- Hipparchus
**Hipparchus**(Greek polytonic|Ἵππαρχος; ca. 190 BC – ca. 120 BC) was a Greekastronomer ,geographer , andmathematician of theHellenistic period.Hipparchus was born in Nicaea (now

Iznik ,Turkey ), and probably died on the island ofRhodes . He is known to have been a working astronomer at least from 147 BC to 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity. He was the first Greek whose quantitative and accurate models for the motion of theSun andMoon survive. For this he certainly made use of the observations and perhaps the mathematical techniques accumulated over centuries by theChaldea ns fromBabylonia . He possessed a trigonometric table, and appears to have solved some problems ofspherical trigonometry . With his solar and lunar theories and his trigonometry, he may have been the first to develop a reliable method to predictsolar eclipse s. His other reputed achievements include the discovery ofprecession , the compilation of the first comprehensivestar catalog of the western world, and possibly the invention of theastrolabe , also of thearmillary sphere which first appeared during his century and was used by him during the creation of much of the star catalogue. It would be three centuries before Claudius Ptolemaeus' synthesis of astronomy would supersede the work of Hipparchus; it is heavily dependent on it in many areas.**Life and work**Relatively little of Hipparchus' direct work survived 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 fromPtolemy 's (2nd century) "Almagest ", with additional references to him byPappus of Alexandria andTheon of Alexandria (ca. 4th century AD) in their commentaries on the "Almagest"; fromStrabo 's "Geographia" ("Geography"), and fromPliny the Elder 's "Naturalis historia" ("Natural history") (1st century AD). [*For general information on Hipparchus see the following biographical articles: G. J. Toomer, "Hipparchus" (1978); and A. Jones, "Hipparchus."*]There is a strong tradition that Hipparchus was born in Nicaea (Greek "Νικαία"), in the ancient district of

Bithynia (modern-day Iznik in province Bursa), in what today isTurkey the country.The exact dates of his life are not known, but Ptolemy attributes to him astronomical observations in the period from 147 BC to 127 BC, and some of these are stated as made in Rhodes; earlier observations since 162 BC might also be made by him. The date of his birth (ca. 190 BC) 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 latest observations. Hipparchus obtained information from

Alexandria as well asBabylon , but it is not known when or if he visited these places.It is not known what Hipparchus' economic means were and how he supported his scientific activities. Also, his appearance is unknown: there are no contemporary portraits. In the 2nd and 3rd centuries

coin s were made in his honour inBithynia that bear his name and show him with aglobe ; this supports the tradition that he was born there.Hipparchus is believed to have died on the island of

Rhodes , where he seems to have spent most of his later life.Hipparchus' only preserved work is "Tōn Aratou kai Eudoxou Fainomenōn exēgēsis" ("Commentary on the Phaenomena of Eudoxus and Aratus"). This is a highly critical commentary in the form of two books on a popular

poem byAratus based on the work by Eudoxus. [*Modern edition:*] Hipparchus also made a list of his major works, which 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.Karl Manitius ("In Arati et Eudoxi Phaenomena", Leipzig, 1894).Hipparchus was in the national news in 2005, when it was again proposed (as in 1898) that the data on the

celestial globe of Hipparchus or in his star catalog may have been preserved in the only surviving large ancient celestial globe which depicts the constellations with moderate accuracy, the globe carried by the Farnese Atlas. There are a variety of mis-steps [*[*] in the more ambitious 2005 paper, thus no specialists in the area accept its widely publicized speculation. [*http://www.dioi.org/ggg.htm D.Rawlins*] , "Farnese Atlas Celestial Globe, Proposed Astronomical Origins", 2005.*[*]*http://www.phys.lsu.edu/farnese/ B. E. Schaefer*] , "Epoch of the Constellations on the Farnese Atlas and their Origin in Hipparchus's Lost Catalog", "Journal for the History of Astronomy", May, 2005 versus [*http://www.people.scs.fsu.edu/~dduke/farnese Dennis Duke*] "Journal for the History of Astronomy", February, 2006.There is evidence, based on references in non-scientific writers such as Plutarch, that Hipparchus was aware of some physical ideas that we consider Newtonian, and some claim that Newton knew this. [

]Lucio Russo , "The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had To Be Reborn", (Berlin: Springer, 2004). ISBN 3-540-20396-6.**Babylonian sources**Earlier Greek astronomers and mathematicians were influenced by Babylonian astronomy to some extent, for instance the period relations of the

Metonic cycle andSaros cycle may have come from Babylonian sources. Hipparchus seems to have been the first to exploit Babylonian astronomical knowledge and techniques systematically [*For more information see G. J. Toomer, "Hipparchus and Babylonian astronomy."*] . Except forTimocharis andAristillus , he was the first Greek known to divide the circle in 360 degrees of 60arc minute s (Eratosthenes before him used a simplersexagesimal system dividing a circle into 60 parts). He also used the Babylonian unit "pechus" ("cubit") of about 2° or 2.5°.Hipparchus probably compiled a list of Babylonian astronomical observations; G. Toomer, a historian of astronomy, has suggested that Ptolemy's knowledge of eclipse records and other Babylonian observations in the "Almagest" came from a list made by Hipparchus. Hipparchus' use of Babylonian sources has always been known in a general way, because of Ptolemy's statements. However,

Franz Xaver Kugler demonstrated that the synodic and anomalistic periods that Ptolemy attributes to Hipparchus had already been used in Babylonianephemerides , specifically the collection of texts nowadays called "System B" (sometimes attributed toKidinnu ). [*Franz Xaver Kugler, "Die Babylonische Mondrechnung" ("The Babylonian lunar computation"), Freiburg im Breisgau, 1900.*]Hipparchus's long draconitic lunar period (5458 months = 5923 draconitic months) also appears a few times in Babylonian records. But the only such tablet explicitly dated is post-Hipparchus so the direction of transmission is not secured.

**Geometry, trigonometry, and other mathematical techniques**Hipparchus is recognised as the first mathematician known to have possessed a

trigonometry table, which he needed when computing the eccentricity of theorbit s of the Moon and Sun. He tabulated values for the chord function, which gives the length of the chord for each angle. He did this for a circle with a circumference of 21,600 and a radius (rounded) of 3438 units: this circle has a unit length of 1 arc minute along its perimeter. He tabulated the chords for angles with increments of 7.5°. In modern terms, the chord of an angle equals twice the sine of half of the angle, i.e.::chord("A") = 2 sin("A"/2).

He described the chord table in a work, now lost, called "Toon en kuklooi eutheioon" ("Of Lines Inside a Circle") by

Theon of Alexandria (4th century) in his commentary on the "Almagest" I.10; some claim his table may have survived in astronomical treatises inIndia , for instance 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. [*Toomer, "The Chord Table of Hipparchus" (1973).*]For his chord table Hipparchus must have used a better approximation for π than the one from

Archimedes of between 3 + 1/7 and 3 + 10/71; perhaps he had the one later used by Ptolemy: 3;8:30 (sexagesimal ) ("Almagest" VI.7); but it is not known if he computed an improved value himself.Hipparchus could construct his chord table using the

Pythagorean theorem and atheorem known to Archimedes. He also might have developed and used the theorem inplane geometry calledPtolemy's theorem , because it was proved by Ptolemy in his "Almagest" (I.10) (later elaborated on by Carnot).Hipparchus was the first to show that the

stereographic projection is conformal, and that it transforms circles on thesphere that do not pass through the center of projection to circles on the plane. This was the basis for theastrolabe .Besides geometry, Hipparchus also used

arithmetic techniques developed by theChaldea ns. 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 the first surviving text of it is that of

Menelaus of Alexandria in the 1st century, who on that basis is now commonly credited with its discovery. (Previous to the finding of the proofs of Menelaus a century ago, professedly conservative scholars' enthusiastic abhorrence of any attributive vacuum - an abhorrence evident throughout the present article - similarly credited Ptolemy with the invention of spherical trigonometry.) Ptolemy later used spherical trigonometry to compute things like the rising and setting points of theecliptic , or to take account of the lunarparallax . 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. Or perhaps he used spherical trigonometry.**Lunar and solar theory****Motion of the Moon**Hipparchus also studied the motion of the

Moon and confirmed the accurate values for two periods of its motion that Chaldean astronomers certainly possessed before him, whatever their ultimate origin. The traditional value (from Babylonian System B) for the meansynodic month is 29 days;31,50,8,20 (sexagesimal) = 29.5305941... d. Expressed as 29 days + 12 hours + 793/1080 hours this value has been used later in theHebrew calendar (possibly from Babylonian sources). The Chaldeans also knew that 251synodic month s = 269anomalistic month s. Hipparchus used an extension of this period by a factor of 17, because after that interval the Moon also would have a similar latitude, and it is close to an integer number of years (345). Therefore, eclipses would reappear under almost identical circumstances. The period is 126007 days 1 hour (rounded). Hipparchus could confirm his computations by comparing eclipses from his own time (presumably27 January 141 BC and26 November 139 BC according to [Toomer 1980] ), with eclipses from Babylonian records 345 years earlier ("Almagest" IV.2; [A.Jones, 2001] ). Alreadyal-Biruni ("Qanun" VII.2.II) andCopernicus ("de revolutionibus" IV.4) noted that the period of 4,267 lunations is actually about 5 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 less than 8 minutes [*Stephenson & Fatoohi 1993; Steele "et al." 1997*] . 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 try to derive an improved value from his own observations. From modern ephemerides [*Chapront "et al." 2002*] and taking account of 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 seconds in the 4th century BC and less than 0.1 seconds in Hipparchus' time**Orbit of the Moon**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 a table giving the daily motion of the Moon according to the date within a long period. The Greeks however preferred to think in geometrical models of the sky.Apollonius of Perga had at the end of the 3rd century BC proposed two models for lunar and planetary motion:

# In the first, the Moon would move uniformly along a circle, but the Earth would be eccentric, i.e., at some distance of the center of the circle. So the apparent angular speed of the Moon (and its distance) would vary.

# The Moon itself would move uniformly (with some mean motion in anomaly) on a secondary circular orbit, called an "epicycle", that itself would move uniformly (with some mean motion in longitude) over the main circular orbit around the Earth, called "deferent"; seedeferent and epicycle .Apollonius demonstrated that these two models were in fact mathematically equivalent. However, all this was theory and had not been put to practice. Hipparchus was the first astronomer we know attempted to determine the relative proportions and actual sizes of theseorbit s.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, which he carefully selected to satisfy 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, and11 September 200 BC.

* For the eccentric model, Hipparchus found for the ratio between the radius of theeccenter and the distance between the center of the eccenter and the center of the ecliptic (i.e., the observer on Earth): 3144 : 327+2/3 ;

* and for the epicycle model, the ratio between the radius of the deferent and the epicycle: 3122+1/2 : 247+1/2 . The somewhat weird numbers are due to the cumbersome unit he used in his chord table according to one group of historians, who explain their reconstruction's inability to agree with these four numbers as partly due to some sloppy rounding and calculation errors by Hipparchus, for which Ptolemy criticised him (he himself made rounding errors too). A simpler alternate reconstruction [*Summarized in Hugh Thurston (2002): "Isis"*] agrees with all four numbers. Anyway, Hipparchus found inconsistent results; he later used the ratio of the epicycle model (3122+1/2 : 247+1/2), which is too small (60 : 4;45 hexadecimal). Ptolemy established a ratio of 60 : 5+1/4. [**93**, 58-69.*Toomer, 1967*] .**Apparent motion of the Sun**Before Hipparchus,

Meton ,Euctemon , and their pupils atAthens had made a solstice observation (i.e., timed the moment of the summersolstice ) onJune 27 , 432 BC (proleptic Julian calendar ).Aristarchus of Samos is said to have done so in 280 BC, and Hipparchus also had an observation byArchimedes . Hipparchus himself observed the summer solstice in 135 BC, but he found observations of the moment ofequinox more accurate, and he made many during his lifetime. Ptolemy gives an extensive discussion of Hipparchus' work on the length of the year in the "Almagest" III.1, and quotes many observations that Hipparchus made or used, spanning 162 BC to 128 BC.Ptolemy quotes an equinox timing by Hipparchus (at

24 March 146 BC at dawn) that differs by 5h from the observation made onAlexandria 's large publicequatorial ring that same day (at 1h 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). He could have used the equatorial ring of his armillary sphere or another equatorial ring for these observations, but Hipparchus (and Ptolemy) knew that observations with these instruments are sensitive to a precise alignment with theequator , so if he were restricted to an armillary, it would make more sense to use its meridian ring as a transit instrument. The problem with an equatorial ring (if an observer is naive enough to trust it very near dawn or dusk) is that atmosphericrefraction lifts the Sun significantly above the horizon: so for a northern hemisphere observer its apparentdeclination is too high, which changes the observed time when the Sun crosses the equator. (Worse, the refraction decreases as the Sun rises and increases as it sets, so it may appear to move in the wrong direction with respect to the equator in the course of the day - as Ptolemy mentions. Ptolemy and Hipparchus apparently did not realize that refraction is the cause.) However, such numbing details have doubtful relation to the data of either man, since there is no textual, scientific, or statistical ground for believing that their equinoxes were taken on an equatorial ring, which is useless for solstices in any case. Not one of two centuries of mathematical investigations of their solar errors has claimed to have traced them to refraction's effect on use of an equatorial ring. And Ptolemy claims his solar observations were on a transit instrument set in the meridian.At the end of his career, Hipparchus wrote a book called "Peri eniausíou megéthous" ("On the Length of the Year") about his results. The established value for the

tropical year , introduced byCallippus in or before 330 BC was 365 + 1/4 days. (Possibly from Babylonian sources, see above [???] . Speculating a Babylonian origin for the Callippic year is hard to defend, since Babylon did not observe solstices thus the only extant System B yearlength was based on Greek solstices. See below.) Hipparchus' equinox observations gave varying results, but he himself points out (quoted in "Almagest" III.1(H195)) that the observation errors by himself and his predecessors may have been as large as 1/4 day. He used old solstice observations, and determined a difference of about one day in about 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 actual value (modern estimate) of 365.24219... days = 365 days 5 hours 48 min 45 s by only about 6 min).Between the solstice observation of Meton and his own, there were 297 years spanning 108,478 days. D.Rawlins noted that this implies a tropical year of 365.24579... days = 365 days;14,44,51 (sexagesimal; = 365 days + 14/60 + 44/60

^{2}+ 51/60^{3}) and that this exact yearlength has been found on one of the few Babylonian clay tablets which explicitly specifies the System B month. [*[*] This is an indication that Hipparchus' work was known to Chaldeans.*http://www.dioi.org/vols/w11.pdf "DIO"*] , volume 1, number 1, pages 49-66; A.Jones, 2001; Thurston, "op. cit.", page 62Another value for the year that is attributed to Hipparchus (by the astrologer

Vettius Valens in the 1st century) is 365 + 1/4 + 1/288 days (= 365.25347... days = 365 days 6 hours 5 min), but this may be 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 if this would be a value for thesidereal year (actual value at his time (modern estimate) ca. 365.2565 days), but the difference with Hipparchus' value for the tropical year is consistent with his rate ofprecession (see below).**Orbit of the Sun**Before Hipparchus, astronomers knew that the lengths of the

season s 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 94½ days, and summer (from summer solstice to autumn equinox) 92½ days. This is inconsistent with a premise of the Sun moving around the Earth in a circle at uniform speed. Hipparchus' 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 today that theplanet s (including the Earth) move inellipse s around the Sun, but this was not discovered untilJohannes Kepler published his first two laws of planetary motion in 1609). The value for 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 theapogee would be at longitude 65.5° from thevernal 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 95¾ and 91¼ days. [*Thurston, "op. cit.", page 67, note 16*] His other triplet of solar positions is consistent with 94¼ and 92½ days, [*"Ibid", note 14; Jones 2001*] an improvement on the results (94½ and 92½ days) attributed to Hipparchus by Ptolemy, which a few scholars still question the authorship of. 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).**Distance, parallax, size of the Moon and Sun**Hipparchus also undertook to find the distances and sizes of the Sun and the Moon. He published his results in a work of two books called "Peri megethoon kai 'apostèmátoon" ("On Sizes and Distances") by Pappus in his commentary on the "Almagest" V.11;

Theon of Smyrna (2nd century) mentions the work with the addition "of the Sun and Moon".Hipparchus measured 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

distance of the Moon, the Sun and Moon had the same apparent diameter; at that distance, the Moon's diameter fits 650 times into the circle, i.e., the mean apparent diameters are 360/650 = 0°33'14".mean 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 orstar s), and the difference is greater when closer to the horizon. He knew that this is because in the then-current models the Moon circles 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 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 (against the opinion of over a century of astronomers) presumes to be the eclipse of

14 March 190 BC. It was total in the region of theHellespont (and in fact in his birth place Nicaea); at the time Toomer proposes the Romans were preparing for war withAntiochus III in the area, and the eclipse is mentioned byLivy in his "Ab Urbe Condita " VIII.2. It 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 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 is 71 (from this eclipse), and the greatest 81 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, 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+½ 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 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 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 the first method is very sensitive to the accuracy of the observations and parameters. (In fact, modern calculations show that the size of the 190 B.C. solar eclipse at Alexandria must have been closer to 9/10ths and not the reported 4/5ths, a fraction more closely matched by the degree of totality at Alexandria of eclipses occurring in 310 B.C. and 129 B.C. which were also nearly total in 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' strategy to establish limits consistent with 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' 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 tovolume s, notdiameter s. From the geometry of book 2 it follows that the Sun is at 2,550 Earth radii, and the mean distance of the Moon is 60½ 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 a more detailed discussion.

**Eclipses**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 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 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 "Peri tes kata platos meniaias tes selenes kineseoos" ("On 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 (1980) argued that this must refer to the large total lunar eclipse of26 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' lunar theory. We do not know what "exact reason" Hipparchus found for seeing 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.**Astronomical instruments and astrometry**Hipparchus and his predecessors mostly used simple instruments for astronomical calculations and observations, such as the

gnomon , theastrolabe , and thearmillary sphere .Hipparchus is credited with 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 anarmillary sphere (which Ptolemy however says he constructed, in "Almagest" V.1); or the predecessor of the planar instrument calledastrolabe (also mentioned byTheon of Alexandria ). With an astrolabe Hipparchus was the first to be able to measure the geographicallatitude andtime by observing stars. Previously this was done at daytime by measuring the shadow cast by a "gnomon ", 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 the apparent diameter of the Sun and Moon.Pappus of Alexandria described it (in his commentary on the "Almagest" of that chapter), as didProclus ("Hypotyposis" IV). It was a 4-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

equinox es, which may be done with anequatorial ring : its shadow falls on itself when the Sun is on theequator (i.e., in one of the equinoctial points on theecliptic ), 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 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 thelatitude andlongitude of places on the Earth. Hipparchus wrote a critique in three books on the work of the geographerEratosthenes of Cyrene (3rd century BC), called "Pròs tèn 'Eratosthénous geografían" ("Against the Geography of Eratosthenes"). It is known to us fromStrabo of Amaseia, who in his turn criticised Hipparchus in his own "Geografia". 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 atlunar eclipse s (Strabo "Geografia" 1.1.12). A lunar eclipse 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.**Star catalog**Late in his career (possibly about 135 BC) Hipparchus compiled his star catalog, the original of which does not survive. He also constructed a celestial globe depicting the constellations, based on his observations. His interest in the

fixed star s may have been inspired by the observation of asupernova (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 byTimocharis andAristillus ; for more information seeDiscovery of precession ).Previously,

Eudoxus of Cnidus in the 4th 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' 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 the constellations, and these are likely to have been based on his own measurements.Hipparchus made his measurements with an

armillary sphere , and obtained the positions of at least 850 stars. It is disputed which coordinate system(s) he used. Ptolemy's catalog in the "Almagest ", which is derived from Hipparchus' catalog, is given in ecliptic coordinates. However Delambre in his "Histoire de l'Astronomie Ancienne" (1817) concluded that Hipparchus knew and used theequatorial coordinate system , a conclusion challenged byOtto Neugebauer in his "A History of Ancient Mathematical Astronomy" (1975). Hipparchus seems to have used a mix of ecliptic coordinates and equatorial coordinates: in his commentary on Eudoxos he provides stars' polar distance (equivalent to thedeclination in the equatorial system), right ascension (equatorial), longitude (ecliptical), polar longitude (hybrid), but not celestial latitude.As with most of his work, Hipparchus' star catalog was adopted and perhaps expanded by Ptolemy. Up until recently, it was heatedly disputed whether the star catalog in the "

Almagest " is due to Hipparchus, but 1976-2002 statistical and spatial analyses (by [*http://de.wikipedia.org/wiki/Robert_Russell_Newton R. R. Newton*] ,Dennis Rawlins , [*http://de.wikipedia.org/wiki/Almagest Gerd Grasshoff*] , [*http://www.dioi.org/vols/wc0.pdf Keith Pickering*] and [*http://people.scs.fsu.edu/~dduke Dennis Duke*] ) have shown conclusively that the "Almagest " star catalog is almost entirely Hipparchan. Ptolemy has even (since Brahe, 1598) been accused by astronomers of fraud for stating ("Syntaxis" book 7 chapter 4) that he observed all 1025 stars: for almost every star he used Hipparchus' data and precessed it to his own epoch 2⅔ centuries later by adding 2°40' to the longitude, using an erroneous (too small) precession constant of 1° per century.In any case the work started by Hipparchus has had a lasting heritage, and was much later updated by Al Sufi (964) and Copernicus (1543).

Ulugh Beg reobserved all the Hipparchus stars he could see from Samarkand in 1437 to about the same accuracy as Hipparchus's. The catalog was superseded only in the late sixteenth 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.**tellar magnitude**Hipparchus ranked stars in six magnitude classes according to their brightness: he assigned the value of one to the twenty brightest stars, to weaker ones a value of two, and so forth to the stars with a class of six, which can be barely seen with the naked eye. A similar system is still used today.

**Precession of the equinoxes (146 BC–130 BC)**:"See also

precession (astronomy) "Hipparchus is perhaps most famous for being almost universally recognized as discoverer of the

precession of theequinox es. His two books on precession, "On the Displacement of the Solsticial and Equinoctial Points" and "On the Length of the Year", are both mentioned in the "Almagest " of ClaudiusPtolemy . According to Ptolemy, Hipparchus measured the longitude ofSpica and other bright stars. Comparing his measurements with data from his predecessors,Timocharis andAristillus , he concluded that Spica had moved 2° relative to theautumnal equinox . He also compared the lengths of thetropical year (the time it takes the Sun to return to an equinox) and thesidereal year (the time it takes the Sun to return to a fixed star), and found a slight discrepancy. Hipparchus concluded that the equinoxes were moving ("precessing") through the zodiac, and that the rate of precession was not less than 1° in a century.Ptolemy followed up on Hipparchus' work in the 2nd century. He confirmed that precession affected the entire sphere of fixed stars (Hipparchus had speculated that only the stars near the zodiac were affected), and concluded that 1° in 100 years was the correct rate of precession. The modern value is 1° in 72 years.

**Named after Hipparchus**The ESA's Hipparcos Space Astrometry Mission was named after him, as are the Hipparchus lunar crater and the

asteroid 4000 Hipparchus .**Monument**The Astronomer's Monument at Griffith Observatory in Los Angeles, California, USA features a relief of Hipparchus as one of six of the greatest astronomers of all time and the only one from Antiquity.

**See also**

*Antikythera mechanism

*Apparent magnitude

*Astrometry

*History of astrology

*Geminus (of Rhodes) (10 BC - circa 60)

*Mira

*Mithraism

*Star catalog s**Notes****References***Edition and translation: Karl Manitius: "In Arati et Eudoxi Phaenomena", Leipzig, 1894.

*J. Chapront, M. Chapront Touze, G. Francou (2002): "A new determination of lunar orbital parameters, precession constant, and tidal acceleration from LLR measurements". "Astronomy and Astrophysics"

**387**, 700-709.

*Duke, Dennis W. (2002). Associations between the ancient star catalogs. "Archive for the History of Exact Sciences" 56(5):435-450.

*A. Jones: "Hipparchus." In "Encyclopedia of Astronomy and Astrophysics". Nature Publishing Group, 2001.

*Patrick Moore (1994): "Atlas of the Universe", Octopus Publishing Group LTD (Slovene translation and completion by Tomaž Zwitter and Savina Zwitter (1999): "Atlas vesolja"), 225.

*Newton, R.R. (1977). "The Crime of Claudius Ptolemy." Baltimore: Johns Hopkins University Press.

*Rawlins, Dennis (1982). An Investigation of the Ancient Star Catalog. "Proceedings of the Astronomical Society of the Pacific" 94, 359-373. Has been updated several times: [*http://www.dioi.org/vols/w82.pdf "DIO"*] , volume 8, number 1 (1998), page 2, note 3, and [*http://www.dioi.org/vols/wa0.pdf "DIO"*] , volume 10 (2000), page 79, note 177.

*J.M.Steele, F.R.Stephenson, L.V.Morrison (1997): "The accuracy of eclipse times measured by the Babylonians". "Journal for the History of Astronomy"**xxviii**, 337..345

*F.R. Stephenson, L.J.Fatoohi (1993): "Lunar Eclipse Times Recorded in Babylonian History". "Journal for the History of Astronomy"**xxiv**, 255..267

*N.M. Swerdlow (1969): "Hipparchus on the distance of the sun." "Centaurus"**14**, 287-305.

*G.J. Toomer (1967): "The Size of the Lunar Epicycle According to Hipparchus." "Centaurus"**12**, 145-150.

*G.J. Toomer (1973): "The Chord Table of Hipparchus and the Early History of Greek Trigonometry." "Centaurus"**18**, 6-28.

*G.J. Toomer (1974): "Hipparchus on the Distances of the Sun and Moon." "Archives for the History of the Exact Sciences"**14**, 126-142.

*G.J. Toomer (1978): "Hipparchus." In "Dictionary of Scientific Biography"**15**: 207-224.

*G.J. Toomer (1980): "Hipparchus' Empirical Basis for his Lunar Mean Motions," "Centaurus"**24**, 97-109.

*G.J. Toomer (1988): "Hipparchus and Babylonian Astronomy." In "A Scientific Humanist: Studies in Memory of Abraham Sachs", ed. Erle Leichty, Maria deJ. Ellis, and Pamel Gerardi. Philadelphia: Occasional Publications of the Samuel Noah Kramer Fund, 9.**External links****General***

* [*http://www.hps.cam.ac.uk/starry/hipparchus.html Biographical page at the*]University of Cambridge

* [*http://www.hps.cam.ac.uk/starry/hipppoem.html University of Cambridge's Page about Hipparchus' sole surviving work*]

* [*http://zebu.uoregon.edu/~js/glossary/hipparchus.html Biographical page at the*]University of Oregon

* [*http://fermatslasttheorem.blogspot.com/2006/04/hipparchus-of-rhodes.html Biography of Hipparchus on Fermat's Last Theorem Blog*]

* [*http://www.giovannipastore.it/ISTRUZIONI.htm Pastore, Giovanni, "ANTIKYTHERA E I REGOLI CALCOLATORI", Rome, 2006, privately published*]

* [*http://www.giovannipastore.it/ANTIKYTHERA.htm The Antikythera Calculator (Italian and English versions)*]**Precession*** [

*http://www.well.com/user/davidu/appendix4.html David Ulansey about Hipparchus's understanding of the precession*]**Celestial bodies*** M44 Praesepe at SEDS (

University of Arizona ): http://www.seds.org/messier/m/m044.html**Star catalog*** [

*http://ottawa.rasc.ca/astronomy/astro_facts/hipparcus.html A brief view by Carmen Rush on Hipparchus' stellar catalog*]

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