Which gets me to part three. And I stay with this for a moment. I'll turn to the figure in just a second. Kepler had a reason for wanting the eccentricity to be bisected for everything, and it's his physics. Here's the picture. I've already given it. A rotating Sun is pushing Mars in its orbit.

Now, the further away Mars is, it's gonna push it weaker, because the flux coming off of the Sun attenuates. Actually, just as an aside, it's an interesting twist here, because in that book on optics I passed around, Kepler announces light, the intensity of light, diminishes in an inverse square of distance.

And here, he has to face the fact he said that about light. He uses an analogy with light and has to argue the push on the planet varies, not as an inverse square, but as one over the distance. Now, what's his basis for saying that? That's what the equant automatically gives you at the two extreme points.

The velocity at the two extreme points on the line of apsides is inversely proportional to the distance from the Sun, if you're using bisected eccentricity. And he liked that, because he never liked the equant. The equant is an empty point in space. How can that be physical? So, he wants the Sun to be the thing dominating the physics.

And a rule of velocity is proportional to one over the distance to the Sun is a lovely looking rule. Okay? Problem. The Earth-Sun orbit doesn't have an equant. It's a counter example to this whole physical picture. So part three takes on the problem of removing this counter example.

He's gonna say maybe they've been wrong for 2000 years about the Earth-Sun orbit. That's back to Hipparchus. Maybe it has bisected eccentricity too. Now that's a much more dramatic claim than it sounds, because as I said quickly at the end last week, cuz I was running out of time, Teco had come up with up with an extraordinarily accurate account of the longitudes of the sun from day after day observing meridianal passings.

And his eccentricity with no equant, that is, the equant was at the center of the circle, and the Earth was off by 0.036 out of a unit radius. That is, that was the eccentricity, 0.036. If it's bisected eccentricity, it should be 0.018. So, he wanted to show that that's the case, and now the issue is how do you show it?

Well, here are two ways. They're both based on the following thought. Let Mars go around the Sun. That means this doesn't work for Ptolemy. We'll come back to Ptolemy in just a moment. Let Mars go around the Sun. Every 687 days therefore, it should be in the same place in the sky, cuz that's its period.

But then, we can observe Mars at that location from different places on the Earth-Sun orbit, and it would be like flipping the angles and observing the Earth from Mars from the same spot in Mars as the Earth goes around the Sun. So, the one on the left, C is the mean sun.

That's the equant, the mean sun, the center of the mean sun is automatically the equant. That's what the mean sun is, uniform angular motion. A is the true sun, actual sun. And he identifies observations where Mars should be at its greatest distance from the line of apsides of the Earth-Sun orbit.

He then identifies two places where the GNF, where we can observe Mars at that position from the Earth. And we can ask what the angles are, f, c, n and m, c, g, and whether or not they are properly symmetric for that circle to be centered at c, the point of equal angular motion.

And the conclusion is no, the center has to be between a and c. In other words, it looks like bisected eccentricity. That's his first way of doing it. Now, unfortunately, this way has a calculation error in it, and when you run the calculations back through correctly, the argument largely collapses.

That is, he had a worse argument than he thought he had, cuz he had made a calculation error. And I trust everybody here who has ever done a lot of calculations, when you get the result you want, you don't go back and check whether you made an error to get the result you want.

Okay, that's essentially what happened here. So it ends up the second way is the more interesting way. Of course, Kepler didn't know the first way didn't work. So, from his point of view we had two complimentary ways of establishing this. The second way is really quite beautiful. Again, Mars is gonna be in the same place every 687 days.

So, choose four spots where the Earth observes Mars on it's orbit, on the Earth-Sun orbit. We're working Copernicus now, we can turn it around to Tycho. Don't worry about that. And, we're actually going to do it two ways, using Tycho's theory for Mars to get the heliocentric longitudes of Mars, and then using Kepler's vicarious hypothesis to get it, to get the heliocentric longitudes around a.

In both cases, we have triangles, therefore we have relative distances. So the points here are e1, e2, e3, and e4. Three of those are enough to determine a circle. The fourth can then be used as a cross-check. What he actually did was several combinations of three, seeing if they gave the same thing.

And the numbers range, I have to look this up. I'm not gonna depend on my memory. The numbers for the location of the center, range from 0.015 to 0.025 versus Tycho's 0.03584. 0.036 is what I gave it before. In other words, the center of this circle as observed in these different ways lies more or less midway between a point of equal angular motion and the actual Sun.

The Earth-Sun orbit is in fact got bisected eccentricity. Now the exact bisect is an extrapolation. It's an idealization off of observations that simply can't be precise enough to do better than a range. So, all he can really conclude, it's pretty much in the middle but it's enough in the middle that he can actually conclude that a 2,000 year tradition of modeling the sun has been wrong all along.

Then of course he's got a problem, he's got two problems. The first problem is how could Tycho's theory of the Sun be so good, and not have the correct location of the center. And there's a whole chapter showing it's the distances that vary, not the longitudes. The distances are very different whether you do the bisection or not.

So he explains that away. Then the other problem he has is he has to replace Tycho's theory. Tycho's theory is no longer any good. So he goes back and simply redoes Tycho's theory, now with bisected eccentricity giving entirely new distances between the Earth at any moment and the Sun at any moment, but on a circle.

Fair enough? Bisected eccentricity on a circle, refined Tycho orbit, within observational accuracy of Tycho's original one, for all the heliocentric longitudes of the Earth. Okay?