how did hipparchus discover trigonometry

It is unknown who invented this method. Like most of his predecessorsAristarchus of Samos was an exceptionHipparchus assumed a spherical, stationary Earth at the centre of the universe (the geocentric cosmology). Hipparchus of Nicea (l. c. 190 - c. 120 BCE) was a Greek astronomer, geographer, and mathematician regarded as the greatest astronomer of antiquity and one of the greatest of all time. https://www.britannica.com/biography/Hipparchus-Greek-astronomer, Ancient History Encyclopedia - Biography of Hipparchus of Nicea, Hipparchus - Student Encyclopedia (Ages 11 and up). How did Hipparchus contribute to trigonometry? For more information see Discovery of precession. He also discovered that the moon, the planets and the stars were more complex than anyone imagined. Eratosthenes (3rd century BC), in contrast, used a simpler sexagesimal system dividing a circle into 60 parts. According to Ptolemy, Hipparchus measured the longitude of Spica and Regulus and other bright stars. Ptolemy cites more than 20 observations made there by Hipparchus on specific dates from 147 to 127, as well as three earlier observations from 162 to 158 that may be attributed to him. Corrections? "Hipparchus on the Distances of the Sun and Moon. Hipparchus (/hprks/; Greek: , Hipparkhos; c.190 c.120BC) was a Greek astronomer, geographer, and mathematician. Hipparchus discovered the wobble of Earth's axis by comparing previous star charts to the charts he created during his study of the stars. [47] Although the Almagest star catalogue is based upon Hipparchus's one, it is not only a blind copy but enriched, enhanced, and thus (at least partially) re-observed.[15]. This was the basis for the astrolabe. [50] ???? It was only in Hipparchus's time (2nd century BC) when this division was introduced (probably by Hipparchus's contemporary Hypsikles) for all circles in mathematics. In the practical part of his work, the so-called "table of climata", Hipparchus listed latitudes for several tens of localities. Hipparchus apparently made similar calculations. Aristarchus of Samos (/?r??st? As shown in a 1991 Delambre in his Histoire de l'Astronomie Ancienne (1817) concluded that Hipparchus knew and used the equatorial coordinate system, a conclusion challenged by Otto Neugebauer in his A History of Ancient Mathematical Astronomy (1975). He also introduced the division of a circle into 360 degrees into Greece. [51], He was the first to use the grade grid, to determine geographic latitude from star observations, and not only from the Sun's altitude, a method known long before him, and to suggest that geographic longitude could be determined by means of simultaneous observations of lunar eclipses in distant places. Ch. His interest in the fixed stars may have been inspired by the observation of a supernova (according to Pliny), or by his discovery of precession, according to Ptolemy, who says that Hipparchus could not reconcile his data with earlier observations made by Timocharis and Aristillus. ?, Aristarkhos ho Samios; c. 310 c. . Late in his career (possibly about 135BC) Hipparchus compiled his star catalog. This opinion was confirmed by the careful investigation of Hoffmann[40] who independently studied the material, potential sources, techniques and results of Hipparchus and reconstructed his celestial globe and its making. (1997). Most of Hipparchuss adult life, however, seems to have been spent carrying out a program of astronomical observation and research on the island of Rhodes. A simpler alternate reconstruction[28] agrees with all four numbers. However, by comparing his own observations of solstices with observations made in the 5th and 3rd centuries bce, Hipparchus succeeded in obtaining an estimate of the tropical year that was only six minutes too long. All thirteen clima figures agree with Diller's proposal. In the first book, Hipparchus assumes that the parallax of the Sun is 0, as if it is at infinite distance. Hipparchus also adopted the Babylonian astronomical cubit unit (Akkadian ammatu, Greek pchys) that was equivalent to 2 or 2.5 ('large cubit'). Anyway, Hipparchus found inconsistent results; he later used the ratio of the epicycle model (3122+12: 247+12), which is too small (60: 4;45 sexagesimal). There are stars cited in the Almagest from Hipparchus that are missing in the Almagest star catalogue. Hipparchus thus had the problematic result that his minimum distance (from book 1) was greater than his maximum mean distance (from book 2). Hipparchus must have used a better approximation for than the one from Archimedes of between 3+1071 (3.14085) and 3+17 (3.14286). [65], Johannes Kepler had great respect for Tycho Brahe's methods and the accuracy of his observations, and considered him to be the new Hipparchus, who would provide the foundation for a restoration of the science of astronomy.[66]. That would be the first known work of trigonometry. Hipparchus's draconitic lunar motion cannot be solved by the lunar-four arguments sometimes proposed to explain his anomalistic motion. For the Sun however, there was no observable parallax (we now know that it is about 8.8", several times smaller than the resolution of the unaided eye). The first trigonometric table was apparently compiled by Hipparchus, who is consequently now known as "the father of trigonometry". Hipparchus wrote a critique in three books on the work of the geographer Eratosthenes of Cyrene (3rd centuryBC), called Prs tn Eratosthnous geographan ("Against the Geography of Eratosthenes"). Hipparchus (/ h p r k s /; Greek: , Hipparkhos; c. 190 - c. 120 BC) was a Greek astronomer, geographer, and mathematician.He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hipparchus's only preserved work is ("Commentary on the Phaenomena of Eudoxus and Aratus"). Note the latitude of the location. He considered every triangle as being inscribed in a circle, so that each side became a chord. [29] (The maximum angular deviation producible by this geometry is the arcsin of 5+14 divided by 60, or approximately 5 1', a figure that is sometimes therefore quoted as the equivalent of the Moon's equation of the center in the Hipparchan model.). 2 - What are two ways in which Aristotle deduced that. (He similarly found from the 345-year cycle the ratio 4,267 synodic months = 4,573 anomalistic months and divided by 17 to obtain the standard ratio 251 synodic months = 269 anomalistic months.) Using the visually identical sizes of the solar and lunar discs, and observations of Earths shadow during lunar eclipses, Hipparchus found a relationship between the lunar and solar distances that enabled him to calculate that the Moons mean distance from Earth is approximately 63 times Earths radius. This is a highly critical commentary in the form of two books on a popular poem by Aratus based on the work by Eudoxus. (Previous to the finding of the proofs of Menelaus a century ago, Ptolemy was credited with the invention of spherical trigonometry.) Alternate titles: Hipparchos, Hipparchus of Bithynia, Professor of Classics, University of Toronto. Hipparchus produced a table of chords, an early example of a trigonometric table. The shadow cast from a shadow stick was used to . [40], Lucio Russo has said that Plutarch, in his work On the Face in the Moon, was reporting some physical theories that we consider to be Newtonian and that these may have come originally from Hipparchus;[57] he goes on to say that Newton may have been influenced by them. The epicycle model he fitted to lunar eclipse observations made in Alexandria at 22 September 201BC, 19 March 200BC, and 11 September 200BC. I. While every effort has been made to follow citation style rules, there may be some discrepancies. One evening, Hipparchus noticed the appearance of a star where he was certain there had been none before. Isaac Newton and Euler contributed developments to bring trigonometry into the modern age. Omissions? Hipparchus compiled a table of the chords of angles and made them available to other scholars. Dovetailing these data suggests Hipparchus extrapolated the 158 BC 26 June solstice from his 145 solstice 12 years later, a procedure that would cause only minuscule error. Comparing both charts, Hipparchus calculated that the stars had shifted their apparent position by around two degrees. (1934). Theon of Smyrna wrote that according to Hipparchus, the Sun is 1,880 times the size of the Earth, and the Earth twenty-seven times the size of the Moon; apparently this refers to volumes, not diameters. However, Strabo's Hipparchus dependent latitudes for this region are at least 1 too high, and Ptolemy appears to copy them, placing Byzantium 2 high in latitude.) Hipparchus is generally recognized as discoverer of the precession of the equinoxes in 127BC. Previously this was done at daytime by measuring the shadow cast by a gnomon, by recording the length of the longest day of the year or with the portable instrument known as a scaphe. Not much is known about the life of Hipp archus. Hipparchus devised a geometrical method to find the parameters from three positions of the Moon at particular phases of its anomaly. Hipparchus had good reasons for believing that the Suns path, known as the ecliptic, is a great circle, i.e., that the plane of the ecliptic passes through Earths centre. This would correspond to a parallax of 7, which is apparently the greatest parallax that Hipparchus thought would not be noticed (for comparison: the typical resolution of the human eye is about 2; Tycho Brahe made naked eye observation with an accuracy down to 1). The catalog was superseded only in the late 16th century by Brahe and Wilhelm IV of Kassel via superior ruled instruments and spherical trigonometry, which improved accuracy by an order of magnitude even before the invention of the telescope. . the radius of the chord table in Ptolemy's Almagest, expressed in 'minutes' instead of 'degrees'generates Hipparchan-like ratios similar to those produced by a 3438 radius. In this only work by his hand that has survived until today, he does not use the magnitude scale but estimates brightnesses unsystematically. [52] [15] However, Franz Xaver Kugler demonstrated that the synodic and anomalistic periods that Ptolemy attributes to Hipparchus had already been used in Babylonian ephemerides, specifically the collection of texts nowadays called "System B" (sometimes attributed to Kidinnu).[16]. Hipparchus seems to have been the first to exploit Babylonian astronomical knowledge and techniques systematically. [2] Hipparchus was born in Nicaea, Bithynia, and probably died on the island of Rhodes, Greece. A solution that has produced the exact .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}5,4585,923 ratio is rejected by most historians although it uses the only anciently attested method of determining such ratios, and it automatically delivers the ratio's four-digit numerator and denominator. Trigonometry was probably invented by Hipparchus, who compiled a table of the chords of angles and made them available to other scholars. Hipparchus seems to have used a mix of ecliptic coordinates and equatorial coordinates: in his commentary on Eudoxus he provides stars' polar distance (equivalent to the declination in the equatorial system), right ascension (equatorial), longitude (ecliptic), polar longitude (hybrid), but not celestial latitude. This model described the apparent motion of the Sun fairly well. Similarly, Cleomedes quotes Hipparchus for the sizes of the Sun and Earth as 1050:1; this leads to a mean lunar distance of 61 radii. Ptolemy quotes (in Almagest III.1 (H195)) a description by Hipparchus of an equatorial ring in Alexandria; a little further he describes two such instruments present in Alexandria in his own time. Hipparchus was born in Nicaea, Bithynia (now Iznik, Turkey) and most likely died on the island of Rhodes. What fraction of the sky can be seen from the North Pole. Alexander Jones "Ptolemy in Perspective: Use and Criticism of his Work from Antiquity to the Nineteenth Century, Springer, 2010, p.36. He was then in a position to calculate equinox and solstice dates for any year. He had two methods of doing this. The term "trigonometry" was derived from Greek trignon, "triangle" and metron, "measure".. [4][5] He was the first whose quantitative and accurate models for the motion of the Sun and Moon survive. Hipparchuss most important astronomical work concerned the orbits of the Sun and Moon, a determination of their sizes and distances from Earth, and the study of eclipses. Hipparchus's use of Babylonian sources has always been known in a general way, because of Ptolemy's statements, but the only text by Hipparchus that survives does not provide sufficient information to decide whether Hipparchus's knowledge (such as his usage of the units cubit and finger, degrees and minutes, or the concept of hour stars) was based on Babylonian practice. As with most of his work, Hipparchus's star catalog was adopted and perhaps expanded by Ptolemy. In this way it might be easily discovered, not only whether they were destroyed or produced, but whether they changed their relative positions, and likewise, whether they were increased or diminished; the heavens being thus left as an inheritance to any one, who might be found competent to complete his plan. Scholars have been searching for it for centuries. Expressed as 29days + 12hours + .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}793/1080hours this value has been used later in the Hebrew calendar. Bowen A.C., Goldstein B.R. Later al-Biruni (Qanun VII.2.II) and Copernicus (de revolutionibus IV.4) noted that the period of 4,267 moons is approximately five minutes longer than the value for the eclipse period that Ptolemy attributes to Hipparchus. Hipparchus, also spelled Hipparchos, (born, Nicaea, Bithynia [now Iznik, Turkey]died after 127 bce, Rhodes? (2nd century bc).A prolific and talented Greek astronomer, Hipparchus made fundamental contributions to the advancement of astronomy as a mathematical science. How did Hipparchus discover and measure the precession of the equinoxes? His theory influence is present on an advanced mechanical device with code name "pin & slot". Trigonometry developed in many parts of the world over thousands of years, but the mathematicians who are most credited with its discovery are Hipparchus, Menelaus and Ptolemy. Hipparchus could confirm his computations by comparing eclipses from his own time (presumably 27 January 141BC and 26 November 139BC according to [Toomer 1980]), with eclipses from Babylonian records 345 years earlier (Almagest IV.2; [A.Jones, 2001]). Ptolemy later measured the lunar parallax directly (Almagest V.13), and used the second method of Hipparchus with lunar eclipses to compute the distance of the Sun (Almagest V.15). [26] Modern scholars agree that Hipparchus rounded the eclipse period to the nearest hour, and used it to confirm the validity of the traditional values, rather than to try to derive an improved value from his own observations. [3], Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity. In combination with a grid that divided the celestial equator into 24 hour lines (longitudes equalling our right ascension hours) the instrument allowed him to determine the hours. Ptolemy mentions (Almagest V.14) that he used a similar instrument as Hipparchus, called dioptra, to measure the apparent diameter of the Sun and Moon. Unlike Ptolemy, Hipparchus did not use ecliptic coordinates to describe stellar positions. "Hipparchus and the Stoic Theory of Motion". Trigonometry, which simplifies the mathematics of triangles, making astronomy calculations easier, was probably invented by Hipparchus. Review of, "Hipparchus Table of Climata and Ptolemys Geography", "Hipparchos' Eclipse-Based Longitudes: Spica & Regulus", "Five Millennium Catalog of Solar Eclipses", "New evidence for Hipparchus' Star Catalog revealed by multispectral imaging", "First known map of night sky found hidden in Medieval parchment", "Magnitudes of Thirty-six of the Minor Planets for the first day of each month of the year 1857", "The Measurement Method of the Almagest Stars", "The Genesis of Hipparchus' Celestial Globe", Hipparchus "Table of Climata and Ptolemys Geography", "Hipparchus on the Latitude of Southern India", Eratosthenes' Parallel of Rhodes and the History of the System of Climata, "Ptolemys Latitude of Thule and the Map Projection in the Pre-Ptolemaic Geography", "Hipparchus, Plutarch, Schrder, and Hough", "On the shoulders of Hipparchus: A reappraisal of ancient Greek combinatorics", "X-Prize Group Founder to Speak at Induction", "A new determination of lunar orbital parameters, precession constant, and tidal acceleration from LLR measurements", "The Epoch of the Constellations on the Farnese Atlas and their Origin in Hipparchus's Lost Catalogue", Eratosthenes Parallel of Rhodes and the History of the System of Climata, "The accuracy of eclipse times measured by the Babylonians", "Lunar Eclipse Times Recorded in Babylonian History", Learn how and when to remove this template message, Biography of Hipparchus on Fermat's Last Theorem Blog, Os Eclipses, AsterDomus website, portuguese, Ancient Astronomy, Integers, Great Ratios, and Aristarchus, David Ulansey about Hipparchus's understanding of the precession, A brief view by Carmen Rush on Hipparchus' stellar catalog, "New evidence for Hipparchus' Star Catalogue revealed by multispectral imaging", Ancient Greek and Hellenistic mathematics, https://en.wikipedia.org/w/index.php?title=Hipparchus&oldid=1141264401, Short description is different from Wikidata, Articles with unsourced statements from September 2022, Articles with unsourced statements from March 2021, Articles containing Ancient Greek (to 1453)-language text, Wikipedia articles incorporating a citation from the 1911 Encyclopaedia Britannica with Wikisource reference, Wikipedia external links cleanup from May 2017, Creative Commons Attribution-ShareAlike License 3.0. "Associations between the ancient star catalogs". Hipparchus also studied the motion of the Moon and confirmed the accurate values for two periods of its motion that Chaldean astronomers are widely presumed to have possessed before him,[24] whatever their ultimate origin. Ptolemy quotes an equinox timing by Hipparchus (at 24 March 146BC at dawn) that differs by 5 hours from the observation made on Alexandria's large public equatorial ring that same day (at 1 hour before noon): Hipparchus may have visited Alexandria but he did not make his equinox observations there; presumably he was on Rhodes (at nearly the same geographical longitude). View three larger pictures Biography Little is known of Hipparchus's life, but he is known to have been born in Nicaea in Bithynia. Earth's precession means a change in direction of the axis of rotation of Earth. [citation needed] Ptolemy claims his solar observations were on a transit instrument set in the meridian. In the second book, Hipparchus starts from the opposite extreme assumption: he assigns a (minimum) distance to the Sun of 490 Earth radii. It had been known for a long time that the motion of the Moon is not uniform: its speed varies. Part 2 can be found here. This is the first of three articles on the History of Trigonometry. Trigonometry was probably invented by Hipparchus, who compiled a table of the chords of angles and made them available to other scholars. Input the numbers into the arc-length formula, Enter 0.00977 radians for the radian measure and 2,160 for the arc length: 2,160 = 0.00977 x r. Divide each side by 0.00977. For his astronomical work Hipparchus needed a table of trigonometric ratios. Recalculating Toomer's reconstructions with a 3600' radiusi.e. Toomer, "The Chord Table of Hipparchus" (1973). He made observations of consecutive equinoxes and solstices, but the results were inconclusive: he could not distinguish between possible observational errors and variations in the tropical year. Astronomy test. [15] Right ascensions, for instance, could have been observed with a clock, while angular separations could have been measured with another device. His famous star catalog was incorporated into the one by Ptolemy and may be almost perfectly reconstructed by subtraction of two and two-thirds degrees from the longitudes of Ptolemy's stars. He computed this for a circle with a circumference of 21,600 units and a radius (rounded) of 3,438 units; this circle has a unit length of 1 arcminute along its perimeter. The result that two solar eclipses can occur one month apart is important, because this can not be based on observations: one is visible on the northern and the other on the southern hemisphereas Pliny indicatesand the latter was inaccessible to the Greek. An Investigation of the Ancient Star Catalog. Hipparchus may also have used other sets of observations, which would lead to different values. ), Italian philosopher, astronomer and mathematician. [40] He used it to determine risings, settings and culminations (cf. were probably familiar to Greek astronomers well before Hipparchus. Therefore, his globe was mounted in a horizontal plane and had a meridian ring with a scale. [15], Nevertheless, this system certainly precedes Ptolemy, who used it extensively about AD 150. He was also the inventor of trigonometry. He . Get a Britannica Premium subscription and gain access to exclusive content. Even if he did not invent it, Hipparchus is the first person whose systematic use of trigonometry we have documentary evidence. [note 1] What was so exceptional and useful about the cycle was that all 345-year-interval eclipse pairs occur slightly more than 126,007 days apart within a tight range of only approximately 12 hour, guaranteeing (after division by 4,267) an estimate of the synodic month correct to one part in order of magnitude 10 million. He had immense in geography and was one of the most famous astronomers in ancient times. Hipparchus opposed the view generally accepted in the Hellenistic period that the Atlantic and Indian Oceans and the Caspian Sea are parts of a single ocean. Hipparchus was recognized as the first mathematician known to have possessed a trigonometric table, which he needed when computing the eccentricity of the orbits of the Moon and Sun. common errors in the reconstructed Hipparchian star catalogue and the Almagest suggest a direct transfer without re-observation within 265 years. The globe was virtually reconstructed by a historian of science. Hipparchus adopted values for the Moons periodicities that were known to contemporary Babylonian astronomers, and he confirmed their accuracy by comparing recorded observations of lunar eclipses separated by intervals of several centuries. how did hipparchus discover trigonometry. He was intellectually honest about this discrepancy, and probably realized that especially the first method is very sensitive to the accuracy of the observations and parameters. . (Parallax is the apparent displacement of an object when viewed from different vantage points). Parallax lowers the altitude of the luminaries; refraction raises them, and from a high point of view the horizon is lowered. 2 - How did Hipparchus discover the wobble of Earth's. Ch. Galileo was the greatest astronomer of his time. It is believed that he computed the first table of chords for this purpose. ), Greek astronomer and mathematician who made fundamental contributions to the advancement of astronomy as a mathematical science and to the foundations of trigonometry. The Greeks were mostly concerned with the sky and the heavens. Thus, by all the reworking within scientific progress in 265 years, not all of Hipparchus's stars made it into the Almagest version of the star catalogue. To do so, he drew on the observations and maybe mathematical tools amassed by the Babylonian Chaldeans over generations. Others do not agree that Hipparchus even constructed a chord table. Hipparchus was born in Nicaea (Greek ), in Bithynia. 2 (1991) pp. to number the stars for posterity and to express their relations by appropriate names; having previously devised instruments, by which he might mark the places and the magnitudes of each individual star. 2 - Why did Copernicus want to develop a completely. Hipparchus also wrote critical commentaries on some of his predecessors and contemporaries. Alexandria is at about 31 North, and the region of the Hellespont about 40 North. "The Chord Table of Hipparchus and the Early History of Greek Trigonometry. Etymology. [58] According to one book review, both of these claims have been rejected by other scholars. Aristarchus of Samos is said to have done so in 280BC, and Hipparchus also had an observation by Archimedes. He had immense in geography and was one of the most famous astronomers in ancient times. One of his two eclipse trios' solar longitudes are consistent with his having initially adopted inaccurate lengths for spring and summer of 95+34 and 91+14 days. Hipparchus was not only the founder of trigonometry but also the man who transformed Greek astronomy from a purely theoretical into a practical predictive science. Ptolemy made no change three centuries later, and expressed lengths for the autumn and winter seasons which were already implicit (as shown, e.g., by A. Aaboe). [59], A line in Plutarch's Table Talk states that Hipparchus counted 103,049 compound propositions that can be formed from ten simple propositions. Emma Willard, Astronography, Or, Astronomical Geography, with the Use of Globes: Arranged Either for Simultaneous Reading and Study in Classes, Or for Study in the Common Method, pp 246, Denison Olmsted, Outlines of a Course of Lectures on Meteorology and Astronomy, pp 22, University of Toronto Quarterly, Volumes 1-3, pp 50, Histoire de l'astronomie ancienne, Jean Baptiste Joseph Delambre, Volume 1, p lxi; "Hipparque, le vrai pre de l'Astronomie"/"Hipparchus, the true father of Astronomy", Bowen A.C., Goldstein B.R. Definition. How did Hipparchus discover trigonometry? Prediction of a solar eclipse, i.e., exactly when and where it will be visible, requires a solid lunar theory and proper treatment of the lunar parallax. Hipparchus apparently made many detailed corrections to the locations and distances mentioned by Eratosthenes. The two points at which the ecliptic and the equatorial plane intersect, known as the vernal and autumnal equinoxes, and the two points of the ecliptic farthest north and south from the equatorial plane, known as the summer and winter solstices, divide the ecliptic into four equal parts. Like others before and after him, he found that the Moon's size varies as it moves on its (eccentric) orbit, but he found no perceptible variation in the apparent diameter of the Sun. Roughly five centuries after Euclid's era, he solved hundreds of algebraic equations in his great work Arithmetica, and was the first person to use algebraic notation and symbolism. According to Synesius of Ptolemais (4th century) he made the first astrolabion: this may have been an armillary sphere (which Ptolemy however says he constructed, in Almagest V.1); or the predecessor of the planar instrument called astrolabe (also mentioned by Theon of Alexandria). The earlier study's M found that Hipparchus did not adopt 26 June solstices until 146 BC, when he founded the orbit of the Sun which Ptolemy later adopted. [12] Hipparchus also made a list of his major works that apparently mentioned about fourteen books, but which is only known from references by later authors. Trigonometry Trigonometry simplifies the mathematics of triangles, making astronomy calculations easier. (It has been contended that authors like Strabo and Ptolemy had fairly decent values for these geographical positions, so Hipparchus must have known them too.

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