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Geocentrism is the idea that the earth is at the centre of all things. The idea often includes geostatism, the earth's being motionless (no daily rotation on its axis, as well as no yearly rotation around the sun), at least in some cosmic sense.
Geocentrism has existed since time immemorial as part of the ubiquitous pre-scientific flat-earth view, but even after the discovery of the Earth's approximate sphericity, geocentrism was still almost universally accepted. Aristotle believed that there was a fundamental difference between the physics of the terrestrial or sublunary realm and the physics of the celestial realm. Terrestrial objects prefer to travel in straight lines, and to arrange themselves from bottom to top as earth, water, air, and fire. Celestial objects prefer to travel in circles -- and to travel endlessly. Claudius Ptolemy, in ancient Roman times, improved on this circular-motion view by introducing epicycles, circles attached to the primary circles, to account for how:
- Mercury and Venus move from one side of the Sun to the other and then back again
- Mars, Jupiter, and Saturn move backwards relative to the stars at "opposition", near the Sun's celestial antipode
Strictly speaking, with enough epicycles, one can account for any motion whatsoever; that is the principle of the Fourier Series. But a theory with too many free parameters is unfalsifiable, and therefore pointless.
Heliocentrism was first proposed by Aristarchus of Samos around the early third century BCE. He may have gotten that idea from having worked out the sizes of the Sun and the Moon. He found that the Moon is smaller than the Earth -- but that the Sun is larger than the Earth! This hypothesis was half-forgotten for nearly 1800 years, but in 1543 a Polish astronomer named Nicholaus Copernicus revived it. It attracted little attention at first, but in the early 1600's, Galileo discovered new phenomena that were strongly suggestive of heliocentrism:
- The Moon's Mountains -- the Moon is a very Earthlike place, contrary to Aristotelian views
- Venus's Phases -- consistent with Venus moving around the Sun
- Jupiter's Moons -- objects moving around something other than the Earth
- Sunspots -- clear evidence of the Sun's imperfection
Galileo's friend Johannes Kepler had analyzed Tycho Brahe's excellent observations, and showed that they much more easily fit heliocentrism:
- First Law -- the planet-Sun lines sweep out areas at constant rates
- Second Law -- elliptical orbits with the Sun at one focus
- Third Law -- period^2 ~ distance^3, assuming heliocentrism
Later in the 1600's, Sir Isaac Newton developed a comprehensive theoretical framework, Newtonian mechanics, that accounted for the Solar System's dynamics -- and that stated that the terrestrial and celestial realms have the same physics. Kepler's first law was a consequence of the conservation of angular momentum, and the remaining two of the inverse-square law of gravity. An interesting result of Newtonianism is that the Sun is not exactly at a point which can reasonably be called the Solar System's center: its barycenter, a.k.a. center of mass or center of gravity. But it stays much closer to that point than most other Solar-System macroscopic objects.
It is sometimes argued that Albert Einstein's general relativity makes heliocentrism a meaningless concept, but I (LP) believe that to be a rather gross mistake. The usual counterargument is that one can always choose a coordinate system in which the Earth is the center of the Universe, but one can do the same for any arbitrary point. However, physically meaningful quantities are independent of what coordinate system one chooses, so that argument ultimately fails. One can nevertheless choose coordinates that reflect some underlying mathematical features; such coordinates often offer convenient shortcuts to coordinate-independent results. Thus, in flat space, one can always set up a Cartesian coordinate system, where the coordinates are related to the underlying geometry in some nice ways. Straight lines in coordinate values are "true" straight lines, which are geodesics (curves with extremal length), a very special kind of curve.
For Newtonian mechanics, with its flat space-time and universal time coordinate, the question of heliocentrism is easily resolved, as stated above; the Solar System's barycenter has a coordinate-system-independent definition. However, for GR, life becomes much trickier, because one has to take into account space-time curvature, something which makes distances, times, and barycenters more difficult to define. But curvature effects are relatively weak in the Solar System; the Earth's velocity around the Sun is 30 km/s or 10^-4 c, making curvature effects ~ 10^-8. Thus, one can easily "cheat" in the Solar System and treat it as purely Newtonian; doing a more careful GR-based treatment will change one's results only slightly.
However, heliocentrism runs contrary to a literal reading of the Bible, especially such passages as Joshua 10:13, where he tells the Sun and not the Earth to stop moving. For that reason, both Catholics and Protestants rejected heliocentrism from its inception. Galileo was tried by the Catholic Inquisition and his views were placed upon the index of forbidden books. Similarly, various Lutheran synods taught geocentrism well into the 19th century.
Eventually, both Catholics and Protestants have accepted heliocentrism, except for a few holdovers to this day. Those are the modern-day geocentrists. They are young-earth creationists, but with the additional belief of geocentrism. Most YECs are heliocentrists, but they show more or less tolerance to geocentric views. Creationist heliocentrists sometimes criticise the geocentrists for being dogmatic about geocentrism, but they always qualify their statements by saying that the Bible and science permit both geocentrism and heliocentrism.
Modern geocentrism is not very significant in the antievolution movement, but some geocentrists, such as Paul Ellwanger and Tom Willis, have been behind the equal time bills. Modern geocentrists include Dr G D Bouw, Marshall Hall, Malcolm Bowden (author of "The Rise of the Evolution Fraud"), James Hanson, Tom Willis, Paul Ellwanger, R G Elmendorf, Paula Haigh, and Robert Sungenis (the last four being Catholics).
- Talk.Origins special about geocentrism
- Geocentricity  by Gerardus D. Bouw
- The non-moving Earth  by Marshall Hall
- The Geocentrism Challenge  : Postings by Robert Sungenis
(The last three links are deadly serious sites belonging to geocentrists. They are not a joke)