation of the apogee in the eccentric model); or the rotation of *P* may be in the same sense as *C*, as in Ptolemy’s planetary theory. The general model is, therefore, extremely versatile.

From his observations of solstics and equinoxes Ptolemy found the same length of seasons as Hipparchus. He therefore concluded that the apogee of the sun is tropically fixed and that its motion can be represented by the simple eccentric of Figure 2. He determined its parameters, as Hipparchus had, from the observed length of the seasons. In Figure 4the sun is in *T* at spring equinox, in *X* at summer solstice, and in *Y* at autumn equinox. Ptolemy states that it moves from *T* to *X* in 941/2 days and from *X* to *Y* in 92 1/2 days. Thus the angles at *M* (the center of uniform motion), *TMX* and *XMY*, can be calculated from the mean motion of the sun; and the angles at *O*, the earth, are right. Hence one can determine the

eccentricity OM and the longitude of the apogee, *TOA.* Since Ptolemy uses the same data and method as Hipparchusm he gets the same results: an eccentricity of 2;30 (where R, the standard radius, is 60)and an apogee longitude of 65;30°. In fact the eccentricity had decreased and the apogee longitude increased since Hipparchis’s time, but this could not be detected through equinox and solstice observations made with crude ancient instruments.

Using the parameters determined, Ptolemy shows how the equation σ can be calculated trigonometrically for a given mean anomaly and sets up a table giving σ as a function of. The only thing now lacking in order to calculate the position of the sun at any date is its mean position at some given date. From his equinox observation of a.d. 132 Ptolemy calculates its position at the date he has chosen as epoch, Thoth 1 (first day of the Egyptian year), year 1 of the Babylonian king Nabonassar (26 February 747 b.c.). To complete the solar theory he discusses the equation of time. Since the sun travels in a pate inclined to the equator and with varying velocity, the interval between two successive meridian transits of the sun (the true solar day) will not be uniform but will vary slightly throughout the year. Since astronomical calculations employ uniform units (that is, mean solar days), whereas local time, in an age when the sundial is the main chronometers is reckoned according to the true solar day, one must be able to convert one to the other. This conversion is done by means of the equation of time, the calculation of which Ptolemy explains.

The lunar theory is the subject of books IV and V. According to Ptolemy one must distinguish three periods connected with the moon: the time in which it returns to the same longitude, the time in which it returns to the same velocity (period of anomaly), and the time in which it returns to the same latitude. In addition one must consider the synodic month, the time between successive conjunctions or oppositions of the sun and moon. He quotes a number of previous attempts to find a period containing an integer number of each of the above (such a period would, clearly, be an eclipse cycle); in particular, from Hipparchus:

(1) In, 126,007 days, 1 hour, there occur 4,267 syndic months, 4,573 returns in anomaly, and 4,612 sidereal revolutions less 7 1/2° (hence the length of a mean syndic month is 29;31, 50, 8, 20 days).

(2) In 5,458 synodic months there occur 5,923 returns in latitude.

Ptolemy says that Hipparchus established these “from Babylonian and his own observations”. We now know from cuneiform documents that these parameters and period relations had all been established by Babylonian astronomers. ^{17} At best Hipparchus could have “confirmed” them from his own observations. Such confirmation could be carried out only by comparison of the circumstances of eclipses separated by a long interval. Ptolemy gives an acute analysis of the conditions that must then obtain in order for one to detect an exact period of return in anomaly. On the basis of his own comparison of eclipses he accepts the above relations, with very slight modifications to the parameters for returns in anomaly and latitude (justified later). Thus he is able to construct tables of
the lunar mean motion in longitude, anomaly, argument of latitude (motion with respect to the nodes in which the orbit of the moon intersects the ecliptic), and elongation (motion with respect to the mean sun).

The next task is to determine the numerical parameters of the lunar model. Ptolemy first assumes (although he knows better) that the moon has a single anomaly, that is, that its motion can be represented by a simple epicycle model (or eccenter with rotating apogee); this was the system of Hipparchus. To determine the size of the epicycle he adopts a method invented by Hipparchus. He takes a set of three lunar eclipses: the time of the middle of each eclipse can be calculated from the observed circumstances. Hence the true longitude of the moon at eclipse middle is known, since it is exactly 180ō different from the true longitude of the sun (calculated from the solar theory). Furthermore, the time intervals between the three eclipses are known; hence one can calculate the travel in mean longitude and mean anomaly between the three points. Thus one has the situation of Figure 5: *P*_{1,}*P*_{2,}*P*_{3} represent the position of the moon on the epicycle at the three eclipses. The angles (the equational differences as seen from the earth O are found by comparing the intervals in true longitude with the intervals in mean longitude; the angles are found by taking the travel in mean anomaly modulo 360 degree. From these one can calculate trigonometrically the size of the epicycle radius*r* in terms of the deferent radius R=OC, and the angle ACP (which gives an epoch value for the anomaly). Ptolemy makes calculations for two sets of eclipses—the first early Babylonian, the second observed by himself—and gets almost identical results. He finally adopts the value r-5; 15, and on this basis he constructs an equation table. Although he borrowed the above procedure from Hipparchus, Ptolemy’s result seems to be a distinct improvement on his predecessor’s. He tells us that through small slips in calculating intervals Hipparchus found two discrepant results from two eclipse triples—namely, 327 2/3: 3144 and 247 1/2: 3122 1/2. Independent evidence shows that Hipparchus adopted the latter eccentricity, although it is a good deal too small.^{18}

In the preceding calculations the moon has been treated as if it lay in the plane of the ecliptic. In fact its orbit is inclined to that one is justified in neglecting it in longitude calculations. Accurate knowledge of the latitude is, however, essential for eclipse calculations. The inclination of 5° which Ptolemy accepts was probably an established value. But his procedure for finding the epoch value and mean motion in argument of latitude from two carefully selected eclipses is both original and a great improvement over Hipparchus’ method, which involved estimating the apparent diameter of the moon and of the shadow of the earth at eclipse, both of which are diffcult to measure accurately.

The simple lunar model of book IV is essentially that of Hipparchus. When Ptolemy compared observed positions of the moon with those calculated from the model, he found good agreement at conjunction and opposition (when elongation of the moon

from the sun is 0° or 180°); this agreement was to be expected, since the model was derived from eclipse observations. But he found serious discrepancies at intermediate elongations. Hipparchus had already mentioned such discrepancies but failed to account for them. In book V Ptolemy develops his own lunar theory. Analyzing his observations, he finds that they seem to indicate an increase in the size of the epicycle between opposition and conjunction that reaches a maximum at quadrature (90° elongation). This increase he represents by incorporating in the model a “crank” mechanism that “pulls in” the epicycle as it approaches quadrature, thus making it appear larger. Compare Figures 6A and 6B, which depict the same situation according to the simple and refined models, respectively. In the latter the epicycle center *C* continues to move uniformly about the earth *O*, but it now moves on a circle the center of which is not *O* but *M. M* moves about *O* in the opposite sense to *C*, so that its elongation form the mean sun is equal to that of *C.* Since *OM + MC* of Figure 6B is equal to *OC* of Figure 6A, it is clear that the two models are identical when or 180, that is, at mean conjunction and opposition. At intermediate elongations, however, the refined model pulls the epicycle closer to *O*, thus increasing the effect of the anomaly. This increase is greatest at quadrature. From two observations of the moon near quadrature by himself and Hipparchus, Ptolemy finds that the maximum equation increases from about 5°at conjunction to 7;40° at quadrature, and hence e in Figure 6B is 10;19 (where *R*=60). As a further refinement he shows that one can obtain better agreement with observation if one reckons the anomaly not from the true epicycle apogee *Â* but from a mean apogee *A* opposite the point *B* (*B* in turn being opposite *M* on the small circle about *O*). Thus a third inequality is introduced, also varying with the elongation but reaching its maximum near the octants (=45° and 135°). Ptolemy can now construct a table to compute the position of the moon. in contract with previous tables, the tabulated function depends on two variables Ptolemy’s solution to tabulating such a function (which may have been his own invention) became standard: he computes the equation at extreme points (in this case at conjunction and quadrature) and introduces an interpolation function (here varying with 2) to be used as a coefficient for intermediate positions.

Ptolemy’s refined lunar model represents the longitudes of the moon excellently. It is a major improvement on Hipparchus’ model, and yet it does not disturb it at the points where it was successful—namely, where eclipses occur. But one effect of the crank mechanism is to increase greatly the variation in the distance of the moon from the earth, so that its minimum distance is little more than half its maximum. If correct, this should be reflected by a similar variation in the apparent size of the moon, whereas the observable variation is much smaller. This objection

seems not to have occurred to Ptolemy: he treats the crank mechanism not merely as a convenient device for predicting the longitude but also as a real feature of the model (for instance, in his parallax computations). Fortunately the apparent size of the moon is of consequence only at eclipses, when the crank-mechanism has no effect.

Having established a complete theory of the motion of sun and moon, Ptolemy can proceed to eclipse theory. But first he must deal with parallax, the angular difference between the true and apparent positions of a body that stems from the fact that we observe it not from the center of the earth but from a point on the surface. In practice only the parallax of the moon is significant, but the misconceptions of the ancients about the distance of the sun led them to estimate solar parallax as well. By comparing a suitable observed position of the moon with its computed position, Ptolemy obtains a value for the parallax. Since this value is equivalent to the angle under which the radius of the earth is seen from the moon in that position, he can position, he can immediately calculate the distance of the moon in earth radii, first at that position and then (from his model) at mean distance. His final figure, fifty-nine earth radii, is close to the truth but is reached by a combination of multiple errors in his data and model, which, by pure chance, cancel each other.

Ptolemy knew that solar parallax is too small to measure directly; but he calculates the distance of the sun, and hence its parallax, by a method invented by Hipparchus. The data required are the apparent diameters of moon and sun at distances such that their apparent diameters are equal, that distance for the moon, and the apparent diameter of the shadow of the earth at that distance of the moon. Ptolemy assumes that moon and sun have exactly the same apparent diameter when the moon is at greatest distance (which implies that annular eclipses cannot occur); he determines the apparent diameter of moon and shadow from two pairs of lunar eclipses by an ingenious method of his own (greatly improving on the figures which Hipparchus had obtained by direct measurement). Then, in Figure 7 he knows EM (= Eσ), the distance of the moon; d_{m}, the apparent diameter of moon and sun; and d_{o} the apparent diameter of the shadow. From these data it is simple to calculate r_{m} and r_{s}, the radii of moon and sun, and *ES*, the distance of the sun. His value for the latter is 1,210 earth radii, too small by a factor of twenty. In fact, the method requires much too accurate a measurement of the apparent diameter of the sun to produce a reliable solar distance, but it continued to be used up to and beyond Copernicus. Ptolemy now constructs a table of solar and lunar parallaxes and explains how to compute the parallax for a given situation. This computation is both laborious and mathematically unsatisfactory. Ptolemaic parallax theory is perhaps the most faulty part of the *Almagest.* But it was not significantly improved until the late sixteenth century.

Eclipse theory, the topic of book VI, is easily derived from what precedes. Ptolemy sets up a table for calculating mean syzygies (conjunctions and oppositions), with the corresponding lunar anomaly and argument of latitude. He then determines (from the apparent sizes of the bodies) the eclipse limits, that is, how far from the node the moon can be at mean syzygy for an eclipse still to take place. The eclipse tables proper give the size in digits and duration of eclipses as a function of the distance of the moon from the node. Ptolemy explains minutely how to compute the size, duration, and other circumstances of both lunar and solar eclipses for any given place. But his method does not allow one to compute the path of a solar eclipse (a development of the late seventeenth century).^{19}

Books VII and VIII deal with the fixed stars. The order of treatment is a logical one, since it is necessary to establish the coordinates of ecliptic stars to observe planetary positions. Ptolemy compares his own observations with those of Hipparchus and earlier Greeks to show that the relative positions of the fixed stars have not changed and that the sphere of the fixed
stars moves about the pole of the ecliptic from east to west 1 ° in 100 years with respect to the tropical points. He ascribes the discovery of the latter motion (the precession of the equinoxes) to Hipparchus, who had estimated it as *not less than* 1° in 100 years. This figure is too low (1° in seventy years would be more accurate); the error is mostly due to Ptolemy’s wrong figure for the mean motion of the sun.^{20} The bulk of these two books is composed of the “Star Catalog”, a list of 1,022 stars, arranged under forty-eight constellations, with the longitude, latitude, and magnitude (from 1 to 6) of each. To compile this entirely from personal observation would be a gigantic task, and Ptolemy has often been denied the credit. Delambtem for instance, maintained that Ptolemy merely added 2;40° to the longitudes of “Hiopparchus’ catalog.”^{21} This particular hypothesis has been disproved.^{22} In fact, the evidence suggests that no starcatalog in this form bad been composed by Hipparchus or anyone else before Ptolemy (the quotations from Hippaqrchs in *Almagest* VII, 1, l show that Ptolemy had before him not a catalog but a description of the constellations with some numerical data concerning distances between stars). Modern computations have revealed numerous errors in Ptolemy’s coordinates.^{23} In general, the longitudes tend to be too small. This too is explained by the error in his solar mean motion, which is embedded in the lunar theory: the moon was used to fix the potion of principal stars (the only practical method for an ancient astronomer).^{24} Book VIII ends with discussion of certain traditional Greek astronomical problems, such as the heliacal raisings and settings of stars.

The last five books are devoted to planetary theory. Here, in contrast with the moon and sun, Ptolemy had no solid theoretical foundation to build upon and much less in the way of a body of observations. The most striking phenomenon of planetary motion is the frequent occurrence of retro gradation, which had been explained at least as early as Apollonius by a simple epicyclic model (see Figure 3) in which the sense of rotation of planet and epicycle is the same. Such a model, however, would produce a retrogradation arc of unvarying length and occurring at regular intervals, whereas observation soon shows that both are and time of retro gradation vary. No geometric model had been proposed that would satisfactorily account for this phenomenon. Certain planetary periods, however, were well established, and as was the law for outer planets that

*Y* = *L* + *A*,

where (in integer numbers) *Y* stands for years, *L* for returns to the same longitude, and *A* for returns in anomaly (Venus and Mercury have the same period of return in longitude as the sun, hence for them *Y* = *L*). Ptolemy quotes from Hipparchus such a period relation for each planet—for instance, for Saturn; “In 59 years occur 57 returns in anomaly and 2 returns in longitude.” We now know that all the period relations quoted are in fact Babylonian in origin. From these Ptolemy constructs tables of mean motion in longitude and anomaly, first applying small corrections; it turns out, however, that the latter are in part based on the models he is going to develop, so he must have used the uncorrected period relations in the original development.

Analysis of observations revealed that each planet has two anomalies; the first varying according to the planet’s elongation from the sun and a second varying according to its position in the ecliptic. Ptolemy isolated the first by comparing different planet-sun configurations in the same part of the ecliptic and the second by comparing the same planet-sun configurations in different parts of the ecliptic. He thus found that the first could be represented by an epicycle model in which the sense of rotation of planet and epicycle are the same, while the second was best represented by an eccenter (to avoid a double epicycle). The model he finally evolved is depicted in Figure 8. The planet *P* moves on

epicycle, center *C. C* moves in the same sense on a circle the center of which, *M*, is distant from the earth *O* by the eccentricity *e.* The uniform motion of *C*, however, takes place not about *M* but about another point E on the opposite side of *M* from *O*and distant from *M* by the same amount *e*. The mean motion in anomaly is counted from a point *F* on the epicycle opposite to *E.* Figure 8 depicts the situation for an outer planet, in which the line *CP* always remains parallel to the line from *O* to the mean sum (thus preserving the law *y* = *L* + *A*). Except for this features the model for Venus is identical. The Mercury model has an additional mechanism to produce a varying eccentricity but is otherwise the same as that for Venus.

The most original element in this model is the introduction of the point *E* (known as the equant from medieval times). This element distrubed later theoreticians (including Copernicus), since it violated the philosophical principle that uniform motion should take place about the center of the circle of revolution. But it enabled Ptolemy to construct the first satisfactory planetary model. In heliocentric theory a Ptolemaic equant model with eccentricity *e* produces planetary longitudes differing from those of a Kepler ellipse with the same eccentricity by less than 10’ (even for the comparatively large eccentricity of Mars).^{25} Ptolemy deduces the existence of the eccenter directly from observations only in the case of Venus. For the other planets he merely assumes it and regards this as justified by the resulting agreement between observed and computed positions.

The determination of the parameters of the model for the individual planets, which occupies most of books IX-XI, reveals Ptolemy’s brilliance. In his mastery of the choice and analysis of observations in conjunction with theory he has no peer until Kepler. For Venus and Mercury the problem is comparatively simple: the center of their epicycles coincides with the mean sun, and so one “sees” the epicycle by observing maximum elongations. The position of the apsidal line (*OMEA* in Figure 8) can be determined by observing symmetrical positions of the epicycle, and then it is simple to calcultate the size of epicycle and eccentricity. For the outer planets such “direct observation” of the epicycle is not possible. So Ptolemy eliminates the effect of the epicycle by choosing three observed oppositions; this choice gives him three positions of *C* as seen from *O*. The problem of finding the eccentricity and apogee then resembles the problem of finding the size of the epicycle of the moon in book IV, but it is complicated by the existence of the equant. Ptolemy meets this complication with an ingenious iterative process: he first assumes that the equant coincides with the center of the accentric; this produces an approximative apsidal line and eccentricity, which are used to compute corrections to the initial data; then the whole process is repeated as many times as is necessary for the results to converge. Such a procedure is alien to classical Greek mathematics, although not unique in the *Almagest.*

Book XII is devoted to establishing tables for computing the arcs and times of retrogradation and (for Venus and Mercury) the greatest elongation. In connection with the former, a theorem of Apollonius on stationary points is adapted to fit Ptolemy’s refined planetary model. In book XIII he deals with planetary latitudes. All five planets have orbits slightly inclined to the ecliptic. From a heliocentric point of view the situation is simple, but a geocentric theory encounters considerable difficulties. The simplest approximation in Ptolemaic terms would be as follows: For an outer planet set the plane of the deferent at an inclination to the ecliptic equal to that of the heliocentric orbit of the planet set the plane of the deferent at an inclination to the ecliptic equal to that of the heliocentric orbit of the planet, and keep the plane of the epicycle parallel to the ecliptic; for an inner planet keep the plane of the deferent in the ecliptic, and set the inclination of the epicycle to the deferent equal to the inclination of the heliocentric orbit of the planet. Ptolemy eventually reaches that solution in the *Planetary Hypotheses* (except that the deferents of Venus and Mercury are inclined to the ecliptic at a very small angle); but in the *Almagest*, misled by faulty observations and the eccentricity of the orbit of the earth, he devises a much more complicated theory in which the epicycles of outer planets and both epicycles and deferents of inner planets undergo varying inclinations as the epicycle moves round the deferent. The resultant tables do, however, represent the actual changes in latitude fairly well and are no mean achievement. The work ends with a discussion of the traditional problem of the heliacal risings and settings of the planets.

As a didactic work the *Almagest* is a masterpiece of clarity and method, superior to any ancient scientific textbook and with few peers from any period. But it is much more than that. Far from being a mere “systematization” of earlier Greek astronomy, as it is somethimes described, it is in many respects an original work. Without minimizing Ptolemy’s debt to Hipparchus (which Ptolemy himself admiringly acknowledges), we may say confidently that Hipparchus’ pioneering work would have had very little effect had it not found its completion in the *Almagest.* The Ptolemaic system is indeed named after the right man.

The *Almagest* contains all the tables necessary for astronomical computations, but they are scattered throughout the work. AT a later date Ptolemy published the tables separately, together with an introduction explaining their use, under the title (*Handy Tables*). The tables themselves are extant only in the revised version of Theon of Alexandria (*fl.* 360), but from Ptolemy’s introduction it is clear that Theon changed nothing essential. Ptolemy
himself is responsible for the numerous differences from the *Almagest*. There are additions (such as a table of the longitudes and latitudes of principal cities, to enable one to convert from the meridian and latitude of Alexandria), changes in layout to facilitate computation, and even occasional improvements in basic parameters. The epoch is changed from era Nabonassar to era Philip (Thoth 1 = 12 November 324 b.c.). The tables became the standard manual in the ancient and Byzantine worlds, and their form persisted beyond the Middle Ages.

Later still Ptolemy published a “popular” résumé of the results of the Almagest under the title ̒ыπоθέσεις τω̑ν πλανωμένων (*Planetary Hypotheses*), in two books. Only the first part of book I survives in Greek, but the whole work is available in Arabic translation. It goes beyond the *Almages*t in several respects. First, it introduces changes in some parameters and even in the models, notably in the theory of planetary latitude already mentioned Second, in accordance with Ptolemy’s declaration in the introduction that one purpose of the work is to help those who aim to represent the heavenly motions mechanically {that is, with a planetarium), the models are made “physical,” whereas in the *Almagest* they had been purely geometric. Ptolemy describes these physical models in detail in book II (most of book I is devoted to listing the numerical parameters). He argues that instead of assigning a whole sphere to each planet, it is sufficient to suppose that the mechanism is contained in a segment of a sphere consisting of a drum-shaped band extending either side of its equator. The most portentous innovation, however, is the system proposed at the end of book I for determining the absolute distances of the planets.

In the *Almagest* Ptolemy had adopted the (traditional) ascending order: moon, Mercury, Venus, sun, Mars, Jupiter, Saturn; but he admitted that this order was arbitrary as far as the planets are concerned, since they have no discernible parallax.^{26} This order has no consequences, since the parameters of each planet are determined independently in terms of a conventional deferent radius of 60. In the *Planetary Hypotheses* Ptolemy proposes a system whereby the greatest distance from the earth attained by each body is exactly equal to the least distance attained by the body next in order outward (that is, the planetary spheres are touching, and there is no space wasted in the universe; this system conforms to Aristotelian thinking). He takes the distance of the moon in earth radii derived in Almagest V: its greatest distance is equal to the least distance of Mercury (if one assumes the above order of the planets). Using the previously determined parameters of the model for Mercury, he now computes the greatest distance of Mercury, which is equal to the least distance of Venus, and so on. By an extraordinary coincidence the greatest distance of Venus derived by this procedure comes out very close to the least distance of the sun derived by an independent procedure in *Almagest* V. Ptolemy takes this finding as a striking proof of the correctness of his system (and incidentally of the assumption that Mercury and Venus lie below the sun). He goes on to compute the exact distances of all the bodies right out to the sphere of the fixed stars in earth radii and stades (assuming the circumference of the earth to be 180,000 stades). Furthermore, taking some “observations“by Hipparchus of the apparent diameters of planets and first-magnitude stars, he computes their true diameters and volumes. This method of determining the exact dimensions of the universe became one of the most popular features of the Ptolemaic system in later times.

A work in two books named *Phases of the Fixed Stars* (*άάσεις ἀστέρων*) dealt in detail with a topic not fully elaborated in the *Almagest* the heliacal risings and settings of bright stars. Only book II survives; and the greater part of this book consists of a “calendar,“listing for every day of the year the heliacal risings and settings, as well as the weather prognostications associated with them by various authorities. Predicting the weather from the “phases” of well-known stars long predates scientific astronomy in Greece, and calendars like this were among the earliest astronomical publications (Ptolemy quotes from authorities as early as Meton and Euctemon). The chief value of the *άάσεις* today is the information it contains on the history of this kind of literature.

Much greater scientific interest attaches to two small works applying mathematics to astronomical problems. The first is the *Analemma* (*περὶ ἀναλήμματоς*), surviving, apart from a few palimpsest fragments, only in William of Moerbeke’s Latin translation from the Greek. It is an explanation of a method for finding angles used in the construction of sundials, involving projection onto the plane of the meridian and swinging other planes into that plane. The actual determination of the angles is achieved not by trigonometry (although Ptolemy shows how that is theoretically possible) but by an ingenious graphical technique which in modern terms would be classified as nomographic. Although the basic idea was not new (Ptolemy criticizes his predecessors, and a similar procedure is described by Vitruvius ca. 30 b.c.),^{27} the sophisticated development is probably Ptolemy’s. The other treatise is the *Planisphaerium* (the Greek title was probably *‘Απλωσις ἐπμρφανείας σφ;αίρας*).^{28} This
treatise survives only in Arabic translation; a revision of this translation was made by the Spanish Islamic astronomer Maslama al-Majr̄t̄ (d. 1007/1008) and was in turn translated into Latin by Hermann of Carinthia in 1143. It treats the problem of mapping circles on the celestial sphere onto a plane. Ptolemy projects them from the south celestial pole onto the plane of the equator. This projection is the mathematical basis of the plane astrolabe, the most popular of medieval astronomical instruments. Since the work explains how to use the mapping to calculate rising times, one of the main uses of the astrolabe, it is highly likely that the instrument itself goes back to Ptolemy (independent evidence suggests that it goes back to Hipparchus).^{29} These two treatises are an important demonstration that Greek mathematics consisted of more than “classical” geometry.

To modern eyes it may seem strange that the same man who wrote a textbook of astronomy on strictly scientific principles should also compose a textbook of astrology (*̓ Απоτελεσματικά*, meaning “astrological influences,” or *τετράβιβλоς*, from its four books). Ptolemy, however, regards the *Tetrabiblos* as the natural complement to the *Almagest*: as the latter enables one to predict the positions of the heavenly bodies, so the former expounds the theory of their influences on terrestrial things. The introductory chapters are devoted to a defense of astrology against charges that it cannot achieve what it claims and that even if it can, it is useless. Ptolemy regards the influence of heavenly bodies as purely physical. From the obvious terrestrial physical effects of the sun and moon, he infers that all heavenly bodies must produce physical effects (that such an argument could be (that such an argument could be seriously advanced reflects the poverty of ancient physical science). By careful observation of the terrestrial manifestations accompanying the various recurring combinations of celestial bodies, he believes it possible to erect a system which, although not mathematically certain, will enable one to make useful predictions. Ptolemy is not a fatalist: at least he regards the influence of the heavenly bodies as only one of the determinants of terrestrial events. But, plausible as this introduction might appear to an ancient philosopher, the rest of the treatise shows it to be a specious “scientific” justification for crude superstition. It is difficult to see how most of the astrological doctrines propounded could be explained “physically” even in ancient terms, and Ptolemy’s occasional attempts to do so are ludicrous. Astrology was almost universally accepted in the Roman empire, and even superior intellects like Ptolemy and Galen could not escape its dominance.

Book I explains the technical concepts of astrology, book II deals with influences on the earth in general (“astrological geography” and weather prediction), and books III and IV with influences on human life. Although dependent on earlier authorities, Ptolemy often develops his own dogma. The discussion in books III and IV is confined to what can be deduced from a man’s horoscope: Ptolemy ignores altogether the branch of astrology known as catarchic, which answers questions about the outcome of events or the right time to do something by consulting the aspect of the heavens at the time of the question. This omission helps to explain why the *Tetrabiblos* never achieved an authority in its field comparable with that of the *Almagest* in astronomy.

The *Geography (Γεωγραφ;ικὴ υ̒φήγησις*), in eight books, is an attempt to map the known world. The bulk of it consists of lists of places with longitude and latitude, accompanied by very brief descriptions of the chief topographical features of the larger land areas. It was undoubtedly accompanied in Ptolemy’s own publication by maps like those found in several of the manuscripts. But knowing how easily maps are corrupted in copying, Ptolemy takes pains to ensure that the reader will be able to reconstruct the maps on the basis of the text alone: he describes in book I how to draw a map of the inhabited world and lists longitudes and latitudes of principal cities and geographical features in books II—VII. Book VIII describes the breakdown of the world map into twenty-six individual maps of smaller areas. Ptolemy tells us that the *Geography* is based, for its factual content, on a similar recent work by Marinus of Tyre. But it seems to have improved on Marinus’ work (for which the *Geography* is the sole source of our knowledge) in several ways. From I, 7–17 (in which various factual errors of Marinus are corrected), it appears that the bulk of Marinus’ text was topographical description (giving, for instance, distances and directions between places), and that this was supplemented by lists of places with the same longest daylight and of places the same distance (in hours) from some standard meridian (book VIII of the Geography, which looks as if it is a remnant of an earlier version, uses a system similar to the latter). Ptolemy was probably the first to employ systematically listings by latitude and longitude. Here, as always, he shows a sound sense of what would be of most practical use to the reader.

Ptolemy also criticizes Marinus’ map projection, a system of rectangular coordinates in which the ratio of the unit of longitude to that of latitude was 4:5. Ptolemy objects that this system distorts distances except near the latitude of Rhodes (36°). While accepting such a system for maps covering a small

area, for the world map he proposes two alternative projection systems (I, 21–24). The known world, according to Ptolemy, covers 180° in longitude from his zero meridian (the Blessed Isles) and in latitude stretches from 16;25° south to 63° north. In his first projection (see Figure 9) the meridians are mapped as radii meeting in a point *H* (*not* the north pole), the parallels of latitude as circular arcs with *H* as center. Distances are preserved along the meridians and along the parallel of Rhodes, and the ratio of distances along the parallel of 63° to those along the parallel of the equator is preserved. These conditions completely determine the projection. Ptolemy modifies it south of the equator by dividing the parallel *MZN* as if it lay at 16;25° *north*, thus avoiding distortion at the expense of mathematical consistency. The second projection aims to achieve more of the appearance of a globe (see Figure 10). The parallels of latitude are again constructed as circular arcs, but now distances are preserved along three parallels: 63° north, 23;50° north, and 16;25° south. The meridians are constructed by drawing circular arcs through the points on these three parallels representing the same angular distance from the central meridian, which is mapped as a straight line along which distances are preserved. The first projection is (except for the modification south of the equator) a true conic projection; the second is not, but for the segment covered by the map is a remarkably good approximation to the true conic projection that was later developed from it (the Bonne projection, which preserves distances along all parallels). Ptolemy took a giant step in the science of mapmaking, but he had no successor for nearly 1,400 years.

The factual content of the *Geography* naturally is inaccurate. Only the Roman empire was well known; and Ptolemy’s idea of the outline of, for instance, southern Africa or India is grossly wrong. Even within the boundaries of the empire there are some serious distortions This was inevitable: although the latitude of a place could be fixed fairly accurately by astronomical observation, very few such observations had been made. Ptolemy says (1,4) that of his predecessors only Hipparchus had given the latitudes of a few cities. Longitudinal differences could be determined from simultaneous observations of an eclipse at two places, but again almost no such observations existed. Ptolemy seems to have known only one, the lunar eclipse of 20 September 331 b.c., observed at Arbela in Assyria and at Carthage. Unfortunately an error in the observation at Arbela led Ptolemy to assume a time difference of about three hours between the two places instead of about two; this error was probably a major factor in the most notorious distortion in his map, the excessive length of the Mediterranean. In the absence of anything resembling a modern survey, Ptolemy, like his predecessors, had to rely on “itineraries“derived from milestones along the main roads and the reports of merchants and soldiers. In view of the inaccuracies in lengths, and especially directions, inevitable in such works, the *Geography* is a remarkable factual as well as scientific achievement.

The *Optics*, in five books, is lost in Greek. An Arabic translation was made from a manuscript lacking book I and the end of book V; from this translation, which is also lost, Eugenics of Sicily produced the extant Latin translation in the twelfth century. Despite the incompleteness and frequent obscurity of the text, the outlines of Ptolemy’s optical theory are clear enough. The lost book I dealt with the general theory of vision. Like most ancient theoreticians Ptolemy believed that vision takes place by means of a “visual flux” emanating from the eye in the form of a cone-shaped bundle of “visual rays,“the apex of which lies within the eyeball; this flux produces sensations in the observer when it strikes colored objects. References back in the surviving books show that besides enunciating the above theory, Ptolemy demonstrated that vision is propagated in a straight line and determined the size of the visual field (both probably by experiments).

In book II he deals with the role of light and color in vision (he believes that the presence of light is a “necessary condition” and that color is an inherent quality of objects); with the perception of the position, size, shape, and movement of objects; and with various types of optical illusion.

Books III and IV treat the theory of reflection (“catoptrics,” to use the ancient term). First, three laws are enunciated: (1) the image appears at some point along the (infinite) line joining the eye to the point of reflection on the mirror. (2) The image appears on the perpendicular from the object to the surface of the mirror. (3) Visual rays are reflected at equal angles. The laws are then demonstrated experimentally. There follows a remarkable discussion on the propriety of assimilating binocular to monocular vision in geometric proofs. Ptolemy incidentally determines the relationships between the images seen by the left and right eyes and the composite image seen by both, using an ingenious experimental apparatus with lines of different colors. He then develops from the three laws a series of theorems on the location, size, and appearance of images, first for plane mirrors, then for spherical convex mirrors, then (in book IV) for spherical concave mirrors and various types of “composite” (such as cylindrical) mirrors.

Book V deals with refraction. The phenomenon is demonstrated by the experiment (at least as old as Archimedes) of the coin in the vessel that appears when water is poured in. There follows a very interesting experiment to determine the magnitude of refraction from air to glass, from water to glass, and from air to water. Ptolemy does this experiment by means of a disk with a graduated circumference. For each pair of media he tabulates the angles of refraction corresponding to angles of incidence of 10 °, 20 °,…,80 °. The results cannot be raw observations, since the second differences are constant in all cases; but they are remarkably close to what one can derive from Snell’s law , with a suitable index of refraction. The rest of the surviving part of the book is devoted to discussion of astronomical refraction, the relationship between the amount of refraction and the density of the media, and the appearance of refracted images.

It is difficult to evaluate Ptolemy’s achievement in the *optics* because so little remains of his predecessor’s work. In “pure optics” we have only the work of Euclid(ca. 300 b.c.), consisting of some elementary geometrical theorems derived from a few postulates that are a crude simplification of the facts of vision. In catoptrics we have a corrupted Latin version of the work of Hero (ca. A.D. 60) and a treatise compiled in late antiquity from authors of various dates, falsely attributed to Euclid. From these we get only an occasional glimpse of Archimedes’ catoptrics, which was probably highly original, particularly in its use of experiment. The establishment of theory by experiment, frequently by constructing special apparatus, is the most striking feature of Ptolemy’s *Optics*. Whether the subject matter is largely derived or original, the *Optics* is an impressive example of the development of a mathematical science with due regard to the physical data, and is worthy of the author of the *Almagest*.

A work on music theory, (*̒ Αρμоνικά*), in three books, deals with the mathematical intervals (on a stretched string) between notes and their classification according to various traditional Greek systems. It seeks a middle ground between the two schools of the Pythagoreans and the followers of Aristoxenus, of whom the former, according to Ptolemy (1, 2), stressed mathematical theory at the expense of the ear’s evidence, while the latter did the reverse. Again we see Ptolemy’s anxiety to erect a theory that is mathematically satisfactory but also takes due account of the Porphyry. According to the commentary of Porphyry(late third century), the *Harmonica* is mostly derivative, especially from the work of one Didymus (first century) ^{30}. We have no means of checking this statement.

Ptolemy’s philosophical standpoint is Aristotelian, as is immediately clear from the preface to the *Almagest*. But it is clear from his astronomy alone that he does not regard Aristotle as holy writ; and influences from later philosophy, notably Stoicism, have been detected. An insignificant philosophical work entitled *περὶ κριоυ καὶ η̒γεμоνικоυ̑* (“On the Facilities of judgement and Command”) goes under his name. There is nothing in its contents conflicting with Ptolemy’s general philosophical position, but the style bears little resemblance to his other works; and the ascription, while generally accepted, seems dubious. Ptolemy’s high repution in later times caused a number of spurious works, mostly astrological, to be foisted on him. An example is the *καρπо́ς* (*Centiloquium*, in its Latin translation), a collection of 100 astrological aphorisms.

Ptolemy was active in almost every mathematical science practiced in antiquity, but several of his works are known to us from references in ancient authors. These are a work on mechanics (*περὶ ρ́оπω̑ν*), in three books; a work in which he attempted to prove Euclid’s parallel postulate; *On Dimension* (*περὶ διαστάσεως*), in one book, in which he “proved” that there can be no more than three dimensions; and *On the Elements* (*περὶ τω̑ν στоιχείων*).

In estimating Ptolemy’s stature and achievement as a scientist, it is unfortunately still necessary to react against the general tendency of nineteenth-century scholarship to denigrate him as a mere compiler of the scientific work of his predecessors. This extreme view, exemplified in the writings of Delambre ^{31}, is no longer held by anyone competent but still persists in hand-books. It need not be refuted in detail. With a candor unusual in ancient authors Ptolemy freely acknowledges what parts of his theory he owes to Hipparchus. To say that he is lying when he claims other parts as his own work is a gratuitous slander, which, when it can be tested by independent evidence (which is rare), has proved false. It is certain that a great part of the theory in the *Almagest* is his personal contribution, and it is unlikely that the situation was radically different in all his other scientific work. On the other hand, his was not an original genius: his method was to take existing theory and to modify and extend it so as to get good agreement with observed facts. In this method, however, Ptolemy was no different from the vast majority of scientists of all periods; and he in no way deserves the reputation of a hack. His work is remarkable for its blend of knowledge, ingenuity, judgement, and clarity. The authority that it achieved in several fields is not surprising.

The *Almagest* was the dominant influence in theoretical astronomy until the end of the sixteenth century. In antiquity it became the standard textbook almost immediately. Commentaries to it were composed by Pappus (fl. 320) and Theon of Alexandria, but neither they nor any other Greek advanced the science beyond it. It was translated into Arabic about 800; and improved translations were made during the ninth century, notably in connection with the astronomical activity patronized by Caliph al-Ma’mun. The Islamic astronomers soon recognized its superiority to what they had derived from Persian and Hindu sources; but since they practiced observation, they also recognized the deficiencies in its solar theory. An example of the influence of Ptolemy on Islamic astronomy and its improvements on his theory is the *Zij* of al-Battani (ca. 880). The first part of this work is closely modeled on the *Almagest*, the second on the *Handy Tables* (also translated earlier). Al-Battani greatly improves on Ptolemy’s values for the obliquity of the ecliptic, the solar mean motion (and hence precession), the eccentricity of the sun, and the longitude of its apogee. He substitutes the sine function (derived from India) for the chord function. Otherwise his work is mostly a restatement of Ptolemy’s. This is typical of Islamic astronomy: the solar theory was refined (so that even the proper motion of the apogee of the sun was enunciated by al-Zarqal ca. 1080), but Ptolemy’s lunar and planetary theories were accepted as they stood. Such attempts as were made to revise them were based not on observation but on philosophical objections, principally to the equant. Alternative systems, preserving uniform circular rotation, were devised by Nasir al-Din al-Tusi (fl. 1250) and his followers at Maragha, and also by Ibn al-Shatir (fl. 1350). The Latter’s lunar model is in one sense a real improvement over Ptolemy’s, since it avoids the exaggerated variation in the lunar distance. But the influence of these reformers was very small (except for a hypothetical transmission from Ibn ash-Shatir to Copernicus, who adopted almost identical models).

The *Almagest* became known in western Europe through Gerard of Cremona’s Latin translation from the Arabic in 1175 (a version made in Sicily from the Greek ca. 1160 seems to have been little known). The arrival from Islamic sources of this and other works based on Ptolemy led to a rise in the level of Western astronomy in the thirteenth century, but until the late fifteenth serious attempts to make independent progress were sporadic and insignificant. The first major blow at the Ptolemaic system was Copernicus’ De *revolutionibus* (1543). Yet even this work betrays in its form and in much of its content the overwhelming influence of the *Almagest*. However great the cosmological implication of the change to a heliocentric system, Copernican astronomy is cast in a firm Ptolemaic mold. This influence was not broken until Tycho Brahe realized that the “reform of astronomy” must be based on systematic accurate observation. So he compiled the first star catalog since Ptolemy based entirely on independent observation (earlier star catalogs, such as the famous one of al-Sufi, epoch 964, had been mostly repetitions of Ptolemy’s with the addition of a constant of precession; star coordinates independent of Ptolemy, such as those in the list of the *Zij al-Mumtahan* compiled at the order of al-Ma’mun, are very few in number); Brahe was also able to make the first real improvements in lunar theory since Ptolemy. Most important, he provided Kepler with the essential material for his treatise on Mars, justifiably entitled *Astronomia nova*. Kepler’s work finally made the *Almagest* obsolete expect as a source of ancient observations.

Ptolemy’s system for establishing the exact distances and sizes of the heavenly bodies was also enthusiastically adopted in Islamic astronomy. One of the numerous adaptations of it found its way to the West in the twelfth-century Latin translation of the epitome of the *Almagest* by al-Farghani (fl. 850). In this version it became part of the standard world picture of the later Middle Ages (for instance, in Dante’s *Divina commedia*) and persisted into the sixteenth century, although its basis was removed by Copernicus.

The *Tetrabiblos* went through a process of transmission to the Islamic world and then to western Europe similar to that of the *Almagest* and also enjoyed an immense reputation in both, although never so unrivaled. The *Geography* was also translated into Arabic early in the ninth century. Its inadequacy for much of the territory subject to Islam was immediately obvious, and so it was soon replaced by revisions more accurate for those parts. The earliest is the *Kitab Surat al-Ard* of al-Khwarizmi (ca. 820). The original Arabic translation of the *Geography* is lost, but its direct and indirect influence on Islamic geographical works is considerable. The *Geography* did not reach western Europe until the fifteenth century (in a Latin translation from the Greek by Jacobus Angelus, ca. 1406), when exploration was already making it obsolete. Nevertheless it became very popular, since it was still the best guide to much of the known world; most cartographic publications of the fifteenth and sixteenth centuries are based on it. Ptolemy’s two map projections were an important stimulus to the development of cartography in the sixteenth century. The first is reflected in Mercator’s map of Europe (1554), the second in various maps
beginning with one by Bernardus Sylvanus in his 1511 edition of the *Geography*, which uses an equivalent of the Bonne projection.^{32}

The *optics* had little direct influence on western Europe. Eugenius’ version was known to Roger Bacon and probably to Witelo in the later thirteenth century, but they are exceptions. It did however, inspire the great optical work of Ibn al-Haytham (d. 1039). The latter was an original scientist who made some notable advances (of which the most important was his correct explanation of the role of light in vision), but the form and much of the content of his *Kitab al-Manazir* are taken from Ptolemy’s work; and the inspiration for his remarkable experiments with light must surely also be attributed to Ptolemy. His treatise was translated into Latin and is the basis of the *Perspectiva* of Witelo (ca. 1270), which became the standard optical treatise of the later Middle Ages. The indirect influence of the *Optics* persisted until the early seventeenth century.

Ptolemy’s treatise on musical theory never attained the authority of his other major works, since rival theories continued to flourish. But it was extensively used by Boethius, whose work was the main source of knowledge of the subject in the Latin West; and hence some of Ptolemy’s musical doctrine was always known. When it became available to western Europe in Greek, it was no more than a historical curiosity. But it had a strange appeal for one great scientist. The last three chapters of book III of the *Harmonica* are missing; they dealt with the relationships between the planetary spheres and musical intervals. Kepler intended to publish a translation of book III, with a “restoration “ of the last chapters and a comparison with his own kindred speculations, as an appendix to his *Harmonica mundi.* The appendix never appeared, but the whole work is tribute to his predecessor.

## NOTES

1. Almagest XI, 5; IX, 7 (Manitius, II, 228, 131).

2. Boll, “Studien” p. 53.

3.*Opera astronomica minora*, p. 155. An alternative MS reading is “the fifteenth year” (152/153).

4. Olympiodorus,…*In Platonis Phaedonem…*, Norwin ed., p. 59, 1. 9.

5. Boll, *op. cit.*, pp. 54-55.

6. Buttmann, “Ueber den Ptolemaus in der Anthologie…” pp. 483 ff.

7. II, 13 (Manitius, I, 129).

8. I, 3 (Manitius, I, 9).

9. III, 59; Lejeune ed., p.116.

10. V, 23-30; Lejeune ed., pp. 237-242.

11. E.g., X,1 (Manitius, II, 156).

12. See Pappus, *Collectio*, VI, intro.; Hultsch ed., p. 474, with Hultsch’s note *ad. loc.*

13. Aristotle, *Metaphysica, A.* 1073b17 ff.; Simplicius, *In Aristotelis De caelo*, Heiberg ed., pp. 491 ff.

14.*Almagest* XII, 1 (Manitius, II, 267 ff.).

15.*Almagest* IX, 2 (Manitius, II, 96-97).

16. Theon, *Commentary on the Almagest*, I, 10; Rome ed., p. 451. See Toomer, “The Chord Table…,” pp. 6-16, 19-20.

17. Kugler, *Die Babylonische Mondrechnung*, pp. 4 ff.

18. See Toomer, “The Size of the Lunar Epicycle…,” pp. 145 ff.

19. The first to compute and draw the path of an eclipse seems to have been Cassini I in 1664; see Lalande, *Astronomie*, II, 358.

20. Shown by A. Ricius, *De motu octarvae sphaerae*, f. 39 (following Levi ben Gerson); see also Laplace, *Exposition*, p. 383.

21. Delambre, *Histoire de l’astronomie ancienne*, II, 250 ff.

22. By Vogt in “Versuch einer Wiederherstellung von Hipparchs Fixsternverzeichnis.”

23. E. g., Peters and Knobel, *Ptolemy’s Catalogue of Stars, passim.*

24. See *Almagest* VII, 4 (Manitius, II, 30).

25. See Caspar’s intro. to his trans. of Kepler’s *Neue Astronomie*, pp. 60*-61*.

26. IX, 1 (Manitius, II, 93-94).

27. Vitruvius, De architectura IX, 7.

28. Suidas,“πτоλεμαι̑оς о́ Κλαύδιоς” Alder ed., IV, 254.

29. Synesius, *Opuscula*,Terzaghi ed., p.138.

30. Porphyry, *Commentary on the Harmonica;* During ed., p. 5.

31. E. g., *Histoire de l’astronomie, II, passim.*

32. See Hopfner in Mzik, *Des Klaudios Ptolemaios Einfuhrung*, p. 105.

## BIBLIOGRAPHY

I. Original Works. Almagest: The standard text is that of J. L. Heiberg, *Claudii Ptolemaei Opera exstant omnia*, I, *Syntaxis mathematica*, 2 pts. (Leipzig, 1898-1903). The only reliable trans. is the German one by K. Manitius:*Ptolemaus, Handbuch der Astronomie*, 2nd ed. (with corrections by O. Neugebauer), 2 vols. (Leipzig, 1963). For detailed discussion of the contents the best guide is still J. B. J. Delambre, *Histoire de l’astronomie ancienne* II (Paris, 1817; repr. New York-London, 1965). Delambre’s notes in the otherwise antiquated edition by N. Halma, *Composition mathematique de Claude Ptolemee*, 2 vols. (Paris, 1813-1816), are also useful. Pappu’s lemmas to the *Мικρо̀ς ἀστρоνоύμενоς* are in Bk. VI (474-633) of his *συναγωγή, Pappi Alexandrini Collectionis quae supersunt*, F. Hultsch, ed., 3 vols. (Berlin,1875-1878; repr. Amsterdam, 1965). No satisfactory account exists of the early history of Greek astronomy; the best available is P. Tannery, *Recherches sur I’historie de l’astronomie ancienne* (Paris, 1893). On the astronomical system of Eudoxus see G. V. Schiaparelli, “Le sfere omocentriche di Eudosso, di Calippo e di Aristotele,” in *Memorie dell’Istituto lombardo di scienze e lettere*, Classe di Scienze Matematiche e Naturali, **13** , no. 4 (1877), repr. in his *Scritti sulla storia della astronomia antica*, II (Bologna, 1926), 5-112. The account by Simplicius is in his commentary on Aristotle’s *De Caelo. Commentaria in Aristotelem Graeca*, J. L. Heiberg, ed., VII (Berlin, 1894), 491-510. Examples of early works on “spherics” are those by Autolycus (late fourth century B. C.), edited by J. Mogenet. *Autolycus de Pitane*, which is *Universite de Louain, Recueil de travaux**d’ histoire et de philologie, 3rd ser., fasc.* 37 (Lounvain, 1950); and Menelaus’ Spherics’ (available only in Arabic translation), M.Krause, ed., in *Abhandlungen der Akademie der Wissenschaften zu Gottingen, Phil.*-hist. KI., 3rd ser., **17** (1936).

On the trigonometry of Ptolemy and Hipparchus see G.J. Toomer, “The Chord Table of Hipparchus and the Early History of Greek Trigonometry,” in Centaurus, **18** (1973), 6-28. The cuneiform evidence for the Babylonian origin of Hipparchus’ lunar periods is given in F.X. Kugler, Die *Babylonische Mondrechunce* (Freiburg im Breisgau, 1900), pp. 4-46. On the difference between the maximum lunar equations of Ptolemy and Hipparchus see G.J. Toomer, “The Size of the Lunar Epicycle According to Hipparchus, “ in Centaurus, **12** (1967), 145-150. On Hipparchus’ determination of the lunar and solar distances see N. Swerdlow, “Hipparchus on the Distance of the Sun, “ in Centaurus, **14** (1969), 287-305. Information on the earliest mappings of the path of a solar eclipse is in Jerome le Francais (Lalande), Astronomie, 3rd ed., **II** (Paris, 1792), 357 ff. Augustinus Ricius gave the correct explanation of the error in Ptolemy’s determination of the amount of precession in his *De motu octavae shaerae*, 2nd ed. (Paris, 1521), f. 39. The same explanation was given independently by P.S. Laplace in his Exposition du systeme du monde, 4th ed. (Paris, 1813), p.383.

A critical trans. of Ptolemy’s star catalog containing detailed comparisons with modern computed positions was made by C. H. F. Peters and E.B. Knobel, *Ptolemy’s Catalogue of Stars, a Revision of the Almagest* (Washington, D.C., 1915). This adopts Dolamber’ erroneous conclusion about Ptolemy’ dependence on the (hypothetical) star catalog of Hipparchus, which was refuted by H. Vogt in “Versuch einer Widerherstllung von Hipparchs Fixstern verzeichnis,“in Astronomische Nachrichten, **224** (1925), 17-54. This star catalog of al-Sufi was published by the Osmania Oriental Publications Bureau : Abu’l-Husayn ’Abd al-Rahman al-Sufi,Suwaru’ l-Kawakib (Hyderbad Deccan, 1954).A French trnch teans. was mand by H. C. F. C. Schjellerup, *Description des etoiles fixes* (St. Petersburg, 1874). The star list Zij al-Muntahan, containing 24 items, was puplised in J.Verned, “Las “Tabulae probatae,” in Homenaje a Millas -Vallicrosa, **II** (Barceloona, 1956), For derivatives from this list see P. Kunitzsch, “Die arabische Herkunft von zwei Sternverzeichnissen,” Gesellschaft,**120** (1970), 218-287.

A description of the Ptolemaic system is given by O. Neugebauer, *The Exact Sciences in Antiquity*, 2nd ed. (Providence, R.I., 1957), pp. 191-206. This should be consulted for such omissions from my account as the details of the Mercury model. Literature comparing Ptolemaic with modern theory will be found ibid., pp. 182-183. A good mathematical presentation of Ptolemy’ theory of celestial motion is given by N. Herz, Geschichte *der Behnbestimmung von Planeten und Kometen,***I** (Leipzig, 1887), 86-169. A mathematical comparison between the Kepler by M.Caspar in the intro. to his trans. of Kepler’s Astronomia nova: Johannes Kepler, Neue Astronomie (Munich-Berlin, 1929). For an analysis of the problem of determining the eccentricity from three observations in an equant model see G. W. Hill, “Ptolemy’ Problem,” in Astronomical Journal, **21** (1900), 33-35. An account of the astronomical instrument described, is given by D.J.Price in the Almagest, and also of the astrology, C. Singer and E.J.Holmyard, eds., III (Oxford, 1957), 582-609.

For details of the Arabic translations of the Almagest and other works of Ptolemy see M. Steinschneider, Die *arabischen Ubersetzungen aus dem Griechishen*(Graz, 1960), (191)-(211). An extensive work by P.Kunitzsch on the history of the Almagest in Arabic and Latin translation is Der Almagest— Die Syntaxis Mathematica des Claudius Ptolemaus in arabische-lateinischer Uberlieferung (Wiesbaden,1974). The astronomical treatise of al-Battani was published in a masterly ed, by C.A. Nallino, “Al-Battani sive Albatentenii Opus astronomicum,” Pubblicazioni del R. Osservatorio astronomico di Brera in Milano, **40,** 3 pts. (1899-1907). An account of the “improvements” to the Ptolemaic system made by al-Tusi and other late Islamic astronomers, with references to further literature, is given by E. S. Kennedy in The Cambridge History of Iran, V (Cambridge, 1968), 668-670. The earliest Latin version of the Almagest is discussd in C. H. Haskins, Studise in the History of Mediaeval Scienaval Science (Cambridge, Mass., 1924 repr. New York 1960), 157-165. Other versions are mentioned ibid., pp. 103-112. An ed. of Copernicus major work was published by F. and C. Zeller, Nikolaus Kopernikus Gesamtausgabe, 2 vols. (Munich, 1944-1949) — vol.**I** is a facsimile of the MS of the De revolutionibus in Copernicus’ own hand; vol, **II** an unreliable “critical” ed of the text. Brahe’ star catalog of 1598 (published only after his death) is in Tychonis Brahe Opera omnia, J.L.E. Dreyer, ed., **III** (Copenhagen, 1916), 335-389. For details of Johannes Kepler, Gesammelte Werke, W. von Dyck and Caspar, eds. (Munich, 1937).

The remains of Pappus’ commentary on the Almagest were published by A. Rome, Commentary on the Almagest were published by A. Rome, *Commentaires de Pappus et de Thieon d’ Alexandrie sur l’ Almageste*, I (Rome, 1931), Studie Testi, no. 54. He also published Theon’s commentary to Bks. I and II, that to Bks. III and IV, in same series: no. 72 (1936)and no. 106 (1943). For the remainder of Theon’ commentary one must still consult the editio princeps of the Greek text of the Almagest: Claudii Ptolemaei Magnae constuctionis . . . lib. xiii, Theonis Alexandrini in eosdem commentariorum lib. xi (Basel, 1538).

Handy Tables: The only printed printed version — except for Ptolemy’ intro., which is on pp. 159-185 of Heiberg’ ed. of *Claudii Ptolemaei Opera quae exsttant omnia,***II** Opera astronomical minora (Leipzig, 1907) — is the execrable ed. by N. Halma, Tables manuelles astronomiques de Ptolemee et de Theon, 3 pts. (Paris, 1822-1825). Pt. I contains Theon’s
smaller commentary on the tables (a larger one exists only in MS). B.L. van der Waerdern makes some useful remarks on the parameters and use of the tables in “Bermerkuongen zu den Handlichen Tageln des Ptolemaios,” in *Sitzungs berichte der Bayerischen Akademie der Wissenschaften zu Munchen*, Math.-nat. Kl., 23 (1953), 261-272; and “Die Handlichen Tafeln des Ptolemaios,” in *Osiris*, 13 (1958), 54-78.

*Planetary Hypotheses:* The Greek text was published by Heiberg in Ptolemy’ *Opera astormonomoica minora*, pp. 70-106, together with a a German trans. of the Arabic by L.Nix, *ibid*., pp. 71-145. By a strange oversight this omits the end of Bk.I(missing from the Greek), which contains Ptolemy’s theory of the absolute distances of the heavently bodies; and thus Ptolemy’s authorship of the latter theory remained unknown in modern times until the complete Arbic text(with English trans. of the part omitted by Nix) was published by B. Goldstein, “The Arabic Version of Ptolemy’s Planetary Hypotheses,” in *Transactions of the American Philosophical Society*, n.s. 57, no. 4(1967), 3-55. For an account of that theory and its medieval developments see N. Swerdlow, “Ptolemy’s Theory of the Distances and Sizes of the Planets” Ph.D. thesis (Yale University, 1968). The text through which it reached the West was published, in its most widespread version, by F. J. Carmody, “Alfragani differentie in quibusdam collectis scientie astrorum” (Berkeley, 1943), multigraphed.

*Phaseis, Analemma, Planishaerium:* The texts are printed in *Opera astronomica minora*, pp. 3-67, 189-223, and 227-259, respectively. On the *Phaseis* see H. Vogt, “Der Kalender des Claudius Prolemaus (Griechische Kalender, ed. F. Boll, V)” in SB Heridelberg. Akad. d. Wiss., in Phil.-hist. k1. (1920), 15. Because of its excellent commentary, the *editio princeps of the liber de Analemmate a Federico Commandino Urbinate instruratus* (Rome, 1562). See especialy P. Luckey. “Das Analemma von Ptolemaus,” in Astronomische Nachichten, 230(1927), 17-46. C\Vitruvius’ description ofo an *Analemma* construction is in his De architectura IX, 7, V. Rose, ed. (Leipzig, 1899), pp. 230-233. The Arabic text of the *Planisphaerium* has never been printed (the *Opera astronomica minora* contains only the medieval Latin trans.); but it exists, in a version prior to Maslama’s revision, in MS Istanbul, Aya Sofya 2671 _{3}. There are related texts in MO Paris, Bibilotheque Nationale Ar, 4821, 69v ff.; see G. Vajda, “Quelques note sur le fornds des manuscrits arabes de la Bibliotheque nationale,” in *Rivista degli studi orientali*, 25 (1950), 7-9. On the relevance of the *Planisphaerium* to the astrolabe see O. Neugebauer, “The Early History of the Astrobale,” in *Isis*, 40 (1949), 240-256. The work of Synesius referring to Hipparchus and the astrolabe is in *Synesii Cyrenensis Opuscula*, N. Terzaghi, ed. (Rome, 1944), pp. 132-142.

*Tetrabiblos:* Critical text by F. Boll and A. Boer, *Claudii Ptolemaei Opera quae exastant omina*, III, 1 AIIOTEAEEMATIKA(Leipzig, 1957). There is a text with English trans. by F, E. Robbins in Loeb Classical Library (Cambridge, Mass., 1940). On various aspects of the *Tetrabiblos, Boll, “Studien,” pp. 111-218, is useful. For the content A. Bouche-Leclereq, L’astrologie grecque* (Paris, 1899; repr. Brussels, 1963), remain unsurpassed.

*Geography*: There is no complete modern ed. For a complete text one must still use C. F. A. Nobbe, *Claudii Ptolemaei Geographia*, 2 vols. (Leipzig 1843-1845; repr. in 1 vol., Hildesheim, 1966); but more satisfactory for the parts theyo contain are the eds. by F. W. Wilberg and and by C.Muller, 2 vols (Paris, 1883-1845), Bks. I-VI; and by C. Muller, 2 vols. (Paris, 1883-1901), Bks. I-V. For other partial eds. see the article by E. Polaschek in Pauly Wissowa (See below) and W.H. Stahl, *Ptolemy’s Geography: A Select Bibliography* (New York, 2953). A German trans. of Bk. I by H. von Mzik, *Des Klaudios ptolemaios Einfuhrung in die darstellende Erdkunde*, Klotho 5 (Vienna, 193), contains a good study of the projections by F. Hopfner, pp. 87-105. The geographical work of alKhwarizmi was edited by H. vn Mzik, *Das Kitab Surat al-Ard des abu Ga far Muhammad ibn Musa al-huwarizmi* (Leipzig, 1926). C.A. Nallino gave a classic account of its relationship to Ptolemy in “Al-Khuwarizmi e il suo rifacimento della Geografia di Tolomeo,” in *Atti dell’ Accademia nazionale dei Lincei. Memorie*, Classe di Scienze Morali, Storiche e Fiologiche, 5th ser., 2, no. 1 (1894), 3-53, repr. in his *Raccolta di scritti editi e inediti*, V(Rome, 1944), 458-532. Excellent reproductions of maps found in MSS of the *Geography* were published in *Clauddi Ptolemaei Geographiae codex Urbinas graecus 82 phototypice odepictus*, pts. I, 2, and II (Leiden-Leipzig, 1932). Pt. I, I. by J. Fischer, contains (pp 290-415) an account of early European cartographic publications influenced by the *Geography*. Bernardus Sylvanus’ map is at the end of his *Claudii Ptholemaei Alexandrini liber geographiae* (Venice, 1511).

*Optics*: This was first printed by G. Govi, L’ *Ottica di Claudio Tolomeo*(Turin, 1885), A. Lejeune provided an excellent critical ed.: L’Optique de Clude Prolemee, Universite de Louvain, Receuil de travauz d’ *histoire et de philologie*, 4th ser., fasc. 8 (Louvain, 1956). The same author gave a good analysis of most parts of the work in two publications: *Euclide et Ptolemme, deux stades de l’optique geometrique grecque, ibid*, 3rd ser., fasc. 31 (Louvain, 1948); and “Recherches sur la catoptrique grecque,” in *Memoires de l’Academie r. de Belgique*. *Classe des sciences*. 52, no. 2(1957). Euclid’s *Optics* and the pseude-Euclidean *Catoptrics* are in *Euclidis opera omnia*, J.L. Heiberg, ed., VII(Leipzig, 1895). The latin trans. of Heroo’ *Catoptrics was edited by W. Schmidt in Heronis Alexandrini Opera quae supersunt omnia*, II, I (Leipzig, 1900). The Arabic text of Ibn al-Haytham’s *Optics* has never been printed (an English trans. and ed. are being prepared by A. Sabra). The Latin trans. is in *Opticae thesaurus*, F.Risner, ed. (Basel, 1572), which also contains Witelo’s optical work. For *Ibn al-Haythams Weg zur Physik*, which is *Boethius*, I (Wiesbaden, 1963). References to Ptolemy’s *Optics* by Roger Bacon are found in several part of his *Opus maius*, J.H. Bridges, ed., 2 vols. (Oxford, 1900), sec index under “Ptolemaeus.”

*Harmonica:* Text published by I. During, “Die Harmo
nielehre des Klaudios Ptolemaios,”which is Guteborgs *hogskolas arskrift***36** no.1 (1930). The same author published Porphry’s commentary, ibid., **38,** no. 2 (1932); and a German trans. of Ptolemy’s work, with commentary on both works, “Ptoklemaios und Porphyrios uber die Musik,“ibid., **40** no. 1 (1934). Boethius’ Dee *institutione musica* was edited by G. Friedlein together with his De *institutione arithmetica* (Leipzig, 1877; repr. Frankfurt, 1966). Kepler’s *Harmonice mundi* was published as vol. **VI** of his Gesamelte Werke by M. Caspar (Munich, 1940).

*Doubtful, spurious, and lost works: περὶ κριτηρίоυ καὶη̒γεμоνικоυ̑* was published by F. Lamert in Claudu *Prolemaei Opera quae exstant omina*, **III** 2 (Leipzig, 1961). The same vol. contains an ed. of the *Καρπо́ς* by A. Boer. On Ptolemy’s philosophical position see Bol, “Studien,“p.p. 66-111. Fragments and testimonia to the lost works are printed in Heiberg’s ed. of the *Opera astronomica minora*, p.p. 263-270.

II. Secondary Literature. General:**B.L.** van der Waerden et al., “Ptolemaios 66,” in Pauly-Wissowa, **XXIII,2** (Stutgart, 1959), 1788-1859, is a good guide. The supplementary article on the Geography by e. Polaschek, ibid., supp. X (1965), 680-83, is useful only for its bibliography.

Life: The evidence is assembled and discussed by F. Bol, “studien uber Claudius Ptolemaus,7rdquo; in *Jahrbucher fur classiche Philologie*, supp. **21** (1894), 53-6. This takes some note of the Arabic sources, which **I** omitted since they add nothing crediable to the Greek evidence. The only formal biographical notice (wretchedly incomplete) is in the tenth-century Byzantine lexicon of Suidas (“the Suda”), Suidae Lexicon, Ada Adler, ed., **IV** (Leipzig, 1935), **254,** no.3033. The “Canobic Inscription“is printed in Heiberg’s ed. of the *Opera astronomica minora*, pp. 149-15. The work of Olympiodorus was published by W. Norvin, *Olympiodori philosophi in Platonis Phaedonem comentaria* (Leipzig, 1913). The origin of the appelation “Phelud(i)ensis” was first correctly explained by **J.J.** Reiske in the German trans. of B. d’sHerbelor’s Bibliotheque orientale: Orientalische Bibliothek, **II** (Hale,1787), 375. It was thence repeated by Philip Buttmann, “Ueber den Ptolemaus in der Anthologie und den Klaudius Ptolemaus,” in*Museum der Alkterthums-Wissenschaft*, ,**2** (1810), 455-506. An exhaustive discussion is given by P. Kunitzsch, Der Almagest (above).

G. J. Toomer

# Ptolemy (or Claudius Ptolemaeus)

# 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, *AΠOTEΛEΣMATIKA*. 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*

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