All right, I'm starting with a slide that corresponds to the one I ended with last week, where I listed the four Galilean claims. And raised the question of the evidence for them. The difference this time Is I'm simply listing the sources, I am listing where the evidence is, almost more than describing the evidence itself, but it's a list, so just remember what they are.
Dissent and the absence of air resistance is the same for all bodies, regardless of weight, shape and density, that's a remarkable claim. Remarkable in at least two respects. Remarkable historically because one would have thought not, but also remarkable because it will continue right down to Einstein. It's a very, very important claim.
It's a fundamental principle of Einstein's general relativity. The second one then is the uniform acceleration in absence of air resistance. The third is the path wise independence principal as I'd like to call it. In the absence of air resistance you acquire the same speed, regardless we'll come back to that tonight, and the fourth is the speed acquired in descent is exactly sufficient to raise it back to the original height.
All of these, you probably weren't told about the third one when you were in high school and first learned these things. But it's actually the one that's most important up there in many respects because it's a fundamental energy proposition. What I want to call your attention to is there's a list of so-called evidence up here, but think about for a moment what the evidence really consisted in.
It consisted of claims made by Galileo and Riccioli. Galileo says nothing about anybody else being present when he does the experiments. He gives us no data. He simply has this remark within a heartbeat, etc. And Riccioli's case, he says something that gives his results a slight bit more credibility, namely, Grimaldi was involved as well.
Grimaldi is a major scientist in history. He's the one who discovered the phenomenon of optical diffraction, and is no small minor figure himself. He was of course a colleague at Bologna, but regardless, picture yourself as not in Italy and you read these and you're told there's evidence. Do you just take people's word for it that there's evidence?
And that what particularly bothers me here after Riccioli published his results, which I pointed out last time are distressingly perfect. Doesn't mean he didn't get them, and remember he was kind of disapproved Galileo, so there's a credence that lent him. And he also says those are his two best of many experiments.
Which may be the two in which he got exact agreement and maybe the others it wasn't so good. But the bothersome thing to me is, once you know the distance of fall in the first second is 15 Roman feet, a very natural experiment suggests itself. Go back to Galileo inclined plane experiments.
You now know the distance of fall in the first second. Therefore, you can predict the exact amount of time it's supposed to take to go down any incline plane. And if you were to do the experiment, you would be off by somewhere around 20%. Okay. Or more, actually.
You could be off by as much as 28%. And, nobody seems to have done that experiment. When I put out that $200 prize for anybody who could show an experiment distinguishing between rolling and falling, that's the one that I just thought was natural. Now there are two possibilities, nobody did it, which is a comment about experimental practice at the time or people did it, got a result they didn't expect and decided not to publish.
You'll see reason to believe the latter later tonight, okay, but either way it's a comment of sorts about experimental practice at the time. When Kepler put out the Ruldophian tables everybody was in a position to cross check them. And he made it even easier for people by putting out the affirmarities giving predictions to various things over the next few years.
What's the cost? What's the check of the claims made by Riccioli and Galileo? Well the natural thing to do is to do follow on experiments of some sort. That you expect yield interesting results, but when they don't yield interesting results, they will expose something is wrong. In fact, I'll make a rather extreme statement.
Successful experimental programs almost always lead on to further experiments that are generating new information, but in the process, checking past experiments, and that just seems not to have happened here. Or if it happened, it didn't make the public literature. Or at least I haven't found it anywhere in the public literature.
One should be cautious here. But the fact that Berteloni mainly was at Dibner the first year I was director, and he spent the whole year reviewing 17th century literature. We had the best library in the western hemisphere I think for that subject, and he couldn't find anything, makes me think it's gonna be very hard to find in the literature.
At any rate, just a reminder, the next slide's kind of minor, and you've seen it before. The three problems with doing experiments in mechanics at the time. The times were very short. Now, five seconds is not that short. But a half second error in five seconds is a substantial error.
You know, it's a 10% error all by itself. So, with the time short you need very high precision not to have substantial percent error. No way to measure velocities. You're gonna find Galileo, you've already read it. Galileo proposes a way to measure velocities in the fourth day and it became the preferred way to measure velocities for the next roughly 100 years.
I'll let you judge for yourself when we get to it. And then there's this question of air resistance and what you do to control for it. Now the point I'm making and repeating these three points is the upshot. Namely, the way you, anytime you look at an experiment and you want to ask yourself critically, you're going to be the outside person criticizing the experiment.
The right move to make I claim, is to ask yourself, what are they going to make of discrepancies, of violations of expectations from the experiment. And, in particular, they have predictions and get discrepancies. What are the discrepancies supposed to be telling them? Well, my comment is, because of these three factors, discrepancies are gonna be ambiguous.
It may be insufficient control of external effects. It may be measurement error. It may be bad theory. Now, your theory that needs revision. Anywhere you have that ambiguity you equally have an ambiguity if there's no discrepancies. What's the absence of discrepancies telling you when you know you've got experimental problems?
Are you just being lucky? Or what? Now one of the many beautiful things about Galileo, and I'm now gonna be really praising of him tonight because, this. His protege Torricelli remarked that day four of two new sciences was the best thing Galileo ever did. That's a paraphrase, but I'm inclined to think in many respects that way too.
So Galileo's approach to dealing with the problems of doing experiment, is first to develop a very rich mathematical theory from hypotheses that at least appear reasonable on their surface, but more important, they're mathematically tractable. Their mathematical tractability allows you to drive consequences but not just any old consequences.
What you are looking for, is the term I come to use, striking. I can't find a word that I really like, but predictions from the theory that are at least a little bit counterintuitive but they're so distinctive in their numerical character that you can almost confirm them quasi-numerically, that is, almost qualitatively.
And the one, three, five, seven progression is a nice example of that, because suppose you get almost 1, 3, 5, 7, 9, 11, 13, 15, you run it out. I did it wrong, no I did it right, you run it out and you just keep getting it all over the place at least to very high accuracy.
The fact that the pattern’s occurring at all is enough to be very compelling evidence even if there are discrepancies as long as they’re not systematic. So that’s his approach, and to do it, what it means is he's got a design experiment to test the striking predictions, hoping at the very least that the results don't clearly falsify.
This is a pop period remark now. What you're doing is designing experiments that if they don't clearly falsify, they pass the test, and that's what you're looking for. But again it's fairly different from Kepler and the striking thing, I'll come back to this at the end, the striking thing about it the experiments tend to be very very contrived.
Even dropping objects off of 300 foot, 312 foot tower in Bologne is not something one just does unless, you are trying to drop a piano on somebody down below or something. But what you're doing is not intervening in nature. You're doing something much more radical. You're contriving a circumstance that usually doesn't occur in nature.
Now, the quote at the end I put here, you'll come back to it many times. It's Christian Hoygans after the Principia in a work that includes his response to the Principia. It's a famous statement I've often said in other courses. This is the clearer statement of the hypothetic deductive method of testing that's ever been put in the literature.
This is a fairly flowery translation by a Australian chap whose name is escaping me at the moment, who focuses on history of science and education. But let me just quote it. One finds in the subject the kind of demonstration which does not carry with it, so higher degree of certainty is that employed in geometry, and which differs distinctly from the method employed by geometers in that they prove their propositions by well established incontrovertible principles.
While here principles are tested by the inferences which are derivable from them. The nature of the subject permits no other treatment. It is possible however this way to establish a probability which is a little short of certainty. Just a moment there. He's using the word probability the way we now use it.
He wrote the first textbook on probability and the way we now use it. We'll come to what he did, he did that in 1661 at his father's request. That story's for later but he's using the word probability here in a way that we should be totally comfortable. This is the case when the consequences of the assumed principles are in perfect accord with the observed phenomenon, and especially when these verifications are numerous.
But above all when one employs the hypothesis to predict new phenomena and finds his expectations realized. It's this last item that describes Galileo as predicting things that nobody had ever thought of and then going out and confirming them at least qualitatively, or I prefer slightly quasi-quantitatively. And it's important because Huygens follows exactly the same method.