The approach Kepler takes in Astronomia Nova is to start with various approaches, all three. The Prutenic tables, Tycho's theories, and Copernicus, from the Alfonsine tables, Prutenic tables and Tycho's. And look at the discrepancies between calculation and observation. And use those to refine, use Tycho's observations and those discrepancies to refine the original trajectories he starts with in a sequence of successive approximations.

But now, you have to ask yourself, how straightforward is that to do? Well, you're gonna be looking at discrepancies, right? You're gonna have a calculation and you're gonna look at an observation. But what you're really concerned with is the difference between the two. So what do those differences depend on?

In particular, what could be making them not accurate? Well, the answer is an awful lot. First of all, the observations themselves are imprecise. So you can't push a discrepancy past a certain point. Second, he's generally having to interpolate between Tycho's observations and interpolation is not precise. On top of that, he knows Tycho's got uncertain corrections for parallax and atmospheric refraction.

In fact, Kepler worried about the atmospheric refraction so much that in the middle of his work on Mars, he stopped and did an extensive investigation of optics. It came out as a book, which I'll pass around. The original title is called, The Optical Part of Astronomy. So and in it, there's a long discussion of refraction.

There's also the first discussion of how the eye works, including that the image is inverted on the retina. So this is really a quite classic book. He did another book on optics in 1611. We'll talk about that next week. But he was worried enough about atmospheric refraction and whether Tycho's data were reliable.

That he starts trying to figure out what to do about atmospheric refraction, and he ends up basically, for his whole life, throwing up his hands. He's never confident. He's almost sure the parallax is wrong. And he finally says it late in life in the Rudolphine tables, you can't trust it.

But, it's left to others to correct it. So, that's a second source in which the discrepancies can't be relied on. A third source is in order to judge the discrepancies. You're observing from the Earth. And you're observing relative to the sun on all three systems. On all three systems, the planet has a relationship to the Sun, relative to the Earth.

So that means the Earth Sun orbit has to enter in. Where's the Earth Sun orbit coming from? Well, he has a great one from Tycho to start with. But is it reliable? That's a possible source of discrepancy, starting from an incorrect theoretical orbit that will help you. Finally, place after place, he can't do the math, the rigorous math.

So he has to replace it by an approximate method. And everywhere he uses an approximate method, the question is how much more error is being introduced from it, okay. So all those are problems. I'll make a quick remark. I love to make this remark. My training background made me an engineer is in effect in applied math.

And I've never read anybody more resourceful as an applied mathematician than Kepler. He did not have Calculus. But he virtually invented the modern numerical approximation to Calculus, among many other things, and he did it in Astronomia Nova. He was really ingenious when he couldn't solve something rigorously of finding an approximate way to do it and come up again later.

But then there's one other problem in all of this. The book is filled with calculations and place after place, the calculations are wrong. This is a place where he simply failed to subtract accurately. And then that effect is that error propagates through many, many pages. And the net effect, when Bill Donahue decided to translate Astronomia Nova.

I don't think he realized what he was getting into, it was 10 years of his life translating this. Because you have to go through every bloody calculation to check it. You have to go through every step. Last remark about this, and it's a capstone to things I said earlier.

I'm gonna show you other respects in which this book is difficult to read in just a moment. The reason I don't teach the book and just give it to you to read. I tried to make clear and I never did say it as well as I'm about to say it, what it was to be trained in doing mathematical astronomy.

I stressed it was to be able to do the calculations at this point in all three systems. In the Ptolemaic system, in the Tychonic system, and the Copernican system. But that's not quite the right description of it. A description that will make a better impression on you is what people would do is send in mail, problems to other people to work out.

Like within the Ptolemaic system, when will Jupiter and Saturn next be in In conjunction with one another? What's the shortest solution to yield the answer to that, okay? So what you were trained to do was problem solving in these systems, and they're three different sets of mathematics. They're related, but they're different.

What Kepler does through this book is, for the first half of it, everywhere he does anything, he does it in all three systems to avoid making questions. So for you to read it, for me to read it, what I do is skip over a lot of the calculations for just that reason, and read the conclusions from it.

You really have to be adept at doing this kind of mathematics. And so I would be really quite punishing you. If we did a course on Astronomia Nova, half the semester would be spent learning to do problems within these systems, okay? That's just the problem. And nicely enough, Newton just doesn't pose the same difficulty.