nhaliday + error + mechanics   8

GALILEO'S STUDIES OF PROJECTILE MOTION
During the Renaissance, the focus, especially in the arts, was on representing as accurately as possible the real world whether on a 2 dimensional surface or a solid such as marble or granite. This required two things. The first was new methods for drawing or painting, e.g., perspective. The second, relevant to this topic, was careful observation.

With the spread of cannon in warfare, the study of projectile motion had taken on greater importance, and now, with more careful observation and more accurate representation, came the realization that projectiles did not move the way Aristotle and his followers had said they did: the path of a projectile did not consist of two consecutive straight line components but was instead a smooth curve. [1]

Now someone needed to come up with a method to determine if there was a special curve a projectile followed. But measuring the path of a projectile was not easy.

Using an inclined plane, Galileo had performed experiments on uniformly accelerated motion, and he now used the same apparatus to study projectile motion. He placed an inclined plane on a table and provided it with a curved piece at the bottom which deflected an inked bronze ball into a horizontal direction. The ball thus accelerated rolled over the table-top with uniform motion and then fell off the edge of the table Where it hit the floor, it left a small mark. The mark allowed the horizontal and vertical distances traveled by the ball to be measured. [2]

By varying the ball's horizontal velocity and vertical drop, Galileo was able to determine that the path of a projectile is parabolic.

https://www.scientificamerican.com/author/stillman-drake/

Galileo's Discovery of the Parabolic Trajectory: http://www.jstor.org/stable/24949756

Galileo's Experimental Confirmation of Horizontal Inertia: Unpublished Manuscripts (Galileo
Gleanings XXII): https://sci-hub.tw/https://www.jstor.org/stable/229718
- Drake Stillman

MORE THAN A DECADE HAS ELAPSED since Thomas Settle published a classic paper in which Galileo's well-known statements about his experiments on inclined planes were completely vindicated.' Settle's paper replied to an earlier attempt by Alexandre Koyre to show that Galileo could not have obtained the results he claimed in his Two New Sciences by actual observations using the equipment there described. The practical ineffectiveness of Settle's painstaking repetition of the experiments in altering the opinion of historians of science is only too evident. Koyre's paper was reprinted years later in book form without so much as a note by the editors concerning Settle's refutation of its thesis.2 And the general literature continues to belittle the role of experiment in Galileo's physics.

More recently James MacLachlan has repeated and confirmed a different experiment reported by Galileo-one which has always seemed highly exaggerated and which was also rejected by Koyre with withering sarcasm.3 In this case, however, it was accuracy of observation rather than precision of experimental data that was in question. Until now, nothing has been produced to demonstrate Galileo's skill in the design and the accurate execution of physical experiment in the modern sense.

Pant of a page of Galileo's unpublished manuscript notes, written late in 7608, corroborating his inertial assumption and leading directly to his discovery of the parabolic trajectory. (Folio 1 16v Vol. 72, MSS Galileiani; courtesy of the Biblioteca Nazionale di Firenze.)

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(The same skeptical historians, however, believe that to show that Galileo could have used the medieval mean-speed theorem suffices to prove that he did use it, though it is found nowhere in his published or unpublished writings.)

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Now, it happens that among Galileo's manuscript notes on motion there are many pages that were not published by Favaro, since they contained only calculations or diagrams without attendant propositions or explanations. Some pages that were published had first undergone considerable editing, making it difficult if not impossible to discern their full significance from their printed form. This unpublished material includes at least one group of notes which cannot satisfactorily be accounted for except as representing a series of experiments designed to test a fundamental assumption, which led to a new, important discovery. In these documents precise empirical data are given numerically, comparisons are made with calculated values derived from theory, a source of discrepancy from still another expected result is noted, a new experiment is designed to eliminate this, and further empirical data are recorded. The last-named data, although proving to be beyond Galileo's powers of mathematical analysis at the time, when subjected to modern analysis turn out to be remarkably precise. If this does not represent the experimental process in its fully modern sense, it is hard to imagine what standards historians require to be met.

The discovery of these notes confirms the opinion of earlier historians. They read only Galileo's published works, but did so without a preconceived notion of continuity in the history of ideas. The opinion of our more sophisticated colleagues has its sole support in philosophical interpretations that fit with preconceived views of orderly long-term scientific development. To find manuscript evidence that Galileo was at home in the physics laboratory hardly surprises me. I should find it much more astonishing if, by reasoning alone, working only from fourteenth-century theories and conclusions, he had continued along lines so different from those followed by profound philosophers in earlier centuries. It is to be hoped that, warned by these examples, historians will begin to restore the old cautionary clauses in analogous instances in which scholarly opinions are revised without new evidence, simply to fit historical theories.

In what follows, the newly discovered documents are presented in the context of a hypothetical reconstruction of Galileo's thought.

...

As early as 1590, if we are correct in ascribing Galileo's juvenile De motu to that date, it was his belief that an ideal body resting on an ideal horizontal plane could be set in motion by a force smaller than any previously assigned force, however small. By "horizontal plane" he meant a surface concentric with the earth but which for reasonable distances would be indistinguishable from a level plane. Galileo noted at the time that experiment did not confirm this belief that the body could be set in motion by a vanishingly small force, and he attributed the failure to friction, pressure, the imperfection of material surfaces and spheres, and the departure of level planes from concentricity with the earth.5

It followed from this belief that under ideal conditions the motion so induced would also be perpetual and uniform. Galileo did not mention these consequences until much later, and it is impossible to say just when he perceived them. They are, however, so evident that it is safe to assume that he saw them almost from the start. They constitute a trivial case of the proposition he seems to have been teaching before 1607-that a mover is required to start motion, but that absence of resistance is then sufficient to account for its continuation.6

In mid-1604, following some investigations of motions along circular arcs and motions of pendulums, Galileo hit upon the law that in free fall the times elapsed from rest are as the smaller distance is to the mean proportional between two distances fallen.7 This gave him the times-squared law as well as the rule of odd numbers for successive distances and speeds in free fall. During the next few years he worked out a large number of theorems relating to motion along inclined planes, later published in the Two New Sciences. He also arrived at the rule that the speed terminating free fall from rest was double the speed of the fall itself. These theorems survive in manuscript notes of the period 1604-1609. (Work during these years can be identified with virtual certainty by the watermarks in the paper used, as I have explained elsewhere.8)

In the autumn of 1608, after a summer at Florence, Galileo seems to have interested himself in the question whether the actual slowing of a body moving horizontally followed any particular rule. On folio 117i of the manuscripts just mentioned, the numbers 196, 155, 121, 100 are noted along the horizontal line near the middle of the page (see Fig. 1). I believe that this was the first entry on this leaf, for reasons that will appear later, and that Galileo placed his grooved plane in the level position and recorded distances traversed in equal times along it. Using a metronome, and rolling a light wooden ball about 4 3/4 inches in diameter along a plane with a groove 1 3/4 inches wide, I obtained similar relations over a distance of 6 feet. The figures obtained vary greatly for balls of different materials and weights and for greatly different initial speeds.9 But it suffices for my present purposes that Galileo could have obtained the figures noted by observing the actual deceleration of a ball along a level plane. It should be noted that the watermark on this leaf is like that on folio 116, to which we shall come presently, and it will be seen later that the two sheets are closely connected in time in other ways as well.

The relatively rapid deceleration is obviously related to the contact of ball and groove. Were the ball to roll right off the end of the plane, all resistance to horizontal motion would be virtually removed. If, then, there were any way to have a given ball leave the plane at different speeds of which the ratios were known, Galileo's old idea that horizontal motion would continue uniformly in the absence of resistance could be put to test. His law of free fall made this possible. The ratios of speeds could be controlled by allowing the ball to fall vertically through known heights, at the ends of which it would be deflected horizontally. Falls through given heights … [more]
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august 2017 by nhaliday
Noise: dinosaurs, syphilis, and all that | West Hunter
Generally speaking, I thought the paleontologists were a waste of space: innumerate, ignorant about evolution, and simply not very smart.

None of them seemed to understand that a sharp, short unpleasant event is better at causing a mass extinction, since it doesn’t give flora and fauna time to adapt.

Most seemed to think that gradual change caused by slow geological and erosion forces was ‘natural’, while extraterrestrial impact was not. But if you look at the Moon, or Mars, or the Kirkwood gaps in the asteroids, or think about the KAM theorem, it is apparent that Newtonian dynamics implies that orbits will be perturbed, and that sometimes there will be catastrophic cosmic collisions. Newtonian dynamics is as ‘natural’ as it gets: paleontologists not studying it in school and not having much math hardly makes it ‘unnatural’.

One of the more interesting general errors was not understanding how to to deal with noise – incorrect observations. There’s a lot of noise in the paleontological record. Dinosaur bones can be eroded and redeposited well after their life times – well after the extinction of all dinosaurs. The fossil record is patchy: if a species is rare, it can easily look as if it went extinct well before it actually did. This means that the data we have is never going to agree with a perfectly correct hypothesis – because some of the data is always wrong. Particularly true if the hypothesis is specific and falsifiable. If your hypothesis is vague and imprecise – not even wrong – it isn’t nearly as susceptible to noise. As far as I can tell, a lot of paleontologists [ along with everyone in the social sciences] think of of unfalsifiability as a strength.

Done Quickly: https://westhunt.wordpress.com/2011/12/03/done-quickly/
I’ve never seen anyone talk about it much, but when you think about mass extinctions, you also have to think about rates of change

You can think of a species occupying a point in a many-dimensional space, where each dimension represents some parameter that influences survival and/or reproduction: temperature, insolation, nutrient concentrations, oxygen partial pressure, toxin levels, yada yada yada. That point lies within a zone of habitability – the set of environmental conditions that the species can survive. Mass extinction occurs when environmental changes are so large that many species are outside their comfort zone.

The key point is that, with gradual change, species adapt. In just a few generations, you can see significant heritable responses to a new environment. Frogs have evolved much greater tolerance of acidification in 40 years (about 15 generations). Some plants in California have evolved much greater tolerance of copper in just 70 years.

As this happens, the boundaries of the comfort zone move. Extinctions occur when the rate of environmental change is greater than the rate of adaptation, or when the amount of environmental change exceeds the limit of feasible adaptation. There are such limits: bar-headed geese fly over Mt. Everest, where the oxygen partial pressure is about a third of that at sea level, but I’m pretty sure that no bird could survive on the Moon.

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Paleontologists prefer gradualist explanations for mass extinctions, but they must be wrong, for the most part.
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september 2016 by nhaliday