A secret society known as the "Illuminati" figures prominently in Angels & Demons. In the book, the Illuminati started in Rome in the 1500s as a group of scientists and others who were opposed to the teachings of the church, and one of their most prominent founding members was Galileo. Well, that's according to the book. According to Wikipedia, the Illuminati had nothing to do with either the church or Galileo, was founded in Bavaria (Germany) not Rome, and was founded in 1776 — 134 years after Galileo died. [But, of course, the Illuminati would have taken care to make sure that the Wikipedia article is misleading, right? Silly me.]
The second example takes more explaining, but is I think more interesting.
In Angels & Demons, the protagonist, symbologist Robert Langdon, finds an important clue in a fictitious text by Galileo which is kept in the library at the Vatican. While reading this fictitious text looking for the clue, Langdon comes across a section on planetary orbits (p. 211 in my paperback edition):
Elliptical orbits. Langdon recalled that much of Galileo's legal trouble [with the church] had begun when he described planetary motion as elliptical. The Vatican exalted the perfection of the circle and insisted heavenly motion must be only circular.
I could be wrong, but I don't think this is the way it was.
I think Galileo got in trouble with the church because of his text titled "Dialogue Concerning the Two Chief World Systems." The two systems were the ancient Ptolemaic system in which everything orbited the Earth, and the newer Copernican system in which all the planets, including Earth, orbited the sun. Galileo's "Dialogue" favored the Copernican system, which angered the church. The church favored the Ptolemaic system, where the Earth is at the center of everything.
But there's more to the story. Galileo in fact argued for circular orbits, not elliptical orbits. The irony is that Galileo should have known better. While it is true that planetary orbits were circular in the Copernican system (as in the Ptolemaic system), Kepler had discovered some 23 years before Galileo published his "Dialogue" that planetary orbits were elliptical. Galileo knew about Kepler's work, and even corresponded with Kepler, but did not believe that planetary orbits were elliptical.
Here are some relevant dates:
1543 Copernicus published his theory
1601 Tycho Brahe died, Kepler got his data
1609 Kepler's laws of planetary motion
1632 Galileo's "Dialogue"
1687 Newton's laws of motion and gravitation
I mention Brahe (pronounced brah-hee) in this chronology because it was his astronomical data that Kepler used to discover the laws of planetary motion. Kepler's first law of planetary motion says that planetary orbits are elliptical. But Galileo ignored Kepler, and went back to the earlier Copernican theory of circular orbits around the sun, which we now know was wrong.
I mention Newton because it was his laws which provided a mathematical proof for Kepler's laws. Kepler had derived his laws empirically—from observational data, without any underlying mathematical framework. Newton provided the mathematical framework.
I think it is common knowledge that the trouble between Galileo and the church was about whether the Earth or the sun was at the center of things. I don't think it is so commonly known that Galileo argued for circular orbits when he should have known from Kepler's work that planetary orbits are elliptical.
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