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Mathematics
While Galileo's application of mathematics to experimental physics was innovative, his mathematical methods were the standard ones of the day, including dozens of examples of an inverse proportion square root method passed down from Fibonacci and Archimedes. The analysis and proofs relied heavily on the Eudoxian theory of proportion, as set forth in the fifth book of Euclid's Elements. This theory had become available only a century before, thanks to accurate translations by Tartaglia and others; but by the end of Galileo's life, it was being superseded by the algebraic methods of Descartes. The concept now named Galileo's paradox was not original with him. His proposed solution, that infinite numbers cannot be compared, is no longer considered useful.[218] biography by Galileo's pupil Vincenzo Viviani stated that Galileo had dropped balls of the same material, but different masses, from the Leaning Tower of Pisa to demonstrate that their time of descent was independent of their mass.[202] This was contrary to what Aristotle had taught: that heavy objects fall faster than lighter ones, in direct proportion to weight.[203][204] While this story has been retold in popular accounts, there is no account by Galileo himself of such an experiment, and it is generally accepted by historians that it was at most a thought experiment which did not actually take place.[205] An exception is Stillman Drake,[206] who argues that the experiment did take place, more or less as Viviani described it. However, most of Galileo's experiments with falling bodies were carried out using inclined planes where both the issues of timing and air resistance were much reduced.[207]
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