All right, what I wanna do now is turn to Galileo's own test of this theory. It's a fantastic test, it's again, this whole night is just things of Galileo being extraordinarily brilliant, and this is a great, great test. It's come to be known as the ski jump experiment.
It is a single folio in Galileo's notebook. You see the diagram as it's drawn, I simply have a replica, I could have brought it in, I have a copy in the other room. I have a facsimilie edition scaled down. The notebook is folio pages, so these are scaled down to 8 and a half, 11, so it's not really a facsimilie, it's reduced facsimile.
Stillman Drake put out a few years ago, I reference in the notes of Galileo's notebooks, these are all notebooks from Padua, long before he goes to Florence and many, many years before he wrote Two New Sciences. All of the work reported in Two New Sciences dates from before 1610 in Padua.
But what he's got here is a set of numbers. He's got trajectories, and he's got numbers on a vertical axis. And he's got some calculations down here, including one of these calculations on a circle and incline plane in a vertical. And the question is, what is he doing?
Stillman Drake was the first to make a proposal about this. It's attracted a lot of attention since then. So let me, before I start turning and showing you results from their interpreting it, let's see what he's done. He knows how to control and then experiment the sublimity, right?
Start the object rolling down an inclined plane from a known height. Okay, so he's gonna know the sublimity when it comes to the table. It's gonna roll off the table. And if the acceleration of gravity is the same along the incline plane, as it is after, it becomes unconstrained.
Then he can calculate the exact parably, he can predict it. He knows, think of it this way. He takes a nice big bath of mud and he moves the mud along. He starts a rolling ball from different heights. It rolls off the table, goes off the table and he measures where it lands.
That's the amplitude. And he knows the height. Right? Because he just moves the baths of mud to different heights, or he keeps it at the same height and moves it laterally. So he knows everything here. Okay, he can determine the actual height, he can determine the actual amplitude, it is a parabola, he's right about that.
And he knows the sublimity, so he can predict which, given the height at which he puts the mud bath, and given the sublimity, he can predict what the amplitude is going to be. And that, presumably, is what these numbers are, predictions. Guess what happens, rolling down the incline plane, you've got five sevenths of the acceleration you have after it rolls off the table.
Okay. He's assuming the acceleration the vertical acceleration is the same in both cases. So his predictions are gonna fail and they do. This is my calculated table, here I note, the sublimity numbers are his, 300, 600, 800, 828, 1000. The altitude is always the same. 828, that's the height at his mud table.
The mean proportional between those, that's just the square root of the product of those two. 498, you can use, that's just a calculation. The theoretical distance is twice that. Right? One half a equals the square root of hp. So there are the theoretical distances. 996, 1,410, 1,628, 1,656, 1,820.
What are his major values shown on that plot? 800, 1,172, 1,328, 1,340 and 1,500. Percent difference ranges from 16.9 to 19.7%. Fair enough? With p from a rolling sphere in a groove of width four ninths the diameter of the sphere, so picture a v groove, or for that matter, semicircular groove, in which it's riding on the edges, the percent difference should be 18.3%.
So from this, we conclude the following. Number one, Galileo in his incline plane experiments, used a groove smaller than the diameter of the rolling ball, okay? Whether it's this, that's my number, by the way, the 18.3 and the four ninths you're gonna see in the next pages. Other people have proposed slightly different numbers.
But I consider the errors in these consistent with just little things going on. Bouncing or something like that. So I think it really does tend to show, very strikingly, the ratio of the acceleration going down the inclined plane, after you adjust for the angle of the plane. Versus, we don't even care about the angle of the plane.
All we care about is the height from which it's falling. So given the height at which it's falling and rolling, we get the ratio of the acceleration of gravity for that, to the acceleration of gravity when we're falling off the table. Okay, it's a beautiful experiment. First thing, it's exactly the right experiment given his development of the parabola.
He knows how to control, the sublimity, roll it down in incline plane. He's got the incline planes, at least at some points, that we should be able to control the height from which it falls. You know, everything is exactly right to do this. It doesn't occur to him there's a distinction between rolling and falling.
What's it screaming at you? Discrepancy here. Something has gone wrong. What does Galileo do with it? He suppresses it. It was discovered in his notebooks in the 20th century. Stillman Drake, as somebody who wants to defend Galileo as a modern science at every opportunity he has, looks at this and says, well Galileo must have attributed the discrepancy to some form of air resistance.
Now, it makes no real sense to attribute it to air resistance. It would have been trivially easy for Galileo to do the following, to move that height up and down and verify that it's actually a parabola. Now, keep going from the same sublimity, see if it's a parabola, it's a parabola, okay?
To a very, very high precision over these kind of distances, short distances on a laboratory tabletop with a lot of weight in the ball. So he could have known it's the parabola. And his conclusion has to be, it's the wrong parabola. Why is it the wrong parabola? And blaming it on air resistance looks very, very unlikely.
So another possibility, he concludes there's something wrong with the theory. But what are you supposed to conclude, what's wrong with the theory? Now, the right answer is, what he's being told here is there is a difference between rolling and falling, but if you don't recognize that this is between rolling and falling, you're not going to see that immediately.
I've got enough time, I wanna protect my time tonight. But I'll give you the analogous problem. In 1860, and repeated then in 1870, James Clark Maxwell reviews what we now call kinetic theory as developed by. And concludes from the inability of the theory to give, remotely, the correct values of the ratio's specific.
He concludes there's something absolutely, fundamentally, wrong with statistically mechanics. That's his first conclusion in 1860. By 1875, he's calling it the worst problem we have. Because he's become so enamored with kinetic theory. And now he wants the problem solved. Well Fineman in describing this in his lectures. You know, the famous three volume lectures of the lectures he gave freshman year at at Caltech.
He gives a very moving comment about it. That Maxwell had come upon the fact that there's something absolutely fundamentally wrong with classical physics. And he had no idea whatever how to respond to it. So all he did was point to it as a problem. He did point to it as a problem quite publicly, and it stayed a problem, stayed a problem, right up to quantum theory finally came out and removed the problem.
Really removed it finally in 1925. But Fineman's right. Was Maxwell come to the view that energy is not continuous quantity? That you have to get over a threshold, for any degree of freedom to start absorbing energy? Which is the solution by the way. That is, the solution is a dumbbell molecule to start rotating has to get a certain level of temperature.
Otherwise it behaves like a single, no dumbbell shape. That's a discovery that was very, very hard to come to by. And for 40 years from the time Maxwell announced the problem, nobody had a clue what to say about it, okay? The same situation is in a way true here.
Why should he immediately recognize rolling versus falling? It's not obvious. On the other hand now, and the point I do wannamake in keeping with question. Is his failure to publish this, means it did not become a part of science. It's only a part of Galileo's biography. And that's unfortunate, because this is exactly the kind of follow on experiment by which historically science has really learned the most.
And had he published it, it's not impossible that others would have picked it up, started fooling around with it, and come to the realization there's a difference between rolling and falling. And I've already told you a good way to do it. If you know the distance of fall in the first second from Riccioli, just measure what it is along an inclined plane.
It's 28% different if you're at five-sevenths. And in this case 18% difference, that's gonna be dramatic. So had he published it, science might have proceeded to a systematic realization of the distinction between rolling and falling in a classic way that science does. By holding it back and keeping it private, nobody else mentioned it.
It looks like nobody else was clever enough to do this experiment. Which is also a comment about how well they understood them, day four and what he had actually managed to achieve with it, day three and day four together. As things actually worked out, we're simply stuck with the fact that rolling versus falling remains confounded throughout this period.
Even though there's an experiment that began to expose. I'll say it differently. I'm making a point that I made with Kepler, about assimilating the community picking up and doing things with it. By not publishing this, Galileo preventing the community from doing anything about this. And a whole question of rolling versus falling had to be discovered in a totally different way.
A way that comes out of Huygens and has nothing to do with rolling versus falling, has to do with real bobs of pendulums. The pendulum actually changes angular position as it goes. And he figured out that that's crucial, and that started him off on a distinction between rotational motion and translational motion, and got him theoretically to the right result independently of any experiment.
So we're learning something fairly significant about experimentation here. We saw a bunch of earlier possible experimental things where air resistance would be involved, we finally get the most brilliant possible experiment here. And what do we end up with? We end up with just the sort of result that would have taught us something profoundly deep and it's suppressed.
And we don't know why. Okay, we just do not know what he concluded. And again, that's one of the problems working in isolation. When you don't understand an experiment and tell nobody else, the chances of getting a lesson from it are much slimmer than if you put it out to the public and start doing things.
Is that clear to people? It's fairly striking. The next four pages, I'm giving you the whole paper. This is Hans' analysis of folio 116V. For people who don't know what that means, v is verso, the reverse page, so folio sheets occur recto and verso, so main page, reverse page.
And this is, I just summarized, I'm not gonna go through this handout, it's four pages, and I put the whole paper under supplements. I haven't done it yet, I'll do it tomorrow morning. The whole paper under supplementary material. But he's the one who's taken the trouble to analyze.
You'll be able to look at his diagram, where he's reconstructed all of the numbers, all of the calculations. You can see what's going on, he's explained it in a fair amount of detail. And, indeed, gives the formula for when it's rolling, not on it's bottom. This is the formula rolling on it's bottom.
The top there is the formula when it's rolling in a groove, that the dimensions of which are smaller than the diameter of the sphere. So you can read this. This is a different experiment. It's much less interesting to me. The ski jump experiment has become pretty famous now.
And the reason it's famous is because it is so extraordinarily ingenious. It's a real test of the theory. He saw exactly how to test the theory. The test failed. And his conclusion was something's wrong with the experiment. What did I say early on? When you have discrepancies and you have problems measuring etc., discrepancies are ambiguous, fair enough?
If so, I'm gonna spend the last 15 minutes finishing Galileo, so we can turn to Descartes. I'm finished with what I have to say about the day four, any questions on that? It's a very rich thing to study to watch people learning how to do science. But equally rich, is to come up with exactly the right experiment and suppress it.