Hipparchus (astronomer)

Hipparchus' equatorial ring

Hipparchus (Greek Ἳππαρχος) (ca. 190 B.C.E. - ca. 120 B.C.E.) was a Greek, astronomer, geographer, and mathematician of the Hellenistic period. He is known to have been active at least from 147 B.C.E. to 127 B.C.E. Hipparchus is considered the greatest astronomical observer, and by some the greatest astronomer of classical antiquity. He was the first Greek to develop quantitative and accurate models for the motion of the Sun and Moon, making 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. Based on 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 of the Western world, and probably the invention of the astrolabe. Three centuries later, the work of Claudius Ptolemaeus depended heavily on Hipparchus. Ptolemy’s synthesis of astronomy superseded Hipparchus's work; although Hipparchus wrote at least fourteen books, only his commentary on the popular astronomical poem by Aratus has been preserved by later copyists.

Life

Most of what is known about Hipparchus comes from Ptolemy's (second century C.E.) Almagest, with additional references to him by Pappus of Alexandria and Theon of Alexandria (fourth century) in their commentaries on the Almagest; from Strabo's Geographia ("Geography"), and from Pliny the Elder's Natural History (Naturalis historia) (first century).

There is a strong tradition that Hipparchus was born in Nicaea (Greek Νικαία), in the ancient district of Bithynia (modern-day Iznik in province Bursa), in what today is Turkey. The exact dates of his life are not known, but Ptolemy attributes astronomical observations to him from 147 B.C.E. to 127 B.C.E.; earlier observations since 162 B.C.E. might also have been made by him. The date of his birth (ca. 190 B.C.E.) was calculated by Delambre based on clues in his work. Hipparchus must have lived some time after 127 B.C.E. because he analyzed and published his latest observations then. 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' livelihood was and how he supported his scientific activities. There are no contemporary portraits of him, but in the second and third centuries coins were made in his honor in Bithynia that bear his name and show him with a globe; this supports 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 B.C.E. to 127 B.C.E.

Thought and Works

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"), a critical commentary in two books on a popular poem by Aratus based on the work by Eudoxus of Cnidus. Hipparchus also made a list of his major works, which apparently mentioned about fourteen books, but which is only known from references by later authors. His famous star catalogue probably was incorporated into the one by Ptolemy, and 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.

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

There is evidence, based on references in non-scientific writers such as Plutarch, that Hipparchus was aware of some physical ideas that we consider Newtonian, and that Newton knew this.

The European Space Agency's Hipparcos Space Astrometry Mission was named after Hipparchus, as were the Hipparchus lunar crater and the asteroid 4000 Hipparchus.

Babylonian sources

Earlier Greek astronomers and mathematicians were influenced by Babylonian astronomy to a limited extent, for instance the period relations of the Metonic cycle and Saros cycle may have come from Babylonian sources. Hipparchus seems to have been the first to systematically exploit Babylonian astronomical knowledge and techniques. He was the first Greek known to divide the circle in 360 degrees of 60 arc minutes (Eratosthenes before him used a simpler sexagesimal system dividing a circle into 60 parts). He also used the Babylonian unit pechus ("cubit") of about 2° or 2½°.

Hipparchus probably compiled a list of Babylonian astronomical observations; historian of astronomy G. Toomer has suggested that Ptolemy's knowledge of eclipse records and other Babylonian observations in the Almagest came from a list made by Hipparchus. Hipparchus' use of Babylonian sources has always been known in a general way, because of Ptolemy's statements. However, Franz Xaver Kugler demonstrated that the 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).

Geometry and trigonometry

Hipparchus is recognised as the first mathematician to compile 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 21,600 and a radius of (rounded) 3438 units: this has a unit length of 1 arc minute along its perimeter. He tabulated the chords for angles with increments of 7.5°. In modern terms, the chord of an angle equals twice the sine of half of the angle, i.e.:

chord(A) = 2 sin(A/2).

He described it in a work (now lost), called Toon en kuklooi eutheioon (Of Lines Inside a Circle) by Theon of Alexandria (fourth century) in his commentary on the Almagest I.10; some claim his table may 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.

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 Lazare 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.

Hipparchus was one of the first Greek mathematicians to used Chaldean arithmetic techniques, 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 first 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 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 January 27, 141 B.C.E. and November 26, 139 B.C.E. according to [Toomer 1980]), with eclipses from Babylonian records 345 years earlier (Almagest IV.2; [Jones 2001]). Al-Biruni (Qanun VII.2.II) and Copernicus (de revolutionibus IV.4) noted that the period of 4,267 lunations is actually about 5 minutes longer than the value for the eclipse period that Ptolemy attributes to Hipparchus. However, the 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 and taking account of the change in the length of the day we estimate that the error in the assumed length of the synodic month was less than 0.2 s in the fourth century B.C.E. 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 third century B.C.E. 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 an epicycle, that itself would move uniformly (with some mean motion in longitude) over the main circular orbit around the Earth, called deferent. 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 three positions of the Moon, at particular phases of its anomaly. In fact, he did this separately for the eccentric and the epicycle model. Ptolemy describes the details in the Almagest IV.11. Hipparchus used two sets of three lunar eclipse observations, which he carefully selected to satisfy the requirements. The eccentric model he fitted to these eclipses from his Babylonian eclipse list: 22/23 December 383 B.C.E., 18/19 June 382 B.C.E., and 12/13 December 382 B.C.E. The epicycle model he fitted to lunar eclipse observations made in Alexandria at 22 September 201 B.C.E., 19 March 200 B.C.E., and 11 September 200 B.C.E.

• 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 cumbersome unit he used in his chord table resulted in peculiar numbers, and errors in rounding and calculating (for which Ptolemy criticized) him produced 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 .

Apparent motion of the Sun

Before Hipparchus, Meton, Euctemon, and their pupils at Athens had made a solstice observation (i.e., timed the moment of the summer solstice) on June 27, 432 B.C.E. (proleptic Julian calendar). Aristarchus of Samos is said to have done so in 280 B.C.E., and Hipparchus also had an observation by Archimedes. Hipparchus himself observed the summer solstice in 135 B.C.E., 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 B.C.E. to 128 b.c.e..

Ptolemy quotes an equinox timing by Hipparchus (at March 24, 146 B.C.E. 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. Ptolemy noted this, however, Ptolemy and Hipparchus apparently did not realize that refraction is the cause.

At the end of his career, Hipparchus wrote a book called Peri eniausíou megéthous ("On the Length of the Year") about his results. The established value for the tropical year, introduced by Callippus in or before 330 B.C.E. (possibly from Babylonian sources), was 365 + 1/4 days. 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. He set the length of the tropical year to 365 + 1/4 - 1/300 days (= 365.24666... days = 365 days 5 hours 55 min, which differs from the actual value (modern estimate) of 365.24219... days = 365 days 5 hours 48 min 45 s by only about 6 min).

Between the solstice observation of Meton and his own, there were 297 years spanning 108,478 days. 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], indicating that Hipparchus' work was known to Chaldeans.

Another value for the year that is attributed to Hipparchus (by the astrologer Vettius Valens in the first 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.

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, including 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 three centuries later, and added lengths for the autumn and winter seasons.

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 of Alexandria in his commentary on the Almagest V.11; Theon of Smyrna (second 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 ten times smaller than the resolution of the unaided eye).

In the first book, Hipparchus assumed that the parallax of the Sun was 0, as if it is at infinite distance. He then analyzed a solar eclipse, presumably that of March 14, 190 B.C.E.. 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 4/5 obscured 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 reasonable values for these geographical positions, and presumably Hipparchus knew them too. Hipparchus was able to 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 81 Earth radii.

In the second book, Hipparchus started from the opposite extreme assumption: he assigned 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 minimum distance of the Sun, it was the maximum mean distance possible for the Moon. With his value for the eccentricity of the orbit, he could also compute the least and greatest distances of the Moon. According to Pappus, Hipparchus 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 was 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 Hipparchus’ second method with lunar eclipses to compute the distance of the Sun (Almagest V.15). He criticized 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. Hipparchus’ results were the best at that time: 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 1,880 times the size of the Earth, and the Earth twenty-seven times the size of the Moon; apparently this refers to volumes, not diameters. From the geometry of book 2 it follows that the Sun is at 2,550 Earth radii, and the mean distance of the Moon is 60½ radii. Similarly, Cleomedes quoted Hipparchus’ ratio for the sizes of the Sun and Earth as 1050:1; this leads to a mean lunar distance of 61 radii. Apparently Hipparchus later refined his computations, and derived accurate single values that he could use for predictions of solar eclipses.

See [Toomer 1974] for a more detailed discussion.

Eclipses

Pliny (Naturalis Historia II.X) tells us that Hipparchus demonstrated that lunar eclipses can occur five months apart, and solar eclipses seven months (instead of the usual six months); and the Sun can be hidden twice in thirty days, but as seen by different nations. Ptolemy discussed this a century later at length in Almagest VI.6. The geometry, and the limits of the positions of Sun and Moon when a solar or lunar eclipse is possible, are explained in Almagest VI.5. Hipparchus apparently made similar calculations. The result that two solar eclipses can occur one month apart is important, because this conclusion can not be based on observations: one eclipse is visible on the northern and the other on the southern hemisphere, and the latter was inaccessible to the Greek.

Prediction of exactly when and where a solar eclipse will be visible requires a solid lunar theory and proper treatment of the lunar parallax. Hipparchus was probably the first to make this prediction. In order to do this accurately, spherical trigonometry is required, 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 "he also discovered for what exact reason, although the shadow causing the eclipse must from sunrise onward be below the earth, it happened once in the past that the moon was eclipsed in the west while both luminaries were visible above the earth." (translation H. Rackham (1938), Loeb Classical Library 330 p.207). Toomer (1980) argued that this must refer to the large total lunar eclipse of November 26, 139 B.C.E., when over a clean sea horizon as seen from the citadel of Rhodes, the Moon was eclipsed in the northwest just after the Sun rose in the southeast. This would be the second eclipse of the 345-year interval that Hipparchus used to verify the traditional Babylonian periods, and puts a late date to the development of Hipparchus' lunar theory. We do not know what "exact reason" Hipparchus found for seeing the Moon eclipsed while apparently it was not in exact opposition to the Sun. Parallax lowers the altitude of the luminaries; refraction raises them, and from a high point of view the horizon is lowered.

Astronomical instruments and astrometry

Hipparchus and his predecessors used simple instruments, such as the gnomon, the astrolabe, and the armillary sphere for astronomical calculations and observations. Hipparchus is credited with the invention or improvement of several astronomical instruments, which were used for a long time for naked-eye observations. According to Synesius of Ptolemais (fourth 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 during the day by measuring the shadow cast by a gnomon, or with the portable instrument known as scaphion.

Ptolemy mentions (Almagest V.14) that he an instrument similar to Hipparchus’, called a 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 four-foot rod with a scale, a sighting hole at one end, and a wedge that could be moved along the rod to exactly obscure the disk of Sun or Moon.

Hipparchus also observed solar equinoxes, which may be done with an equatorial ring: its shadow falls on itself when 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 (Messina, Italy), but Hipparchus was the first to apply mathematical rigor to the determination of the latitude and longitude of places on the Earth. Hipparchus wrote a critique in three books on the work of the geographer Eratosthenes of Cyrene (third century B.C.E.), called Pròs tèn 'Eratosthénous geografían ("Against the Geography of Eratosthenes"). It is known to us from Strabo of Amaseia, who in his turn criticized 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

Late in his career (about 135 B.C.E.) Hipparchus compiled a star catalogue. He also constructed a celestial globe depicting the constellations, based on his observations. His interest in the fixed stars may have been inspired by the observation of a supernova (according to Pliny), or by his discovery of precession (according to Ptolemy, who says that Hipparchus could not reconcile his data with earlier observations made by Timocharis and Aristyllos).

Previously, Eudoxus of Cnidus in the fourth century B.C.E. had described the stars and constellations in two books called Phaenomena and Entropon. Aratus wrote a poem called Phaenomena or Arateia based on Eudoxus' work. Hipparchus wrote a commentary on the Arateia, his only preserved work, which contains many stellar positions and times for rising, culmination, and setting of the constellations, and these are likely to have been based on his own measurements.

Hipparchus made his measurements with an equatorial armillary sphere, and obtained the positions of maybe about 850 stars. It is disputed which coordinate system he used. Ptolemy's catalogue in the Almagest, which is derived from Hipparchus' catalogue, is given in ecliptic coordinates. However Delambre in his Histoire de l'Astronomie Ancienne (1817) concluded that Hipparchus knew and used the equatorial coordinate system, a conclusion challenged by Otto Neugebauer in his A History of Ancient Mathematical Astronomy (1975). Hipparchus seems to have used a mix of ecliptic coordinates and equatorial coordinates: in his commentary on Eudoxus of Cnidus he provides the polar distance (equivalent to the declination in the equatorial system) and the ecliptic longitude.

Hipparchus' original catalogue is no longer in existence. However, an analysis of an ancient statue of Atlas (the “Farnese Atlas”) supporting a globe, published in 2005, shows stars at positions that appear to have been determined using Hipparchus' data. [1].

As with most of his work, Hipparchus’ star catalogue was adopted and expanded by Ptolemy. It has been strongly disputed how much of the star catalogue in the Almagest is due to Hipparchus, and how much is original work by Ptolemy. Statistical analysis (by Bradly Schaeffer, and others) shows that the classical star catalogue has a complex origin. Ptolemy has even been accused of fraud for stating that he re-measured all stars; many of his positions are wrong and it appears that in most cases he used Hipparchus' data and precessed them to his own epoch three centuries later, but using an erroneously small precession constant.

The work begun by Hipparchus has had a lasting heritage, and was added to much later by Al Sufi (964), and by Ulugh Beg as late as 1437. It was superseded only by more accurate observations after invention of the telescope.

Stellar magnitude

Hipparchus ranked stars in six magnitude classes according to their brightness: he assigned the value of one to the twenty brightest stars, to weaker ones a value of two, and so forth to the stars with a class of six, which can be barely seen with the naked eye. A similar system is still used today.

Precession of the Equinoxes (146 B.C.E.-130 B.C.E.)

Hipparchus is perhaps most famous for having discovered the precession of the equinoxes. His two books on precession, On the Displacement of the Solsticial and Equinoctial Points and On the Length of the Year, are both mentioned in the [Almagest of Claudius Ptolemy. According to Ptolemy, Hipparchus measured the longitude of Spica and other bright stars. Comparing his measurements with data from his predecessors, Timocharis and Aristillus, he realized that Spica had moved 2° relative to the autumnal equinox. He also compared the lengths of the tropical year (the time it takes the Sun to return to an equinox) and the sidereal year (the time it takes the Sun to return to a fixed star), and found a slight discrepancy. Hipparchus concluded that the equinoxes were moving ("precessing") through the zodiac, and that the rate of precession was not less than 1° in a century.

Ptolemy followed up on Hipparchus' work in the second century C.E. He confirmed that precession affected the entire sphere of fixed stars (Hipparchus had speculated that only the stars near the zodiac were affected), and concluded that 1° in 100 years was the correct rate of precession. The modern value is 1° in 72 years.

Hipparchus and Astrology

As far as is known, Hipparchus never wrote about astrology, the application of astronomy to the practice of divination. Nevertheless the work of Hipparchus dealing with the calculation and prediction of celestial positions would have been very useful to those engaged in astrology. Astrology developed in the Greco-Roman world during the Hellenistic period, borrowing many elements from Babylonian astronomy. Remarks made by Pliny the Elder in his Natural History Book 2.24, suggest that some ancient authors regarded 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."

References

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• Chapront, J.; M. Chapront Touze, and G. Francou, A new determination of lunar orbital parameters, precession constant, and tidal acceleration from LLR measurements. Astron. Astrophys. 387 (2002): 700-709.
• Jones, A. "Hipparchus." In Encyclopedia of Astronomy and Astrophysics. Nature Publishing Group, 2001.
• Moore, Patrick Moore. Atlas of the Universe. Octopus Publishing Group LTD, 1994. (Slovene translation and completion by Tomaž Zwitter and Savina Zwitter (1999): Atlas vesolja), 225.
• Plato, J. Burnet (Editor). Opera: Volume II: Parmenides, Philebus, Symposium, Phaedrus, Alcibiades I and II, Hipparchus, Amatores. Oxford University Press, 1922. ISBN 978-0198145417
• Plato, W. R. M. Lamb (Translator). Plato: Charmides, Alcibiades 1 & 2, Hipparchus, The Lovers, Theages, Minos, Epinomis. Loeb Classical Library, 1927. ISBN 978-0674992214
• Schaefer , B. E. "The Epoch of the Constellations on the Farnese Atlas and their Origin in Hipparchus's Lost Catalogue." Journal for the History of Astronomy 36 (2005): 1-29.
• Swerdlow, N. M.. "Hipparchus on the distance of the sun." Centaurus 14(1969): 287-305
• Toomer, G.J. "The Size of the Lunar Epicycle According to Hipparchus." Centaurus 12 (1967): 145-150.
• Toomer, G.J. "The Chord Table of Hipparchus and the Early History of Greek Trigonometry." Centaurus 18 (1973): 6-28.
• Toomer, G.J. "Hipparchus on the Distances of the Sun and Moon." Archives for the History of the Exact Sciences 14 (1974): 126-142.
• Toomer, G.J. "Hipparchus." In Dictionary of Scientific Biography 15: 207-224, 1978
• Toomer, G.J.: "Hipparchus' Empirical Basis for his Lunar Mean Motions," Centaurus 24 (1980): 97-109.
• Toomer, G.J. "Hipparchus and Babylonian Astronomy." In A Scientific Humanist: Studies in Memory of Abraham Sachs, ed. Erle Leichty, Maria deJ. Ellis, and Pamel Gerardi. Philadelphia: Occasional Publications of the Samuel Noah Kramer Fund, 9, 1988.
• Edition and translation: Karl Manitius: In Arati et Eudoxi Phaenomena. Leipzig, 1894.

External links

All links retrieved July 17, 2024.

• Biography of Hipparchus on Fermat's Last Theorem Blog
• Schaefer's site on the Farnese Atlas

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