A major concept of the book was to connect the science with the actual story around it. This aspect endowed the book with a lot of fascinating science history and biographical information about the scientists. I particularly enjoyed the stories of how Galileo cleverly overcame the many obstacles he encountered when performing his experiments. Enjoy this excerpt from the book.
Galileo Galilei was born the oldest of seven children in Pisa on February 15, 1564 to Vincenzo Galilei and Giulia Ammannati. His father was both a music practitioner and music theorist. Although a number of his compositions were published Vincenzo made only a meager living as a performer and teacher.
From his most significant book Fronimo (containing many compositions for two lutes), we see the true extent of Vincenzo’s passion (or addiction) for music, learning that he played his lute:
“… walking in town, riding a horse, standing at the window, lying in bed.”
Galileo learned several things from his father. He became an accomplished lute player in his own right as a result of many duets (where he played second lute) with his compulsive father. Like his father, Galileo was a freethinker and they both enjoyed remarking how people who invoke authority to win arguments are fools. As a firm believer in empirical validation, Vincenzo carried out experiments to support his musical theories. In particular, he established the fundamental relationship between the pitch and corresponding tension of a string: the pitch (or frequency) of a vibrating string varies as the square root of its tension. His father’s high regard for the necessity of experimental observation to verify theory undoubtedly influenced Galileo, as it would become the cornerstone of all of his scientific inquiries.
A Swinging Chandelier
In 1583, so the story goes, that while attending mass at a cathedral in Pisa, Galileo observed the swinging motion of a chandelier (put in motion by a passing breeze or perhaps the person who had just lit it). It was clear to him, that if left undisturbed, consecutive swings became smaller and smaller, but how much time did it take for each of these consecutive swings to occur? Using his pulse to measure the time (precise clocks were not around yet), he was surprised to find that while the size of each swing gets smaller, the time for it to occur stayed the same; he was intrigued.
Although it’s uncertain if this story is true, Galileo’s first notes on the subject of a swinging pendulum (a good model for a swinging chandelier) occurred in late 1588 or early 1589, although he didn’t begin conducting experiments until 1602. A more likely story is that Galileo noted this type of motion while helping his father conduct experiments of his own involving the tension of musical strings in 1588–9. The experiments in question would have involved suspending varying weights from a string, plucking it and noting the pitch of the resulting sound. Clearly, small swinging motions of the attached weight would occur as an artifact of such experiments. Thereafter, Galileo probably recalled having previously seen the same motion in the swinging of a cathedral chandelier without ever having considered the physics of it. Indeed, this is what he has Sagredo say in Discourses on Two New Sciences:
“A thousand times I have given attention to oscillations, in particular those of lamps in some churches hanging from very long cords, inadvertently set in motion by someone, but the most that I ever got from such observations was the improbability of the opinion of many, who would have it that motions of this kind are maintained and continued by the medium, that is, the air. It would seem to me that the air must have exquisite judgment and little else to do, consuming hours and hours in pushing back and forth a hanging weight with such regularity.”
Rolling Down Inclined Planes
Galileo probably began some initial investigations with objects rolling down an inclined plane in 1602 but then, discouraged by the possibility of getting useful results, changed his focus to the pendulum. However, in 1604 Galileo devised a way to measure the accelerating speed of an object rolling down an inclined plane. The experiments that followed provided Galileo with quite accurate results that he then used to connect to the motions of free fall and a swinging pendulum.
Galileo wasn’t satisfied to simply know that two objects of different masses fall to the ground at the same time. Rather, he wanted to know how long it took for a falling object to reach a certain height above the ground. To be sure, he was looking for a mathematical expression relating height and time. Unfortunately, in attempting to solve this problem, Galileo had some experimental challenges to overcome.
While very accurate means of measuring distance and weight existed, this was not so for measuring time; Galileo needed to create a “stopwatch”. Galileo’s stopwatch consisted of a container filled with water that had a hole in the bottom of it. As the water ran out of the bottom of the container at a constant rate (of approximately three fluid ounce per second), Galileo was provided with an accurate way to measure time. Galileo describes his design and assures of its accuracy in Discourses on Two New Sciences (once again, through Salviati):
“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 decent, … the water thus collected was weighed, after each decent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.”
However, even with his water clock, an object in free fall was simply too fast for Galileo to accurately measure. Instead, Galileo devised a way to slow down the free fall while preserving the key physical results he was after, and to allow him to make accurate measurements with the water clock. Galileo’s plan was simple and elegant: have the object roll down an inclined plane.
The object now “falling” at this slower speed allowed Galileo to perform accurate measurements with his water clock. Galileo was convinced that whether an object rolls down to a certain height (via an inclined plane) or free falls to that exact same height, the underlying physics is still the same. Therefore he anticipated that the mathematical relationship to calculate the time to reach the height – although not exactly the same – was still very similar for both routes.
After all, the only difference between an object free falling to a given height, and one that is rolling down to that same height is that the latter moves in both the vertical (height) and horizontal (length) directions whereas the former only has a vertical direction since the object simply falls straight down to the ground.
It has been suggested that Galileo initially used another means of timing objects rolling down the inclined plane. Both Galileo’s father and brother were musicians, and Galileo himself was a competent lute player; perhaps Galileo made use of his musical inclinations. Placing rubber bands or, in Galileo’s case, gut “frets” around the inclined plane would have caused a sound to be made each time the object rolled over one.
Adjusting the placements of these frets such that the object reached each of them at the same interval of time would then allow an accurate determination of the time. In order to set such a placement, Galileo could have started the object rolling, listened for the “pluck” of each fret, and then using his internal sense of rhythm (perhaps he tapped his foot, or sang a marching tune), made the necessary adjustments in position until each fret represented a set interval of time.
Now all that was needed was to measure the distance from the initial position of the ball to each of the frets, which was something Galileo could determine very accurately. Given that the clocks of the day could not measure times shorter than a second, this method would have most likely been more accurate.