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1045:, which asserted that the scholar fabricated his observations to fit his theories. Newton accused Ptolemy of systematically inventing data or doctoring the data of earlier astronomers, and labelled him "the most successful fraud in the history of science". One striking error noted by Newton was an autumn equinox said to have been observed by Ptolemy and "measured with the greatest care" at 2pm on 25 September 132, when the equinox should have been observed at 9:54am the day prior. Herbert Lewis, who had reworked some of Ptolemy's calculations, agreed with Newton that "Ptolemy was an outrageous fraud", and that "all those results capable of statistical analysis point beyond question towards fraud and against accidental error".
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because of his knowledge of precession, which distinguishes him from all his predecessors. Hipparchus' celestial globe had an ecliptic drawn in, but the coordinates were equatorial. Since
Hipparchus' star catalogue has not survived in its original form, but was absorbed into the Almagest star catalogue (and heavily revised in the 265 years in between), the Almagest star catalogue is the oldest one in which complete tables of coordinates and magnitudes have come down to us.
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great, if not indeed the first, scientific treatise." He continued, "Newton’s work does focus critical attention on the many difficulties and inconsistencies apparent in the fine structure of the
Almagest. In particular, his conclusion that the Almagest is not a historical account of how Ptolemy actually derived his models and parameters is essentially the same as mine, although our reasons for this conclusion and our inferences from it differ radically."
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indeed be traced back to
Hipparchus, but not that the complete star catalogue was simply "copied". Rather, Hipparchus' major errors are no longer present in the Almagest and, on the other hand, Hipparchus' star catalogue had some stars that are entirely absent from the Almagest. It can be concluded that Hipparchus' star catalogue, while forming the basis, has been reobserved and revised.
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Under the scrutiny of modern scholarship, and the cross-checking of observations contained in the
Almagest against figures produced through backwards extrapolation, various patterns of errors have emerged within the work. A prominent example is Ptolemy's use of measurements said to have been taken at
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Ptolemy's comprehensive treatise of mathematical astronomy superseded most older texts of Greek astronomy. Some were more specialized and thus of less interest; others simply became outdated by the newer models. As a result, the older texts ceased to be copied and were gradually lost. Much of what we
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adopted
Hipparchus' solar model, which consisted of a simple eccentric deferent. For the Moon, Ptolemy began with Hipparchus' epicycle-on-deferent, then added a device that historians of astronomy refer to as a "crank mechanism": he succeeded in creating models for the other planets, where Hipparchus
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Subtracting the systematic error leaves other errors that cannot be explained by precession. Of these errors, about 18 to 20 are also found in
Hipparchus' star catalogue (which can only be reconstructed incompletely). From this it can be concluded that a subset of star coordinates in the Almagest can
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Errors in
Ptolemy’s star catalogue: In blue are the stars in the Almagest’s list; in orange, the points of modern measurements transformed to the second century, taking proper motion into account; the orange and blue band is the ecliptic; and the blue curve is the equator. Figure inspired by Hoffmann
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Many of the longitudes and latitudes have been corrupted in the various manuscripts. Most of these errors can be explained by similarities in the symbols used for different numbers. For example, the Greek letters Α and Δ were used to mean 1 and 4 respectively, but because these look similar copyists
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consists of thirteen sections, called books. As with many medieval manuscripts that were handcopied or, particularly, printed in the early years of printing, there were considerable differences between various editions of the same text, as the process of transcription was highly personal. An example
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The catalogue actually contained 1,028 entries, but three of these were deliberate duplicates, because
Ptolemy regarded certain stars as being shared between adjacent constellations. Three other entries were non-stellar, i.e. the Double Cluster in Perseus, M44 (Praesepe) in Cancer, and the globular
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The layout of the catalogue has always been tabular. Ptolemy writes explicitly that the coordinates are given as (ecliptical) "longitudes" and "latitudes", which are given in columns, so this has probably always been the case. It is significant that
Ptolemy chooses the ecliptical coordinate system
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278 years earlier, rejected the qualification of fraud. John
Phillips Britton, Visiting Fellow at Yale University, wrote of R.R. Newton, "I think that his main conclusion with respect to Ptolemy’s stature and achievements as an astronomer is simply wrong, and that the Almagest should be seen as a
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Although no line figures have survived from antiquity, the figures can be reconstructed on the basis of the descriptions in the star catalogue: The exact celestial coordinates of the figures' heads, feet, arms, wings and other body parts are recorded. It is therefore possible to draw the stick
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The star catalogue contains 48 constellations, which have different surface areas and numbers of stars. In Book VIII, Chapter 3, Ptolemy writes that the constellations should be outlined on a globe, but it is unclear exactly how he means this: should surrounding polygons be drawn or should the
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covers motion in latitude, that is, the deviation of planets from the ecliptic. The final topic of this chapter also covers how to determine when a planet first becomes visible after being hidden by the glare of the sun, as well as the last time it is seen before being hidden by the sun's
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Even without the errors introduced by copyists, and even accounting for the fact that the longitudes are more appropriate for 58 AD than for 137 AD, the latitudes and longitudes are not fully accurate, with errors as great as large fractions of a degree. Some errors may be due to
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Of the stars in the catalogue, 108 (just over 10%) were classified by Ptolemy as 'unformed', by which he meant lying outside the recognized constellation figures. These were later absorbed into their surrounding constellations or in some cases used to form new constellations.
981:, was intended to supplant the old translation. The new translation was a great improvement; the new commentary was not, and aroused criticism. The Pope declined the dedication of George's work, and Regiomontanus's translation had the upper hand for over 100 years.
1140:, including detailed historical comments in a 69-page preface. It has been described as "suffer from excessive literalness, particularly where the text is difficult" by Toomer, and as "very faulty" by Serge Jodra. The scanned books are available in full at the
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existed.) This system is believed to have originated with Hipparchus. The stellar positions too are of Hipparchan origin, despite Ptolemy's claim to the contrary. Ptolemy identified 48 constellations: The 12 of the
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as a textbook of mathematical astronomy. It explained geometrical models of the planets based on combinations of circles, which could be used to predict the motions of celestial objects. In a later book, the
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522:, which occurs when planets appear to pause, then briefly reverse their motion against the background of the zodiac. Ptolemy understood these terms to apply to Mercury and Venus as well as the outer planets.
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and the northern zodiac constellations (Aries through Virgo) are in the table at the end of Book VII, while the rest are in the table at the beginning of Book VIII. The brightest stars were marked first
595:, most of the stars of a given zodiac constellation in the catalog fall in the 30-degree range designated by the same name. The ecliptic longitudes are about 26° lower than those of AD 2000 (the
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wrote, "Unfortunately, Newton’s arguments in support of these charges are marred by all manner of distortions, misunderstandings, and excesses of rhetoric due to an intensely polemical style."
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The overall quality of Claudius Ptolemy's scholarship and place as "one of the most outstanding scientists of antiquity" has been challenged by several modern writers, most prominently by
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that all coordinates were calculated from Hipparchus' observations, whereby the precession constant, which was known too inaccurately at the time, led to a summation error (Delambre 1817);
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noon, but which systematically produce readings that are off by half an hour, as if the observations were taken at 12:30pm. However, an explanation for this error was found in 1969.
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which was 265 years earlier (Alm. VII, 2). But calculations show that his ecliptic longitudes correspond more closely to around the middle of the first century CE (+48 to +58).
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Example of a Greek manuscript of the Almagest, showing a table layout for the stars of Ursa Minor. The functions of the columns, colours and rows are labelled in this depiction.
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As mentioned, Ptolemy includes a star catalog containing 1022 stars. He says that he "observed as many stars as it was possible to perceive, even to the sixth magnitude". The
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covers problems associated with the daily motion attributed to the heavens, namely risings and settings of celestial objects, the length of daylight, the determination of
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and the eccentric deferent to astronomy. Hipparchus (2nd century BC) had crafted mathematical models of the motion of the Sun and Moon. Hipparchus had some knowledge of
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includes five main points, each of which is the subject of a chapter in Book I. What follows is a close paraphrase of Ptolemy's own words from Toomer's translation.
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are given in terms of a zodiac constellation and a number of degrees and fractions of a degree. The zodiac constellations each represent exactly 30°, starting with
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sometimes wrote the wrong one. In Arabic manuscripts, there was confusion between for example 3 and 8 (ج and ح). (At least one translator also introduced errors.
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The figure he used is based on Hipparchus' own estimate for precession, which was 1° in 100 years, instead of the correct 1° in 72 years. Dating attempts through
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Vogt, H. (1925). "Versuch einer Wiederherstellung von Hipparchs Fixsternverzeichnis" [Attempting a restoration of Hipparchus' fixed-star catalog].
825:, and he felt that Greek models should match those of the Babylonians in accuracy. He was unable to create accurate models for the remaining five planets.
394:, and other observations that change with the observer's position. There is also a study of the angles made by the ecliptic with the vertical, with tables.
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Ptolemy inherited from his Greek predecessors a geometrical toolbox and a partial set of models for predicting where the planets would appear in the sky.
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representing longitude 0° to 30°. The degrees are added to the lower limit of the 30-degree range to obtain the longitude. Unlike the situation with the
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Geometric construction used by Hipparchus in his determination of the distances to the Sun and Moon, which was later incorporated into Ptolemy's work
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A podcast discussion by Prof. M Heath and Dr A. Chapman of a recent re-discovery of a 14th-century manuscript in the university of Leeds Library
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These modern stick figures as a reconstruction of the historical constellations of the Almagest are available in the free planetarium software
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made a full translation accompanied by a commentary that was as long as the original text. George's translation, done under the patronage of
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Hamilton, N. T.; Swerdlow, N. M. (1987). "The Canobic Inscription: Ptolemy's Earliest Work". In Berggren, J. L.; Goldstein, B. R. (eds.).
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by Robert Van Gent. Positions for any date, translation of dates in calendars used by Ptolemy. Extensive list of references and articles.
657:, who translated an Arabic manuscript into Latin around 1175, put 300° for the latitude of several stars. He had apparently learned from
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made the first Latin translation directly from a Greek copy, but it was not as influential as a later translation into Latin made in
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made in the 12th century from an Arabic translation, which would endure until original Greek copies resurfaced in the 15th century.
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that were formally adopted by the International Astronomical Union in 1922, with official boundaries that were agreed in 1928.
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The Earth, in relation to the distance of the fixed stars, has no appreciable size and must be treated as a mathematical point.
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Although some have described the charges laid by Newton as "erudite and imposing", others have disagreed with the findings.
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Since Tycho Brahe found this offset, astronomers and historians investigated this problem and suggested several causes:
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Dambis, A. K.; Efremov, Yu. N. (2000). "Dating Ptolemy's Star Catalogue through Proper Motions: The Hipparchan Epoch".
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for Hipparchus. In the 13th century a Spanish version was produced, which was later translated under the patronage of
344:'s cosmology: on the spherical form of the heavens, with the spherical Earth lying motionless as the center, with the
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which are now around ecliptic longitude 91° are attributed longitudes in the Almagest of 66° and 64°20' respectively.
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Kunitzsch, Paul (2008) . "Almagest: Its Reception and Transmission in the Islamic World". In Selin, Helaine (ed.).
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that the difference is a sum of individual errors of various kinds, including calibration with outdated solar data;
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In the 15th century, a Greek version appeared in Western Europe. The German astronomer Johannes Müller (known as
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The first translations into Arabic were made in the 9th century, with two separate efforts, one sponsored by the
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1288: = 6). Each numerical magnitude was considered twice the brightness of the following one, which is a
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795:(5th century AD) put Mercury and Venus in motion around the Sun. Ptolemy's authority was preferred by most
689:. These were probably measured by a different person or persons from the others, and in an inaccurate way.
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causing stars that are low in the sky to appear higher than where they really are. A series of stars in
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was organized is given below; it is a Latin edition printed in 1515 at Venice by Petrus Lichtenstein.
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that the data had in fact been observed a century earlier by Menelaus of Alexandria (Björnbo 1901);
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2507:(24 February 1978). "Casting Doubt on Ptolemy: The Crime of Claudius Ptolemy. Robert R. Newton".
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A direct French translation from the Greek text was published in two volumes in 1813 and 1816 by
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in 1984, with a second edition in 1998. The third was a partial translation by Bruce M. Perry in
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cover the motions of the fixed stars, including precession of the equinoxes. They also contain a
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could not have been completed before about 150, a quarter-century after Ptolemy began observing.
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of the stars also appear to date the actual observation to Hipparchus' time instead of Ptolemy.
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1962:. Studies in the History of Mathematics and Physical Sciences. Vol. 14. Berlin: Springer.
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from the Arabic (finished in 1175). Gerard translated the Arabic text while working at the
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165:, in 147 or 148. N. T. Hamilton found that the version of Ptolemy's models set out in the
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Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures
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603:(138 AD) and that he found that the longitudes had increased by 2° 40′ since the time of
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epoch). Ptolemy says that the ecliptic longitudes are for the beginning of the reign of
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PDFs of different resolutions. Edition of Petrus Liechtenstein, Latin translation of
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Gerard of Cremona's Latin translation made in Toledo about 1175 from the Arabic, ...
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Hanson, Norwood Russell (1960). "The Mathematical Power of Epicyclical Astronomy".
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2274:. In Juste, David; van Dalen, Benno; Hasse, Dag Nikolaus; Burnett, Charles (eds.).
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1963:
1284: = 1), while the faintest visible to the naked eye were sixth magnitude (
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3024:. Digitized version of manuscript made in Northern Italy c. 1200–1225 held by the
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cluster Omega Centauri. Ptolemy states that the longitudes (which increase due to
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Maimonides explaining why you need to learn Almagest first to understand science
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know about the work of astronomers like Hipparchus comes from references in the
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Hoskin, Michael (1999). "2. Astronomy in Antiquity". In Hoskin, Michael (ed.).
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figures in the modern sense so that they fit the description in the Almagest.
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that Ptolemy's instrument was wrongly calibrated and had a systematic offset.
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228:'greatest'. The Arabic name was popularized by a Latin translation known as
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1528 copy of a Latin translation of "Almagestum", translated from Greek by
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that was accepted for more than 1,200 years from its origin in Hellenistic
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Models and Precision: The Quality of Ptolemy's Observations and Parameters
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122:. One of the most influential scientific texts in history, it canonized a
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364:(the apparent path of the Sun through the stars); and an introduction to
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2668:(1977). "Scandal in the Heavens: Renowned Astronomer Accused of Fraud".
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French translation, with some diagrams animated, interactive, or random
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figures be sketched or even line figures be drawn? This is not stated.
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addresses general issues associated with creating models for the five
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2278:. Ptolemaeus Arabus et Latinus. Turnhout: Brepols. pp. 305–307.
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Around AD 150, Ptolemy wrote his great handbook of astronomy called
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by Christian Heinrich Friedrich Peters and Edward Ball Knobel, 1915
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Online luni-solar and planetary ephemeris calculator based on the
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1267:(138 AD), whereas the latitudes do not change with time (but see
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3002:. PDF scans of Heiberg's Greek edition, now in the public domain
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2565:"The Star Catalogue Commonly Appended to the Alfonsine Tables"
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4592:
4570:
4226:
3217:
3068:
2268:
George of Trebizond on Planetary Order, Distances, and Orbs (
2179:
Ptolemy's Catalogue of the Stars – A Revision of the Almagest
1006:("The Ultimate Grasp of the Divisions of Spheres", 1138–39).
939:
777:
658:
596:
491:
270:
260:
244:
218:
2215:
The Almagest: Introduction to the Mathematics of the Heavens
1131:
The Almagest: Introduction to the Mathematics of the Heavens
1088:
309:
3302:
1708:
1706:
1368:
685:
are off by a couple of degrees, including the star we call
495:
427:
352:
revolving around the Earth. Then follows an explanation of
94:
63:
57:
5237:
2993:
Syntaxis mathematica in J.L. Heiberg's edition (1898–1903)
2365:"Ptolemy's Determination of the Obliquity of the Ecliptic"
2306:
A Sourcebook for Ancient Greek: Grammar, Poetry, and Prose
60:
1718:
401:
379:
48:
1997:(in French). Vol. 1. Paris: J. Hermann – via
1775:
1773:
1703:
1565:
1563:
546:
The celestial realm is spherical, and moves as a sphere.
2164:– via Astronomy Library of the Vienna University.
1691:
934:
No Latin translation was made before the 12th century.
776:
Other classical writers suggested different sequences.
692:
154:. It is also a key source of information about ancient
2959:
Islamic Science and the Making of European Renaissance
2890:
Early Physics and Astronomy: A Historical Introduction
2714:(in French) (Online ed.) – via Imago Mundi.
2241:
Islamic Science and the Making of European Renaissance
2015:(in French). Vol. 2. Paris: H. Grand – via
1802:
1800:
1344:
5190:
2950:– via MDZ/München, Bayerische Staatsbibliothek.
1824:
1812:
1770:
1646:"How Ptolemy’s spare stars became new constellations"
1560:
1548:
1301:, 21 to the north of the zodiac, and 15 to the south.
841:
had failed, by introducing a third device called the
412:
of the equinoxes and begins explaining the theory of
400:
covers the length of the year, and the motion of the
78:
66:
2113:(2nd ed.). Berlin: Springer. pp. 140–141.
1760:
1758:
1663:
1651:
861:
or partial spheres. In contrast to the mathematical
212:
202:
191:
69:
51:
5274:
2493:
Quarterly Journal of the Royal Astronomical Society
2468:
Quarterly Journal of the Royal Astronomical Society
1848:
1797:
1785:
1745:
1743:
1741:
1739:
1737:
1735:
1733:
705:These constellations form the basis for the modern
462:of 1022 stars, described by their positions in the
54:
45:
1836:
2429:Monthly Notices of the Royal Astronomical Society
2276:Ptolemy's Science of the Stars in the Middle Ages
1884:
1872:
1860:
1755:
791:) placed the Sun second in order after the Moon.
300:
5290:
2023:
1730:
1362:
1163:Ptolemy's catalogue of stars; a revision of the
1103:into English have been published. The first, by
889:became an authoritative work for many centuries.
2424:"On the Origin of Ptolemy's Catalogue of Stars"
931:is thought to be the first Arabic translator.
3940:
3926:
3418:
3114:
2775:Bulletin of the American Mathematical Society
2745:The History and Practice of Ancient Astronomy
2390:
1581:
3077:Animation of Ptolemy's model of the universe
2892:(2nd ed.). Cambridge University Press.
2538:"The Genesis of Hipparchus' Celestial Globe"
2168:
2142:
1625:
1609:
1593:
1538:
1457:
1386:
1001:
741:Ptolemy assigned the following order to the
716:
640:
279:), and the superlative form of this (Greek:
229:
197:
186:
3432:
2801:A History of Ancient Mathematical Astronomy
2771:"Mathematical Methods in Ancient Astronomy"
2213:— (2014). Donahue, William H. (ed.).
2026:From Ancient Omens to Statistical Mechanics
2009:Composition mathématique de Claude Ptolémée
1991:Composition mathématique de Claude Ptolémée
254:
3933:
3919:
3425:
3411:
3121:
3107:
2795:
2765:
2542:Mediterranean Archaeology and Archaeometry
2489:"Ptolemy Revisited - Reply to R.R. Newton"
2084:The Cambridge Concise History of Astronomy
1083:Claudii Ptolemaei opera quae exstant omnia
992:, brought back Arabic disputations of the
142:worlds, and in Western Europe through the
2786:
2580:
2562:
2553:
2503:
2486:
2458:
2441:
2355:
2302:
2283:
2208:. Chicago: Encyclopaedia Brittanica, Inc.
2108:
1918:(1st ed.). New York: Facts on File.
1830:
1818:
1779:
1697:
1374:
1350:
799:and late medieval European astronomers.
725:16th-century representation of Ptolemy's
552:The Earth is at the center of the cosmos.
16:Astronomical treatise by Claudius Ptolemy
2906:
2884:
2535:
2046:
1953:
1526:
1505:
1490:
1263:) are for the beginning of the reign of
1087:
892:
880:
827:
720:
626:
566:
308:
93:treatise on the apparent motions of the
20:
5280:
2858:
2718:
2362:
2212:
2190:
1958:The History of Ptolemy's Star Catalogue
1932:
1854:
1806:
1791:
1641:
1445:
1414:
1398:
1268:
360:; observations of the obliquity of the
161:Ptolemy set up a public inscription at
5291:
4962:Latin translations of the 12th century
2985:(Almagest), original Greek, edited by
2925:
2829:
2421:
2327:
2235:
2081:
1911:
1842:
1724:
1681:
1669:
1657:
1554:
1433:
4692:Straightedge and compass construction
3914:
3406:
3102:
3020:Latin translation from the Arabic by
2954:
2742:
2705:
2600:
2261:
2005:
1984:
1890:
1878:
1866:
1764:
1712:
562:
4657:Incircle and excircles of a triangle
2664:
2633:
2569:Journal for the History of Astronomy
2394:Journal for the History of Astronomy
2170:Peters, Christian Heinrich Friedrich
1749:
1569:
1028:
1003:Muntahā al-idrāk fī taqāsīm al-aflāk
988:, who had been on an embassy to the
869:is sometimes described as a book of
693:Constellations in the star catalogue
317:, translated to Latin from Greek by
313:Table of contents to a 1528 copy of
169:was earlier than the version in the
2601:Lewis, H. A. G. (1979). "Review of
2363:Britton, John Phillips (Dec 1969).
1915:Encyclopedia of Space and Astronomy
1271:). The constellations north of the
1115:, was included in volume 16 of the
533:
198:
187:
13:
2911:(2nd ed.). Berlin: Springer.
2736:
2383:10.1111/j.1600-0498.1969.tb00132.x
2243:(2nd ed.). Cambridge, Mass.:
2184:Carnegie Institution of Washington
1292:. (The ratio was subjective as no
1179:Epytoma Ioannis de Monte Regio in
1085:, vols. 1.1 and 1.2 (1898, 1903).
1064:
14:
5325:
2976:
2724:"Star Tales – Ptolemy's Almagest"
2239:(2011) . Buchwald, Jed Z. (ed.).
5299:Ancient Greek astronomical works
5260:
5248:
5236:
5224:
5212:
5200:
5174:
5161:
3389:
3388:
2217:. Translated by Perry, Bruce M.
2017:Bibliothèque nationale de France
1999:Bibliothèque nationale de France
1188:
1172:
1156:
1118:Great Books of the Western World
745:, beginning with the innermost:
382:is vertical, the shadows of the
263:. The treatise was later titled
41:
2788:10.1090/S0002-9904-1948-09089-9
2051:(in German). Berlin: Springer.
1933:Britton, John Phillips (1992).
1321:
1304:
663:used the letter س (sin) for 300
468:ecliptic longitude and latitude
275:, "The Great Treatise"; Latin:
237:The work was originally titled
4994:A History of Greek Mathematics
4507:The Quadrature of the Parabola
2521:10.1126/science.199.4331.872.a
2146:Claudius Ptolemaeus Almagestum
2119:10.1007/978-1-4020-4425-0_8988
1912:Angelo, Joseph A. Jr. (2006).
1252:
1021:(only fragments survive), and
1:
3823:Inferior and superior planets
3046:Almagest Ephemeris Calculator
2603:The Crime of Claudius Ptolemy
2536:Hoffmann, Susanne M. (2018).
2143:Lichtenstein, Petrus (1515).
2047:Hoffmann, Susanne M. (2017).
2030:Copenhagen University Library
1928:– via books.google.com.
1240:
1043:The Crime of Claudius Ptolemy
807:
781:
112:
107:
4775:Intersecting secants theorem
3129:48 constellations listed by
3008:Toomer's English translation
2866:(Rev. ed.). Cambridge:
2690:10.1126/science.198.4318.707
1363:Hamilton & Swerdlow 1987
1338:
948:Toledo School of Translators
901:'s Latin translation of the
876:
284:
265:
239:
224:
217:being a corruption of Greek
7:
4770:Intersecting chords theorem
4637:Doctrine of proportionality
2730:(Revised, online ed.).
2698:
2205:Ptolemy, Copernicus, Kepler
2194:(1952). "The Almagest". In
1419:Ptolemy’s 48 constellations
1203:
1002:
295:
213:
203:
192:
185:The name comes from Arabic
25:An edition in Latin of the
10:
5330:
4466:On the Sphere and Cylinder
4419:On the Sizes and Distances
3900:Medieval Islamic astronomy
3697:On the Sizes and Distances
2955:Shank, Michael H. (2009).
2591:10.1177/002182868601700202
2415:10.1177/002182860003100202
2387:Reprinted as Chapter 1 of.
2285:10.1484/M.PALS-EB.5.120189
2262:Shank, Michael H. (2020).
2092:Cambridge University Press
1899:
1149:
1099:Three translations of the
965:, after his birthplace of
671:like the Hebrew ס (samekh)
434:, the motion of the lunar
378:, the points at which the
280:
271:
245:
219:
5168:Ancient Greece portal
5157:
5107:
4985:
4972:Philosophy of mathematics
4942:
4935:
4909:
4887:Ptolemy's table of chords
4831:
4813:
4712:
4705:
4561:
4523:
4340:
3948:
3942:Ancient Greek mathematics
3890:Medieval European science
3882:
3861:
3770:
3719:
3658:
3620:Sosigenes the Peripatetic
3440:
3386:
3138:
3026:State Library of Victoria
2637:Astronomische Nachrichten
2619:10.1080/03085697908592494
2422:Dreyer, J. L. E. (1917).
2065:10.1007/978-3-658-18683-8
1968:10.1007/978-1-4612-4468-4
1582:Dambis & Efremov 2000
1448:, Book I, Chapter 5, p.9.
1146:French National library.
984:During the 16th century,
859:three-dimensional spheres
717:Ptolemy's planetary model
641:Errors in the coordinates
4839:Aristarchus's inequality
4412:On Conoids and Spheroids
3079:by Andre Rehak (YouTube)
2909:A Survey of the Almagest
2907:Pedersen, Olaf (2011) .
2658:10.1002/asna.19252240202
2563:Kunitzsch, Paul (1986).
2487:Gingerich, Owen (1981).
2303:Tomarchio, John (2022).
2221:, NM: Green Lion Press.
2196:Hutchins, Robert Maynard
1904:
1626:Peters & Knobel 1915
1610:Peters & Knobel 1915
1594:Peters & Knobel 1915
1539:Peters & Knobel 1915
1310:For instance, the stars
1245:
1121:in 1952. The second, by
973:. Around the same time,
558:The Earth does not move.
426:cover the motion of the
180:
4947:Ancient Greek astronomy
4760:Inscribed angle theorem
4750:Greek geometric algebra
4405:Measurement of a Circle
3615:Sosigenes of Alexandria
3434:Ancient Greek astronomy
2749:Oxford University Press
2049:Hipparchs Himmelsglobus
1954:Graßhoff, Gerd (1990).
996:, such as the works of
942:by the Italian scholar
444:covers solar and lunar
340:contains an outline of
5181:Mathematics portal
4967:Non-Euclidean geometry
4922:Mouseion of Alexandria
4795:Tangent-secant theorem
4745:Geometric mean theorem
4730:Exterior angle theorem
4725:Angle bisector theorem
4429:On Sizes and Distances
3687:On Sizes and Distances
3032:University of Vienna:
2555:10.5281/ZENODO.1477980
2464:"Was Ptolemy a Fraud?"
2443:10.1093/mnras/77.7.528
2356:Journals and magazines
1096:
1073:under the Latin title
909:
890:
833:
823:Mesopotamian astronomy
738:
679:atmospheric refraction
633:
572:
549:The Earth is a sphere.
504:covers the motions of
490:covers the motions of
366:spherical trigonometry
321:
276:
255:
230:
30:
4869:Pappus's area theorem
4805:Theorem of the gnomon
4682:Quadratrix of Hippias
4605:Circles of Apollonius
4553:Problem of Apollonius
4531:Constructible numbers
4355:Archimedes Palimpsest
3798:Deferent and epicycle
3727:Antikythera mechanism
2743:Evans, James (1998).
2706:Jodra, Serge (2013).
2505:Goldstein, Bernard R.
1939:. New York: Garland.
1183:Ptolomei, Latin, 1496
1105:R. Catesby Taliaferro
1091:
896:
884:
831:
819:deferent and epicycle
817:) had introduced the
771:Sphere of fixed stars
724:
630:
570:
538:The cosmology of the
329:illustrating how the
312:
24:
5085:prehistoric counting
4882:Ptolemy's inequality
4823:Apollonius's theorem
4662:Method of exhaustion
4632:Diophantine equation
4622:Circumscribed circle
4439:On the Moving Sphere
3869:Babylonian astronomy
3560:Hippocrates of Chios
2987:Johan Ludvig Heiberg
2983:Syntaxis Mathematica
2932:Syntaxis mathematica
2927:Ptolemaeus, Claudius
2803:. Berlin: Springer.
2605:, by R. R. Newton".
2309:. Washington, D.C.:
2149:(in Latin). Venice.
2135:Mathematike Syntaxis
1403:"Ptolemy’s Almagest"
1210:Abū al-Wafā' Būzjānī
1075:Syntaxis mathematica
1050:Bernard R. Goldstein
1019:Pappus of Alexandria
1009:Commentaries on the
903:Syntaxis Mathematica
867:Planetary Hypotheses
855:Planetary Hypotheses
518:covers stations and
480:, and the motion of
326:Syntaxis Mathematica
303:Syntaxis Mathematica
256:Syntaxis Mathematica
240:Mathēmatikḕ Sýntaxis
5171: •
4977:Neusis construction
4897:Spiral of Theodorus
4790:Pythagorean theorem
4735:Euclidean algorithm
4677:Lune of Hippocrates
4546:Squaring the circle
4302:Theon of Alexandria
3977:Aristaeus the Elder
3640:Theon of Alexandria
2682:1977Sci...198..707W
2650:1925AN....224...17V
2582:1986JHA....17...89K
2480:1980QJRAS..21..253G
2407:2000JHA....31..115D
2174:Knobel, Edward Ball
2057:2017hihi.book.....H
1727:, pp. 217–221.
1715:, pp. 305–307.
1584:, pp. 115–134.
1377:, pp. 140–141.
1221:Book of Fixed Stars
1212:(who also wrote an
1113:Annapolis, Maryland
1094:George of Trebizond
1015:Theon of Alexandria
975:George of Trebizond
899:George of Trebizond
804:Apollonius of Perga
581:ecliptic longitudes
408:' discovery of the
404:. Ptolemy explains
319:George of Trebizond
246:Μαθηματικὴ Σύνταξις
167:Canopic Inscription
85:) is a 2nd-century
4864:Menelaus's theorem
4854:Irrational numbers
4667:Parallel postulate
4642:Euclidean geometry
4610:Apollonian circles
4152:Isidore of Miletus
3874:Egyptian astronomy
3788:Circle of latitude
2333:Ptolemy's Almagest
2200:Adler, Mortimer J.
1127:Ptolemy's Almagest
1109:St. John's College
1097:
971:Cardinal Bessarion
910:
891:
848:Ptolemy wrote the
834:
739:
634:
573:
563:The star catalogue
322:
266:Hē Megálē Sýntaxis
253:, and also called
134:, in the medieval
101:paths, written by
31:
5314:2nd-century books
5188:
5187:
5153:
5152:
4905:
4904:
4892:Ptolemy's theorem
4765:Intercept theorem
4615:Apollonian gasket
4541:Doubling the cube
4514:The Sand Reckoner
3908:
3907:
3783:Celestial spheres
3400:
3399:
3040:Gerard of Cremona
3022:Gerard of Cremona
2961:by George Saliba"
2918:978-1-4939-3922-0
2899:978-0-521-40899-8
2877:978-0-7188-9478-8
2868:Lutterworth Press
2781:(11): 1013–1041.
2758:978-0-19-509539-5
2676:(4318): 707–709.
2515:(4331): 872–873.
2346:978-0-387-91220-2
2320:978-1-949822-20-5
2295:978-2-503-58639-7
2254:978-0-262-51615-0
2228:978-1-88800-943-9
2192:Ptolemy, Claudius
2128:978-1-4020-4559-2
2101:978-0-521-57291-0
2074:978-3-658-18682-1
2039:978-8-7770-9002-8
1925:978-0-8160-5330-8
1572:, pp. 17–54.
1458:Lichtenstein 1515
1387:Lichtenstein 1515
1290:logarithmic scale
1041:in the 1977 book
1029:Modern assessment
944:Gerard of Cremona
793:Martianus Capella
743:planetary spheres
665:(like the Hebrew
655:Gerard of Cremona
520:retrograde motion
478:naked eye planets
272:Ἡ Μεγάλη Σύνταξις
5321:
5309:Works by Ptolemy
5284:
5278:
5265:
5264:
5263:
5253:
5252:
5251:
5241:
5240:
5229:
5228:
5227:
5217:
5216:
5205:
5204:
5196:
5179:
5178:
5166:
5165:
5164:
4940:
4939:
4927:Platonic Academy
4874:Problem II.8 of
4844:Crossbar theorem
4800:Thales's theorem
4740:Euclid's theorem
4710:
4709:
4627:Commensurability
4588:Axiomatic system
4536:Angle trisection
4501:
4491:
4453:
4443:
4433:
4423:
4399:
4389:
4372:
3935:
3928:
3921:
3912:
3911:
3895:Indian astronomy
3848:Sublunary sphere
3818:Hipparchic cycle
3757:Mural instrument
3732:Armillary sphere
3711:
3701:
3691:
3681:
3671:
3427:
3420:
3413:
3404:
3403:
3392:
3391:
3333:Piscis Austrinus
3223:Corona Australis
3123:
3116:
3109:
3100:
3099:
3092:
3084:
3073:
3010:Duckworth, 1984.
2972:
2951:
2949:
2947:
2922:
2903:
2881:
2855:
2826:
2797:Neugebauer, Otto
2792:
2790:
2767:Neugebauer, Otto
2762:
2731:
2715:
2712:cosmovisions.com
2693:
2661:
2630:
2597:
2584:
2559:
2557:
2532:
2500:
2483:
2455:
2445:
2418:
2386:
2350:
2324:
2299:
2287:
2258:
2232:
2209:
2187:
2182:. Vol. 86.
2165:
2163:
2161:
2139:
2105:
2086:(1st ed.).
2078:
2043:
2020:
2014:
2006:— (1816).
2002:
1996:
1981:
1961:
1950:
1929:
1894:
1888:
1882:
1876:
1870:
1864:
1858:
1852:
1846:
1840:
1834:
1828:
1822:
1816:
1810:
1804:
1795:
1789:
1783:
1777:
1768:
1762:
1753:
1747:
1728:
1722:
1716:
1710:
1701:
1695:
1689:
1679:
1673:
1667:
1661:
1655:
1649:
1639:
1633:
1623:
1617:
1607:
1601:
1591:
1585:
1579:
1573:
1567:
1558:
1552:
1546:
1536:
1530:
1524:
1509:
1503:
1494:
1488:
1465:
1455:
1449:
1443:
1437:
1431:
1422:
1412:
1406:
1396:
1390:
1384:
1378:
1372:
1366:
1360:
1354:
1348:
1332:
1325:
1319:
1308:
1302:
1287:
1283:
1269:the star catalog
1256:
1227:Star cartography
1192:
1176:
1160:
1077:, was edited by
1039:Robert R. Newton
1023:Ammonius Hermiae
1013:were written by
1005:
986:Guillaume Postel
936:Henry Aristippus
816:
812:
809:
797:medieval Islamic
790:
786:
783:
727:geocentric model
534:Ptolemy's cosmos
466:, together with
348:and the various
287:
282:
274:
273:
268:
258:
248:
247:
242:
233:
227:
222:
221:
216:
206:
201:
200:
195:
190:
189:
124:geocentric model
117:
114:
111:
109:
103:Claudius Ptolemy
81:
76:
75:
72:
71:
68:
65:
62:
59:
56:
53:
50:
47:
5329:
5328:
5324:
5323:
5322:
5320:
5319:
5318:
5304:Astronomy books
5289:
5288:
5287:
5279:
5275:
5271:
5261:
5259:
5249:
5247:
5235:
5225:
5223:
5211:
5199:
5191:
5189:
5184:
5173:
5162:
5160:
5149:
5115:Arabian/Islamic
5103:
5092:numeral systems
4981:
4931:
4901:
4849:Heron's formula
4827:
4809:
4701:
4697:Triangle center
4687:Regular polygon
4564:and definitions
4563:
4557:
4519:
4499:
4489:
4451:
4441:
4431:
4421:
4397:
4387:
4370:
4336:
4307:Theon of Smyrna
3952:
3944:
3939:
3909:
3904:
3878:
3857:
3843:Spherical Earth
3778:Callippic cycle
3766:
3747:Equatorial ring
3715:
3709:
3699:
3689:
3679:
3669:
3654:
3645:Theon of Smyrna
3436:
3431:
3401:
3396:
3382:
3228:Corona Borealis
3134:
3127:
3090:
3082:
3071:
2979:
2967:(Book review).
2945:
2943:
2919:
2900:
2878:
2823:
2759:
2739:
2737:Further reading
2734:
2701:
2696:
2460:Gingerich, Owen
2358:
2353:
2347:
2337:Duckworth Books
2321:
2296:
2264:"Regiomontanus
2255:
2229:
2159:
2157:
2129:
2102:
2075:
2040:
2012:
1994:
1978:
1947:
1926:
1907:
1902:
1897:
1889:
1885:
1877:
1873:
1865:
1861:
1853:
1849:
1841:
1837:
1829:
1825:
1817:
1813:
1805:
1798:
1790:
1786:
1778:
1771:
1763:
1756:
1748:
1731:
1723:
1719:
1711:
1704:
1696:
1692:
1680:
1676:
1668:
1664:
1656:
1652:
1640:
1636:
1624:
1620:
1608:
1604:
1592:
1588:
1580:
1576:
1568:
1561:
1553:
1549:
1537:
1533:
1525:
1512:
1504:
1497:
1489:
1468:
1456:
1452:
1444:
1440:
1432:
1425:
1413:
1409:
1397:
1393:
1385:
1381:
1373:
1369:
1361:
1357:
1349:
1345:
1341:
1336:
1335:
1326:
1322:
1309:
1305:
1285:
1281:
1257:
1253:
1248:
1243:
1206:
1199:
1193:
1184:
1177:
1168:
1161:
1152:
1067:
1065:Modern editions
1031:
979:Pope Nicholas V
879:
814:
810:
788:
784:
774:
719:
695:
643:
565:
536:
358:table of chords
307:
298:
183:
156:Greek astronomy
115:
106:
79:
44:
40:
17:
12:
11:
5:
5327:
5317:
5316:
5311:
5306:
5301:
5286:
5285:
5272:
5270:
5269:
5257:
5245:
5233:
5221:
5209:
5186:
5185:
5158:
5155:
5154:
5151:
5150:
5148:
5147:
5142:
5137:
5132:
5127:
5122:
5117:
5111:
5109:
5108:Other cultures
5105:
5104:
5102:
5101:
5100:
5099:
5089:
5088:
5087:
5077:
5076:
5075:
5065:
5064:
5063:
5053:
5052:
5051:
5041:
5040:
5039:
5029:
5028:
5027:
5017:
5016:
5015:
5005:
5004:
5003:
4989:
4987:
4983:
4982:
4980:
4979:
4974:
4969:
4964:
4959:
4957:Greek numerals
4954:
4952:Attic numerals
4949:
4943:
4937:
4933:
4932:
4930:
4929:
4924:
4919:
4913:
4911:
4907:
4906:
4903:
4902:
4900:
4899:
4894:
4889:
4884:
4879:
4871:
4866:
4861:
4856:
4851:
4846:
4841:
4835:
4833:
4829:
4828:
4826:
4825:
4819:
4817:
4811:
4810:
4808:
4807:
4802:
4797:
4792:
4787:
4782:
4780:Law of cosines
4777:
4772:
4767:
4762:
4757:
4752:
4747:
4742:
4737:
4732:
4727:
4721:
4719:
4707:
4703:
4702:
4700:
4699:
4694:
4689:
4684:
4679:
4674:
4672:Platonic solid
4669:
4664:
4659:
4654:
4652:Greek numerals
4649:
4644:
4639:
4634:
4629:
4624:
4619:
4618:
4617:
4612:
4602:
4597:
4596:
4595:
4585:
4584:
4583:
4578:
4567:
4565:
4559:
4558:
4556:
4555:
4550:
4549:
4548:
4543:
4538:
4527:
4525:
4521:
4520:
4518:
4517:
4510:
4503:
4493:
4483:
4480:Planisphaerium
4476:
4469:
4462:
4455:
4445:
4435:
4425:
4415:
4408:
4401:
4391:
4381:
4374:
4364:
4357:
4352:
4344:
4342:
4338:
4337:
4335:
4334:
4329:
4324:
4319:
4314:
4309:
4304:
4299:
4294:
4289:
4284:
4279:
4274:
4269:
4264:
4259:
4254:
4249:
4244:
4239:
4234:
4229:
4224:
4219:
4214:
4209:
4204:
4199:
4194:
4189:
4184:
4179:
4174:
4169:
4164:
4159:
4154:
4149:
4144:
4139:
4134:
4129:
4124:
4119:
4114:
4109:
4104:
4099:
4094:
4089:
4084:
4079:
4074:
4069:
4064:
4059:
4054:
4049:
4044:
4039:
4034:
4029:
4024:
4019:
4014:
4009:
4004:
3999:
3994:
3989:
3984:
3979:
3974:
3969:
3964:
3958:
3956:
3950:Mathematicians
3946:
3945:
3938:
3937:
3930:
3923:
3915:
3906:
3905:
3903:
3902:
3897:
3892:
3886:
3884:
3880:
3879:
3877:
3876:
3871:
3865:
3863:
3859:
3858:
3856:
3855:
3850:
3845:
3840:
3835:
3830:
3825:
3820:
3815:
3810:
3805:
3800:
3795:
3790:
3785:
3780:
3774:
3772:
3768:
3767:
3765:
3764:
3759:
3754:
3749:
3744:
3739:
3734:
3729:
3723:
3721:
3717:
3716:
3714:
3713:
3707:On the Heavens
3703:
3693:
3683:
3680:(Eratosthenes)
3673:
3662:
3660:
3656:
3655:
3653:
3652:
3647:
3642:
3637:
3632:
3627:
3622:
3617:
3612:
3607:
3602:
3597:
3592:
3587:
3585:Philip of Opus
3582:
3577:
3572:
3567:
3562:
3557:
3552:
3547:
3542:
3537:
3532:
3527:
3522:
3517:
3512:
3507:
3502:
3497:
3492:
3487:
3482:
3477:
3472:
3467:
3462:
3457:
3452:
3446:
3444:
3438:
3437:
3430:
3429:
3422:
3415:
3407:
3398:
3397:
3387:
3384:
3383:
3381:
3380:
3375:
3370:
3365:
3360:
3355:
3350:
3345:
3340:
3335:
3330:
3325:
3320:
3315:
3310:
3305:
3300:
3295:
3290:
3285:
3280:
3275:
3270:
3265:
3260:
3255:
3250:
3245:
3240:
3235:
3230:
3225:
3220:
3215:
3210:
3205:
3200:
3195:
3190:
3185:
3180:
3175:
3170:
3165:
3160:
3155:
3150:
3145:
3139:
3136:
3135:
3126:
3125:
3118:
3111:
3103:
3097:
3096:
3088:
3080:
3074:
3062:
3057:
3049:
3043:
3029:
3011:
3005:
2995:
2990:
2978:
2977:External links
2975:
2974:
2973:
2952:
2923:
2917:
2904:
2898:
2886:Pedersen, Olaf
2882:
2876:
2856:
2844:10.1086/348869
2838:(2): 150–158.
2827:
2821:
2793:
2763:
2757:
2738:
2735:
2733:
2732:
2728:ianridpath.com
2716:
2702:
2700:
2697:
2695:
2694:
2666:Wade, Nicholas
2662:
2644:(2–3): 17–54.
2631:
2598:
2560:
2548:(4): 281–287.
2533:
2501:
2484:
2456:
2419:
2401:(2): 115–134.
2388:
2359:
2357:
2354:
2352:
2351:
2345:
2325:
2319:
2300:
2294:
2259:
2253:
2237:Saliba, George
2233:
2227:
2210:
2188:
2166:
2140:
2127:
2106:
2100:
2079:
2073:
2044:
2038:
2021:
2003:
1986:Halma, Nicolas
1982:
1976:
1951:
1945:
1930:
1924:
1908:
1906:
1903:
1901:
1898:
1896:
1895:
1883:
1871:
1859:
1847:
1835:
1831:Gingerich 1981
1823:
1819:Gingerich 1980
1811:
1796:
1784:
1780:Goldstein 1978
1769:
1754:
1729:
1717:
1702:
1698:Kunitzsch 1986
1690:
1674:
1662:
1650:
1634:
1618:
1602:
1586:
1574:
1559:
1557:, p. 528.
1547:
1531:
1510:
1495:
1466:
1450:
1438:
1423:
1407:
1391:
1379:
1375:Kunitzsch 2008
1367:
1355:
1351:Tomarchio 2022
1342:
1340:
1337:
1334:
1333:
1320:
1303:
1294:photodetectors
1265:Antoninus Pius
1250:
1249:
1247:
1244:
1242:
1239:
1238:
1237:
1229:
1224:
1217:
1205:
1202:
1201:
1200:
1194:
1187:
1185:
1178:
1171:
1169:
1162:
1155:
1151:
1148:
1138:Nicholas Halma
1066:
1063:
1054:Owen Gingerich
1030:
1027:
990:Ottoman Empire
929:Sahl ibn Bishr
878:
875:
773:
772:
769:
766:
763:
760:
757:
754:
751:
747:
718:
715:
707:constellations
694:
691:
687:Alpha Centauri
642:
639:
625:
624:
621:
618:
615:
601:Antoninus Pius
591:of modern-day
564:
561:
560:
559:
556:
553:
550:
547:
535:
532:
531:
530:
523:
513:
499:
485:
471:
464:constellations
460:star catalogue
449:
439:
417:
395:
369:
306:
299:
297:
294:
277:Magna Syntaxis
188:اَلْمَجِسْطِيّ
182:
179:
163:Canopus, Egypt
15:
9:
6:
4:
3:
2:
5326:
5315:
5312:
5310:
5307:
5305:
5302:
5300:
5297:
5296:
5294:
5282:
5277:
5273:
5268:
5258:
5256:
5246:
5244:
5239:
5234:
5232:
5222:
5220:
5215:
5210:
5208:
5203:
5198:
5197:
5194:
5183:
5182:
5177:
5170:
5169:
5156:
5146:
5143:
5141:
5138:
5136:
5133:
5131:
5128:
5126:
5123:
5121:
5118:
5116:
5113:
5112:
5110:
5106:
5098:
5095:
5094:
5093:
5090:
5086:
5083:
5082:
5081:
5078:
5074:
5071:
5070:
5069:
5066:
5062:
5059:
5058:
5057:
5054:
5050:
5047:
5046:
5045:
5042:
5038:
5035:
5034:
5033:
5030:
5026:
5023:
5022:
5021:
5018:
5014:
5011:
5010:
5009:
5006:
5002:
4998:
4997:
4996:
4995:
4991:
4990:
4988:
4984:
4978:
4975:
4973:
4970:
4968:
4965:
4963:
4960:
4958:
4955:
4953:
4950:
4948:
4945:
4944:
4941:
4938:
4934:
4928:
4925:
4923:
4920:
4918:
4915:
4914:
4912:
4908:
4898:
4895:
4893:
4890:
4888:
4885:
4883:
4880:
4878:
4877:
4872:
4870:
4867:
4865:
4862:
4860:
4857:
4855:
4852:
4850:
4847:
4845:
4842:
4840:
4837:
4836:
4834:
4830:
4824:
4821:
4820:
4818:
4816:
4812:
4806:
4803:
4801:
4798:
4796:
4793:
4791:
4788:
4786:
4785:Pons asinorum
4783:
4781:
4778:
4776:
4773:
4771:
4768:
4766:
4763:
4761:
4758:
4756:
4755:Hinge theorem
4753:
4751:
4748:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4726:
4723:
4722:
4720:
4718:
4717:
4711:
4708:
4704:
4698:
4695:
4693:
4690:
4688:
4685:
4683:
4680:
4678:
4675:
4673:
4670:
4668:
4665:
4663:
4660:
4658:
4655:
4653:
4650:
4648:
4645:
4643:
4640:
4638:
4635:
4633:
4630:
4628:
4625:
4623:
4620:
4616:
4613:
4611:
4608:
4607:
4606:
4603:
4601:
4598:
4594:
4591:
4590:
4589:
4586:
4582:
4579:
4577:
4574:
4573:
4572:
4569:
4568:
4566:
4560:
4554:
4551:
4547:
4544:
4542:
4539:
4537:
4534:
4533:
4532:
4529:
4528:
4526:
4522:
4516:
4515:
4511:
4509:
4508:
4504:
4502:
4498:
4494:
4492:
4488:
4484:
4482:
4481:
4477:
4475:
4474:
4470:
4468:
4467:
4463:
4461:
4460:
4456:
4454:
4450:
4446:
4444:
4440:
4436:
4434:
4430:
4426:
4424:
4422:(Aristarchus)
4420:
4416:
4414:
4413:
4409:
4407:
4406:
4402:
4400:
4396:
4392:
4390:
4386:
4382:
4380:
4379:
4375:
4373:
4369:
4365:
4363:
4362:
4358:
4356:
4353:
4351:
4350:
4346:
4345:
4343:
4339:
4333:
4330:
4328:
4327:Zeno of Sidon
4325:
4323:
4320:
4318:
4315:
4313:
4310:
4308:
4305:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
4240:
4238:
4235:
4233:
4230:
4228:
4225:
4223:
4220:
4218:
4215:
4213:
4210:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4163:
4160:
4158:
4155:
4153:
4150:
4148:
4145:
4143:
4140:
4138:
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4118:
4115:
4113:
4110:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4088:
4085:
4083:
4080:
4078:
4075:
4073:
4070:
4068:
4065:
4063:
4060:
4058:
4055:
4053:
4050:
4048:
4045:
4043:
4040:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3968:
3965:
3963:
3960:
3959:
3957:
3955:
3951:
3947:
3943:
3936:
3931:
3929:
3924:
3922:
3917:
3916:
3913:
3901:
3898:
3896:
3893:
3891:
3888:
3887:
3885:
3881:
3875:
3872:
3870:
3867:
3866:
3864:
3860:
3854:
3851:
3849:
3846:
3844:
3841:
3839:
3836:
3834:
3831:
3829:
3828:Metonic cycle
3826:
3824:
3821:
3819:
3816:
3814:
3813:Heliocentrism
3811:
3809:
3806:
3804:
3801:
3799:
3796:
3794:
3793:Counter-Earth
3791:
3789:
3786:
3784:
3781:
3779:
3776:
3775:
3773:
3769:
3763:
3760:
3758:
3755:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3724:
3722:
3718:
3712:
3708:
3704:
3702:
3700:(Aristarchus)
3698:
3694:
3692:
3688:
3684:
3682:
3678:
3674:
3672:
3668:
3664:
3663:
3661:
3657:
3651:
3648:
3646:
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3583:
3581:
3578:
3576:
3573:
3571:
3568:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3538:
3536:
3533:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3486:
3483:
3481:
3478:
3476:
3473:
3471:
3468:
3466:
3463:
3461:
3458:
3456:
3453:
3451:
3448:
3447:
3445:
3443:
3439:
3435:
3428:
3423:
3421:
3416:
3414:
3409:
3408:
3405:
3395:
3385:
3379:
3376:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3339:
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3291:
3289:
3286:
3284:
3281:
3279:
3276:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3219:
3216:
3214:
3211:
3209:
3206:
3204:
3201:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3140:
3137:
3132:
3124:
3119:
3117:
3112:
3110:
3105:
3104:
3101:
3095:
3089:
3087:
3081:
3078:
3075:
3070:
3066:
3063:
3061:
3058:
3056:
3055:
3050:
3047:
3044:
3041:
3037:
3035:
3030:
3027:
3023:
3019:
3017:
3012:
3009:
3006:
3004:(Koine Greek)
3003:
3001:
3000:De Analemmate
2996:
2994:
2991:
2988:
2984:
2981:
2980:
2970:
2966:
2962:
2960:
2953:
2942:
2938:
2934:
2933:
2928:
2924:
2920:
2914:
2910:
2905:
2901:
2895:
2891:
2887:
2883:
2879:
2873:
2869:
2865:
2861:
2857:
2853:
2849:
2845:
2841:
2837:
2833:
2828:
2824:
2822:3-540-06995-X
2818:
2814:
2810:
2806:
2802:
2798:
2794:
2789:
2784:
2780:
2776:
2772:
2768:
2764:
2760:
2754:
2750:
2746:
2741:
2740:
2729:
2725:
2721:
2717:
2713:
2709:
2708:"L'Almageste"
2704:
2703:
2691:
2687:
2683:
2679:
2675:
2671:
2667:
2663:
2659:
2655:
2651:
2647:
2643:
2640:(in German).
2639:
2638:
2632:
2628:
2624:
2620:
2616:
2612:
2608:
2604:
2599:
2596:
2592:
2588:
2583:
2578:
2575:(49): 89–98.
2574:
2570:
2566:
2561:
2556:
2551:
2547:
2543:
2539:
2534:
2530:
2526:
2522:
2518:
2514:
2510:
2506:
2502:
2498:
2494:
2490:
2485:
2481:
2477:
2473:
2469:
2465:
2461:
2457:
2453:
2449:
2444:
2439:
2435:
2431:
2430:
2425:
2420:
2416:
2412:
2408:
2404:
2400:
2396:
2395:
2389:
2384:
2380:
2376:
2372:
2371:
2366:
2361:
2360:
2348:
2342:
2338:
2334:
2330:
2329:Toomer, G. J.
2326:
2322:
2316:
2312:
2308:
2307:
2301:
2297:
2291:
2286:
2281:
2277:
2273:
2271:
2267:
2260:
2256:
2250:
2246:
2242:
2238:
2234:
2230:
2224:
2220:
2216:
2211:
2207:
2206:
2201:
2197:
2193:
2189:
2185:
2181:
2180:
2175:
2171:
2167:
2156:
2152:
2148:
2147:
2141:
2138:
2136:
2130:
2124:
2120:
2116:
2112:
2107:
2103:
2097:
2093:
2089:
2085:
2080:
2076:
2070:
2066:
2062:
2058:
2054:
2050:
2045:
2041:
2035:
2031:
2027:
2022:
2018:
2011:
2010:
2004:
2000:
1993:
1992:
1987:
1983:
1979:
1977:3-540-97181-5
1973:
1969:
1965:
1960:
1959:
1952:
1948:
1946:0-8153-0215-0
1942:
1938:
1937:
1931:
1927:
1921:
1917:
1916:
1910:
1909:
1892:
1887:
1880:
1875:
1868:
1863:
1856:
1851:
1844:
1839:
1832:
1827:
1820:
1815:
1808:
1803:
1801:
1793:
1788:
1781:
1776:
1774:
1766:
1761:
1759:
1751:
1746:
1744:
1742:
1740:
1738:
1736:
1734:
1726:
1721:
1714:
1709:
1707:
1700:, p. 89.
1699:
1694:
1687:
1683:
1678:
1672:, p. 78.
1671:
1666:
1660:, p. 44.
1659:
1654:
1647:
1643:
1638:
1631:
1627:
1622:
1615:
1611:
1606:
1599:
1595:
1590:
1583:
1578:
1571:
1566:
1564:
1556:
1551:
1544:
1540:
1535:
1528:
1527:Graßhoff 1990
1523:
1521:
1519:
1517:
1515:
1507:
1506:Hoffmann 2018
1502:
1500:
1492:
1491:Hoffmann 2017
1487:
1485:
1483:
1481:
1479:
1477:
1475:
1473:
1471:
1463:
1459:
1454:
1447:
1442:
1435:
1430:
1428:
1420:
1416:
1411:
1404:
1400:
1395:
1388:
1383:
1376:
1371:
1364:
1359:
1353:, p. xv.
1352:
1347:
1343:
1330:
1324:
1317:
1313:
1307:
1300:
1295:
1291:
1279:
1274:
1270:
1266:
1262:
1255:
1251:
1236:
1235:
1230:
1228:
1225:
1223:
1222:
1218:
1215:
1211:
1208:
1207:
1198:, Latin, 1515
1197:
1191:
1186:
1182:
1175:
1170:
1166:
1159:
1154:
1153:
1147:
1145:
1144:
1139:
1134:
1132:
1128:
1124:
1120:
1119:
1114:
1110:
1106:
1102:
1095:
1090:
1086:
1084:
1080:
1079:J. L. Heiberg
1076:
1072:
1062:
1059:
1055:
1051:
1046:
1044:
1040:
1035:
1026:
1024:
1020:
1016:
1012:
1007:
1004:
999:
995:
991:
987:
982:
980:
976:
972:
968:
964:
963:Regiomontanus
959:
957:
953:
949:
945:
941:
937:
932:
930:
926:
923:
918:
916:
908:
904:
900:
895:
888:
883:
874:
872:
868:
864:
860:
856:
851:
846:
844:
839:
830:
826:
824:
820:
815: 190 BC
805:
800:
798:
794:
789: 347 BC
779:
770:
767:
764:
761:
758:
755:
752:
749:
748:
746:
744:
736:
732:
728:
723:
714:
710:
708:
703:
699:
690:
688:
684:
680:
674:
672:
668:
664:
660:
656:
650:
648:
647:proper motion
638:
629:
622:
619:
616:
613:
612:
611:
608:
606:
602:
598:
594:
590:
586:
582:
577:
569:
557:
554:
551:
548:
545:
544:
543:
541:
527:
524:
521:
517:
514:
511:
507:
503:
500:
497:
493:
489:
486:
483:
479:
475:
472:
469:
465:
461:
457:
453:
450:
447:
443:
440:
437:
433:
429:
425:
421:
418:
415:
411:
407:
403:
399:
396:
393:
389:
385:
381:
377:
373:
370:
367:
363:
359:
355:
351:
347:
343:
339:
336:
335:
334:
332:
327:
320:
316:
311:
304:
293:
291:
286:
278:
267:
262:
257:
252:
241:
235:
232:
226:
215:
210:
205:
194:
178:
176:
172:
168:
164:
159:
157:
153:
149:
145:
141:
137:
133:
129:
125:
121:
104:
100:
96:
92:
88:
84:
83:
74:
38:
37:
28:
23:
19:
5281:Britton 1992
5276:
5267:Solar System
5172:
5159:
5001:Thomas Heath
4992:
4875:
4859:Law of sines
4715:
4647:Golden ratio
4512:
4505:
4496:
4490:(Theodosius)
4486:
4478:
4471:
4464:
4457:
4448:
4438:
4432:(Hipparchus)
4428:
4418:
4410:
4403:
4394:
4384:
4376:
4371:(Apollonius)
4367:
4359:
4348:
4347:
4322:Zeno of Elea
4082:Eratosthenes
4072:Dionysodorus
3706:
3696:
3690:(Hipparchus)
3686:
3677:Catasterismi
3676:
3666:
3665:
3525:Eratosthenes
3133:after 150 AD
3065:Star catalog
3053:
3033:
3015:
2999:
2968:
2964:
2958:
2944:. Retrieved
2935:(in Greek).
2931:
2908:
2889:
2863:
2860:Ridpath, Ian
2835:
2831:
2800:
2778:
2774:
2744:
2727:
2720:Ridpath, Ian
2711:
2673:
2669:
2641:
2635:
2610:
2606:
2602:
2594:
2572:
2568:
2545:
2541:
2512:
2508:
2496:
2492:
2471:
2467:
2433:
2427:
2398:
2392:
2374:
2368:
2332:
2305:
2275:
2269:
2265:
2240:
2214:
2204:
2178:
2158:. Retrieved
2145:
2134:
2132:
2110:
2083:
2048:
2025:
2008:
1990:
1957:
1935:
1914:
1886:
1874:
1862:
1855:Ptolemy 2014
1850:
1838:
1826:
1814:
1807:Britton 1992
1792:Britton 1969
1787:
1720:
1693:
1677:
1665:
1653:
1642:Ridpath 2023
1637:
1621:
1605:
1589:
1577:
1550:
1534:
1453:
1446:Ptolemy 1952
1441:
1415:Ridpath 2023
1410:
1399:Ridpath 2023
1394:
1382:
1370:
1358:
1346:
1323:
1306:
1254:
1233:
1219:
1213:
1195:
1180:
1164:
1141:
1135:
1130:
1126:
1123:G. J. Toomer
1116:
1100:
1098:
1082:
1074:
1070:
1068:
1047:
1042:
1036:
1032:
1010:
1008:
993:
983:
960:
951:
933:
919:
914:
911:
906:
902:
886:
866:
862:
854:
849:
847:
837:
835:
801:
775:
740:
735:Cosmographia
734:
711:
704:
700:
696:
675:
651:
644:
635:
609:
578:
574:
539:
537:
525:
515:
501:
487:
473:
455:
451:
441:
423:
419:
397:
371:
337:
330:
325:
323:
314:
302:
289:
236:
184:
174:
173:. Hence the
170:
166:
160:
91:astronomical
87:mathematical
35:
34:
32:
26:
18:
5255:Outer space
5219:Mathematics
5068:mathematics
4876:Arithmetica
4473:Ostomachion
4442:(Autolycus)
4361:Arithmetica
4137:Hippocrates
4067:Dinostratus
4052:Dicaearchus
3982:Aristarchus
3808:Geocentrism
3720:Instruments
3710:(Aristotle)
3515:Cleostratus
3480:Aristarchus
3460:Anaximander
3442:Astronomers
3343:Sagittarius
3198:Capricornus
3193:Canis Minor
3188:Canis Major
3091:(in Hebrew)
3083:(in French)
2613:: 105–107.
2607:Imago Mundi
2090:, England:
1843:Toomer 1984
1725:Saliba 2011
1682:Toomer 1984
1670:Angelo 2006
1658:Hoskin 1999
1596:, pp.
1555:Dreyer 1917
1529:, p. .
1508:, p. .
1493:, p. .
1436:, p. .
1434:Toomer 1984
1389:, p. .
1365:, p. .
1331:since 2019.
897:Picture of
731:Peter Apian
346:fixed stars
251:Koine Greek
148:Renaissance
144:Middle Ages
120:Koine Greek
5293:Categories
5120:Babylonian
5020:arithmetic
4986:History of
4815:Apollonius
4500:(Menelaus)
4459:On Spirals
4378:Catoptrics
4317:Xenocrates
4312:Thymaridas
4297:Theodosius
4282:Theaetetus
4262:Simplicius
4252:Pythagoras
4237:Posidonius
4222:Philonides
4182:Nicomachus
4177:Metrodorus
4167:Menaechmus
4122:Hipparchus
4112:Heliodorus
4062:Diophantus
4047:Democritus
4027:Chrysippus
3997:Archimedes
3992:Apollonius
3962:Anaxagoras
3954:(timeline)
3883:Influenced
3862:Influences
3833:Octaeteris
3762:Triquetrum
3650:Timocharis
3635:Theodosius
3595:Posidonius
3555:Hipparchus
3545:Heraclides
3485:Aristyllus
3470:Apollonius
3465:Andronicus
3373:Ursa Minor
3368:Ursa Major
3363:Triangulum
3203:Cassiopeia
3163:Argo Navis
3072:(in Latin)
3034:Almagestum
2998:Ptolemy's
2965:Aestimatio
2864:Star Tales
2436:(7): 528.
2335:. London:
1891:Halma 1816
1879:Halma 1813
1867:Jodra 2013
1765:Lewis 1979
1713:Shank 2020
1684:, p.
1628:, p.
1612:, p.
1541:, p.
1460:, p.
1329:Stellarium
1316:62 Orionis
1312:nu Orionis
1261:precession
1241:References
1196:Almagestum
1181:Almagestum
1058:Hipparchus
1017:(extant),
998:al-Kharaqī
967:Königsberg
885:Ptolemy's
813: – c.
811: 262
787: – c.
785: 427
605:Hipparchus
410:precession
406:Hipparchus
231:Almagestum
193:al-majisṭī
152:Copernicus
146:and early
132:Alexandria
116: 170
27:Almagestum
5231:Astronomy
4581:Inscribed
4341:Treatises
4332:Zenodorus
4292:Theodorus
4267:Sosigenes
4212:Philolaus
4197:Oenopides
4192:Nicoteles
4187:Nicomedes
4147:Hypsicles
4042:Ctesibius
4032:Cleomedes
4017:Callippus
4002:Autolycus
3987:Aristotle
3967:Anthemius
3737:Astrolabe
3670:(Ptolemy)
3590:Philolaus
3580:Oenopides
3565:Hypsicles
3510:Cleomedes
3505:Callippus
3495:Autolycus
3450:Aglaonice
3308:Ophiuchus
3248:Delphinus
3208:Centaurus
3143:Andromeda
3014:Ptolemy.
2941:767751182
2888:(1993) .
2862:(2018) .
2377:: 29–41.
2370:Centaurus
2311:CUA Press
2245:MIT Press
2155:644484399
2088:Cambridge
1750:Wade 1977
1570:Vogt 1925
1339:Citations
1278:magnitude
1232:Euclid's
1133:in 2014.
956:Alfonso X
925:Al-Ma'mun
877:Influence
871:cosmology
683:Centaurus
593:astrology
526:Book XIII
452:Books VII
414:epicycles
392:solstices
388:equinoxes
342:Aristotle
292:derives.
207:meaning '
136:Byzantine
99:planetary
5145:Japanese
5130:Egyptian
5073:timeline
5061:timeline
5049:timeline
5044:geometry
5037:timeline
5032:calculus
5025:timeline
5013:timeline
4716:Elements
4562:Concepts
4524:Problems
4497:Spherics
4487:Spherics
4452:(Euclid)
4398:(Euclid)
4395:Elements
4388:(Euclid)
4349:Almagest
4257:Serenus
4232:Porphyry
4172:Menelaus
4127:Hippasus
4102:Eutocius
4077:Domninus
3972:Archytas
3838:Solstice
3771:Concepts
3667:Almagest
3610:Seleucus
3570:Menelaus
3530:Euctemon
3394:Category
3348:Scorpius
3273:Hercules
3263:Eridanus
3258:Equuleus
3148:Aquarius
3054:Almagest
3016:Almagest
2971:: 63–72.
2946:10 April
2799:(1975).
2769:(1948).
2722:(2023).
2699:Websites
2529:17757580
2499:: 40–44.
2462:(1980).
2452:10340650
2331:(1984).
2270:Almagest
2219:Santa Fe
2202:(eds.).
2176:(1915).
1988:(1813).
1234:Elements
1214:Almagest
1204:See also
1165:Almagest
1101:Almagest
1071:Almagest
1025:(lost).
1011:Syntaxis
994:Almagest
952:Abrachir
915:Syntaxis
907:Almagest
887:Almagest
863:Syntaxis
850:Syntaxis
838:Syntaxis
667:ש (shin)
540:Syntaxis
516:Book XII
446:eclipses
432:parallax
430:, lunar
420:Books IV
398:Book III
376:latitude
362:ecliptic
331:Syntaxis
315:Almagest
296:Contents
290:Almagest
175:Almagest
171:Almagest
128:Universe
110:AD 100 –
82:-mə-jest
36:Almagest
5193:Portals
5125:Chinese
5080:numbers
5008:algebra
4936:Related
4910:Centers
4706:Results
4576:Central
4247:Ptolemy
4242:Proclus
4207:Perseus
4162:Marinus
4142:Hypatia
4132:Hippias
4107:Geminus
4097:Eudoxus
4087:Eudemus
4057:Diocles
3742:Dioptra
3605:Pytheas
3600:Ptolemy
3550:Hicetas
3540:Geminus
3535:Eudoxus
3490:Attalus
3455:Agrippa
3353:Serpens
3338:Sagitta
3323:Perseus
3318:Pegasus
3213:Cepheus
3131:Ptolemy
2989:, 1898.
2678:Bibcode
2670:Science
2646:Bibcode
2627:1150735
2577:Bibcode
2509:Science
2476:Bibcode
2474:: 253.
2403:Bibcode
2053:Bibcode
1900:Sources
1150:Gallery
1143:Gallica
765:Jupiter
753:Mercury
506:Jupiter
502:Book XI
482:Mercury
474:Book IX
442:Book VI
386:at the
372:Book II
350:planets
285:megístē
281:μεγίστη
225:megístē
220:μεγίστη
214:majisṭī
196:, with
140:Islamic
126:of the
29:in 1515
5140:Indian
4917:Cyrene
4449:Optics
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