Abstract
Galileo Galilei (1564–1642), the famous Italian mathematician at the leading edge of the scientific revolution that was to sweep Europe, was curious about motion. He was an experimentalist who for the first time had the insight and talent to link theory with experiment. He rolled balls down an inclined plane in order to see how things fell toward the Earth. He discovered in this way that objects of any weight fell toward the Earth at the same rate – that they had a uniform acceleration. He surmised that if they fell in a vacuum, where there was no air resistance to slow some objects more than others, even a feather and a cannon ball would descend at the same rate, and reach the ground at the same time. He also explored the motion of pendulums and other phenomena. He is perhaps most famous for his 1610 telescopic discoveries of the moving moons of Jupiter, the phases of Venus, and the craters of the moon, all of which convinced him, against the ages-old wisdom of Aristotle and of the Catholic Church, of the rightness of the Copernican heliocentric view of the solar system.
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Notes
- 1.
It is quite possible that Galileo initially timed the rolling ball by means of musical beats and later confirmed their accuracy by means of the water clock. Galileo was musically inclined, accomplished on the lute, and his father and brother were musicians. Setting adjustable gut “frets” on the inclined plane would enable the ball to make an audible bumping sound as it passed over the frets. Adjusting the spacing of these frets so the bumps occurred at exactly even intervals, according to his internal sense of rhythm, could easily have been done. Indeed, that method was likely far more accurate than any clocks of the day, which could not measure times shorter than a second. This idea was advanced by the late Stillman Drake, Canadian historian of science and Galileo expert. See Stillman Drake, “The Role of Music in Galileo’s Experiments”, Scientific American, June, 1975. The important thing is that the time intervals be deemed to be equal, whatever those intervals may be:
The phrase “measure time” makes us think at once of some standard unit such as the astronomical second. Galileo could not measure time with that kind of accuracy. His mathematical physics was based entirely on ratios, not on standard units as such. In order to compare ratios of times it is necessary only to divide time equally; it is not necessary to name the units, let alone measure them in seconds. The conductor of an orchestra, moving his baton, divides time evenly with great precision over long periods without thinking of seconds or any other standard unit. He maintains a certain even beat according to an internal rhythm, and he can divide that beat in half again with an accuracy rivaling that of any mechanical instrument. Ibid., 98.
- 2.
Dialog Concerning Two New Sciences (translated by Henry Crew and Alfonso de Salvio, Macmillan 1914). This classic translation is also available online: http://galileoandeinstein.physics.virginia.edu/tns_draft/index.html. See also Hawking [1] (Short title) Dialogs, which uses the same translation.
- 3.
Dialogs, 136–37 (my italics).
- 4.
Dialogs, 123 (my italics).
- 5.
The acceleration of a falling body near the surface of the Earth due to gravity is 9.8 m/s per second, written usually as 9.8 m/s2. This means it gains 9.8 m/s in velocity each second of its fall. As noted in Chap. 1, this value for acceleration, or “g” as it is called, will diminish as we move farther away from the Earth’s surface (diminishing, in fact, with the square of the increasing distance from the center of the Earth).
- 6.
Galileo’s first theorem in the section of his Dialogs titled “Naturally Accelerated Motion” states: “The time in which any space is traversed by a body starting from rest and uniformly accelerated is equal to the time in which that same space would be traversed by the same body moving at a uniform speed whose value is the mean of the highest speed and the speed just before acceleration began.” Dialogs, 132.
- 7.
The method of finding the limit of a function to quantitatively derive velocity or acceleration was unknown, and lay three-quarters of a century in the future.
- 8.
Dialogs, 133: Theorem II, Proposition II in “Naturally Accelerated Motion.”
- 9.
Ibid.
- 10.
Galileo is here proving his Theorem I, Proposition I: “The time in which any space is traversed by a body starting from rest and uniformly accelerated is equal to the time in which that same space would be traversed by the same body moving at a uniform speed whose value is the mean of the highest speed and the speed just before acceleration began.” Dialogs, 132.
- 11.
Ibid. 132–33.
- 12.
A calculus course is advised to properly examine the many nuances and variations of this idea to more complex functions, and its inapplicability to certain classes of functions not considered here.
- 13.
As one ventures far from Earth the value of g changes with the inverse square of the object from the center of the Earth, so in those cases the equation must be modified to take that into account, and the equation is PE = −GMm/r.
- 14.
We here adopt the surface of the Earth as the arbitrary zero reference point for potential energy, since the rock can descend no lower.
Reference
Hawking S (ed) (2002) On the shoulders of giants: dialogs concerning two new sciences. Running Press, Philadelphia
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MacDougal, D.W. (2012). Galileo’s Great Discovery: How Things Fall. In: Newton's Gravity. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5444-1_2
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