The IAU hath decreed1:
A planet is a celestial body that
- is in orbit around the Sun,
- has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and
- has cleared the neighbourhood around its orbit.
Fiat. Fiat. Fiat.
Looking up at the lights in the night sky probably predates humanity, but it most likely took a peculiar human compulsion to deduce order from the Universe before anyone wondered what those lights were. Most lights rise in the east, circle about a single point in the sky, and set in the west. Throughout most of human history, these could not be observed to change position with respect to one another and were designated "fixed stars".
Some lights, however, weren't quite so constant. Two big ones, the Sun and the Moon, circled but at different speeds from the stars. Many grew long tails, passed once, and were never seen again, presaging some momentous events on earth. Some fell to the ground. A few wandered drunkenly across the sky, rising and setting at odd times, their paths appearing to shoot forward and double back relative to the fixed stars.
These last were constant enough to always appear, but erratic enough to provide information that had to be explained. The earliest explanations invariably came from religious mythologies created by cultures doing the explaining. The ancient Sumerians named them Enki, Inanna, Zib, Nanna, Utu, Gugulana, Enlil, and Ninurta after major deities. The Egyptians called the same lights Sabgu, Ba'ah, Aten, Aah, Seba-djai, Heru-deshret, Her-wepes-tawy, Her-ka-pet.
The ancient Greeks2 realized that two of the wanderers, Eosphoros ("the morning star") and Hesperos ("the evening star") were really the same body. Sometimes associated with Hera, it became confused with Aphrodite because of a myth where Eosphoros vies with Aphrodite for beauty. Thus, they knew of five such wanderers (planeta): Hermes, Aphrodite, Helios, Selene, Ares, Zeus, and Cronos. When the Romans covered their culture with a Greek veneer these became Mercury, Venus, Sol, Diana Lucifera (or just "Luna"), Mars, Jupiter, and Saturn.
The planets' wanderings were carefully charted by astrologers for casting horoscopes, but no-one attempted to explain their motions until Claudius Ptolemy. In his Almagest, which appeared around 150 AD, the seven planets went around the Earth, but the planets themselves followed curious circles (or "epicycles") imposed upon the major circles.
As more and more observations were made that did not fit Ptolemy's epicycles, new theoretical factors were added into the system, which became more and more complicated. In 1514 a Polish canon and physician, Micolaj Kopernik, gave a hand-written pamphlet to his friends in which he asserts that the Sun us at the center of the Universe, not the Earth. This and the underlying mathematical work developed into De revolutionibus orbium coelestium, published at his death in 1543.
Being the time of the Reformation, a monk named Giordano Bruno interpreted Copernicus' heliocentric theory as a refutation of Papal authority over the Universe. This did not please the Vatican, and they burned him at the stake for it in 1600, incidentally investing the Ptolemaic system as Church dogma and marking the Copernican system as heretical.
But real science was breaking out. In 1575, King Frederik II of (Lutheran) Denmark gave Danish astronomer Tycho Brahe his own little fiefdom on the island of Ven in which to carry out his obsession with astronomical observations. Brahe collected so much data that his student Johannes Kepler was able to realize that planetary orbits (including Earth's, but not the Moon's) were ellipses around the Sun.
Galileo's use of a telescope to discover four moons circling Jupiter expanded the list of planets (yes, they were considered planets at the time), but this also forced Galileo to accept the Copernican view of the Solar System. Galileo was within the Vatican's reach, so he got in Big Trouble.
The real star *cough* of the show is Newton, whose theories of physics resulted in precise mathematical formulae which could calculate planets' orbits and masses. The first big success was the 1758 return of a comet predicted by Edmond Halley. An empirical rule proposed in 1766 by Johann Titius predicted the solar distances of the known planets rather well, except for a gap between Mars and Jupiter. This (and Saturn's deviations from its expected orbit) allowed William Herschel to discover a whole new planet (now named Uranus to extend the Roman mythological system) in 1781.
This got the planet-hunting game going in earnest. Astronomers went looking for new planets in the gap between Mars and Jupiter, and on January 1, 1801, Giuseppe Piazzi found one: Ceres. Heinrich Wilhelm Olbers found another, Pallas, on March 28, 1802. Karl L. Harding found a third, Juno, on Septemer 1, 1804. Olbers found the fourth, Vesta, on March 29, 1807. Ceres, Pallas, Juno, and Vesta were much smaller than the other planets, but no-one worried about it for awhile.
John Couch Adams and Urbain Jean Joseph Le Verrier both used pertubations of Uranus' orbit to posit the existence of a new planet. Using Le Verrier's calculations, Johann Gottfried Galle and Heinrich Louis d'Arrest observed a new planet on September 23, 1846, which was eventually named Neptune after an international dispute over whether Couch or Le Verrier should get credit for predicting its existence. Regardless of that, it was the decisive victory for Newtonian physics.
Karl Ludwig Hencke found Astraea in the gap in 1846, and someone found at least one more every year after that. Starting in 1866, the new terms asteroid and minor planet began to be used for the objects in the gap. It was a good system and reduced the number of planets to something considered manageable.
And then in 1930, a 24-year-old astronomer, Clyde W. Tombaugh, found a dot that moved when comparing two photographic negatives of the sky. The dot was named Pluto as a veiled tribute to Percival Lowell.
About the same time, Frederick C. Leonard (1930) and Kenneth E. Edgeworth (1943) proposed the existence of a cloud of icy objects out beyond Neptune, debris left over from the formation of the Solar System. This was named the Kuiper Belt after Gerard Kuiper, who proposed that this was the source of certain comets in 1951.
There have been plenty of attempts to find more planets. In 1973, Rawlins and Hammerton thought they had discovered a Planet X based upon cometary orbit data, but this was never accepted. The acceptance of the impact theory for the K-T Extinction event, and a rough 26 million year cycle observed for mass extinctions on Earth caused several scientists in 1984 to propose the existence of a red or brown dwarf star named Nemesis which perturbs comets into the inner Solar System. This Nemesis Theory has also failed to gain traction.
For seventy years, schoolchildren were taught about the nine planets of the Solar System. But people began to have problems with Pluto. Pluto was always thought of as odd, with its highly eccentric orbit, and was sometimes considered an "escaped moon of Neptune". Estimates of its size kept shrinking, from the size of Mercury down to about the size of the Moon. Then, Pluto was discovered to have its own moon, Charon, so large and so close that the system's center of gravity is out in space. But things really fell apart when A Kuiper Belt object larger than Pluto was found: 2003 UB313, affectionately named "Xena" by its discoverers.
It is no secret that the definition at the beginning of this article, adopted in August 2006 at the IAU's 26th General Assembly in Prague, was specifically tailored to exclude Pluto and similar- or larger-sized Kuiper Belt Objects such as 2003 UB313 (now officially named Eris), of which there are likely to be dozens, if not hundreds. Eris and Pluto are thrown into a new category of dwarf planet, along with the much smaller bodies Ceres and Charon.
The last criterion of the definition makes a lot of people scratch their heads. Could the Earth clear the Kupier Belt if it were moved there? Has it even cleared its own orbit of pesky things like the Moon, not to mention hundreds of Earth-crossing asteroids?
At any rate, the IAU is the governing body for such categorizations, and we will have to live with this definition, and the eight planets that fall into the category.
1IAU resolution 5,
http://www.iau.org/fileadmin/content/pdfs/Resolution_GA26-5-6.pdf
2Often credited to Pythagoras with other such things, but some texts credit it to Ibycus of Rhegium.
Glossary of Ancient Egyptian terms and names
http://nefertiti.iwebland.com/glossary.htm
Names of the Planets
http://www.aerospaceweb.org/question/astronomy/q0171.shtml
Biographies of Nicolaus Copernicus and Tycho Brahe, MacTutor History of Mathematics archive
http://www-history.mcs.st-andrews.ac.uk/Biographies/Copernicus.html
http://www-history.mcs.st-andrews.ac.uk/Biographies/Brahe.html
Wikipedia, for a few of the dates
Stephen Soter, "What Is a Planet?" Scientific American, January 2007, pp 34-41.