Nicaea (/hɪˈpɑːrkəs/; Greek: Ἵππαρχος,
Hipparkhos; c. 190 – c. 120 BC) was a Greek
astronomer, geographer, and mathematician. He is considered the
founder of trigonometry but is most famous for his incidental
discovery of precession of the equinoxes.
Hipparchus was born in Nicaea,
Bithynia (now İznik, 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.
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
Babylonians and by
Meton of Athens
Meton of Athens (5th century BC), Timocharis,
Aristarchus of Samos
Aristarchus of Samos and Eratosthenes, among others. 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 eclipses. His other reputed
achievements include the discovery and measurement 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.
1 Life and work
1.1 Modern speculation
2 Babylonian sources
3 Geometry, trigonometry, and other mathematical techniques
4 Lunar and solar theory
4.1 Motion of the Moon
Orbit of the Moon
4.3 Apparent motion of the Sun
Orbit of the Sun
4.5 Distance, parallax, size of the
Moon and the Sun
5 Astronomical instruments and astrometry
6.1 Stellar magnitude
Precession of the equinoxes
Precession of the equinoxes (146–127 BC)
11 Editions and translations
14 Further reading
15 External links
15.3 Celestial bodies
Life and work
Relatively little of Hipparchus's direct work survives into modern
times. Although he wrote at least fourteen books, only his commentary
on the popular astronomical poem by
Aratus was preserved by later
copyists. Most of what is known about
Hipparchus comes from Strabo's
Pliny's Natural History
Pliny's Natural History in the 1st century; Ptolemy's
2nd-century Almagest; and additional references to him in the 4th
Pappus of Alexandria
Pappus of Alexandria and
Theon of Alexandria in their
commentaries on the Almagest.
There is a strong tradition that
Hipparchus was born in
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 147–127 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
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
It is not known what Hipparchus's 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 coins were made in his honour in
Bithynia that bear his name
and show him with a globe; this supports the tradition that he was
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 ideas from Aristotle, held sway for nearly 2000 years,
until the heliocentric model of Copernicus.
Hipparchus's 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
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".
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 Atlas. 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.
According to one book review, 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, which counts the
number of ways of adding one or more pairs of parentheses around
consecutive subsequences of two or more items in any sequence of ten
symbols. This has led to speculation that
Hipparchus knew about
enumerative combinatorics, a field of mathematics that developed
independently in modern mathematics.
Further information: Babylonian astronomy
Earlier Greek astronomers and mathematicians were influenced by
Babylonian astronomy to some extent, for instance the period relations
Metonic cycle and
Saros cycle may have come from Babylonian
sources (see "Babylonian astronomical diaries").
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's 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 (5,458 months = 5,923 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 settled by the tablets.
Hipparchus's draconitic lunar motion cannot be solved by the
lunar-four arguments that are sometimes proposed to explain his
anomalistic motion. A solution that has produced the exact 5,458/5,923
ratio is rejected by most historians though it uses the only anciently
attested method of determining such ratios, and it automatically
delivers the ratio's four-digit numerator and denominator. Hipparchus
initially used (
Almagest 6.9) his 141 BC eclipse with a Babylonian
eclipse of 720 BC to find the less accurate ratio 7160 synodic months
= 7,770 draconitic months, simplified by him to 716 = 777 through
division by 10. (He similarly found from the 345-year cycle the ratio
4267 synodic months = 4573 anomalistic months and divided by 17 to
obtain the standard ratio 251 synodic months = 269 anomalistic
months.) If he sought a longer time base for this draconitic
investigation he could use his same 141 BC eclipse with a moonrise
1245 BC eclipse from Babylon, an interval of 13,645 synodic months =
14,8807 1/2 draconitic months ≈ 14,623 1/2 anomalistic months.
Dividing by 5/2 produces 5458 synodic months = 5923 precisely. The
obvious main objection is that the early eclipse is unattested though
that is not surprising in itself and there is no consensus on whether
Babylonian observations were recorded this remotely. Though
Hipparchus's tables formally went back only to 747 BC, 600 years
before his era, the tables were actually good back to before the
eclipse in question because as only recently noted their use in
reverse is no more difficult than forwards.
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 orbits of the
Moon and Sun. He tabulated values
for the chord function, which gives the length of the chord for each
angle. He did this for a circle with a circumference of 21,600 and a
radius (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 the
radius times twice the sine of half of the angle, i.e.:
chord(A) = r(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 in
his 4th-century commentary on the
Almagest I.10; some claim his table
may have survived in astronomical treatises in India, for instance the
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.
But some scholars do not believe
Āryabhaṭa's sine table
Āryabhaṭa's sine table has
anything to do with Hipparchus's chord table which does not exist
today. Some scholars do not agree with this hypothesis that Hipparchus
constructed a chord table. Bo C. Klintberg states "With mathematical
reconstructions and philosophical arguments I show that Toomer's 1973
paper never contained any conclusive evidence for his claims that
Hipparchus had a 3438'-based chord table, and that the Indians used
that table to compute their sine tables. Recalculating Toomer's
reconstructions with a 3600' radius – i.e. the radius of the chord
table in Ptolemy's Almagest, expressed in 'minutes' instead of
'degrees' – generates Hipparchan-like ratios similar to those
produced by a 3438' radius. It is therefore possible that the radius
of Hipparchus's chord table was 3600', and that the Indians
independently constructed their 3438'-based sine table."
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
(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.
Hipparchus also used arithmetic techniques developed
by the Chaldeans. He was one of the first Greek mathematicians to do
this, and in this way expanded the techniques available to astronomers
There are several indications that
Hipparchus knew spherical
trigonometry, but the first surviving text of it is that of Menelaus
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
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
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
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.
Lunar and solar theory
Geometric construction used by
Hipparchus in his determination of the
distances to the sun and moon.
Motion of the Moon
Lunar theory and
Orbit of the Moon
Hipparchus also studied the motion of the
Moon and confirmed the
accurate values for two periods of its motion that Chaldean
astronomers are widely presumed to have 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. The Chaldeans also knew that 251 synodic months
≈ 269 anomalistic months.
Hipparchus used the multiple of this
period by a factor of 17, because that interval is also an eclipse
period, and is also 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). What was so exceptional and useful about the cycle was
that all 345-year-interval eclipse pairs occur slightly over 126,007
days apart within a tight range of only about ±1/2 hour, guaranteeing
(after division by 4267) an estimate of the synodic month correct to
one part in order of magnitude 10 million. The 345 year periodicity is
why the ancients could conceive of a mean month and quantify it so
accurately that it is even today correct to a fraction of a second of
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
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's 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
Moon according to the date within a long period. The Greeks
however preferred to think in geometrical models of the sky.
Apollonius of Perga
Apollonius of Perga had at the end of the 3rd century BC proposed
two models for lunar and planetary motion:
In the first, the
Moon would move uniformly along a circle, but the
Earth would be eccentric, i.e., at some distance of the center of the
circle. So the apparent angular speed of the
Moon (and its distance)
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 epicycle. Apollonius demonstrated that these two models
were in fact mathematically equivalent. However, all this was theory
and had not been put to practice.
Hipparchus was the first astronomer
we know attempted to determine the relative proportions and actual
sizes of these orbits.
Hipparchus devised a geometrical method to find the parameters from
three positions of the Moon, at particular phases of its anomaly. In
fact, he did this separately for the eccentric and the epicycle model.
Ptolemy describes the details in the
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 the arcsin of 5 1/4 divided by 60, 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.)
Apparent motion of the Sun
Before Hipparchus, Meton, Euctemon, and their pupils at
made a solstice observation (i.e., timed the moment of the summer
solstice) on 27 June 432 BC (proleptic Julian calendar).
Aristarchus of Samos
Aristarchus of Samos is said to have done so in 280 BC, and
Hipparchus also had an observation by Archimedes. As shown in a 1991
paper, in 158 BC
Hipparchus computed a very erroneous summer solstice
from Callippus's calendar. He observed the summer solstice in 146 and
135 BC both accurate to a few hours, but observations of the
moment of equinox were simpler, and he made twenty during his
Ptolemy gives an extensive discussion of Hipparchus's work
on the length of the year in the
Almagest III.1, and quotes many
Hipparchus made or used, spanning 162–128 BC.
Analysis of Hipparchus's seventeen equinox observations made at Rhodes
shows that the mean error in declination is positive seven arc
minutes, nearly agreeing with the sum of refraction by air and
Swerdlow's parallax. The random noise is two arc minutes or more
nearly one arcminute if rounding is taken into account which
approximately agrees with the sharpness of the eye.
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,
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
Sun crosses the equator. (Worse, the refraction decreases as
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
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
Recent expert translation and analysis by Anne Tihon of papyrus P.
Fouad 267 A has confirmed the 1991 finding cited above that Hipparchus
obtained a summer solstice in 158 BC But the papyrus makes the date
June 26, over a day earlier than the 1991 paper's conclusion for June
28. The earlier study's §M found that
Hipparchus did not adopt June
26 solstices until 146 BC when he founded the orbit of the sun which
Ptolemy later adopted. Dovetailing these data suggests Hipparchus
extrapolated the 158 BC June 26 solstice from his 145 solstice 12
years later a procedure that would cause only minuscule error. The
papyrus also confirmed that
Hipparchus had used Callippic solar motion
in 158 BC, a new finding in 1991 but not attested directly until P.
Fouad 267 A. Another table on the papyrus is perhaps for sidereal
motion and a third table is for Metonic tropical motion, using a
previously unknown year of 365 1/4 – 1/309 days. This was presumably
found by dividing the 274 years from 432 to 158 BC, into the
corresponding interval of 100077 days and 14 3/4 hours between Meton's
sunrise and Hipparchus's sunset solstices.
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. Speculating a
Babylonian origin for the Callippic year is hard to defend, since
Babylon did not observe solstices thus the only extant System B year
length was based on Greek solstices (see below). Hipparchus's equinox
observations gave varying results, but he himself points out (quoted
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,
including earth spin acceleration) in his time of about 365.2425 days,
an error of about 6 min per year, an hour per decade, 10 hours
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 year length has been found on one of the few Babylonian clay
tablets which explicitly specifies the System B month. This is an
indication that Hipparchus's work was known to Chaldeans.
Another value for the year that is attributed to
Hipparchus (by the
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's
value for the tropical year is consistent with his rate of precession
Orbit of the Sun
Before Hipparchus, astronomers knew that the lengths of the seasons
are not equal.
Hipparchus made observations of equinox and solstice,
and according to
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's 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 planets, including the Earth, move in approximate
ellipses around the Sun, but this was not discovered until Johannes
Kepler published his first two laws of planetary motion in 1609. The
value for the eccentricity attributed to
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 the Sun
Hipparchus on sizes and distances
Diagram used in reconstructing one of Hipparchus's methods of
determining the distance to the moon. This represents the earth-moon
system during a partial solar eclipse at A (Alexandria) and a total
solar eclipse at H (Hellespont).
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
Theon of Smyrna (2nd
century) mentions the work with the addition "of the
Sun and Moon".
Hipparchus measured the apparent diameters of the
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
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 =
Like others before and after him, he also noticed that the
Moon has a
noticeable parallax, i.e., that it appears displaced from its
calculated position (compared to the
Sun or stars), and the difference
is greater when closer to the horizon. He knew that this is because 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
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 his birthplace, 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 Libri VIII.2. It was also observed in Alexandria,
Sun was reported to be obscured 4/5ths by the Moon.
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
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,
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
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 189 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 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
Ptolemy later measured the lunar parallax directly (
and used the second method of
Hipparchus with lunar eclipses to
compute the distance of the
Almagest V.15). He criticizes
Hipparchus for making contradictory assumptions, and obtaining
conflicting results (
Almagest V.11): but apparently he failed to
understand Hipparchus's 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
60.3 Earth radii, within his limits from Hipparchus's 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 volumes, not diameters. From
the geometry of book 2 it follows that the
Sun is at 2,550 Earth
radii, and the mean distance of the
Moon is 60½ radii. Similarly,
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.
Pliny (Naturalis Historia II.X) tells us that
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
Moon when a solar
or lunar eclipse is possible, are explained in
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
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
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's lunar theory. We do not know what "exact
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 gnomon, 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 fixed stars. Previously this was done
at daytime by measuring the shadow cast by a gnomon, by recording the
length of the longest day of the year or with the portable instrument
known as a scaphe.
Equatorial ring of Hipparchus's time.
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
Pappus of Alexandria described it (in his commentary on
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
Hipparchus also observed solar equinoxes, which may be done with an
equatorial ring: its shadow falls on itself when the
Sun is on the
equator (i.e., in one of the equinoctial points on the ecliptic), but
the shadow falls above or below the opposite side of the ring when the
Sun is south or north of the equator.
Ptolemy quotes (in Almagest
III.1 (H195)) a description by
Hipparchus of an equatorial ring in
Alexandria; a little further he describes two such instruments present
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
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
Strabo Geografia 1 January 2012). 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.
Hipparchus holding his celestial globe, in Raphael's The School of
Athens (c. 1510)
Late in his career (possibly about 135 BC)
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 stars may have been
inspired by the observation of a supernova (according to Pliny), or by
his discovery of precession, according to Ptolemy, who says that
Hipparchus could not reconcile his data with earlier observations made
Timocharis and Aristillus. For more information see Discovery of
precession. In Raphael's painting The School of Athens,
depicted holding his celestial globe, as the representative figure for
Eudoxus of Cnidus in the 4th century BC had described
the stars and constellations in two books called Phaenomena and
Aratus wrote a poem called Phaenomena or Arateia based on
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's 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's 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's data and precessed it to his
own epoch 2 2⁄3 centuries later by adding 2°40' to the
longitude, using an erroneously small precession constant of 1° per
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.
Hipparchus is considered the
greatest observational astronomer from classical antiquity until
Hipparchus ranked stars in three magnitude very general classes
according to their brightness but he did not assign a numerical
brightness value to any star. The magnitude system
ranging from 1 (brightest) to 6 (faintest) was established by
Ptolemy. That system by
Ptolemy is effectively still
in use today, though extended and made more precise through the
introduction of a logarithmic scale by
N. R. Pogson
N. R. Pogson in 1856.[citation
Precession of the equinoxes
Precession of the equinoxes (146–127 BC)
Hipparchus is generally recognized as discoverer of the precession of
the equinoxes in 127 BC. 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
Regulus and other bright stars. Comparing his measurements
with data from his predecessors,
Timocharis and Aristillus, he
Spica had moved 2° relative to the autumnal equinox.
He also compared the lengths of the tropical year (the time it takes
Sun to return to an equinox) and the sidereal year (the time it
Sun to return to a fixed star), and found a slight
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.
Hipparchus's treatise "Against the Geography of Eratosthenes" in three
books is not preserved. Most of our knowledge of it comes from
Strabo, according to whom
Hipparchus thoroughly and often unfairly
Eratosthenes mainly for internal contradictions and
inaccuracy in determining positions of geographical localities.
Hipparchus insists that a geographic map must be based only on
astronomical measurements of latitudes and longitudes and
triangulation for finding unknown distances. In geographic theory and
Hipparchus introduced three main innovations. He was the
first to use the grade grid, to determine geographic latitude from
star observations, and not only from the sun’s altitude, a method
known long before him, and to suggest that geographic longitude could
be determined by means of simultaneous observations of lunar eclipses
in distant places. In the practical part of his work, the so-called
"table of climata",
Hipparchus listed latitudes for several tens of
localities. In particular, he improved Eratosthenes' values for the
latitudes of Athens, Sicily, and southern extremity of India. In
calculating latitudes of climata (latitudes correlated with the length
of the longest solstitial day),
Hipparchus used an unexpectedly
accurate value for the obliquity of the ecliptic, 23°40′ (the
actual value in the second half of the 2nd century BC was
approximately 23°43′), whereas all other ancient authors knew only
a roughly rounded value 24°, and even
Ptolemy used a less accurate
Hipparchus opposed the view generally accepted
Hellenistic period that the Atlantic and Indian Oceans and the
Caspian Sea are parts of a single ocean. At the same time he extends
the limits of the oikoumene, i.e. the inhabited part of the land, up
to the equator and the Arctic Circle. Hipparchus’ ideas found
their reflection in the Geography of Ptolemy. In essence, Ptolemy's
work is an extended attempt to realize Hipparchus’ vision of what
geography ought to be.
The rather cumbersome formal name for the ESA's
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
The Astronomer's Monument at the
Griffith Observatory in Los Angeles,
California, United States features a relief of
Hipparchus as one of
six of the greatest astronomers of all time and the only one from
Editions and translations
Berger H. Die geographischen Fragmente des Hipparch. Leipzig: B. G.
Dicks D.R. The Geographical Fragments of Hipparchus. Edited with an
Introduction and Commentary. London: Athlon Press, 1960. Pp. xi + 215.
Manitius K. In Arati et Eudoxi Phaenomena commentariorum libri tres.
Leipzig: B. G. Teubner, 1894. 376 S.
^ C. M. Linton (2004). From Eudoxus to Einstein: a history of
mathematical astronomy. Cambridge University Press. p. 52.
^ G J Toomer's chapter "
Ptolemy and his Greek Predecessors" in
"Astronomy before the Telescope", British Museum Press, 1996,
^ Stephen C. McCluskey (2000). Astronomies and cultures in early
medieval Europe. Cambridge University Press. p. 22.
^ Jones, Alexander Raymond (2017). Hipparchus. Encyclopedia
Brittanica, Inc. Retrieved 25 August 2017.
^ For general information on
Hipparchus see the following biographical
articles: G. J. Toomer, "Hipparchus" (1978); and A. Jones,
Hipparchus of Nicea". Ancient History Encyclopedia. Retrieved June
2016. Check date values in: access-date= (help)
^ Modern edition:
Karl Manitius (In Arati et Eudoxi Phaenomena,
^ D.Rawlins, "
Farnese Atlas Celestial Globe, Proposed Astronomical
^ B. E. Schaefer, "Epoch of the Constellations on the Farnese Atlas
and their Origin in Hipparchus's Lost Catalog", Journal for the
History of Astronomy, May, 2005 versus Dennis Duke Journal for the
History of Astronomy, February, 2006.
^ Lucio Russo, The Forgotten Revolution: How Science Was Born in
300 BCE and Why It Had To Be Reborn, (Berlin: Springer, 2004).
ISBN 3-540-20396-6, pp. 286–293.
^ Lucio Russo, The Forgotten Revolution: How Science Was Born in
300 BCE and Why It Had To Be Reborn, (Berlin: Springer, 2004).
ISBN 3-540-20396-6, pp. 365–379.
^ Mott Greene, "The birth of modern science?" Review of The Forgotten
Revolution, Nature 430 (5 August 2004): 614.
^ Stanley, Richard P. (1997), "Hipparchus, Plutarch, Schröder, and
Hough" (PDF), The American Mathematical Monthly, 104 (4): 344–350,
doi:10.2307/2974582, MR 1450667
^ Acerbi, F. (2003), "On the shoulders of Hipparchus: A reappraisal of
ancient Greek combinatorics" (PDF), Archive for History of Exact
Sciences, 57: 465–502, doi:10.1007/s00407-003-0067-0, archived from
the original (PDF) on 21 July 2011
^ For more information see G. J. Toomer, "
Hipparchus and Babylonian
^ Franz Xaver Kugler, Die Babylonische Mondrechnung ("The Babylonian
lunar computation"), Freiburg im Breisgau, 1900.
^ Aaboe, Asger (1955), "On the Babylonian origin of some Hipparchian
parameters", Centaurus, 4 (2): 122–125, Bibcode:1955Cent....4..122A,
doi:10.1111/j.1600-0498.1955.tb00619.x . On p. 124, Aaboe
identifies the Hipparchian equation 5458 syn. mo. = 5923 drac. mo.
with the equation of 1,30,58 syn. mo. = 1,38,43 drac. mo. (written in
sexagesimal) which he cites to p. 73. of Neugebauer's Astronomical
Cuneiform Texts, London 1955.
^ Pro & con arguments are given at DIO volume 11 number 1 article
3 sections C & D.
^ See demonstration of reverse use of Hipparchus's table for the 1245
^ Toomer, "The Chord Table of Hipparchus" (1973).
^ Reference: Hipparchus's 3600'-Based Chord Table and Its Place in the
Ancient Greek and Indian Trigonometry, Bo C. Klintberg,
Indian Journal of History of Science 40 (2):169-203 (2005)
^ Dennis Rawlins, "Aubrey Diller Legacies", DIO 5 (2009); Shcheglov
D.A. (2002-2007): "Hipparchus’ Table of Climata and Ptolemy’s
Geography", Orbis Terrarum 9 (2003–2007), 177–180.
^ Dennis Rawlins, "Hipparchos' Eclipse-Based Longitudes:
Regulus", DIO 16 (2009).
^ Detailed dissents on both values are presented in DIO volume 11
number 1 articles 1 & 3 and DIO volume 20 article 3 section L. See
also these analyses' summary.
^ These figures are for dynamical time, not the solar time of
Hipparchus's era. E.g., the true 4267 year interval was nearer 126,007
days plus a little over a 1/2 hour.
^ Footnote 18 of DIO 6 (1996).
^ Stephenson & Fatoohi 1993; Steele et al. 1997
^ Chapront et al. 2002
^ Summarized in Hugh Thurston (2002): Isis 93, 58–69.
^ Toomer, 1967
^ Explained at equation 25 of a recent investigation, paper #2.
^ Leverington, David (2003),
Babylon to Voyager and Beyond: A History
of Planetary Astronomy, Cambridge University Press, p. 30,
ISBN 9780521808408 .
^ DIO, volume 1, number 1, pages 49–66; A.Jones, 2001; Thurston, op.
cit., page 62
^ Thurston, op. cit., page 67, note 16. R. Newton proposed that
Hipparchus made an error of a degree in one of the trios' eclipses.
D.Rawlins's theory (Thurston op. cit.) that
Hipparchus analysed the
two trios in pairs not threesomes provides a possible explanation for
the one degree slip. It was a fudge necessitated by inadequacies of
analysing by pairs instead of using the better method
Almagest Book 4 Parts 6 and 11.
^ Ibid, note 14; Jones 2001
^ http://eclipse.gsfc.nasa.gov/SEcat5/SE-0199--0100.html, #04310, Fred
^ Swerdlow, N. M. (August 1992), "The Enigma of Ptolemy's Catalogue of
Stars", Journal for the History of Astronomy, 23: 173–183,
^ Gerd Grasshoff: The history of Ptolemy's star catalogue, Springer,
New York, 1990, ISBN 3-540-97181-5 (Analyse des im "Almagest"
^ "Keith Pickering" (PDF). Retrieved 6 August 2012.
^ "The Measurement Method of the
Almagest Stars", by Dennis Duke, DIO:
the International Journal of Scientific History,12 (2002).
^ Benson Bobrick, The Fated Sky, Simon & Schuster, 2005, p 151
^ Giorgio de Santillana & Hertha von Dechend, "Hamlet's Mill",
David R Godine, Boston, publisher, 1977, p 66
^ Alexander Jones "
Ptolemy in Perspective: Use and Criticism of his
Work from Antiquity to the Nineteenth Century, Springer, 2010, p.36.
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Hipparch. Leipzig: B. G. Teubner, 1869.; Dicks D.R. The Geographical
Fragments of Hipparchus. London: Athlon Press, 1960.
^ On Hipparchus's geography see: Berger H. Die geographischen
Fragmente des Hipparch. Leipzig: B. G. Teubner, 1869.; Dicks D.R. The
Geographical Fragments of Hipparchus. London: Athlon Press, 1960;
Neugebauer O. A History of Ancient Mathematical Astronomy. Pt. 1-3.
Berlin, Heidelberg, New York: Springer Verlag, 1975: 332–338;
Shcheglov D.A. Hipparchus’ "Table of Climata and Ptolemy’s
Geography". Orbis Terrarum 9. 2003–2007: 159–192.
^ Shcheglov D.A. "
Hipparchus on the
Latitude of Southern India".
Greek, Roman, and Byzantine Studies 45. 2005: 359–380; idem.
"Eratosthenes' Parallel of
Rhodes and the History of the System of
Climata". Klio 88. 2006: 351–359.; idem. "Hipparchus’ Table of
Climata and Ptolemy’s Geography". Orbis Terrarum 9. 2003–2007:
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Hipparchus and Posidonius". Klio 27.3: 258–269; cf. Shcheglov D.A.
"Hipparchus’ Table of Climata and Ptolemy’s Geography", 177–180.
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Latitude of Thule and the Map Projection
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O'Connor, John J.; Robertson, Edmund F., "Hipparchus", MacTutor
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Biographical page at the University of Cambridge
University of Cambridge's Page about Hipparchus's sole surviving work
Biographical page at the University of Oregon
Hipparchus on Fermat's Last
Hipparchus (c. 190 - c. 120 B.C.), SEDS
Os Eclipses, AsterDomus website, portuguese
Ancient Astronomy, Integers, Great Ratios, and Aristarchus
David Ulansey about Hipparchus's understanding of the precession
M44 Praesepe at SEDS, University of Arizona
A brief view by Carmen Rush on Hipparchus' stellar catalog
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