4.4 Getting the Swing of Things

Measuring time makes us think of accurate clocks and precisely defined units like seconds. Galileo was thinking in terms of ratios and, for him, it was sufficient to be able to divide time into equal intervals. Stillman Drake, Galileo's biographer, thought that he was able to do that using music.

As we have mentioned, Galileo composed music and is said to have played the lute quite well. Musicians are actually able to keep the time quite accurately for long periods while playing. They do this without external aids and without thinking of standard units like seconds. An internal rhythm allows them to keep an even beat, and to divide that beat in half again and again with a precision rivaling that of any mechanical instrument. Drake notes that, if the cymbalist in an orchestra were to miss his entry even by a 64th note, everyone in the audience would immediately notice it. For a musical person it should be perfectly feasible to accurately divide a period into equal intervals without any external timekeeper [5].

In the 1970s, Drake studied a previously unpublished page of Galileo's working notes. It contained lists of figures scribbled on a page. He says that it looked singularly unpromising at first glance but he slowly realized that it recorded an actual experiment. There were three columns on the page, which are reproduced in Table 4.1. The second column seemed to be an index, containing the integers from one to eight. The first column contained the squares of these integers, but it differed from the others by being written with a different pen and ink. Drake concluded that it must have been added after the other two. The figures in the third column very nearly represented the distances covered by a ball rolling down an inclined plane at the end of eight equal times. Drake points out that these distances must have been measured, because they were not exact multiples of the squares in the first column. Also, they could not have been measured with the water clock, as described by Galileo in the published version of the experiment. That would require dividing a single time into eight equal intervals, which is impracticable using a constant flow of water. It could have been done using a pendulum, but keeping track of the motions of the ball and bob at the same time would have been challenging. Another possibility is to use the sound made by water droplets dripping into a container at a constant rate, but also this requires two external sensory impressions to be coordinated. Drake believed the experiment to have been much simpler and more elegant [5].

Table 4.1 Figures entered in Galileo's notebook, recording an actual experiment. The third column very nearly represents the distances covered by a ball rolling down an inclined plane at the end of eight equal times. The second column is an index. The first column was written after the others, in different pen and ink, and contains the squares of the figures in the second column.

Being a competent lute player, there is little doubt that Galileo could keep a precise beat. If the ball could be made to produce sounds during its descent, they could be used to divide the time into equal intervals. One way to do this is to attach adjustable frets to the plane. Unlike modern string instruments, which have fixed frets, the lutes that Galileo was familiar with would have movable frets around their necks. Attaching such frets to the plane, the ball would make an audible bump each time it passed over one. In his reconstruction of the experiment, Drake could easily find the approximate positions of the frets by firmly singing a slow march while releasing the ball. In just three or four runs he was able to make eight chalk marks on the board that represented the beats of the song. Instead of the gut frets that Galileo would have used, he put rubber bands around the plane at these positions, as indicated in Figure 4.3. It was then easy to adjust the locations of the frets during a few runs to obtain an even beat. The accuracy of the results was better than a 64th of a second.

Figure 4.3 Schematic description of Drake's reproduction of Galileo's experiment. Placing rubber bands on the board, the ball makes audible bumps when passing them. In a few runs the locations of the bands can be adjusted to produce an even beat.

Drake's reconstruction led him to suspect that the occasional plus and minus signs after figures in the third column of Table 4.1 indicated that Galileo thought that those sounds seemed a little early or late, but not enough to require further adjustments. It also allowed him to find a physical explanation for a certain discrepancy in Galileo's data. His second fret seemed to be placed a little earlier than theory would predict. Drake saw that the ball was delayed in climbing the first fret, since it moved slower in the beginning of its descent. To obtain an even beat the second fret had to be moved closer than theory predicted, otherwise the second time would not be equal to the first. Once the ball was underway it passed the other frets without effort.

With this ingeniously simple method it was obviously possible to divide the descent of the ball into equal time intervals more accurately than any water timer could measure, and without having to rely on any other external means. The figures in Table 4.1 also imply another important conclusion: Galileo did not yet know the times-squared law when he performed the experiment. If he did, it would have been pointless to measure the distances. A single run would have been enough to confirm the law and there would have been nothing to write down. Calculations on the manuscript page along with evidence that the first column was written after the third indicated that the measurements led Galileo to the law of free fall, not the other way around. The experiment was an exploration of nature, an attempt to seek an unknown rule from measurements. Drake also pointed out that if his reconstruction was correct it would surely explain why Galileo never published the experiment. Even in his days it would be foolish to state that he had arrived at a precise mathematical law by humming a song while watching a ball roll down a plank [5].

Galileo's notes reveal that he probably performed the experiment essentially as Drake describes it. Regardless of this, the reconstruction teaches us another important thing about the experiment: there are different ways to carry it out. Even without a sophisticated timekeeper, there is more than one way to obtain Galileo's law with the required precision, using little more than an inclined plane and a ball.