Hipparchus

Hipparchus (Greek Ἳππαρχος) (circa 190 BC – circa 120 BC) was a Greek astronomer, geographer, and mathematician. The ESA's Hipparcos Space Astrometry Mission was named after him, as are the Hipparchus lunar crater and the asteroid 4000 Hipparchus.

Hipparchus was born in Nicaea (now Iznik, Turkey) and probably died on the island of Rhodes. He is known to have been active at least from 147 BC to 127 BC. Hipparchus is considered the greatest astronomical observer, and by some the greatest astronomer of antiquity. He was the first Greek to develop quantitative and accurate models for the motion of the Sun and Moon. For this he made use of the observations and knowledge accumulated over centuries by the Chaldeans from Babylonia. He was also the first to compile a trigonometric table, which allowed him to solve any triangle. With his solar and lunar theories and his numerical trigonometry, he was probably the first to develop a reliable method to predict solar eclipses. His other achievements include the discovery of precession, the compilation of the first star catalogue, and probably the invention of the astrolabe. Claudius Ptolemaeus three centuries later depended much on Hipparchus. However, his synthesis of astronomy superseded Hipparchus's work: although Hipparchus wrote at least 14 books, only his commentary on the popular astronomical poem by Aratus has been preserved by later copyists. As a consequence, we know comparatively little about Hipparchus.

Contents

1 Life and work 2 Babylonian sources 3 Geometry and trigonometry 4 Lunar and solar theory 4.1 Motion of the moon 4.2 Orbit of the Moon 4.3 Apparent motion of the Sun 4.4 Orbit of the Sun 4.5 Calendar 4.6 Distance, parallax, size of the Moon and Sun 4.7 Eclipses 5 Astronomical instruments and astrometry 6 Geography 7 Star catalogue 7.1 Celestial bodies 7.2 Celestial coordinate systems 7.3 Brightness of stars 8 Precession of the equinoxes (146 BC-130 BC) 9 Hipparchus and astrology 10 See also 11 Literature 12 External links 12.1 General 12.2 Precession 12.3 Celestial bodies 12.4 Star catalogue

Life and work

Most of what is known about Hipparchus comes from Ptolemy's (2nd century) Almagest ("the great treatise"; ed. [Toomer 1981]), with additional references to him by Pappus of Alexandria and Theon of Alexandria (4th century) in their commentaries on the Almagest; from Strabo's Geographia ("Geography"), and from Pliny the Elder's Naturalis historia ("Natural history") (1st century).

There is a strong tradition that Hipparchus was born in Nicaea (Greek Νικαία), ancient district Bithynia, (modern-day İznik in province Bursa), in what today is Turkey.

The exact dates of his life are not known, but Ptolemy attributes astronomical observations to him from 147 BC to 127 BC; earlier observations since 162 BC might also be made by him. The date of his birth (circa 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 as Babylon, but it is not known if and when 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 coins were made in his honour in Bithynia that bear his name and show him with a globe; this confirms the tradition that he was born there.

Hipparchus is believed to have died on the island of Rhodes, where he spent most of his later life — Ptolemy attributes observations to him from Rhodes in the period from 141 BC to 127 BC.

Hipparchus's main original works are lost. His only preserved work is Toon Aratou kai Eudoxou Fainomenoon exegesis ("Commentary on the Phaenomena of Eudoxus and Aratus"). This is a critical commentary in two books on a popular poem by Aratus based on the work by Eudoxus. It was published by Karl Manitius (In Arati et Eudoxi Phaenomena, Leipzig, 1894). Hipparchus also made a list of his major works, which apparently mentioned about 14 books, but which is only known from references by later authors. His famous star catalogue probably was incorporated into the one by Ptolemy, but cannot be reliably reconstructed. We know he made a celestial globe; a copy of a copy may have been preserved in the oldest surviving celestial globe accurately depicting the constellations: the globe carried by the Farnese Atlas [Schaefer 2005].

Hipparchus is recognized as the originator and father of scientific astronomy. He is believed to be the greatest Greek astronomic observer, and many regard him as the greatest astronomer of ancient times, although Cicero gave preferences to Aristarchus of Samos. Some put in this place also Ptolemy of Alexandria. Hipparchus writings had been mostly superseded by those of Ptolemy, so later copyists have not preserved them for posterity.

Also see the biographical articles by [Toomer 1978] and [Jones 2001].

Babylonian sources

Many of the works of Greek scientists - mathematicians, astronomers, geographers - have been preserved up to the present time, or some aspects of their work and thoughts are still known through later references. However, achievements in these fields by middle-eastern civilizations, notably those in Babylonia, had been forgotten. After the discovery of the archeological sites in the 19th century, many writings on clay tablets have been found, some of them related to astronomy. Most known astronomical tablets have been described by A.Sachs, and later published by Otto Neugebauer in "Astronomical Cuneiform Texts" (3 vol.s; Princeton and London, 1955).

Since the rediscovery of the Babylonian civilization, it has become apparent that Greek astronomers, and in particular Hipparchus, borrowed a lot from the Chaldeans.

F.X. Kugler demonstrated in his book Die Babylonische Mondrechnung ("The Babylonian lunar computation", Freiburg im Breisgau, 1900) the following. Ptolemy had stated in his Almagest IV.2 that Hipparchus improved the values for the Moon's periods known to him from "even more ancient astronomers" by comparing eclipse observations made earlier by "the Chaldeans", and by himself. However Kugler found that the periods that Ptolemy attributes to Hipparchus had already been used in Babylonian ephemerides, specifically the collection of texts nowadays called "System B" (sometimes attributed to Kidinnu). Apparently Hipparchus only confirmed the validity of the periods he learned from the Chaldeans by his newer observations.

It is clear that Hipparchus (and Ptolemy after him) had an essentially complete list of eclipse observations covering many centuries. Most likely these had been compiled from the "diary" tablets: these are clay tablets recording all relevant observations that the Chaldeans routinely made. Preserved examples date from 652 BC to AD 130, but probably the records went back as far as the reign of the Babylonian king Nabonassar: Ptolemy starts his chronology with the first day in the Egyptian calendar of the first year of Nabonassar, i.e. 26 February 747 BC.

This raw material by itself must have been hard to use, and no doubt the Chaldeans themselves compiled extracts of e.g. all observed eclipses (some tablets with a list of all eclipses in a period of time covering a saros have been found). This allowed them to recognise periodic recurrences of events. Among others they used in System B (cf. Almagest IV.2):

• 223 (synodic) months = 239 returns in anomaly (anomalistic month) = 242 returns in latitude (draconic month). This is now known as the saros period which is very useful for predicting eclipses.
• 251 (synodic) months = 269 returns in anomaly
• 5458 (synodic) months = 5923 returns in latitude
• 1 synodic month = 29;31:50:08:20 days (sexagesimal; 29.53059413... days in decimals = 29 days 12 hours 44 min 3⅓ s)

The Babylonians expressed all periods in synodic months, probably because they used a lunisolar calendar. Various relations with yearly phenomena led to different values for the length of the year.

Similarly various relations between the periods of the planets were known. The relations that Ptolemy attributes to Hipparchus in Almagest IX.3 had all already been used in predictions found on Babylonian clay tablets.

All this knowledge was transferred to the Greeks probably shortly after the conquest by Alexander the Great (331 BC). According to the late classical philosopher Simplicius (early 6th century AD), Alexander ordered the translation of the historical astronomical records under supervision of his chronicler Callisthenes of Olynthus, who sent it to his uncle Aristotle. It is worth mentioning here that although Simplicius is a very late source, his account may be reliable. He spent some time in exile at the Sassanide (Persian) court, and may have accessed sources otherwise lost in the West. It is striking that he mentions the title t貥sis (Greek: guard) which is an odd name for a historical work, but is in fact an adequate translation of the Babylonian title massartu meaning "guarding" but also "observing". Anyway, Aristotle's pupil Callippus of Cyzicus introduced his 76-year cycle, which improved upon the 19-year Metonic cycle, about that time. He had the first year of the his first cycle start at the summer solstice of 28 June 330 BC (Julian proleptic date), but later he seems to have counted lunar months from the first month after Alexander's decisive battle at Gaugamela in fall 331 BC. So Callippus may have obtained his data from Babylonian sources and his calendar may have been anticipated by Kidinnu. Also it is known that the Babylonian priest known as Berossus wrote around 281 BC a book in Greek on the (rather mythological) history of Babylonia, the Babyloniaca, for the new ruler Antiochus I; it is said that later he founded a school of astrology on the Greek island of Kos. Another candidate for teaching the Greeks about Babylonian astronomy/astrology was Sudines who was at the court of Attalos Soter late in the 3rd century B.C.

In any case, the translation of the astronomical records required profound knowledge of the cuneiform script, the language, and the procedures, so it seems likely that it was done by some unidentified Chaldeans. Now the Babylonians dated their observations in their lunisolar calendar, in which months and years have varying lengths (29 or 30 days; 12 or 13 months respectively). At the time they did not use a regular calendar (such as based on the Metonic cycle like they did later), but started a new month based on observations of the New Moon. This made it very tedious to compute the time interval between events.

What Hipparchus may have done is transform these records to the Egyptian calendar, which uses a fixed year of always 365 days (consisting of 12 months of 30 days and 5 extra days): this makes computing time intervals much easier. Ptolemy dated all observations in this calendar. He also writes that "All that he (=Hipparchus) did was to make a compilation of the planetary observations arranged in a more useful way" (Almagest IX.2). Pliny states (Naturalis Historia II.IX(53)) on eclipse predictions: "After their time (=Thales) the courses of both stars (=Sun and Moon) for 600 years were prophecied by Hipparchus, ...". This seems to imply that Hipparchus predicted eclipses for a period of 600 years, but considering the enormous amount of computation required, this is very unlikely. Rather, Hipparchus would have made a list of all eclipses from Nabonasser's time to his own.

Other traces of Babylonian practice in Hipparchus work are:

• first Greek known to divide the circle in 360 degrees of 60 arc minutes.
• first consistent use of the sexagesimal number system.
• the use of the unit pechus ("cubit") of about 2? or 2½?.
• use of a short period of 248 days = 9 anomalistic months.

Also see G.J. Toomer (1981?): "Hipparchus and Babylonian Astronomy".

Geometry and trigonometry

Hipparchus is recognised as the first mathematician who compiled a trigonometry table, which he needed when computing the eccentricity of the orbits 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 21600 and a radius of (rounded) 3438 units: this has a unit length of 1 arcminute 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 it 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; his table seems to have survived in astronomical treatises in India, for instance the Surya Sidhanta. This 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. See [Toomer 1973].

For his chord table Hipparchus must have used a better approximation for π than the one from Archimedes (between 3 + 1/7 and 3 + 10/71); maybe 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 a theorem known to Archimedes. He also might have developed and used the theorem in plane geometry called Ptolemy'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 the sphere that do not pass through the center of projection to circles on the plane. This was the basis for the astrolabe.

Besides geometry, Hipparchus also used arithmetic techniques from the Chaldeans. He was one of the first Greek mathematicians to do this, and in this way expanded the techniques available to astronomers and geographers.

There is no indication that Hipparchus knew spherical trigonometry, which was first developed by Menelaus of Alexandria in the 1st century. Ptolemy later used the new technique for computing things like the rising and setting points of the ecliptic, or to take account of the lunar parallax. Hipparchus may have used a globe for this (to read values off the coordinate grids drawn on it), as well as approximations from planar geometry, or arithmetical approximations developed by the Chaldeans.

Lunar and solar theory

Motion of the moon

Hipparchus also studied the motion of the Moon and confirmed the accurate values for some periods of its motion that Chaldean astronomers had obtained before him. The traditional value (from Babylonian System B) for the mean synodic 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 the Hebrew calendar (possibly from Babylonian sources). The Chaldeans also knew that 251 synodic months = 269 anomalistic months. Hipparchus extended 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 (presumably 27 January 141 BC and 26 November 139 BC according to [Toomer 1980]), with eclipses from Babylonian records 345 years earlier (Almagest IV.2; [Jones 2001]). Already al-Biruni (Qanun VII.2.II) and Copernicus (de revolutionibus IV.4) noted that the period of 4267 lunations is actually about 5 minutes longer than the value for the eclipse period that Ptolemy attributes to Hipparchus. However, the best clocks and timing methods of the age had an accuracy of no better than 8 minutes. Modern scholars agree that Hipparchus rounded the eclipse period to the nearest hour, and used it to confirm the validity of the traditional values, rather than 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 s in the 4th century BC and less than 0.1 s 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:

1. 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.
2. The Moon itself would move uniformly (with some mean motion in anomaly) on a secondary circular orbit, called epicycle, that itself would move uniformly (with some mean motion in longitude) over the main circular orbit around the Earth, called deferent; see deferent 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 to attempt to determine the relative proportions and actual sizes of these orbits.

Hipparchus devised a geometrical method to find the parameters from 3 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 2 sets of 3 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, and 11 September 200 BC.

• For the eccentric model, Hipparchus found for the ratio between the radius of the eccenter 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. The results are distinctly different. This is partly due to some sloppy rounding and calculation errors, for which Ptolemy criticised him (he himself made rounding errors too...). 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 . See [Toomer 1967].

Apparent motion of the Sun

Before Hipparchus, Meton, Euktemon, and their pupils at Athens had made a solstice observation (i.e. timed the moment of the summer solstice) on June 27, 432 BC (proleptic julian calendar). Aristarchus is said to have done so in 280 BC, and Hipparchus also had an observation by Archimedes. Hipparchus himself observed the summer solstice in 135 BC, but he found observations of the moment of equinox 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 from the observation made on that day in Alexandria (at 5h after sunrise): Hipparchus may have visited Alexandria but he did not make his equinox observations there; presumably he was on Rhodes (at the same geographical longitude). He may have used his own armillary sphere or an equatorial ring for these observations. Hipparchus (and Ptolemy) knew that observations with these instruments are sensitive to a precise alignment with the equator. The real problem however is that atmospheric refraction lifts the Sun significantly above the horizon: so its apparent declination is too high, which changes the observed time when the Sun crosses the equator. Worse, the refraction decreases as the Sun rises, so it may appear to move in the wrong direction with respect to the equator in the course of the day - as Ptolemy mentions; however, Ptolemy and Hipparchus apparently did not realize that refraction is the cause.

To the end of his career Hipparchus wrote a book called Peri eniaus� meg鴨ous ("On the Length of the Year") about his results. The established value for the tropical year, introduced by Callippus in or before 330 BC (possibly from Babylonian sources, see above), was 365 + 1/4 day. 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. So he used the 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 108478 days. This implies a tropical year of 365.24579... days = 365 days;14,44,51 (sexagesimal; = 365 days + 14/60 + 44/60² + 51/60³), and this value has been found on a Babylonian clay tablet [A.Jones, 2001]. This is an indication that Hipparchus' work was known to Chaldeans.

Another 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 the sidereal 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 of precession (see below).

Orbit of the Sun

Before Hipparchus the Chaldean astronomers knew that the lengths of the seasons are not equal. Hipparchus made equinox and solstice observations, and according to Ptolemy (Almagest III.4) determined that spring (from spring equinox to summer solstice) lasted 94 + 1/2 days, and summer (from summer solstice to autumn equinox) 92 + 1/2 days. This is an unexpected result given 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 (of course today we know that the planets like the Earth move in ellipses around the Sun, but this was not discovered until Johannes Kepler published his first two laws of planetary motion in 1609). The value for the eccentricity attributed to Hipparchus by Ptolemy is that the offset is 1/24 of the radius of the orbit (which is too large), and the direction of the apogee would be at longitude 65.5? from the vernal equinox. Hipparchus may also have used another set of observations (94 + 1/4 and 92 + 3/4 days), which would lead to different values. The question remains if Hipparchus is really the author of the values provided by Ptolemy, who found no change 3 centuries later, and added lengths for the autumn and winter seasons.

Calendar

...to be written ...

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 'apost譡toon ("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 mean distance of the Moon, the Sun and Moon had the same apparent diameter; at that distance, the Moon's diameter fits 650 times into the circle, i.e. the mean apparent diameters are 360/650 = 0?33'14".

Like others before and after him, he also noticed that the Moon has a noticeable parallax, i.e. that it appears displaced from its calculated position (compared to the Sun or stars), and the difference is greater when closer to the horizon. He knew that this is because the Moon circles the center of the Earth, but the observer is at the surface - 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", more than 10 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, presumably that of 14 March 190 BC. It was total in the region of the Hellespont (and in fact in his birth place Nicaea); at the time the Romans were preparing for war with Antiochus III in the area, and the eclipse is mentioned by Livy in his Ab Urbe Condita VIII.2 . It was also observed in Alexandria, where the Sun was reported to be obscured for 4/5 by the Moon. Alexandria and Nicaea are on the same meridian. Alexandria is at about 31? North, and the region of the Hellespont at about 41? North; authors like Strabo and Ptolemy had fairly decent values for these geographical positions, and presumably Hipparchus knew them too. So 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 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 83 Earth radii.

In the second book, Hipparchus starts from the opposite extreme assumption: he assigns a (minimum) distance to the Sun of 470 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 minumum 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 solar eclipse at Alexandria must have been closer to 9/10 than to the reported 4/5).

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 book 2.

Theon of Smyrna wrote that according to Hipparchus, the Sun is 1880 times the size of the Earth, and the Earth 27 times the size of the Moon; apparently this refers to volumes, not diameters. From the geometry of book 2 it follows that the Sun is at 2550 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 5 months apart, and solar eclipses 7 months (instead of the usual 6 months); and the Sun can be hidden twice in 30 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 1 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, but Hipparchus 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 "and 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). Toomer (1980) argued that this must refer to the large total lunar eclipse of 26 November 139 BC, when over a clean sea horizon as seen from the citadel of Rhodes, the Moon was eclipsed in the North-West just after the Sun rose in the South-East. 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 is credited with the invention or improvement of several astronomical instruments, which were used for a long time with naked-eye observations. According to Synesius of Ptolemais (4th century) he made the first astrolabion: this may have been an armillary sphere (which Ptolemy however says he constructed, in Almagest V.1); or the predecessor of the planar instrument called astrolabe (also mentioned by Theon of Alexandria). With an astrolabe Hipparchus was the first to be able to measure the geographical latitude and time by observing stars. Previously this was done at daytime by measuring the shadow cast by a gnomon, or with the portable instrument known as scaphion.

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 did Proclus (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 equinoxes, which may be done with an equatorial ring: its shadow falls on itself when the Sun is on the equator (i.e. in one of the equinoctial points on the ecliptic), but the shadow falls above or below the opposite side of the ring when the Sun is South or North of the equator. Ptolemy quotes (in Almagest III.1 (H195)) a description by Hipparchus of an equatorial ring in Alexandria; a little further he describes two such instruments present in Alexandria in his own time.

Geography

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 the latitude and longitude of places on the earth. Hipparchos wrote a critique in 3 books on the work of the geographer Eratosthenes of Cyrene (3rd century BC), called Pr򳠴讠'Eratosth鮯us geograf� ("Against the Geography of Eratosthenes"). It is known to us from Strabo 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 at lunar eclipses (Strabo Geografia 7). 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 catalogue

After that in 135 BC, enthusiastic about a nova star in the constellation of Scorpius, he measured with an equatorial armillary sphere ecliptical coordinates of about 1000 stars (the exact number is not known) for his star catalogue.

He also knew the works Phainomena (Phenomena). That poem, known as Phaenomena or Arateia, describes the constellations and the stars that form them. Hipparchus' commentary contains many measurements of stellar position and times for rising, culmination, and setting of the constellations treated inn the Phaenomena, and these are likely to have been based on measurements of stellar positions—and Enoptron (Mirror of Nature) of Eudoxus of Cnidus, who had near Cyzicus on the southern coast of the Sea of Marmara his school and through Aratus' astronomical epic poem Phenomena Eudoxus' sphere, which was made from metal or stone and where there were marked constellations, brightest stars, tropic of Cancer and tropic of Capricorn. These comparisons embarrassed him because he couldn't put together Eudoxus' detailed statements with his own observations and observations of that time. From all this he found that coordinates of the stars and the Sun had systematically changed. Their ecliptic latitudes β remained unchanged, but their ecliptic longitudes λ had increased, at a rate which he estimated to be at least 1 degree per century.

This catalog served him to find any changes on the sky but unfortunately it is not preserved today. However, a 2005 analysis of an ancient statue of Atlas shows stars at positions that appear to have been determined using Hipparchus' data. [1] (http://www.phys.lsu.edu/farnese/) His star map was thoroughly modified as late as 1000 years later in 964 by Al Sufi and 1500 years later (1437) by Ulugh Beg. Later, Halley would use his star catalogue to discover proper motions as well.

The system of celestial coordinates used in Hipparchus's star catalog is not known. Since Ptolemy's copy in the Almagest is given in ecliptical coordinates, that system would seem the most likely; although there is evidence that both ecliptic coordinates and equatorial coordinates were used in the original observations.

Celestial bodies

Hipparchus in 130 BC wrote about an open cluster, the M44 Praesepe (NGC 2632) as a "Little Cloud" or "Cloudy Star". Before him the object was known to Aratus circa 260 BC, who wrote about it as a "Little Mist". Hipparchus also included this object in his famous star catalogue. The cluster was also known to Chinese astronomers. [Moore 1994], [10]

...to be extended ...

Celestial coordinate systems

Delambre in his Histoire de l'Astronomie Ancienne (1817) concluded that Hipparchus knew and used a real (celestial) equatorial coordinate system, directly with the right ascension and declination (or with its complement, polar distance). Later Otto Neugebauer (1899-1990) in his A History of Ancient Mathematical Astronomy (1975) rejected Delambre's claims.

Brightness of stars

Hipparchus had in 134 BC ranked stars in six magnitude classes according to their brightness: he assigned the value of 1 to the 20 brightest stars, to weaker ones a value of 2, and so forth to the stars with a class of 6, which can be barely seen with the naked eyes. This scheme was later adopted by Ptolemy and a similar system is still in use today. (See Apparent magnitude.)

Precession of the equinoxes (146 BC-130 BC)

Hipparchus is perhaps most famous for having been the first to measure the precession of the equinoxes. There is some suggestion that the Babylonians may have known about precession but it appears that Hipparchus was to first to really understand it and measure it. According to al-Battani Chaldean astronomers had distinguished the tropical and sidereal year. He stated they had around 330 BC an estimation for the length of the sidereal year to be S_K = 365 days 6 hours 11 min (= 365.258 days) with an error of (about) 2 min. This phenomenon was probably also known to Kidinnu around 314 BC. Yu Xi (fourth century) was the first Chinese astronomer to mention precession.

Hipparchus and his predecessors mostly used simple instruments for astronomical calculations, such as the gnomon, astrolabe, armillary sphere etc.

Additionally, as first in the history he correctly explained this with retrogradical movement of vernal point γ over the ecliptic for about 45", 46" or 47" (36" or 3/4' according to Ptolemy) per annum (today's value is 50.29") and he showed the Earth's axis is not fixed in space. By comparing his own measurements of the position of the equinoxes to the star Spica during a lunar eclipse at the time of equinox with those of Euclid's contemporaries (Timocharis of Alexandria (circa 320 BC–260 BC), Aristyllus 150 years earlier, the records of Chaldean astronomers (especially Kidinnu's records), and observations of a temple in Thebes, Egypt, that was built around 2000 BC) he still later observed that the equinox had moved 2? relative to Spica. He also noticed this motion in other stars. He obtained a value of not less than 1? in a century. The modern value is 1? in 72 years.

After him many Greek and Arab astronomers had confirmed this phenomenon. Ptolemy compared his catalogue with those of Aristyllus, Timocharis, Hipparchus and the observations of Agrippa and Menelaus of Alexandria from the early 1st century and he finally confirmed Hipparchus' empirical fact that the poles of the celestial equator in one Platonic year (approximately 25,777 sidereal years) encircle the ecliptical pole. The diameter of this circle is equal to the inclination of ecliptic relative to the celestial equator. The equinoctial points in this time traverse the whole ecliptic and they move 1? in a century. This velocity is equal to that calculated by Hipparchus. Because of these accordances Delambre, P. Tannery and other French historians of astronomy had wrongly jumped to conclusions that Ptolemy recorded his star catalogue from Hipparchus' with an ordinary extrapolation. It was not known until 1898 when Marcel Boll and others had found that Ptolemy's catalogue differs from Hipparchus' not only in the number of stars but in other respects.

This phenomenon was named by Ptolemy just because the vernal point γ leads the Sun. In Latin praecesse means "to overtake" or "to outpass", and today also means to twist or to turn. Its own name shows this phenomenon was discovered before its theoretical explanation, otherwise it would have been given a better term. Many later astronomers, physicists and mathematicians had occupied themselves with this problem, practically and theoretically. The phenomenon itself had opened many new promising solutions in several branches of celestial mechanics: Thabit ibn Qurra's theory of trepidation and oscillation of equinoctial points, Isaac Newton's general gravitational law (which had explained it in full), Leonhard Euler's kinematic equations and Joseph Lagrange's equations of motion, Jean d'Alembert's dynamical theory of the movement of a rigid body, some algebraic solutions for special cases of precession, John Flamsteed's and James Bradley's difficulties in the making of precise telescopic star catalogues, Friedrich Bessel's and Simon Newcomb's measurements of precession, and finally the precession of perihelion in Albert Einstein's general theory of relativity.

Lunisolar precession causes the vernal point γ to move along the ecliptic in the opposite direction of the Sun's apparent movement and the circulation of the celestial pole. Today's value is ψ = 50.388" per annum along a fixed ecliptic. This circle becomes a spiral as the other planets perturb earth's orbit. The latter is planetary precession where the plane of the ecliptic wobbles like a spinning top (akin to lunisolar precession) around the invariable plane of the solar system (fixed by the jovian planets) with a quasi-period of 100,000 years and an inclination varying between the extremes of 0.1? and 3? (relative to the present ecliptic, the inclination of the moving ecliptic reaches 4? with a quasi-period of 60,000 to 70,000 years). The angle between the ecliptic and celestial equators ε = 23? 26' 21.448" is becoming smaller by 0.468" per annum. Also, the point γ moves along a fixed equator by λ' = 0.1055" per annum in the same direction as the Sun. The sum of these precessions gives an annual general precession in longitude p = 50.291" which causes the tropical year to differ from the sidereal year. These values were adopted by the IAU in 1976 for noon 1 January 2000 (J2000.0) — the values for 1900 or 1950 are still present in many encyclopedias.

Hipparchus and astrology

In addition to his other writings dealing with astronomical topics, the work of Hipparchus (flourished second half of second century BC) dealing with the calculation and prediction of celestial positions would have been very useful to those engaged in the sort of astronomy known as astrology. Astrology developed in the Greco-Roman world during the Hellenistic period, borrowing many elements from Babylonian astronomy; some historians have suggested that Hipparchus played a key role in this. Remarks made by Pliny the Elder (who died 79 AD, following the eruption of the volcano Mount Vesuvius), in his Natural History Book 2. 24, suggest that some ancient authors did regard Hipparchus as an important figure in the history of astrology. Pliny claimed that Hipparchus 'can never be sufficiently praised, no one having done more to prove that man is related to the stars and that our souls are a part of heaven.'

See also

• Apparent magnitude
• Astrometry
• History of astrology
• Geminus (of Rhodes) (10 BC - circa 60)
• Mira
• Mithraism
• Star catalogues

Literature

• Edition and translation: Karl Manitius: In Arati et Eudoxi Phaenomena, Leipzig, 1894.
• G.J.Toomer (1967): The Size of the Lunar Epicycle According to Hipparchus. Centaurus 12(3), 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. Arch.Hist.Exact Sci. 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 (1981?): Hipparchus and Babylonian Astronomy, (?)
• 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.
• A.Jones: Hipparchus in "Encyclopedia of Astronomy and Astrophysics", Nature Publishing Group, 2001.
• J.Chapront, M.Chapront Touze, G.Francou (2002): A new determination of lunar orbital parameters, precession constant, and tidal acceleration from LLR measurements. Astron.Astrophys. 387, 700..709.
• B.E.Schaefer (2005): The Epoch of the Constellations on the Farnese Atlas and their Origin in Hipparchus's Lost Catalogue. J.Hist.Astron. xxxvi, 1..29.

External links

General

• [1] Hipparchus mentioned as Hipparchos in a history of mankind (http://www.hkupasteur.hku.hk/hkuip/causeries/Antiquity.html)
• [3] A lot of original Wikipedian articles (upto 2002-09-20) were transposed here (http://www.malaspina.com/site/person_639.asp)
• [4] Biographical page at the (http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Hipparchus.html) University of St. Andrews
• [5] Biographical page at the (http://www.hps.cam.ac.uk/starry/hipparchus.html) University of Cambridge
• [6] University of Cambridge's Page about Hipparchus' sole surviving work (http://www.hps.cam.ac.uk/starry/hipppoem.html)
• [7] Biographical page at the (http://zebu.uoregon.edu/~js/glossary/hipparchus.html) University of Oregon

Precession

• [8] Cavendish Laboratory (http://www.inference.phy.cam.ac.uk/teaching/dynamics/p0.html) briefly about Hipparchus' celestial dynamics and generally about the precession of the Earth's main axis
• [9] David Ulansey about Hipparchus's understanding of the precession (http://www.well.com/user/davidu/appendix4.html)

Celestial bodies

• [10] M44 Praesepe at SEDS (University of Arizona): http://www.seds.org/messier/m/m044.html

Star catalogue

• [2] A brief view by Carmen Rush on Hipparchus' stellar catalogue (http://ottawa.rasc.ca/astronomy/astro_facts/hipparcus.html)
• [11] Schaefer's site on the (http://www.phys.lsu.edu/farnese/) Farnese Atlas