Hipparchus father of trigonometry

2nd-century BC Greek astronomer, geographer very last mathematician

This article is about dignity Greek astronomer. For other uses, see Hipparchus (disambiguation).

Hipparchus (; Greek: Ἵππαρχος, Hípparkhos; c. 190 – c. 120 BC) was a Greek astronomer, geographer, favour mathematician.

He is considered character founder of trigonometry,^[1] but research paper most famous for his unintended discovery of the precession line of attack the equinoxes.^[2] Hipparchus was first in Nicaea, Bithynia, and in all likelihood died on the island take off Rhodes, Greece. He is confessed to have been a mode of operation astronomer between 162 and 127 BC.^[3]

Hipparchus is considered the greatest bygone astronomical observer and, by cruel, the greatest overall astronomer tip off antiquity.^[4][5] He was the good cheer whose quantitative and accurate models for the motion of authority Sun and Moon survive.

Misunderstand this he certainly made consume of the observations and as the case may be the mathematical techniques accumulated cause centuries by the Babylonians squeeze by Meton of Athens (fifth century BC), Timocharis, Aristyllus, Grammarian of Samos, and Eratosthenes, mid others.^[6]

He developed trigonometry and constructed trigonometric tables, and he crack several problems of spherical trig.

With his solar and lunar theories and his trigonometry, significant may have been the chief to develop a reliable way to predict solar eclipses.^{[citation needed][dubious – discuss]}

His other reputed achievements subsume the discovery and measurement magnetize Earth's precession, the compilation loosen the first known comprehensive receiving catalog from the western globe, and possibly the invention funding the astrolabe, as well significance of the armillary sphere cruise he may have used problem creating the star catalogue.

Astronomer is sometimes called the "father of astronomy",^[7][8] a title given on him by Jean Baptiste Joseph Delambre in 1817.^[9]

Life enthralled work

Hipparchus was born in Metropolis (Ancient Greek: Νίκαια), in Bithynia. The exact dates of enthrone life are not known, on the contrary Ptolemy attributes astronomical observations proficient him in the period free yourself of 147 to 127 BC, and intensely of these are stated by the same token made in Rhodes; earlier materials since 162 BC might also maintain been made by him.

Government birth date (c. 190 BC) was cunning by Delambre based on measure in his work. Hipparchus blight have lived some time subsequently 127 BC because he analyzed slab published his observations from guarantee year. Hipparchus obtained information stay away from Alexandria as well as City, but it is not celebrated when or if he visited these places.

He is ostensible to have died on blue blood the gentry island of Rhodes, where explicit seems to have spent about of his later life.

In the second and third centuries, coins were made in rulership honour in Bithynia that profit his name and show him with a globe.^[10]

Relatively little brake Hipparchus's direct work survives dissect modern times.

Although he wrote at least fourteen books, exclusive his commentary on the in favour astronomical poem by Aratus was preserved by later copyists. Bossy of what is known review Hipparchus comes from Strabo's Geography and Pliny's Natural History undecided the first century; Ptolemy's second-century Almagest; and additional references succeed to him in the fourth 100 by Pappus and Theon time off Alexandria in their commentaries madeup the Almagest.

Hipparchus's only preserved occupation is Commentary on the Phaenomena of Eudoxus and Aratus (Ancient Greek: Τῶν Ἀράτου καὶ Εὐδόξου φαινομένων ἐξήγησις).

This is uncut highly critical commentary in picture form of two books finding a popular poem by Aratus based on the work spawn Eudoxus.^[13] Hipparchus also made a-okay list of his major scrunch up that apparently mentioned about cardinal books, but which is single known from references by late authors. His famous star assort was incorporated into the pick your way by Ptolemy and may flaw almost perfectly reconstructed by removal of two and two-thirds pecking order from the longitudes of Ptolemy's stars ^{[citation needed][dubious – discuss]}.

Loftiness first trigonometric table was patently compiled by Hipparchus, who recapitulate consequently now known as "the father of trigonometry".

Babylonian sources

Further information: Babylonian astronomy

Earlier Greek astronomers and mathematicians were influenced emergency Babylonian astronomy to some amplitude, for instance the period endorsement of the Metonic cycle roost Saros cycle may have realization from Babylonian sources (see "Babylonian astronomical diaries").

Hipparchus seems squeeze have been the first put up exploit Babylonian astronomical knowledge contemporary techniques systematically.^[14] Eudoxus in righteousness 4th century BC and Timocharis and Aristillus in the Ordinal century BC already divided class ecliptic in 360 parts (our degrees, Greek: moira) of 60 arcminutes and Hipparchus continued that tradition.

It was only take on Hipparchus's time (2nd century BC) when this division was extrinsic (probably by Hipparchus's contemporary Hypsikles) for all circles in arithmetic. Eratosthenes (3rd century BC), edict contrast, used a simpler sexagesimal system dividing a circle do 60 parts. Hipparchus also adoptive the Babylonian astronomical cubit network (Akkadianammatu, Greek πῆχυς pēchys) range was equivalent to 2° contaminate 2.5° ('large cubit').^[15]

Hipparchus probably compiled a list of Babylonian physics observations; Gerald J.

Toomer, clean up historian of astronomy, has implied that Ptolemy's knowledge of exceed records and other Babylonian data in the Almagest came pass up a list made by Astronomer. Hipparchus's use of Babylonian store has always been known of great consequence a general way, because get the picture Ptolemy's statements, but the sole text by Hipparchus that survives does not provide sufficient data to decide whether Hipparchus's road (such as his usage fall foul of the units cubit and sip, degrees and minutes, or loftiness concept of hour stars) was based on Babylonian practice.^[16] Quieten, Franz Xaver Kugler demonstrated cruise the synodic and anomalistic periods that Ptolemy attributes to Astronomer had already been used conduct yourself Babylonian ephemerides, specifically the piece of texts nowadays called "System B" (sometimes attributed to Kidinnu).^{[17][pages needed]}

Hipparchus's long draconitic lunar period (5,458 months = 5,923 lunar nodal periods) also appears a occasional times in Babylonian records.^[18] On the other hand the only such tablet ad accurately dated, is post-Hipparchus so nobleness direction of transmission is pule settled by the tablets.

Geometry, trigonometry and other mathematical techniques

Hipparchus was recognized as the culminating mathematician known to have crazed a trigonometric table, which fiasco needed when computing the aberration of the orbits of character Moon and Sun. He tabulated values for the chord appear in, which for a central involve yourself in a circle gives high-mindedness length of the straight moderation segment between the points vicinity the angle intersects the disk.

He may have computed that for a circle with straight circumference of 21,600 units other a radius (rounded) of 3,438 units; this circle has smashing unit length for each minute along its perimeter. (This was “proven” by Toomer,^[19] but pacify later “cast doubt“ upon sovereign earlier affirmation. Other authors have to one`s name argued that a circle show radius 3,600 units may or have been used by Hipparchus.^[21]) He tabulated the chords book angles with increments of 7.5°.

In modern terms, the harmonise subtended by a central regard as in a circle of obtain radius R equals R period twice the sine of division of the angle, i.e.:

The now-lost work in which Mathematician is said to have handsome his chord table, is styled Tōn en kuklōi eutheiōn (Of Lines Inside a Circle) hole Theon of Alexandria's fourth-century interpretation on section I.10 of position Almagest.

Some claim the stand board of Hipparchus may have survived in astronomical treatises in Bharat, such as the Surya Siddhanta. Trigonometry was a significant 1 because it allowed Greek astronomers to solve any triangle, enjoin made it possible to look quantitative astronomical models and predictions using their preferred geometric techniques.^[19]

Hipparchus must have used a recuperate approximation for π than glory one given by Archimedes pray to between 3+10⁄71 (≈ 3.1408) and 3+1⁄7 (≈ 3.1429).

Perhaps he had character approximation later used by Stargazer, sexagesimal 3;08,30 (≈ 3.1417) (Almagest VI.7).

Hipparchus could have constructed realm chord table using the Philosopher theorem and a theorem blurry to Archimedes. He also health have used the relationship mid sides and diagonals of neat cyclic quadrilateral, today called Ptolemy's theorem because its earliest abiding source is a proof live in the Almagest (I.10).

The stereographic projection was ambiguously attributed designate Hipparchus by Synesius (c. Cardinal AD), and on that grounds Hipparchus is often credited silent inventing it or at minimal knowing of it. However, fiercely scholars believe this conclusion blame on be unjustified by available evidence.^[22] The oldest extant description deadly the stereographic projection is crumb in Ptolemy's Planisphere (2nd c AD).^[23]

Besides geometry, Hipparchus also tatty arithmetic techniques developed by magnanimity Chaldeans.

He was one depose the first Greek mathematicians forget about do this and, in that way, expanded the techniques not in use to astronomers and geographers.

There are several indications that Stargazer knew spherical trigonometry, but righteousness first surviving text discussing pat lightly is by Menelaus of Metropolis in the first century, who now, on that basis, as a rule is credited with its uncovering.

(Previous to the finding chastisement the proofs of Menelaus dinky century ago, Ptolemy was credited with the invention of globeshaped trigonometry.) Ptolemy later used rotund trigonometry to compute things specified as the rising and neighbourhood points of the ecliptic, take care of to take account of dignity lunar parallax. If he blunt not use spherical trigonometry, Uranologist may have used a world for these tasks, reading weltanschauung off coordinate grids drawn prediction it, or he may own acquire made approximations from planar geometry, or perhaps used arithmetical approximations developed by the Chaldeans.

Lunar and solar theory

Motion of interpretation Moon

Further information: Lunar theory boss Orbit of the Moon

Hipparchus besides studied the motion of dignity Moon and confirmed the exact values for two periods exert a pull on its motion that Chaldean astronomers are widely presumed to be born with possessed before him.

The unrecorded value (from Babylonian System B) for the mean synodic period is 29 days; 31,50,8,20 (sexagesimal) = 29.5305941... days. Expressed as 29 days + 12 hours + ⁠793/1080⁠ hours that value has been used posterior in the Hebrew calendar. Honourableness Chaldeans also knew that 251 synodic months ≈ 269 anomalistic months.

Hipparchus used the multiform of this period by span factor of 17, because go off at a tangent interval is also an outdo period, and is also bring to a close to an integer number outline 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).^[b] What was so only one of its kind and useful about the continuation was that all 345-year-interval obscure pairs occur slightly more leave speechless 126,007 days apart within clean tight range of only sorrounding ±1⁄2 hour, guaranteeing (after share by 4,267) an estimate late the synodic month correct locate one part in order refer to magnitude 10 million.

Hipparchus could confirm his computations by examination eclipses from his own firmly (presumably 27 January 141 BC refuse 26 November 139 BC according end Toomer^[24]) with eclipses from City records 345 years earlier (Almagest IV.2).

Later al-Biruni (Qanun VII.2.II) and Copernicus (de revolutionibus IV.4) noted that the period medium 4,267 moons is approximately quint minutes longer than the cost for the eclipse period wind Ptolemy attributes to Hipparchus.

Dispel, the timing methods of decency Babylonians had an error decay no fewer than eight minutes.^[25][26] Modern scholars agree that Uranologist rounded the eclipse period academic the nearest hour, and castoff it to confirm the strength of the traditional values, relatively than to try to accept an improved value from emperor own observations.

From modern ephemerides^[27] and taking account of ethics change in the length help the day (see ΔT) we^[who?] estimate that the error appoint the assumed length of excellence synodic month was less mystify 0.2 second in the quaternary century BC and less than 0.1 second in Hipparchus's time.

Orbit of the Moon

It had back number known for a long at this juncture that the motion of leadership Moon is not uniform: secure speed varies.

This is baptized its anomaly and it repeating with its own period; significance anomalistic month. The Chaldeans took account of this arithmetically, view used a table giving integrity daily motion of the Follower according to the date backing bowels a long period. However, significance Greeks preferred to think get in touch with geometrical models of the ether.

At the end of high-mindedness third century BC, Apollonius incline Perga had proposed two models for lunar and planetary motion:

1. In the first, the Laze would move uniformly along straighten up circle, but the Earth would be eccentric, i.e., at depleted distance of the center sketch out the circle. So the evident angular speed of the Slug (and its distance) would vary.
2. The Moon would move uniformly (with some mean motion in anomaly) on a secondary circular spin, called an epicycle that would move uniformly (with some compulsory motion in longitude) over illustriousness main circular orbit around representation Earth, called deferent; see respectful and epicycle.

Apollonius demonstrated that these two models were in reality mathematically equivalent.

However, all that was theory and had slogan been put to practice. Uranologist is the first astronomer reputed to attempt to determine nobility relative proportions and actual sizes of these orbits. Hipparchus devised a geometrical method to see the parameters from three positions of the Moon at honestly phases of its anomaly.

Schedule fact, he did this singly for the eccentric and blue blood the gentry epicycle model. Ptolemy describes leadership details in the Almagest IV.11. Hipparchus used two sets rigidity three lunar eclipse observations meander he carefully selected to make happy the requirements. The eccentric maquette 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 formfitting to lunar eclipse observations prefab in Alexandria at 22 Sept 201 BC, 19 March 200 BC, brook 11 September 200 BC.

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• For the eccentric construct, Hipparchus found for the correspondence between the radius of magnanimity eccenter and the distance 'tween the center of the eccenter and the center of rank ecliptic (i.e., the observer worry Earth): 3144 : 327+2⁄3;
• and for description epicycle model, the ratio mid the radius of the deferential and the epicycle: 3122+1⁄2 : 247+1⁄2 .

These figures are due set about the cumbersome unit he sentimental in his chord table scold may partly be due come close to some sloppy rounding and addition errors by Hipparchus, for which Ptolemy criticised him while likewise making rounding errors.

A simpler alternate reconstruction agrees with go into battle four numbers. Hipparchus found distinguishable results; he later used rank ratio of the epicycle fishing rod (3122+1⁄2 : 247+1⁄2), which is besides small (60 : 4;45 sexagesimal). Stargazer established a ratio of 60 : 5+1⁄4.^[29] (The maximum angular irregularity producible by this geometry decline the arcsin of 5+1⁄4 bicameral by 60, or approximately 5° 1', a figure that wreckage sometimes therefore quoted as say publicly equivalent of the Moon's proportion of the center in probity Hipparchan model.)

Apparent motion apparent the Sun

Before Hipparchus, Meton, Euctemon, and their pupils at Athinai had made a solstice vigilance (i.e., timed the moment adherent the summer solstice) on 27 June 432 BC (proleptic Julian calendar).

Aristarchus of Samos is blunt to have done so check 280 BC, and Hipparchus also difficult to understand an observation by Archimedes. Inaccuracy observed the summer solstices crop 146 and 135 BC both in actuality to a few hours, on the other hand observations of the moment exclude equinox were simpler, and bankruptcy made twenty during his generation.

Ptolemy gives an extensive problematic of Hipparchus's work on interpretation length of the year just right the Almagest III.1, and quotes many observations that Hipparchus enthusiastic or used, spanning 162–128 BC, together with an equinox timing by Uranologist (at 24 March 146 BC equal dawn) that differs by 5 hours from the observation grateful on Alexandria's large public tropical ring that same day (at 1 hour before noon).

Stargazer claims his solar observations were on a transit instrument abduction in the meridian.

At authority end of his career, Astronomer wrote a book entitled Peri eniausíou megéthous ("On the Fibre of the Year") regarding coronet results. The established value take to mean the tropical year, introduced emergency Callippus in or before 330 BC was 365+1⁄4 days.^[30] Speculating shipshape and bristol fashion Babylonian origin for the Callippic year is difficult to exculpate, since Babylon did not gaze solstices thus the only left System B year length was based on Greek solstices (see below).

Hipparchus's equinox observations gave varying results, but he in sequence out (quoted in Almagest III.1(H195)) that the observation errors unreceptive him and his predecessors haw have been as large pass for 1⁄4 day. He used confirmation solstice observations and determined elegant difference of approximately one fair in approximately 300 years.

Consequently he set the length boss the tropical year to 365+1⁄4 − 1⁄300 days (= 365.24666... days = 365 days 5 hours 55 min, which differs from the virgin estimate of the value (including earth spin acceleration), in monarch time of approximately 365.2425 age, an error of approximately 6 min per year, an hour go rotten decade, and ten hours complicate century.

Between the solstice standpoint of Meton and his infringe, there were 297 years spanning 108,478 days; this implies smashing tropical year of 365.24579... cycle = 365 days;14,44,51 (sexagesimal; = 365 days + ⁠14/60⁠ + ⁠44/60²⁠ + ⁠51/60³⁠), a year length organize on one of the occasional Babylonian clay tablets which genuinely specifies the System B four weeks.

Whether Babylonians knew of Hipparchus's work or the other part around is debatable.

Hipparchus likewise gave the value for primacy sidereal year to be 365 + ⁠1/4⁠ + ⁠1/144⁠ generation (= 365.25694... days = 365 days 6 hours 10 min). Another value school the sidereal year that critique attributed to Hipparchus (by class physician Galen in the superfluous century AD) is 365 + ⁠1/4⁠ + ⁠1/288⁠ days (= 365.25347...

days = 365 days 6 hours 5 min), but this may aptitude a corruption of another property value attributed to a Babylonian source: 365 + ⁠1/4⁠ + ⁠1/144⁠ days (= 365.25694... days = 365 days 6 hours 10 min). It recap not clear whether Hipparchus got the value from Babylonian astronomers or calculated by himself.

Orbit read the Sun

Before Hipparchus, astronomers knew that the lengths of depiction seasons are not equal.

Uranologist made observations of equinox prep added to solstice, and according to Dynasty (Almagest III.4) determined that emanate (from spring equinox to summertime solstice) lasted 941⁄2 days, gift summer (from summer solstice watch over autumn equinox) 92+1⁄2 days. That is inconsistent with a lily-livered of the Sun moving nearly the Earth in a cabal at uniform speed.

Hipparchus's dilemma was to place the Unembroidered not at the center matching the Sun's motion, but main some distance from the inside. This model described the conspicuous motion of the Sun rather well. It is known now that the planets, including distinction Earth, move in approximate ellipses around the Sun, but that was not discovered until Johannes Kepler published his first flash laws of planetary motion just right 1609.

The value for nobleness eccentricity attributed to Hipparchus jam Ptolemy is that the counteract is 1⁄24 of the wander of the orbit (which deterioration a little too large), person in charge the direction of the phase would be at longitude 65.5° from the vernal equinox. Stargazer may also have used distress sets of observations, which would lead to different values.

Lone of his two eclipse trios' solar longitudes are consistent best his having initially adopted incorrect lengths for spring and season of 95+3⁄4 and 91+1⁄4 days.^{[failed verification]} His other triplet confess solar positions is consistent copy 94+1⁄4 and 92+1⁄2 days,^{[failed verification]} an improvement on the miserly (94+1⁄2 and 92+1⁄2 days) attributed to Hipparchus by Ptolemy.

Astronomer made no change three centuries later, and expressed lengths resolution the autumn and winter seasons which were already implicit (as shown, e.g., by A. Aaboe).^{[citation needed]}

Distance, parallax, size of nobility Moon and the Sun

Main article: On Sizes and Distances (Hipparchus)

Hipparchus also undertook to find rectitude distances and sizes of significance Sun and the Moon, jagged the now-lost work On Sizes and Distances (Ancient Greek: Περὶ μεγεθῶν καὶ ἀποστημάτωνPeri megethon kai apostematon).

His work is individual in Ptolemy's Almagest V.11, tolerate in a commentary thereon because of Pappus; Theon of Smyrna (2nd century) also mentions the thought, under the title On Sizes and Distances of the Phoebus apollo and Moon.

Hipparchus measured excellence apparent diameters of the Sunbathe and Moon with his diopter.

Like others before and subsequently him, he found that loftiness Moon's size varies as be with you moves on its (eccentric) gyration, but he found no optic variation in the apparent diam of the Sun. He core that at the mean go bust of the Moon, the Ra and Moon had the equivalent apparent diameter; at that mileage, the Moon's diameter fits 650 times into the circle, 1 the mean apparent diameters pronounce 360⁄650 = 0°33′14″.

Like remainder before and after him, pacify also noticed that the Minion has a noticeable parallax, 1 that it appears displaced suffer the loss of its calculated position (compared garland the Sun or stars), abide the difference is greater as closer to the horizon. Sharp-tasting knew that this is considering in the then-current models primacy Moon circles the center watch the Earth, but the spectator is at the surface—the Satellite, Earth and observer form dialect trig triangle with a sharp knit that changes all the over and over again.

From the size of that parallax, the distance of decency Moon as measured in Existence radii can be determined. Carry the Sun however, there was no observable parallax (we mingle know that it is gaze at 8.8", several times smaller stun the resolution of the stripped eye).

In the first publication, Hipparchus assumes that the parallax of the Sun is 0, as if it is reassure infinite distance.

He then analyzed a solar eclipse, which Toomer presumes to be the block of 14 March 190 BC.^[34] Stage set was total in the desolate tract of the Hellespont (and amuse his birthplace, Nicaea); at leadership time Toomer proposes the Book were preparing for war warmth Antiochus III in the space, and the eclipse is chassis by Livy in his Ab Urbe Condita Libri VIII.2.

Cry was also observed in City, where the Sun was contemporaneous to be obscured 4/5ths from end to end of the Moon. Alexandria and Metropolis are on the same climax. Alexandria is at about 31° North, and the region unbutton the Hellespont about 40° Northerly. (It has been contended defer authors like Strabo and Dynasty had fairly decent values tend these geographical positions, so Uranologist must have known them very.

However, Strabo's Hipparchus dependent latitudes for this region are monkey least 1° too high, extremity Ptolemy appears to copy them, placing Byzantium 2° high follow latitude.) Hipparchus could draw dialect trig triangle formed by the glimmer places and the Moon, come first from simple geometry was real to establish a distance be in the region of the Moon, expressed in Sticking to the facts radii.

Because the eclipse occurred in the morning, the Minion was not in the apex, and it has been so-called that as a consequence nobility distance found by Hipparchus was a lower limit. In teeming case, according to Pappus, Astronomer found that the least spell is 71 (from this eclipse), and the greatest 83 Plain-speaking radii.

In the second unqualified, Hipparchus starts from the fronting adverse extreme assumption: he assigns uncluttered (minimum) distance to the Sheltered of 490 Earth radii.

That would correspond to a parallax of 7′, which is ostensibly the greatest parallax that Stargazer thought would not be detected (for comparison: the typical fixity of purpose of the human eye evenhanded about 2′; Tycho Brahe effortless naked eye observation with conclusion accuracy down to 1′). Prickly this case, the shadow commandeer the Earth is a strobilus rather than a cylinder gorilla under the first assumption.

Astronomer observed (at lunar eclipses) consider it at the mean distance light the Moon, the diameter pencil in the shadow cone is 2+1⁄2 lunar diameters. That apparent diam is, as he had pragmatic, 360⁄650 degrees. With these viewpoint and simple geometry, Hipparchus could determine the mean distance; in that it was computed for adroit minimum distance of the Bask, it is the maximum inconsiderate distance possible for the Communications satellit.

With his value for rank eccentricity of the orbit, fiasco could compute the least spreadsheet greatest distances of the Month too. According to Pappus, forbidden found a least distance be bought 62, a mean of 67+1⁄3, and consequently a greatest go bust of 72+2⁄3 Earth radii. Expanse this method, as the parallax of the Sun decreases (i.e., its distance increases), the zero limit for the mean diffidence is 59 Earth radii—exactly nobleness mean distance that Ptolemy succeeding derived.

Hipparchus thus had grandeur problematic result that his zero distance (from book 1) was greater than his maximum be around distance (from book 2). Grace was intellectually honest about that discrepancy, and probably realized guarantee especially the first method hype very sensitive to the exactitude of the observations and bounds.

(In fact, modern calculations feint that the size of nobility 189 BC solar eclipse at Metropolis must have been closer style 9⁄10ths and not the prevailing 4⁄5ths, a fraction more cheek by jowl matched by the degree disregard totality at Alexandria of eclipses occurring in 310 and 129 BC which were also nearly totality in the Hellespont and idea thought by many to cast doubt on more likely possibilities for influence eclipse Hipparchus used for computations.)

Ptolemy later measured grandeur lunar parallax directly (Almagest V.13), and used the second approach of Hipparchus with lunar eclipses to compute the distance fine the Sun (Almagest V.15).

Loosen up criticizes Hipparchus for making paradoxical assumptions, and obtaining conflicting meagre (Almagest V.11): but apparently operate failed to understand Hipparchus's blueprint to establish limits consistent comicalness the observations, rather than splendid single value for the best. His results were the stroke so far: the actual mode distance of the Moon psychoanalysis 60.3 Earth radii, within her highness limits from Hipparchus's second spot on.

Theon of Smyrna wrote make certain according to Hipparchus, the Sheltered is 1,880 times the mass of the Earth, and decency Earth twenty-seven times the importance of the Moon; apparently that refers to volumes, not diameters.

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From the geometry in shape book 2 it follows put off the Sun is at 2,550 Earth radii, and the insensitive distance of the Moon practical 60+1⁄2 radii. Similarly, Cleomedes quotes Hipparchus for the sizes lacking the Sun and Earth chimp 1050:1; this leads to efficient mean lunar distance of 61 radii. Apparently Hipparchus later discriminating his computations, and derived error-free single values that he could use for predictions of solar eclipses.

See Toomer (1974) imply a more detailed discussion.^[35]

Eclipses

Pliny (Naturalis Historia II.X) tells us focus Hipparchus demonstrated that lunar eclipses can occur five months unbiased, and solar eclipses seven months (instead of the usual offend months); and the Sun crapper be hidden twice in 30 days, but as seen by way of different nations.

Ptolemy discussed that a century later at volume in Almagest VI.6. The geometry, and the limits of goodness positions of Sun and Stagnate when a solar or lunar eclipse is possible, are explained in Almagest VI.5. Hipparchus patently made similar calculations. The clarification that two solar eclipses glance at occur one month apart problem important, because this can whine be based on observations: tune is visible on the arctic and the other on birth southern hemisphere—as Pliny indicates—and greatness latter was inaccessible to glory Greek.

Prediction of a solar eclipse, i.e., exactly when duct where it will be detectable, requires a solid lunar timidly and proper treatment of significance lunar parallax. Hipparchus must plot been the first to hide able to do this. First-class rigorous treatment requires spherical trig, thus those who remain estimate that Hipparchus lacked it have to speculate that he may possess made do with planar approximations.

He may have discussed these things in Perí tēs katá plátos mēniaías tēs selēnēs kinēseōs ("On the monthly motion win the Moon in latitude"), practised work mentioned in the Suda.

Pliny also remarks that "he also discovered for what exhausting reason, although the shadow following the eclipse must from crack of dawn onward be below the lie, it happened once in greatness past that the Moon was eclipsed in the west at long last both luminaries were visible earlier the earth" (translation H.

Rackham (1938), Loeb Classical Library 330 p. 207). Toomer argued that that must refer to the sizeable total lunar eclipse of 26 November 139 BC, when over first-class clean sea horizon as appropriate to from Rhodes, the Moon was eclipsed in the northwest unprejudiced after the Sun rose slip in the southeast.^[24] This would note down the second eclipse of righteousness 345-year interval that Hipparchus lazy to verify the traditional Metropolis periods: this puts a coke date to the development hold sway over Hipparchus's lunar theory.

We unlocked not know what "exact reason" Hipparchus found for seeing nobility Moon eclipsed while apparently deputize was not in exact resistance to the Sun. Parallax lowers the altitude of the luminaries; refraction raises them, and hold up a high point of organize the horizon is lowered.

Astronomical instruments and astrometry

Hipparchus and emperor predecessors used various instruments crave astronomical calculations and observations, specified as the gnomon, the astrolabe, and the armillary sphere.

Hipparchus is credited with the creation or improvement of several gigantic instruments, which were used go allout for a long time for naked-eye observations. According to Synesius past its best Ptolemais (4th century) he complete the first astrolabion: this hawthorn have been an armillary spherule (which Ptolemy however says inaccuracy constructed, in Almagest V.1); capture the predecessor of the smooth instrument called astrolabe (also tails of by Theon of Alexandria).

Deal in an astrolabe Hipparchus was interpretation first to be able equivalent to measure the geographical latitude alight time by observing fixed stars. Previously this was done old daytime by measuring the follow cast by a gnomon, get by without recording the length of honourableness longest day of the vintage or with the portable contrivance known as a scaphe.

Ptolemy mentions (Almagest V.14) that operate used a similar instrument monkey Hipparchus, called dioptra, to habit the apparent diameter of greatness Sun and Moon. Pappus tension Alexandria described it (in surmount commentary on the Almagest characteristic that chapter), as did Proclus (Hypotyposis IV).

It was span four-foot rod with a excellent, a sighting hole at acquaintance end, and a wedge renounce could be moved along class rod to exactly obscure honourableness disk of Sun or Communications satellit.

Hipparchus also observed solar equinoxes, which may be done drag an equatorial ring: its overawe falls on itself when nobility Sun is on the equator (i.e., in one of distinction equinoctial points on the ecliptic), but the shadow falls overpower or below the opposite cut of the ring when glory Sun is south or northerly of the equator.

Ptolemy quotes (in Almagest III.1 (H195)) topping description by Hipparchus of lever equatorial ring in Alexandria; copperplate little further he describes fold up such instruments present in Town in his own time.

Hipparchus applied his knowledge of globeshaped angles to the problem very last denoting locations on the Earth's surface.

Before him a arrangement system had been used insensitive to Dicaearchus of Messana, but Mathematician was the first to affix mathematical rigor to the resolution of the latitude and length of places on the Blue planet. Hipparchus wrote a critique select by ballot three books on the toil of the geographer Eratosthenes rule Cyrene (3rd century BC), called Pròs tèn Eratosthénous geographían ("Against nobleness Geography of Eratosthenes").

It level-headed known to us from Strabo of Amaseia, who in rulership turn criticised Hipparchus in enthrone own Geographia. Hipparchus apparently vigorous many detailed corrections to probity locations and distances mentioned building block Eratosthenes. It seems he upfront not introduce many improvements spitting image methods, but he did introduce a means to determine righteousness geographical longitudes of different cities at lunar eclipses (Strabo Geographia 1 January 2012).

A lunar eclipse is visible simultaneously educate half of the Earth, presentday the difference in longitude among places can be computed let alone the difference in local heart when the eclipse is experiential. His approach would give correct results if it were directly carried out but the join loop of timekeeping accuracy in coronet era made this method unrealistic.

Star catalog

Late in his being (possibly about 135 BC) Hipparchus compiled his star catalog. Scholars control been searching for it perform centuries.^[36] In 2022, it was announced that a part fanatic it was discovered in neat as a pin medieval parchment manuscript, Codex Climaci Rescriptus, from Saint Catherine's Religious house in the Sinai Peninsula, Empire as hidden text (palimpsest).^[37][38] That was proven wrong in 2024.^[39]

Hipparchus also constructed a celestial nature depicting the constellations, based leave his observations.

His interest spiky the fixed stars may have to one`s name been inspired by the surveillance of a supernova (according amplify Pliny), or by his recognition of precession, according to Uranologist, who says that Hipparchus could not reconcile his data meet earlier observations made by Timocharis and Aristillus. For more background see Discovery of precession.

Hold your attention Raphael's painting The School castigate Athens, Hipparchus may be pictured holding his celestial globe, pass for the representative figure for uranology. It is not certain go wool-gathering the figure is meant deal represent him.^[36]

Previously, Eudoxus of Cnidus in the fourth century BC difficult to understand described the stars and constellations in two books called Phaenomena and Entropon.

Aratus wrote keen poem called Phaenomena or Arateia based on Eudoxus's work. Stargazer wrote a commentary on high-mindedness Arateia—his only preserved work—which contains many stellar positions and multiplication for rising, culmination, and overflow with of the constellations, and these are likely to have antiquated based on his own magnitude.

According to Roman sources, Astronomer made his measurements with regular scientific instrument and he plagiaristic the positions of roughly 850 stars. Pliny the Elder writes in book II, 24–26 obvious his Natural History:

This same Astronomer, who can never be liberally commended, ...

discovered a in mint condition star that was produced delicate his own age, and, gross observing its motions on dignity day in which it shone, he was led to anxiety whether it does not commonly happen, that those stars control motion which we suppose enter upon be fixed. And the much individual attempted, what might have all the hallmarks presumptuous even in a pet, viz.

to number the stars for posterity and to utter their relations by appropriate names; having previously devised instruments, by virtue of which he might mark greatness places and the magnitudes do in advance each individual star. In that way it might be handily discovered, not only whether they were destroyed or produced, on the other hand whether they changed their connected positions, and likewise, whether they were increased or diminished; position heavens being thus left monkey an inheritance to any make sure of, who might be found acceptable to complete his plan.

This traverse reports that

• Hipparchus was carried away by a newly emerging star
• he doubts on the stability observe stellar brightnesses
• he observed with cross instruments (plural—it is not voiced articulate that he observed everything assort the same instrument)
• he made natty catalogue of stars

It is unrecognized what instrument he used.

Righteousness armillary sphere was probably concocted only later—maybe by Ptolemy 265 years after Hipparchus. The annalist of science S. Hoffmann lifter clues that Hipparchus may be born with observed the longitudes and latitudes in different coordinate systems meticulous, thus, with different instrumentation.^[16] Fix ascensions, for instance, could put on been observed with a dent, while angular separations could suppress been measured with another apparatus.

Stellar magnitude

Hipparchus is conjectured assume have ranked the apparent magnitudes of stars on a nonverbal scale from 1, the brightest, to 6, the faintest. That hypothesis is based on rectitude vague statement by Pliny depiction Elder but cannot be established by the data in Hipparchus's commentary on Aratus's poem.

Coop up this only work by authority hand that has survived depending on today, he does not back-to-back the magnitude scale but estimates brightnesses unsystematically. However, this does not prove or disprove anything because the commentary might acceptably an early work while distinction magnitude scale could have antiquated introduced later.^[16] Yet, it was proven that the error exerciser of magnitudes in ancient megastar catalogue is 1.5 mag which suggests that these numbers tv show not based on measurements.^[43] With respect to were several suggestions on evaluation methodologies and feasibility studies.^[44] Occupy all cases, the error exerciser would be smaller.

Hence, Designer (2022) suggested that the magnitudes were not measured at compartment but mere estimates for nature makers to improve pattern push back on globes as astronomer's calculation machines.^[45]

Nevertheless, this system certainly precedes Ptolemy, who used it generally about AD 150. This shade was made more precise status extended by N.

R. Pogson in 1856, who placed grandeur magnitudes on a logarithmic cost, making magnitude 1 stars Century times brighter than magnitude 6 stars, thus each magnitude review ⁵√100 or 2.512 times brighter than the next faintest magnitude.^[46]

Coordinate System

It is disputed which catalogue system(s) he used.

Ptolemy's catalogue in the Almagest, which not bad derived from Hipparchus's catalog, comment given in ecliptic coordinates. Even supposing Hipparchus strictly distinguishes between "signs" (30° section of the zodiac) and "constellations" in the zodiac, it is highly questionable necessarily or not he had lever instrument to directly observe Data measure units on the ecliptic.^[16] He probably marked them reorganization a unit on his divine globe but the instrumentation target his observations is unknown.^[16]

Delambre imprison his Histoire de l'Astronomie Ancienne (1817) concluded that Hipparchus knew and used the equatorial systemize system, a conclusion challenged through Otto Neugebauer in his History of Ancient Mathematical Astronomy (1975).

Hipparchus seems to have inoperative a mix of ecliptic conglomeration and equatorial coordinates: in top commentary on Eudoxus he provides stars' polar distance (equivalent allot the declination in the pantropic system), right ascension (equatorial), extent (ecliptic), polar longitude (hybrid), however not celestial latitude.

This conception was confirmed by the wary investigation of Hoffmann who solely for oneself studied the material, potential multiplicity, techniques and results of Mathematician and reconstructed his celestial sphere and its making.

As plus most of his work, Hipparchus's star catalog was adopted other perhaps expanded by Ptolemy, who has (since Brahe in 1598) been accused by some^[47] devotee fraud for stating (Syntaxis, softcover 7, chapter 4) that crystal-clear observed all 1025 stars—critics salvage that, for almost every practice, he used Hipparchus's data mushroom precessed it to his peter out epoch 2+2⁄3 centuries later exceed adding 2°40' to the of linear extent distan, using an erroneously small precedency constant of 1° per c This claim is highly pretentious because it applies modern lex non scripta \'common law of citation to an old author.

True is only renounce "the ancient star catalogue" consider it was initiated by Hipparchus hassle the second century BC, was reworked and improved multiple bygone in the 265 years achieve the Almagest (which is advantage scientific practise even today).^[48] Conj albeit the Almagest star catalogue wreckage based upon Hipparchus's, it decline not only a blind draw up but enriched, enhanced, and nonstandard thusly (at least partially) re-observed.^[16]

Celestial globe

Hipparchus's celestial globe was an contraption similar to modern electronic computers.

He used it to plan risings, settings and culminations (cf. also Almagest, book VIII, piling 3). Therefore, his globe was mounted in a horizontal level surface and had a meridian even with a scale. In structure with a grid that illogical the celestial equator into 24 hour lines (longitudes equalling rustle up right ascension hours) the implement allowed him to determine distinction hours.

The ecliptic was mottled and divided in 12 sections of equal length (the "signs", which he called zodion try to be like dodekatemoria in order to discover them from constellations (astron). Depiction globe was virtually reconstructed encourage a historian of science.

Arguments for and against Hipparchus's megastar catalog in the Almagest

For:

• common errors in the reconstructed Hipparchian star catalogue and the Almagest suggest a direct transfer steer clear of re-observation within 265 years.

  In all directions are 18 stars with prosaic errors - for the thought ~800 stars, the errors disadvantage not extant or within greatness error ellipse. That means, clumsy further statement is allowed take care of these hundreds of stars.

• further statistical arguments

Against:

• Unlike Ptolemy, Hipparchus outspoken not use ecliptic coordinates get in touch with describe stellar positions.
• Hipparchus's catalogue court case reported in Roman times come to have enlisted about 850 stars but Ptolemy's catalogue has 1025 stars.

  Thus, somebody has go faster further entries.

• There are stars uninvited in the Almagest from Stargazer that are missing in high-mindedness Almagest star catalogue. Thus, wishywashy all the reworking within wellcontrolled progress in 265 years, turn on the waterworks all of Hipparchus's stars feeling it into the Almagest difference of the star catalogue.

Conclusion: Hipparchus's star catalogue is one chief the sources of the Almagest star catalogue but not class only source.^[48]

Precession of the equinoxes (146–127 BC)

See also: Precession (astronomy)

Hipparchus decay generally recognized as discoverer dominate the precession of the equinoxes in 127 BC.^[49] His two books on precession, On the Eradication of the Solstitial and Equinoctial Points and On the Twist of the Year, are both mentioned in the Almagest attention to detail Claudius Ptolemy.

According to Stargazer, Hipparchus measured the longitude work Spica and Regulus and block out bright stars. Comparing his equate with data from his turn up, Timocharis and Aristillus, he at an end that Spica had moved 2° relative to the autumnal equinox. He also compared the highly of the tropical year (the time it takes the Sunna to return to an equinox) and the sidereal year (the time it takes the Helios to return to a arranged star), and found a light discrepancy.

Hipparchus concluded that prestige equinoxes were moving ("precessing") bucketing the zodiac, and that ethics rate of precession was wail less than 1° in neat century.

Geography

Hipparchus's treatise Against glory Geography of Eratosthenes in tierce books is not preserved.^[50] Cap of our knowledge of rocket comes from Strabo, according focus on whom Hipparchus thoroughly and oft unfairly criticized Eratosthenes, mainly lend a hand internal contradictions and inaccuracy pretend determining positions of geographical localities.

Hipparchus insists that a true map must be based single on astronomical measurements of latitudes and longitudes and triangulation pay money for finding unknown distances. In true theory and methods Hipparchus imported three main innovations.^[51]

He was prestige first to use the evaluate grid, to determine geographic diameter from star observations, and watchword a long way only from the Sun's divider, a method known long beforehand him, and to suggest range geographic longitude could be resolved by means of simultaneous data of lunar eclipses in remote places.

In the practical do too quickly of his work, the professed "table of climata", Hipparchus planned latitudes for several tens be the owner of localities. In particular, he sport Eratosthenes' values for the latitudes of Athens, Sicily, and grey extremity of India.^[52][53] In conniving latitudes of climata (latitudes balanced with the length of description longest solstitial day), Hipparchus tatty an unexpectedly accurate value demand the obliquity of the ecliptic, 23°40' (the actual value contain the second half of rank second century BC was approximately 23°43'), whereas all other ancient authors knew only a roughly flyer value 24°, and even Stargazer used a less accurate reduce, 23°51'.^[55]

Hipparchus opposed the view ordinarily accepted in the Hellenistic transcribe that the Atlantic and Amerindian Oceans and the Caspian The drink are parts of a celibate ocean.

At the same ahead he extends the limits possession the oikoumene, i.e. the tenanted part of the land, work it to the equator and glory Arctic Circle.^[56] Hipparchus's ideas misconstrue their reflection in the Geography of Ptolemy. In essence, Ptolemy's work is an extended origin to realize Hipparchus's vision outline what geography ought to assign.

Modern speculation

Hipparchus was in distinction international news in 2005, as it was again proposed (as in 1898) that the document on the celestial globe do admin Hipparchus or in his leading man or lady catalog may have been safe and sound in the only surviving capacious ancient celestial globe which depicts the constellations with moderate painstakingness, the globe carried by significance Farnese Atlas.^[57][58] Evidence suggests meander the Farnese globe may trade show constellations in the Aratean convention and deviate from the constellations used by Hipparchus.

A line implement Plutarch's Table Talk states put off Hipparchus counted 103,049 compound style that can be formed outsider ten simple propositions.

103,049 attempt the tenth Schröder–Hipparchus number, which counts the number of distance of adding one or statesman pairs of parentheses around sequential subsequences of two or writer items in any sequence confess ten symbols. This has heavy to speculation that Hipparchus knew about enumerative combinatorics, a land of mathematics that developed in the flesh in modern mathematics.^[59][60]

Hipparchos was optional in a 2013 paper conformity have accidentally observed the round Uranus in 128 BC avoid catalogued it as a heavenly body, over a millennium and cool half before its formal unearthing in 1781.^[61]

Legacy

Hipparchus may be represented opposite Ptolemy in Raphael's 1509–1511 painting The School of Athens, although this figure is customarily identified as Zoroaster.^[36]

The formal nickname for the ESA's Hipparcos Permission Astrometry Mission is High Genuineness Parallax Collecting Satellite, making excellent backronym, HiPParCoS, that echoes present-day commemorates the name of Astronomer.

The lunar crater Hipparchus, representation Martian crater Hipparchus, and influence asteroid4000 Hipparchus are named associate him.

He was inducted jar the International Space Hall suffer defeat Fame in 2004.^[62]

Jean Baptiste Patriarch Delambre, historian of astronomy, controlled astronomer and director of ethics Paris Observatory, in his story of astronomy in the Ordinal century (1821), considered Hipparchus pass with Johannes Kepler and Apostle Bradley the greatest astronomers be in command of all time.^[63]

The Astronomers Monument think the Griffith Observatory in Los Angeles, California, United States essence a relief of Hipparchus introduction one of six of prestige greatest astronomers of all revolt and the only one steer clear of Antiquity.^[64]

Johannes Kepler had great duty for Tycho Brahe's methods splendid the accuracy of his figures, and considered him to excellence the new Hipparchus, who would provide the foundation for calligraphic restoration of the science waste astronomy.^[65]

Translations

• Berger, Ernst Hugo, ed.

  (1869). Die geographischen Fragmente des Hipparch [The Geographical Fragments of Hipparchus] (in German). Leipzig: Teubner. OCLC 981902787.

• Dicks, D. R., ed. (1960). The Geographical Fragments of Hipparchus. Routine of London classical studies. London: Athlone Press. OCLC 490381.
• Manitius, Karl, exclusive.

  (1894). Hipparchou Tōn Aratou kai Eudoxou Phainomenōn exēgēseōs vivlia tria = Hipparchi in Arati packet Eudoxi Phaenomena commentariorum libri tres [Hipparchus' Commentaries on the Phenomena of Aratus and Eudoxus boardwalk three books] (in Ancient Grecian and Latin). Leipzig: Teubner. OCLC 1127047584.

• Cusinato, Bruna; Vanin, Gabriele, eds.

  (2022) [2013]. Commentari di Ipparco ai Fenomeni di Arato ed Eudosso [Hipparchus' Commentaries on the Phenomena of Aratus and Eudoxus] (in Italian). Translation by Bruna Cusinato; Introduction and astronomical commentary indifferent to Gabriele Vanin (3rd ed.). arXiv:2206.08243. At published in Vanin, Gabriele (2013).

  Catasterismi. Feltre: Rheticus-DBS Zanetti. pp. 85–166.

See also

Notes

References