I'm only gonna go about five more minutes before I take a break, but I wanna look briefly at Apollonius' Theorem. So the basic idea Apollonius came up with is we'll start with retrograde motion and figure in the upper left. If a planet is moving on a sphere and that sphere itself the center of it is fixed to another sphere.
And they rotate separately from one another then it would be possible to account for retrograde motion when the motion in the epicycle as it's labelled up there where the motion in the epicycle goes in the opposite direction from the motion in the center. So the epicycle then becomes a basic way of accounting for retrograde motion instead of having multiple spheres.
And of course, it looks much simpler because we don't need so many spheres. Okay? The other thing Apollonius noticed however and this is his theorem, is there are two ways to account we'll do it with the sun, two ways of explaining why the sun appears to speed up and slow down.
One way at the bottom is the observer is not in the center, therefore when the sun is farthest away from us is going to appear to be going slower and when nearest us is appears to be going faster. But in real fact, it's going in same speed all the time.
That's one possibility. A second possibility is no, no the sun is sitting on an epicycle. And that epicycle is moving in just such a way as to give us, you see it over here the exact same thing as our being off center. But the difference between the two is in the one case, the sun is moving uniformly and there's only one motion, in the upper case the sun is a compound of two circular motions.
So the actual motions are different for the two but the two are totally observationally equivalent. It's more than that the position of the planet S at all times is the same relative to O on both of these systems. Now, that's a deep point. It's a point Ptolemy comes back to time and again announcing, thanks to Apollonius' Theorem there are two different ways of representing this.
We can do it with an epicycle. We can do it with what's called an e center, a circle in which the observer is off-center. Okay, starting to see a problem here, right? Two different ways of representing exactly the same thing, which leads me to a general point. There are two fundamental evidence problems in astronomy, really in all astronomy.
But it's particularly true, it goes right back to the Babylonians. The first problem is with the exception of the moon, we sitting on the Earth have no way to determine how far any object is away from us. The way to do it is shown in the diagram. Take observations of the object from two different positions on the earth.
Get the difference in angle, that's called the parallactic angle, and that will tell you how far away it is. The parallax of the sun is around nine seconds of arc in fact, way less than they could measure. So for the sun, they were off by about a factor of 20 in their estimate of how far the sun is away.
They had it about 20 times closer than it is and for all the planets, they had no way of saying whatsoever, how far away they are. Now, I just wanna make a point about that. Suppose you knew the distances from the Earth of the sun, the moon, and all the five planets at all times.
You wouldn't know what's going around what once and for all, in a sense that it could be that the sun would be going around the Earth and the planets around the sun, which is the way we do navigation. It's called the Tychonic system or it could be the Earth going around the sun.
But the thing is, we would know one thing. We would know whether the five planets are going around the earth or around the sun, if we knew those distances. We do not know those distances. The first time those distances finally emerge and we'll see in the next class.
And even then they're essentially based on an assumption. So that's the first problem. We know everything else, we know angular position other than being deceived by appearances like atmospheric refraction, we know angular positions. If we knew distances, we would have everything in spherical coordinates and we would know where everything is relative to everything else at all times.
By the way, the first time they successfully measured how far a star is away, is 1839. It's a long time before we can measure these distances. But getting the distances to the sun and Mars etc. That turns out to be a very big deal and this course will feature it.
Okay, it's one of the errors Newton made, he too had the sun way too close. The other problem we've just seen from Apollonius distinguishing between real and apparent motion. Is the speeding up and slowing down of the sun a consequence of real motion or is it merely apparent motion because we're off center.
Now that's, now a days thought of as relative versus absolute motion, that's not the way they thought of it. They thought of it as apparent versus true motion. True motion is motion that's really generated, apparent motion is motion that only appears to be the case. And separating those two is principal thing the claims to be doing successfully for the first time.
Whether he succeeds or not, we'll worry about next semester, but that's what caused him to write the book. He thought he had a solution for true versus apparent motion, once and for all. But I can't emphasize this too much. We have huge amount of information. We can look up there every night that it's dark and not cloudy and record where objects are, relative to the stars, when they're not near the sun.
We can do that for centuries and it still leaves us not knowing how far away they are, which means we don't know what their trajectories are relative to the Earth. I'll say that a different way. If we knew how far away they were, knew their angular position, we would have their trajectory relative to the Earth.
And we would see immediately that the five planets are not going in anything like a circle around the Earth. They're going in essentially a circle around the sun. That was not known and they had no way to get at it whatsoever. The problem Apollonius faced is if you simply do it with an epicycle and try to do retrograde motion what you get is a totally symmetric pattern.
Every retrograde loop looks the same, what James Evans has shown down here, this is reading for next week. Is if you do that, and you look at the actual pattern of the loops, you get total failure to fit. So you're not recovering the appearances with a simple epicyclic model.
Aparcus went a step further and moved the observer of the center and still didn't match. You'll see that after break, so that's the problem inherited. And now in this f Noigabower, Harold Noigabower was a professor at Brown University. Who did more to bring ancient astronomy into the modern world than any single person.
He has a three volume work called the History of Ancient Astronomy that does China, India, Japan, and all of the Islamic, all of the Greek and some of the Islamic. He divides the entire history of Orbital Astronomy into three periods. There's the roughly 1000 years before Ptolemy is where things were done qualitatively or you saw patterns.
But nobody tried to construct the model that would let you calculate longitude and latitude of every object, every night, into the indefinite past and indefinite future. The first person to succeed at doing that is Ptolemy. And from Ptolemy to Newton, that's what people did. So called Cinematic models of the motion, that includes Copernicus, it includes Kepler it includes just a whole lot of people.
What happens with Newton's Principia? Well not right away, he shows how to start this, but what happens is we quit calculating the motions from observation and we start calculating them from gravitational forces. That is we use the physics to generate the motions rather than infer the motions from observation and that's the third period.
It occurred to me, and thanks to Boris Hasselblatt, he's the one who drove it home to me, we're presently entering probably what's gonna be viewed centuries from now as the fourth era in this. The combination of being able to put up satellites and take advantage of what's called chaos.
Highly sensitive points in the gravitational field surrounding our planetary system. We've managed to take satellites, get them at the right point and with a very small burst, throw them into a totally different orbit, taking advantage of three body and four body effects. This is called Nonlinear Dynamics one of the three Field's Prizes was just given to an Iranian woman working in the area.
Boris Hasselblatt works in the area. But it's actually turned into a practical activity of being able to move satellites around on minimal energy taking advantage of so called, infinite sensitivities within our planetary system. And needless to say it's only recently that we put up satellites and only since 1990, that we've actually figured out how to manipulate them in this way.