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Geocentrism is the belief that the sun, the planets and all the stars revolve around the Earth. You can thank the Old Testament for the lingering belief in this one. It has at least four verses claiming that the earth "cannot be moved," and one stating that the sun goes around the earth.
 Modern geocentrism
There are possible variations on this. "Geocentrism", taken literally, means that the earth is at the center, which is consistent with daily rotation of the earth, without the earth orbiting the sun annually. However, this is not how geocentrism was traditionally understood, and is not how it persists today. Geocentric models claim the earth has does not move at all, which means that the entire universe revolves around the earth daily. (This would mean that the stars and galaxies revolve around the Earth once every 24 hours, despite modern astronomy demonstrating that they are many light-years away and would therefore have to race round faster than light.) A person can take a relativistic view, that "all motion is relative", and claim one cannot tell the difference between the earth being in motion and it being fixed. This has the fault that,the Bible makes false statements about the fixity of the earth, but instead makes meaningless statements; and therefore it rarely comes up.
21st century believers in a fixed earth may distance themselves from the Ptolemaic system of epicycles. Some call their model geocentricity in an attempt to make that clear. They may prefer a "compromise" position first suggested by Tycho Brahe: the Sun revolves around the earth, but the planets revolve around the Sun. The Bible has no statements about the motions of the planets (and the stars), so any description of the movements of the heavenly bodies can fit the Bible, as long as the earth is fixed and the sun is in a closed path around the earth.
A minority of fairly dense fundamentalists and apathetic people believe modern geocentrism. There are more than one might assume, in the United States, Great Britain and Germany between 16% and 19% of people actually believe the sun goes round the earth.  Some geocentrists are rather vocal, have websites,  and would like you to know that at least they're not as dumb as those heretical Flat Earthers.
The geocentric model was accepted as standard throughout the Western world through the Middle Ages and the Renaissance, Scripture supported it and supported Ptolemy. Ptolemy's theory of the solar system placed all celestial bodies orbiting around the Earth, with epicycles introduced to compensate for the retrograde motion of planets such as Mars.
The ancient Greeks noted that, if the Earth did move around the sun, the stars should appear to shift their positions from one part of the year to the next, which the stars don't do. Either that, or the stars would have to be so ridiculously far away that the shift in position was too small to be seen by the naked eye. Hey, that would make for a ludicrously large empty space between the most distant planet, Saturn, and the "fixed stars". (Measurements dependent on naked-eye instruments could detect a parallax of something like one arc-minute, which would give a distance roughly 300 times the distance to Saturn.)
The geocentric model was not formally challenged until the works of Nicolaus Copernicus, who suggested a heliocentric universe in 1514. Probably due to the Galileo controversy, his book was still listed on the Index (the list of books banned by the Catholic Church) between 1616 and 1758.
 For the Bible tells us so
In order of appearance, the verses are:
- 1 Chronicles 16:30: Fear before him, all the earth: the world also shall be stable, that it be not moved.
- Psalm 93:1: The LORD reigneth, he is clothed with majesty; the LORD is clothed with strength, wherewith he hath girded himself: the world also is stablished (sic), that it cannot be moved.
- Psalm 96:10: Say among the heathen that the LORD reigneth: the world also shall be established that it shall not be moved: he shall judge the people righteously.
- Psalm 104:5: (Bless the LORD . . .) Who laid the foundations of the earth, that it should not be removed for ever.
- Ecclesiastes 1:5: The sun also ariseth, and the sun goeth down, and hasteth to his place where he arose.
Take note the psalms were songs sung in praise of Jehovah. Psalms 93 seems to thank Jehovah for the floods (of Noah?). Psalms 96 commands the listener to pray to Jehovah because the gods of other nations are false (idols). Psalms 104 thanks Jehovah for all he created and for his protection against outsiders.
However—and this is a point that bible fundamentalists overlook—nowhere does the psalmist or any other biblical writer claim to be an astrophysicist. On the other hand, they point out that the psalmist is telling us that we have as much trust in the Lord as we have in the fixity of the earth. If the earth is not fixed, then what reliance can we have in the Lord?
Yet it is difficult to argue with a convinced person over the proper interpretation of the Bible, what is to be taken literally and what is figurative. We can say that nobody before the rise of modern science thought that the Bible was consistent with the Earth being a planet of the Sun. A couple of late medievals (Nicholas of Cusa and Nicolas Oresme) brought up the possibility that the Earth moves.
It is clear that no one has ever arrived at heliocentrism from enlightened Scriptural understanding. Even today only from scientific observations and human reasoning or accepting common opinion enable people to accept the motion of the Earth. When believers interpret the Bible in a heliocentric way this is because they accept what astronomy has discovered.
The people who wrote the Old Testament believed the earth is fixed while the sun moves and wrote what they believed. This simple explanation makes sense since people living in in Judea then didn't know anything nearer the truth. Complex figurative interpretations of the Bible texts are theoretically possible  but Occam's razor shows these suggestions are less likely to be correct.
 The sun travels around the earth
Ecclesiastes 1:5 says "The sun rises and the sun sets, and hurries back to where it rises." This is understood by geocentrists to mean that the sun revolves around the earth. However, the most common and contextually-informed understanding of the verse is that it is describing the visible path of the sun in the sky, not the path of the sun in the universe. The point of the biblical passage that contains the verse is that life on earth amounts only to endless, monotonous, and meaningless cycles-- such as the endless cycle of sunrise and sunset. The true spirit of the passage is contained in verse 9 of the same chapter, which reads "What has been will be again, what has been done will be done again." It is clear from this perspective that the verse makes no astrophysical implications whatsoever.
A similar verse is Psalm 19:6 which reads "It [the sun] rises at one end of the heavens and makes its circuit to the other; nothing is hidden from its heat.".
 Ceasing the revolution of the sun
Joshua 10:12-13 reads "On the day the LORD gave the Amorites over to Israel, Joshua said to the LORD in the presence of Israel: "O sun, stand still over Gibeon, O moon, over the Valley of Aijalon." So the sun stood still, and the moon stopped, till the nation avenged itself on its enemies, as it is written in the Book of Jashar. The sun stopped in the middle of the sky and delayed going down about a full day."
Aside for the purely mechanical difficulties of having the sun and moon stand still this would only work if the sun were traveling around the earth. For the sun to appear above one city for an extended amount of time the earth would have to stop rotating, it would not matter if the sun were moving or not. (See this video for the terrible things that would happen if the Earth suddenly stopped rotating.)
Galileo, in his Letter to Castelli pointed out that in the Ptolemaic system of the universe, the heavens in general are moving around the earth from east to west, while the motion of the sun in the heavens was a motion from west to east; and therefore, if the Bible were describing a Ptolemaic universe, and saying literally that the sun ceased its motion, the result would be less daylight (by about four minutes).
 See also
- ↑ Actually, the problem of faster-than-light occurs at the position of the planet Neptune.
- ↑ New Poll Gauges Americans' General Knowledge Levelsmodern medievalism: Geocentrism
- ↑ The non-moving Earth & anti-evolution web page of The Fair Education Foundation, Inc. Static Earth, Ready to have your world view shaken
- ↑ Psalm 104:26 also mentions the leviathan.
- ↑ 
- ↑ page 108, in "Letter to Castelli (1630)", pages 103-109, in: Maurice A. Finocchiaro, ed. & trans, The Essential Galileo, Indianapolis/Cambridge: Hackett Publishing, 2008, ISBN 978-0-87220-937-4