Galileo on the Acceleration of Falling Bodies
Galileo's Dialogue, Day 2, Part 3
As I wrote in a recent post, I intend to write posts for each of the four days in the Dialogue Comparing the Ptolemaic and Copernican Systems1, the book that got Galileo in trouble with Rome. This is the third post about Day Two, the longest day in the Dialogue. Here are the previous posts:
Day One: Galileo Dismantles Aristotle's Separation of Earth from the Heavens.
Day Two, Part 1: Galileo Attacks Aristotle’s Followers.
Day Two, Part 2: Galileo Insists the Earth is Spinning on its Axis.
This specific post focuses on a specific statement by Salviati about the acceleration of falling bodies:
First of all, it is necessary to reflect that the movement of descending bodies is not uniform, but that starting from rest they are continually accelerated. This fact is known and observed by all, except the modern author mentioned, who, saying nothing about acceleration, makes the motion uniform. But this general knowledge is of no value unless one knows the ratio according to which the increase in speed takes place, something which has been unknown to all philosophers down to our time. It was discovered by our friend the Academician, who, on some of his yet unpublished writings, showing in confidence to me and to some other friends of his, proves the following.
The acceleration of straight motion in heavy bodies proceeds according to the odd numbers beginning from one. That is, marking off whatever equal times you wish, and as many of them, then if the moving body leaving a state of rest shall have passed during the first time such a space as, say, an ell, then in the second time it will go three ells; in the third, five; in the fourth, seven, and it will continue thus according to the successive odd numbers. In sum, this is the same as to say that the spaces passed over by the body starting from rest have to each other the ratios of the squares of the times in which such spaces were traversed. Or we may say that the spaces passed over are to each other as the squares of the times. [pp.221-222]
The “modern author” is Johannes Gregorius Locher, a student of Christof Scheiner. The book in question is the Disquisitiones mathematicae de controversiis ac novitatibus astronomicis (Mathematical discussions about astronomical controversies and innovations), Ingolstadt, 1614. The Academician is, of course, Galileo.
Here is a table showing the development that Salviati describes:
As far as I know, this is the first time that appears in print the relation of the distance travelled by a falling body with the square of the time elapsed. This shows the evolution of Galileo’s thought since the time that he was collaborating with Paolo Sarpi. See my previous post Sarpi and Galileo Drop Aristotle, which points out that at the time they thought that “[t]he velocity of a body in free fall depends on the distance travelled, as opposed to the time elapsed.” The formal presentation of this topic appears at the beginning of Day Three in the Discourses and Mathematical Demonstrations Relating to Two New Sciences, Galileo’s last work, published in 1638.
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Galileo Galilei. Dialogue concerning the two chief world systems — Ptolemaic and Copernican. Translated by Stillman Drake, foreword by Albert Einstein. University of California Press, 2nd ed., 1967.