Galileo proposed that a falling body would fall with a uniform acceleration, as long as the resistance of the medium through which it was falling remained negligible, or in the limiting case of its falling through a vacuum.[140] He also derived the correct kinematical law for the distance travelled during a uniform acceleration starting from rest—namely, that it is proportional to the square of the elapsed time ( d ∝ t 2 ).[141] Prior to Galileo, Nicole Oresme, in the 14th century, had derived the times-squared law for uniformly accelerated change,[142] and Domingo de Soto had suggested in the 16th century that bodies falling through a homogeneous medium would be uniformly accelerated.[143] Galileo expressed the time-squared law using geometrical constructions and mathematically precise words, adhering to the standards of the day. (It remained for others to re-express the law in algebraic terms).
He also concluded that objects retain their velocity unless a force—often friction—acts upon them, refuting the generally accepted Aristotelian hypothesis that objects "naturally" slow down and stop unless a force acts upon them. Philosophical ideas relating to inertiahad been proposed by John Philoponus centuries earlier, as had Jean Buridan, and according to Joseph Needham, Mo Tzu had proposed it centuries before either of them; nevertheless, Galileo was the first to express it mathematically, verify it experimentally, and introduce the idea of frictional force, the key breakthrough in validating the concept. Galileo's Principle of Inertia stated: "A body moving on a level surface will continue in the same direction at constant speed unless disturbed." This principle was incorporated intoNewton's laws of motion (first law).
See also: Equations for a falling body
Mathematics
While Galileo's application of mathematics to experimental physics was innovative, some of his mathematical methods were the standard ones of the day, including dozens of examples of an inverse proportion square rootmethod 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century before, thanks to accurate translations by Tartaglia and others; but by the end of Galileo's life, it was being superseded by the algebraic methods of Descartes.
Galileo contributed to the method of indivisibles, now known as Cavalieri's principle, which relates to forerunners of infinitesimal calculus.[144] In 1604, while at the University of Padua, he was experimenting with indivisibles in formulating his law of falling bodies.[145] From 1621 onward, he and Cavalieri exchanged a series of letters exploring the hypothesis that a continuum is composed of indivisibles, a hypothesis fiercely opposed by the Jesuits as contrary to Aristotelian dogma (Alexander, p. 86).
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.
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