Hipparchus.
Hipparchus or
Hipparch ( ; c. 190
BC – c. 120 BC) was a
Greek astronomer,
geographer,
and
mathematician of the
Hellenistic period.
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 147 BC 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
Chaldeans from
Babylonia.
He developed
trigonometry and
constructed
trigonometric tables, and
he has 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 of
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 first appeared
during his century and was used by him 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.
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
Aratus 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 (ca. 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
Νικαία), 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 (ca. 190 BC) was calculated by
Delambre based on clues in his
work. Hipparchus must have lived some time after 127 BC because he
analyzed and published his latest observations.
Hipparchus obtained
information from Alexandria as well as Babylon, but it is
not known 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
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 born there.
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.
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.
There is evidence, based on references by non-scientific writers
such as Plutarch, that Hipparchus was aware of some physical ideas
that we consider
Newtonian, and
some claim that Newton knew this.
Babylonian sources
Earlier Greek astronomers and mathematicians were influenced by
Babylonian astronomy to some extent, for instance the period
relations of the
Metonic cycle and
Saros cycle may have come from
Babylonian sources. Hipparchus seems to have been the first to
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. 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 draconitic months) 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.
Geometry, trigonometry, and other mathematical techniques
Hipparchus is recognized as the first mathematician known to have
possessed a
trigonometry table, which
he needed when computing the
eccentricity of the
orbits of the Moon and Sun. He tabulated values for
the
chord function, which gives the
length of the chord for each angle. He did this for a circle with a
circumference of 21,600 and a radius (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.:
- chord(A) = 2 sin(A/2).
He
described the chord table in a work, now lost, called Tōn en
kuklō_{i} 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
Chaldeans. He was one of the first Greek
mathematicians to do this, and in this way expanded the techniques
available to astronomers and geographers.
There 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. Or perhaps
he used spherical trigonometry.
Lunar and solar theory
Motion 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 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... d. Expressed as 29 days +
12 hours + 793/1080 hours this value has been used later
in the
Hebrew calendar (possibly
from Babylonian sources). The Chaldeans also knew that 251
synodic months = 269
anomalistic months. Hipparchus used an
extension of this period by a factor of 17, because after that
interval the Moon also would have a similar latitude, and it is
close to an integer number of years (345). Therefore, eclipses
would reappear under almost identical circumstances. The period is
126007 days 1 hour (rounded). Hipparchus could confirm
his computations by comparing eclipses from his own time
(presumably
27 January 141 BC and
26 November 139 BC according to [Toomer
1980]), with eclipses from Babylonian records 345 years earlier
(
Almagest IV.2; [A.Jones, 2001]). Already
al-Biruni (
Qanun VII.2.II) and
Copernicus (
de revolutionibus IV.4)
noted that the period of 4,267 lunations is actually about 5
minutes longer than the value for the eclipse period that Ptolemy
attributes to Hipparchus. However, the 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
Orbit of the Moon
It had been known for a long time that the motion of the Moon is
not uniform: its speed varies. This is called its
anomaly,
and it repeats with its own period; the
anomalistic month. The Chaldeans took
account of this arithmetically, and used a table giving the daily
motion of the Moon according to the date within a long period. The
Greeks however preferred to think in geometrical models of the sky.
Apollonius of Perga had at the
end of the 3rd century BC proposed two models for lunar and
planetary motion:
- In the first, the Moon would move uniformly along a circle, but
the Earth would be eccentric, i.e., at some distance of the center
of the circle. So the apparent angular speed of the Moon (and its
distance) would vary.
- The Moon itself would move uniformly (with some mean motion in
anomaly) on a secondary circular orbit, called 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
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 \arcsin ( 5.25/60 ), 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 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 5h from the observation
made on Alexandria's large public equatorial ring that same day (at 1h 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 numbing 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 refraction's effect on
use of an equatorial ring. And 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 + 44/60
^{2} +
51/60
^{3}) 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) ca. 365.2565 days), but the difference
with Hipparchus' value for the tropical year is consistent with his
rate of
precession (see below).
Orbit of the Sun
Before Hipparchus, astronomers knew that the lengths of the
seasons 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
planets,
including the Earth, move in
ellipses around
the Sun, but this was not discovered until
Johannes Kepler published his first two laws
of planetary motion in 1609). The value for the
eccentricity attributed to Hipparchus
by Ptolemy is that the offset is 1/24 of the radius of the orbit
(which is 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
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
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 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 B.C. 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 B.C. and 129
B.C. 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
volumes, not
diameters. From the geometry of book 2 it follows
that the Sun is at 2,550 Earth radii, and the mean distance of the
Moon is 60½ radii. Similarly,
Cleomedes
quotes Hipparchus for the sizes of the Sun and Earth as 1050:1;
this leads to a mean lunar distance of 61 radii. Apparently
Hipparchus later refined his computations, and derived accurate
single values that he could use for predictions of solar
eclipses.
See [Toomer 1974] for a more detailed discussion.
Eclipses
Pliny (
Naturalis Historia
II.X) tells us that Hipparchus demonstrated that lunar eclipses can
occur 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.
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 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
equinoxes,
which may be done with an
equatorial
ring: its shadow falls on itself when the Sun is on the
equator (i.e., in one of the equinoctial
points on the
ecliptic), but the shadow
falls above or below the opposite side of the ring when the Sun is
south or north of the equator. Ptolemy quotes (in
Almagest
III.1 (H195)) a description by Hipparchus of an equatorial ring in
Alexandria; a little further he describes two such instruments
present in Alexandria in his own time.
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 eclipses (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 stars may have been inspired by the
observation of a
supernova (according to
Pliny), or by his discovery of precession (according to Ptolemy,
who says that Hipparchus could not reconcile his data with earlier
observations made by
Timocharis and
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. Up until recently, it was heatedly
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 sixteenth 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
magnitude classes according to their
brightness: he assigned the value of one to the twenty brightest
stars, to weaker ones a value of two, and so forth to the stars
with a class of six, which can be barely seen with the naked eye. A
similar system is still used today.
Precession of the equinoxes (146–130 BC)
- See also Precession
Hipparchus is perhaps most famous for being
almost universally recognized as
discoverer of the
precession of the
equinoxes. His two books on precession,
On the Displacement of the Solsticial and Equinoctial
Points and
On the Length of the Year, are both
mentioned in the
Almagest of
Claudius
Ptolemy. According to Ptolemy,
Hipparchus measured the longitude of
Spica and
other bright stars. Comparing his measurements with data from his
predecessors,
Timocharis and
Aristillus, he 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.
Named after Hipparchus
The
ESA's Hipparcos Space Astrometry
Mission was named after him, as are the lunar crater Hipparchus and the asteroid 4000 Hipparchus.
Monument
The Astronomer's Monument at Griffith Observatory 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.
Notes
References
- Edition and translation: Karl Manitius: In Arati et Eudoxi
Phaenomena, Leipzig, 1894.
- J. Chapront, M. Chapront Touze, G. Francou (2002): "A new determination of lunar orbital parameters,
precession constant, and tidal acceleration from LLR
measurements". Astronomy and Astrophysics
387, 700-709.
- Duke, Dennis W. (2002). Associations between the ancient star
catalogs. Archive for the History of Exact Sciences
56(5):435-450. (Author's draft here.)
- A. Jones: "Hipparchus." In Encyclopedia of Astronomy and
Astrophysics. Nature Publishing Group, 2001.
- Patrick Moore (1994): Atlas of the Universe, Octopus
Publishing Group LTD (Slovene translation and completion by Tomaž
Zwitter and Savina Zwitter (1999): Atlas vesolja),
225.
- Newton, R.R. (1977). The Crime of Claudius Ptolemy.
Baltimore: Johns Hopkins University Press.
- Rawlins, Dennis (1982). An Investigation of the Ancient Star
Catalog. Proceedings of the Astronomical Society of the
Pacific 94, 359-373. Has been updated several times: DIO, volume 8, number 1 (1998), page 2, note
3, and DIO, volume 10 (2000), page 79, note
177.
- B.E. Schaefer (2005): "The Epoch of the Constellations on the Farnese
Atlas and their Origin in Hipparchus's Lost Catalogue".
Journal for the History of Astronomy
xxxvi, 1..29.
- J.M.Steele, F.R.Stephenson, L.V.Morrison (1997): "The accuracy of eclipse times measured by the
Babylonians". Journal for the History of Astronomy
xxviii, 337..345
- F.R. Stephenson, L.J.Fatoohi (1993): "Lunar Eclipse Times Recorded in Babylonian
History". Journal for the History of Astronomy
xxiv, 255..267
- N.M. Swerdlow (1969): "Hipparchus on the distance of the sun."
Centaurus 14, 287-305.
- G.J. Toomer (1967): "The Size of the Lunar Epicycle According
to Hipparchus." Centaurus 12,
145-150.
- G.J. Toomer (1973): "The Chord Table of Hipparchus and the
Early History of Greek Trigonometry." Centaurus
18, 6-28.
- G.J. Toomer (1974): "Hipparchus on the Distances of the Sun and
Moon." Archives for the History of the Exact Sciences
14, 126-142.
- G.J. Toomer (1978): "Hipparchus." In Dictionary of
Scientific Biography 15: 207-224.
- G.J. Toomer (1980): "Hipparchus' Empirical Basis for his Lunar
Mean Motions," Centaurus 24, 97-109.
- G.J. Toomer (1988): "Hipparchus and Babylonian Astronomy." In
A Scientific Humanist: Studies in Memory of Abraham Sachs,
ed. Erle Leichty, Maria deJ. Ellis, and Pamel Gerardi.
Philadelphia: Occasional Publications of the Samuel Noah Kramer
Fund, 9.
External links
General
Precession
Celestial bodies
Star catalog