All right. How to get rid of the equant. I'm gonna be quick on this. These are pages, again, from Jim Evans' book. The basic idea, it doesn't work perfectly, but let's try to picture it. We have Ptolemy with an eccentricity, an equant here, an observer here. It turns out that if you reduce the eccentricity to three-halves of Ptolemy's, put the center at three-halves away from the e of Ptolemy's total and do one-half of the e on an epicycle, you get the same effect as the equant from the point of view of equant.

The planet is at all times at the same angular position relative to the equant this way. That's an Ibn al-Shatir. It's probably not an Ibn al-Shatir discovery. It probably precedes Ibn al-Shatir, but it's in Ibn al-Shatir as a solution for getting rid of the equant. Now that's explained in these two pages very nicely.

So here you see the actual way it's done. That's the full eccentricity. We end up dividing it in four parts, and we end up up there with a circle that's e over two, and then here the eccentricity three-halves of e. And as you see here from the point of view of the equant, the planet appears to be the same.

From the point of view of the observer, there's a very small difference in longitudinal position. And the actual trajectory is not a circle anymore. It's a fattened circle. The dotted line gives you the result of the transformation versus the circle. But it does manage to get rid of the equant.

And as I say, it was inherited, and does succeed. Everything we know about Copernicus from his own statement, that's what he most wanted to get rid of is the equant, cuz then he could preserve the solid crystalline spheres, and that very much seems to be what he was going for.

This is a diagram from a wonderful book by Giorgio de Santillana, a professor at MIT in the 1960s, who wrote a book called, The Crime of Galileo, which is a defense of Galileo. But one of the points he makes in it, this diagram, is people talk about the Copernican system being simpler than the Ptolemaic.

They have the same number of epicycles. In the Copernican they are minor epicycles to replace the equant. In the Ptolemaic, they are large epicycles, because they represent in the outer planets the earth sun orbit. But as far as complexity, there's the one on the left, there's the one on the right, and the argument is in no way is the one on the right simpler.

Why do think Copernican astronomy is simpler? Because when Galileo presents it, he removes all the complications and just has wonderful spheres going around one another. It's one way to make a case. Just drop off, be honest. Lawyers are good at being dishonest. Galileo was very good at that.

But there's just in no way is it a simpler system. If you look on the inside, it's actually more complicated because of the peculiarities of the earth-sun orbit that are probably just a reflection of bad data, and he's trying to match data that shouldn't have been matched at all.

This is an example I'm gonna to take you through it, I'm going to have to do it a little bit quickly. This is the first place Copernicus in De Rev works out, what the distance of Saturn is from the earth. And he does it by triangulation. So this is a little bit complicated.

This is the total eccentricity. I'm sorry, yeah, this, BC, I misspoke. BC is the mean distance of Saturn, From e. E is the center of the system near the Sun. L is the earth, going around at this time, on a circular orbit, around the sun. D is the center of b and c.

It's the center of the circle, of Saturn assuming it, and he did assume it's in a circular orbit. This is the little epicycle that handles the equin. And there is, at F, the planet, in this case, Saturn. What does he know? He knows, from observation at l, where f is, relative to the fixed stars.

He knows from theory, the heliocentric longitude, in this case, relative to point e of f. And he stipulates, I think the number is here, it's 100,000. No, it's 10,000. He stipulates that ad this distance from Saturn to the center of its circle is 10,000 units. And he now argues by a triangulation to the conclusion that el is 1 over 090 in units, where ad equals bd is 10,000.

Now that is purely triangulation knowing the two longitudes, geocentric and heliocentric. From observation, and if you read the whole thing he tells you what observations he's using. And then he makes a wonderful remark, and there is very little difference between that and what Ptolemy gave. In other words, he's doing a triangulation and getting the same radius ratios as Ptolemy got.

And it looks like a meaningful test, it's not really because the heliocentric longitudes are built, he's built in the Ptolemaic model. That's how he got it. He just did a transformation of Ptolemaic model to get the heliocentric longitudes. So, it's really a cross check that the complications he's introduced are still totally compatible with Ptolemy.

But it is an example of his doing triangulation. And the important point, suppose you could get heliocentric longitudes without presupposing, in effect, Ptolemaic theory. There was an independent way of getting them. Then, if you had the geocentric longitudes, if you also had the heliocentric longitudes without begging any questions relative to Ptolemy.

Then you could compute these distances and argue with Ptolemy, see we got a circle here and the circle Saturn describes in not around the earth, the circle is around the Sun. Okay? It would be very compelling. Now the problem is he can't quite do that. So the most he can do is simply show consistency with Ptolemy, though he presents it this way, it's nothing more than consistency.

But there's a potential here. If you can get heliocentric longitudes of any planet without having to resort to some form of Ptolemaic theory transposed, you may not only be able to confirm that the planets are going around the sun. If you get accurate enough observations and accurate enough heliocentric longitudes, you can distinguish between it being a circle and that distorted circle.

You've got real potential here. Okay, and that was something that could never be done in Ptolemaic astronomy. There's no such triangulation relative to the sun, cuz you have no idea where the sun is distant from the earth versus Mars. Okay? So, there's a gain here. The gain can be put in several different ways, and I'll come to that in a moment, but I simply want you to understand the triangulation, because when we get to Kepler next week, you're gonna find Kepler makes enormous use of triangulation and he has a quasi independent way of getting heliocentric longitudes.

That, I'll have to show you next week. Now, one last remark and then I'll go to Corey's question. Kepler couldn't understand this diagram, and in 1596 when he was trying to understand it, he wrote his teacher, Michael Maestlin, to explain this to him. And Maestlin wrote a wonderful piece back that appeared as an appendix in Kepler's first book explaining how all this works.

As the diagram indicates to you, this is very messy. People didn't pick up on the idea of using triangulation and didn't know whether what Copernicus was doing here was in effect giving a very strong confirmation of Copernican astronomy, or simply showing his math was consistent with having actually transformed Ptolemy.

It just wasn't clear till Maestlin published that. And I will put Maestlin's essay up next week, for anybody who wants to see it.