Michael Fowler, University of Virginia
Galileo and Einstein is an introductory physics course designed for first year nonscientists given each Spring at the University of Virginia with an average enrollment of just over a hundred students.
About ten years ago, the University of Virginia began encouraging faculty to give University Seminars, which could be rather informal one-semester two credit hours minicourses, intended mainly for first year students, with a maximum enrollment of twenty. I had been reading and enjoying Galileo's Two New Sciences at the time, and it occurred to me that this text could perhaps be the basis of a University Seminar for nonscience students. The book covers a wide variety of topics. Admittedly, some sections are hopelessly obscure, but then whole stretches of the book are beautifully written and very illuminating. For example, Galileo gives a clear analysis of why there can be no giants, and the arguments he presents are as relevant as ever—easily applied to demolish many Hollywood fantasies, from giant ants to shrunken kids. His discussion of falling motion is of course a classic: why Aristotle had to be wrong, and, more practically, how to figure out the range of a cannon. When I began to put the Seminar together, I decided to add some selections from The Starry Messenger: Galileo's memorable description of his discovery of the Moons of Jupiter, the phases of Venus, and mountains on the Moon, and how well it all fitted into Copernicus' scheme.
I used these books for the Seminar because they were written almost four hundred years ago. Their insights and conclusions followed from simple observation and careful reasoning, not dependent on the abstractions and subtleties of modern science. Even calculus hadn't been invented when they were written. Galileo aimed for a wide audience, and wrote in a breezy style (I used the Crew translation, more accessible than Drake). I felt that once our students got used to the approach, they would be able to follow the arguments pretty well.
As I soon discovered, though, Galileo's audience knew a lot more than our students. He assumed some knowledge of elementary Greek geometry, and of Greek astronomy. I found my students knew almost no geometry, and many of them did not understand, for example, why there are seasons. I was obliged to back up and give some introductory lectures on Greek geometry and astronomy. These proved very enlightening for the students. They were amazed to discover that the Greeks had a fairly accurate measure of the size of the earth—they thought that before Columbus, no one really believed the earth was round. They were dumbfounded that the Greeks had measured the distance to the Moon, and got it close to right.
To give the students some insight into how the Greeks reached all these remarkable conclusions, I had to get them thinking geometrically, at least in a semiquantitative fashion. I began by giving out practical homeworks, which they did in pairs. We began with some very simple surveying around the University Grounds, estimating the heights of buildings by pacing out distances on the ground, measuring an angle, then doing a scale drawing (no calculation, and certainly no trigonometry). Then they measured the angle of the Pole Star and another star, and found on returning a couple of hours later that the Pole Star hadn't moved, the other star had. We built up a picture of how the stars move through the sky, and how that movement would be perceived at different places, for example at the North Pole or at Quito, on the equator. We next discussed sundials at different places on earth, and why clocks go clockwise. I used a white styrofoam ball illuminated by a spotlight, against a black background, to illustrate phases of the Moon. By putting a small video camera on the surface of a globe (the Earth) and using it to "observe the Moon" (the styrofoam ball) it was possible to show how the crescent Moon appears at different angles to the horizon when observed from different places on Earth (the video camera was small enough to move around on the globe). The aim of course was for them to build a clear three-dimensional picture of the earth-moon-sun system, a necessary preparation for understanding just how the Greeks figured out the distance to the Moon by watching eclipses and drawing diagrams. This minireview of Greek science proved to be an effective preparation for Galileo.
Along with reading Two New Sciences, we did some of the experiments. We rolled a ball down a ramp, and timed it with a water clock. This was done by a group of students, who did indeed discover that the ball rolled four times as far in twice the time (contrary to their expectations!) Perhaps it's worth noting here that whenever we did any kind of demonstration or experiment, I asked the students to think first what they expected to see. For the ramp, I would ask the class to predict how far the ball would roll in twice the time, and have them write down their prediction. I gave them two or three minutes to do that. Then I asked them to consult with their neighbor and try to reach a consensus. Finally, they got together in groups of four. I would pick a group at random, ask them what they predicted, and invite comment where appropriate. After some discussion, we rolled the ball down the ramp to find out if they were right. Finally, I presented Galileo's argument as to why the ramp motion should be a good representation of falling motion, and mentioned its shortcomings as a way to measure acceleration of a falling body.
Next, we did an experiment Galileo would have liked to have done—dropping a ball in front of a measuring stick, making a video, and playing it back frame by frame to see how far the ball falls between successive frames. We spent a great deal of time (as did Galileo) analyzing the distances fallen in successive time intervals, and verified Galileo's prediction that the intervals go as the odd integers. We plotted graphs and defined acceleration. I found it was also well worth doing the slightly more tricky experiment of videoing a ball thrown directly upwards, and having the students plot the position, average velocity and acceleration at a sequence of points including the turnaround, because, despite everything taught so far, they still confuse acceleration and velocity, and will tell you there is zero acceleration at the topmost point. (This is a good question to ask them in class, and have them consult with neighbors, etc.) Another neat video experiment we did was to project a ball horizontally, and discover from the video that the vertical motion was the same as for a falling ball, but the horizontal motion was at uniform speed. This is of course a direct representation of the famous "Compound Motion" Galileo used to find the range of a cannon as a function of muzzle speed and angle of projection, and his diagram from Two New Sciences can be placed next to the video. It's an excellent match.
Having worked so hard to understand acceleration in projectile motion, I couldn't resist taking the next step, beyond Galileo. It takes very little work at this point to extend the analysis, following Newton, to a projectile fired from a high mountain, a cannonball moving so fast that the earth's curvature must be taken into account. This was the crucial step that Galileo missed: it unifies the familiar motion of a thrown stone with the celestial motion of the moon. From there to the planets and Universal Gravitation is straightforward. I later made an applet of Newton's cannon on a mountain, with the help of a graduate student programmer. It brings the famous image in the Principia to life. You can easily find it on the web by googling Newton mountain.
After teaching the seminar this way in my first year, I decided to drop some of the more obscure sections of Two New Sciences I had been covering, and instead to give lectures on Kepler and Brahe. These proved popular, and at that point I felt I had a complete course, beginning with Thales and Pythagoras, and ending with Newton's laws and Universal Gravitation. In fact, as must be evident from the above, I found in practice that working through Galileo's arguments in Two New Sciences was an excellent preparation for Newton's Laws. Galileo already states the First Law, in essence, and the sticking point in the Second Law is understanding the very difficult concept of acceleration. Galileo's thorough discussion of one-dimensional acceleration, and his "Compound Motion" generalization to parabolic motion in two dimensions, leads naturally via the Newton Mountain example to circular motion. Following this trail makes it much more plausible that steady circular motion means acceleration towards the center. (It is also helpful to rediscuss parabolic motion, taking the example of a cannonball projected upwards at an angle, and think about the acceleration at the topmost point, comparing that with circular motion). To round out the discussion, this is an appropriate place to introduce the idea of velocity as a vector (admittedly a postNewtonian concept) and acceleration as the rate of change of that vector quantity.
It is perhaps not evident from the above that I used countless demonstrations, most of them involving the students, to illustrate these concepts. I tried to stay away from fancy equipment. For example, in discussing the Second Law, to convince them that a centripetal force was necessary for circular motion, students attempted to make a bowling ball go in a circular path on a smooth floor by guiding it with a broom. They almost always tried by pushing it from behind, reminiscent of Kepler's theory of planetary motion, and of course classically Aristotelian. However, they quickly discovered that they really did have to push the ball "sideways" to get circular motion. We tested the Third Law by having students of very different masses collide in carts, using old kitchen spring scales to bounce off each other and at the same time measure the force felt be each. Another "weighing" experiment illustrated Galileo's assertion that one does not feel the weight of a load on one's shoulder if one is falling (incidentally, a precursor of Einstein's "happiest thought" that led to General Relativity). I had the students weigh themselves on a spring scale in an elevator, and to note how their weight varied as the elevator started and stopped.
The problem with the course as described above was that although it received very positive student evaluations, and I believe gave them some sense of Newtonian mechanics and how the ideas evolved historically, it did not attract enough nonscience majors, at least in the eyes of the administration. This is why I brought in Einstein. I don't feel too apologetic, in many ways it was a natural extension of the course—Newton was the first major revolution in Physics, and Einstein the second. Furthermore, it made evident the limits of validity of Newton's Laws. And, actually, it is easier for students to grasp the essential points of Special Relativity that it is for them to understand Newton's Laws, at least in my experience.
So devoting the last third of the course to Einstein has led to a much more popular course, with over a hundred students enrolled. This has necessitated cutting out some of the earlier historical material, but there is still time for a reasonably thorough coverage of many of Galileo's arguments, and of Newton's Laws.
All the course materials are readily available on the web, including full lecture notes. My website was visited 375,000 times in the year 2000. To visit, just google Galileo Einstein.
The Basic Books
Galileo Galilei, Dialogues Concerning Two New Sciences. Translated by Henry Crew and Alfonso de Salvio, Dover NY, 1954 but still in print. ISBN 0-486-60099-8.
Galileo Galilei, Sidereus Nuncius (The Starry Messenger). Translated by Albert Van Helden, Chicago, 1989. >ISBN 0-225-27903-0
November 8, 2001