Teaching Calculus Using History (Part 2: Galileo Brings the Reality Check)
Link to the first post in this sequence
We appreciate the contribution that Aristotle left us, but as we leave him we discover something disturbing. At what point do people start to check Aristotle's claims and find them false? Well there are some rumblings here and there in the history. But over the next thousand years, his teachings spread to the Christian and Muslim world, and become the philosophical and even religious orthodoxy of the western world.
The historical events leading up to the Scientific Revolution are interesting, but they are many and complex. So I'm skipping over all of the Classical Period, Dark Ages, Medieval Period, and all the background of the Scientific Revolution.
Here we arrive at Galileo, unhappy with the teachings of Aristotle. The difficulty of figuring out celestial mechanics (the motions of the planets through the sky) was always a headache from the start. Galileo made important contributions to this study, but I will largely ignore them and his other contributions. Instead I will only describe his work on free fall.
As a more "terrestrial" issue, Galileo and others had noted that objects in free fall do not seem to fall at a constant speed. They fall slowly at the start and faster at the end, and continuously increase their speed along the way, or so it seems. But how to measure and prove this? Time measurement was not very precise, and objects fall too fast to be measured well by sand clocks anyway.
So ... Tower of Pisa right? That's the Galileo building right? He dropped things off the tower?
Yes, it's a neat story, and really is important in all the experiments that Galileo did. If I recall from memory, that particular experiment was meant to test whether objects of different masses fell at different rates. Intuition says heavier objects fall faster, but observation proved shockingly that they don't. Whenever we get around to building a theory of physics, we should remember to come back and see if our theory agrees with this empirical result.
But that doesn't really resolve the problem of doing precise time measurements in order to clock whether an object's speed at the beginning of a free fall is faster or slower at the end. To resolve this measurement problem, Galileo had a brilliant resolution! It's so simple, but I bet most people wouldn't think of it: He slowed the object down during its descent, so that a rougher measurement of time would become accurate enough!
Ok ... but how did he slow down the free fall of an object? Great question, whoever asked it! He used the inclined plane. Oh the inclined plane, the scourge of the high school physics student! These poor little kids do inclined plane problem after inclined plane problem. And yet little do they know that it has its incredibly noble origin in the very experiments that Galileo used to overturn Aristotelian physics.
Galileo had the idea of setting a round or cylindrical object on an inclined plane and letting it roll down. This effectively was like freefall but in slow-motion. This gave him the time necessary to measure the speed of the object at the beginning of its descent, and then also at the end of its descent, and to prove that its speed in fact increases throughout.
This was not without controversy. For one thing it overturned Aristotelian teaching, which by this point had become part of the official philosophy of the Catholic Church. This made people motivated to deny Galileo's findings.
I can hear you now, dear reader. "But so what if they wanted to deny his findings! His findings were measurable, reproducible, quantitative! You can't argue with facts!" What are you from the 2000's? We now know that people will find ways to argue with the plainly observable and measurable, and do so with gusto. That is no more true now than it was in the 17th century.
One interesting objection was ... that this was an experiment! Yep, they thought experiments were bad! But they actually had an argument, tracing back to Aristotle, which was actually somewhat sympathetic. Aristotle believed in pure observation. No putting your hands on it, no interfering with nature. If you get involved, you are necessarily biasing the observation with your own input to the system! You might bias it in ways that we can explain, but you might also bias it in ways that nobody can quite understand. So fundamentally, these Aristotelians thought, you should never study nature by looking at things that humans had built.
Of course this is a fundamentally ancient way of thinking, and it makes very little sense to a person born in the modern world. We are so used to thinking of a materialistic world, and we are so used to thinking that experiments are the gold standard of scientific conduct, that we cannot quite make sense of this ancient way of thinking. From the ancient view, I think there was a belief that human interaction with something necessarily imbued it with an invisible influence and memory of the interaction which is not purely material but perhaps spiritual, or something like it.
But we should so much the more appreciate Galileo and his predecessors for breaking us out of this old way of thinking. It took something like 180,000 years to do it, but guys ... we finally did it!
As revolutionary as this breakthrough was, in some sense, the hard part is yet to come. It's one thing to burn an ancient theory down with new and better measurement tools and techniques. It's another thing to build a better theory in its place.
Galileo knew that an object in free fall would increase its speed continuously throughout its descent. But ... what did that mean for the motion of the object?
Although the Aristotelian physics was wrong about an object having a constant rate of descent, it did have one thing going for it: Mathematical simplicity. If your velocity is constant, any high schooler can describe the change in the position of the object. If an object falls at, say, 3 feet per second, then if you let it fall for 5 seconds it will move a total distance of ... you guessed it ... 15 feet.
But what about in the new Galilean paradigm, in which objects do not have constant velocity? The velocity increases. That is to say, the rate of change changes. We call this the acceleration. That is to say, the acceleration of an object is the rate of change of the velocity. In the Aristotelian paradigm, although Aristotle didn't think of it this way, he basically said that the acceleration was zero (that objects had constant speed throughout their free fall). In the Galilean paradigm they have positive acceleration during free fall.
But to just say that it's positive is not enough information. What exactly IS the acceleration? Is it constant? Does it vary? If it varies, what makes it vary? The height from which it's dropped? Some other factors? Oh man, Galilean physics is stressful.
Let's just start at the easiest possible hypothesis. Let's assume that the acceleration is constant, and see if there is some specific constant number which would match our experimental data.
But here is where Galileo reached the limits of his mathematical abilities. This is a math problem! And he didn't have the mathematical tools to be able to solve it.
So sad to be Galileo, not knowing math. Well, I guess that's the end of the story, and we will never understand the motion of bodies in free fall ...
Or will we?