Misplaced Pages

#690309

89-473: Star map is another name for a star chart, a map of the night sky. Star map ( s ) or starmap(s) may also refer to: Star map A variety of archaeological sites and artifacts found are thought to indicate ancient made star charts. The oldest known star chart may be a carved ivory Mammoth tusk, drawn by early people from Asia who moved into Europe, that was discovered in Germany in 1979. This artifact

178-495: A cyclic quadrilateral , today called Ptolemy's theorem because its earliest extant source is a proof in the Almagest (I.10). The stereographic projection was ambiguously attributed to Hipparchus by Synesius (c. 400 AD), and on that basis Hipparchus is often credited with inventing it or at least knowing of it. However, some scholars believe this conclusion to be unjustified by available evidence. The oldest extant description of

267-672: A corruption of another value attributed to a Babylonian source: 365 + ⁠ 1 / 4 ⁠ + ⁠ 1 / 144 ⁠ days (= 365.25694... days = 365 days 6 hours 10 min). It is not clear whether Hipparchus got the value from Babylonian astronomers or calculated by himself. Before Hipparchus, astronomers knew that the lengths of the seasons are not equal. Hipparchus made observations of equinox and solstice, and according to Ptolemy ( Almagest III.4) determined that spring (from spring equinox to summer solstice) lasted 94 1 ⁄ 2 days, and summer (from summer solstice to autumn equinox) 92 + 1 ⁄ 2 days. This

356-460: A difference of approximately one day in approximately 300 years. So he set the length of the tropical year to 365 + 1 ⁄ 4 − 1 ⁄ 300 days (= 365.24666... days = 365 days 5 hours 55 min, which differs from the modern estimate of the value (including earth spin acceleration), in his time of approximately 365.2425 days, an error of approximately 6 min per year, an hour per decade, and ten hours per century. Between

445-502: A more detailed discussion. Pliny ( Naturalis Historia II.X) tells us that Hipparchus demonstrated that lunar eclipses can occur five months apart, and solar eclipses seven months (instead of the usual six months); and the Sun can be hidden twice in thirty days, but as seen by different nations. Ptolemy discussed this a century later at length in Almagest VI.6. The geometry, and the limits of

534-619: A panel in the same caves depicting a charging bison, a man with a bird's head and the head of a bird on top of a piece of wood, together may depict the Summer Triangle , which at the time was a circumpolar formation . Rappenglueck also discovered a drawing of the Northern Crown constellation in the cave of El Castillo (North of Spain), made in the same period as the Lascaux chart. Another star chart panel, created more than 21,000 years ago,

623-488: A plate of the southern constellations and two plates showing the entire northern and southern hemispheres in stereographic polar projection. Polish astronomer Johannes Hevelius published his Firmamentum Sobiescianum star atlas posthumously in 1690. It contained 56 large, double page star maps and improved the accuracy in the position of the southern stars. He introduced 11 more constellations, including Scutum , Lacerta , and Canes Venatici . In 1824 Sidney Hall produced

712-439: A popular poem by Aratus based on the work by Eudoxus . 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. 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 . The first trigonometric table

801-413: A set of star charts called Urania's Mirror . They are illustrations based on Alexander Jamieson 's A Celestial Atlas , but the addition of holes punched in them allowed them to be held up to a light to see a depiction of the constellation's stars. Hipparchus Hipparchus ( / h ɪ ˈ p ɑːr k ə s / ; Greek : Ἵππαρχος , Hípparkhos ; c.  190  – c.  120  BC)

890-499: A simpler sexagesimal system dividing a circle into 60 parts. Hipparchus also adopted the Babylonian astronomical cubit unit ( Akkadian ammatu , Greek πῆχυς pēchys ) that was equivalent to 2° or 2.5° ('large cubit'). Hipparchus probably compiled a list of Babylonian astronomical observations; Gerald J. Toomer , a historian of astronomy, has suggested that Ptolemy's knowledge of eclipse records and other Babylonian observations in

979-604: A sun or full moon, a lunar crescent, several stars including the Pleiades cluster and possibly the Milky Way. The oldest accurately dated star chart appeared in ancient Egyptian astronomy in 1534 BC. The earliest known star catalogues were compiled by the ancient Babylonian astronomers of Mesopotamia in the late 2nd millennium BC, during the Kassite Period ( ca. 1531–1155 BC). The oldest records of Chinese astronomy date to

SECTION 10

#1732899096691

1068-532: A table giving the daily motion of the Moon according to the date within a long period. However, the Greeks preferred to think in geometrical models of the sky. At the end of the third century BC, Apollonius of Perga had proposed two models for lunar and planetary motion: Apollonius demonstrated that these two models were in fact mathematically equivalent. However, all this was theory and had not been put to practice. Hipparchus

1157-525: A tight range of only approximately ± 1 ⁄ 2 hour, guaranteeing (after division by 4,267) an estimate of the synodic month correct to one part in order of magnitude 10 million. Hipparchus could confirm his computations by comparing eclipses from his own time (presumably 27 January 141 BC and 26 November 139 BC according to Toomer ) with eclipses from Babylonian records 345 years earlier ( Almagest IV.2 ). Later al-Biruni ( Qanun VII.2.II) and Copernicus ( de revolutionibus IV.4) noted that

1246-449: A triangle formed by the two places and the Moon, and from simple geometry was able to establish a distance of the Moon, expressed in Earth radii. Because the eclipse occurred in the morning, the Moon was not in the meridian , and it has been proposed that as a consequence the distance found by Hipparchus was a lower limit. In any case, according to Pappus, Hipparchus found that the least distance

1335-642: Is 32,500 years old and has a carving that resembles the constellation Orion , although it could not be confirmed and could also be a pregnancy chart. German researcher Dr Michael Rappenglueck, of the University of Munich, has suggested that drawing on the wall of the Lascaux caves in France could be a graphical representation of the Pleiades open cluster of stars. This is dated from 33,000 to 10,000 years ago. He also suggested

1424-485: Is 71 (from this eclipse), and the greatest 83 Earth radii. In the second book, Hipparchus starts from the opposite extreme assumption: he assigns a (minimum) distance to the Sun of 490 Earth radii. 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′). In this case,

1513-400: Is also close to an integer number of years (4,267 moons : 4,573 anomalistic periods : 4,630.53 nodal periods : 4,611.98 lunar orbits : 344.996 years : 344.982 solar orbits : 126,007.003 days : 126,351.985 rotations). 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

1602-401: Is consistent with 94 + 1 ⁄ 4 and 92 + 1 ⁄ 2 days, an improvement on the results ( 94 + 1 ⁄ 2 and 92 + 1 ⁄ 2 days) attributed to Hipparchus by Ptolemy. 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 ). Hipparchus also undertook to find

1691-512: Is inconsistent with a premise of the Sun moving around the Earth in a circle at uniform speed. Hipparchus's solution was to place the Earth not at the center of the Sun's motion, but at some distance from the center. This model described the apparent motion of the Sun fairly well. It is known today that the planets , including the Earth, move in approximate ellipses around the Sun, but this was not discovered until Johannes Kepler published his first two laws of planetary motion in 1609. The value for

1780-411: Is post-Hipparchus so the direction of transmission is not settled by the tablets. 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. He tabulated values for the chord function, which for a central angle in a circle gives the length of the straight line segment between

1869-646: Is sometimes called the "father of astronomy", a title conferred on him by Jean Baptiste Joseph Delambre in 1817. Hipparchus was born in Nicaea ( Ancient Greek : Νίκαια ), in Bithynia . The exact dates of his life are not known, but Ptolemy attributes astronomical observations to him in the period from 147 to 127 BC, and some of these are stated as made in Rhodes ; earlier observations since 162 BC might also have been made by him. His birth date ( c.  190  BC)

SECTION 20

#1732899096691

1958-526: Is the first astronomer known to attempt to determine the relative proportions and actual sizes of these orbits. Hipparchus devised a geometrical method to find the parameters from three positions of the Moon at particular phases of its anomaly. In fact, he did this separately for the eccentric and the epicycle model. Ptolemy describes the details in the Almagest IV.11. Hipparchus used two sets of three lunar eclipse observations that he carefully selected to satisfy

2047-630: Is uncertain, but is estimated as 705–10 AD. During the Song dynasty (960–1279), the Chinese astronomer Su Song wrote a book titled Xin Yixiang Fa Yao (New Design for the Armillary Clock) containing five maps of 1,464 stars. This has been dated to 1092. In 1193, the astronomer Huang Shang prepared a planisphere along with explanatory text. It was engraved in stone in 1247, and this chart still exists in

2136-485: Is visible simultaneously on half of the Earth, and the difference in longitude between places can be computed from the difference in local time when the eclipse is observed. His approach would give accurate results if it were correctly carried out but the limitations of timekeeping accuracy in his era made this method impractical. Late in his career (possibly about 135 BC) Hipparchus compiled his star catalog. Scholars have been searching for it for centuries. In 2022, it

2225-474: The Almagest came from a list made by Hipparchus. 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. However, Franz Xaver Kugler demonstrated that

2314-533: The Almagest . Some claim the table of Hipparchus may have survived in astronomical treatises in India, such as the Surya Siddhanta . Trigonometry was a significant innovation, because it allowed Greek astronomers to solve any triangle, and made it possible to make quantitative astronomical models and predictions using their preferred geometric techniques. Hipparchus must have used a better approximation for π than

2403-508: The Chinese constellations by name and does not show individual stars. The Farnese Atlas is a 2nd-century AD Roman copy of a Hellenistic era Greek statue depicting the Titan Atlas holding the celestial sphere on his shoulder. It is the oldest surviving depiction of the ancient Greek constellations, and includes grid circles that provide coordinate positions. Because of precession ,

2492-617: The Dutch East Indies . Their compilations resulted in the 1601 globe of Jodocus Hondius , who added 12 new southern constellations. Several other such maps were produced, including Johann Bayer 's Uranometria in 1603. The latter was the first atlas to chart both celestial hemispheres and it introduced the Bayer designations for identifying the brightest stars using the Greek alphabet. The Uranometria contained 48 maps of Ptolemaic constellations,

2581-616: The Tang dynasty (618–907) and discovered in the Mogao Caves of Dunhuang in Gansu , Western China along the Silk Road . This is a scroll 210 cm in length and 24.4 cm wide showing the sky between declinations 40° south to 40° north in twelve panels, plus a thirteenth panel showing the northern circumpolar sky. A total of 1,345 stars are drawn, grouped into 257 asterisms . The date of this chart

2670-514: The Warring States period (476–221 BC), but the earliest preserved Chinese star catalogues of astronomers Shi Shen and Gan De are found in the 2nd-century BC Shiji by the Western Han historian Sima Qian . The oldest Chinese graphical representation of the night sky is a lacquerware box from the 5th-century BC Tomb of Marquis Yi of Zeng , although this depiction shows the positions of

2759-572: The eccentricity attributed to Hipparchus by Ptolemy is that the offset is 1 ⁄ 24 of the radius of the orbit (which is a little too large), and the direction of the apogee would be at longitude 65.5° from the vernal equinox . Hipparchus may also have used other sets of observations, which would lead to different values. One of his two eclipse trios' solar longitudes are consistent with his having initially adopted inaccurate lengths for spring and summer of 95 + 3 ⁄ 4 and 91 + 1 ⁄ 4 days. His other triplet of solar positions

Star map (disambiguation) - Misplaced Pages Continue

2848-575: The ecliptic . This may have served as a prototype for the oldest European printed star chart, a 1515 set of woodcut portraits produced by Albrecht Dürer in Nuremberg , Germany . During the European Age of Discovery , expeditions to the southern hemisphere began to result in the addition of new constellations. These most likely came from the records of two Dutch sailors, Pieter Dirkszoon Keyser and Frederick de Houtman , who in 1595 traveled together to

2937-476: The 4th century BC and Timocharis and Aristillus in the 3rd century BC already divided the ecliptic in 360 parts (our degrees , Greek: moira) of 60 arcminutes and Hipparchus continued this tradition. 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. Eratosthenes (3rd century BC), in contrast, used

3026-528: The Geography of Eratosthenes"). It is known to us from Strabo of Amaseia, who in his turn criticised Hipparchus in his own Geographia . Hipparchus apparently made many detailed corrections to the locations and distances mentioned by Eratosthenes. It seems he did not introduce many improvements in methods, but he did propose a means to determine the geographical longitudes of different cities at lunar eclipses (Strabo Geographia 1 January 2012). A lunar eclipse

3115-546: The Greek. 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 must have been the first to be able to do this. A rigorous treatment requires spherical trigonometry , thus those who remain certain that Hipparchus lacked it must speculate that he may have made do with planar approximations. He may have discussed these things in Perí tēs katá plátos mēniaías tēs selēnēs kinēseōs ("On

3204-578: The Hellespont and are thought by many to be more likely possibilities for the eclipse Hipparchus used for his computations.) 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). He criticizes Hipparchus for making contradictory assumptions, and obtaining conflicting results ( Almagest V.11): but apparently he failed to understand Hipparchus's strategy to establish limits consistent with

3293-409: The Moon eclipsed while apparently it was not in exact opposition to the Sun. Parallax lowers the altitude of the luminaries; refraction raises them, and from a high point of view the horizon is lowered. Hipparchus and his predecessors used various instruments for astronomical calculations and observations, such as the gnomon , the astrolabe , and the armillary sphere . Hipparchus is credited with

3382-452: The Moon's equation of the center in the Hipparchan model.) Before Hipparchus, Meton , Euctemon , and their pupils at Athens had made a solstice observation (i.e., timed the moment of the summer solstice ) on 27 June 432 BC ( proleptic Julian calendar ). Aristarchus of Samos is said to have done so in 280 BC, and Hipparchus also had an observation by Archimedes . He observed

3471-564: The Sun is on the equator (i.e., in one of the equinoctial points on the ecliptic ), but the shadow falls above or below the opposite side of the ring when the Sun is south or north of the equator. 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 applied his knowledge of spherical angles to

3560-548: The Wen Miao temple in Suzhou . In Muslim astronomy , the first star chart to be drawn accurately was most likely the illustrations produced by the Persian astronomer Abd al-Rahman al-Sufi in his 964 work titled Book of Fixed Stars . This book was an update of parts VII.5 and VIII.1 of the 2nd century Almagest star catalogue by Ptolemy . The work of al-Sufi contained illustrations of

3649-448: The apparent diameter of the Sun and Moon. Pappus of Alexandria described it (in his commentary on the Almagest of that chapter), as did Proclus ( Hypotyposis IV). It was a four-foot rod with a scale, a sighting hole at one end, and a wedge that could be moved along the rod to exactly obscure the disk of Sun or Moon. Hipparchus also observed solar equinoxes , which may be done with an equatorial ring : its shadow falls on itself when

Star map (disambiguation) - Misplaced Pages Continue

3738-407: The apparent diameters of the Sun and Moon with his diopter . 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. He found that at the mean distance of the Moon, the Sun and Moon had the same apparent diameter; at that distance, the Moon's diameter fits 650 times into

3827-413: The center of the Earth, but the observer is at the surface—the Moon, Earth and observer form a triangle with a sharp angle that changes all the time. From the size of this parallax, the distance of the Moon as measured in Earth radii can be determined. 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). In

3916-467: The change in the length of the day (see ΔT ) we estimate that the error in the assumed length of the synodic month was less than 0.2 second in the fourth century BC and less than 0.1 second in Hipparchus's time. It had been known for a long time that the motion of the Moon is not uniform: its speed varies. This is called its anomaly and it repeats with its own period; the anomalistic month . The Chaldeans took account of this arithmetically, and used

4005-462: The chords for angles with increments of 7.5°. In modern terms, the chord subtended by a central angle in a circle of given radius R equals R times twice the sine of half of the angle, i.e.: The now-lost work in which Hipparchus is said to have developed his chord table, is called Tōn en kuklōi eutheiōn ( Of Lines Inside a Circle ) in Theon of Alexandria 's fourth-century commentary on section I.10 of

4094-403: The circle, i.e., the mean apparent diameters are 360 ⁄ 650 = 0°33′14″. Like others before and after him, he also noticed that the Moon has a noticeable parallax , i.e., that it appears displaced from its calculated position (compared to the Sun or stars ), and the difference is greater when closer to the horizon. He knew that this is because in the then-current models the Moon circles

4183-538: The constellations and portrayed the brighter stars as dots. The original book did not survive, but a copy from about 1009 is preserved at the Oxford University . Perhaps the oldest European star map was a parchment manuscript titled De Composicione Spere Solide . It was most likely produced in Vienna , Austria in 1440 and consisted of a two-part map depicting the constellations of the northern celestial hemisphere and

4272-463: The distances and sizes of the Sun and the Moon, in the now-lost work On Sizes and Distances ( Ancient Greek : Περὶ μεγεθῶν καὶ ἀποστημάτων Peri megethon kai apostematon ). His work is mentioned in Ptolemy's Almagest V.11, and in a commentary thereon by Pappus ; Theon of Smyrna (2nd century) also mentions the work, under the title On Sizes and Distances of the Sun and Moon . Hipparchus measured

4361-563: The first book, Hipparchus assumes that the parallax of the Sun is 0, as if it is at infinite distance. He then analyzed a solar eclipse, which Toomer presumes to be the eclipse of 14 March 190 BC. It was total in the region of the Hellespont (and in his birthplace, Nicaea); at the time Toomer proposes the Romans were preparing for war with Antiochus III in the area, and the eclipse is mentioned by Livy in his Ab Urbe Condita Libri VIII.2. It

4450-467: The first century; Ptolemy's second-century Almagest ; and additional references to him in the fourth century by Pappus and Theon of Alexandria in their commentaries on the Almagest . Hipparchus's only preserved work is Commentary on the Phaenomena of Eudoxus and Aratus ( Ancient Greek : Τῶν Ἀράτου καὶ Εὐδόξου φαινομένων ἐξήγησις ). This is a highly critical commentary in the form of two books on

4539-419: The first method is very sensitive to the accuracy of the observations and parameters. (In fact, modern calculations show that the size of the 189 BC solar eclipse at Alexandria must have been closer to 9 ⁄ 10 ths and not the reported 4 ⁄ 5 ths, a fraction more closely matched by the degree of totality at Alexandria of eclipses occurring in 310 and 129 BC which were also nearly total in

SECTION 50

#1732899096691

4628-429: The first surviving text discussing it is by Menelaus of Alexandria in the first century, who now, on that basis, commonly is credited with its discovery. (Previous to the finding of the proofs of Menelaus a century ago, Ptolemy was credited with the invention of spherical trigonometry.) Ptolemy later used spherical trigonometry to compute things such as the rising and setting points of the ecliptic , or to take account of

4717-400: The first to develop a reliable method to predict solar eclipses . His other reputed achievements include the discovery and measurement of Earth's precession, the compilation of the first known comprehensive star catalog from the western world, and possibly the invention of the astrolabe , as well as of the armillary sphere that he may have used in creating the star catalogue. Hipparchus

4806-440: The geometry of book 2 it follows that the Sun is at 2,550 Earth radii, and the mean distance of the Moon is 60 + 1 ⁄ 2 radii. 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. Apparently Hipparchus later refined his computations, and derived accurate single values that he could use for predictions of solar eclipses. See Toomer (1974) for

4895-725: The greatest overall astronomer of antiquity . He was the first whose quantitative and accurate models for the motion of the Sun and Moon survive. For this he certainly made use of the observations and perhaps the mathematical techniques accumulated over centuries by the Babylonians and by Meton of Athens (fifth century BC), Timocharis , Aristyllus , Aristarchus of Samos , and Eratosthenes , among others. He developed trigonometry and constructed trigonometric tables , and he solved several problems of spherical trigonometry . With his solar and lunar theories and his trigonometry, he may have been

4984-447: The invention or improvement of several astronomical instruments, which were used for a long time for naked-eye observations. 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 ). With an astrolabe Hipparchus

5073-449: The large total lunar eclipse of 26 November 139 BC, when over a clean sea horizon as seen from Rhodes, the Moon was eclipsed in the northwest just after the Sun rose in the southeast. This would be the second eclipse of the 345-year interval that Hipparchus used to verify the traditional Babylonian periods: this puts a late date to the development of Hipparchus's lunar theory. We do not know what "exact reason" Hipparchus found for seeing

5162-633: The lunar parallax . If he did not use spherical trigonometry, Hipparchus may have used a globe for these tasks, reading values off coordinate grids drawn on it, or he may have made approximations from planar geometry, or perhaps used arithmetical approximations developed by the Chaldeans. 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. The traditional value (from Babylonian System B) for

5251-475: The mean synodic month is 29 days; 31,50,8,20 (sexagesimal) = 29.5305941... days. Expressed as 29 days + 12 hours + ⁠ 793 / 1080 ⁠  hours this value has been used later in the Hebrew calendar . The Chaldeans also knew that 251 synodic months ≈ 269 anomalistic months . Hipparchus used the multiple of this period by a factor of 17, because that interval is also an eclipse period, and

5340-511: The monthly motion of the Moon in latitude"), a work mentioned in the Suda . Pliny also remarks that "he also discovered for what exact reason, although the shadow causing the eclipse must from sunrise onward be below the earth, it happened once in the past that the Moon was eclipsed in the west while both luminaries were visible above the earth" (translation H. Rackham (1938), Loeb Classical Library 330 p. 207). Toomer argued that this must refer to

5429-531: The observation made on Alexandria 's large public equatorial ring that same day (at 1 hour before noon). Ptolemy claims his solar observations were on a transit instrument set in the meridian. At the end of his career, Hipparchus wrote a book entitled Peri eniausíou megéthous ("On the Length of the Year") regarding his results. The established value for the tropical year , introduced by Callippus in or before 330 BC

SECTION 60

#1732899096691

5518-419: The observations, rather than a single value for the distance. His results were the best so far: the actual mean distance of the Moon is 60.3 Earth radii, within his limits from Hipparchus's second book. 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 . From

5607-482: The one given by Archimedes of between 3 + 10 ⁄ 71 (≈ 3.1408) and 3 + 1 ⁄ 7 (≈ 3.1429). Perhaps he had the approximation later used by Ptolemy, sexagesimal 3;08,30 (≈ 3.1417) ( Almagest VI.7). Hipparchus could have constructed his chord table using the Pythagorean theorem and a theorem known to Archimedes. He also might have used the relationship between sides and diagonals of

5696-510: The other way around is debatable. Hipparchus also gave the value for the sidereal year to be 365 + ⁠ 1 / 4 ⁠ + ⁠ 1 / 144 ⁠ days (= 365.25694... days = 365 days 6 hours 10 min). Another value for the sidereal year that is attributed to Hipparchus (by the physician Galen in the second century AD) is 365 + ⁠ 1 / 4 ⁠ + ⁠ 1 / 288 ⁠ days (= 365.25347... days = 365 days 6 hours 5 min), but this may be

5785-411: The parallax of the Sun decreases (i.e., its distance increases), the minimum limit for the mean distance is 59 Earth radii—exactly the mean distance that Ptolemy later derived. Hipparchus thus had the problematic result that his minimum distance (from book 1) was greater than his maximum mean distance (from book 2). He was intellectually honest about this discrepancy, and probably realized that especially

5874-557: The period of 4,267 moons is approximately five minutes longer than the value for the eclipse period that Ptolemy attributes to Hipparchus. However, the timing methods of the Babylonians had an error of no fewer than eight minutes. 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. From modern ephemerides and taking account of

5963-441: The points where the angle intersects the circle. He may have 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 for each arcminute along its perimeter. (This was “proven” by Toomer, but he later “cast doubt“ upon his earlier affirmation. Other authors have argued that a circle of radius 3,600 units may instead have been used by Hipparchus. ) He tabulated

6052-403: The positions of Sun and Moon when a solar or lunar eclipse is possible, are explained in Almagest VI.5. Hipparchus apparently made similar calculations. 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 hemisphere—as Pliny indicates—and the latter was inaccessible to

6141-421: The positions of the constellations slowly change over time. By comparing the positions of the 41 constellations against the grid circles, an accurate determination can be made of the epoch when the original observations were performed. Based upon this information, the constellations were catalogued at 125 ± 55 BC . This evidence indicates that the star catalogue of the 2nd-century BC Greek astronomer Hipparchus

6230-438: The problem of denoting locations on the Earth's surface. Before him a grid system had been used by Dicaearchus of Messana , but Hipparchus was the first to apply mathematical rigor to the determination of the latitude and longitude of places on the Earth. Hipparchus wrote a critique in three books on the work of the geographer Eratosthenes of Cyrene (3rd century BC), called Pròs tèn Eratosthénous geographían ("Against

6319-406: The ratio of the epicycle model ( 3122 + 1 ⁄ 2  : 247 + 1 ⁄ 2 ), which is too small (60 : 4;45 sexagesimal). Ptolemy established a ratio of 60 : 5 + 1 ⁄ 4 . (The maximum angular deviation producible by this geometry is the arcsin of 5 + 1 ⁄ 4 divided by 60, or approximately 5° 1', a figure that is sometimes therefore quoted as the equivalent of

6408-546: The representative figure for astronomy. It is not certain that the figure is meant to represent him. Previously, Eudoxus of Cnidus in the fourth century BC had described the stars and constellations in two books called Phaenomena and Entropon . Aratus wrote a poem called Phaenomena or Arateia based on Eudoxus's work. Hipparchus wrote a commentary on the Arateia —his only preserved work—which contains many stellar positions and times for rising, culmination, and setting of

6497-687: The requirements. The eccentric model he fitted to these eclipses from his Babylonian eclipse list: 22/23 December 383 BC, 18/19 June 382 BC, and 12/13 December 382 BC. The epicycle model he fitted to lunar eclipse observations made in Alexandria at 22 September 201 BC, 19 March 200 BC, and 11 September 200 BC. These figures are due to the cumbersome unit he used in his chord table and may partly be due to some sloppy rounding and calculation errors by Hipparchus, for which Ptolemy criticised him while also making rounding errors. A simpler alternate reconstruction agrees with all four numbers. Hipparchus found inconsistent results; he later used

6586-504: The second and third centuries, coins were made in his honour in Bithynia that bear his name and show him with a globe . Relatively little of Hipparchus's direct work survives into modern times. Although he wrote at least fourteen books, only his commentary on the popular astronomical poem by Aratus was preserved by later copyists. Most of what is known about Hipparchus comes from Strabo 's Geography and Pliny 's Natural History in

6675-417: The shadow of the Earth is a cone rather than a cylinder as under the first assumption. Hipparchus observed (at lunar eclipses) that at the mean distance of the Moon, the diameter of the shadow cone is 2 + 1 ⁄ 2 lunar diameters. That apparent diameter is, as he had observed, 360 ⁄ 650 degrees. With these values and simple geometry, Hipparchus could determine the mean distance; because it

6764-505: The solstice observation of Meton and his own, there were 297 years spanning 108,478 days; this implies a tropical year of 365.24579... days = 365 days;14,44,51 (sexagesimal; = 365 days + ⁠ 14 / 60 ⁠ + ⁠ 44 / 60 ⁠ + ⁠ 51 / 60 ⁠ ), a year length found on one of the few Babylonian clay tablets which explicitly specifies the System B month. Whether Babylonians knew of Hipparchus's work or

6853-503: The stereographic projection is found in Ptolemy 's Planisphere (2nd century AD). Besides geometry, Hipparchus also used arithmetic techniques developed by the Chaldeans . He was one of the first Greek mathematicians to do this and, in this way, expanded the techniques available to astronomers and geographers. There are several indications that Hipparchus knew spherical trigonometry, but

6942-473: The summer solstices in 146 and 135 BC both accurately to a few hours, but observations of the moment of equinox were simpler, and he made twenty during his lifetime. Ptolemy gives an extensive discussion of Hipparchus's work on the length of the year in the Almagest III.1, and quotes many observations that Hipparchus made or used, spanning 162–128 BC, including an equinox timing by Hipparchus (at 24 March 146 BC at dawn) that differs by 5 hours from

7031-462: 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 ). Hipparchus's long draconitic lunar period (5,458 months = 5,923 lunar nodal periods) also appears a few times in Babylonian records . But the only such tablet explicitly dated,

7120-551: Was 365 + 1 ⁄ 4 days. Speculating a Babylonian origin for the Callippic year is difficult to defend, since Babylon did not observe solstices thus the only extant System B year length was based on Greek solstices (see below). Hipparchus's equinox observations gave varying results, but he points out (quoted in Almagest III.1(H195)) that the observation errors by him and his predecessors may have been as large as 1 ⁄ 4 day. He used old solstice observations and determined

7209-502: 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 was born in Nicaea , Bithynia , and probably died on the island of Rhodes , Greece. He is known to have been a working astronomer between 162 and 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some,

7298-572: Was also observed in Alexandria, where the Sun was reported to be obscured 4/5ths by the Moon. Alexandria and Nicaea are on the same meridian. Alexandria is at about 31° North, and the region of the Hellespont about 40° North. (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. 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 could draw

7387-824: Was announced that a part of it was discovered in a medieval parchment manuscript, Codex Climaci Rescriptus , from Saint Catherine's Monastery in the Sinai Peninsula , Egypt as hidden text ( palimpsest ). Hipparchus also constructed a celestial globe depicting the constellations, based on his observations. 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 . For more information see Discovery of precession . In Raphael 's painting The School of Athens , Hipparchus may be depicted holding his celestial globe, as

7476-537: Was apparently compiled by Hipparchus, who is consequently now known as "the father of trigonometry". Earlier Greek astronomers and mathematicians were influenced by Babylonian astronomy to some extent, for instance the period relations of the Metonic cycle and Saros cycle may have come from Babylonian sources (see " Babylonian astronomical diaries "). Hipparchus seems to have been the first to exploit Babylonian astronomical knowledge and techniques systematically. Eudoxus in

7565-417: Was calculated by Delambre based on clues in his work. Hipparchus must have lived some time after 127 BC because he analyzed and published his observations from that year. Hipparchus obtained information from Alexandria as well as Babylon , but it is not known when or if he visited these places. He is believed to have died on the island of Rhodes, where he seems to have spent most of his later life. In

7654-404: Was computed for a minimum distance of the Sun, it is the maximum mean distance possible for the Moon. With his value for the eccentricity of the orbit, he could compute the least and greatest distances of the Moon too. According to Pappus, he found a least distance of 62, a mean of 67 + 1 ⁄ 3 , and consequently a greatest distance of 72 + 2 ⁄ 3 Earth radii. With this method, as

7743-566: Was found in the La Tête du Lion cave ( fr ). The bovine in this panel may represent the constellation Taurus , with a pattern representing the Pleiades just above it. A star chart drawn 5000 years ago by the Indians in Kashmir, which also depict a supernova for the first time in human history. The Nebra sky disk , a 30 cm wide bronze disk dated to 1600 BC, bears gold symbols generally interpreted as

7832-405: Was the first to be able to measure the geographical latitude and time by observing fixed stars. 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 . Ptolemy mentions ( Almagest V.14) that he used a similar instrument as Hipparchus, called dioptra , to measure

7921-612: Was used. A Roman era example of a graphical representation of the night sky is the Ptolemaic Egyptian Dendera zodiac , dating from 50 BC. This is a bas relief sculpting on a ceiling at the Dendera Temple complex . It is a planisphere depicting the zodiac in graphical representations. However, individual stars are not plotted. The oldest surviving manuscript star chart was the Dunhuang Star Chart , dated to

#690309