Okay that's the background. I'm now gonna present Evans' approach to this. So what Evans does is start out with indeed the Hipparchian model. Where the Earth is off-center, you've got an epicycle. You see the line pointing to the vernal equinox around which that's the, as it were, the zero axis.

And you see for the, and I think it's in the 1970s, you see the failure of the model to reproduce the actual retrograde loops for Mars, and it's fairly dramatic. And now, instead, you go to this equant with equiangular motion about a point off-center, that would ask how a sphere manages to revolve equiangularly off-center.

That's one of the reasons it gets criticized. But equiangular motion, about a point off center. And lo and behold, you've got the Mars pattern, and you could end up getting the pattern for all five planets. So, this discovery, he calls it bisection of eccentricity, meaning the center of the deferent circle is midway between the point of equiangular motion and the observer.

This was his great discovery and what turned him into a legend for 14 centuries. He succeeded where nobody before him had at reproducing the retrograde loops. And I'm absolutely confident that when he got it, he saw this as, in effect, a secret of the universe, because all four of these planets fall into place.

Mercury we'll see a little later. But all four of them fall right into place. And all four of them you get complete reproductions of the retrograde loops and success in predicting the stationary points. Okay, so it's an enormous success. The Islamic astronomers will, I'm gonna give you a list at the next to last slide tonight of things the Islamic astronomers pointed out were inadequacies in Ptolemy.

Now, they did that very late in the history of Islamic astronomy, after 1200 largely. But they picked up on this in ways that are very striking, as we'll see next week. But yeah, they're looking at this and saying we want to restore Aristotle while achieving what Ptolemy achieved.

And it's not obvious how to do that. In fact, I'll say it differently. Why did Ptolemy survive 1400 years? Because to replace him, you had to do this well. And nobody could see any way of achieving that til around 1350 when one Islamic astronomer saw it. And we only discovered his monograph in 1957.

Copernicus was highly influenced by it, but we only discovered the actual monograph and discovered what the Islamic astronomers managed to do in my lifetime. I'll tell that story next week. It's a nice story. It shows you how much we still have to learn about ancient astronomy. Understand there were no books.

There were just handwritten manuscripts. And they're scattered all over the place. All you have to do is find a new one and things change. I can tell good stories about that from my years at the Dibner Institute cuz we had some of those manuscripts. At any rate, that's the key to Ptolemaic Astronomy.

This is a drawing to scale for Mars. And that will give you a very different impression than the one I just gave you, not to scale. The epicycle of Mars is approximately 60%, I don't, you'll see the number in just a moment, the size of the deferent. Mars comes relatively close to us, closer than the sun ever comes and of course goes much further away.

So this is to scale. This is from the Noel Swerdlow paper. When you read the paper, you'll see these same words, but you'll see them in print. In 1991, second time I taught this course, I had Noel Swerdlow come and give a lecture. It is his lecture going public for the first time.

On how Ptolemy discovered the equant and for years I've simply put on reserve the typescript of his paper and he finally got talked into publishing it in 2004. So now I can give you the original paper, but I like his typescript diagrams better than the ones in the paper.

So I show it here. But he here describes the original Hipparchian model in which the observer and the equal angular motion are at two positions. You see the kind of calculation that's involved. Take the point M and its straight line, that's out to an axis on the epicycle.

And you're gonna measure where the planet is, which is parallel to the line from the observer to the mean sun. You're gonna measure from that angle. But we're not observing from that position, we're observing from O. And as a result, there's another angle that has to be put in.

Little D1 in there. That's called the equation of center and it adjusts things from, as it were, the real motion to the observer in the off center position. Now this is the Hipparchian model. The next figure shows you the same thing, but now with the equant at the other side of the midpoint of the circle and, as a result, the angle of D1 changes significantly.

It's gotta be referenced relative to the equant and the size of the correction is noticeably larger. If you look at the size of the correction of the two points at the upper right-hand corner, prior slide versus this slide, I think you will find it's significantly larger. At any rate, that's the actual scheme.

Now, what I stressed in talking about the scheme was, all of this things, all of the elements, they're called, the elements of the orbit ,like the eccentricity and the location of the center, the size of the epicycle radius relative deferent. All of these things are determined from observation.

We start with a general model. We identify a series of observations. It gives us the actual values of all of these. And they can be done century after century. And what happens if they stay the same century after century? You start deciding there must be something really right about this model because the model's being assumed to do measurements.

Some model mediated measurements. But the measurements when repeated, at totally different times, keep giving you essentially the same value. If the models, if the model is totally wrong, you would not expect it to do that. It's another way of saying it. I'll say it a different way, what's the principle evidence we have for Newton's Law of Gravity?

Our measurements of the constant of proportionality, capital G for it. Okay, so it's the same thing here. We're measuring constants and the constants are coming out good.