# Hipparchus

**Hipparchus**(

**Hipparhcos**, 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.

Hipparchus was born in Nicaea (now in 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 altogether, 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 However, his synthesis of astronomy superseded Hipparchus's work: although Hipparchus wrote at least 14 books, only his commentary on the popular astronomical poem by Aratus has been preserved by later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.

## 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 B.C. 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 A.D. 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' 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.

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 O.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 130 A.D., 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 -746 (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;31:50:8:20 days (sexagesimal; 29.53059413... days in decimals = 29d12h44m3+1/3s)

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.

It is not known how this knowledge was transferred to the Greeks, in particular Hipparchus. Since the conquest by Alexander the Great (331 BC), Babylonia had been part of the Hellenistic Seleucid Empire, so no doubt exchange between the various groups of sages intensified. For instance 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.

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 unknown Chaldean. 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.

## 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).

*toon en kuklooi eutheioon*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 Siddhanta. 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 &pi 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.

## 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

Hipparchos wrote a critique in 3 books on the work of the geographer Eratosthenes of Cyrene (3rd century BC), called *Pròs tèn 'Eratosthénous geografían* ("Against the Geography of Eratosthenes"). It is known to us from Strabo of Amaseia, who in his turn critised 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 proposed 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 method would give the most accurate data as would any previous one, if it would be correctly carried out. However, it was never properly applied, and for this reason maps remained rather inaccurate until modern times.

## 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^{d};31,50,28,20 (sexagesimal) = 29.53059429... d . Expressed as 29

^{d}+ 12

^{h}+ 793/1080

^{h}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

^{d}1

^{h}(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 Delta-T) we estimate that the error in the assumed length of the synodic month was less than 0.2

^{s}in the 4

^{th}cy. B.C 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:

- In the first, the Moon would move uniformly along a circle, but the Earth would be eccentric,
*i.e.*at some distance of the center of the circle. So the apparent angular speed of the Moon (and its distance) would vary. - The Moon itself would move uniformly (with some mean motion in anomaly) on a secondary circular orbit, called
**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.

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 .

### 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.

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 eniausíou megéthous* ("On the Length of the Year") about his results. The established value for the tropical year, determined by Kalippus in or before 330 BC, 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^{d} 5^{h} 55 ^{m}, which differs from the actual value (modern estimate) of 365.24219... days = 365^{d} 5^{h} 48^{m} 45^{s} by only about 6^{m}).

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^{d};14,44,51 (sexagesimal; = 365^{d} + 14/60 + 44/60^{2} + 51/60^{3}), 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^{d} 6^{h} 5^{m}), but this may be a corruption of another value attributed to a Babylonian source: 365 + 1/4 + 1/144 days (= 365.25694... days = 365^{d} 6^{h} 10^{m}). 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 lenghts for the autumn and winter seasons.

### calendar

### distance, parallax, size of the Moon and Sun

Hipparchus also undertook to find the distances and sizes of the Sun and the Moon. He published his results in a work of two books called*peri megethoon kai 'apostèmátoon*("On Sizes and Distances") by Pappus in his commentary on the

*Almagest*V.11; Theon of Smyrna (2nd century) mentions the work with the addition "of the Sun and Moon".

Hipparchus measured the apparent diameters of the Sun and Moon with his *diopter*. Like others before and after him, he found that the Moon's size varies as it moves on its (eccentric) orbit, but he found no perceptible variation in the apparent diameter of the Sun. He found that at the **mean** distance of the Moon, the Sun and Moon had the same apparent diameter; at that distance, the Moon's diamater 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 observeable 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 downto 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 excentricity 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 nortern 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 known from the Suda lexicon.

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.

## 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 850 (falsely quoted elsewhere as 1600 or 1080) and in 129 BC he made first big 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 celestial latitudes λ remained unchanged, but their celestial longitudes β had reduced as would equinoctial points, intersections of ecliptic and celestial equator, move with progressive velocity every year for 1/100'.

This map served him to find any changes on the sky but unfortunately it is not preserved today. 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. His work speaks for itself. Another loss is that we know almost nothing about Hipparchus' life (this was stressed by Fred Hoyle).

In his star map Hipparchus drew the position of every star on the basis of its celestial latitude (its angular distance from the ecliptic plane) and its celestial longitude (its angular distance from an arbitrary point, for instance as is custom in astronomy from vernal equinox). The system from his star map was also transferred to maps for Earth. Before him longitudes and latitudes were used by Dicaearchus of Messana, but they got their meanings in coordinate net not until Hipparchus.

### 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^{d} 6^{h} 11^{m} (= 365.2576388^{d}) with an error of (about) 110^{s}. This phenomenon was probably also known to Kidinnu around 314 BC. A. Biot and Delambre attribute the discovery of precession also to old Chinese astronomers.

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 Ψ'=50.387", 50.26") 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 in 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 poles of the celestial equator in one Platonic year (approximately 25,777 sidereal years) encircle the ecliptical pole. The diameter of these cicles is equal to the inclination of ecliptic. 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. This 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 practically 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 the 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 motion of point γ by the ecliptic in the opposite direction of the apparent solar year's movement and the circulation of celestial pole. This circle becomes a spiral because of additional ascendancy of the planets. This is planetary precession where the ecliptical plane swings from its central position for ±4° in 60,000 years. The angle between ecliptic and celestial equator ε = 23° 26' is reducing for 0.47" per annum. Also, the point γ slides by equator for *p* = 0.108" per annum now in the same direction as the Sun. The sum of precessions gives an annual *general precession in longitude* Ψ = 50.288" which causes the origination of tropical year.

## See also:

- Apparent magnitude
- Astrometry
- 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 .

## External links

### general:

- [1] Hipparchus mentioned as Hipparchos in a history of mankind
- [2] A brief view by Carmen Rush on Hipparchus' stellar catalogue
- [3] A lot of original Wikipedian articles (upto 2002-09-20) were transposed here (
*This may be encouraging for us Wikipedians, but we must consider Hoyle's note that many defects and/or inaccuracies may still be present here*) - [4] Page at the University of St. Andrews
- [5] Page at the University of Cambridge
- [6] University of Cambridge's Page about Hipparchus' sole surviving work
- [7] Page at the University of Oregon

### precession:

- [8] Cavendish Laboratory 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

### celestial bodies: