Abstract
The Gravity-Powered Calculator is an exhibit of the Exploratorium in San Francisco. It is presented by its American creators as an amazing device that extracts the square roots of numbers, using only the force of gravity. But if you analyze his concept construction one can not help but recall the research of Galileo on falling bodies, the inclined plane and the projectile motion; exactly what the American creators did not put into prominence with their exhibit. Considering the equipment only for what it does, in my opinion, is very reductive compared to the historical roots of the Galilean mathematical physics contained therein. Moreover, if accurate deductions are contained in the famous study of S. Drake on the Galilean drawings and, in particular on Folio 167 v, the parabolic paths of the ball leaping from its launch pad after descending a slope really actualize Galileo’s experiments. The exhibit therefore may be best known as a ‘Galilean calculator’.
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Notes
The distance from the end of the short pad.
The distance of the initial position of the ball along the graduated track, from its lower extreme, taken as origin.
“When we began the APE project, we weren’t entirely sure what active prolonged engagement meant. We knew that we wanted active visitors to feel as if they were in the driver’s seat, deciding from themselves what to try next, rather than following a set of instructions from the museum. By prolonged, we knew we wanted visitors to spend more time with the exhibits, getting more involved with the phenomena than they might at other exhibits. And finally, we had the idea that an engaged visitor group would be trying a variety of things at the exhibit, and that each action they took would somehow build on their previous actions”.
To avoid the psychological difficulty of ordinary visitors who might feel culturally distant from the great men.
The photo was obtained thanks to a reworking of my short movie performed by Maurizio Recchi whom I met at the AIF Congress in Mantua.
The Géomètre Cabri II plus is a dynamic geometry software widely used for teaching applications.
http://gallica.bnf.fr/ark:/12148/bpt6k94907p.r=edizione+nazionale+galileo+Galilei.langFR; Translated from the Italian and Latin into English (Galilei 1914, p. 244) at the site: http://www.archive.org/stream/cu31924012322701#page/n7/mode/2up.
“Continuance of motion at uniform speed in a straight line ultimately became the cornerstone of Newtonian physics. It is now called ‘inertial motion’: which Galileo allowed only for heavy bodies moving through relatively short distances near the earth’s surface. In his Physics, a heavy body must gain or lose speed if it approaches or moves away from the earth’s centre; that is, if it falls or rises at all. Over short horizontal distances, as in his 1608 experiment, the body could be considered as remaining at the same distance from the earth’s centre…” (Drake 1980).
“… the fact remains that in finding the essence of the motion of fall Galileo committed an error. For the ‘principle’ which he accepts as being sufficiently evident and natural the speed of the moving body (in free fall) is proportional to the distance covered does not at all lead to the law of the fall as he just formulated it” (Koyré 1978, p. 68).
In Italian at the site: http://gallica.bnf.fr/ark:/12148/bpt6k94909c/f118.image.r=edizione+nazionale+galileo+Galilei.langFR. Translated in English (Koyré 1978, p. 67): “Reflecting on the problems for which, for the demonstration of the accidents which I have observed, I lacked an utterly indubitable principle that could take as an axiom, I have arrived at a proposition which is most natural and evident; and with it being assumed I can demonstrate the rest, namely, that the spaces traversed in natural motion are in the squared [doppia, i.e. double] proportion of the times, and consequently the spaces traversed in equal times are as the odd numbers ab unitate, etc.”
In Italian: «Et il principio è questo: che il mobile naturale vadia crescendo di velocità con quella proporzione che si discosta dal principio suo moto…» (Galilei 1890–1909b). In English: “And the principle is this, that the natural moving body increases its speed in the proportion that it is distant from the beginning of its motion…” (Koyré 1978, p. 67). A meaningful figure helps the Author to explain this concept in the original document.
“Alexandre Koyré, a professor at the Sorbonne, managed to struggle to hold back his sarcasm: A bronze ball rolling in a ‘well polished and smooth’ gutter of wood! A bucket of water with a small hole from which the liquid flows out is collected in a glass and then weighed to measure the times of descent…How many sources of errors and accumulated inaccuracies ! It is obvious that these Galilean experiments are worthless” (Johnson 2008, p. 21).
“…what is offered by Galilean epistemology, which is both a priorist and experimentalist at one and the same time (one could even say that it is the latter because it is the former) are experiments which are designed on a theoretical basis and of which the function is to confirm or to refute the application to reality of the laws deduced from principles which themselves have a quite different basis” (Koyré 1978, p. 68).
“Had all of these been published by Favaro, it is probable that physicists would long ago have arrived at the same analysis of the experimental procedures employed by Galileo that will offered here” (Drake 1990, p. XVIII).
The site http://www.mpiwg-berlin.mpg.de/Galileo_Prototype/ contains a well organized collection of Galileo’s “Notes on motion”, which include the Folio 116 v and the other Galileo’s graphical documents.
Galileo exhorts us to consider this situation in the forementioned text of Two New Sciences: “Imagine any particle projected along a horizontal plane …”.
Vergara Caffarelli also draws attention to the similarity of the Folio 116 v with the figure reported by Galileo about 30 years later in his Discorsi, Fourth Day, in relation to Proposition VI on the amplitude of the semiparabola. This figure may be chosen as a key to interpret the geometry used by Galileo since 1608, for which the calculation of the horizontal distance reached depends only on the height of the table and the starting height of the ball. It suggests that, for Galileo, does count these data and not the inclination of the plane is actually used, so it can very well be neglected (Vergara Caffarelli 2009, pp. 173–174).
An example of a curved deflector was found in Folio 175 of Galileo (Drake 1973, p. 295).
“He expresses quantitative relationships between time, distance, velocity, and acceleration, in terms of proportions … In particular, he does not have the real number system and decimal notation at his disposal and does not formulate his conclusions in terms of equations involving variables and constants” (Hahn 2002).
In reality, when a ball rolls in an inclined channel, part of its kinetic energy is absorbed by the rotation, for which the exit velocity is slower of that of the translatory motion on a smooth plane. The ball behaves as if it were subjected to an acceleration of gravity 5/7ths of the whole, without considering friction. This observation, however, is not crucial for us to study for this exhibit. In regard to scientific and historical accuracy of this argument, see: Bonera (1995), Drake (1973, p. 300), Teichmann (1988, p. 130), Teichmann (1999, p. 126) and Vergara Caffarelli (2009, pp. 289–290).
“The phenomenon of the fall has always been a subject of meditation and astonishment for physics. It is, therefore, not surprising that Galileo … should continue to be concerned with this at Padua. He understood very well that a fundamental theorem, the fundamental theorem even, of the new science was involved” (Koyré 1978, p. 67).
Until recently it was called Institute and Museum of the History of Science.
But also by a pendulum.
This inset could be enriched by putting a diagram in it, like the one contained in Fig. 10; it ideally merges the old sketch outlined by Galileo with the modern historical interpretation of the elements in the actual device.
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Cerreta, P. The Gravity-Powered Calculator, a Galilean Exhibit. Sci & Educ 23, 747–760 (2014). https://doi.org/10.1007/s11191-012-9549-2
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DOI: https://doi.org/10.1007/s11191-012-9549-2