Ptolemy's geocentric model of the solar system was so successful it remained the standard in mathematical astronomy until the 16th century. Spurred by the printing press and the relatively easy duplication and dissemination of ideas, science was being reinvented in Europe. Just as Columbus "discovered" the Americas, and Europe goes on an orgy of exploration and colonization, mathematical astronomers began to question the make up of the solar system. This led to the discarding of the geocentric model, and the development of the classical basis for our current understanding of the working of the solar system.

In chronological order, here are the key people in the development of the heliocentric theory of the solar system, with links to more detailed descriptions.

Nicholas Copernicus (1473-1543)
For the first time in nearly 1400 years, Copernicus develops a heliocentric theory of the solar system that rivals the work of Ptolemy. While mathematically very similar to Ptolemy's model, and still relying on circles and epicycles, the Copernican model is the first one that is a coherent system; the parameters describing the orbits of the planets are all interconnected. His work becomes controversial, and is eventually banned by the Catholic Church, largely because of Galileo.
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Tycho Brahe (1546-1601)
Brahe was fascinated with the Ptolemy vs. Copernicus debate. He realized that one could not prove either model correct based on the observational data of his time. He sets up a massive observatory, called Uraniborg, and makes the most accurate naked-eye observations in history. He is able to show that a comet and a nova were farther than the moon. He also develops a third model of the solar system, which is a hybrid of Ptolemy and Copernicus.
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Johannes Kepler (1571-1630)
Kepler was the most gifted mathematical astronomer of his day. He uses the data from Brahe to tease out the true nature of the orbits of the planets. He discovers three laws of planetary motion, now called Kepler's Laws. Though his discoveries were based only on empirical data, and not from a central theory, it marks the first time that in human histroy that we understood how the solar system actually works.
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Galileo Galilei (1564-1642)
Galileo is often called the father of modern science, due to his extensive experimentation. He is the first to use a telescope and look at the sky, and makes a number of discoveries which he publishes in Siderius Nuncius. This causes him to go public in his advocacy of the heliocentric theory, which causes some problems with the Catholic Church. He also begins to straighten out the basic physics of motion that Aristotle had messed up.
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Issac Newton (1643-1727)
Newton developes calculus, develops the basic laws of motion (now called classical physics,) and proposes his theory of universal gravitation. Starting from his fundamental ideas, Newton is able to derive Kepler's Laws. His |