Ptolemy

Claudius Ptolemy (/ˈtɒləmi/; Koinē Greek: Κλαύδιος Πτολεμαῖος, Klaúdios Ptolemaîos [kláwdios ptolɛmɛ́os]; Latin: Claudius Ptolemaeus; c. AD 100 – c.  170)[2] was a Greco-Roman[3] mathematician, astronomer, geographer and astrologer. He lived in the city of Alexandria in the Roman province of Egypt, wrote in Koine Greek, and held Roman citizenship.[4] The 14th-century astronomer Theodore Meliteniotes gave his birthplace as the prominent Greek city Ptolemais Hermiou (Greek: Πτολεμαΐς ‘Ερμείου) in the Thebaid (Greek: Θηβαΐδα [Θηβαΐς]). This attestation is quite late, however, and, according to Gerald Toomer, the translator of his Almagest into English, there is no reason to suppose he ever lived anywhere other than Alexandria.[5] He died there around AD 168.[6]

Ptolemy wrote several scientific treatises, three of which were of importance to later Byzantine, Islamic and Western European science. The first is the astronomical treatise now known as the Almagest, although it was originally entitled the Mathematical Treatise (Μαθηματικὴ Σύνταξις, Mathēmatikē Syntaxis) and then known as the Great Treatise (Ἡ Μεγάλη Σύνταξις, Hē Megálē Syntaxis). The second is the Geography, which is a thorough discussion of the geographic knowledge of the Greco-Roman world. The third is the astrological treatise in which he attempted to adapt horoscopic astrology to the Aristotelian natural philosophy of his day. This is sometimes known as the Apotelesmatika (Ἀποτελεσματικά) but more commonly known as the Tetrabiblos from the Greek (Τετράβιβλος) meaning "Four Books" or by the Latin Quadripartitum.

Ptolemy
PSM V78 D326 Ptolemy
Early Baroque artist's rendition[1]
Bornc. AD 100[2]
Egypt, Roman Empire
Diedc. AD 170 (aged 69–70)[2]
Alexandria, Egypt, Roman Empire
Occupation

Background

Ptolemy urania
Engraving of a crowned Ptolemy being guided by the muse of Astronomy, Urania, from Margarita Philosophica by Gregor Reisch, 1508. Although Abu Ma'shar believed Ptolemy to be one of the Ptolemies who ruled Egypt after the conquest of Alexander the title ‘King Ptolemy’ is generally viewed as a mark of respect for Ptolemy's elevated standing in science.

Ptolemaeus (Πτολεμαῖος Ptolemaios) is a Greek name. It occurs once in Greek mythology, and is of Homeric form.[7] It was common among the Macedonian upper class at the time of Alexander the Great, and there were several of this name among Alexander's army, one of whom made himself pharaoh in 323 BC: Ptolemy I Soter, the first king of the Ptolemaic Kingdom. All male kings of Hellenistic Egypt, until Egypt became a Roman province in 30 BC ending the Macedonian family's rule, were also Ptolemies.

The name Claudius is a Roman nomen; the fact that Ptolemy bore it indicates he lived under the Roman rule of Egypt with the privileges and political rights of Roman citizenship. It would have suited custom if the first of Ptolemy's family to become a citizen (whether he or an ancestor) took the nomen from a Roman called Claudius who was responsible for granting citizenship. If, as was common, this was the emperor, citizenship would have been granted between AD 41 and 68 (when Claudius, and then Nero, were Roman emperors). The astronomer would also have had a praenomen, which remains unknown.

The ninth-century Persian astronomer Abu Maʿshar presents Ptolemy as a member of Egypt's royal lineage, stating that the descendants of Alexander's general Ptolemy I, who ruled Egypt, were wise "and included Ptolemy the Wise, who composed the book of the Almagest". Abu Maʿshar recorded a belief that a different member of this royal line "composed the book on astrology and attributed it to Ptolemy". We can evidence historical confusion on this point from Abu Maʿshar's subsequent remark "It is sometimes said that the very learned man who wrote the book of astrology also wrote the book of the Almagest. The correct answer is not known."[8] There is little evidence on the subject of Ptolemy's ancestry, apart from what can be drawn from the details of his name (see above); however, modern scholars refer to Abu Maʿshar's account as erroneous,[9] and it is no longer doubted that the astronomer who wrote the Almagest also wrote the Tetrabiblos as its astrological counterpart.[10]

Ptolemy wrote in Koine Greek and can be shown to have utilized Babylonian astronomical data.[11][12] He was a Roman citizen, but was ethnically either a Greek[2][13][14] or a Hellenized Egyptian.[13][15][16] He was often known in later Arabic sources as "the Upper Egyptian",[17] suggesting he may have had origins in southern Egypt.[18] Later Arabic astronomers, geographers and physicists referred to him by his name in Arabic: بَطْلُمْيوسBaṭlumyus.[19]

Astronomy

Ptolemy 1476 with armillary sphere model
Ptolemy with an armillary sphere model, by Joos van Ghent and Pedro Berruguete, 1476, Louvre, Paris

Ptolemy's Almagest is the only surviving comprehensive ancient treatise on astronomy. Babylonian astronomers had developed arithmetical techniques for calculating astronomical phenomena; Greek astronomers such as Hipparchus had produced geometric models for calculating celestial motions. Ptolemy, however, claimed to have derived his geometrical models from selected astronomical observations by his predecessors spanning more than 800 years, though astronomers have for centuries suspected that his models' parameters were adopted independently of observations.[20] Ptolemy presented his astronomical models in convenient tables, which could be used to compute the future or past position of the planets.[21] The Almagest also contains a star catalogue, which is a version of a catalogue created by Hipparchus. Its list of forty-eight constellations is ancestral to the modern system of constellations, but unlike the modern system they did not cover the whole sky (only the sky Hipparchus could see). Across Europe, the Middle East and North Africa in the Medieval period, it was the authoritative text on astronomy, with its author becoming an almost mythical figure, called Ptolemy, King of Alexandria.[22] The Almagest was preserved, like most of extant Classical Greek science, in Arabic manuscripts (hence its familiar name). Because of its reputation, it was widely sought and was translated twice into Latin in the 12th century, once in Sicily and again in Spain.[23] Ptolemy's model, like those of his predecessors, was geocentric and was almost universally accepted until the appearance of simpler heliocentric models during the scientific revolution.

His Planetary Hypotheses went beyond the mathematical model of the Almagest to present a physical realization of the universe as a set of nested spheres,[24] in which he used the epicycles of his planetary model to compute the dimensions of the universe. He estimated the Sun was at an average distance of 1,210 Earth radii, while the radius of the sphere of the fixed stars was 20,000 times the radius of the Earth.[25]

Ptolemy presented a useful tool for astronomical calculations in his Handy Tables, which tabulated all the data needed to compute the positions of the Sun, Moon and planets, the rising and setting of the stars, and eclipses of the Sun and Moon. Ptolemy's Handy Tables provided the model for later astronomical tables or zījes. In the Phaseis (Risings of the Fixed Stars), Ptolemy gave a parapegma, a star calendar or almanac, based on the appearances and disappearances of stars over the course of the solar year.[26]

The Geography

La Cosmographie de Claude Ptolemée.djvu&page=9
Geography by Ptolemy, Latin manuscript of the early 15th century

Ptolemy's other main work is his Geography (also called the Geographia), a compilation of geographical coordinates of the part of the world known to the Roman Empire during his time. He relied somewhat on the work of an earlier geographer, Marinos of Tyre, and on gazetteers of the Roman and ancient Persian Empire. He also acknowledged ancient astronomer Hipparchus for having provided the elevation of the north celestial pole[27] for a few cities.[28]

The first part of the Geography is a discussion of the data and of the methods he used. As with the model of the Solar System in the Almagest, Ptolemy put all this information into a grand scheme. Following Marinos, he assigned coordinates to all the places and geographic features he knew, in a grid that spanned the globe. Latitude was measured from the equator, as it is today, but Ptolemy preferred[29] to express it as climata, the length of the longest day rather than degrees of arc: the length of the midsummer day increases from 12h to 24h as one goes from the equator to the polar circle. In books 2 through 7, he used degrees and put the meridian of 0 longitude at the most western land he knew, the "Blessed Islands", often identified as the Canary Islands, as suggested by the location of the six dots labelled the "FORTUNATA" islands near the left extreme of the blue sea of Ptolemy's map here reproduced.

PtolemyWorldMap
A 15th-century manuscript copy of the Ptolemy world map, reconstituted from Ptolemy's Geography (circa AD 150), indicating the countries of "Serica" and "Sinae" (China) at the extreme east, beyond the island of "Taprobane" (Sri Lanka, oversized) and the "Aurea Chersonesus" (Malay Peninsula).
Prima Europe tabula
Prima Europe tabula. A 15th-century copy of Ptolemy's map of Britain and Ireland.

Ptolemy also devised and provided instructions on how to create maps both of the whole inhabited world (oikoumenè) and of the Roman provinces. In the second part of the Geography, he provided the necessary topographic lists, and captions for the maps. His oikoumenè spanned 180 degrees of longitude from the Blessed Islands in the Atlantic Ocean to the middle of China, and about 80 degrees of latitude from Shetland to anti-Meroe (east coast of Africa); Ptolemy was well aware that he knew about only a quarter of the globe, and an erroneous extension of China southward suggests his sources did not reach all the way to the Pacific Ocean.

The maps in surviving manuscripts of Ptolemy's Geography, however, only date from about 1300, after the text was rediscovered by Maximus Planudes. It seems likely that the topographical tables in books 2–7 are cumulative texts – texts which were altered and added to as new knowledge became available in the centuries after Ptolemy.[30] This means that information contained in different parts of the Geography is likely to be of different dates.

Claudius Ptolemy- The World
A printed map from the 15th century depicting Ptolemy's description of the Ecumene, (1482, Johannes Schnitzer, engraver).

Maps based on scientific principles had been made since the time of Eratosthenes, in the 3rd century BC, but Ptolemy improved map projections. It is known from a speech by Eumenius that a world map, an orbis pictus, doubtless based on the Geography, was on display in a school in Augustodunum, Gaul in the third century.[31] In the 15th century, Ptolemy's Geography began to be printed with engraved maps; the earliest printed edition with engraved maps was produced in Bologna in 1477, followed quickly by a Roman edition in 1478 (Campbell, 1987). An edition printed at Ulm in 1482, including woodcut maps, was the first one printed north of the Alps. The maps look distorted when compared to modern maps, because Ptolemy's data were inaccurate. One reason is that Ptolemy estimated the size of the Earth as too small: while Eratosthenes found 700 stadia for a great circle degree on the globe, Ptolemy uses 500 stadia in the Geography. It is highly probable that these were the same stadion, since Ptolemy switched from the former scale to the latter between the Syntaxis and the Geography, and severely readjusted longitude degrees accordingly. See also Ancient Greek units of measurement and History of geodesy.

Because Ptolemy derived many of his key latitudes from crude longest day values, his latitudes are erroneous on average by roughly a degree (2 degrees for Byzantium, 4 degrees for Carthage), though capable ancient astronomers knew their latitudes to more like a minute. (Ptolemy's own latitude was in error by 14'.) He agreed (Geography 1.4) that longitude was best determined by simultaneous observation of lunar eclipses, yet he was so out of touch with the scientists of his day that he knew of no such data more recent than 500 years before (Arbela eclipse). When switching from 700 stadia per degree to 500, he (or Marinos) expanded longitude differences between cities accordingly (a point first realized by P. Gosselin in 1790), resulting in serious over-stretching of the Earth's east-west scale in degrees, though not distance. Achieving highly precise longitude remained a problem in geography until the application of Galileo's Jovian moon method in the 18th century. It must be added that his original topographic list cannot be reconstructed: the long tables with numbers were transmitted to posterity through copies containing many scribal errors, and people have always been adding or improving the topographic data: this is a testimony to the persistent popularity of this influential work in the history of cartography.

Astrology

Ptolemy 16century
The mathematician Claudius Ptolemy 'the Alexandrian', as depicted by a 16th-century engraving[1]

Ptolemy has been referred to as "a pro-astrological authority of the highest magnitude".[32] His astrological treatise, a work in four parts, is known by the Greek term Tetrabiblos, or the Latin equivalent Quadripartitum: "Four Books". Ptolemy's own title is unknown, but may have been the term found in some Greek manuscripts: Apotelesmatika, roughly meaning "Astrological Outcomes", "Effects" or "Prognostics".[33][34]

As a source of reference, the Tetrabiblos is said to have "enjoyed almost the authority of a Bible among the astrological writers of a thousand years or more".[35] It was first translated from Arabic into Latin by Plato of Tivoli (Tiburtinus) in 1138, while he was in Spain.[36] The Tetrabiblos is an extensive and continually reprinted treatise on the ancient principles of horoscopic astrology. That it did not quite attain the unrivaled status of the Almagest was, perhaps, because it did not cover some popular areas of the subject, particularly electional astrology (interpreting astrological charts for a particular moment to determine the outcome of a course of action to be initiated at that time), and medical astrology, which were later adoptions.

The great popularity that the Tetrabiblos did possess might be attributed to its nature as an exposition of the art of astrology, and as a compendium of astrological lore, rather than as a manual. It speaks in general terms, avoiding illustrations and details of practice. Ptolemy was concerned to defend astrology by defining its limits, compiling astronomical data that he believed was reliable and dismissing practices (such as considering the numerological significance of names) that he believed to be without sound basis.

Much of the content of the Tetrabiblos was collected from earlier sources; Ptolemy's achievement was to order his material in a systematic way, showing how the subject could, in his view, be rationalized. It is, indeed, presented as the second part of the study of astronomy of which the Almagest was the first, concerned with the influences of the celestial bodies in the sublunary sphere. Thus explanations of a sort are provided for the astrological effects of the planets, based upon their combined effects of heating, cooling, moistening, and drying.

Ptolemy's astrological outlook was quite practical: he thought that astrology was like medicine, that is conjectural, because of the many variable factors to be taken into account: the race, country, and upbringing of a person affects an individual's personality as much as, if not more than, the positions of the Sun, Moon, and planets at the precise moment of their birth, so Ptolemy saw astrology as something to be used in life but in no way relied on entirely.

A collection of one hundred aphorisms about astrology called the Centiloquium, ascribed to Ptolemy, was widely reproduced and commented on by Arabic, Latin and Hebrew scholars, and often bound together in medieval manuscripts after the Tetrabiblos as a kind of summation. It is now believed to be a much later pseudepigraphical composition. The identity and date of the actual author of the work, referred to now as Pseudo-Ptolemy, remains the subject of conjecture.

Criticism

Despite Ptolomy's prominence as a philosopher, the Dutch historian of science Eduard Jan Dijksterhuis criticizes the Tetrabiblos, stating that "it only remain puzzling that the very writer of the Almagest, who had taught how to develop astronomy from accurate observations and mathematical constructions, could put together such a system of superficial analogies and unfounded assertions."[37]

Music

Ptolemy also wrote an influential work, Harmonics, on music theory and the mathematics of music.[38] After criticizing the approaches of his predecessors, Ptolemy argued for basing musical intervals on mathematical ratios (in contrast to the followers of Aristoxenus and in agreement with the followers of Pythagoras), backed up by empirical observation (in contrast to the overly theoretical approach of the Pythagoreans). Ptolemy wrote about how musical notes could be translated into mathematical equations and vice versa in Harmonics. This is called Pythagorean tuning because it was first discovered by Pythagoras. However, Pythagoras believed that the mathematics of music should be based on the specific ratio of 3:2, whereas Ptolemy merely believed that it should just generally involve tetrachords and octaves. He presented his own divisions of the tetrachord and the octave, which he derived with the help of a monochord. His Harmonics never had the influence of his Almagest or Planetary Hypotheses, but a part of it (Book III) did encourage Kepler in his own musings on the harmony of the world (Kepler, Harmonice Mundi, Appendix to Book V).[39] Ptolemy's astronomical interests also appeared in a discussion of the "music of the spheres". See: Ptolemy's intense diatonic scale.

Optics

His Optics is a work that survives only in a poor Arabic translation and in about twenty manuscripts of a Latin version of the Arabic, which was translated by Eugene of Palermo (c. 1154). In it, Ptolemy writes about properties of light, including reflection, refraction, and colour. The work is a significant part of the early history of optics[40] and influenced the more famous 11th-century Book of Optics by Alhazen (Ibn al-Haytham). It contains the earliest surviving table of refraction from air to water, for which the values (with the exception of the 60° angle of incidence), although historically praised as experimentally derived, appear to have been obtained from an arithmetic progression.[41]

The work is also important for the early history of perception. Ptolemy combined the mathematical, philosophical and physiological traditions. He held an extramission-intromission theory of vision: the rays (or flux) from the eye formed a cone, the vertex being within the eye, and the base defining the visual field. The rays were sensitive, and conveyed information back to the observer's intellect about the distance and orientation of surfaces. Size and shape were determined by the visual angle subtended at the eye combined with perceived distance and orientation. This was one of the early statements of size-distance invariance as a cause of perceptual size and shape constancy, a view supported by the Stoics.[42] Ptolemy offered explanations for many phenomena concerning illumination and colour, size, shape, movement and binocular vision. He also divided illusions into those caused by physical or optical factors and those caused by judgmental factors. He offered an obscure explanation of the sun or moon illusion (the enlarged apparent size on the horizon) based on the difficulty of looking upwards.[43][44]

Named after Ptolemy

There are several characters or items named after Ptolemy, including:

See also

Footnotes

  1. ^ a b Since no contemporary depictions or descriptions of Ptolemy are known to have existed, later artist's impressions are unlikely to have reproduced his appearance accurately
  2. ^ a b c d Ptolemy at Encyclopædia Britannica
  3. ^ Heath, Sir Thomas (1921). A History of Greek Mathematics. Oxford: Clarendon Press. pp. vii, 273.
  4. ^ See 'Background' section on his status as a Roman citizen
  5. ^ G. J. Toomer, "Ptolemy (or Claudius Ptolemaeus). " Complete Dictionary of Scientific Biography. 2008. Retrieved from Encyclopedia.com. 21 Jan, 2013. Concerning the possibility that Ptolemy might have been born in Ptolemais Hermiou, Toomer writes: "The statement by Theodore Meliteniotes that he was born in Ptolemais Hermiou (in Upper Egypt) could be correct, but it is late (ca. 1360) and unsupported."
  6. ^ Jean Claude Pecker (2001), Understanding the Heavens: Thirty Centuries of Astronomical Ideas from Ancient Thinking to Modern Cosmology, p. 311, Springer, ISBN 3-540-63198-4.
  7. ^ Πτολεμαῖος, Georg Autenrieth, A Homeric Dictionary, on Perseus
  8. ^ Abu Maʿshar, De magnis coniunctionibus, ed.-transl. K. Yamamoto, Ch. Burnett, Leiden, 2000, 2 vols. (Arabic & Latin text); 4.1.4.
  9. ^ Jones (2010). "Ptolemy's Doctrine of the Terms and Its Reception" by Stephan Heilen, p. 68.
  10. ^ Robbins, Ptolemy Tetrabiblos "Introduction"; p. x.
  11. ^ Asger Aaboe, Episodes from the Early History of Astronomy, New York: Springer, 2001, pp. 62–65.
  12. ^ Alexander Jones, "The Adaptation of Babylonian Methods in Greek Numerical Astronomy", in The Scientific Enterprise in Antiquity and the Middle Ages, p. 99.
  13. ^ a b

    But what we really want to know is to what extent the Alexandrian mathematicians of the period from the first to the fifth centuries C.E. were Greek. Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted ... So should we assume that Ptolemy and Diophantus, Pappus and Hypatia were ethnically Greek, that their ancestors had come from Greece at some point in the past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definitively. But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities ... And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from the founding of Alexandria, small numbers of Egyptians were admitted to the privileged classes in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the Egyptians to become "Hellenized", to adopt Greek habits and the Greek language. Given that the Alexandrian mathematicians mentioned here were active several hundred years after the founding of the city, it would seem at least equally possible that they were ethnically Egyptian as that they remained ethnically Greek. In any case, it is unreasonable to portray them with purely European features when no physical descriptions exist.

    — Victor J. Katz (1998). A History of Mathematics: An Introduction, p. 184. Addison Wesley, ISBN 0-321-01618-1
  14. ^ "Ptolemy". Britannica Concise Encyclopedia. Encyclopædia Britannica, Inc., 2006.
  15. ^ George Sarton (1936). "The Unity and Diversity of the Mediterranean World", Osiris 2, p. 406–463 [429].
  16. ^ John Horace Parry (1981). The Age of Reconnaissance, p. 10. University of California Press. ISBN 0-520-04235-2.
  17. ^ J. F. Weidler (1741). Historia astronomiae, p. 177. Wittenberg: Gottlieb. (cf. Martin Bernal (1992). "Animadversions on the Origins of Western Science", Isis 83 (4), p. 596–607 [606].)
  18. ^ Martin Bernal (1992). "Animadversions on the Origins of Western Science", Isis 83 (4), p. 596–607 [602, 606].
  19. ^ Shahid Rahman; Tony Street; Hassan Tahiri, eds. (2008). "The Birth of Scientific Controversies, The Dynamics of the Arabic Tradition and Its Impact on the Development of Science: Ibn al-Haytham's Challenge of Ptolemy's Almagest". The Unity of Science in the Arabic Tradition. 11. Springer Netherlands. pp. 183–225 [183]. doi:10.1007/978-1-4020-8405-8. ISBN 978-1-4020-8404-1.
  20. ^ "Dennis Rawlins". The International Journal of Scientific History. Retrieved 2009-10-07.
  21. ^ Goldstein, Bernard R. (1997). "Saving the Phenomena: The Background to Ptolemy's Planetary Theory". Journal for the History of Astronomy. 28 (1): 1–12. Bibcode:1997JHA....28....1G. doi:10.1177/002182869702800101.
  22. ^ S. C. McCluskey, Astronomies and Cultures in Early Medieval Europe, Cambridge: Cambridge Univ. Pr. 1998, pp. 20–21.
  23. ^ Charles Homer Haskins, Studies in the History of Mediaeval Science, New York: Frederick Ungar Publishing, 1967, reprint of the Cambridge, Mass., 1927 edition
  24. ^ Dennis Duke, Ptolemy's Cosmology
  25. ^ Bernard R. Goldstein, ed., The Arabic Version of Ptolemy's Planetary Hypotheses, Transactions of the American Philosophical Society 57, no. 4 (1967), pp. 9–12. Note that the Sun is actually on average at a distance of over 46,000 Earth radii from the Earth, so Ptolemy's estimate of 1210 was off by a factor of almost 40.
  26. ^ Evans, James; Berggren, J. Lennart (2018-06-05). Geminos's Introduction to the Phenomena: A Translation and Study of a Hellenistic Survey of Astronomy. Princeton University Press. ISBN 9780691187150.
  27. ^ The north celestial pole is the point in the sky lying at the common centre of the circles which the stars appear to people in the northern hemisphere to trace out during the course of a sidereal day.
  28. ^ Shcheglov D.A. (2002–2007): "Hipparchus’ Table of Climata and Ptolemy’s Geography", Orbis Terrarum 9 (2003–2007), 177–180.
  29. ^ "DIO". www.dioi.org.
  30. ^ Bagrow 1945.
  31. ^ Talbert, Richard J. A. (2012). "Urbs Roma to Orbis Romanus". In Talbert (ed.). Ancient Perspectives: Maps and Their Places in Mesopotamia, Egypt, Greece and Rome. Chicago. pp. 170–72. ISBN 978-0-226-78940-8.
  32. ^ Jones (2010). The Use and Abuse of Ptolemy's Tetrabiblos in Renaissance and Early Modern Europe by H. Darrel Rutkin, p. 135.
  33. ^ Robbins, Ptolemy Tetrabiblos, "Introduction" p. x.
  34. ^ Jones (2010) p. xii.
  35. ^ Robbins, Ptolemy Tetrabiblos, 'Introduction' p. xii.
  36. ^ F. A. Robbins, 1940; Thorndike 1923
  37. ^ Dijksterhuis, Eduard Jan (1969). The mechanization of the world picture. Translated by C. Dikshoorn. Princeton, NJ: Princeton University Press. p. 88.
  38. ^ Wardhaugh, Benjamin (2017-07-05). Music, Experiment and Mathematics in England, 1653–1705. London and New York: Routledge. p. 7. ISBN 9781351557085.
  39. ^ Hetherington, Norriss S. Encyclopedia of Cosmology (Routledge Revivals): Historical, Philosophical, and Scientific Foundations of Modern Cosmology Routledge, 8 apr. 2014 ISBN 978-1317677666 p 527
  40. ^ Smith, A. Mark (1996). Ptolemy's Theory of Visual Perception– An English translation of the Optics. The American Philosophical Society. ISBN 0-87169-862-5. Retrieved 27 June 2009.
  41. ^ Carl Benjamin Boyer, The Rainbow: From Myth to Mathematics (1959)
  42. ^ H. W. Ross and C. Plug, "The History of Size Constancy and Size Illusions", in V. Walsh & J. Kulikowski (eds.) Perceptual Constancy: Why Things Look as They Do. Cambridge: Cambridge University Press, 1998, pp. 499–528.
  43. ^ H. E. Ross and G. M. Ross, "Did Ptolemy Understand the Moon Illusion?", Perception 5 (1976): 377–395.
  44. ^ A. I. Sabra, "Psychology Versus Mathematics: Ptolemy and Alhazen on the Moon Illusion", in E. Grant & J. E. Murdoch (eds.) Mathematics and Its Application to Science and Natural Philosophy in the Middle Ages. Cambridge: Cambridge University Press, 1987, pp. 217–247.
  45. ^ Mars Labs. Google Maps.

References

Ptolemaeus - Quadripartitum, 1622 - 4658973
Quadripartitum, 1622

Texts and translations

  • Bagrow, L. (January 1, 1945). "The Origin of Ptolemy's Geographia". Geografiska Annaler. Geografiska Annaler, Vol. 27. 27: 318–387. doi:10.2307/520071. ISSN 1651-3215. JSTOR 520071.
  • Berggren, J. Lennart, and Alexander Jones. 2000. Ptolemy's Geography: An Annotated Translation of the Theoretical Chapters. Princeton and Oxford: Princeton University Press. ISBN 0-691-01042-0.
  • Campbell, T. (1987). The Earliest Printed Maps. British Museum Press.
  • Hübner, Wolfgang, ed. 1998. Claudius Ptolemaeus, Opera quae exstant omnia Vol III/Fasc 1: ΑΠΟΤΕΛΕΣΜΑΤΙΚΑ (= Tetrabiblos). De Gruyter. ISBN 978-3-598-71746-8 (Bibliotheca scriptorum Graecorum et Romanorum Teubneriana). (The most recent edition of the Greek text of Ptolemy's astrological work, based on earlier editions by F. Boll and E. Boer.)
  • Lejeune, A. (1989) L'Optique de Claude Ptolémée dans la version latine d'après l'arabe de l'émir Eugène de Sicile. [Latin text with French translation]. Collection de travaux de l'Académie International d'Histoire des Sciences, No. 31. Leiden: E.J.Brill.
  • Neugebauer, Otto (1975). A History of Ancient Mathematical Astronomy. I-III. Berlin and New York: Sprnger Verlag.
  • Nobbe, C. F. A., ed. 1843. Claudii Ptolemaei Geographia. 3 vols. Leipzig: Carolus Tauchnitus. (The most recent edition of the complete Greek text)
  • Ptolemy. 1930. Die Harmonielehre des Klaudios Ptolemaios, edited by Ingemar Düring. Göteborgs högskolas årsskrift 36, 1930:1. Göteborg: Elanders boktr. aktiebolag. Reprint, New York: Garland Publishing, 1980.
  • Ptolemy. 2000. Harmonics, translated and commentary by Jon Solomon. Mnemosyne, Bibliotheca Classica Batava, Supplementum, 0169-8958, 203. Leiden and Boston: Brill Publishers. ISBN 90-04-11591-9
  • Robbins, Frank E. (ed.) 1940. Ptolemy Tetrabiblos. Cambridge, Massachusetts: Harvard University Press (Loeb Classical Library). ISBN 0-674-99479-5.
  • Smith, A.M. (1996) Ptolemy's theory of visual perception: An English translation of the Optics with introduction and commentary. Transactions of the American Philosophical Society, Vol. 86, Part 2. Philadelphia: The American Philosophical Society.
  • Stevenson, Edward Luther (trans. and ed.). 1932. Claudius Ptolemy: The Geography. New York: New York Public Library. Reprint, New York: Dover, 1991. (This is the only complete English translation of Ptolemy's most famous work. Unfortunately, it is marred by numerous mistakes and the placenames are given in Latinised forms, rather than in the original Greek).
  • Stückelberger, Alfred, and Gerd Graßhoff (eds). 2006. Ptolemaios, Handbuch der Geographie, Griechisch-Deutsch. 2 vols. Basel: Schwabe Verlag. ISBN 978-3-7965-2148-5. (Massive 1018 pp. scholarly edition by a team of a dozen scholars that takes account of all known manuscripts, with facing Greek and German text, footnotes on manuscript variations, color maps, and a CD with the geographical data)
  • Taub, Liba Chia (1993). Ptolemy's Universe: The Natural Philosophical and Ethical Foundations of Ptolemy's Astronomy. Chicago: Open Court Press. ISBN 0-8126-9229-2.
  • Ptolemy's Almagest, Translated and annotated by G. J. Toomer. Princeton University Press, 1998
  • Sir Thomas Heath, A History of Greek Mathematics, Oxford : Clarendon Press, 1921.

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Alexander IV of Macedon

Alexander IV (Greek: Ἀλέξανδρος Δ΄; 323–309 BC), erroneously called sometimes in modern times Aegus, was the son of Alexander the Great (Alexander III of Macedon) and Princess Roxana of Bactria.

Caesarion

Ptolemy XV Philopator Philometor Caesar (23 June, 47 BC – 23 August, 30 BC), better known by the nicknames Caesarion and Ptolemy Caesar, was the last Pharaoh of Egypt, reigning with his mother Cleopatra VII from 2 September 44 BC until her death by 12 August 30 BC and as sole ruler until his death was ordered by Octavian, the later Roman emperor Augustus. Caesarion was the eldest son of Cleopatra and possibly the only biological son of Julius Caesar, after whom he was named. He was the last sovereign member of the Ptolemaic dynasty of Egypt. In addition to being co-ruler of Egypt as Pharaoh with his mother, he was expected to be his father's successor as the Roman emperor.

Cleopatra

Cleopatra VII Philopator (Ancient Greek: Κλεοπᾰ́τρᾱ Φιλοπάτωρ, translit. Kleopátrā Philopátōr; 69 – 10 or 12 August 30 BC) was the last active ruler of the Ptolemaic Kingdom of Egypt, nominally survived as pharaoh by her son Caesarion. As a member of the Ptolemaic dynasty, she was a descendant of its founder Ptolemy I Soter, a Macedonian Greek general and companion of Alexander the Great. After the death of Cleopatra, Egypt became a province of the Roman Empire, marking the end of the Hellenistic period that had lasted since the reign of Alexander (336–323 BC). While her native language was Koine Greek, she was the first Ptolemaic ruler to learn the Egyptian language.In 58 BC, Cleopatra presumably accompanied her father Ptolemy XII during his exile to Rome, after a revolt in Egypt allowed his eldest daughter Berenice IV to claim the throne. The latter was killed in 55 BC when Ptolemy XII returned to Egypt with Roman military assistance. When Ptolemy XII died in 51 BC, he was succeeded by Cleopatra and her younger brother Ptolemy XIII as joint rulers, but a falling-out between them led to open civil war. After losing the 48 BC Battle of Pharsalus in Greece against his rival Julius Caesar in Caesar's Civil War, the Roman statesman Pompey fled to Egypt, a Roman client state. Ptolemy XIII had Pompey killed while Caesar occupied Alexandria in pursuit of Pompey. Caesar, a consul of the Roman Republic, attempted to reconcile Ptolemy XIII with Cleopatra. Ptolemy XIII's chief adviser Potheinos viewed Caesar's terms as favoring Cleopatra, and so his forces, which eventually fell under the control of Cleopatra's younger sister, Arsinoe IV, besieged Caesar and Cleopatra at the palace. The siege was lifted by reinforcements in early 47 BC and Ptolemy XIII died shortly thereafter in the Battle of the Nile. Arsinoe IV was exiled to Ephesus, and Caesar, now an elected dictator, declared Cleopatra and her younger brother Ptolemy XIV as joint rulers of Egypt. However, Caesar maintained a private affair with Cleopatra that produced a son, Caesarion (Ptolemy XV). Cleopatra traveled to Rome as a client queen in 46 and 44 BC, staying at Caesar's villa. When Caesar was assassinated in 44 BC, Cleopatra attempted to have Caesarion named as his heir, but this fell instead to Caesar's grandnephew Octavian (known as Augustus by 27 BC, when he became the first Roman emperor). Cleopatra then had Ptolemy XIV killed and elevated Caesarion as co-ruler.

In the Liberators' civil war of 43–42 BC, Cleopatra sided with the Roman Second Triumvirate formed by Octavian, Mark Antony, and Marcus Aemilius Lepidus. After their meeting at Tarsos in 41 BC, Cleopatra had an affair with Antony that would eventually produce three children: Alexander Helios, Cleopatra Selene II, and Ptolemy Philadelphus. Antony used his authority as a triumvir to carry out the execution of Arsinoe IV at Cleopatra's request. He became increasingly reliant on Cleopatra for both funding and military aid during his invasions of the Parthian Empire and Kingdom of Armenia. In the Donations of Alexandria, Cleopatra's children with Antony were declared rulers over various erstwhile territories under Antony's authority. This event, along with his marriage to Cleopatra and divorce of Octavian's sister Octavia Minor, led to the Final War of the Roman Republic. After engaging in a war of propaganda, Octavian forced Antony's allies in the Roman Senate to flee Rome in 32 BC and declared war on Cleopatra. The naval fleet of Antony and Cleopatra was defeated at the 31 BC Battle of Actium by Octavian's general Agrippa. Octavian's forces invaded Egypt in 30 BC and defeated those of Antony, leading to his suicide. When Cleopatra learned that Octavian planned to bring her to Rome for his triumphal procession, she committed suicide by poisoning, with the popular belief being that she was bitten by an asp.

Cleopatra's legacy survives in numerous works of art, both ancient and modern. Roman historiography and Latin poetry produced a generally polemic and negative view of the queen that pervaded later Medieval and Renaissance literature. In the visual arts, ancient depictions of Cleopatra include Roman and Ptolemaic coinage, statues, busts, reliefs, cameo glass, cameo carvings, and paintings. She was the subject of many works in Renaissance and Baroque art, which included sculptures, paintings, poetry, theatrical dramas such as William Shakespeare's Antony and Cleopatra (1608), and operas such as George Frideric Handel's Giulio Cesare in Egitto (1724). In modern times Cleopatra has appeared in both the applied and fine arts, burlesque satire, Hollywood films such as Cleopatra (1963), and brand images for commercial products, becoming a pop culture icon of Egyptomania since the Victorian era.

Library of Alexandria

The Great Library of Alexandria in Alexandria, Egypt, was one of the largest and most significant libraries of the ancient world. The Library was part of a larger research institution called the Mouseion, which was dedicated to the Muses, the nine goddesses of the arts. The idea of a universal library in Alexandria may have been proposed by Demetrius of Phalerum, an exiled Athenian statesman living in Alexandria, to Ptolemy I Soter, who may have established plans for the Library, but the Library itself was probably not built until the reign of his son Ptolemy II Philadelphus. The Library quickly acquired a large number of papyrus scrolls, due largely to the Ptolemaic kings' aggressive and well-funded policies for procuring texts. It is unknown precisely how many such scrolls were housed at any given time, but estimates range from 40,000 to 400,000 at its height.

Alexandria came to be regarded as the capital of knowledge and learning, in part because of the Great Library. Many important and influential scholars worked at the Library during the third and second centuries BC, including, among many others: Zenodotus of Ephesus, who worked towards standardizing the texts of the Homeric poems; Callimachus, who wrote the Pinakes, sometimes considered to be the world's first library catalogue; Apollonius of Rhodes, who composed the epic poem the Argonautica; Eratosthenes of Cyrene, who calculated the circumference of the earth within a few hundred kilometers of accuracy; Aristophanes of Byzantium, who invented the system of Greek diacritics and was the first to divide poetic texts into lines; and Aristarchus of Samothrace, who produced the definitive texts of the Homeric poems as well as extensive commentaries on them. During the reign of Ptolemy III Euergetes, a daughter library was established in the Serapeum, a temple to the Greco-Egyptian god Serapis.

Despite the widespread modern belief that the Library was "burned" once and cataclysmically destroyed, the Library actually declined gradually over the course of several centuries, starting with the purging of intellectuals from Alexandria in 145 BC during the reign of Ptolemy VIII Physcon, which resulted in Aristarchus of Samothrace, the head librarian, resigning from his position and exiling himself to Cyprus. Many other scholars, including Dionysius Thrax and Apollodorus of Athens, fled to other cities, where they continued teaching and conducting scholarship. The Library, or part of its collection, was accidentally burned by Julius Caesar during his civil war in 48 BC, but it is unclear how much was actually destroyed and it seems to have either survived or been rebuilt shortly thereafter; the geographer Strabo mentions having visited the Mouseion in around 20 BC and the prodigious scholarly output of Didymus Chalcenterus in Alexandria from this period indicates that he had access to at least some of the Library's resources.

The Library dwindled during the Roman Period, due to lack of funding and support. Its membership appears to have ceased by the 260s AD. Between 270 and 275 AD, the city of Alexandria saw a rebellion and an imperial counterattack that probably destroyed whatever remained of the Library, if it still existed at that time. The daughter library of the Serapeum may have survived after the main Library's destruction. The Serapeum was vandalized and demolished in 391 AD under a decree issued by Coptic Christian Pope Theophilus of Alexandria, but it does not seem to have housed books at the time and was mainly used as a gathering place for Neoplatonist philosophers following the teachings of Iamblichus.

Ptolemaic Kingdom

The Ptolemaic Kingdom (; Koinē Greek: Πτολεμαϊκὴ βασιλεία, translit. Ptolemaïkḕ basileía) was a Hellenistic kingdom based in ancient Egypt. It was ruled by the Ptolemaic dynasty, which started with Ptolemy I Soter's accession after the death of Alexander the Great in 323 BC and which ended with the death of Cleopatra and the Roman conquest in 30 BC.

The Ptolemaic Kingdom was founded in 305 BC by Ptolemy I Soter, a diadochus originally from Macedon in northern Greece who declared himself pharaoh of Egypt and created a powerful Macedonian Greek dynasty that ruled an area stretching from southern Syria to Cyrene and south to Nubia. Scholars also argue that the kingdom was founded in 304 BC because of different use of calendars: Ptolemy crowned himself in 304 BC on the ancient Egyptian calendar, but in 305 BC on the ancient Macedonian calendar; to resolve the issue, the year 305/4 was counted as the first year of Ptolemaic Kingdom in Demotic papyri.Alexandria, a Greek polis founded by Alexander the Great, became the capital city and a major center of Greek culture and trade. To gain recognition by the native Egyptian populace, the Ptolemies named themselves as pharaohs. The later Ptolemies took on Egyptian traditions by marrying their siblings per the Osiris myth, had themselves portrayed on public monuments in Egyptian style and dress, and participated in Egyptian religious life. The Ptolemies were involved in foreign and civil wars that led to the decline of the kingdom and its final conquest by Rome. Their rivalry with the neighboring Seleucid Empire of West Asia led to a series of Syrian Wars in which both powers jockeyed for control of the Levant. Hellenistic culture continued to thrive in Egypt throughout the Roman and Byzantine periods until the Muslim conquest.

Ptolemaic dynasty

The Ptolemaic dynasty (; Ancient Greek: Πτολεμαῖοι, Ptolemaioi), sometimes also known as the Lagids () or Lagidae (; Λαγίδαι, Lagidai, after Lagus, Ptolemy I's father), was a Macedonian Greek royal family, which ruled the Ptolemaic Kingdom in Egypt during the Hellenistic period. Their rule lasted for 275 years, from 305 to 30 BC. They were the last dynasty of ancient Egypt.

Ptolemy, one of the seven somatophylakes (bodyguards) who served as Alexander the Great's generals and deputies, was appointed satrap of Egypt after Alexander's death in 323 BC. In 305 BC, he declared himself Ptolemy I, later known as Sōter "Saviour".

The Egyptians soon accepted the Ptolemies as the successors to the pharaohs of independent Egypt. Ptolemy's family ruled Egypt until the Roman conquest of 30 BC.

All the male rulers of the dynasty took the name Ptolemy. Ptolemaic queens regnant, some of whom were married to their brothers, were usually called Cleopatra, Arsinoe or Berenice. The most famous member of the line was the last queen, Cleopatra VII, known for her role in the Roman political battles between Julius Caesar and Pompey, and later between Octavian and Mark Antony. Her apparent suicide at the conquest by Rome marked the end of Ptolemaic rule in Egypt.

Ptolemy III Euergetes

Ptolemy III Euergetes (Greek: Πτολεμαῖος Εὐεργέτης, Ptolemaĩos Euergétēs "Ptolemy the Benefactor"; 284–222 BC) was the third king of the Ptolemaic dynasty in Egypt from 246 to 222 BC.

Ptolemy II Philadelphus

Ptolemy II Philadelphus (Greek: Πτολεμαῖος Φιλάδελφος, Ptolemaîos Philádelphos "Ptolemy, lover of his sister"; 308/9–246 BCE) was the king of Ptolemaic Egypt from 283 to 246 BCE. He was the son of Ptolemy I Soter, the Macedonian Greek general of Alexander the Great who founded the Ptolemaic Kingdom after the death of Alexander, and queen Berenice I, originally from Macedon in northern Greece.

During Ptolemy II's reign, the material and literary splendour of the Alexandrian court was at its height. He promoted the Museum and Library of Alexandria. He erected a commemorative stele, the Great Mendes Stela. He also led the Ptolemaic Kingdom against the rival Seleucid Empire in the first of a series of Syrian Wars that witnessed periodic territorial changes between the two powers in West Asia.

Ptolemy IV Philopator

Ptolemy IV Philopator (Greek: Πτολεμαῖος Φιλοπάτωρ, Ptolemaĩos Philopátōr "Ptolemy, lover of his Father"; 245/4–204 BC), son of Ptolemy III and Berenice II, was the fourth Pharaoh of Ptolemaic Egypt from 221 to 204 BC. The decline of the Ptolemaic dynasty began under the reign of Ptolemy IV.

Ptolemy IX Lathyros

Ptolemy IX Soter II (Greek: Πτολεμαῖος Σωτήρ, Ptolemaĩos Sōtḗr "Ptolemy the Saviour"), commonly nicknamed Lathyros (Λάθυρος, Láthuros "chickpea"), reigned twice as king of Ptolemaic Egypt. He took the throne after the death of his father Ptolemy VIII in 116 BC, in joint rule with his mother Cleopatra III.

He was deposed in 107 BC by his mother and brother, Ptolemy X. He ruled Egypt once more from his brother's death in 88 BC to his own death in 81 BC. The legitimate Ptolemaic line in Egypt ended shortly after the death of Ptolemy IX with the death of his nephew Ptolemy XI. Ptolemy IX's illegitimate son Ptolemy XII then took the throne of Egypt.

Ptolemy I Soter

Ptolemy I Soter (; Greek: Πτολεμαῖος Σωτήρ, Ptolemaĩos Sōtḗr "Ptolemy the Savior"; c. 367 BC – January 282 BC) was a companion and historian of Alexander the Great of the Kingdom of Macedon in northern Greece who became ruler of Egypt, part of Alexander's former empire. Ptolemy was pharaoh of Ptolemaic Egypt from 305/304 to 282 BC. He was the founder of the Ptolemaic dynasty which ruled Egypt until the death of Cleopatra in 30 BC, turning the country into a Hellenistic kingdom and Alexandria into a center of Greek culture.Ptolemy I was the son of Arsinoe of Macedon by either her husband Lagus or Philip II of Macedon, the father of Alexander. Ptolemy was one of Alexander's most trusted companions and military officers. He had been an intimate friend of Alexander since childhood. After the death of Alexander in 323 BC, Ptolemy retrieved his body as it was en route to be buried in Macedon, placing it in Memphis instead, where it was later moved to Alexandria in a new tomb. Afterwards he joined a coalition against Perdiccas, the royal regent over Philip III of Macedon. The latter invaded Egypt but was assassinated by his own officers in 320 BC, allowing Ptolemy I to consolidate his control over the country.

Ptolemy I may have married Thaïs, his mistress during the life of Alexander; he is known to have married the Persian noblewoman Artakama on Alexander's orders. He later married Eurydice, daughter of the Macedonian regent Antipater; their sons Ptolemy Keraunos and Meleager ruled in turn as kings of the kingdom their maternal grandfather had governed. Ptolemy's final marriage was to Eurydice's cousin and lady-in-waiting, Berenice I. Their son Ptolemy II, Ptolemy I's successor, ruled jointly with his sister-wife Arsinoe II, who had previously been married to their father's political enemy Lysimachus and their half-brother Ptolemy Keraunos.

Ptolemy VIII Physcon

Ptolemy VIII Euergetes II (Greek: Πτολεμαῖος Εὐεργέτης, Ptolemaĩos Euergétēs "Ptolemy the Benefactor"; c. 182 BC – June 26, 116 BC), nicknamed Physcon (Φύσκων "the Fat"), was a king of the Ptolemaic dynasty in Egypt.

Ptolemy VII Neos Philopator

Ptolemy VII Neos Philopator (Greek: Πτολεμαῖος Νέος Φιλοπάτωρ, Ptolemaĩos Néos Philopátōr "Ptolemy the New Beloved of his Father") was an Egyptian king of the Ptolemaic period. His reign is controversial, and it is possible that he did not reign at all, but was only granted royal dignity posthumously.

Ptolemy VI Philometor

Ptolemy VI Philometor (Greek: Πτολεμαῖος Φιλομήτωρ, Ptolemaĩos Philomḗtōr "Ptolemy, lover of his Mother"); c. 186–145 BC) was a king of Egypt from the Ptolemaic period. He reigned from 180 to 164 BC and from 163 to 145 BC.

Ptolemy V Epiphanes

Ptolemy V Epiphanes (Greek: Πτολεμαῖος Ἐπιφανής, Ptolemaĩos Epiphanḗs "Ptolemy the Illustrious"); 210–181 BC), son of Ptolemy IV Philopator and Arsinoe III of Egypt, was the fifth ruler of the Ptolemaic dynasty from 204 to 181 BC. He inherited the throne at the age of five, and under a series of regents, the kingdom was paralyzed. The Rosetta Stone was produced during his reign as an adult.

Ptolemy XIII Theos Philopator

Ptolemy XIII Theos Philopator (Greek: Πτολεμαῖος Θεός Φιλοπάτωρ, Ptolemaĩos Theós Philopátōr "Ptolemy, God Beloved of his Father"; 62 BC/61 BC – prob. January 13, 47 BC, reigned from 51 BC) was one of the last members of the Ptolemaic dynasty (305–30 BC) of Egypt. He was the son of Ptolemy XII and the brother of and co-ruler with Cleopatra VII. Cleopatra's exit from Egypt caused a civil war to break out between the pharaohs. Ptolemy later ruled jointly with his other sister, Arsinoe IV.

Ptolemy XII Auletes

Ptolemy Neos Dionysos Theos Philopator Theos Philadelphos (Ancient Greek: Πτολεμαῖος Νέος Διόνυσος Θεός Φιλοπάτωρ Θεός Φιλάδελφος, Ptolemaios Néos Diónysos Theós Philopátōr Theós Philádelphos "Ptolemy New Dionysus, God Beloved of his Father, God Beloved of his Brother"; uncertain–before 22 March 51 BC) was a pharaoh of the Ptolemaic dynasty of Ancient Egypt. He was commonly known as Auletes (Αὐλητής, Aulētḗs "the Flutist"), referring to the king's love of playing the flute in Dionysian festivals.

He ruled from 80 to 58 BC and again from 55 to 51 BC, with an interregnum of forced exile to Rome as his eldest daughter Berenice IV claimed the Ptolemaic throne. With the funding and military assistance of the Roman Republic, which officially viewed Ptolemy XII as one of its client rulers, he was able to recapture Ptolemaic Egypt and have his rival daughter Berenice IV killed. On his death he was succeeded by his daughter Cleopatra VII and son Ptolemy XIII as joint rulers, as stipulated in his will and testament.

Ptolemy XI Alexander II

Ptolemy XI Alexander II (Greek: Πτολεμαῖος Ἀλέξανδρος, Ptolemaĩos Aléxandros) was a member of the Ptolemaic dynasty who ruled Egypt for a few days in 80 BC.

Ptolemy X Alexander I

Ptolemy X Alexander I (Greek: Πτολεμαῖος Ἀλέξανδρος, Ptolemaĩos Aléxandros) was King of Egypt from 110 BC to 109 BC and 107 BC till his death in 88 BC, in co-regency with his mother Cleopatra III until 101 BC, and then possibly with his niece-wife Berenice III.

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