I'm just going to work through the slides. This is not going to be so much a technical lecture as last week was. What we have up here and the present moment are the five innovations in planetary theory that got introduced in Astronomia Nova. They are in order. The line of apsides goes through the actual true Sun, not through the mean Sun.

The Earth-Sun orbit should be bisected in its eccentricity. That, of course, is working still under a circle with an equant. And later in Astronomia Nova, it goes without saying that Kepler came back and replaced it with an ellipse. But that one, as you'll see later, is so near a circle that the difference between the equant and the area rule doesn't make much difference.

That's the second one. Then replace the equant with the area rule or its equivalent the component of the arch length velocity normal to a radius drawn from the central body varies inversely with the distance from the central body. Kepler never quite says it that way. I'm not sure anybody says it quite that way before Newton, but it's sitting right there in the Principia right at the beginning.

Then take the trajectory of the orbiting body to as defining an ellipse. And finally have the orbital motion as occurring in a plane that passes through the true Sun has a distinct inclination in that inclination, for all they could tell, remains constant over time. None of those five have ever been proposed in astronomy before.

All five are still with us. But they didn't enter astronomy immediately. They entered Kepler's work immediately. Others were slow to pick it up. But once they resisted, the one that people held back on, interestingly enough, was really just the area rule, which we'll get to a little later.

Okay. Those are major innovations all in one work. Tycho had said that what he wanted to do was reconstruct all of astronomy from the ground up. He did. That is, he did it in the name of Kepler, it is. Kepler took it over, and Kepler did the very thing.

It's an interesting question, how happy Tycho would've been. Not he would have been with very unhappy with the Copernican aspect, but how content he would have been with ellipses, et cetera, but my guess is he would have been thrilled. We can't tell, but he might very well have been thrilled.

And, of course, the product you get out of this is this figure from Voelkel and Gingerich. It's the top figure I want you to look at. What they've done is take modern knowledge and reconstruct the very best Ptolelmny possible theory. That's the dotted line with those errors in it for Mars.

And then they've taken the Rudolphine Tables, Kepler's own shot at it, not even an optimal shot in all respects, for reasons I'll make clear in a few minutes. And just look at the difference. Look at the difference at the top. You can barely see that we're talking about an error.

Now the top figure is in degrees, the bottom figure is in minutes, but the spacing is half. So, that's a factor of 30 difference in scale going from the top figure to the bottom figure. And it works out that the total improvement is about a factor of 50.

That is one and half orders of magnitude improvement. And it's pretty hard to find other cases in the history of science where you get that kind of improvement, and by one person in a very short period of time. That's very spectacular. And this plot, this table, I'll throw it up one last time.

This time intentionally doing the Latin, rather than the English. You have it from last time in both ways. 28 all reasonably near opposition, but 28 sets of observations. Observations is a slight adjustment needed here on timing that is, these are re-worked observations that Kepler has re-worked Tycho. But the impressive thing is when you compare calculated and observed longitudes, the errors are very, very small.

One cluster, and one bit as we talked about last time. Also notice the distances being given to six significant figures of Mars from the Sun and the Sun-Earth distance. Sun from the Earth, he puts it, and Mars from the Sun. I'll come right back. This table occurs earlier in the book than he has for the final ellipse for both the Mars and the Sun.

In fact though, all the numbers in it are based on the end of the book. He just doesn't tell you that. So Donahue, working through the numbers, realized where the numbers were coming from, and then started verifying it. So this is actually his final results that he's presenting as interim results.

But, be that as it may. My point about this table is, it set a new standard, period. And the rest of this semester, we're gonna see person after person offering an alternative to Kepler, giving exactly the same table up to a point. Why? Because nobody else had access to Tycho's observations, without going off to Denmark and writing them down.

So the tendency was just to take Kepler's table and see how they did in comparison. And if they didn't do comparably in comparison, out. That was the standard. So that's the other thing I'm trying to thrust before you. You'll see the tables later, fair amount later in the course.

I guess it's tenth week of the course. But the point is, overnight, he set a totally new standard in astronomy. And it's a two-fold standard. One is the accuracy. But it's the accuracy all the time, not just for stationary points, size of retrograde loops, etc. And that was not Kepler's initiating, that was Tycho.

It is Tycho's view. If we're gonna do this right, we set up orbits that are right all the time, not just occasionally.