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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Two New Sciences | 4/6 | https://en.wikipedia.org/wiki/Two_New_Sciences | reference | science, encyclopedia | 2026-05-05T08:47:11.405455+00:00 | kb-cron |
=== The motion of objects === Galileo expresses clearly for the first time the constant acceleration of a falling body which he was able to measure accurately by slowing it down using an inclined plane. In Two New Sciences, Galileo (Salviati speaks for him) used a wood molding, "12 cubits long, half a cubit wide and three finger-breadths thick" as a ramp with a straight, smooth, polished groove to study rolling balls ("a hard, smooth and very round bronze ball"). He lined the groove with "parchment, also smooth and polished as possible". He inclined the ramp at various angles, effectively slowing down the acceleration enough so that he could measure the elapsed time. He would let the ball roll a known distance down the ramp, and use a water clock to measure the time taken to move the known distance. This clock was
a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length. The water collected was weighed, and after each descent on a very accurate balance, the differences and ratios of these weights gave him the differences and ratios of the times. This was done with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.
==== The law of falling bodies ==== While Aristotle had observed that heavier objects fall more quickly than lighter ones, in Two New Sciences Galileo postulated that this was due not to inherently stronger forces acting on the heavier objects, but to the countervailing forces of air resistance and friction. To compensate, he conducted experiments using a shallowly inclined ramp, smoothed so as to eliminate as much friction as possible, on which he rolled down balls of different weights. In this manner, he was able to provide empirical evidence that matter accelerates vertically downward at a constant rate, regardless of mass, due to the effects of gravity. The unreported experiment found in folio 116V tested the constant rate of acceleration in falling bodies due to gravity. This experiment consisted of dropping a ball from specified heights onto a deflector in order to transfer its motion from vertical to horizontal. The data from the inclined plane experiments were used to calculate the expected horizontal motion. However, discrepancies were found in the results of the experiment: the observed horizontal distances disagreed with the calculated distances expected for a constant rate of acceleration. Galileo attributed the discrepancies to air resistance in the unreported experiment, and friction in the inclined plane experiment. These discrepancies forced Galileo to assert that the postulate held only under "ideal conditions", i.e., in the absence of friction and/or air resistance.
==== Bodies in motion ==== Aristotelian physics argued that the Earth must not move as humans are unable to perceive the effects of this motion. A popular justification of this is the experiment of an archer shooting an arrow straight up into the air. If the Earth were moving, Aristotle argued, the arrow should fall in a different location than the launch point. Galileo refuted this argument in Dialogues Concerning the Two Chief World Systems. He provided the example of sailors aboard a boat at sea. The boat is obviously in motion, but the sailors are unable to perceive this motion. If a sailor were to drop a weighted object from the mast, this object would fall at the base of the mast rather than behind it (due to the ship's forward motion). This was the result of simultaneously the horizontal and vertical motion of the ship, sailors, and ball.
==== Relativity of motions ====
One of Galileo's experiments regarding falling bodies was that describing the relativity of motions, explaining that, under the right circumstances, "one motion may be superimposed upon another without effect upon either...". In Two New Sciences, Galileo made his case for this argument and it would become the basis of Newton's first law, the law of inertia. He poses the question of what happens to a ball dropped from the mast of a sailing ship or an arrow fired into the air on the deck. According to Aristotle's physics, the ball dropped should land at the stern of the ship as it falls straight down from the point of origin. Likewise the arrow when fired straight up should not land in the same spot if the ship is in motion. Galileo offers that there are two independent motions at play. One is the accelerating vertical motion caused by gravity while the other is the uniform horizontal motion caused by the moving ship which continues to influence the trajectory of the ball through the principle of inertia. The combination of these two motions results in a parabolic curve. The observer cannot identify this parabolic curve because the ball and observer share the horizontal movement imparted to them by the ship, meaning only the perpendicular, vertical motion is perceivable. Surprisingly, nobody had tested this theory with the simple experiments needed to gain a conclusive result until Pierre Gassendi published the results of said experiments in his letters entitled De Motu Impresso a Motore Translato (1642).
== Infinity ==
The book also contains a discussion of infinity. Galileo considers the example of numbers and their squares. He starts by noting that:
It cannot be denied that there are as many [squares] as there are numbers because every number is a [square] root of some square: 1 ↔ 1, 2 ↔ 4, 3 ↔ 9, 4 ↔ 16, and so on. But he notes what appears to be a contradiction: