Hipparchus (astronomer)
Encyclopedia
Hipparchus, or more correctly Hipparchos , was a Greek
astrologer
, astronomer
, geographer
, and mathematician
of the Hellenistic period. He is considered the founder of trigonometry
.
Hipparchus was born in Nicaea (now Iznik
, Turkey
), and probably died on the island of Rhodes
. He is known to have been a working astronomer at least from 162 to 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity
. He was the first whose quantitative and accurate models for the motion of the Sun
and Moon
survive. For this he certainly made use of the observations and perhaps the mathematical techniques accumulated over centuries by the Chaldea
ns from Babylonia
. He developed trigonometry
and constructed trigonometric
tables, and he solved several problems of spherical trigonometry
. With his solar and lunar
theories and his trigonometry, he may have been the first to develop a reliable method to predict solar eclipse
s. His other reputed achievements include the discovery of Earth's precession, the compilation of the first comprehensive star catalog of the western world, and possibly the invention of the astrolabe
, also of the armillary sphere
, which he used during the creation of much of the star catalogue. It would be three centuries before Claudius Ptolemaeus
' synthesis of astronomy would supersede the work of Hipparchus; it is heavily dependent on it in many areas.
was preserved by later copyists. Most of what is known about Hipparchus comes from Ptolemy
's (2nd century) Almagest
, with additional references to him by Pappus of Alexandria
and Theon of Alexandria
(c. 4th century AD) in their commentaries on the Almagest; from Strabo
's Geographia ("Geography"), and from Pliny the Elder
's Naturalis historia ("Natural history") (1st century AD).
There is a strong tradition that Hipparchus was born in Nicaea (Greek Νίκαια), in the ancient district of Bithynia
(modern-day Iznik in province Bursa
), in what today is the country Turkey
.
The exact dates of his life are not known, but Ptolemy attributes to him astronomical observations in the period from 147 BC to 127 BC, and some of these are stated as made in Rhodes; earlier observations since 162 BC might also have been made by him. His birth date (c. 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 observations from that year. Hipparchus obtained information from Alexandria
as well as Babylon
, but it is not known when or if he visited these places. He is believed to have died on the island of Rhodes
, where he seems to have spent most of his later life.
It is not known what Hipparchus' economic means were nor how he supported his scientific activities. His appearance is likewise unknown: there are no contemporary portraits. In the 2nd and 3rd centuries coin
s were made in his honour in Bithynia
that bear his name and show him with a globe
; this supports the tradition that he was born there.
Hipparchus is thought to be the first to calculate a heliocentric system, but he abandoned his work because the calculations showed the orbits were not perfectly circular as believed to be mandatory by the science of the time. As an astronomer of antiquity his influence, supported by Aristotle, held sway for nearly 2000 years, until the heliocentric model of Copernicus
.
Hipparchus' only preserved work is Τῶν Ἀράτου καὶ Εὐδόξου φαινομένων ἐξήγησις ("Commentary on the Phaenomena of Eudoxus and Aratus"). This is a highly critical commentary in the form of two books on a popular poem by Aratus
based on the work by Eudoxus
. Hipparchus also made a list of his major works, which apparently mentioned about fourteen books, but which is only known from references by later authors. His famous star catalog was incorporated into the one by Ptolemy, and may be almost perfectly reconstructed by subtraction of two and two thirds degrees from the longitudes of Ptolemy's stars.The first trigonometric table was apparently compiled by Hipparchus, who is now consequently known as "the father of trigonometry.
. There are a variety of mis-steps in the more ambitious 2005 paper, thus no specialists in the area accept its widely publicized speculation.
Lucio Russo
has said that Plutarch
, in his work On the Face in the Moon, was reporting some physical theories that we consider to be Newtonian and that these may have come originally from Hipparchus; he goes on to say that Newton may have been influenced by them. Both of these claims have been rejected by other scholars.
A line in Plutarch
's Table Talk states that Hipparchus counted 103049 compound propositions that can be formed from ten simple propositions; 103049 is the tenth Schröder–Hipparchus number
and this line has led to speculation that Hipparchus knew about enumerative combinatorics
, a field of mathematics that developed independently in modern mathematics.
and Saros cycle
may have come from Babylonian sources. Hipparchus seems to have been the first to exploit Babylonian astronomical knowledge and techniques systematically. Except for Timocharis
and Aristillus
, 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 2.5°.
Hipparchus probably compiled a list of Babylonian astronomical observations; G. J. Toomer, a historian of astronomy, has suggested that Ptolemy's knowledge of eclipse records and other Babylonian observations in the Almagest came from a list made by Hipparchus. Hipparchus' use of Babylonian sources has always been known in a general way, because of Ptolemy's statements. However, Franz Xaver Kugler
demonstrated that the synodic and anomalistic periods that Ptolemy attributes to Hipparchus had already been used in Babylonian ephemerides, specifically the collection of texts nowadays called "System B" (sometimes attributed to Kidinnu
).
Hipparchus's long draconitic
lunar period (5458 months = 5923 lunar nodal periods) also appears a few times in Babylonian records. But the only such tablet explicitly dated is post-Hipparchus so the direction of transmission is not secured
.
s of the Moon and Sun. He tabulated values for the chord
function, which gives the length of the chord for each angle. He did this for a circle with a circumference of 21,600 and a radius (rounded) of 3438 units: this circle has a unit length of 1 arc minute along its perimeter. He tabulated the chords for angles with increments of 7.5°. In modern terms, the chord of an angle equals twice the sine
of half of the angle, i.e.:
He described the chord table in a work, now lost, called Tōn en kuklōi eutheiōn (Of Lines Inside a Circle) by Theon of Alexandria
(4th century) in his commentary on the Almagest I.10; some claim his table may have survived in astronomical treatises in India
, for instance the Surya Siddhanta
. Trigonometry was a significant innovation, because it allowed Greek astronomers to solve any triangle, and made it possible to make quantitative astronomical models and predictions using their preferred geometric techniques.
For his chord table Hipparchus must have used a better approximation for π
than the one from Archimedes
of between 3 + 1/7 and 3 + 10/71; perhaps he had the one later used by Ptolemy: 3;8:30 (sexagesimal) (Almagest VI.7); but it is not known if he computed an improved value himself.
Hipparchus could construct his chord table using the Pythagorean theorem
and 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 developed by the Chaldea
ns. He was one of the first Greek mathematicians to do this, and in this way expanded the techniques available to astronomers and geographers.
There are several indications that Hipparchus knew spherical trigonometry, but the first surviving text of it is that of Menelaus of Alexandria
in the 1st century, who on that basis is now commonly credited with its discovery. (Previous to the finding of the proofs of Menelaus a century ago, Ptolemy was credited with the invention of spherical trigonometry.) Ptolemy later used spherical trigonometry to compute 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 these tasks, reading values off coordinate grids drawn on it, or he may have made approximations from planar geometry, or perhaps used arithmetical approximations developed by the Chaldeans. He might have used spherical trigonometry.
Aubrey Diller has shown that the clima calculations which Strabo
preserved from Hipparchus were performed by spherical trigonometry with the sole accurate obliquity known to have been used by ancient astronomers, 23°40'. All thirteen clima figures agree with Diller's proposal. Further confirming his contention is the finding that the big errors in Hipparchus's longitude of Regulus
and both longitudes of Spica
agree to a few minutes in all three instances with a theory that he took the wrong sign for his correction for parallax when using eclipses for determining stars' positions.
and confirmed the accurate values for two periods of its motion that Chaldean astronomers certainly possessed before him, whatever their ultimate origin
. The traditional value (from Babylonian System B) for the mean synodic month is 29 days;31,50,8,20 (sexagesimal) = 29.5305941... days. 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 used a multiple of this period by a factor of 17, because that interval is also an eclipse period. The Moon also is close to an integer number of years (4267 moons : 4573 anomalistic periods : 4630.53 nodal periods : 4611.98 lunar orbits : 344.996 years : 344.982 solar orbits : 126,007.003 days : 126,351.985 rotations). The 345-year eclipses reoccur with almost identical time of day, elevation, and celestial position.
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; [A.Jones, 2001]). Already al-Biruni
(Qanun VII.2.II) and Copernicus
(de revolutionibus IV.4) noted that the period of 4,267 moons is actually about 5 minutes longer than the value for the eclipse period that Ptolemy attributes to Hipparchus. However, the timing methods of the Babylonians had an error of no less than 8 minutes. 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 (see ΔT
) we estimate that the error in the assumed length of the synodic month was less than 0.2 seconds in the 4th century BC and less than 0.1 seconds in Hipparchus' time.
had at the end of the 3rd century BC proposed two models for lunar and planetary motion:
Apollonius demonstrated that these two models were in fact mathematically equivalent. However, all this was theory and had not been put to practice. Hipparchus was the first astronomer we know attempted to determine the relative proportions and actual sizes of these orbit
s.
Hipparchus devised a geometrical method to find the parameters from three positions of the Moon, at particular phases of its anomaly. In fact, he did this separately for the eccentric and the epicycle model. Ptolemy describes the details in the Almagest IV.11. Hipparchus used two sets of three lunar eclipse observations, which he carefully selected to satisfy the requirements. The eccentric model he fitted to these eclipses from his Babylonian eclipse list: 22/23 December 383 BC, 18/19 June 382 BC, and 12/13 December 382 BC. The epicycle model he fitted to lunar eclipse observations made in Alexandria at 22 September 201 BC, 19 March 200 BC, and 11 September 200 BC.
The somewhat weird numbers are due to the cumbersome unit he used in his chord table according to one group of historians, who explain their reconstruction's inability to agree with these four numbers as partly due to some sloppy rounding and calculation errors by Hipparchus, for which Ptolemy criticised him (he himself made rounding errors too). A simpler alternate reconstruction agrees with all four numbers. Anyway, Hipparchus found inconsistent results; he later used the ratio of the epicycle model (3122+1/2 : 247+1/2), which is too small (60 : 4;45 sexagesimal). Ptolemy established a ratio of 60 : 5+1/4. (The maximum angular deviation producible by this geometry is
, or about 5° 1', a figure that is sometimes therefore quoted as the equivalent of the Moon's equation of the center
in the Hipparchan model.)
, 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 of Samos
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 by 5 hours from the observation made on Alexandria
's large public equatorial ring
that same day (at 1 hour before noon): Hipparchus may have visited Alexandria but he did not make his equinox observations there; presumably he was on Rhodes (at nearly the same geographical longitude). He could have used the equatorial ring of his armillary sphere or another equatorial ring for these observations, but Hipparchus (and Ptolemy) knew that observations with these instruments are sensitive to a precise alignment with the equator
, so if he were restricted to an armillary, it would make more sense to use its meridian ring as a transit instrument. The problem with an equatorial ring (if an observer is naive enough to trust it very near dawn or dusk) is that atmospheric refraction
lifts the Sun significantly above the horizon: so for a northern hemisphere observer 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 and increases as it sets, so it may appear to move in the wrong direction with respect to the equator in the course of the day – as Ptolemy mentions. Ptolemy and Hipparchus apparently did not realize that refraction is the cause.) However, such details have doubtful relation to the data of either man, since there is no textual, scientific, or statistical ground for believing that their equinoxes were taken on an equatorial ring, which is useless for solstices in any case. Not one of two centuries of mathematical investigations of their solar errors has claimed to have traced them to the effect of refraction on use of an equatorial ring. Ptolemy claims his solar observations were on a transit instrument set in the meridian.
At the end of his career, Hipparchus wrote a book called Peri eniausíou megéthous ("On the Length of the Year") about his results. The established value for the tropical year
, introduced by Callippus
in or before 330 BC was 365 + 1/4 days. (Possibly from Babylonian sources, see above [???]. Speculating a Babylonian origin for the Callippic year is hard to defend, since Babylon did not observe solstices thus the only extant System B yearlength was based on Greek solstices. See below. Hipparchus' equinox observations gave varying results, but he himself points out (quoted in Almagest III.1(H195)) that the observation errors by himself and his predecessors may have been as large as 1/4 day. He used old solstice observations, and determined a difference of about one day in about 300 years. So he set the length of the tropical year to 365 + 1/4 - 1/300 days (= 365.24666... days = 365 days 5 hours 55 min, which differs from the actual value (modern estimate) of 365.24219... days = 365 days 5 hours 48 min 45 s by only about 6 min).
Between the solstice observation of Meton and his own, there were 297 years spanning 108,478 days. D.Rawlins noted that this implies a tropical year of 365.24579... days = 365 days;14,44,51 (sexagesimal; = 365 days + 14/60 + 44/602 + 51/603) and that this exact yearlength has been found on one of the few Babylonian clay tablets which explicitly specifies the System B month. This is an indication that Hipparchus' work was known to Chaldeans.
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) about 365.2565 days), but the difference with Hipparchus' value for the tropical year is consistent with his rate of precession
(see below).
s are not equal. Hipparchus made observations of equinox and solstice, and according to Ptolemy (Almagest III.4) determined that spring (from spring equinox to summer solstice) lasted 94½ days, and summer (from summer solstice to autumn equinox) 92½ days. This is inconsistent with a premise of the Sun moving around the Earth in a circle at uniform speed. Hipparchus' solution was to place the Earth not at the center of the Sun's motion, but at some distance from the center. This model described the apparent motion of the Sun fairly well. It is known today that the planet
s, including the Earth, move in appoximate ellipse
s 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 a little too large), and the direction of the apogee would be at longitude 65.5° from the vernal equinox. Hipparchus may also have used other sets of observations, which would lead to different values. One of his two eclipse trios' solar longitudes are consistent with his having initially adopted inaccurate lengths for spring and summer of 95¾ and 91¼ days. His other triplet of solar positions is consistent with 94¼ and 92½ days, an improvement on the results (94½ and 92½ days) attributed to Hipparchus by Ptolemy, which a few scholars still question the authorship of. Ptolemy made no change three centuries later, and expressed lengths for the autumn and winter seasons which were already implicit (as shown, e.g., by A. Aaboe).
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 Perí megethōn kaí apostēmátōn ("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 star
s), and the difference is greater when closer to the horizon. He knew that this is because in the then-current models the Moon circles the center of the Earth, but the observer is at the surface—the Moon, Earth and observer form a triangle with a sharp angle that changes all the time. From the size of this parallax, the distance of the Moon as measured in Earth radii
can be determined. For the Sun however, there was no observable parallax (we now know that it is about 8.8", several times smaller than the resolution of the unaided eye).
In the first book, Hipparchus assumes that the parallax of the Sun is 0, as if it is at infinite distance. He then analyzed a solar eclipse, which Toomer (against the opinion of over a century of astronomers) presumes to be the eclipse of 14 March 190 BC. It was total in the region of the Hellespont (and in fact in his birth place Nicaea); at the time Toomer proposes 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 4/5ths by the Moon. Alexandria and Nicaea are on the same meridian. Alexandria is at about 31° North, and the region of the Hellespont about 40° North. (It has been contended that authors like Strabo and Ptolemy had fairly decent values for these geographical positions, so Hipparchus must have known them too. However, Strabo's Hipparchus dependent latitudes for this region are at least 1° too high, and Ptolemy appears to copy them, placing Byzantium 2° high in latitude.) Hipparchus could draw a triangle formed by the two places and the Moon, and from simple geometry was able to establish a distance of the Moon, expressed in Earth radii. Because the eclipse occurred in the morning, the Moon was not in the meridian
, and it has been proposed that as a consequence the distance found by Hipparchus was a lower limit. In any case, according to Pappus, Hipparchus found that the least distance is 71 (from this eclipse), and the greatest 81 Earth radii.
In the second book, Hipparchus starts from the opposite extreme assumption: he assigns a (minimum) distance to the Sun of 490 Earth radii. This would correspond to a parallax of 7', which is apparently the greatest parallax that Hipparchus thought would not be noticed (for comparison: the typical resolution of the human eye is about 2'; Tycho Brahe
made naked eye observation with an accuracy down to 1'). In this case, the shadow of the Earth is a cone
rather than a cylinder
as under the first assumption. Hipparchus observed (at lunar eclipses) that at the mean distance of the Moon, the diameter of the shadow cone is 2+½ lunar diameters. That apparent diameter is, as he had observed, 360/650 degrees. With these values and simple geometry, Hipparchus could determine the mean distance; because it was computed for a minimum distance of the Sun, it is the maximum mean distance possible for the Moon. With his value for the eccentricity of the orbit, he could compute the least and greatest distances of the Moon too. According to Pappus, he found a least distance of 62, a mean of 67+1/3, and consequently a greatest distance of 72+2/3 Earth radii. With this method, as the parallax of the Sun decreases (i.e., its distance increases), the minimum limit for the mean distance is 59 Earth radii – exactly the mean distance that Ptolemy later derived.
Hipparchus thus had the problematic result that his minimum distance (from book 1) was greater than his maximum mean distance (from book 2). He was intellectually honest about this discrepancy, and probably realized that especially the first method is very sensitive to the accuracy of the observations and parameters. (In fact, modern calculations show that the size of the 190 BC solar eclipse at Alexandria must have been closer to 9/10ths and not the reported 4/5ths, a fraction more closely matched by the degree of totality at Alexandria of eclipses occurring in 310 BC and 129 BC which were also nearly total in the Hellespont and are thought by many to be more likely possibilities for the eclipse Hipparchus used for his computations.)
Ptolemy later measured the lunar parallax directly (Almagest V.13), and used the second method of Hipparchus with lunar eclipses to compute the distance of the Sun (Almagest V.15). He criticizes Hipparchus for making contradictory assumptions, and obtaining conflicting results (Almagest V.11): but apparently he failed to understand Hipparchus' strategy to establish limits consistent with the observations, rather than a single value for the distance. His results were the best so far: the actual mean distance of the Moon is 60.3 Earth radii, within his limits from Hipparchus' second book.
Theon of Smyrna
wrote that according to Hipparchus, the Sun is 1,880 times the size of the Earth, and the Earth twenty-seven times the size of the Moon; apparently this refers to volume
s, not diameter
s. From the geometry of book 2 it follows that the Sun is at 2,550 Earth radii, and the mean distance of the Moon is 60½ radii. Similarly, Cleomedes
quotes Hipparchus for the sizes of the Sun and Earth as 1050:1; this leads to a mean lunar distance of 61 radii. Apparently Hipparchus later refined his computations, and derived accurate single values that he could use for predictions of solar eclipses.
See [Toomer 1974] for a more detailed discussion.
(Naturalis Historia II.X) tells us that Hipparchus demonstrated that lunar eclipses can occur five months apart, and solar eclipses seven months (instead of the usual six months); and the Sun can be hidden twice in thirty days, but as seen by different nations. Ptolemy discussed this a century later at length in Almagest VI.6. The geometry, and the limits of the positions of Sun and Moon when a solar or lunar eclipse is possible, are explained in Almagest VI.5. Hipparchus apparently made similar calculations. The result that two solar eclipses can occur one month apart is important, because this can not be based on observations: one is visible on the northern and the other on the southern hemisphere – as Pliny indicates – and the latter was inaccessible to the Greek.
Prediction of a solar eclipse, i.e., exactly when and where it will be visible, requires a solid lunar theory and proper treatment of the lunar parallax. Hipparchus must have been the first to be able to do this. A rigorous treatment requires spherical trigonometry
, thus those who remain certain that Hipparchus lacked it must speculate that he may have made do with planar approximations. He may have discussed these things in Perí tēs katá plátos mēniaías tēs selēnēs kinēseōs ("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 26 November 139 BC, when over a clean sea horizon as seen from Rhodes, the Moon was eclipsed in the northwest just after the Sun rose in the southeast. This would be the second eclipse of the 345-year interval that Hipparchus used to verify the traditional Babylonian periods: this puts a late date to the development of Hipparchus' lunar theory. We do not know what "exact reason" Hipparchus found for seeing the Moon eclipsed while apparently it was not in exact opposition to the Sun. Parallax lowers the altitude of the luminaries; refraction raises them, and from a high point of view the horizon is lowered.
, the astrolabe
, and the armillary sphere
.
Hipparchus is credited with the invention or improvement of several astronomical instruments, which were used for a long time for naked-eye observations. According to Synesius
of Ptolemais (4th century) he made the first astrolabion: this may have been 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 a scaphe.
Ptolemy mentions (Almagest V.14) that he used a similar instrument as Hipparchus, called dioptra
, to measure the apparent diameter of the Sun and Moon. Pappus of Alexandria
described it (in his commentary on the Almagest of that chapter), as 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 equinox
es, 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.
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. Hipparchus wrote a critique in three 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 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 eclipse
s (Strabo Geografia 1.1.12). A lunar eclipse is visible simultaneously on half of the Earth, and the difference in longitude between places can be computed from the difference in local time when the eclipse is observed. His approach would give accurate results if it were correctly carried out but the limitations of timekeeping accuracy in his era made this method impractical.
s 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 Aristillus
. For more information see Discovery of precession.
Previously, Eudoxus of Cnidus
in the 4th century BC had described the stars and constellations in two books called Phaenomena and Entropon. Aratus
wrote a poem called Phaenomena or Arateia based on Eudoxus' work. Hipparchus wrote a commentary on the Arateia – his only preserved work – which contains many stellar positions and times for rising, culmination, and setting of the constellations, and these are likely to have been based on his own measurements.
Hipparchus made his measurements with an armillary sphere
, and obtained the positions of at least 850 stars. It is disputed which coordinate system(s) he used. Ptolemy's catalog in the Almagest
, which is derived from Hipparchus' catalog, is given in ecliptic coordinates
. However Delambre in his Histoire de l'Astronomie Ancienne (1817) concluded that Hipparchus knew and used 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 Eudoxos he provides stars' polar distance (equivalent to the declination
in the equatorial system), right ascension (equatorial), longitude (ecliptical), polar longitude (hybrid), but not celestial latitude.
As with most of his work, Hipparchus' star catalog was adopted and perhaps expanded by Ptolemy. Delambre, in 1817, cast doubt on Ptolemy's work. It was disputed whether the star catalog in the Almagest
is due to Hipparchus, but 1976–2002 statistical and spatial analyses (by R. R. Newton
, Dennis Rawlins
, Gerd Grasshoff, Keith Pickering and Dennis Duke) have shown conclusively that the Almagest
star catalog is almost entirely Hipparchan. Ptolemy has even (since Brahe, 1598) been accused by astronomers of fraud for stating (Syntaxis, book 7, chapter 4) that he observed all 1025 stars: for almost every star he used Hipparchus' data and precessed it to his own epoch 2⅔ centuries later by adding 2°40' to the longitude, using an erroneously small precession constant of 1° per century.
In any case the work started by Hipparchus has had a lasting heritage, and was much later updated by Al Sufi (964) and Copernicus (1543). Ulugh Beg
reobserved all the Hipparchus stars he could see from Samarkand in 1437 to about the same accuracy as Hipparchus's. The catalog was superseded only in the late 16th century by Brahe and Wilhelm IV of Kassel via superior ruled instruments and spherical trigonometry, which improved accuracy by an order of magnitude even before the invention of the telescope.
classes according to their brightness: he assigned the value of one to the twenty brightest stars, to fainter 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.
Hipparchus is known for being almost
universally recognized as discoverer of the precession
of the equinox
es. His two books on precession, On the Displacement of the Solsticial and Equinoctial Points and On the Length of the Year, are both mentioned in the Almagest
of Claudius Ptolemy
. According to Ptolemy, Hipparchus measured the longitude of Spica
and Regulus
and other bright stars. Comparing his measurements with data from his predecessors, Timocharis
and Aristillus
, he concluded 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.
's Hipparcos Space Astrometry Mission
was High Precision Parallax Collecting Satellite; it was deliberately named in this way to give an acronym, HiPParCoS, that echoed and commemorated the name of Hipparchus. The lunar crater Hipparchus
and the asteroid
4000 Hipparchus
are more directly named after him.
in Los Angeles, California, USA features a relief of Hipparchus as one of six of the greatest astronomers of all time and the only one from Antiquity.
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
astrologer
Astrologer
An astrologer practices one or more forms of astrology. Typically an astrologer draws a horoscope for the time of an event, such as a person's birth, and interprets celestial points and their placements at the time of the event to better understand someone, determine the auspiciousness of an...
, astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
, geographer
Geographer
A geographer is a scholar whose area of study is geography, the study of Earth's natural environment and human society.Although geographers are historically known as people who make maps, map making is actually the field of study of cartography, a subset of geography...
, and mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
of the Hellenistic period. He is considered the founder of trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
.
Hipparchus was born in Nicaea (now Iznik
Iznik
İznik is a city in Turkey which is primarily known as the site of the First and Second Councils of Nicaea, the first and seventh Ecumenical councils in the early history of the Church, the Nicene Creed, and as the capital city of the Empire of Nicaea...
, Turkey
Turkey
Turkey , known officially as the Republic of Turkey , is a Eurasian country located in Western Asia and in East Thrace in Southeastern Europe...
), and probably died on the island of Rhodes
Rhodes
Rhodes is an island in Greece, located in the eastern Aegean Sea. It is the largest of the Dodecanese islands in terms of both land area and population, with a population of 117,007, and also the island group's historical capital. Administratively the island forms a separate municipality within...
. He is known to have been a working astronomer at least from 162 to 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity
Classical antiquity
Classical antiquity is a broad term for a long period of cultural history centered on the Mediterranean Sea, comprising the interlocking civilizations of ancient Greece and ancient Rome, collectively known as the Greco-Roman world...
. He was the first whose quantitative and accurate models for the motion of the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
and Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
survive. For this he certainly made use of the observations and perhaps the mathematical techniques accumulated over centuries by the Chaldea
Chaldea
Chaldea or Chaldaea , from Greek , Chaldaia; Akkadian ; Hebrew כשדים, Kaśdim; Aramaic: ܟܐܠܕܘ, Kaldo) was a marshy land located in modern-day southern Iraq which came to briefly rule Babylon...
ns from Babylonia
Babylonia
Babylonia was an ancient cultural region in central-southern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in central-southern Mesopotamia (present-day Iraq), with Babylon as its capital. Babylonia emerged as...
. He developed trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
and constructed trigonometric
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
tables, and he solved several problems of spherical trigonometry
Spherical trigonometry
Spherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...
. With his solar and lunar
Lunar theory
Lunar theory attempts to account for the motions of the Moon. There are many irregularities in the Moon's motion, and many attempts have been made over a long history to account for them. After centuries of being heavily problematic, the lunar motions are nowadays modelled to a very high degree...
theories and his trigonometry, he may have been the first to develop a reliable method to predict solar eclipse
Solar eclipse
As seen from the Earth, a solar eclipse occurs when the Moon passes between the Sun and the Earth, and the Moon fully or partially blocks the Sun as viewed from a location on Earth. This can happen only during a new moon, when the Sun and the Moon are in conjunction as seen from Earth. At least...
s. His other reputed achievements include the discovery of Earth's precession, the compilation of the first comprehensive star catalog of the western world, and possibly the invention of the astrolabe
Astrolabe
An astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and longitude, surveying, triangulation, and to...
, also of the armillary sphere
Armillary sphere
An armillary sphere is a model of objects in the sky , consisting of a spherical framework of rings, centred on Earth, that represent lines of celestial longitude and latitude and other astronomically important features such as the ecliptic...
, which he used during the creation of much of the star catalogue. It would be three centuries before Claudius Ptolemaeus
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
' synthesis of astronomy would supersede the work of Hipparchus; it is heavily dependent on it in many areas.
Life and work
Relatively little of Hipparchus' direct work survives into modern times. Although he wrote at least fourteen books, only his commentary on the popular astronomical poem by AratusAratus
Aratus was a Greek didactic poet. He is best known today for being quoted in the New Testament. His major extant work is his hexameter poem Phaenomena , the first half of which is a verse setting of a lost work of the same name by Eudoxus of Cnidus. It describes the constellations and other...
was preserved by later copyists. Most of what is known about Hipparchus comes from Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
's (2nd century) Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
, with additional references to him by Pappus of Alexandria
Pappus of Alexandria
Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
and Theon of Alexandria
Theon of Alexandria
Theon was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's Elements and Ptolemy's Handy Tables, as well as writing various commentaries...
(c. 4th century AD) in their commentaries on the Almagest; from Strabo
Strabo
Strabo, also written Strabon was a Greek historian, geographer and philosopher.-Life:Strabo was born to an affluent family from Amaseia in Pontus , a city which he said was situated the approximate equivalent of 75 km from the Black Sea...
's Geographia ("Geography"), and from Pliny the Elder
Pliny the Elder
Gaius Plinius Secundus , better known as Pliny the Elder, was a Roman author, naturalist, and natural philosopher, as well as naval and army commander of the early Roman Empire, and personal friend of the emperor Vespasian...
's Naturalis historia ("Natural history") (1st century AD).
There is a strong tradition that Hipparchus was born in Nicaea (Greek Νίκαια), in the ancient district of Bithynia
Bithynia
Bithynia was an ancient region, kingdom and Roman province in the northwest of Asia Minor, adjoining the Propontis, the Thracian Bosporus and the Euxine .-Description:...
(modern-day Iznik in province Bursa
Bursa Province
Bursa Province is a province in western Turkey, along the Sea of Marmara. Its adjacent provinces are Balıkesir to the west, Kütahya to the south, Bilecik and Sakarya to the east, Kocaeli to the northeast and Yalova to the north. The province has an area of 11,043 km2 and a population of 3,187,000...
), in what today is the country Turkey
Turkey
Turkey , known officially as the Republic of Turkey , is a Eurasian country located in Western Asia and in East Thrace in Southeastern Europe...
.
The exact dates of his life are not known, but Ptolemy attributes to him astronomical observations in the period from 147 BC to 127 BC, and some of these are stated as made in Rhodes; earlier observations since 162 BC might also have been made by him. His birth date (c. 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 observations from that year. Hipparchus obtained information from Alexandria
Alexandria
Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
as well as Babylon
Babylon
Babylon was an Akkadian city-state of ancient Mesopotamia, the remains of which are found in present-day Al Hillah, Babil Province, Iraq, about 85 kilometers south of Baghdad...
, but it is not known when or if he visited these places. He is believed to have died on the island of Rhodes
Rhodes
Rhodes is an island in Greece, located in the eastern Aegean Sea. It is the largest of the Dodecanese islands in terms of both land area and population, with a population of 117,007, and also the island group's historical capital. Administratively the island forms a separate municipality within...
, where he seems to have spent most of his later life.
It is not known what Hipparchus' economic means were nor how he supported his scientific activities. His appearance is likewise unknown: there are no contemporary portraits. In the 2nd and 3rd centuries coin
Coin
A coin is a piece of hard material that is standardized in weight, is produced in large quantities in order to facilitate trade, and primarily can be used as a legal tender token for commerce in the designated country, region, or territory....
s were made in his honour in Bithynia
Bithynia
Bithynia was an ancient region, kingdom and Roman province in the northwest of Asia Minor, adjoining the Propontis, the Thracian Bosporus and the Euxine .-Description:...
that bear his name and show him with a globe
Globe
A globe is a three-dimensional scale model of Earth or other spheroid celestial body such as a planet, star, or moon...
; this supports the tradition that he was born there.
Hipparchus is thought to be the first to calculate a heliocentric system, but he abandoned his work because the calculations showed the orbits were not perfectly circular as believed to be mandatory by the science of the time. As an astronomer of antiquity his influence, supported by Aristotle, held sway for nearly 2000 years, until the heliocentric model of Copernicus
Copernican heliocentrism
Copernican heliocentrism is the name given to the astronomical model developed by Nicolaus Copernicus and published in 1543. It positioned the Sun near the center of the Universe, motionless, with Earth and the other planets rotating around it in circular paths modified by epicycles and at uniform...
.
Hipparchus' only preserved work is Τῶν Ἀράτου καὶ Εὐδόξου φαινομένων ἐξήγησις ("Commentary on the Phaenomena of Eudoxus and Aratus"). This is a highly critical commentary in the form of two books on a popular poem by Aratus
Aratus
Aratus was a Greek didactic poet. He is best known today for being quoted in the New Testament. His major extant work is his hexameter poem Phaenomena , the first half of which is a verse setting of a lost work of the same name by Eudoxus of Cnidus. It describes the constellations and other...
based on the work by Eudoxus
Eudoxus of Cnidus
Eudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy...
. Hipparchus also made a list of his major works, which apparently mentioned about fourteen books, but which is only known from references by later authors. His famous star catalog was incorporated into the one by Ptolemy, and may be almost perfectly reconstructed by subtraction of two and two thirds degrees from the longitudes of Ptolemy's stars.The first trigonometric table was apparently compiled by Hipparchus, who is now consequently known as "the father of trigonometry.
Modern speculation
Hipparchus was in the international news in 2005, when it was again proposed (as in 1898) that the data on the celestial globe of Hipparchus or in his star catalog may have been preserved in the only surviving large ancient celestial globe which depicts the constellations with moderate accuracy, the globe carried by the Farnese AtlasFarnese Atlas
The Farnese Atlas is a 2nd-century Roman marble copy of a Hellenistic sculpture of Atlas kneeling with a globe weighing heavily on his shoulders. It is the oldest extant statue of the Titan of Greek mythology, who is represented in earlier vase-painting, and more important, the oldest known...
. There are a variety of mis-steps in the more ambitious 2005 paper, thus no specialists in the area accept its widely publicized speculation.
Lucio Russo
Lucio Russo
Lucio Russo is an Italian physicist, mathematician and historian of science. Born in Venice, he teaches at the Mathematics Department of the Science College in the University of Rome Tor Vergata....
has said that Plutarch
Plutarch
Plutarch then named, on his becoming a Roman citizen, Lucius Mestrius Plutarchus , c. 46 – 120 AD, was a Greek historian, biographer, essayist, and Middle Platonist known primarily for his Parallel Lives and Moralia...
, in his work On the Face in the Moon, was reporting some physical theories that we consider to be Newtonian and that these may have come originally from Hipparchus; he goes on to say that Newton may have been influenced by them. Both of these claims have been rejected by other scholars.
A line in Plutarch
Plutarch
Plutarch then named, on his becoming a Roman citizen, Lucius Mestrius Plutarchus , c. 46 – 120 AD, was a Greek historian, biographer, essayist, and Middle Platonist known primarily for his Parallel Lives and Moralia...
's Table Talk states that Hipparchus counted 103049 compound propositions that can be formed from ten simple propositions; 103049 is the tenth Schröder–Hipparchus number
Schröder–Hipparchus number
In number theory, the Schröder–Hipparchus numbers form an integer sequence that can be used to count the number of plane trees with a given set of leaves, the number of ways of inserting parentheses into a sequence, and the number of ways of dissecting a convex polygon into smaller polygons by...
and this line has led to speculation that Hipparchus knew about enumerative combinatorics
Enumerative combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations...
, a field of mathematics that developed independently in modern mathematics.
Babylonian sources
Earlier Greek astronomers and mathematicians were influenced by Babylonian astronomy to some extent, for instance the period relations of the Metonic cycleMetonic cycle
In astronomy and calendar studies, the Metonic cycle or Enneadecaeteris is a period of very close to 19 years which is remarkable for being very nearly a common multiple of the solar year and the synodic month...
and Saros cycle
Saros cycle
The saros is a period of 223 synodic months , that can be used to predict eclipses of the Sun and Moon. One saros after an eclipse, the Sun, Earth, and Moon return to approximately the same relative geometry, and a nearly identical eclipse will occur, in what is referred to as an eclipse cycle...
may have come from Babylonian sources. Hipparchus seems to have been the first to exploit Babylonian astronomical knowledge and techniques systematically. Except for Timocharis
Timocharis
Timocharis of Alexandria was a Greek astronomer and philosopher. Likely born in Alexandria, he was a contemporary of Euclid....
and Aristillus
Aristillus
Aristillus was a Greek astronomer, presumably of the school of Timocharis . He was among the earliest meridian-astronomy observers....
, he was the first Greek known to divide the circle in 360 degrees
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
of 60 arc minutes (Eratosthenes
Eratosthenes
Eratosthenes of Cyrene was a Greek mathematician, poet, athlete, geographer, astronomer, and music theorist.He was the first person to use the word "geography" and invented the discipline of geography as we understand it...
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 2.5°.
Hipparchus probably compiled a list of Babylonian astronomical observations; G. J. Toomer, a historian of astronomy, has suggested that Ptolemy's knowledge of eclipse records and other Babylonian observations in the Almagest came from a list made by Hipparchus. Hipparchus' use of Babylonian sources has always been known in a general way, because of Ptolemy's statements. However, Franz Xaver Kugler
Franz Xaver Kugler
Franz Xaver Kugler was a German chemist, mathematician, Assyriologist, and Jesuit priest.Kugler was born in Königsbach, Palatinate, then part of the Kingdom of Bavaria. He earned a Ph.D. in chemistry in 1885, and the following year he entered the Jesuits. By 1893 he had been ordained as a priest...
demonstrated that the synodic and anomalistic 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
Kidinnu
Kidinnu was a Chaldean astronomer and mathematician. Strabo of Amaseia called him Kidenas, Pliny the Elder Cidenas, and Vettius Valens Kidynas....
).
Hipparchus's long draconitic
Orbital period
The orbital period is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.There are several kinds of...
lunar period (5458 months = 5923 lunar nodal periods) also appears a few times in Babylonian records. But the only such tablet explicitly dated is post-Hipparchus so the direction of transmission is not secured
Dennis Rawlins
Dennis Rawlins is an American astronomer, historian, and publisher.-Polar controversies:While studying historical magnetic declination data in polar regions, Rawlins was surprised to find that there were no such data from the 1909 expedition of Robert E. Peary, eventually leading him to become...
.
Geometry, trigonometry, and other mathematical techniques
Hipparchus was recognized as the first mathematician known to have possessed a trigonometric table, which he needed when computing the eccentricity of the orbitOrbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
s of the Moon and Sun. He tabulated values for the chord
Chord (geometry)
A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...
function, which gives the length of the chord for each angle. He did this for a circle with a circumference of 21,600 and a radius (rounded) of 3438 units: this circle has a unit length of 1 arc minute along its perimeter. He tabulated the chords for angles with increments of 7.5°. In modern terms, the chord of an angle equals twice the sine
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
of half of the angle, i.e.:
- chord(A) = 2 sin(A/2).
He described the chord table in a work, now lost, called Tōn en kuklōi eutheiōn (Of Lines Inside a Circle) by Theon of Alexandria
Theon of Alexandria
Theon was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's Elements and Ptolemy's Handy Tables, as well as writing various commentaries...
(4th century) in his commentary on the Almagest I.10; some claim his table may have survived in astronomical treatises in India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
, for instance the Surya Siddhanta
Surya Siddhanta
The Surya Siddhanta is one of the earliest siddhanta in archeo-astronomy of the Hindus by an unknown author. It describes the archeo-astronomy theories, principles and methods of the ancient Hindus. This siddhanta is supposed to be the knowledge that the Sun god gave to an Asura called Maya. Asuras...
. Trigonometry was a significant innovation, because it allowed Greek astronomers to solve any triangle, and made it possible to make quantitative astronomical models and predictions using their preferred geometric techniques.
For his chord table Hipparchus must have used a better approximation for π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
than the one from Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
of between 3 + 1/7 and 3 + 10/71; perhaps he had the one later used by Ptolemy: 3;8:30 (sexagesimal) (Almagest VI.7); but it is not known if he computed an improved value himself.
Hipparchus could construct his chord table using the Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
and a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
known to Archimedes. He also might have developed and used the theorem in plane geometry called Ptolemy's theorem
Ptolemy's theorem
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral . The theorem is named after the Greek astronomer and mathematician Ptolemy...
, because it was proved by Ptolemy in his Almagest (I.10) (later elaborated on by Carnot
Lazare Carnot
Lazare Nicolas Marguerite, Comte Carnot , the Organizer of Victory in the French Revolutionary Wars, was a French politician, engineer, and mathematician.-Education and early life:...
).
Hipparchus was the first to show that the stereographic projection
Stereographic projection
The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...
is conformal
Conformal map
In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...
, and that it transforms circles on the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
that do not pass through the center of projection to circles on the plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
. This was the basis for the astrolabe
Astrolabe
An astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and longitude, surveying, triangulation, and to...
.
Besides geometry, Hipparchus also used arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
techniques developed by the Chaldea
Chaldea
Chaldea or Chaldaea , from Greek , Chaldaia; Akkadian ; Hebrew כשדים, Kaśdim; Aramaic: ܟܐܠܕܘ, Kaldo) was a marshy land located in modern-day southern Iraq which came to briefly rule Babylon...
ns. He was one of the first Greek mathematicians to do this, and in this way expanded the techniques available to astronomers and geographers.
There are several indications that Hipparchus knew spherical trigonometry, but the first surviving text of it is that of Menelaus of Alexandria
Menelaus of Alexandria
Menelaus of Alexandria was a Greek mathematician and astronomer, the first to recognize geodesics on a curved surface as natural analogs of straight lines.-Life and Works:...
in the 1st century, who on that basis is now commonly credited with its discovery. (Previous to the finding of the proofs of Menelaus a century ago, Ptolemy was credited with the invention of spherical trigonometry.) Ptolemy later used spherical trigonometry to compute things like the rising and setting points of the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...
, or to take account of the lunar parallax
Parallax
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines. The term is derived from the Greek παράλλαξις , meaning "alteration"...
. Hipparchus may have used a globe for these tasks, reading values off coordinate grids drawn on it, or he may have made approximations from planar geometry, or perhaps used arithmetical approximations developed by the Chaldeans. He might have used spherical trigonometry.
Aubrey Diller has shown that the clima calculations which Strabo
Strabo
Strabo, also written Strabon was a Greek historian, geographer and philosopher.-Life:Strabo was born to an affluent family from Amaseia in Pontus , a city which he said was situated the approximate equivalent of 75 km from the Black Sea...
preserved from Hipparchus were performed by spherical trigonometry with the sole accurate obliquity known to have been used by ancient astronomers, 23°40'. All thirteen clima figures agree with Diller's proposal. Further confirming his contention is the finding that the big errors in Hipparchus's longitude of Regulus
Regulus
Regulus is the brightest star in the constellation Leo and one of the brightest stars in the night sky, lying approximately 77.5 light years from Earth. Regulus is a multiple star system composed of four stars which are organized into two pairs...
and both longitudes of Spica
Spica
Spica is the brightest star in the constellation Virgo, and the 15th brightest star in the nighttime sky. It is 260 light years distant from Earth...
agree to a few minutes in all three instances with a theory that he took the wrong sign for his correction for parallax when using eclipses for determining stars' positions.
Lunar and solar theory
Motion of the Moon
Hipparchus also studied the motion of the MoonMoon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
and confirmed the accurate values for two periods of its motion that Chaldean astronomers certainly possessed before him, whatever their ultimate origin
Aristarchus of Samos
Aristarchus, or more correctly Aristarchos , was a Greek astronomer and mathematician, born on the island of Samos, in Greece. He presented the first known heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe...
. The traditional value (from Babylonian System B) for the mean synodic month is 29 days;31,50,8,20 (sexagesimal) = 29.5305941... days. Expressed as 29 days + 12 hours + 793/1080 hours this value has been used later in the Hebrew calendar
Hebrew calendar
The Hebrew calendar , or Jewish calendar, is a lunisolar calendar used today predominantly for Jewish religious observances. It determines the dates for Jewish holidays and the appropriate public reading of Torah portions, yahrzeits , and daily Psalm reading, among many ceremonial uses...
(possibly from Babylonian sources). The Chaldeans also knew that 251 synodic months = 269 anomalistic months. Hipparchus used a multiple of this period by a factor of 17, because that interval is also an eclipse period. The Moon also is close to an integer number of years (4267 moons : 4573 anomalistic periods : 4630.53 nodal periods : 4611.98 lunar orbits : 344.996 years : 344.982 solar orbits : 126,007.003 days : 126,351.985 rotations). The 345-year eclipses reoccur with almost identical time of day, elevation, and celestial position.
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; [A.Jones, 2001]). Already al-Biruni
Al-Biruni
Abū al-Rayḥān Muḥammad ibn Aḥmad al-BīrūnīArabic spelling. . The intermediate form Abū Rayḥān al-Bīrūnī is often used in academic literature...
(Qanun VII.2.II) and Copernicus
Nicolaus Copernicus
Nicolaus Copernicus was a Renaissance astronomer and the first person to formulate a comprehensive heliocentric cosmology which displaced the Earth from the center of the universe....
(de revolutionibus IV.4) noted that the period of 4,267 moons is actually about 5 minutes longer than the value for the eclipse period that Ptolemy attributes to Hipparchus. However, the timing methods of the Babylonians had an error of no less than 8 minutes. 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 (see ΔT
Delta T
ΔT, Delta T, delta-T, deltaT, or DT is the time difference obtained by subtracting Universal Time from Terrestrial Time : ΔT=TT−UT....
) we estimate that the error in the assumed length of the synodic month was less than 0.2 seconds in the 4th century BC and less than 0.1 seconds in Hipparchus' time.
Orbit of the Moon
It had been known for a long time that the motion of the Moon is not uniform: its speed varies. This is called its anomaly, and it repeats with its own period; the anomalistic month. The Chaldeans took account of this arithmetically, and used a table giving the daily motion of the Moon according to the date within a long period. The Greeks however preferred to think in geometrical models of the sky. Apollonius of PergaApollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...
had at the end of the 3rd century BC proposed two models for lunar and planetary motion:
- In the first, the Moon would move uniformly along a circle, but the Earth would be eccentric, i.e., at some distance of the center of the circle. So the apparent angular speed of the Moon (and its distance) would vary.
- The Moon itself would move uniformly (with some mean motion in anomaly) on a secondary circular orbit, called an epicycle, that itself would move uniformly (with some mean motion in longitude) over the main circular orbit around the Earth, called deferent; see deferent and epicycleDeferent and epicycleIn the Ptolemaic system of astronomy, the epicycle was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets...
.
Apollonius demonstrated that these two models were in fact mathematically equivalent. However, all this was theory and had not been put to practice. Hipparchus was the first astronomer we know attempted to determine the relative proportions and actual sizes of these orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
s.
Hipparchus devised a geometrical method to find the parameters from three positions of the Moon, at particular phases of its anomaly. In fact, he did this separately for the eccentric and the epicycle model. Ptolemy describes the details in the Almagest IV.11. Hipparchus used two sets of three lunar eclipse observations, which he carefully selected to satisfy the requirements. The eccentric model he fitted to these eclipses from his Babylonian eclipse list: 22/23 December 383 BC, 18/19 June 382 BC, and 12/13 December 382 BC. The epicycle model he fitted to lunar eclipse observations made in Alexandria at 22 September 201 BC, 19 March 200 BC, 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 according to one group of historians, who explain their reconstruction's inability to agree with these four numbers as partly due to some sloppy rounding and calculation errors by Hipparchus, for which Ptolemy criticised him (he himself made rounding errors too). A simpler alternate reconstruction agrees with all four numbers. Anyway, Hipparchus found inconsistent results; he later used the ratio of the epicycle model (3122+1/2 : 247+1/2), which is too small (60 : 4;45 sexagesimal). Ptolemy established a ratio of 60 : 5+1/4. (The maximum angular deviation producible by this geometry is
, or about 5° 1', a figure that is sometimes therefore quoted as the equivalent of the Moon's equation of the centerEquation of the center
For further closely related mathematical developments see also Two-body problem, also Gravitational two-body problem, also Kepler orbit, and Kepler problem...
in the Hipparchan model.)
Apparent motion of the Sun
Before Hipparchus, Meton, EuctemonEuctemon
Euctemon was an Athenian astronomer. He was a contemporary of Meton and worked closely with this astronomer. Little is known of his work apart from his partnership with Meton and what is mentioned by Ptolemy...
, and their pupils at Athens
Athens
Athens , is the capital and largest city of Greece. Athens dominates the Attica region and is one of the world's oldest cities, as its recorded history spans around 3,400 years. Classical Athens was a powerful city-state...
had made a solstice observation (i.e., timed the moment of the summer solstice
Solstice
A solstice is an astronomical event that happens twice each year when the Sun's apparent position in the sky, as viewed from Earth, reaches its northernmost or southernmost extremes...
) on June 27, 432 BC (proleptic Julian calendar
Proleptic Julian calendar
The proleptic Julian calendar is produced by extending the Julian calendar to dates preceding AD 4 when its quadrennial leap year stabilized. The leap years actually observed between its official implementation in 45 BC and AD 4 were erratic, see the Julian calendar article for details.A calendar...
). Aristarchus of Samos
Aristarchus of Samos
Aristarchus, or more correctly Aristarchos , was a Greek astronomer and mathematician, born on the island of Samos, in Greece. He presented the first known heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe...
is said to have done so in 280 BC, and Hipparchus also had an observation by Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
. Hipparchus himself observed the summer solstice in 135 BC, but he found observations of the moment of equinox
Equinox
An equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator...
more accurate, and he made many during his lifetime. Ptolemy gives an extensive discussion of Hipparchus' work on the length of the year in the Almagest III.1, and quotes many observations that Hipparchus made or used, spanning 162 BC to 128 BC.
Ptolemy quotes an equinox timing by Hipparchus (at 24 March 146 BC at dawn) that differs by 5 hours from the observation made on Alexandria
Alexandria
Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
's large public equatorial ring
Equatorial ring
An equatorial ring was an astronomical instrument used in the Hellenistic world to determine the exact moment of the spring and autumn equinoxes...
that same day (at 1 hour before noon): Hipparchus may have visited Alexandria but he did not make his equinox observations there; presumably he was on Rhodes (at nearly the same geographical longitude). He could have used the equatorial ring of his armillary sphere or another equatorial ring for these observations, but Hipparchus (and Ptolemy) knew that observations with these instruments are sensitive to a precise alignment with the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....
, so if he were restricted to an armillary, it would make more sense to use its meridian ring as a transit instrument. The problem with an equatorial ring (if an observer is naive enough to trust it very near dawn or dusk) is that atmospheric refraction
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...
lifts the Sun significantly above the horizon: so for a northern hemisphere observer its apparent declination
Declination
In astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Declination in astronomy is comparable to geographic latitude, but projected onto the celestial sphere. Declination is measured in degrees north and...
is too high, which changes the observed time when the Sun crosses the equator. (Worse, the refraction decreases as the Sun rises and increases as it sets, so it may appear to move in the wrong direction with respect to the equator in the course of the day – as Ptolemy mentions. Ptolemy and Hipparchus apparently did not realize that refraction is the cause.) However, such details have doubtful relation to the data of either man, since there is no textual, scientific, or statistical ground for believing that their equinoxes were taken on an equatorial ring, which is useless for solstices in any case. Not one of two centuries of mathematical investigations of their solar errors has claimed to have traced them to the effect of refraction on use of an equatorial ring. Ptolemy claims his solar observations were on a transit instrument set in the meridian.
At the end of his career, Hipparchus wrote a book called Peri eniausíou megéthous ("On the Length of the Year") about his results. The established value for the tropical year
Tropical year
A tropical year , for general purposes, is the length of time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice...
, introduced by Callippus
Callippus
Callippus or Calippus was a Greek astronomer and mathematician.Callippus was born at Cyzicus, and studied under Eudoxus of Cnidus at the Academy of Plato. He also worked with Aristotle at the Lyceum, which means that he was active in Athens prior to Aristotle's death in 322...
in or before 330 BC was 365 + 1/4 days. (Possibly from Babylonian sources, see above [???]. Speculating a Babylonian origin for the Callippic year is hard to defend, since Babylon did not observe solstices thus the only extant System B yearlength was based on Greek solstices. See below. Hipparchus' equinox observations gave varying results, but he himself points out (quoted in Almagest III.1(H195)) that the observation errors by himself and his predecessors may have been as large as 1/4 day. He used old solstice observations, and determined a difference of about one day in about 300 years. So he set the length of the tropical year to 365 + 1/4 - 1/300 days (= 365.24666... days = 365 days 5 hours 55 min, which differs from the actual value (modern estimate) of 365.24219... days = 365 days 5 hours 48 min 45 s by only about 6 min).
Between the solstice observation of Meton and his own, there were 297 years spanning 108,478 days. D.Rawlins noted that this implies a tropical year of 365.24579... days = 365 days;14,44,51 (sexagesimal; = 365 days + 14/60 + 44/602 + 51/603) and that this exact yearlength has been found on one of the few Babylonian clay tablets which explicitly specifies the System B month. This is an indication that Hipparchus' work was known to Chaldeans.
Another value for the year that is attributed to Hipparchus (by the astrologer Vettius Valens
Vettius Valens
Vettius Valens was a 2nd-century Hellenistic astrologer, a somewhat younger contemporary of Claudius Ptolemy.Valens' major work is the Anthology, ten volumes in Greek written roughly within the period 150 to 175. The Anthology is the longest and most detailed treatise on astrology which has...
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
Sidereal year
A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the fixed stars. Hence it is also the time taken for the Sun to return to the same position with respect to the fixed stars after apparently travelling once around the ecliptic. It was equal to at noon 1 January...
(actual value at his time (modern estimate) about 365.2565 days), but the difference with Hipparchus' value for the tropical year is consistent with his rate of precession
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
(see below).
Orbit of the Sun
Before Hipparchus, astronomers knew that the lengths of the seasonSeason
A season is a division of the year, marked by changes in weather, ecology, and hours of daylight.Seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earth's axis relative to the plane of revolution...
s are not equal. Hipparchus made observations of equinox and solstice, and according to Ptolemy (Almagest III.4) determined that spring (from spring equinox to summer solstice) lasted 94½ days, and summer (from summer solstice to autumn equinox) 92½ days. This is inconsistent with a premise of the Sun moving around the Earth in a circle at uniform speed. Hipparchus' solution was to place the Earth not at the center of the Sun's motion, but at some distance from the center. This model described the apparent motion of the Sun fairly well. It is known today that the planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
s, including the Earth, move in appoximate ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
s around the Sun, but this was not discovered until Johannes Kepler
Johannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
published his first two laws of planetary motion in 1609. The value for the eccentricity attributed to Hipparchus by Ptolemy is that the offset is 1/24 of the radius of the orbit (which is a little too large), and the direction of the apogee would be at longitude 65.5° from the vernal equinox. Hipparchus may also have used other sets of observations, which would lead to different values. One of his two eclipse trios' solar longitudes are consistent with his having initially adopted inaccurate lengths for spring and summer of 95¾ and 91¼ days. His other triplet of solar positions is consistent with 94¼ and 92½ days, an improvement on the results (94½ and 92½ days) attributed to Hipparchus by Ptolemy, which a few scholars still question the authorship of. Ptolemy made no change three centuries later, and expressed lengths for the autumn and winter seasons which were already implicit (as shown, e.g., by A. Aaboe).
Distance, parallax, size of the Moon and Sun
Theon of Smyrna
Theon of Smyrna was a Greek philosopher and mathematician, whose works were strongly influenced by the Pythagorean school of thought. His surviving On Mathematics Useful for the Understanding of Plato is an introductory survey of Greek mathematics.-Life:Little is known about the life of Theon of...
(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
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population 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 star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s), and the difference is greater when closer to the horizon. He knew that this is because in the then-current models the Moon circles the center of the Earth, but the observer is at the surface—the Moon, Earth and observer form a triangle with a sharp angle that changes all the time. From the size of this parallax, the distance of the Moon as measured in Earth radii
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
can be determined. For the Sun however, there was no observable parallax (we now know that it is about 8.8", several times smaller than the resolution of the unaided eye).
In the first book, Hipparchus assumes that the parallax of the Sun is 0, as if it is at infinite distance. He then analyzed a solar eclipse, which Toomer (against the opinion of over a century of astronomers) presumes to be the eclipse of 14 March 190 BC. It was total in the region of the Hellespont (and in fact in his birth place Nicaea); at the time Toomer proposes the Romans were preparing for war with Antiochus III in the area, and the eclipse is mentioned by Livy
Livy
Titus Livius — known as Livy in English — was a Roman historian who wrote a monumental history of Rome and the Roman people. Ab Urbe Condita Libri, "Chapters from the Foundation of the City," covering the period from the earliest legends of Rome well before the traditional foundation in 753 BC...
in his Ab Urbe Condita
Ab urbe condita
Ab urbe condita is Latin for "from the founding of the City ", traditionally set in 753 BC. AUC is a year-numbering system used by some ancient Roman historians to identify particular Roman years...
VIII.2. It was also observed in Alexandria, where the Sun was reported to be obscured 4/5ths by the Moon. Alexandria and Nicaea are on the same meridian. Alexandria is at about 31° North, and the region of the Hellespont about 40° North. (It has been contended that authors like Strabo and Ptolemy had fairly decent values for these geographical positions, so Hipparchus must have known them too. However, Strabo's Hipparchus dependent latitudes for this region are at least 1° too high, and Ptolemy appears to copy them, placing Byzantium 2° high in latitude.) Hipparchus could draw a triangle formed by the two places and the Moon, and from simple geometry was able to establish a distance of the Moon, expressed in Earth radii. Because the eclipse occurred in the morning, the Moon was not in the meridian
Meridian (astronomy)
This article is about the astronomical concept. For other uses of the word, see Meridian.In the sky, a meridian is an imaginary great circle on the celestial sphere. It passes through the north point on the horizon, through the celestial pole, up to the zenith, through the south point on the...
, and it has been proposed that as a consequence the distance found by Hipparchus was a lower limit. In any case, according to Pappus, Hipparchus found that the least distance is 71 (from this eclipse), and the greatest 81 Earth radii.
In the second book, Hipparchus starts from the opposite extreme assumption: he assigns a (minimum) distance to the Sun of 490 Earth radii. This would correspond to a parallax of 7', which is apparently the greatest parallax that Hipparchus thought would not be noticed (for comparison: the typical resolution of the human eye is about 2'; Tycho Brahe
Tycho Brahe
Tycho Brahe , born Tyge Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations...
made naked eye observation with an accuracy down to 1'). In this case, the shadow of the Earth is a cone
Conical surface
In geometry, a conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex...
rather than a cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
as under the first assumption. Hipparchus observed (at lunar eclipses) that at the mean distance of the Moon, the diameter of the shadow cone is 2+½ lunar diameters. That apparent diameter is, as he had observed, 360/650 degrees. With these values and simple geometry, Hipparchus could determine the mean distance; because it was computed for a minimum distance of the Sun, it is the maximum mean distance possible for the Moon. With his value for the eccentricity of the orbit, he could compute the least and greatest distances of the Moon too. According to Pappus, he found a least distance of 62, a mean of 67+1/3, and consequently a greatest distance of 72+2/3 Earth radii. With this method, as the parallax of the Sun decreases (i.e., its distance increases), the minimum limit for the mean distance is 59 Earth radii – exactly the mean distance that Ptolemy later derived.
Hipparchus thus had the problematic result that his minimum distance (from book 1) was greater than his maximum mean distance (from book 2). He was intellectually honest about this discrepancy, and probably realized that especially the first method is very sensitive to the accuracy of the observations and parameters. (In fact, modern calculations show that the size of the 190 BC solar eclipse at Alexandria must have been closer to 9/10ths and not the reported 4/5ths, a fraction more closely matched by the degree of totality at Alexandria of eclipses occurring in 310 BC and 129 BC which were also nearly total in the Hellespont and are thought by many to be more likely possibilities for the eclipse Hipparchus used for his computations.)
Ptolemy later measured the lunar parallax directly (Almagest V.13), and used the second method of Hipparchus with lunar eclipses to compute the distance of the Sun (Almagest V.15). He criticizes Hipparchus for making contradictory assumptions, and obtaining conflicting results (Almagest V.11): but apparently he failed to understand Hipparchus' strategy to establish limits consistent with the observations, rather than a single value for the distance. His results were the best so far: the actual mean distance of the Moon is 60.3 Earth radii, within his limits from Hipparchus' second book.
Theon of Smyrna
Theon of Smyrna
Theon of Smyrna was a Greek philosopher and mathematician, whose works were strongly influenced by the Pythagorean school of thought. His surviving On Mathematics Useful for the Understanding of Plato is an introductory survey of Greek mathematics.-Life:Little is known about the life of Theon of...
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 volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
s, not diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
s. From the geometry of book 2 it follows that the Sun is at 2,550 Earth radii, and the mean distance of the Moon is 60½ radii. Similarly, Cleomedes
Cleomedes
Cleomedes was a Greek astronomer who is known chiefly for his book On the Circular Motions of the Celestial Bodies.-Placing his work chronologically:...
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
PlinyPliny the Elder
Gaius Plinius Secundus , better known as Pliny the Elder, was a Roman author, naturalist, and natural philosopher, as well as naval and army commander of the early Roman Empire, and personal friend of the emperor Vespasian...
(Naturalis Historia II.X) tells us that Hipparchus demonstrated that lunar eclipses can occur five months apart, and solar eclipses seven months (instead of the usual six months); and the Sun can be hidden twice in thirty days, but as seen by different nations. Ptolemy discussed this a century later at length in Almagest VI.6. The geometry, and the limits of the positions of Sun and Moon when a solar or lunar eclipse is possible, are explained in Almagest VI.5. Hipparchus apparently made similar calculations. The result that two solar eclipses can occur one month apart is important, because this can not be based on observations: one is visible on the northern and the other on the southern hemisphere – as Pliny indicates – and the latter was inaccessible to the Greek.
Prediction of a solar eclipse, i.e., exactly when and where it will be visible, requires a solid lunar theory and proper treatment of the lunar parallax. Hipparchus must have been the first to be able to do this. A rigorous treatment requires spherical trigonometry
Spherical trigonometry
Spherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...
, thus those who remain certain that Hipparchus lacked it must speculate that he may have made do with planar approximations. He may have discussed these things in Perí tēs katá plátos mēniaías tēs selēnēs kinēseōs ("On the monthly motion of the Moon in latitude"), a work mentioned in the Suda
Suda
The Suda or Souda is a massive 10th century Byzantine encyclopedia of the ancient Mediterranean world, formerly attributed to an author called Suidas. It is an encyclopedic lexicon, written in Greek, with 30,000 entries, many drawing from ancient sources that have since been lost, and often...
.
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
Loeb Classical Library
The Loeb Classical Library is a series of books, today published by Harvard University Press, which presents important works of ancient Greek and Latin Literature in a way designed to make the text accessible to the broadest possible audience, by presenting the original Greek or Latin text on each...
330 p. 207). 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 Rhodes, the Moon was eclipsed in the northwest just after the Sun rose in the southeast. This would be the second eclipse of the 345-year interval that Hipparchus used to verify the traditional Babylonian periods: this puts a late date to the development of Hipparchus' lunar theory. We do not know what "exact reason" Hipparchus found for seeing the Moon eclipsed while apparently it was not in exact opposition to the Sun. Parallax lowers the altitude of the luminaries; refraction raises them, and from a high point of view the horizon is lowered.
Astronomical instruments and astrometry
Hipparchus and his predecessors used various instruments for astronomical calculations and observations, such as the gnomonGnomon
The gnomon is the part of a sundial that casts the shadow. Gnomon is an ancient Greek word meaning "indicator", "one who discerns," or "that which reveals."It has come to be used for a variety of purposes in mathematics and other fields....
, the astrolabe
Astrolabe
An astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and longitude, surveying, triangulation, and to...
, and the armillary sphere
Armillary sphere
An armillary sphere is a model of objects in the sky , consisting of a spherical framework of rings, centred on Earth, that represent lines of celestial longitude and latitude and other astronomically important features such as the ecliptic...
.
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
Synesius
Synesius , a Greek bishop of Ptolemais in the Libyan Pentapolis after 410, was born of wealthy parents, who claimed descent from Spartan kings, at Balagrae near Cyrene between 370 and 375.-Life:...
of Ptolemais (4th century) he made the first astrolabion: this may have been an armillary sphere
Armillary sphere
An armillary sphere is a model of objects in the sky , consisting of a spherical framework of rings, centred on Earth, that represent lines of celestial longitude and latitude and other astronomically important features such as the ecliptic...
(which Ptolemy however says he constructed, in Almagest V.1); or the predecessor of the planar instrument called astrolabe
Astrolabe
An astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and longitude, surveying, triangulation, and to...
(also mentioned by Theon of Alexandria
Theon of Alexandria
Theon was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's Elements and Ptolemy's Handy Tables, as well as writing various commentaries...
). With an astrolabe Hipparchus was the first to be able to measure the geographical latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
and time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
by observing stars. Previously this was done at daytime by measuring the shadow cast by a gnomon
Gnomon
The gnomon is the part of a sundial that casts the shadow. Gnomon is an ancient Greek word meaning "indicator", "one who discerns," or "that which reveals."It has come to be used for a variety of purposes in mathematics and other fields....
, or with the portable instrument known as a scaphe.
Dioptra
A dioptra is a classical astronomical and surveying instrument, dating from the 3rd century BCE. The dioptra was a sighting tube or, alternatively, a rod with a sight at both ends, attached to a stand...
, to measure the apparent diameter of the Sun and Moon. Pappus of Alexandria
Pappus of Alexandria
Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
described it (in his commentary on the Almagest of that chapter), as did Proclus
Proclus
Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
(Hypotyposis IV). It was a 4-foot rod with a scale, a sighting hole at one end, and a wedge that could be moved along the rod to exactly obscure the disk of Sun or Moon.
Hipparchus also observed solar equinox
Equinox
An equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator...
es, which may be done with an equatorial ring
Equatorial ring
An equatorial ring was an astronomical instrument used in the Hellenistic world to determine the exact moment of the spring and autumn equinoxes...
: its shadow falls on itself when the Sun is on the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....
(i.e., in one of the equinoctial points on the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...
), but the shadow falls above or below the opposite side of the ring when the Sun is south or north of the equator. Ptolemy quotes (in Almagest III.1 (H195)) a description by Hipparchus of an equatorial ring in Alexandria; a little further he describes two such instruments present in Alexandria in his own time.
Hipparchus applied his knowledge of spherical angles to the problem of denoting locations on the Earth's surface. Before him a grid system had been used by Dicaearchus
Dicaearchus
Dicaearchus of Messana was a Greek philosopher, cartographer, geographer, mathematician and author. Dicaearchus was Aristotle's student in the Lyceum. Very little of his work remains extant. He wrote on the history and geography of Greece, of which his most important work was his Life of Greece...
of Messana
Messina, Italy
Messina is the third largest city on the island of Sicily, Italy and the capital of the province of Messina. It has a population of about 250,000 inhabitants in the city proper and about 650,000 in the province...
, but Hipparchus was the first to apply mathematical rigor to the determination of the latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
and longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....
of places on the Earth. Hipparchus wrote a critique in three books on the work of the geographer Eratosthenes
Eratosthenes
Eratosthenes of Cyrene was a Greek mathematician, poet, athlete, geographer, astronomer, and music theorist.He was the first person to use the word "geography" and invented the discipline of geography as we understand it...
of Cyrene (3rd century BC), called Pròs tèn 'Eratosthénous geografían ("Against the Geography of Eratosthenes"). It is known to us from Strabo
Strabo
Strabo, also written Strabon was a Greek historian, geographer and philosopher.-Life:Strabo was born to an affluent family from Amaseia in Pontus , a city which he said was situated the approximate equivalent of 75 km from the Black Sea...
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
Geographic coordinate system
A geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position...
of different cities
City
A city is a relatively large and permanent settlement. Although there is no agreement on how a city is distinguished from a town within general English language meanings, many cities have a particular administrative, legal, or historical status based on local law.For example, in the U.S...
at lunar eclipse
Lunar eclipse
A lunar eclipse occurs when the Moon passes behind the Earth so that the Earth blocks the Sun's rays from striking the Moon. This can occur only when the Sun, Earth, and Moon are aligned exactly, or very closely so, with the Earth in the middle. Hence, a lunar eclipse can only occur the night of a...
s (Strabo Geografia 1.1.12). A lunar eclipse is visible simultaneously on half of the Earth, and the difference in longitude between places can be computed from the difference in local time when the eclipse is observed. His approach would give accurate results if it were correctly carried out but the limitations of timekeeping accuracy in his era made this method impractical.
Star catalog
Late in his career (possibly about 135 BC) Hipparchus compiled his star catalog, the original of which does not survive. He also constructed a celestial globe depicting the constellations, based on his observations. His interest in the fixed starFixed star
The fixed stars are celestial objects that do not seem to move in relation to the other stars of the night sky. Hence, a fixed star is any star except for the Sun. A nebula or other starlike object may also be called a fixed star. People in many cultures have imagined that the stars form pictures...
s may have been inspired by the observation of a supernova
Supernova
A supernova is a stellar explosion that is more energetic than a nova. It is pronounced with the plural supernovae or supernovas. Supernovae are extremely luminous and cause a burst of radiation that often briefly outshines an entire galaxy, before fading from view over several weeks or months...
(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
Timocharis
Timocharis of Alexandria was a Greek astronomer and philosopher. Likely born in Alexandria, he was a contemporary of Euclid....
and Aristillus
Aristillus
Aristillus was a Greek astronomer, presumably of the school of Timocharis . He was among the earliest meridian-astronomy observers....
. For more information see Discovery of precession.
Previously, Eudoxus of Cnidus
Eudoxus of Cnidus
Eudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy...
in the 4th century BC had described the stars and constellations in two books called Phaenomena and Entropon. Aratus
Aratus
Aratus was a Greek didactic poet. He is best known today for being quoted in the New Testament. His major extant work is his hexameter poem Phaenomena , the first half of which is a verse setting of a lost work of the same name by Eudoxus of Cnidus. It describes the constellations and other...
wrote a poem called Phaenomena or Arateia based on Eudoxus' work. Hipparchus wrote a commentary on the Arateia – his only preserved work – which contains many stellar positions and times for rising, culmination, and setting of the constellations, and these are likely to have been based on his own measurements.
Hipparchus made his measurements with an armillary sphere
Armillary sphere
An armillary sphere is a model of objects in the sky , consisting of a spherical framework of rings, centred on Earth, that represent lines of celestial longitude and latitude and other astronomically important features such as the ecliptic...
, and obtained the positions of at least 850 stars. It is disputed which coordinate system(s) he used. Ptolemy's catalog in the Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
, which is derived from Hipparchus' catalog, is given in ecliptic coordinates
Ecliptic coordinate system
The ecliptic coordinate system is a celestial coordinate system that uses the ecliptic for its fundamental plane. The ecliptic is the path that the sun appears to follow across the celestial sphere over the course of a year. It is also the intersection of the Earth's orbital plane and the celestial...
. However Delambre in his Histoire de l'Astronomie Ancienne (1817) concluded that Hipparchus knew and used the equatorial coordinate system
Equatorial coordinate system
The equatorial coordinate system is a widely-used method of mapping celestial objects. It functions by projecting the Earth's geographic poles and equator onto the celestial sphere. The projection of the Earth's equator onto the celestial sphere is called the celestial equator...
, 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
Ecliptic coordinate system
The ecliptic coordinate system is a celestial coordinate system that uses the ecliptic for its fundamental plane. The ecliptic is the path that the sun appears to follow across the celestial sphere over the course of a year. It is also the intersection of the Earth's orbital plane and the celestial...
and equatorial coordinates
Equatorial coordinate system
The equatorial coordinate system is a widely-used method of mapping celestial objects. It functions by projecting the Earth's geographic poles and equator onto the celestial sphere. The projection of the Earth's equator onto the celestial sphere is called the celestial equator...
: in his commentary on Eudoxos he provides stars' polar distance (equivalent to the declination
Declination
In astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Declination in astronomy is comparable to geographic latitude, but projected onto the celestial sphere. Declination is measured in degrees north and...
in the equatorial system), right ascension (equatorial), longitude (ecliptical), polar longitude (hybrid), but not celestial latitude.
As with most of his work, Hipparchus' star catalog was adopted and perhaps expanded by Ptolemy. Delambre, in 1817, cast doubt on Ptolemy's work. It was disputed whether the star catalog in the Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
is due to Hipparchus, but 1976–2002 statistical and spatial analyses (by R. R. Newton
Robert Russell Newton
Robert Russell Newton, also R. R. Newton was an American physicist, astronomer, and historian of science.Newton was Supervisor of the Applied Physics Laboratory at Johns Hopkins University. Newton was known for his book The Crime of Claudius Ptolemy . In Newton's view, Ptolemy was "the most...
, Dennis Rawlins
Dennis Rawlins
Dennis Rawlins is an American astronomer, historian, and publisher.-Polar controversies:While studying historical magnetic declination data in polar regions, Rawlins was surprised to find that there were no such data from the 1909 expedition of Robert E. Peary, eventually leading him to become...
, Gerd Grasshoff, Keith Pickering and Dennis Duke) have shown conclusively that the Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
star catalog is almost entirely Hipparchan. Ptolemy has even (since Brahe, 1598) been accused by astronomers of fraud for stating (Syntaxis, book 7, chapter 4) that he observed all 1025 stars: for almost every star he used Hipparchus' data and precessed it to his own epoch 2⅔ centuries later by adding 2°40' to the longitude, using an erroneously small precession constant of 1° per century.
In any case the work started by Hipparchus has had a lasting heritage, and was much later updated by Al Sufi (964) and Copernicus (1543). Ulugh Beg
Ulugh Beg
Ulugh Bek was a Timurid ruler as well as an astronomer, mathematician and sultan. His commonly-known name is not truly a personal name, but rather a moniker, which can be loosely translated as "Great Ruler" or "Patriarch Ruler" and was the Turkic equivalent of Timur's Perso-Arabic title Amīr-e...
reobserved all the Hipparchus stars he could see from Samarkand in 1437 to about the same accuracy as Hipparchus's. The catalog was superseded only in the late 16th century by Brahe and Wilhelm IV of Kassel via superior ruled instruments and spherical trigonometry, which improved accuracy by an order of magnitude even before the invention of the telescope.
Stellar magnitude
Hipparchus ranked stars in six magnitudeApparent magnitude
The apparent magnitude of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere...
classes according to their brightness: he assigned the value of one to the twenty brightest stars, to fainter 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 (146–130 BC)
- See also Precession (astronomy)
Hipparchus is known for being almost
Aristarchus of Samos
Aristarchus, or more correctly Aristarchos , was a Greek astronomer and mathematician, born on the island of Samos, in Greece. He presented the first known heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe...
universally recognized as discoverer of the precession
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
of the equinox
Equinox
An equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator...
es. His two books on precession, On the Displacement of the Solsticial and Equinoctial Points and On the Length of the Year, are both mentioned in the Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
of Claudius Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
. According to Ptolemy, Hipparchus measured the longitude of Spica
Spica
Spica is the brightest star in the constellation Virgo, and the 15th brightest star in the nighttime sky. It is 260 light years distant from Earth...
and Regulus
Regulus
Regulus is the brightest star in the constellation Leo and one of the brightest stars in the night sky, lying approximately 77.5 light years from Earth. Regulus is a multiple star system composed of four stars which are organized into two pairs...
and other bright stars. Comparing his measurements with data from his predecessors, Timocharis
Timocharis
Timocharis of Alexandria was a Greek astronomer and philosopher. Likely born in Alexandria, he was a contemporary of Euclid....
and Aristillus
Aristillus
Aristillus was a Greek astronomer, presumably of the school of Timocharis . He was among the earliest meridian-astronomy observers....
, he concluded that Spica had moved 2° relative to the autumnal equinox. He also compared the lengths of the tropical year
Tropical year
A tropical year , for general purposes, is the length of time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice...
(the time it takes the Sun to return to an equinox) and the sidereal
Sidereal year
A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the fixed stars. Hence it is also the time taken for the Sun to return to the same position with respect to the fixed stars after apparently travelling once around the ecliptic. It was equal to at noon 1 January...
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.
Named after Hipparchus
The rather cumbersome formal name for the ESAEuropean Space Agency
The European Space Agency , established in 1975, is an intergovernmental organisation dedicated to the exploration of space, currently with 18 member states...
's Hipparcos Space Astrometry Mission
Hipparcos
Hipparcos was a scientific mission of the European Space Agency , launched in 1989 and operated between 1989 and 1993. It was the first space experiment devoted to precision astrometry, the accurate measurement of the positions of celestial objects on the sky...
was High Precision Parallax Collecting Satellite; it was deliberately named in this way to give an acronym, HiPParCoS, that echoed and commemorated the name of Hipparchus. The lunar crater Hipparchus
Hipparchus (lunar crater)
Hipparchus is the degraded remnant of a lunar crater. It is located to the southeast of Sinus Medii, near the center of the visible Moon. To the south is the prominent crater Albategnius, and to the southwest lies Ptolemaeus, a feature of comparable dimensions to Hipparchus. Horrocks lies entirely...
and the asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
4000 Hipparchus
4000 Hipparchus
4000 Hipparchus is a main-belt asteroid discovered on January 4, 1989 by Seiji Ueda and Hiroshi Kaneda at Kushiro.- External links :*...
are more directly named after him.
Monument
The Astronomer's Monument at the Griffith ObservatoryGriffith Observatory
Griffith Observatory is in Los Angeles, California, United States. Sitting on the south-facing slope of Mount Hollywood in L.A.'s Griffith Park, it commands a view of the Los Angeles Basin, including downtown Los Angeles to the southeast, Hollywood to the south, and the Pacific Ocean to the southwest...
in Los Angeles, California, USA features a relief of Hipparchus as one of six of the greatest astronomers of all time and the only one from Antiquity.
General
- Biographical page at the University of CambridgeUniversity of CambridgeThe University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
- University of Cambridge's Page about Hipparchus' sole surviving work
- Biographical page at the University of OregonUniversity of Oregon-Colleges and schools:The University of Oregon is organized into eight schools and colleges—six professional schools and colleges, an Arts and Sciences College and an Honors College.- School of Architecture and Allied Arts :...
- Biography of Hipparchus on Fermat's Last Theorem Blog
- Pastore, Giovanni, ANTIKYTHERA E I REGOLI CALCOLATORI, Rome, 2006, privately published
- The Antikythera Calculator (Italian and English versions)
- Os Eclipses, AsterDomus website, portuguese
- Ancient Astronomy, Integers, Great Ratios, and Aristarchus
Precession
Celestial bodies
- M44 Praesepe at SEDS (University of ArizonaUniversity of ArizonaThe University of Arizona is a land-grant and space-grant public institution of higher education and research located in Tucson, Arizona, United States. The University of Arizona was the first university in the state of Arizona, founded in 1885...
): http://www.seds.org/messier/m/m044.html