All right a couple of final slides, we're going to leave Kepler almost for good but I want to just, given what we've had from him, I just want to see we've learned. So once again I come back, but now expand, the fundamental evidence problems in Orbital Astronomy. The first one I said, the very first class, how to determine the distances of celestial objects from the Earth?

In particular, the distance of Mercury, Venus, Mars, Jupiter and Saturn, as well as the Sun and Moon and even comets. Not only on average but also over the course of time. And if you can do that, if you can plot the distance at all times from the Earth of all those things, you can of course can observe the angular positions modulo corrections for parallax and refraction.

You wouldn't need to do anything more, you could just describe the trajectories relative to the Earth and then notice hey, it's a lot simpler if they go around the Sun. They describe really smooth curves around the Sun, around the Earth it's a mess, et cetera. You just start picking up things right away.

So that I call the fundamental problem and I still call it the fundamental problem. Second, the other fundamental problem I singled out the first class, how to determine which apparent celestial motions and changes of motion, as observed from the Earth, are merely apparent and which are real, at least relative to the fixed stars?

That's the picture you have. If only we could go after the fixed stars and observe the whole system from there. Or doing a Tom Meggle move, if only we could go just on the edge of the outside, in a nice, convenient spot, and view the whole thing, because that's what we're sort of trying to figure out, what it looks like from there.

But we're having to do it from inside, and from inside, distinguishing between apparent and real motion is very, very messy. And he knew that of course, because he could show that the Tollemache agreed with everything even though it had totally different account of real versus apparent motion. But now, the viewants are didn't come up earlier in the course.

How to determine whether such calculational rules as Kepler's area rule and and ellipse strikingly accurate as they seem to be, are merely one amount many comparably accurate, though still only approximate ways of representing Tiko's observations, or are instead something more than this. If they're more than this, you get the question hold exactly or would hold exactly were it not for external factors.

Whether they project beyond the given observations indefinitely into the past for the celestial objects in question, and whether they project beyond these objects to other possible celestial objects. So, it's both projection in time and projection to other objects. And finally, and, of course, this is the one I hope the main reason I had you read the epitome, Kepler's epitome was to appreciate how much of a problem it is to determine what physical principles are governing the motions of the various celestial objects especially the planets.

We can't go out there and intervene and do experiments. We've gotta do things without any form of intervention on it. And how to do that in a very real sense is the central problem of the two semesters of this course. Because that's what Newton basically achieved. He managed to get at least a reasonable approximation.

To the physics that's occurring. But getting there was not overnight. It's a very, very hard problem, getting there. So those are the problems. Last slide for the night, to what extent did Kepler appreciate these problems? Well, I want to claim he appreciated them a lot. If Owen Gingrich were here, I wonder how we would react.

It's the sort of thing where one reads somebody like Kepler from the point of view of your own background. Owen has done a lot of calculations within Astronomy, but I don't know that he's ever worked from raw data not knowing what it was saying. I have, and this had a very large effect on how I think about extracting evidence from raw data because I've seen how really difficult it is to do on relatively things we know a lot about, but let's look at the six things I want to suggest that Kepler at least appreciated the challenge posed by them.

The first the possibility not just of systematic error in the raw data, but of the corrections introduced to compensate for the systematic error Now successfully compensating for them or maybe doing worse than that may be increasing the error. How do you detect systematic error and correct for it?

You can detect random error because it fluctuates about things. So systematic error is another story. Second, the possibility of being led down a garden path by the need to use theoretical assumptions to get anything out of observation whatsoever. What you want, you have to use theoretical assumptions to do it, how can you protect yourself from being done in by those theoretical assumptions?

We watched Kepler try to do it by cross checking things all over the place. If he got it from one assumption, he tried to see if it would hold the same way with a slightly different assumption, et cetera. Third, the fact that any number of distinct trajectories can in principle agree with imprecise data to more or less, the same level of accuracy.

We are gonna end up by the time we get to Newton of seeing seven different ways of doing the orbits of the planets. They're all within the same level of accuracy, they all differ from one another. And that just a fact of life. If data are imprecise, you can fit a lot of curves to them.

And how you choose among those, he was acutely aware that's the resistance on the ellipse, he knew it was an oval, but he knew there were a lot of different ovals. And he was very aware that any one of them could conceivably be just as good as any of the others with respect to the data he had with its imprecision.

Fourth possibility being led down a garden path, what I mean by that is a long path of research, all of which gets thrown out as being a loser. An example, if you want one from the history of Physics is the 40 years of electromagnetism based on the ether, and twists in the ether, et cetera.

But at any rate, by the need to appeal to underlying physics in one way or another in reaching conclusions about the physically true trajectory, while at the same time having the evidence for the physics come from the trajectory. That is, there's a built in circularity. We want the physics to pick out the right trajectory, we want the right trajectory to confirm the physics.

How do you avoid begging the question there, and constructing castle in the sky? Fifth, how to decide whether residual discrepancies between theory and observation should be dismissed, at least for the moment, as arising from observational error or extraneous secondary physical effects, or whether they should instead be regarded as falsifying the theory.

Kepler knew, by the time he died, there were a large number of systematic discrepancies going on. I think his attitude was he can't do more with the you give up, and the reason you can't do more is the fifth question. How do you figure out what they're telling, the discrepancies are telling you, when they can be telling you many different things?

And finally, how to reach conclusions especially in the face of the prior difficulties about orbital motion in the remote past, and in the long term future? And about whether ether orbiting bodies would have the same trajectories as though now accessible to investigation. Okay, I think he appreciated all six of those problems, and I'm not sure, and I say this carefully, I'm not sure anybody prior in history, appreciated it in the same way.

Because I don't think anybody other then perhaps, Ptolemy ever worked from so much data trying to work so carefully. But we don't know that much about Ptolemy, so it's a conjecture on my part. Ptolemy's the one candidate from the entire past, who might have done this same sort of working through on things.

And begun to appreciate how much play the evidence actually gave you for alternatives to the theory you decided on. And do remember, that's a good closing thought for the day, do remember how many times Ptolemy tells you there's more than one way to represent this. All right, we're through with Kepler.

Next week, we start for three weeks Galileo. One week on his astronomy, then we shift away from astronomy for a while and start doing mechanics. The other thing, it's going to come together. Mechanics and astronomy are kinda gonna come together by the end.