You've seen this slide before, when we reviewed astronomy between 16 between Descartes and Newton. We came up with these seven different ways of calculating orbits that were developed during the century. The key thing here is that Newton knew about all of them, and in most cases, knew about them from primary sources.

But he knew all seven. And knew them as of 1680. One of the things that came up a couple weeks ago when I was at Johns Hopkins and I think I shocked the people in the room. I know I shocked one of the people in the room who asked, well, wasn't the area rule established?

And my comment was, at the time, nobody was calculating orbits using the area rule at all. Not one person was doing it that way. But at any rate, this slide, you've seen all but the bottom last time. This is just my summary of what got accomplished by De motu corporum in gyrum.

What we're going to be doing tonight is watching the picture that all of you were writing about for today. The picture of what Newton had achieved come apart entirely at his own hand. And come apart almost immediately at his own hand. And that's what sets up the Principia as he turns what seems to be a fairly straightforward evidence problem into a horrendous evidence problem, that the Principia then addresses and in that resolving.

So the only thing new here is the remark at the bottom, this is a list of what was accomplished, and at the bottom the Kepler Horrocks orbital rules have a prima facie claim to being at least essentially exact. So the point is, going back to that prior slide, you look at these and what it says is, if you proceed under the assumption that there's an inverse square centripetal force.

Centripetal acceleration is slightly more accurate, directed toward the Sun, then the conclusion is the appropriate way to calculate orbits is fairly clearly Horrocks' way. That's the one Newton is authorizing. Which is, of course, just a modification of Kepler, but an interesting modification of Kepler. So, he's choosing among these.

And that was, probably, part of what Halley saw. Because Halley was acutely aware of all seven. Looks like Halley may have even read Astronomia Nova. We can't verify that, but he's either read it or managed to come up with something on his own independently when he was an undergraduate.

But either way, that's one feature. The other feature, of course, is that everything in here is about either sufficient conditions or necessary conditions for something to hold exactly. And one nice thing about having, particularly having sufficient and necessary conditions for anything holding exactly, is whenever it doesn't hold exactly in the world, it's telling you one of those conditions is being violated.

It's particularly true of necessary conditions that it's being violated and that then becomes a research direction. But the important thing, it's said at the bottom, if De motu corporum in gyrum taken at face value, or equally, have prima facie claim, is the phrase used here, you're answering a question about the alternative models for the calculation of orbits.

And at the same time, you're answering a question about Galilean free fall and parabolic projection, provided again, there is an assumption that there's an inverse square centrifugal force towards the Earth, namely gravity, and nothing in De motu corporum in gyrum, says that, okay? So that's the background. What I didn't say last time, and I won't even be saying as I've got a couple papers so far.

I got three in before tonight, and we'll have those back at the latest by Thursday morning. What I didn't stress last time and I'm not gonna stress that much in my comments on the papers, because there's no offhand way to see much of this. There are really serious loose ends in De motu corporum in gyrum.

A couple of them I did list, and I'm picking it up on the papers I've seen so far, what's the basis of reasoning from the phenomenon of the three hash power rule for theorem two, and then to the ellipse in the scolium to problem three. That is, in two places Newton concludes there's an inverse square in the case of the planets because of uniform motion in circular orbits, which it's not.

And there's a conclusion that there's an ellipse when, in fact, what the theorem says is if there's an ellipse, there's inverse squared centripetal force. So one loose end is just what exactly is the reasoning of that, and that I'll comment on all the papers. Second one is, is there any independent evidence for an inverse square centripetal tendency extending throughout the space around the Sun, other than the yet to be substantiated potential evidence from the trajectories of comets?

Third, what evidence is there contrary to the findings of Galileo and Huygens that terrestrial gravity is inverse square? Equally, what evidence is there that air resistance varies linearly with velocity? These are all loose ends. He's doing a pure mathematical theory under assumptions and paying very little attention to the empirical world.

Now more serious ones, and the ones that are gonna dominate us for the next few minutes. Actually the next hour. Insofar is at least three centers of inverse square forces have been identified, Sun, Jupiter and Saturn, and probably a fourth, the Earth. Perhaps a fourth. And at least Jupiter and Saturn are in motion around the Sun.

How can the motions of the planetary satellites be referred to the planets as their centers? The centers are not points in space, they are moving. Indeed, to what point in space, taken to be at rest, should all the orbital motions be referred? That's not a modern way of phrasing the question, that was a way of phrasing the question at the time.

Because what was the Copernican question? To what point should the motions be referred, Sun or Earth? So there's just another way of saying it. Equally, it's tied to the one just preceding it, how far do the centripetal tendencies towards Jupiter and Saturn extend outward from them, all the way to the Sun?

If so, why isn't the Sun contrary to Copernicanism itself in motion as well, okay? Those two are sort of tied together. We've got multiple centers. But the De motu corporum in gyrum treats one center at a time. There's nothing about multiple centers in there at all. And that's a loose end, because the world doesnt work that way unless Jupiter and Saturn are in isolation from the Sun.

You've got one system here, you've got another system, but then how could that be, because the centripetal acceleration towards the Sun acts on Jupiter, it must be acting on the satellites as well. So they're not isolated systems. Finally the last one, the deceleration from resistance depends on the weight of the body as does Huygens' centrifugal tension in a string retaining a body of circular motion.

Yet the centripetal forces of theorems two and four in problem five appear to be independent of the weight of the body in the manner of Galilean motion. What justifies this difference? I don't know if any of you noticed that, but there's no comment about weights, or much less with the term mass he hadn't got yet.

The quantity of matter in these bodies, he is in effect creating them, it's most noticeable in problem four, when he sets up what the trajectory is going to be. It doesn't depend in any way at all on the body itself, right? So it's got to be like gravity, all bodies are gonna respond the same way to these centripetal forces.

Well, what's the basis for that? We'll come back to those. It's those three that are gonna occupy us the whole night. Because I'm gonna be, what I'm gonna be proposing is that in the weeks after De motu corporum was sent to London, that's November of 1684, Newton started worrying about the loose ends, and it's the last three that he worried about most of all.