Now part four of the Horologium Oscillatorium, the last difficult part of the thing. Mersan had pointed out a problem, and it challenged people to solve it. It became known as the center of oscillation problem. Descartes failed at it, we don't know how many other people failed at it, I think Furmat failed at it.
Let me state the problem. Take a pendulum with, and it can be circular, this is not an issue about circle versus cycloidal and have it have two, three, or four bobs on it. What's the length of the pendulum with a single bob on it that keeps perfect time at the same arc with one with multiple bobs?
And the obvious thought, take the center of gravity of all of the bobs. Some were saying they'd done the obvious thought and found that it doesn't work, it's wrong. And the problem became, okay, what is this length have to be to go completely synchronous with one with multiple bobs?
Everybody see what the problem is? It turns out, this is the fundamental problem of the 18th Century mathematics for an interesting reason, you can't solve it with Newton's equations, Newton's laws of motion, you need something more. The first solution is by. It's based on two principles. They're both generalization of Galilean principles, so once again, we're gonna have a Galilean solution to a problem.
If any number of weights begin to move by the force of their own gravity their center of gravity cannot rise higher than its location at the beginning of the motion. That's a form of the Torricelli Principle, but now talking about the center of gravity relative to weathers. Abstracting from the air and every other impediment, the center of gravity of a pendulum crosses through equal arcs in descending and ascending.
I'll give you the solution on the next page, but this is of course his diagram, these are both his diagrams, and he's working out where the distance has to be. And it turns out, if I take a two inch diameter spherical bob, and that's about that big, how much longer is the length of a single pendulum once I don't represent it as a point mass, I represent it as a physical body?
How much longer is it than to the center of the bob? The answer is one-tenth of the line on two inch diameter. So the right way to measure the length of a pendulum, and he tells you this very carefully in part four, is you have to make a correction because the real length is to a point different from the center of the spherical bob.
Those are the assumptions. He derives two key propositions that didn't give you the solution, the solution is here. Proposition three, if any magnitudes all descend or ascend, albeit through unequal intervals. Remember on a pendulum, the ones at the shorter length are gonna descend less than the ones further out.
Albeit unequal intervals, the heights of descent or ascent of each multiplied by the magnitude of itself, that's mass times height, in our words. Mass, he doesn't have, I've said that before. Yield a sum of products equal to that, which results from the multiplication of the height of descent or ascent of the center of gravity of all the magnitudes times all the magnitudes, okay?
So take all the magnitudes and the height they fall. Okay. The sum of those is the same as the height the center of gravity falls times the total sum of the masses. That's a proved proposition off of this. Proposition four, assume that a pendulum is composed of many weights and beginning from rest, has completed any part of its whole oscillation.
Imagine next that the common bond between the weights has been broken and that each weight converts its acquired velocity upwards and rises as high as it can. Granting all this the common center of gravity will return to the same height which it had before the oscillation. Now what that is in our terms is a conservation of energy principle.
The solution you can see in modern terms it's the integral of Y squared times the sum of the masses over the interval of 'Y'. This of course, the upper one, is what we call the moment of inertia now. This is fully anticipating moment of inertia. He's got this sum here, L squared over L, etc.
As I say, it, the nice thing about this, there's several nice things about it. Once you have it, this is trivial to test. You do a six bob pendulum, you work out what the one bob is supposed to be and you test it. So there's no real issue about whether he had a solution.
He had the solution to a long standing problem. What is interesting about it is people started complaining about the solution. The complaint was, we don't like his principles. They're not intuitively sound. This became a controversy. I'll state the controversy the way it developed. What are the right intuitively sound principles from which to derive this solution?
Huygens' is not. Leibniz defends this solution. You see what Leibniz says about it. Forces are proportional jointly to bodies of the same specific gravity or solidity and to heights, which produce their velocity and from which their velocities can be acquired. More generally since no velocities may actually be produced, the forces are proportional to the heights, which might be produced by these velocities.
They are not generally proportional to their own velocities, but to the heights, therefore, their velocities square. Many errors have arisen from this latter view. I believe this error is also the reason why a number of scholars have recently questioned Huygens law for the center of oscillation of a pendulum which is completely true.
This principle came to be known as conservation of you know, the conservation of height. Think of it this way, height times mass, velocity squared times mass, those two are always in balance with one another. The one is what we call potential energy, the other is what we call kinetic energy, the sum of the two is always a constant.
That's a modern way of putting this. And that was the complaint starting right away, that's not an intuitively clear principle, we want better principles for it. We'll look at those principles next semester. There are two really striking aspects of this solution beyond the fact that he solved an outstanding problem.
First striking aspect, well, actually, it's three. When he gives you now, his measurements of surface gravity, he tells you you have to correct for the center of oscillation. Why? Because you can't get a point mass bob. You've got to have a physical body. Whatever that physical body is, it is slightly different to the length to it, from the center.
In fact what he did, and what you, I guess what all of you complained about, the hardest reading I've given you so far. I thought you would sort of skip over it. Not having what we call the calculus hadn't been invented from his point of view yet. Newton had invented it but note he hadn't published and Leibniz had yet to invent it.
Huygens starts doing the equivalent of working out what the moment of inertia is for different sized bodies. So he's not only gonna do it for multiple bobs, he's gonna do it for any shape a bob whatsoever and get the center of oscillation. Okay, that's part you got stuck with reading.
Now, that's first point. So whatever, however, we're gonna measure gravity with a pendulum, we've gotta pay attention to the center of oscillation. That's the real length of the pendulum. It's not the length to the center of the body. Okay? And different bobs make a difference. Point number two, now that we know this, it's trivial to tune a clock.
If I have a clock that's ten seconds slow for a day, I can calculate exactly what additional mass I have to add on to it, small mass at the end to tune it. And he tells you, in Horologium Oscillatorium, this is how you tune clocks. Which we still do with pendulum clocks, is we add small weights to them to re-tune them.
Okay? The third thing about it, though, and this is, now I'm gonna get very sophisticated on you, this is a new form of evidence. Think of it this way. I take Galilean principles and I make an idealization, the bob is a single point and I derive a law.
And then I find out, hey, for real bobs it's not quite the same as if all the bob are concentrated at its center of gravity. So therefore I have to do some additional refinement of the theory removing that idealization. Now one way to do it is an ad hoc correction.
That's not what he does. He takes the very Galilean principles he has before, he generalizes them to multiple box. Go back and read, sorry, yeah. Read these two and have it be one bob, and we're right back to the Galilean principles we started with. These new principles collapse to the old one if there's only one bob.
So all we're now doing is saying we understated the original Galilean principles by not paying attention to the case of multiple masses. Put the multiple masses in, we have the same principles. We do that, and we get rid of the discrepancies without any new principles. The same theory worked, for the idealized case, and then for the non-idealized case.
I'll give you a different example of it. Newton derives from the Law of Gravity, Keplerian motion. The Law of Gravity then entails Keplerian motion cannot hold in our solar system because of the planets are interacting with one another. So what's the principle evidence for the Principia? The success of getting the deviations from Keplerian motion over the next three centuries.
It's exactly the same move. The same theory gives you the idealized case without changing anything, it let's you do the discrepancies. That's exactly what Newton probably picked up from seeing Huygens do this in part four. Take the same theory and extend it to a richer problem without doing anything significant.
Now that says something about the original theory, right? Since the original theory is more robust than you realize 'cause it can handle something beyond the idealized case. And that too is a new form of evidence and that one Huygans' recognized, he knew he had done something really important when he generalized these principles and got what he wanted.