An Elementary Illustration of Theoretical Understanding

Galileo’s Law of Falling Bodies

Not only does the empirical scientist select aspects of data and measure them; [s/]he also relates the measurements to one another. With that last step science moves totally outside the viewpoint of common sense.1Op. cit. (CWL 10) 1997, 139-40.

How is a scientific insight generated? “[I]t is that movement from sensible data to law that we have to consider.”2“The idea of obtaining a law is the idea of obtaining a formula that holds for not only those cases but all intermediate cases, and all cases that would be obtained by going beyond the amount of time involved.” Ibid. (CWL 10) 1997, 134. We are drawing attention, in a preliminary fashion, to the reach of the dynamics of knowing (and its elements identified in the first four boxes) introduced in Journeyism 10. Please note the elements are identical to the “MAC” process discussed in Journeyism 9. “A basic example of scientific insight is Galileo’s discovery of the law of falling bodies.”3Ibid. (CWL 10) 1997, 131. “What Galileo did was select measurable aspects or elements in what he wished to investigate, that is, the free fall of a body.”4Ibid. (CWL 10) 1997, 131.

When Galileo moved from measuring distances and times to correlating distances and times, he was bringing together two objective, measurable features of objects. He was relating things to one another … .5Op. cit. (CWL 10) 1997, 139-40.

Exercise6This exercise (including footnotes 7-11, 13-15) is taken directly from Terrance Quinn, Invitation to Generalized Empirical Method, World Scientific Publishing Co. Pte. Ltd., 2017 [in Chapter 1.7, “Space and Time”], 21-26.

Now let us recall and re-enact some of Galileo’s discoveries, and along the way make a few elementary observations about what we are doing. First, let’s begin with Galileo’s description of the apparatus he used in his experiments.

A piece of wooden moulding or scantling, about 12 cubits7A cubit: ancient measurement, elbow to finger tips, ancient unit of measure based on the forearm from elbow to fingertip, usually from 18 to 22 inches, early 14c., from Latin cubitum “the elbow,” from PIE keu(b)- “to bend.” Such a measure, known by a word meaning “forearm” or the like, was known to many peoples (e.g. Greek pekhys, Hebrew ammah, English ell). long [about 7 m], half a cubit [about 30 cm] wide and three finger-breadths [about 5 cm] thick, was taken; on its edge was cut a channel a little more than one finger in breadth; having made this groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished a possible, we rolled along it a hard, smooth, and very round bronze ball.8Galileo Galilei, Dialogues Concerning Two New Sciences [1638], trans. Henry Crew and Alfonso De Salvio (New York: McGraw-Hill Book Company, 1963), “Naturallly Accelerated Motion.” Online, see, for example, Galileo Galilei, Dialogues Concerning Two New Sciences [1638], Online Library of Liberty,

What is speed? Galileo considered the ratios of distance traversed to time elapsed. But, how might such a ratio be obtained? It is one thing to measure distance. How might time be measured?

For the measurement of time, we employed 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… the water thus collect was weighed, after each descent, on a very accurate balance; the difference and ratios of these weights gave us the differences and ratios of the times …9Galileo Galilei, Dialogues. See note 8.

Galileo also described details of his experiment.

Having placed this board in a sloping position, by lifting one end some one or two cubits above the other, we rolled the ball, as I was just saying, along the channel noting, in a manner presently to be described, the time required to make the descent. We … now rolled the ball only one-quarter the length of the channel; and having measured the time of its descent, we found it precisely one-half of the former. Next we tried other distances, comparing the time for the whole length with that for the half, or with that for two-thirds, or three-fourth, or indeed for any fraction; in such experiments, repeated a full hundred times, we always found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the plane, i.e., of the channel, along which we rolled the ball.10Galileo Galilei. Dialogues. See note 8.

Provided in the table below are some of Galileo’s measured times and distances. Galileo measured shorter distances with a unit called points, which today is about 29/30 mm. As shown on the chart, in total, the ball rolled about 2104 points, or 2104 x 29/30 mm, or about 2,034 mm (2.34 m), less than 7 feet in eight time intervals.


Time Distance (points) Distance divided by 33
1 33 1
2 130 3.94
3 298 9.03
4 526 15.94
5 824 24.97
6 1192 36.12
7 1620 49.09
8 2104 63.76

In one column we have the measurements for time: 1, 2, 3, …, 8. How far does the ball roll in the first unit of time? In the right column, this distance is labeled 1. What about the measured distance for the second unit of time? In points, the distance is 130. But, compared to the first distance over the first unit of time, it is 3.94 x 33, that is, 3.94 times the first distance. And so on. Now, what is each measured distance compared to the measured distance traversed in the first unit of time (that is, if all the measured distances are divided by 33)? The resulting ratios 1, 3.94, 9.03, … are (approximately) equal to the squares of the measured times 1, 2, 3,…!

We have measured distances and measured times. Let us start with distances. How are measured distances obtained? Let’s observe what is happening when obtaining measurements with an elementary ruler. A ruler may be a length of wood, say, with markings that one can see (or, for the blind, touch). It will help to get an actual ruler. Markings also are numbered: 1, 2, 3, … . How, though, are locations of markings and their numberings determined?

One has some kind of elementary length – a finger’s breadth, or perhaps a small length of wood such as what in Galileo’s time was called a “point” (or for a larger measuring beam, a cubit, a unit length that is the distance from one’s elbow to one’s finger tips)11See note 8.. To make a ruler, one places one end of the unit length at the beginning of the ruler; a mark is made where the unit length ends. Then, continue in this way, until a series of markings is produced. Along the now marked ruler, we have visible lengths that we correlate with markings.12We have added four images (two for distance and two for time) to aid elementary insight into what is meant by a series of correlations of correlations. In that context, we would also note further advanced implications for scientific insight derived from the field of mathematics with respect to the meaning of “abstraction.” See Op. cit. (CWL 10) 1997, “2.3 Abstraction: What is Abstracted From,” “2.4 What One Reaches by Abstraction,” and “2.5 Abstraction and Operations: Group Theory,” 124-132.

Note, here, there are no numbers yet. How do we go on to include numbers, such as when one measures, say, “5 points”?

We do something more here, yes? Do we not also correlate the number ‘5’ with one of the markings, which, as above, one also correlates with a visible multiple of a seen length along the ruler? In reaching elementary measurements there is, then, a nesting of various insights. In saying “5 points,” one correlates ‘5’ with what one already is correlating. And, a table of such measurements … represents a series of correlations of correlations… .

What about time measurements? … The challenge is to describe how one measures time in the law of falling bodies. Keeping to that context, we may recall that Galileo used a water-clock to measure time,13See note 10. weighing volumes of water that flowed from a vessel of water, and that water

collected was weighed, after each descent, on a very accurate balance.14See note 10.

All along there are assumptions about rates of flow, as well as a relationship between volume and weight.15Galileo was aware of Archimedes’ principle of displacement (Laura Fermi and Gilberta Bernardini, Galileo and the Scientific Revolution (Mineola NY: Dover Pubs., 200; first published, New York: Basic Books, 1961) 22, 23, 69, 114, 118). The exercise is somewhat easier if, instead of following Galileo, one uses a modern graduated cylinder with markings at regular heights along its edge (corresponding to measured equal increments of volume), or some kind of elementary spring or pendulum.

In any case, it is through further exercises in self-attention that one can find that, just as for distance measurements, series of time measurements are obtained though a series of correlations of correlations.

What, now, of Galileo’s insight pointed to above – that the ratios 1, 3.94, 9.03, … are (approximately) equal to the squares of the measured time 1, 2, 3, …? In that insight, do we not, in fact, reach yet a further correlation? [The resulting ratios 1, 3.94, 9.03, … are (approximately) equal to the squares of the measured times 1, 2, 3, … .] In other words, in understanding Galileo’s Law of Falling Bodies, one reaches a correlation of correlations of correlations.16Quinn’s exercise ends here. “[T]he distance is proportional to the time squared, or in more modern notation s + vt = gt2 / 22.” Op. cit. (CWL 10) 1997, 133. The notation and our exercise in Galileo’s elementary discovery is put into perspective by Quinn in an earlier footnote: For the physicist, what are speed and acceleration? And in what way, precisely, is the speed of a free-falling object accelerated? [F]or later classical physics, velocity and acceleration are defined as first and second derivatives. “What does a (classical) physicist mean by velocity? He means ds/dt. What does he mean by an acceleration? He means d2s/dt2. If you know what is meant by those symbols from the differential calculus, you know exactly what is meant by acceleration and velocity, and if you do not know what those symbols mean, you do not understand acceleration and velocity” (Bernard Lonergan, Topics in Education, Collected Works of Bernard Lonergan, vol. 10, ed. By R. M. Doran and F. E. Crowe (Toronto: University of Toronto Press, 1993). For an introduction to elementary calculus, see Terrance J. Quinn, “The Calculus Campaign,” Journal of Macrodynamic Analysis, 2 (2002): 8-36.

And so, the aim of scientific insight (or explanatory understanding) “is to turn attention away from all directly perceptible aspects and direct it to a non-imaginable term that can be reached only through a series of correlations of correlations of correlations.”17Op. cit. (CWL 3) 1993, 271. “It is the fundamental step in an empirical science.”18Op. cit. (CWL 10) 1997, 133.


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