Introduction
The book is divided into four days, each addressing different areas of physics. Galileo dedicates ''Two New Sciences'' to Lord Count of Noailles.Summary
Page numbers at the start of each paragraph are from the 1898 version, presently adopted as standard, and are found in the Crew and Drake translations.Day one: Resistance of bodies to separation
0Preliminary discussions. Sagredo (taken to be the younger Galileo) cannot understand why with machines one cannot argue from the small to the large: "I do not see that the properties of circles, triangles and...solid figures should change with their size". Salviati (speaking for Galileo) says the common opinion is wrong. Scale matters: a horse falling from a height of 3 or 4 cubits will break its bones whereas a cat falling from twice the height won't, nor will a grasshopper falling from a tower. 6The first example is a hemp rope which is constructed from small fibres which bind together in the same way as a rope round a windlass to produce something much stronger. Then the vacuum that prevents two highly polished plates from separating even though they slide easily gives rise to an experiment to test whether water can be expanded or whether a vacuum is caused. In fact, Sagredo had observed that a suction pump could not lift more than 18 cubits of water and Salviati observes that the weight of this is the amount of resistance to a void. The discussion turns to the strength of a copper wire and whether there are minute void spaces inside the metal or whether there is some other explanation for its strength. 8This leads into a discussion of infinites and the continuum and thence to the observation that the number of squares equal the number of roots. He comes eventually to the view that "if any number can be said to be infinite, it must be unity" and demonstrates a construction in which an infinite circle is approached and another to divide a line. 5The difference between a fine dust and a liquid leads to a discussion of light and how the concentrated power of the sun can melt metals. He deduces that light has motion and describes an (unsuccessful) attempt to measure its speed. 06Aristotle believed that bodies fell at a speed proportional to weight but Salviati doubts that Aristotle ever tested this. He also did not believe that motion in a void was possible, but since air is much less dense than water Salviati asserts that in a medium devoid of resistance (a vacuum) all bodies—a lock of wool or a bit of lead—would fall at the same speed. Large and small bodies fall at the same speed through air or water providing they are of the same density. Since ebony weighs a thousand times as much as air (which he had measured), it will fall only a very little more slowly than lead which weighs ten times as much. But shape also matters—even a piece of gold leaf (the densest of all substances sserts Salviati floats through the air and a bladder filled with air falls much more slowly than lead. 28Measuring the speed of a fall is difficult because of the small time intervals involved and his first way round this used pendulums of the same length but with lead or cork weights. The period of oscillation was the same, even when the cork was swung more widely to compensate for the fact that it soon stopped. 39This leads to a discussion of the vibration of strings and he suggests that not only the length of the string is important for pitch but also the tension and the weight of the string.Day two: Cause of cohesion
51Salviati proves that a balance can be used not only with equal arms but with unequal arms with weights inversely proportional to the distances from the fulcrum. Following this he shows that the moment of a weight suspended by a beam supported at one end is proportional to the square of the length. The resistance to fracture of beams of various sizes and thicknesses is demonstrated, supported at one or both ends. 69He shows that animal bones have to be proportionately larger for larger animals and the length of a cylinder that will break under its own weight. He proves that the best place to break a stick placed upon the knee is the middle and shows how far along a beam that a larger weight can be placed without breaking it. 78He proves that the optimum shape for a beam supported at one end and bearing a load at the other is parabolic. He also shows that hollow cylinders are stronger than solid ones of the same weight.Day three: Naturally accelerated motion
91He first defines uniform (steady) motion and shows the relationship between speed, time and distance. He then defines uniformly accelerated motion where the speed increases by the same amount in increments of time. Falling bodies start very slowly and he sets out to show that their velocity increases in simple proportionality to time, not to distance which he shows is impossible. 08He shows that the distance travelled in naturally accelerated motion is proportional to the square of the time. He describes an experiment in which a steel ball was rolled down a groove in a piece of wooden moulding 12 cubits long (about 5.5m) with one end raised by one or two cubits. This was repeated, measuring times by accurately weighing the amount of water that came out of a thin pipe in a jet from the bottom of a large jug of water. By this means he was able to verify the uniformly accelerated motion. He then shows that whatever the inclination of the plane, the square of the time taken to fall a given vertical height is proportional to the inclined distance. 21He next considers descent along the chords of a circle, showing that the time is the same as that falling from the vertex, and various other combinations of planes. He gives an erroneous solution to the brachistochrone problem, claiming to prove that the arc of the circle is the fastest descent. 16 problems with solutions are given.Day four: The motion of projectiles
Additional day: The force of percussion
23What is the weight of water falling from a bucket hanging on a balance arm onto another bucket suspended to the same arm? 25Piling of wooden poles for foundations; hammers and the force of percussion. 36Speed of fall along inclined planes; again on the principle of inertia.Methodology
Many contemporary scientists, such as Gassendi, dispute Galileo's methodology for conceptualizing his law of falling bodies. Two of the main arguments are that his epistemology followed the example of Platonist thought or hypothetico-deductivist. It has now been considered to be ''ex suppositione'', or knowing the how and why effects from past events in order to determine the requirements for the production of similar effects in the future. Galilean methodology mirrored that of Aristotelian and Archimedean epistemology. Following a letter from Cardinal Bellarmine in 1615 Galileo distinguished his arguments and Copernicus' as natural suppositions as opposed to the "fictive" that are "introduced only for the sake of astronomical computations," such asThe two new sciences
The two sciences mentioned in the title are the strength of materials and the motion of objects (the forebears of modern material engineering andThe science of materials
The discussion begins with a demonstration of the reasons that a large structure proportioned in exactly the same way as a smaller one must necessarily be weaker known as the square–cube law. Later in the discussion this principle is applied to the thickness required of the bones of a large animal, possibly the first quantitative result inThe 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 "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
Infinity
The book also contains a discussion ofIt cannot be denied that there are as many quaresas there are numbers because every number is a quareroot of some square: 1 ↔ 1, 2 ↔ 4, 3 ↔ 9, 4 ↔ 16, and so on.But he notes what appears to be a contradiction:
Yet at the outset we said there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers.(In modern language, there is a
We can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal", greater", and "less", are not applicable to infinite, but only to finite, quantities.This conclusion, that ascribing sizes to infinite sets should be ruled impossible, owing to the contradictory results obtained from these two ostensibly natural ways of attempting to do so, is a resolution to the problem that is consistent with, but less powerful than, the methods used in modern mathematics. The resolution to the problem may be generalized by considering Galileo's first definition of what it means for sets to have equal sizes, that is, the ability to put them in one-to-one correspondence. This turns out to yield a way of comparing the sizes of infinite sets that is free from contradictory results. These issues of infinity arise from problems of rolling circles. If two concentric circles of different radii roll along lines, then if the larger does not slip it appears clear that the smaller must slip. But in what way? Galileo attempts to clarify the matter by considering hexagons and then extending to rolling 100 000-gons, or n-gons, where he shows that a finite number of finite slips occur on the inner shape. Eventually, he concludes "the line traversed by the larger circle consists then of an infinite number of points which completely fill it; while that which is traced by the smaller circle consists of an infinite number of points which leave empty spaces and only partly fill the line," which would not be considered satisfactory now.
Reactions by commentators
Part of ''Two New Sciences'' was pure mathematics, as has been pointed out by the mathematician Alfréd Rényi, who said that it was the most significant book on mathematics in over 2000 years: Greek mathematics did not deal with motion, and so they never formulated mathematical laws of motion, even though Archimedes developed differentiation and integration. ''Two New Sciences'' opened the way to treating physics mathematically by treating motion mathematically for the first time. The Greek mathematicianGassendi's thoughts
Koyré's thoughts
The law of falling bodies was published by Galileo in 1638. But in the 20th century some authorities challenged the reality of Galileo's experiments. In particular, the French historian of science Alexandre Koyré bases his doubt on the fact that the experiments reported in ''Two New Sciences'' to determine the law of acceleration of falling bodies, required accurate measurements of time which appeared to be impossible with the technology of 1600. According to Koyré, the law was created deductively, and the experiments were merely illustrative thought experiments. In fact, Galileo's water clock (described above) provided sufficiently accurate measurements of time to confirm his conjectures. Later research, however, has validated the experiments. The experiments on falling bodies (actually rolling balls) were replicated using the methods described by Galileo, and the precision of the results was consistent with Galileo's report. Later research into Galileo's unpublished working papers from 1604 clearly showed the reality of the experiments and even indicated the particular results that led to the time-squared law.See also
* '' De Motu Antiquiora'' (Galileo's earliest investigations of the motion of falling bodies)Notes
References
* Drake, Stillman, translator (1974). ''Two New Sciences'', University of Wisconsin Press, 1974. . A new translation including sections on centers of gravity and the force of percussion. * * Henry Crew and Alfonso de Salvio, translators, 914(1954).External links
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