# User:Hephaestos/hipu

Hipparchus (Greek Ἳππαρχος) (circa 194 B.C. - circa 120 B.C.) was a Greek astronomer, mathematician and geographer.

An astrometric project, the Hipparcos Space Astrometry Mission of the European Space Agency (ESA), was named after him.

### Life and work

Hipparchus was born in Nicaea (Greek Νικαία), ancient district Bithynia, (modern-day Iznik) in province Bursa, in what today is Turkey.

The exact dates of his life are not known for sure, but he is believed to have made his observations from 162 B.C. to 127 B.C.. The date of his birth (circa 190 B.C.) was calculated by Delambre, based on clues in his work. We don't know anything about his youth either. Most of what is known about Hipparchus is from Ptolemy's Almagest (The Clock), from Strabo's Geographia (Geography) and from Pliny the Elder's Naturalis historia (Natural history). He probably studied in Alexandria.

Hipparchus is believed to have died on the island of Rhodes, where he spent most of his later life.

His main original works are lost. His only major preserved work is the (Commentary) on the Phaenomena of Eudoxus and Aratus or Commentary on Aratus, a critical commentary in 2 books on a popular poem by Aratus, which describes the constellations and the stars that form them. This work contains many measurements of stellar positions and was translated by Karl Manitius (In Arati et Eudoxi Phaenomena, Leipzig, 1894). Aside from this survives the list he made of his complete works, and apparently he wrote fourteen other major books, all presumably lost in the burning of the Great Royal Alexandrian Library in 642.

Hipparchus is recognized as the originator and father of scientific astronomy. He is believed to be the greatest Greek astronomic observer, and many regard him at the same time as the greatest astronomer of ancient times, although Cicero still gave preferences to Aristarchus of Samos. Some put in this place also Ptolemy of Alexandria.

### Brightness of stars

Hipparchus had in 134 B.C. ranked stars in six magnitude classes according to their brightness: he assigned the value of 1 to the 20 brightest stars, to weaker ones a value of 2, and so forth to the stars with a class of 6, which can be barely seen with the naked eyes. This was later adopted by Ptolemy, and modern astronomers with telescopes, photographic plates and other measuring devices for the light as they extended a luminosity with a density of light current j of a star on the Earth and put it on a qunatitative base. (See Apparent magnitude.) Observations with measuring devices for the light had shown that the density of light current of a star with a apparent magnitude 1m is hundred times greater of a star with a magnitude 6m. If we consider a property of an eye that a response is proportional with a logarithm of irritation, we get Pogson's physiological law (also called Pogson's ratio) from 1854 (other sources 1858):

${\displaystyle m-m_{0}=-2.5\log _{10}{\left({\frac {j}{j_{0}}}\right)}}$

### Astronomical instruments and astrometry

Hipparchus is credited with the invention or improvement of several astronomical instruments, which were used for a long time with naked-eye observations. About 150 B.C. he made the first astrolabion, which may have been an armillary sphere or the predecessor of a planar instrument astrolabe, which was improved in 3rd century by Arab astronomers and brought by them in Europe in 10th century. With an astrolabe Hipparchus was among the first 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, but the way this was used changed during his time. They put it in a metallic hemisphere, which was divided inside in concentric circles, and used it as a portable instrument, named scaphion, for determination of geographical coordinates from measured solar altitudes. With this instrument Eratosthenes of Cyrene 220 B.C. had measured the length of Earth's meridian, and after that they used this instrument to survey smaller regions as well. Ptolemy reported that Hipparchus invented an improved type of theodolite with which to measure angles.

Hipparchus had proposed to determine the geographical longitudes of several cities at solar eclipses. An eclipse does not occur simultaneously at all places on Earth, and their difference in longitude can be computed from the difference in time when the eclipse is observed. His method would give the most accurate data as would any previous one, if it would be correctly carried out. However, it was never properly applied, and for this reason maps remained rather inaccurate until modern times.

### Geometry and trigonometry

It is known that that Hipparchus compiled one of the first catalogue of stars. He also thought to have compiled the first trigonometry tables, when computing the eccentricity of the orbit of the Sun. He tabulated values for the chord function, which gave the length of the chord for each angle. In modern terms, the chord of an angle equals twice the sine of half of the angle, e.g., chord(A) = 2 sin(A/2).

He had a method of solving spherical triangles. The theorem in plane geometry called "Ptolemy's theorem" was developed by Hipparchus. This theorem was 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.

## Motion of the Earth

### Precession of the equinoxes (146 B.C.-130 B.C.)

Hipparchus is perhaps most famous for having been the first to measure the precession of the equinoxes. There is some suggestion that the Babylonians may have known about precession but it appears that Hipparchus was to first to really understand it and measure it. According to al-Battani Chaldean astronomers had distinguished the tropical and sidereal year. He stated they had around 330 B.C. an estimation for the length of the sidereal year to be SK = 365d 6h 11m (= 365.2576388d) with an error of (about) 110s. This phenomenon was probably also known to Kidinnu around 314 B.C.. A. Biot and Delambre attribute the discovery of precession also to old Chinese astronomers.

Hipparchus and his predecessors mostly used simple instruments for astronomical calculations, such as the gnomon, astrolabe, armillary sphere etc.

Before him Meton, Euktemon and their students had determined 440 B.C. (431 B.C., or June 27, 432 BC proleptic julian calendar) the two points of the solstice. Aristarchus of Samos is said to have done so in 280 B.C. Hipparchus found observations of the moment of equinox more accurate, and he observed many during his lifetime. Hipparchus on his own in Alexandria 146 B.C. determined the equinoctial point, using Archimedes' observations of solstices. Hipparchus himself made several observations of the solstices and equinoxes. From these observations a year later in 145 B.C. he also on his own determined the length of the tropical year to be TH = 365.24666...d = 365d 5h 55 m 12 s (365d + 1/4 - 1/300 = 365.24666...d = 365d 5h 55 m), which differs from the actual value (modern estimate) T = 365.24219...d = 365d 5h 48m 45s by only 6m 27s (6m 15s) (365.2423d = 365d 5h 48m, by only 7m).

We do not know the correct order for the precision of this value but most probably he was not able to make measurements within seconds so the correct value of his discovery was 365d 5h 55 m.

Between the solstice observation of Meton and his own, there were 297 years spanning 108478 days. This implies a tropical year of 365.24579... days = 365d;14,44,51 (sexagesimal; = 365d + 14/60 + 44/602 + 51/603), and this value has been found on a Babylonian clay tablet [A.Jones, 2001]. This is an indication that Hipparchus' work was known to Chaldeans.

Another value for the year that is attributed to Hipparchos is 365 + 1/4 + 1/288 days (= 365.253472... days = 365d6h5m), but this may be a corruption of another value attributed to a Babylonian source: 365 + 1/4 + 1/144 days (= 365.25694... days = 365d6h10m). It is not clear if this would be a value for the sidereal year (actual value at his time (modern estimate) 365.2565d).

Before him the Chaldean astronomers knew the lengths of seasons are not equal. Hipparchus made equinox and solstice observations, and according to Ptolemy (Almagest III.4) determined that spring (from spring equinox to summer solstice) lasted 94 + 1/2 days, and summer (from summer solstice to autumn equinox) 92 1/2 days. This is an unexpected result given a premise of the Sun moving around the Earth in a circle at uniform speed. His solution was to place the Earth not at the center of the Sun's motion, but at some distance from the center. Thus he introduced the concept of eccentricity. This model described the apparent motion of the Sun fairly well (of course today we know that the planets like the Earth move in ellipses around the Sun, but this was not discovered until Johannes Kepler published his first two laws of planetary motion in 1609). The value for the eccentricity attributed to him by Ptolemy is that the offset is 1/24 of the radius of the orbit (which is too large), and the direction of the apogee would be at longitude 65.5&degree; from the vernal equinox. Hipparchus may also have used another set of observations (94 1/4 and 92 3/4 days), which would lead to different values. The question remains if Hipparchus is really the author of the values provided by Ptolemy. With his estimation of the length of seasons he tried to determine, as of today, the eccentricity of Earth's orbit, and according to John Dreyer he got the incorrect value e = 0.04166 (which is too large).

After that from 141 B.C. to 126 B.C. he lived mostly on the island of Rhodes, again in Alexandria and in Siracuse, and around 130 B.C. in Babylon, during which period he made a lot of precise and lasting observations. When he measured the length of gnomon shadow at solstice he determined the length of tropical year and he was finding times of the known bright star sunsets and times of sunrises. From all of these measurements he found in 134 B.C. the length of sidereal year to be SH = 365d 6h 10m (365.2569444...d), which differs from today's S = 365.2563657...d = 365d 6h 9m 10s for 50s. Hipparchus also had measurements of the times of solstices from Aristarchus dating from 279 B.C. and from the school of Meton and Euctemon dating from 431 B.C.. This was a long enough period of time to allow him to calculate the difference between the length of the sidereal year and the tropical year, and led him to the discovery of precession. When he compared both lengths, he saw the tropical year is shorter for about 20 minutes from sidereal.

Additionally, as first in the history he correctly explained this with retrogradical movement of vernal point γ over the ecliptic for about 45", 46" or 47" (36" or 3/4' according to Ptolemy) per annum (today's value is Ψ'=50.387", 50.26") and he showed the Earth's axis is not fixed in space.

### Star catalogue

After that in 135 B.C., enthusiastic about a nova star in the constellation of Scorpius, he measured with an equatorial armillary sphere ecliptical coordinates of about 850 (falsely quoted elsewhere as 1600 or 1080) and in 129 B.C. he made first big star catalogue.

This map served him to find any changes on the sky but unfortunately it is not preserved today. His star map was thoroughly modified as late as 1000 years later in 964 by Al Sufi and 1500 years later (1437) by Ulugh Beg. Later, Halley would use his star catalogue to discover proper motions as well. His work speaks for itself. Another loss is that we know almost nothing about Hipparchus' life (this was stressed by Fred Hoyle).

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

By comparing his own measurements of the position of the equinoxes to the star Spica during a lunar eclipse at the time of equinox with those of Euclid's contemporaries (Timocharis of Alexandria (circa 320 B.C.-260 B.C.), Aristyllus 150 years earlier, the records of Chaldean astronomers (especially Kidinnu's records), and observations of a temple in Thebes, Egypt that was built in around 2000 BC) he still later observed that the equinox had moved 2° relative to Spica. He also noticed this motion in other stars. He obtained a value of not less than 1° in a century. The modern value is 1° in 72 years.

He also knew the works Phainomena (Phenomena)—Hipparchus' commentary contains many precise times for rising, culmination, and setting of the constellations treated inn the Phaenomena, and these are likely to have been based on measurements of stellar positions—and Enoptron (Mirror of Nature) of Eudoxus of Cnidus, who had near Cyzicus on the southern coast of the Sea of Marmara his school and through Aratus' astronomical epic poem Phenomena Eudoxus' sphere, which was made from metal or stone and where there were marked constellations, brightest stars, tropic of Cancer and tropic of Capricorn. These comparisons embarrassed him because he couldn't put together Eudoxus' detailed statements with his own observations and observations of that time. From all this he found that coordinates of the stars and the Sun had systematically changed. Their celestial latitudes λ remained unchanged, but their celestial longitudes β had reduced as would equinoctial points, intersections of ecliptic and celestial equator, move with progressive velocity every year for 1/100'.

After him many Greek and Arab astronomers had confirmed this phenomenon. Ptolemy compared his catalogue with those of Aristyllus, Timocharis, Hipparchus and the observations of Agrippa and Menelaus of Alexandria from the early 1st century and he finally confirmed Hipparchus' empirical fact that poles of the celestial equator in one Platonic year (approximately 25,777 sidereal years) encircle the ecliptical pole. The diameter of these cicles is equal to the inclination of ecliptic. The equinoctial points in this time traverse the whole ecliptic and they move 1° in a century. This velocity is equal to that calculated by Hipparchus. Because of these accordances Delambre, P. Tannery and other French historians of astronomy had wrongly jumped to conclusions that Ptolemy recorded his star catalogue from Hipparchus' with an ordinary extrapolation. This was not known until 1898 when Marcel Boll and others had found that Ptolemy's catalogue differs from Hipparchus' not only in the number of stars but in other respects.

This phenomenon was named by Ptolemy just because the vernal point γ leads the Sun. In Latin praecesse means "to overtake" or "to outpass", and today also means to twist or to turn. Its own name shows this phenomenon was discovered practically before its theoretical explanation, otherwise it would have been given a better term. Many later astronomers, physicists and mathematicians had occupied themselves with this problem, practically and theoretically. The phenomenon itself had opened many new promising solutions in several branches of celestial mechanics: Thabit ibn Qurra's theory of trepidation and oscillation of equinoctial points, Isaac Newton's general gravitational law (which had explained it in full), Leonhard Euler's kinematic equations and Joseph Lagrange's equations of motion, Jean d'Alembert's dynamical theory of the movement of the rigid body, some algebraic solutions for special cases of precession, John Flamsteed's and James Bradley's difficulties in the making of precise telescopic star catalogues, Friedrich Bessel's and Simon Newcomb's measurements of precession, and finally the precession of perihelion in Albert Einstein's General Theory of Relativity.

Lunisolar precession causes the motion of point γ by the ecliptic in the opposite direction of the apparent solar year's movement and the circulation of celestial pole. This circle becomes a spiral because of additional ascendancy of the planets. This is planetary precession where the ecliptical plane swings from its central position for ±4° in 60,000 years. The angle between ecliptic and celestial equator ε = 23° 26' is reducing for 0.47" per annum. Also, the point γ slides by equator for p = 0.108" per annum now in the same direction as the Sun. The sum of precessions gives an annual general precession in longitude Ψ = 50.288" which causes the origination of tropical year.

### Apparent motion of the Sun

Hipparchus also undertook to find the distances and sizes of the Sun and the Moon. He determined the Moon's horizontal parallax, which leads to a distance for the Moon.

### Distance to the Sun

...to be written ...

### Size of the Moon

...to be written ...

### Motion of the Moon

Hipparchus also studied the motion of the Moon and confirmed the accurate values for some periods of its motion that Chaldean astronomers (especially Kidinnu) had obtained before him, while obtaining more accurate measurents of other periods than had previously existed. The traditional value for the mean synodic month is 29d;31,50,28,20 (sexagesimal) = 29.53059429... d . Expressed as 29d + 12h + 793/1080 h this value has been used later in the Hebrew calendar (possibly from Babylonian sources). The Chaldeans also knew that 251 synodic months = 269 anomalystic months. Hipparchus extended this period by a factor of 17, because after that interval the Moon also would have a similar latitude, and it is close to an integer number of years (345). Therefore eclipses would reappear under almost identical circumstances. The period is 126007d1h (rounded). Hipparchus could confirm his computations by comparing eclipses from his own time (139 BC), with eclipses from Babylonian records 345 years earlier (Ptolemy, Almagest IV.2; [G.J.Toomer 1981]; [A.Jones 2001]). From modern ephemerides [Chapront et al. 2002] and taking account of the change in the length of the day (see Delta-T) we estimate that the error in the assumed length of the synodic month was less than 0.2s in the 4th BC and less than 0.1s in Hipparchus' time.

While undertaking to find the distances and sizes of the Sun and the Moon, Hipparchus determined the length of synodic month to 23/50s = 0.46s about 139 B.C. in Babylon according to Strabo of Amaseia in Pontus. He determined the Moon's horizontal parallax. He discovered the irregularity in lunar movement, which changes medium lunar longitude and today is called the equalization of the center with a value:

I = 377' sin m + 13' sin 2m,

where m is medium anomaly of the Moon.

### Celestial coordinate systems

Delambre in his Histoire de l'Astronomie Ancienne (1817) concluded that Hipparchus knew and used a real (celestial) equatorial coordinate system, directly with the right ascension and declination (or with its complement, polar distance). Later Otto Neugebauer (1899-1990) in his A History of Ancient Mathematical Astronomy (1975) rejected Delambre's claims.