All right, with that as background, let's look at the commentary, all it's very clean. Start at the top. Just the first sentence. I understand that our predecessors assumed a large number of celestial spheres. Principally, in order to account for the apparent motion of the planets through uniform motion.

Or it seemed highly unreasonable that a heavenly body should not always move uniformly in a perfect circular figure. I'm gonna jump down now to the part I've marked out. Nevertheless, the theories concerning these matters that have been put forth far and wide by Ptolemy and most others, although they correspond numerically with the apparent motions, that's where those insert, also seem quite doubtful.

For these theories were inadequate unless they also envisioned certain equant circles. On account of which it appeared that the planet never moves with the uniform velocity either in it's deference sphere or with respect to it's proper center. Therefore, a theory of this kind seams neither perfect enough or sufficiently in accordance with reason.

Therefore, when I noticed these difficulties. By the way, the reason you get things like the inserts and difficulties, in Latin you have gender features to nouns. So you can use a pronoun and know what it links to before very clearly what you can't in English. And in translating you have to insert words.

I've often pondered whether perhaps a more reasonable model composed of circles could be found from which every apparent irregularity would follow. While everything in itself moved uniformly. Just as the principle of perfect motion requires. After I had attacked this exceedingly difficult and nearly insolvable problem it at last occurred to me how it could be done with fewer and far more suitable devices and had formally been put forth if some postulates called axioms are granted to us.

Which follow in this order. First postulate. There is no one center of all the celestial spheres or spheres. Now, orbium and sphaerarum. Orbium are solid crystalline spheres on which spherical planets sit. Okay, and it's orbium, the title of De Revolutionibus. Is coelestium orbium, namely revolutions of the heavenly, solid crystalline spheres.

The more interesting ones, third postulate, all spheres surround the sun as though it were in the middle of all of them, and therefore, the center of the universe is near the sun. I'll jump now, whatever motion appears in the sphere of the fix stars, belongs not to it, but to the earth.

The earth rotates daily. Thus, the entire earth along with the nearby elements rotates with a daily motion on its fixed poles, while the sphere of the fixed star remains immovable, and the outer most heaven. Sixth postulate. Whatever motions appear to us to belong to the sun are not due to the motion of the sun, but to the motion of the earth and our sphere.

With which we revolve around the sun just as any other planet. And thus the earth is carried by more than one motion. Pause a moment. It is a fundamental principle of Aristotle, that no body can ever engage in more than one natural motion at a time. He's going to have three, we've got two so far.

I'll talk about the third later. The retrograde and direct motion that appears in a planet, belongs not to them, but to the motion of the earth. Thus, the motion of the earth by itself accounts for a considerable number of apparently irregular motions in the heavens jumping down. In the same way, in case anyone believes that we have asserted the movement of the earth for no good reason along with the Pythagoreans.

Pause a moment, Pythagoreans was the dirtiest word the Church was using for anybody proposing innovations versus Church learning at the time, okay. So he's denying he's a Pythagorean. Kepler denies he's a Pythagorean. But they thought of it as a Pagan religion. Or at least that's what they said.

He will also receive considerable evidence for this in the explanation of the circles. And in fact, the evidence by which natural philosophers attempt so very hard to confirm the immobility of the earth, depends for the most part, upon appearances. All their evidence falls apart here in the first place since we overthrow the immobility of the earth.

Also by means of an appearance. The reason I block that off and make it so important, that's gigantic revolution brought on by Copernicus. Appearances can't be trusted. Okay. Saving appearances is fine, but the appearances should be looked on with great suspicion and in particular, the very appearance of retrograde motion is nothing but that.

It's nothing but an appearance. There is no such motion at all. That's part of the revolution that we're going to be seeing here. Now, what follows this arc in words, almost no diagrams, except the one Swerdlow has added. In words a description of the suns orbit, the moons orbit, followed by Venus, followed by the four planets that Ptolemy handled one way, followed by Mercury.

And what is stunning is everyone of those orbits is identical to the orbit in Ibn al-Shatir. To use just the first example of this, this is orbit in Ibn al-Shatir where the earth is now moved into the center and what we've got replacing the equine is a epicycle that carries, well it's more then that.

We have an epicycle that does retrograde motion. That's the old epicycle. So that's the second circle. Look inside that and you will see an R2 and an R3. That is, there are two epicycles inside of that one, so that we end up with an epicycle inside of an epicycle inside of an epicycle.

We've got four radii for the planets. If you do that, you get the same effect as the equant as I'll show you in the second half tonight. You get the same effect as the eccentricity and you get a complete reproduction of Ptolemy's longitudinal positions for Venus. Mars, Jupiter, and Saturn.

This just carries right forward removing all elements of non uniform circular motion and putting the sun in the middle. And that's what Ibn al-Shatir was trying to do. Was to restore originality in principles to Ptolemy astronomy. And he in effect did a transformation. Geometric transformation of Ptolemaic astronomy to eliminate all the objection feature for this four orbits.

He did it for all of them, this is from the same paper comparing all. All of the bodies. Not the sun. The sun is uninteresting in this regard. So the moon, Ibn al-Shatir introduced two circles, R2 and R3, and with the principle, obvious error in Ptolemy astronomy, the moon coming too close to the earth, he eliminated it.

But he captured all of the inequalities that Ptolemy had discovered with a fairly simple three circle motion. This is the exact same theory that's in De Rev for the moon. There's no difference. Each of the planets what you're given here is a comparison between Copernicus's values for the radii and for Mars, Jupiter, Venus, and Saturn, Ibn al-Shatir's values for the same radii you will see how close they are and all the radii stay the same.

But the really spectacular case is down here with Mercury where we've got six circles to achieve Mercury. And the values between the two are not only Copernicus used the same six circles, but the values are almost identical. Now I'll give you Noel Swerdlow's argument for why this can't be coincidence.

It could be coincidence that Copernicus discovered a way of eliminating the equant and eccentricity. Which, by the way, he restores in De Rev. He brings eccentricity back. But, in Commentariolus, the eccentricity's being handled by a circle. It could be coincidence that Copernicus happened upon the same thing. It's starting to stretch the imagination that he happened upon the same way of solving the problem of the moon.

It stretches the imagination even more that he found the same way of handling mercury the most complicated. But now the clincher, his explanation for I don't off hand remember which one. I think its R5 or R4. His explanation for that circle in it is wrong. He misunderstood what he was talking about at the time of com materials, so somebody had given him the scheme yet and he totally understood it.

He got this scheme right. He could calculate and solve what is given the right values but now he actually doesn't understand it well enough to explain. That's Swerdlow's argument that there's no possible way Copernicus got this anywhere but Ibn al-Shatir. I'll let you judge that for yourself. Owen Gingrich has never been happy with how outspoken Swerdlow is on this.

But then Copernicus is Owen's great hero. And that does complicate things a little bit. Okay that's the case. But look. That leaves totally unanswered a key question. Where's heliocentrism in this? What Ibn al-Shatir did was a geometric transformation of Ptolemaic astronomy leaving it geocentric, but eliminating all the classic objections to it.

And it was a real achievement. And as I said before, it had to be the culmination of many, many people before him. Some of whom we've identified like Erdi and Tusi in particular.