Claudius Ptolemy - biography of the philosopher. G.E.

Astronomer Claudius Ptolemy, who worked in Alexandria in the 2nd century AD. e., summed up the work of ancient Greek astronomers, the main images of Hipparchus, as well as his own observations and built a perfect theory of planetary motion based on geocentric system of the world of Aristotle.

Claudius Ptolemy (Κλαύδιος Πτολεμαῖος , lat. Ptolemaeus), less often Ptolome (Πτολομαῖος, Ptolomaeus) (c. 87-c. 165) - ancient Greek astronomer, astrologer, mathematician, optician, music theorist and geographer. In the period from 127 to 151 he lived in Alexandria, where he carried out astronomical observations.

Despite the fact that Claudius Ptolemy is one of the largest figures in late Hellenistic astronomy, there is no mention of his life and work by contemporary authors.

The collection of astronomical knowledge of ancient Greece and Babylon, Ptolemy outlined in his work "The Great Construction", better known as Almagest(the Arabs brought his work to the Europeans, so it sounds in translation from Greek “megistos” - the greatest) - a work of 13 books.

In "Almagest" it is stated geocentric system of the world, according to which the Earth is at the center of the universe, and all celestial bodies revolve around it.

This model is based on mathematical calculations made by Eudoxus of Cnidus, Hipparchus, Apollonius of Perga and Ptolemy himself. And the astronomical tables of Hipparchus served as practical material, which, in addition to Greek observations, relied on the records of Babylonian astronomers.

Key provisions on which the Ptolemaic system is built

• The firmament is a rotating sphere.
• The earth is a sphere placed in the center of the world.
• The earth can be considered a point compared to the distance to the sphere of fixed stars.
• The earth is motionless.

Ptolemy confirms his position with experiments. Does not recognize other opinions and views.

About the movement of the luminaries

Each planet, according to Ptolemy, moves uniformly in a circle (epicycle), the center of which moves in another circle (deferent). This allows us to explain the apparent uneven motion of the planets and, to some extent, the change in their brightness.

For the Moon and planets, Ptolemy introduces additional deferents, epicycles, eccentrics and latitudinal oscillations of the orbits, as a result of which the position of all the luminaries was determined with an error that was negligible at that time - about 1 °. This ensured the reliability of the calculation of planetary ephemeris for a long time (stellar ephemeris - tables of apparent positions of stars). But according to Ptolemy's theory, the distance to the Moon and its apparent size should have changed greatly, which is not really observed. In addition, within the framework of geocentrism, it was inexplicable why the basic period of revolution along the first epicycle for the upper planets was exactly equal to a year and why Mercury and Venus never move far from the Sun, rotating around the Earth in synchronism with it.

Ptolemy considered the motion of the planet along the deferent to be uniform not with respect to the center of the deferent, but with respect to a special point symmetrical with the center of the Earth relative to the center of the deferent.

Star catalog

Ptolemy supplemented Hipparchus' star catalog; the number of stars in it is increased to 1022. Ptolemy apparently corrected the positions of the stars from the catalog of Hipparchus, taking for precession ( precession- a phenomenon in which the angular momentum of a body changes its direction in space under the action of an external force moment) an inaccurate value of 1˚ per century (the correct value is ~1˚ for 72 years).

Deviation of the Moon's motion

The Almagest contains a description of the phenomenon discovered by Ptolemy of the deviation of the moon's motion from an exact circular one. He gives the astrological characteristics of the so-called "fixed stars".

Ptolemy's astronomical instruments

The astronomical instruments used by Ptolemy are also described here: armillary sphere (astrolabon)- a tool for determining the ecliptic coordinates of celestial bodies, triquetrum to measure angular distances in the sky, diopter for measuring the angular diameters of the Sun and the Moon, quadrant and meridian circle to measure the height of the luminaries above the horizon, and the equinoctial ring to observe the time of the equinoxes

Mathematical problems for astronomical calculations

In the Almagest, some mathematical problems were solved that were of practical importance for astronomical calculations: a table of chords was built with a step of half a degree, a theorem on the properties of a quadrilateral, now known as Ptolemy's theorem (a circle can be circumscribed around a quadrilateral if and only if the product of its diagonals is equal to the sum of the products of its opposite sides).

Calculation methods of Ptolemy of Babylonian origin: sexagesimal fractions are used, the full angle is divided into 360 degrees, a special zero character is introduced for empty digits, etc.

For astronomical calculations, a mobile ancient Egyptian calendar with a fixed year length of 365 days is used.

Before the advent of the heliocentric system, the Almagest remained the most important astronomical work; Ptolemy's book was studied and commented on throughout the civilized world. In the 8th century it was translated into Arabic, and a century later it reached medieval Europe. The heliocentric system of the world of Ptolemy dominated astronomy until the 16th century, i.e. almost 15 centuries.

But his work was repeatedly criticized, and in 1977 the American physicist Robert Russell Newton published the book The Crime of Claudius Ptolemy, in which he accused Ptolemy of falsifying data, as well as of passing off Hipparchus' achievements as his own.

But scientists consider these accusations to be unfounded, since an analysis of the data presented by Ptolemy in the Almagest shows that a significant part of them, especially for the brightest stars, belongs to Ptolemy himself.

Other writings of Ptolemy

He wrote a treatise on music « Harmonic" , in which he created the theory of harmony, in a treatise "Optics" experimentally investigated the refraction of light at the air-water and air-glass interface and proposed his law of refraction (which is approximately valid only for small angles), for the first time correctly explained the apparent increase in the Sun and Moon on the horizon as a psychological effect. In the book "Tetrabook" Ptolemy summed up his statistical observations about the life expectancy of people: for example, a person aged 56 to 68 was considered old, and only after that he was considered old. In labor "Geography" he left a detailed guide to compiling an atlas of the world with the exact coordinates of each point.

Claudius Ptolemy occupies one of the most honorable places in the history of world science. His writings played a huge role in the development of astronomy, mathematics, optics, geography, chronology, and music. The literature dedicated to him is truly enormous. And at the same time, his image to this day remains unclear and contradictory. Among the figures of science and culture of bygone eras, one can hardly name many people about whom such contradictory judgments would be expressed and such fierce disputes among specialists as about Ptolemy.

This is explained, on the one hand, by the most important role played by his works in the history of science, and on the other hand, by the extreme scarcity of biographical information about him.

Ptolemy owns a number of outstanding works in the main areas of ancient natural science. The largest of them, and the one that left the greatest mark on the history of science, is the astronomical work published in this edition, usually called the Almagest.

Almagest is a compendium of ancient mathematical astronomy, which reflects almost all of its most important areas. Over time, this work supplanted the earlier works of ancient authors on astronomy and thus became a unique source on many important issues in its history. For centuries, until the era of Copernicus, the Almagest was considered a model of a strictly scientific approach to solving astronomical problems. Without this work, it is impossible to imagine the history of medieval Indian, Persian, Arabic and European astronomy. The famous work of Copernicus "On rotations", which marked the beginning of modern astronomy, was in many respects a continuation of the "Almagest".

Other works of Ptolemy, such as "Geography", "Optics", "Harmonics", etc., also had a great influence on the development of the relevant areas of knowledge, sometimes no less than the "Almagest" on astronomy. In any case, each of them marked the beginning of a tradition of exposition of a scientific discipline, which has been preserved for centuries. In terms of the breadth of scientific interests, combined with the depth of analysis and the rigor of the presentation of the material, few people can be placed next to Ptolemy in the history of world science.

However, Ptolemy paid the most attention to astronomy, to which, in addition to the Almagest, he devoted other works. In "Planetary Hypotheses" he developed the theory of planetary motion as an integral mechanism within the framework of the geocentric system of the world adopted by him, in "Handy Tables" he gave a collection of astronomical and astrological tables with explanations necessary for a practicing astronomer in his daily work. Special treatise "Tetrabook" in which also great importance attached to astronomy, he dedicated to astrology. Several of Ptolemy's writings are lost and known only by their titles.

Such a variety of scientific interests gives full reason to classify Ptolemy among the most prominent scientists known to the history of science. World fame, and most importantly, the rare fact that his works for centuries were perceived as timeless sources of scientific knowledge, testify not only to the breadth of the author’s outlook, the rare generalizing and systematizing power of his mind, but also to the high skill of presenting the material. In this regard, the writings of Ptolemy, and above all the Almagest, have become a model for many generations of scholars.

Very little is known about the life of Ptolemy. The little that has been preserved in ancient and medieval literature on this issue is presented in the work of F. Boll. The most reliable information concerning the life of Ptolemy is contained in his own writings. In the Almagest, he gives a number of his observations, which date back to the era of the reign of the Roman emperors Hadrian (117-138) and Antoninus Pius (138-161): the earliest - March 26, 127 AD, and the latest - February 2 141 AD In the Canopic Inscription dating back to Ptolemy, in addition, the 10th year of the reign of Antoninus is mentioned, i.e. 147/148 AD Trying to assess the limits of Ptolemy's life, it must also be borne in mind that after the Almagest he wrote several more large works, various in subject matter, of which at least two ("Geography" and "Optics") are encyclopedic in nature, which, according to the most conservative estimate would have taken at least twenty years. Therefore, it can be assumed that Ptolemy was still alive under Marcus Aurelius (161-180), as reported by later sources. According to Olympiodorus, an Alexandrian philosopher of the 6th century. AD, Ptolemy worked as an astronomer in the city of Canope (now Abukir), located in the western part of the Nile Delta, for 40 years. This report, however, is contradicted by the fact that all of Ptolemy's observations given in the Almagest were made in Alexandria. The name Ptolemy itself testifies to the Egyptian origin of its owner, who probably belonged to the number of Greeks, adherents of the Hellenistic culture in Egypt, or descended from the Hellenized local inhabitants. The Latin name "Claudius" suggests that he had Roman citizenship. The ancient and medieval sources also contain a lot of less reliable evidence about the life of Ptolemy, which can neither be confirmed nor refuted.

Almost nothing is known about Ptolemy's scientific environment. "Almagest" and a number of his other works (except "Geography" and "Harmonics") is dedicated to a certain Cyrus (Σύρος). This name was quite common in Hellenistic Egypt during the period under review. We have no other information about this person. It is not even known whether he was engaged in astronomy. Ptolemy also uses planetary observations of a certain Theon (kn.ΙΧ, ch.9; book X, ch.1), made in the period 127-132. AD He reports that these observations were “left” to him by “the mathematician Theon” (book X, ch. 1, p. 316), which, apparently, suggests a personal contact. Perhaps Theon was Ptolemy's teacher. Some scholars identify him with Theon of Smyrna (first half of the 2nd century AD), a Platonic philosopher who paid attention to astronomy [HAMA, p.949-950].

Ptolemy undoubtedly had employees who helped him in making observations and calculating tables. The amount of calculations that needed to be done to build astronomical tables in the Almagest is truly enormous. In Ptolemy's time, Alexandria was still a major scientific center. It operated several libraries, the largest of which was located in the Alexandrian Museion. Apparently, personal contacts existed between the library staff and Ptolemy, as is often the case even now with scientific work. Someone helped Ptolemy in the selection of literature on issues of interest to him, brought manuscripts or led him to the shelves and niches where the scrolls were stored.

Until recently, it was assumed that the Almagest is the earliest extant astronomical work of Ptolemy. However, recent research has shown that the Canopic Inscription preceded the Almagest. Mentions of the "Almagest" are contained in the "Planetary Hypotheses", "Handy Tables", "Tetrabooks" and "Geography", which makes their later writing undoubted. This is also evidenced by the analysis of the content of these works. In the Handy Tables, many tables are simplified and improved compared to similar tables in the Almagest. The "Planetary Hypotheses" uses a different system of parameters to describe the movements of the planets and solves a number of issues in a new way, for example, the problem of planetary distances. In "Geography" the zero meridian is transferred to the Canary Islands instead of Alexandria, as is customary in the "Almagest". "Optics" was also created, apparently, later than "Almagest"; it deals with astronomical refraction, which does not play a prominent role in the Almagest. Since the "Geography" and "Harmonics" do not contain a dedication to Cyrus, it can be argued with a certain degree of risk that these works were written later than other works of Ptolemy. We have no other more precise landmarks that would allow us to chronologically record the works of Ptolemy that have come down to us.

To appreciate the contribution of Ptolemy to the development of ancient astronomy, it is necessary to clearly understand the main stages of its previous development. Unfortunately, most of the works of Greek astronomers relating to the early period (V-III centuries BC) have not come down to us. We can judge their content only from quotations in the writings of later authors, and above all from Ptolemy himself.

At the origins of the development of ancient mathematical astronomy are four features of the Greek cultural tradition, clearly expressed already in the early period: a penchant for philosophical understanding of reality, spatial (geometric) thinking, adherence to observations and the desire to harmonize the speculative image of the world and the observed phenomena.

In the early stages, ancient astronomy was closely connected with the philosophical tradition, from where it borrowed the principle of circular and uniform motion as the basis for describing the apparent uneven movements of the luminaries. The earliest example of the application of this principle in astronomy was the theory of homocentric spheres by Eudoxus of Cnidus (c. 408-355 BC), improved by Callippus (4th century BC) and adopted with certain changes by Aristotle (Metaphys. XII, 8).

This theory has qualitatively reproduced the features of the movement of the sun, moon and five planets: the daily rotation of the heavenly sphere, the movement of luminaries along the ecliptic from west to east various speeds, changes in latitude and backward motions of the planets. The movements of the luminaries in it were controlled by the rotation of the heavenly spheres, to which they were attached; The spheres turned around a single center (center of the world), coinciding with the center of the fixed land, had the same radius, zero thickness and were considered consisting of the air. Visible changes in the shine of the luminaries and the related changes in their distances regarding the observer within the framework of this theory could not get a satisfactory explanation.

The principle of circular and uniform movement was also successfully applied in the sphere - the section of ancient mathematical astronomy, in which the tasks associated with the daily rotation of the heavenly sphere and its most important circles, primarily equator and ecliptic, sunrises and sunset, zodiac signs regarding the horizon at various latitudes were solved. . These problems were solved using the methods of spherical geometry. In the previous Ptolemy time, a number of treatises appeared on the sphere, including the autolic (c. 310 BC), Euclid (second half of the 4th century BC), theodosius (second half of the 2nd century BC. AD), Hypsicles (II century BC), Menelaus (I century AD) and others [Matvievskaya, 1990, p.27-33].

An outstanding achievement of ancient astronomy was the theory of the heliocentric movement of the planets, proposed by Aristarchus by Samo's (c. 320-250 BC). However, this theory, as far as our sources allow us to judge, did not have any noticeable influence on the development of mathematical astronomy itself, i.e. It did not lead to the creation of an astronomical system, which has not only philosophical, but also practical significance and allows you to determine the positions of the luminaries in the sky with the necessary degree of accuracy.

an important step The invention of the excenters and epicycles, which made it possible to qualitatively explain at the same time, on the basis of uniform and circular movements, the observed uneven movement of the luminaries and changes in their distances relative to the observer, was the invention. The equivalence of the epicyclic and eccentric models for the case of the Sun proved Apollonius Pergian (III-II centuries BC). He also applied an epicyclic model to explain the backward movements of the planets. New mathematical means made it possible to move from qualitative to a quantitative description of the movements of the luminaries. For the first time, apparently, this problem was successfully solved by Hipparchus (II century BC). He created on the basis of eccentric and epicyclic models of the theory of the movement of the sun and moon, which made it possible to determine their current coordinates for any point in time. However, he was not able to develop a similar theory for planets due to the lack of observations.

Gypparech also owns a number of other outstanding achievements in astronomy: opening a precession, creating a star catalog, measuring lunar parallax, determining distances to the sun and moon, developing the theory of lunar eclipses, the design of astronomical instruments, in particular the armillary sphere, conducting a large number of observations that did not lose partly of its significance to the present day, and much more. The role of Hipparchus in the history of ancient astronomy is truly enormous.

Making observations was a special trend in ancient astronomy long before Hipparchus. In the early period, observations were mainly qualitative in nature. With the development of kinematic-geometric modeling, observations are mathematized. The main purpose of observations is to determine the geometric and velocity parameters of the accepted kinematic models. At the same time, astronomical calendars are being developed that allow fixing the dates of observations and determining the intervals between observations on the basis of a linear uniform time scale. When observing, the positions of the luminaries were fixed relative to the selected points of the kinematic model at the current moment, or the time of passage of the luminary through the selected point of the scheme was determined. Among such observations: determining the moments of equinoxes and solstices, the height of the Sun and Moon when passing through the meridian, the temporal and geometric parameters of eclipses, the dates of the Moon's coverage of stars and planets, the positions of the planets relative to the Sun, Moon and stars, the coordinates of stars, etc. The earliest observations of this kind date back to the 5th century BC. BC. (Meton and Euctemon in Athens); Ptolemy was also aware of the observations of Aristillus and Timocharis, made in Alexandria at the beginning of the 3rd century. BC, Hipparchus on Rhodes in the second half of the II century. BC, Menelaus and Agrippa, respectively, in Rome and Bithynia at the end of the 1st century. BC, Theon in Alexandria at the beginning of the 2nd century. AD At the disposal of Greek astronomers there were also (already, apparently, in the 2nd century BC) the results of observations of Mesopotamian astronomers, including lists of lunar eclipses, planetary configurations, etc. The Greeks were also familiar with the lunar and planetary periods, accepted in the Mesopotamian astronomy of the Seleucid period (IV-I centuries BC). They used this data to test the accuracy of the parameters of their own theories. Observations were accompanied by the development of theory and the construction of astronomical instruments.

A special direction in ancient astronomy was the observation of stars. Greek astronomers identified about 50 constellations in the sky. It is not known exactly when this work was done, but by the beginning of the 4th century. BC. it was, apparently, already completed; there is no doubt that the Mesopotamian tradition played an important role in this.

Descriptions of constellations constituted a special genre in ancient literature. The starry sky was depicted clearly on celestial globes. Tradition associates the earliest samples of this kind of globes with the names of Eudoxus and Hipparchus. However, ancient astronomy went much further than simply describing the shape of the constellations and the arrangement of the stars in them. An outstanding achievement was the creation by Hipparchus of the first stellar catalog containing the ecliptic coordinates and brightness estimates of each star included in it. The number of stars in the catalog, according to some sources, did not exceed 850; according to another version, it included about 1022 stars and was structurally similar to Ptolemy's catalog, differing from it only in the longitudes of the stars.

The development of ancient astronomy took place in close connection with the development of mathematics. The solution of astronomical problems was largely determined by the mathematical means that astronomers had at their disposal. A special role in this was played by the works of Eudoxus, Euclid, Apollonius, Menelaus. The appearance of the Almagest would have been impossible without the previous development of logistics methods - a standard system of rules for performing calculations, without planimetry and the basics of spherical geometry (Euclid, Menelaus), without plane and spherical trigonometry (Hipparchus, Menelaus), without the development of methods for kinematic-geometric modeling movements of the luminaries using the theory of eccentres and epicycles (Apollonius, Hipparchus), without developing methods for setting functions of one, two and three variables in tabular form (Mesopotamian astronomy, Hipparchus?). For its part, astronomy directly influenced the development of mathematics. Such, for example, sections of ancient mathematics as trigonometry of chords, spherical geometry, stereographic projection, etc. developed only because they were given special importance in astronomy.

In addition to geometric methods for modeling the movements of the stars, ancient astronomy also used arithmetic methods of Mesopotamian origin. Greek planetary tables have come down to us, calculated on the basis of Mesopotamian arithmetic theory. The data of these tables were apparently used by ancient astronomers to substantiate the epicyclic and eccentric models. In the time preceding Ptolemy, approximately from the 2nd century BC. BC, a whole class of special astrological literature became widespread, including lunar and planetary tables, which were calculated based on the methods of both Mesopotamian and Greek astronomy.

Ptolemy's work was originally entitled Mathematical Work in 13 Books (Μαθηματικής Συντάξεως βιβλία ϊγ). In late antiquity, it was referred to as the "great" (μεγάλη) or "greatest (μεγίστη) work", as opposed to the "Small Astronomical Collection" (ό μικρός αστρονομούμενος) - a collection of small treatises on the sphere and other sections of ancient astronomy. In the ninth century when translating the “Mathematical Essay” into Arabic, the Greek word ή μεγίστη was reproduced in Arabic as “al-majisti”, from which the currently generally accepted Latinized form of the name of this work “Almagest” comes from.

The Almagest consists of thirteen books. The division into books undoubtedly belongs to Ptolemy himself, while the division into chapters and their titles were introduced later. It can be stated with certainty that during the time of Pappus of Alexandria at the end of the 4th century. AD this kind of division already existed, although it differed significantly from the current one.

The Greek text that has come down to us also contains a number of later interpolations that do not belong to Ptolemy, but were introduced by scribes for various reasons [RA, p.5-6].

The Almagest is a textbook mainly of theoretical astronomy. It is intended for the already prepared reader familiar with Euclid's geometry, spherics and logistics. The main theoretical problem solved in the Almagest is the prediction of the apparent positions of the luminaries (the Sun, the Moon, planets and stars) on the celestial sphere at an arbitrary moment in time with an accuracy corresponding to the possibilities of visual observations. Another important class of problems solved in the Almagest is the prediction of dates and other parameters of special astronomical phenomena associated with the movement of the stars - lunar and solar eclipses, heliacal risings and setting of planets and stars, determination of parallax and distances to the Sun and Moon and etc. In solving these problems, Ptolemy follows a standard methodology that includes several steps.

1. On the basis of preliminary rough observations, the characteristic features in the motion of the star are clarified and a kinematic model is selected that best suits the observed phenomena. The procedure for choosing one model from several equally possible ones must satisfy the "principle of simplicity"; Ptolemy writes about this: “We consider it appropriate to explain the phenomena with the help of the simplest assumptions, unless the observations contradict the hypothesis put forward” (book III, ch. 1, p. 79). Initially, the choice is made between a simple eccentric and a simple epicyclic model. At this stage, questions are being resolved about the correspondence of the circles of the model to certain periods of the movement of the luminary, about the direction of movement of the epicycle, about the places of acceleration and deceleration of movement, about the position of the apogee and perigee, etc.

2. Based on the adopted model and using observations, both his own and those of his predecessors, Ptolemy determines the periods of motion of the luminary with the maximum possible accuracy, the geometric parameters of the model (radius of the epicycle, eccentricity, longitude of the apogee, etc.), the moments of passage of the luminary through the selected points of the kinematic scheme to tie the movement of the star to the chronological scale.

This technique works most simply when describing the motion of the Sun, where a simple eccentric model is sufficient. In studying the motion of the moon, however, Ptolemy had to modify the kinematic model three times in order to find such a combination of circles and lines that would best fit the observations. Significant complications also had to be introduced into the kinematic models for describing the motions of the planets in longitude and latitude.

A kinematic model that reproduces the movements of the luminary must satisfy the "principle of uniformity" of circular motions. “We believe,” writes Ptolemy, “that for a mathematician the main task is ultimately to show that celestial phenomena are obtained with the help of uniform circular motions” (book III, ch. 1, p. 82). This principle, however, is not strictly followed. He refuses it every time (without, however, explicitly stipulating this) when observations require it, for example, in the lunar and planetary theories. Violation of the principle of uniformity of circular motions in a number of models later became the basis for criticism of the Ptolemaic system in the astronomy of the countries of Islam and medieval Europe.

3. After determining the geometric, velocity and time parameters of the kinematic model, Ptolemy proceeds to the construction of tables, with the help of which the coordinates of the luminary at an arbitrary moment of time should be calculated. Such tables are based on the idea of ​​a linear homogeneous time scale, the beginning of which is taken to be the beginning of the era of Nabonassar (-746, February 26, true noon). Any value recorded in the table is the result of complex calculations. Ptolemy at the same time shows a virtuoso mastery of Euclid's geometry and the rules of logistics. In conclusion, rules for using tables are given, and sometimes also examples of calculations.

The presentation in the Almagest is strictly logical. At the beginning of book I, general questions concerning the structure of the world as a whole, its most general mathematical model, are considered. It proves the sphericity of the sky and the Earth, the central position and immobility of the Earth, the insignificance of the size of the Earth compared to the size of the sky, two main directions in the celestial sphere are distinguished - the equator and the ecliptic, parallel to which the daily rotation of the celestial sphere and periodic movements of the luminaries occur, respectively. The second half of Book I deals with chord trigonometry and spherical geometry, methods for solving triangles on a sphere using Menelaus' theorem.

Book II is entirely devoted to questions of spherical astronomy, which do not require knowledge of the coordinates of the luminaries as a function of time for their solution; it considers the tasks of determining the times of sunrise, sunset and passage through the meridian of arbitrary arcs of the ecliptic at different latitudes, the length of the day, the length of the shadow of the gnomon, the angles between the ecliptic and the main circles of the celestial sphere, etc.

In book III, a theory of the motion of the Sun was developed, which contains the definition of the duration of the solar year, the choice and justification of the kinematic model, the determination of its parameters, the construction of tables for calculating the longitude of the Sun. The final section explores the concept of the equation of time. The theory of the Sun is the basis for studying the motion of the Moon and stars. The longitudes of the Moon at the moments of lunar eclipses are determined from the known longitude of the Sun. The same goes for determining the coordinates of stars.

Books IV-V are devoted to the theory of the motion of the Moon in longitude and latitude. The motion of the Moon is studied approximately in the same way as the motion of the Sun, with the only difference that Ptolemy, as we have already noted, successively introduces here three kinematic models. An outstanding achievement was the discovery by Ptolemy of the second inequality in the motion of the moon, the so-called evection, associated with the location of the moon in quadratures. In the second part of book V, the distances to the Sun and Moon are determined and the theory of solar and lunar parallax is constructed, which is necessary for predicting solar eclipses. Parallax tables (book V, ch.18) are perhaps the most complex of all that are contained in the Almagest.

Book VI is devoted entirely to the theory of lunar and solar eclipses.

Books VII and VIII contain a stellar catalog and deal with a number of other fixed star issues, including the theory of precession, the construction of a celestial globe, heliacal rising and setting of stars, and so on.

Books IX-XIII set out the theory of planetary motion in longitude and latitude. In this case, the motions of the planets are analyzed independently of each other; movements in longitude and latitude are also considered independently. When describing the movements of the planets in longitude, Ptolemy uses three kinematic models, differing in detail, respectively for Mercury, Venus and the upper planets. They implement an important improvement known as the equant, or eccentricity bisector, that improves the accuracy of planetary longitudes by about three times over the simple eccentric model. In these models, however, the principle of uniformity of circular rotations is formally violated. Kinematic models for describing the motion of planets in latitude are particularly complex. These models are formally incompatible with the kinematic models of motion in longitude accepted for the same planets. Discussing this problem, Ptolemy expresses several important methodological statements that characterize his approach to modeling the movements of the stars. In particular, he writes: “And let no one ... consider these hypotheses too artificial; one should not apply human concepts to the divine ... But to celestial phenomena one should try to adapt as simple as possible assumptions ... Their connection and mutual influence in various movements seem to us very artificial in the models we arrange, and it is difficult to make sure that the movements do not interfere each other, but in the sky none of these movements will meet with obstacles from such a connection. It would be better to judge the very simplicity of heavenly things not on the basis of what seems to us so ... ”(book XIII, ch. 2, p. 401). Book XII analyzes the backward motions and magnitudes of the maximum elongations of the planets; at the end of book XIII, the heliacal risings and setting of the planets are considered, which require, for their determination, knowledge of both the longitude and latitude of the planets.

The theory of planetary motion, set forth in the Almagest, belongs to Ptolemy himself. In any case, there are no serious grounds indicating that anything like this existed in the time preceding Ptolemy.

In addition to the Almagest, Ptolemy also wrote a number of other works on astronomy, astrology, geography, optics, music, etc., which were very famous in antiquity and the Middle Ages, including:

"Kanope inscription",

"Handy tables",

"Planet Hypotheses"

"Analemma"

"Planispherium"

"Tetrabook"

"Geography",

"Optics",

"Harmonics", etc. For the time and order of writing these works, see section 2 of this article. Let's briefly review their content.

The Canopic Inscription is a list of the parameters of the Ptolemaic astronomical system, which was carved on a stele dedicated to the Savior God (possibly Serapis) in the city of Canope in the 10th year of the reign of Antoninus (147/148 AD). The stele itself has not survived, but its contents are known from three Greek manuscripts. Most of the parameters adopted in this list coincide with those used in the Almagest. However, there are discrepancies not related to scribal errors. The study of the text of the Canopic Inscription showed that it dates back to a time earlier than the time of the creation of the Almagest.

"Handy Tables" (Πρόχειροι κανόνες), the second largest after the "Almagest" astronomical work of Ptolemy, is a collection of tables for calculating the positions of the stars on the sphere at an arbitrary moment and for predicting some astronomical phenomena, primarily eclipses. The tables are preceded by Ptolemy's "Introduction" which explains the basic principles of their use. "Hand-tables" have come down to us in the arrangement of Theon of Alexandria, but it is known that Theon changed little in them. He also wrote two commentaries on them - the Great Commentary in five books and the Small Commentary, which were supposed to replace Ptolemy's Introduction. "Handy tables" are closely related to the "Almagest", but also contain a number of innovations, both theoretical and practical. For example, they adopted other methods for calculating the latitudes of the planets, a number of parameters of kinematic models were changed. The era of Philip (-323) is taken as the initial era of the tables. The tables contain a star catalog, including about 180 stars in the vicinity of the ecliptic, in which longitudes are measured sidereal, with Regulus ( α Leo) is taken as the origin of sidereal longitude. There is also a list of about 400 "Most Important Cities" with geographical coordinates. The "Handy Tables" also contains the "Royal Canon" - the basis of Ptolemy's chronological calculations (see Appendix "Calendar and Chronology in the Almagest"). In most tables, the values ​​of functions are given with an accuracy of minutes, the rules for their use are simplified. These tables had an undeniably astrological purpose. In the future, "Hand-held tables" were very popular in Byzantium, Persia and in the medieval Muslim East.

"Planetary hypotheses" (Ύποτέσεις τών πλανωμένων) _ small, but having importance in the history of astronomy, the work of Ptolemy, consisting of two books. Only part of the first book has survived in Greek; however, a complete Arabic translation of this work, belonging to Thabit ibn Koppe (836-901), has come down to us, as well as a translation into Hebrew of the 14th century. The book is devoted to the description of the astronomical system as a whole. The "planetary hypotheses" differ from the "Almagest" in three respects: a) they use a different system of parameters to describe the movements of the luminaries; b) simplified kinematic models, in particular, a model for describing the motion of planets in latitude; c) the approach to the models themselves has been changed, which are considered not as geometric abstractions designed to “save phenomena”, but as parts of a single mechanism that is physically implemented. The details of this mechanism are built from ether, the fifth element of Aristotelian physics. The mechanism that controls the movements of the luminaries is a combination of a homocentric model of the world with models built on the basis of eccentrics and epicycles. The movement of each luminary (Sun, Moon, planets and stars) takes place inside a special spherical ring of a certain thickness. These rings are successively nested in each other in such a way that there is no room for emptiness. The centers of all rings coincide with the center of the motionless Earth. Inside the spherical ring, the luminary moves according to the kinematic model adopted in the Almagest (with minor changes).

In the Almagest, Ptolemy defines absolute distances (in units of the Earth's radius) only to the Sun and Moon. For planets, this cannot be done due to their lack of noticeable parallax. In The Planetary Hypotheses, however, he finds absolute distances for the planets as well, on the assumption that the maximum distance of one planet is equal to the minimum distance of the planet following it. The accepted sequence of the arrangement of the luminaries: Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, fixed stars. The Almagest defines the maximum distance to the Moon and the minimum distance to the Sun from the center of the spheres. Their difference closely corresponds to the total thickness of the spheres of Mercury and Venus obtained independently. This coincidence in the eyes of Ptolemy and his followers confirmed the correct location of Mercury and Venus in the interval between the Moon and the Sun and testified to the reliability of the system as a whole. At the end of the treatise, the results of determining the apparent diameters of the planets by Hipparchus are given, on the basis of which their volumes are calculated. "Planetary hypotheses" enjoyed great fame in late antiquity and in the Middle Ages. The planetary mechanism developed in them was often depicted graphically. These images (Arabic and Latin) served as a visual expression of the astronomical system, which was usually defined as the "Ptolemaic system."

The Phases of the Fixed Stars (Φάσεις απλανών αστέρων) is a small work by Ptolemy in two books devoted to weather predictions based on observations of the dates of synodic stellar phenomena. Only book II has come down to us, containing a calendar in which a weather prediction is given for each day of the year, assuming that on that day one of the four possible synodic phenomena occurred (heliacal rising or setting, acronic rising, cosmic setting). For example:

Thoth 1 141/2 hours: [star] in the tail of Leo (ß Leo) rises;

according to Hipparchus, the north winds are ending; according to Eudoxus,

rain, thunderstorm, northern winds end.

Ptolemy uses only 30 stars of the first and second magnitude and gives predictions for five geographical climates for which the maximum

The duration of the day changes from 13 1/2 H to 15 1/2 H through 1/2 H. Dates are given in the Alexandrian calendar. The dates of the equinoxes and solstices are also indicated (I, 28; IV, 26; VII, 26; XI, 1), which makes it possible to approximately date the time of writing the work as 137-138 years. AD Weather predictions based on star-rise observations seem to reflect a pre-scientific stage in the development of ancient astronomy. However, Ptolemy introduces an element of science into this not quite astronomical area.

"Analemma" (Περί άναλήμματος) is a treatise that describes a method for finding, by geometric construction in a plane, arcs and angles that fix the position of a point on a sphere relative to selected great circles. Fragments of the Greek text and a complete Latin translation of this work by Willem of Meerbeke (13th century AD) have survived. In it, Ptolemy solves the following problem: to determine the spherical coordinates of the Sun (its height and azimuth), if the geographic latitude of the place φ, the longitude of the Sun λ and the time of day are known. To fix the position of the Sun on the sphere, he uses a system of three orthogonal axes that form an octant. Relative to these axes, the angles on the sphere are measured, which are then determined in the plane by construction. The applied method is close to those currently used in descriptive geometry. Its main area of ​​application in ancient astronomy was the construction of sundials. An exposition of the content of the "Analemma" is contained in the writings of Vitruvius (On Architecture IX, 8) and Heron of Alexandria (Dioptra 35), who lived half a century earlier than Ptolemy. But although the basic idea of ​​the method was known long before Ptolemy, his solution is distinguished by a completeness and beauty that we do not find in any of his predecessors.

"Planispherium" (probable Greek name: "Άπλωσις επιφανείας σφαίρας) is a small work by Ptolemy devoted to the use of the theory of stereographic projection in solving astronomical problems. It has survived only in Arabic; the Spanish-Arabic version of this work, which belonged to Maslama al-Majriti (Χ-ΧΙ cc. . AD), was translated into Latin by Herman from Carinthia in 1143. The idea of ​​a stereographic projection is as follows: the points of a ball are projected from any point on its surface onto a plane tangent to it, while the circles drawn on the surface of the ball, pass into circles on the plane and the angles retain their magnitude.The basic properties of the stereographic projection were already known, apparently, two centuries before Ptolemy.In the Planisphere, Ptolemy solves two problems: celestial sphere and (2) determine the times of rising of the ecliptic arcs in the direct and oblique spheres (i.e., at ψ \u003d O and ψ ≠ O, respectively) purely geometrically. This work is also related in its content to the problems being solved at the present time in descriptive geometry. The methods developed in it served as the basis for the creation of the astrolabe, an instrument that played an important role in the history of ancient and medieval astronomy.

"Tetrabook" (Τετράβιβλος or "Αποτελεσματικά, i.e. "Astrological Influences") is the main astrological work of Ptolemy, also known under the Latinized name "Quadripartitum". It consists of four books.

In the time of Ptolemy, belief in astrology was widespread. Ptolemy was no exception in this regard. He sees astrology as a necessary complement to astronomy. Astrology predicts earthly events, taking into account the influence of heavenly bodies; astronomy provides information about the positions of the stars, necessary for making predictions. Ptolemy, however, was not a fatalist; he considers the influence of celestial bodies to be only one of the factors determining events on Earth. In works on the history of astrology, four types of astrology, common in the Hellenistic period, are usually distinguished - world (or general), genetlialogy, katarchen and interrogative. In the work of Ptolemy, only the first two types are considered. Book I gives general definitions of the basic astrological concepts. Book II is entirely devoted to world astrology, i.e. methods of predicting events concerning large earthly regions, countries, peoples, cities, large social groups etc. Here questions of so-called "astrological geography" and weather predictions are considered. Books III and IV are devoted to methods of predicting individual human destinies. The work of Ptolemy is characterized by a high mathematical level, which distinguishes it favorably from other astrological works of the same period. This is probably why the "Tetrabook" enjoyed great prestige among astrologers, despite the fact that it did not contain katarchen astrology, i.e. methods of determining the favorable or unfavorable moment for any case. During the Middle Ages and the Renaissance, Ptolemy's fame was sometimes determined by this particular work rather than by his astronomical works.

Ptolemy's "Geography" or "Geographical Manual" (Γεωγραφική ύφήγεσις) in eight books was very popular. In terms of volume, this work is not much inferior to the Almagest. It contains a description of the part of the world known in Ptolemy's time. However, Ptolemy's work differs significantly from similar writings of his predecessors. The descriptions themselves take up little space in it; the main attention is paid to the problems of mathematical geography and cartography. Ptolemy reports that he borrowed all the factual material from the geographical work of Marinus of Tire (dated approximately from PO AD), which, apparently, was a topographic description of regions indicating directions and distances between points. The main task of mapping is to display the spherical surface of the Earth on a flat map surface with minimal distortion.

In Book I, Ptolemy critically analyzes the projection method used by Marinus of Tyre, the so-called cylindrical projection, and rejects it. He proposes two other methods, equidistant conic and pseudoconic projections. He takes the dimensions of the world in longitude equal to 180 °, counting longitude from the zero meridian passing through the Isles of the Blessed (Canary Islands), from west to east, in latitude - from 63 ° north to 16; 25 ° south of the equator (which corresponds to parallels through the Fule and through a point symmetrical to the Meroe with respect to the equator).

Books II-VII provide a list of cities with geographical longitude and latitude and brief descriptions. In compiling it, apparently, lists of places with the same length of the day, or places located at a certain distance from the prime meridian, were used, which may have been part of the work of Marin of Tirsky. similar look the lists are contained in book VIII, which also gives a division of the world map into 26 regional maps. The composition of Ptolemy's work also included the maps themselves, which, however, have not come down to us. The cartographic material commonly associated with Ptolemy's Geography is actually of later origin. Ptolemy's "Geography" played an outstanding role in the history of mathematical geography, no less than the "Almagest" in the history of astronomy.

"Optics" of Ptolemy in five books has come down to us only in a Latin translation of the XII century. from Arabic, and the beginning and end of this work are lost. It is written in line with the ancient tradition represented by the works of Euclid, Archimedes, Heron and others, but, as always, Ptolemy's approach is original. Books I (which has not survived) and II deal with general theory vision. It is based on three postulates: a) the process of vision is determined by the rays that come from the human eye and, as it were, feel the object; b) color is a quality inherent in the objects themselves; c) color and light are equally necessary to make an object visible. Ptolemy also states that the process of vision occurs in a straight line. AT books III and IV deals with the theory of reflection from mirrors - geometric optics, or catoptrics, to use the Greek term. The presentation is carried out with mathematical rigor. Theoretical positions are proved experimentally. The problem of binocular vision is also discussed here, mirrors of various shapes, including spherical and cylindrical, are considered. Book V is about refraction; it investigates the refraction during the passage of light through the media air-water, water-glass, air-glass with the help of a device specially designed for this purpose. The results obtained by Ptolemy are in good agreement with Snell's law of refraction -sin α / sin β = n 1 / n 2, where α is the angle of incidence, β is the angle of refraction, n 1 and n 2 are the refractive indices in the first and second media, respectively. Astronomical refraction is discussed at the end of the surviving part of Book V.

The Harmonics (Αρμονικά) is a short work by Ptolemy in three books on musical theory. It deals with the mathematical intervals between notes, according to various Greek schools. Ptolemy compares the teachings of the Pythagoreans, who, in his opinion, emphasized the mathematical aspects of theory to the detriment of experience, and the teachings of Aristoxenus (4th century AD), who acted in the opposite way. Ptolemy himself seeks to create a theory that combines the advantages of both directions, i.e. strictly mathematical and at the same time taking into account the data of experience. Book III, which has come down to us incompletely, deals with the applications of musical theory in astronomy and astrology, including, apparently, the musical harmony of the planetary spheres. According to Porfiry (3rd century AD), Ptolemy borrowed the content of the Harmonica for the most part from the works of the Alexandrian grammarian of the second half of the 1st century. AD Didyma.

The name of Ptolemy is also associated with a number of less famous works. Among them is a treatise on philosophy "On the powers of judgment and decision-making" (Περί κριτηρίον και ηγεμονικού) , which outlines the ideas of mainly Peripatetic and Stoic philosophy, a small astrological work "Fruit" (Καρπός), known in Latin translation under the name "Centiloquium ” or “Fructus”, which included one hundred astrological positions, a treatise on mechanics in three books, from which two fragments have been preserved - “Heavy” and “Elements”, as well as two purely mathematical works, in one of which the postulate of parallel is proved, and in the other, that there are no more than three dimensions in space. Pappus of Alexandria, in a commentary on book V of the Almagest, credits Ptolemy with the creation of a special instrument called the "meteoroscope", similar to the armillary sphere.

Thus, we see that there is, perhaps, not a single area in ancient mathematical natural science where Ptolemy did not make a very significant contribution.

The work of Ptolemy had a huge impact on the development of astronomy. The fact that its significance was immediately appreciated is evidenced by the appearance already in the 4th century. AD comments - essays devoted to explaining the content of the Almagest, but often having independent significance.

The first known commentary was written around 320 by one of the most prominent representatives of the Alexandrian scientific school - Pappus. Most of this work has not come down to us - only comments on books V and VI of the Almagest have survived.

The second commentary, compiled in the 2nd half of the 4th c. AD Theon of Alexandria, has come down to us in a more complete form (books I-IV). The famous Hypatia (c. 370-415 AD) also commented on the Almagest.

In the 5th century Neoplatonist Proclus Diadochus (412-485), who headed the Academy in Athens, wrote an essay on astronomical hypotheses, which was an introduction to astronomy by Hipparchus and Ptolemy.

The closure of the Academy of Athens in 529 and the resettlement of Greek scientists in the countries of the East served as the rapid spread of ancient science here. The teachings of Ptolemy were mastered and significantly affected the astronomical theories that were formed in Syria, Iran and India.

In Persia, at the court of Shapur I (241-171), the Almagest became known, apparently, already around 250 AD. and was then translated into Pahlavi. There was also a Persian version of Ptolemy's Hand Tables. Both of these works had a great influence on the content of the main Persian astronomical work of the pre-Islamic period, the so-called Shah-i-Zij.

The Almagest was translated into Syriac, apparently, at the beginning of the 6th century. AD Sergius of Reshain (d. 536), famous physicist and a philosopher, a disciple of Philopon. In the 7th century a Syriac version of Ptolemy's Hand Tables was also in use.

From the beginning of the ninth century "Almagest" was also distributed in the countries of Islam - in Arabic translations and commentaries. It is listed among the first works of Greek scholars translated into Arabic. The translators used not only the Greek original, but also the Syriac and Pahlavi versions.

The most popular among the astronomers of the countries of Islam was the name "The Great Book", which sounded in Arabic as "Kitab al-majisti". Sometimes, however, this work was called the "Book of Mathematical Sciences" ("Kitab at-ta "alim"), which more accurately corresponded to its original Greek name "Mathematical Essay".

There were several Arabic translations and many adaptations of the Almagest made in different time. Their approximate list, which in 1892 numbered 23 names, is gradually being refined. At present, the main issues related to the history of the Arabic translations of the Almagest, in in general terms clarified. According to P. Kunitsch, "Almagest" in the countries of Islam in the IX-XII centuries. was known in at least five different versions:

1) Syriac translation, one of the earliest (not preserved);

2) a translation for al-Ma "mun of the beginning of the 9th century, apparently from Syriac; its author was al-Hasan ibn Quraish (not preserved);

3) another translation for al-Ma "mun, made in 827/828 by al-Hajjaj ibn Yusuf ibn Matar and Sarjun ibn Khiliya ar-Rumi, apparently also from Syriac;

4) and 5) translation of Ishaq ibn Hunayn al-Ibadi (830-910), the famous translator of Greek scientific literature, made in 879-890. directly from Greek; came to us in the processing of the largest mathematician and astronomer Sabit ibn Korra al-Harrani (836-901), but in the XII century. was also known as an independent work. According to P. Kunitsch, later Arabic translations more accurately conveyed the content of the Greek text.

At present, many Arabic writings have been thoroughly studied, which in essence represent comments on the Almagest or its processing, performed by astronomers of Islamic countries, taking into account the results of their own observations and theoretical research [Matvievskaya, Rosenfeld, 1983]. Among the authors are prominent scientists, philosophers and astronomers of the medieval East. The astronomers of the countries of Islam made changes of greater or lesser degree of importance in almost all sections of the Ptolemaic astronomical system. First of all, they specified its main parameters: the angle of inclination of the ecliptic to the equator, the eccentricity and longitude of the apogee of the Sun's orbit, and the average velocities of the Sun, Moon and planets. They replaced the tables of chords with sines and also introduced a whole set of new trigonometric functions. They developed more precise methods for determining the most important astronomical quantities, such as parallax, the equation of time, and so on. Old ones were improved and new astronomical instruments were developed, on which observations were regularly made, significantly exceeding in accuracy the observations of Ptolemy and his predecessors.

A significant part of the Arabic-language astronomical literature was ziji. These were collections of tables - calendar, mathematical, astronomical and astrological, which astronomers and astrologers used in their daily work. The zijs included tables that made it possible to record observations chronologically, find the geographical coordinates of a place, determine the moments of sunrise and sunset of the stars, calculate the positions of the stars on the celestial sphere for any moment in time, predict lunar and solar eclipses, and determine parameters that have astrological significance. The zijs provided rules for using tables; sometimes more or less detailed theoretical proofs of these rules were also placed.

Ziji VIII-XII centuries. were created under the influence, on the one hand, of Indian astronomical works, and on the other, of Ptolemy's Almagest and Hand Tables. An important role was also played by the astronomical tradition of pre-Muslim Iran. Ptolemaic astronomy in this period was represented by the “Proven Zij” by Yahya ibn Abi Mansur (9th century AD), two Zijs of Habash al-Khasib (IX century AD), “Sabaean Zij” by Muhammad al-Battani (c. . 850-929), "Comprehensive zij" by Kushyar ibn Labban (c. 970-1030), "Canon Mas "ud" by Abu Rayhan al-Biruni (973-1048), "Sanjar zij" by al-Khazini (first half of the 12th century .) and other works, especially the Book on the Elements of the Science of the Stars by Ahmad al-Farghani (IX century), which contains an exposition of Ptolemy's astronomical system.

In the XI century. The Almagest was translated by al-Biruni from Arabic into Sanskrit.

During late antiquity and the Middle Ages, the Greek manuscripts of the Almagest continued to be preserved and copied in the regions under the rule of the Byzantine Empire. The earliest Greek manuscripts of the Almagest that have come down to us date back to the 9th century AD. . Although astronomy in Byzantium did not enjoy the same popularity as in the countries of Islam, however, the love for ancient science did not fade away. Byzantium therefore became one of the two sources from which information about the Almagest penetrated into Europe.

Ptolemaic astronomy first became known in Europe thanks to the translations of the zijs al-Farghani and al-Battani into Latin. Separate quotations from the Almagest in the works of Latin authors are already found in the first half of the 12th century. However, this work became available to scholars of medieval Europe in its entirety only in the second half of the 12th century.

In 1175 the eminent translator Gerardo of Cremona, working in Toledo in Spain, completed the Latin translation of the Almagest, using the Arabic versions of Hajjaj, Ishaq ibn Hunayn and Thabit ibn Korra. This translation has become very popular. It is known in numerous manuscripts and already in 1515 was printed in Venice. In parallel or a little later (c. 1175-1250), an abbreviated version of the Almagest (Almagestum parvum) appeared, which was also very popular.

Two (or even three) other medieval Latin translations of the Almagest, made directly from the Greek text, have remained less known. The first of these (the name of the translator is unknown), entitled "Almagesti geometria" and preserved in several manuscripts, is based on a Greek manuscript of the 10th century, which was brought in 1158 from Constantinople to Sicily. The second translation, also anonymous and even less popular in the Middle Ages, is known in a single manuscript.

A new Latin translation of the Almagest from the Greek original was carried out only in the 15th century, when, from the beginning of the Renaissance, a heightened interest in the ancient philosophical and natural scientific heritage appeared in Europe. On the initiative of one of the propagandists of this heritage of Pope Nicholas V, his secretary George of Trebizond (1395-1484) translated the Almagest in 1451. The translation, which was very imperfect and full of errors, was nevertheless printed in Venice in 1528 and reprinted in Basel in 1541 and 1551.

The shortcomings of the translation of George of Trebizond, known from the manuscript, caused sharp criticism of astronomers who needed a full-fledged text of Ptolemy's capital work. The preparation of a new edition of the Almagest is associated with the names of two of the greatest German mathematicians and astronomers of the 15th century. - Georg Purbach (1423-1461) and his student Johann Müller, known as Regiomontanus (1436-1476). Purbach intended to publish the Latin text of the Almagest, corrected from the Greek original, but did not have time to finish the work. Regiomontanus also failed to complete it, although he spent a lot of effort studying Greek manuscripts. But he published Purbach's work "The New Theory of the Planets" (1473), which explained the main points of Ptolemy's planetary theory, and he himself compiled summary"Almagest", published in 1496. These publications, which appeared before the appearance of the printed edition of the translation of George of Trebizond, played a major role in popularizing the teachings of Ptolemy. According to them, Nicolaus Copernicus also got acquainted with this doctrine [Veselovsky, Bely, pp. 83-84].

The Greek text of the Almagest was first printed in Basel in 1538.

We also note the Wittenberg edition of book I of the Almagest, as presented by E. Reinhold (1549), which served as the basis for its translation into Russian in the 80s of the 17th century. unknown translator. The manuscript of this translation was recently discovered by V.A. Bronshten in the Moscow University Library [Bronshten, 1996; 1997].

New edition of the Greek text, together with French translation carried out in 1813-1816. N. Alma. In 1898-1903. an edition of the Greek text by I. Geiberg was published that meets modern scientific requirements. It served as the basis for all subsequent translations of the Almagest into European languages: German, which was published in 1912-1913. K. Manitius [NA I, II; 2nd ed., 1963], and two English ones. The first of them belongs to R. Tagliaferro and is of low quality, the second - to J. Toomer [RA]. Annotated edition of the Almagest on English language J. Toomer is currently considered the most authoritative among historians of astronomy. During its creation, in addition to the Greek text, a number of Arabic manuscripts in the versions of Hajjaj and Ishak-Sabit were also used [RA, p.3-4].

I.N.'s translation is also based on I. Geiberg's edition. Veselovsky published in this edition. I.N. Veselovsky, in the introduction to his comments on the text of N. Copernicus's book "On the Rotations of the Celestial Spheres", wrote: I had at my disposal the edition of Abbé Alma (Halma) with notes by Delambre (Paris, 1813-1816)” [Copernicus, 1964, p.469]. From this it seems to follow that the translation of I.N. Veselovsky was based on an outdated edition by N. Alma. However, in the archives of the Institute of the History of Natural Science and Technology of the Russian Academy of Sciences, where the manuscript of the translation is stored, a copy of the edition of the Greek text by I. Geiberg, which belonged to I.N. Veselovsky. A direct comparison of the text of the translation with the editions of N. Alm and I. Geiberg shows that I.N. Veselovsky revised further in accordance with the text of I. Geiberg. This is indicated, for example, by the accepted numbering of chapters in books, the designations in the figures, the form in which the tables are given, and many other details. In his translation, in addition, I.N. Veselovsky took into account most of the corrections made to the Greek text by K. Manitius.

Of particular note is the critical English edition of Ptolemy's star catalog published in 1915, undertaken by H. Peters and E. Noble [R. - TO.].

A large amount of scientific literature, both astronomical and historical-astronomical in nature, is associated with the Almagest. First of all, it reflected the desire to comprehend and explain the theory of Ptolemy, as well as attempts to improve it, which were repeatedly undertaken in antiquity and in the Middle Ages and culminated in the creation of the teachings of Copernicus.

Over time, the interest in the history of the emergence of the Almagest, in the personality of Ptolemy himself, that has manifested since antiquity, does not decrease - and perhaps even increases. It is impossible to give any satisfactory overview of the literature on the Almagest in a short article. This is a big independent work, which is beyond the scope of this study. Here we have to confine ourselves to pointing out a small number of works, mostly modern ones, which will help the reader navigate the literature about Ptolemy and his work.

First of all, mention should be made of the most numerous group of studies (articles and books) devoted to the analysis of the content of the Almagest and the determination of its role in the development of astronomical science. These problems are considered in writings on the history of astronomy, starting with the oldest ones, for example, in the two-volume History of Astronomy in Antiquity, published in 1817 by J. Delambre, Studies in the History of Ancient Astronomy by P. Tannery, History of Planetary Systems from Thales to Kepler" by J. Dreyer, in the fundamental work of P. Duhem "Systems of the World", in O. Neugebauer's masterfully written book "Exact Sciences in Antiquity" [Neugebauer, 1968]. The content of the Almagest is also studied in works on the history of mathematics and mechanics. Among the works of Russian scientists, the works of I.N. Idelson devoted to Ptolemy's planetary theory [Idelson, 1975], I.N. Veselovsky and Yu.A. Bely [Veselovsky, 1974; Veselovsky, Bely, 1974], V.A. Bronshten [Bronshten, 1988; 1996] and M.Yu. Shevchenko [Shevchenko, 1988; 1997].

The results of numerous studies carried out by the beginning of the 70s concerning the Almagest and the history of ancient astronomy in general are summarized in two fundamental works: History of Ancient Mathematical Astronomy by O. Neugebauer [NAMA] and Review of the Almagest by O. Pedersen . Anyone who wishes to take up the Almagest seriously cannot do without these two outstanding works. Big number valuable comments on various aspects of the content of the Almagest - the history of the text, computational procedures, Greek and Arabic manuscript tradition, the origin of parameters, tables, etc., can be found in the German [HA I, II] and English [RA] editions of the translation "Almagest".

Research on the Almagest continues at the present time with no less intensity than in the previous period, in several main areas. The greatest attention is paid to the origin of the parameters of Ptolemy's astronomical system, the kinematic models and computational procedures adopted by him, and the history of the star catalog. Much attention is also paid to the study of the role of Ptolemy's predecessors in the creation of the geocentric system, as well as the fate of Ptolemy's teachings in the medieval Muslim East, in Byzantium and Europe.

See also in this regard. A detailed analysis in Russian of biographical data on the life of Ptolemy is presented in [Bronshten, 1988, p.11-16].

See kn.XI, ch.5, p.352 and kn.IX, ch.7, p.303, respectively.

A number of manuscripts indicate the 15th year of the reign of Antoninus, which corresponds to 152/153 AD. .

Cm. .

It is reported, for example, that Ptolemy was born in Ptolemaida Hermia, located in Upper Egypt, and that this explains his name "Ptolemy" (Theodore of Miletus, XIV century AD); according to another version, he was from Pelusium, a border town east of the Nile Delta, but this statement is most likely the result of an erroneous reading of the name "Claudius" in Arabic sources [NAMA, p.834]. In late antiquity and the Middle Ages, Ptolemy was also credited with royal origin [NAMA, p.834, p.8; Toomer, 1985].

The opposite point of view is also expressed in the literature, namely, that in the time preceding Ptolemy there already existed a developed heliocentric system based on epicycles, and that Ptolemy's system is only a reworking of this earlier system [Idelson, 1975, p. 175; Rawlins, 1987]. However, in our opinion, such assumptions do not have sufficient grounds.

On this issue, see [Neigebauer, 1968, p.181; Shevchenko, 1988; Vogt, 1925], as well as [Newton, 1985, Ch.IX].

For a more detailed overview of the methods of pre-Ptolemaic astronomy, see.

Or in other words: "Mathematical collection (construction) in 13 books."

The existence of "Small Astronomy" as a special direction in ancient astronomy is recognized by all historians of astronomy with the exception of O. Neigenbauer. See on this issue [NAMA, p.768-769].

See on this issue [Idelson, 1975: 141-149].

For the Greek text, see (Heiberg, 1907, s.149-155]; for the French translation, see ; for descriptions and studies, see [HAMA, p.901,913-917; Hamilton etc., 1987; Waerden, 1959, Col. 1818- 1823; 1988(2), S.298-299].

The only more or less complete edition of Hand Tables belongs to N. Alma; the Greek text of Ptolemy's "Introduction" see; studies and descriptions, see .

For Greek text, translation and commentary, see .

For Greek text, see ; parallel German translation, including those parts that have been preserved in Arabic, see [ibid., S.71-145]; for the Greek text and a parallel translation into French, see ; Arabic text with an English translation of the part missing from the German translation, see ; studies and comments, see [NAMA, p.900-926; Hartner, 1964; Murschel, 1995; SA, pp. 391-397; Waerden, 1988(2), pp. 297-298]; description and analysis of Ptolemy's mechanical model of the world in Russian, see [Rozhanskaya, Kurtik, p. 132-134].

For the Greek text of the surviving part, see ; for Greek text and French translation, see ; see studies and comments.

For fragments of the Greek text and Latin translation, see; see studies.

The Arabic text has not yet been published, although several manuscripts of this work are known, earlier than the era of al-Majriti .; see Latin translation; German translation, see ; studies and comments, see [NAMA, p.857-879; Waerden, 1988(2), S.301-302; Matvievskaya, 1990, p.26-27; Neugebauer, 1968, pp. 208-209].

For Greek text, see ; for the Greek text and parallel English translation, see ; full translation into Russian from English, see [Ptolemy, 1992]; translation into Russian from ancient Greek of the first two books, see [Ptolemy, 1994, 1996); for an outline of the history of ancient astrology, see [Kurtik, 1994]; see studies and comments.

Description and analysis of Ptolemy's methods of cartographic projection, see [Neigebauer, 1968, p.208-212; NAMA, r.880-885; Toomer, 1975, pp. 198-200].

For Greek text, see ; collection of ancient maps, see; English translation see ; for the translation of individual chapters into Russian, see [Bodnarsky, 1953; Latyshev, 1948]; for a more detailed bibliography concerning Ptolemy's Geography, see [NAMA; Toomer, 1975, p.205], see also [Bronshten, 1988, p. 136-153]; about the geographical tradition in the countries of Islam, dating back to Ptolemy, see [Krachkovsky, 1957].

For a critical edition of the text, see ; for descriptions and analysis, see [NAMA, p.892-896; Bronshten, 1988, p. 153-161]. For a more complete bibliography, see .

For Greek text, see ; German translation with comments, see ; astronomical aspects of Ptolemy's musical theory, see [NAMA, p.931-934]. For a brief outline of the musical theory of the Greeks, see [Zhmud, 1994: 213-238].

For Greek text, see ; more detailed description cm. . For a detailed analysis of the philosophical views of Ptolemy, see.

For Greek text, see ; however, according to O. Neugebauer and other researchers, there are no serious grounds for attributing this work to Ptolemy [NAMA, p.897; Haskins, 1924, p. 68 et seq.].

For Greek text and German translation, see ; see French translation.

The version of Hajjaj ibn Matar is known in two Arabic manuscripts, of which the first (Leiden, cod. or. 680, complete) dates from the 11th century. AD, the second (London, British Library, Add.7474), partially preserved, dates back to the 13th century. . Ishak-Sabit's version has come down to us in a larger number of copies of various completeness and safety, of which we note the following: 1) Tunis, Bibl. Nat. 07116 (XI century, complete); 2) Teheran, Sipahsalar 594 (XI century, the beginning of book 1, tables and catalog of stars are missing); 3) London, British Library, Add.7475 (beginning of the 13th century, book VII-XIII); 4) Paris, Bible. Nat.2482 (beginning of the 13th century, book I-VI). For a complete list of currently known Arabic manuscripts of the Almagest, see. For a comparative analysis of the content of various versions of the translations of the Almagest into Arabic, see.

For an overview of the content of the most famous zijs of astronomers in Islamic countries, see.

The Greek text in I. Geiberg's edition is based on seven Greek manuscripts, of which the following four are the most important: A) Paris, Bibl. Nat., gr.2389 (complete, 9th century); C) Vaticanus, gr.1594 (complete, IX century); C) Venedig, Marc, gr.313 (complete, 10th century); D) Vaticanus gr.180 (complete, X century). Letter designations of manuscripts were introduced by I. Geiberg.

In this regard, the works of R. Newton [Newton, 1985, etc.], who accuse Ptolemy of falsifying the data of astronomical observations and concealing the astronomical (heliocentric?) system that existed before him, have gained great fame. Most historians of astronomy reject the global conclusions of R. Newton, while recognizing that some of his results regarding observations cannot but be recognized as fair.

According to which the central place in the Universe is occupied by the planet Earth, which remains motionless. The Moon, the Sun, all the stars and planets are already gathering around it. It was first formulated in Ancient Greece. It became the basis for ancient and medieval cosmology and astronomy. An alternative later became the heliocentric system of the world, which became the basis for the current

The emergence of geocentrism

The Ptolemaic system was considered fundamental for all scientists for many centuries. Since ancient times, the Earth has been considered the center of the universe. It was assumed that there is a central axis of the Universe, and some kind of support keeps the Earth from falling.

Ancient people believed that it was some mythical giant creature, such as an elephant, a turtle, or several whales. Thales of Miletus, who was considered the father of philosophy, suggested that the world ocean itself could be such a natural support. Some have suggested that the Earth, located in the center of space, does not need to move in any direction, it simply rests in the very center of the universe without any support.

World system

Claudius Ptolemy sought to give his own explanation for all the visible movements of the planets and other celestial bodies. The main problem was that all observations were carried out at that time exclusively from the surface of the Earth, because of this it was impossible to reliably determine whether our planet is in motion or not.

In this regard, astronomers of antiquity had two theories. According to one of them, the Earth is at the center of the universe and remains motionless. Mostly the theory was based on personal impressions and observations. And according to the second version, which was based solely on speculative conclusions, the Earth rotates around its own axis and moves around the Sun, which is the center of the whole world. However, this fact clearly contradicted the existing opinions and religious views. That is why the second point of view did not receive a mathematical justification, for many centuries in astronomy the opinion about the immobility of the Earth was approved.

Proceedings of the astronomer

In the book of Ptolemy, entitled "The Great Construction", the main ideas of ancient astronomers about the structure of the Universe were summarized and outlined. The Arabic translation of this work was widely used. It is known under the name "Almagest". Ptolemy based his theory on four main assumptions.

The Earth is located directly in the center of the Universe and is motionless, all celestial bodies move around it in circles at a constant speed, that is, evenly.

The Ptolemaic system is called geocentric. In a simplified form, it is described as follows: the planets move in circles at a uniform speed. In the common center of everything is the motionless Earth. The Moon and the Sun revolve around the Earth without epicycles, but along the deferents that lie inside the sphere, and "fixed" stars remain on the surface.

The daily movement of any of the luminaries was explained by Claudius Ptolemy as the rotation of the entire Universe around the motionless Earth.

planetary motion

Interestingly, for each of the planets, the scientist selected the sizes of the radii of the deferent and epicycle, as well as the speed of their movement. This could only be done under certain conditions. For example, Ptolemy took it for granted that the centers of all the epicycles of the lower planets are located in a certain direction from the Sun, while the radii of the epicycles of the upper planets in the same direction are parallel.

As a result, the direction to the Sun in the Ptolemaic system became predominant. It was also concluded that the periods of revolution of the corresponding planets are equal to the same sidereal periods. All this in Ptolemy's theory meant that the system of the world includes the most important features of the actual and real movements of the planets. Much later, another brilliant astronomer, Copernicus, managed to fully reveal them.

One of important issues within the framework of this theory, there was a need to calculate the distance, how many kilometers from the Earth to the Moon. It has now been reliably established that it is 384,400 kilometers.

Merit of Ptolemy

The main merit of Ptolemy was that he managed to give a full and exhaustive explanation of the apparent movements of the planets, and also made it possible to calculate their position in the future with an accuracy that would correspond to observations made by the naked eye. As a result, although the theory itself was fundamentally wrong, it did not cause serious objections, and any attempts to contradict it were immediately severely suppressed by the Christian church.

Over time, serious discrepancies between theory and observations were discovered, which arose as accuracy improved. They were finally eliminated only by significantly complicating the optical system. For example, certain irregularities in the apparent motion of the planets, which were discovered as a result of later observations, were explained by the fact that it is no longer the planet itself that revolves around the center of the first epicycle, but the so-called center of the second epicycle. And now a celestial body is moving along its circumference.

If such a construction turned out to be insufficient, additional epicycles were introduced until the position of the planet on the circle correlated with the observational data. As a result, at the beginning of the 16th century, the system developed by Ptolemy turned out to be so complex that it did not meet the requirements that were imposed on astronomical observations in practice. First of all, it concerned navigation. New methods were needed to calculate the motion of the planets, which were supposed to be easier. They were developed by Nicolaus Copernicus, who laid the foundation for the new astronomy on which modern science is based.

Representations of Aristotle

Aristotle's geocentric system of the world was also popular. It consisted in the postulate that the Earth is a heavy body for the Universe.

As practice has shown, all heavy bodies fall vertically, as they are in motion towards the center of the world. The earth itself was located in the center. On this basis, Aristotle refuted the orbital motion of the planet, coming to the conclusion that it leads to a parallactic displacement of the stars. He also sought to calculate how much from the Earth to the Moon, having managed to achieve only approximate calculations.

Biography of Ptolemy

Ptolemy was born around 100 AD. The main sources of information about the biography of the scientist are his own writings, which modern researchers have managed to arrange in chronological order through cross-references.

Fragmentary information about his fate can also be gleaned from the works of Byzantine authors. But it should be noted that this is unreliable information that is not trustworthy. It is believed that he owes his wide and versatile erudition to the active use of the volumes stored in the Library of Alexandria.

Works of a scientist

The main works of Ptolemy are related to astronomy, but he also left a mark in other scientific fields. In particular, in mathematics he deduced Ptolemy's theorem and inequality, based on the theory of the product of the diagonals of a quadrilateral inscribed in a circle.

Five books make up his treatise on optics. In it, he describes the nature of vision, considers various aspects of perception, describes the properties of mirrors and the laws of reflections, and discusses for the first time in world science a detailed and fairly accurate description of atmospheric refraction.

Many people know Ptolemy as a talented geographer. In eight books, he details the knowledge inherent in the man of the ancient world. It was he who laid the foundations of cartography and mathematical geography. He published the coordinates of eight thousand points located from Egypt to Scandinavia and from Indo-China to the Atlantic Ocean.

Name: Claudius Ptolemy

Years of life: about 100 years - about 170 years

State: Ancient Greece

Field of activity: Astronomy, astrology, mathematics

Greatest Achievement: He brought together almost all the knowledge of astronomy of Ancient Greece, became the forefather of the mechanics of the planets, astrophysics.

Claudius Ptolemy was a famous scientist, mathematician, philosopher, theologian, geographer, astronomer and astrologer.

He lived and worked around 90-168 AD in Alexandria.

Most of all in history, his works on the geocentric model of the world were remembered, which, although they were erroneous, had rather strong mathematical justifications.

The Ptolemaic system was one of the most influential and enduring intellectual-scientific achievements in human history.

Unfortunately, apart from his writings about the life of Ptolemy, about his family and appearance, there is almost no information.

Works of Ptolemy

The first and largest of them was originally called "Mathematical collection in thirteen books", but the Arabic version of the name "Almagest" has survived to our time.

He also wrote the treatise Tetrabiblos (or "Four Books") on astronomy, in which he suggests that it is possible to predict events from the behavior of celestial bodies.

The first chapter of the Almagest contains a discussion of epistemology and philosophy. Two themes are of central importance in this chapter: the structure of philosophy - and in the ancient world this term included all human knowledge and wisdom - and the reasons for studying mathematics.

The only philosopher Ptolemy relies on in his work is Aristotle.

He agrees with him in the division of philosophy into practical and theoretical. And also in the division of theoretical philosophy into three branches: physics, mathematics and theology, understanding by theology the science that studies the root cause of the creation of the Universe.

And yet, by placing theology on a par with natural science and mathematics, these philosophers differed from their contemporaries, secular philosophers.

Ptolemaic world system

In the Almagest, Ptolemy collected all the astronomical knowledge of the Greek and Babylonian world. The development of the mathematical basis of this theory was carried out at one time by such scientists as Eudoxus of Cnidus, Hipparchus and Ptolemy himself.

Based mainly on the observations of Hipparchus, the scientist gives an idea of ​​the geocentric system. This theory was so well proven that it was popular until the sixteenth century, when it was disproved by Copernicus and replaced by the heliocentric system of the world.

According to Ptolemaic cosmology, the Earth is the center of the universe and is stationary, while other celestial bodies revolve around it in the following order: Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn.

Ptolemy gave many reasons why the Earth is in the center.

One of them was that if this is not the case, then things will not fall to the Earth, but the Earth will be pulled towards the center of the universe.

Ptolemy proved the theory of the immobility of the planet by the argument that a thing thrown vertically in one place cannot fall in the same place if the Earth moves.

Ptolemy's computational methods were accurate enough to meet the demands of the astronomers, astrologers, and navigators of the time.

Geography of Ptolemy

The second of Ptolemy's significant works was "Geography", which provides detailed geographical knowledge about the Greco-Roman world. It consisted of eight books.

This work is also a compilation of the information about geography that was known at that time. Mostly the work of Marinos of Tyre, an earlier geographer, is used.

The first part of this treatise is a description of the data and methods used by Ptolemy and introduced by him into grandiose schemes, as in the case of the Almagest. This book defines the concepts of longitude and latitude, the globe, tells how geography differs from country studies.

He also gave instructions on how to create maps of the world and Roman provinces.

The rest of the books give a description of the entire world known to Ptolemy, although, probably, these works were supplemented by someone, centuries after Ptolemy, since information about countries was entered that the scientist could not have.

For the same reason, Ptolemy's original topographic lists have not survived to this day, as they were constantly corrected and improved. This, by the way, speaks of the constant popularity of the treatise.

It is authentically known that in the XIII century the Byzantine monk Maxim Planud discovered the "Geography", but without the geographical maps that Ptolemy compiled.

In the middle of the 15th century, the maps were restored by the cosmographer Nikolai Germanus.

Astrology of Ptolemy

For several centuries, Ptolemy's treatise "Tetrabiblos" was the most authoritative manual on astrology, it was reprinted many times, as it was very popular. Ptolemy described in it the important provisions of this science, correlating them with the Aristotelian natural philosophy of that time.

In general terms, the scientist defined the limits of astronomy, citing astronomical data that is beyond doubt, and discarding, in his opinion, erroneous practices like numerology.

The astrological worldview of Ptolemy was quite rational. He believed that astrology could be used in life, since the personality of people was influenced not only by upbringing or the environment of birth, but also by the location of celestial bodies at the time of birth.

He did not call to rely on astrology completely, but considered it possible to use it in life.

Ptolemy's theorems

Ptolemy was also an eminent mathematician and geometer who introduced new geometric proofs and theorems, such as Ptolemy's inequality.

In one work, he studied the projections of points on the celestial sphere, in another, the forms of solid objects presented on a plane.

In the Pentateuch "Optics" Ptolemy was the first to write about some of the properties of light - reflection, refraction and color.

In honor of this outstanding scientist and philosopher, craters on the Moon and Mars were named.

* 1. Introduction - p. 5 * 2. About the sequence of presentation - p. 7 * 3. About the fact that the sky has a spherical movement - p. 7 * 4. About the fact that the Earth as a whole has the form of a sphere - p. 9 * 5. About the fact that the Earth is in the middle of the sky - p.10 * 6. About the fact that in comparison with the heavens the Earth is a point - p.11 * 7. About the fact that the Earth does not make any forward movement - p. 12 * 8. That there are two different kind of the first movements - p.14 * 9. On special concepts - p.15 * 10. On the magnitudes of straight lines in a circle - p.16 * 11. Table of straight lines in a circle - p. .21 * 13. Preliminary theorems for proofs of the sphere - p.27 * 14. On the arcs enclosed between the equinoctial and oblique circles - p.30 * 15. Declension table - p.31 * 16. On the times of rising in the right sphere - p. .31*

Notes pages 464 - 479

* 1. About general position inhabited part of the Earth - p.34 * 2. About how, according to a given value greatest day the arcs of the horizon cut off by the equinoctial and oblique circles are determined - p. 35 * 3. How, under the same assumptions, the height of the pole is determined, and vice versa - p. 36 * 4. How the Sun is calculated, where, when and how often happens directly overhead - p. sunrises in the oblique sphere of parts of the circle passing through the midpoints of the zodiac constellations and the equinoctial circle - p.45 * 8. Table of times of rising along arcs of ten degrees - p. * 10. About the angles formed by a circle passing through the midpoints of the zodiac constellations, and the midday circle - p.57 * 11. About the angles formed by the same inclined circle with the horizon - p. inclined circle and a circle drawn through the poles of the horizon - page 62 * 13. Values ​​of angles and arcs for various parallels - page 67 *

Notes pages 479 - 494

* 1. On the duration of the annual period of time - p.75 * 2. Tables of average movements of the Sun - p.83 * 3. On hypotheses regarding uniform circular motion - p.85 * 4. On the apparent inequality of the movement of the Sun - p.91 * 5. On determining the values ​​of inequality for various positions - p.94 * 6. Table of the solar anomaly - p.94 * 7. On the epoch of the average motion of the Suns - p.98 * 8. On calculating the position of the Sun - p. inequality of the day - page 100 *

Notes pages 494 - 508

* 1. On what observations should the theory of the Moon be built - p.103 * 2. On the periods of lunar movements - p.104 * 3. On particular values ​​of the average movements of the Moon - p.108 * 4. Tables of the average movements of the Moon - p.109 * 5. About the fact that with a simple hypothesis about the motion of the Moon, it will be an eccentric or epicycle hypothesis, the visible phenomena will be the same - p. 109 * 6. Definition of the first, or simple lunar inequality - p. 117 * 7. About correction average movements of the Moon in longitude and anomalies - p.126 * 8. On the epoch of average movements of the Moon in longitude and anomalies - p.127 * 9. On the correction of average movements of the Moon in latitude and their epochs - p. , or simple, inequality of the Moon - p.131 * 11. That the difference between the value of the lunar inequality accepted by Hipparchus and that found by us is obtained not from the difference in the assumptions made, but as a result of calculations - p.131 *

Notes pages 509 - 527

* 1. On the device of the astrolabe - p.135 * 2. On the hypotheses of the double inequality of the Moon - p.137 * 3. On the magnitude of the inequality of the Moon, depending on the position relative to the Sun - p.139 * 4. On the magnitude of the ratio for the eccentricity of the lunar orbit - p.141 * 5. On the "inclination" of the lunar epicycle - p.141 * 6. On how the true position of the Moon is determined geometrically by periodic movements - p.146 * 7. Building a table for the complete inequality of the Moon - p.147 * 8 Table of complete lunar inequality - p.150 * 9. On the calculation of the motion of the Moon as a whole - p.151 * 10. On the fact that the eccentric circle of the Moon does not produce any noticeable difference in syzygies - p.151 * 11. On the parallaxes of the Moon - p.154 * 12. On the construction of a parallax instrument - p.155 * 13. Determining the distances of the Moon - p. about what is determined together with it - p. 162 * 16. About the magnitudes of the Sun, Moon and Earth - p.163 * 17. On the particular values ​​of the parallaxes of the Sun and the Moon - p.164 * 18. Table of parallaxes - p.168 * 19. On the definition of parallaxes - p.168 *

Notes pp. 527 - 547

* 1. About new moons and full moons - p.175 * 2. Compiling tables of average syzygies - p.175 * 3. Tables of new moons and full moons - p.177 * 4. About how to determine the average and true syzygies - p.180 * 5. About the limits for eclipses of the Sun and the Moon - p.181 * 6. About the intervals between months in which eclipses occur - p.184 * 7. Building tables of eclipses - p.190 * 8. Eclipse tables - p.197 * 9. Calculation of lunar eclipses - p. 199 * 10. Calculation of solar eclipses - p. 201 * 11. About the angles of "inclinations" in eclipses - p. inclinations" - p.208 *

Notes pages 547 - 564

* 1. That the fixed stars always maintain the same position in relation to each other - p. p.214 * 3. About the fact that the sphere of fixed stars moves around the poles of the zodiac in the direction of the sequence of signs - p.216 * 4. About the method of compiling a catalog of fixed stars - p.223 * 5. Catalog of the constellations of the northern sky - p.224 *

Notes pages 565 - 579

* 1. Catalog of constellations of the southern sky - p.245 * 2. On the position of the circle of the Milky Way - p.264 * 3. On the structure of the celestial globe - p. fixed stars configurations - p.269 * 5. About the simultaneous risings, culminations and setting of fixed stars - p.273 * 6. About heliacal risings and setting of fixed stars - p.274 *

Notes pages 580 - 587

* 1. On the sequence of the spheres of the Sun, the Moon and the five planets - p.277 * 2. On the presentation of hypotheses regarding the planets - p.278 * 3. On the periodic returns of the five planets - p.280 * 4. Tables of average movements in longitude and anomalies for the five planets - p. 282 * 5. Basic provisions regarding the hypotheses about the five planets - p. 298 * 6. On the nature and differences between the hypotheses - p. * 8. About the fact that the planet Mercury also, during one revolution, twice becomes in the position closest to the Earth - p.306 * 9. About the ratio and magnitude of Mercury's anomalies - p. * 11. About the era of periodic movements of Mercury - p. 315 *

Notes pp. 587 - 599

* 1. Determining the position of the apogee of the planet Venus - p.316 * 2. About the magnitude of the epicycle of Venus - p.317 * 3. About the relationship of the eccentricities of the planet Venus - p.318 * 4. About correcting the periodic movements of Venus - p.320 * 5. On the epoch of periodic motions of Venus - p.323 * 6. Preliminary information concerning the rest of the planets - p.324 * 7. Determination of the eccentricity and position of the apogee of Mars - p.325 * 8. Determination of the magnitude of the epicycle of Mars - p.335 * 9. About correction of the periodic movements of Mars - p.336 * 10. About the era of his periodic movements of Mars - p.339 *

Notes pages 599 - 609

* 1. Determining the eccentricity and position of Jupiter's apogee - p.340 * 2. Determining the magnitude of Jupiter's epicycle - p.348 * 3. About correcting Jupiter's periodic motions - p.349 * 4. About the era of Jupiter's periodic motions - p.351 * 5 Determination of the eccentricity and position of the apogee of Saturn - p.352 * 6. Determination of the magnitude of the epicycle of Saturn - p.360 * 7. About the correction of periodic movements of Saturn - p.361 * 8. About the era of periodic movements of Saturn - p.363 * 9. O how the true positions are geometrically determined from periodic movements - p.364 * 10. Construction of tables of anomalies - p.364 * 11. Tables for determining the longitudes of five planets - p. *

Notes pages 610 - 619

* 1. About the preliminary provisions concerning retrograde motions - p.373 * 2. Determination of backward motions of Saturn - p.377 * 3. Determination of backward motions of Jupiter - p.381 * 4. Definition of backward motions of Mars - p.382 * 5. Determination of backward movements of Venus - p.384 * 6. Determination of backward movements of Mercury - p.386 * 7. Construction of a table of positions - p.388 * 8. Table of positions. Values ​​of the corrected anomaly - p.392 * 9. Determination of the greatest distances of Venus and Mercury from the Sun - p.393 * 10. Table of the greatest distances of the planets from the true position from the Sun - p.397 *

Notes pages 620 - 630

* 1. On the hypotheses concerning the movement of five planets in latitude - p.398 * 2. On the nature of the movement in the alleged inclinations and appearances according to the hypotheses - p.400 * 3. On the magnitude of inclinations and appearances for each planet - p.402 * 4 Construction of tables for partial values ​​of deviations in latitude - p.404 * 5. Tables for calculating latitude - p.419 * 6. Calculation of deviations of five planets in latitude - p. 422 * 8. About the fact that the features of the risings and setting of Venus and Mercury are consistent with the accepted hypotheses - p. five planets - p.428 * 11. Epilogue of the composition - p.428 *

Notes pages 630 - 643

Applications

Ptolemy and his astronomical work, - G.E. Kurtik, G.P. Matvievskaya

The translator of "Almagest" I.N. Veselovsky, - S.V. Zhytomyr

Calendar and chronology in the Almagest, - G.E. Kurtik