So to end the course, five minutes, it's a series of three comments. I'll pass this out to you. It's already in the handout, but I want to make sure everybody has a copy of this. For my purpose, when I look back at this course, one way to look at it, is what are the milestones that get us to Newton's Principia, and the ones I choose to list others might list different ones.

But the ones that stood out in this course, Copernicus' challenge to the Ptolemaic tradition subsequently reinforced by Tycho. Then, Tycho's more than a decade and a half body of positional data, that's a major step, milestone from prior times, Galileo's telescopic observations, especially of Jupiter's satellites, phases of Venus.

Then Kepler's orbital reforms, plus the three as power rule, culminating in the Rudolphine tables and the new standard for mathematical astronomy. Galileo's development of a fragment of mathematical motion under uniform parallel gravity by idealizing motion without air resistance, Descartes singling out the conatus to recede from the center.

Huygens's extension of Galileo's mathematical theory of motion to cover evolute control pendular motion, yielding a standard measure of the strength of surface gravity. Huygen's development of Descartes conatus a centro, into a mathematical theory of circular motions and centrifugal forces and a second measure of the strength of surface gravity.

Newton's discovery, prompted by Hooke, that the theory of uniform circular motion is a special case, of a more general mathematical theory of motion under centripetal forces. A theory that links Kepler's orbital reforms to inverse square centripetal forces and finally, Newton's finding a measure of the strength of inverse square four centers, and then allowing those centers to interact.

Those are the steps that open the way to the Principia, but the further thing I showed you tonight, which is highly speculative of universal gravity reaching beyond that. Why? Because otherwise, there's a conceptual problem. How do forces adjust themselves if they're external to a body to the body on which they act, and the way to get rid of that is say they're not external.

But now you're saying that every particle of matter in the universe interacts at all the time with every other particle of matter? I'll do it the way Konk does it. Konk introduces in addition, three categories here, in the third is community. Community is everything interacting with everything else and he sites Newton's, for this.

All right, that's one way to think of the course. I'll just run quickly. What I passed out is 20 questions. You have them so I won't sit on them. They're 20 questions, that if you want to judge whether you understand this course, and what you got out of it, here's a test for yourselves.

Take each of these 20 questions and ask yourself how did this question arise historically and why did it become increasingly important? All 20 questions are answered in the Principia. They were all open at the time the Principia began. They were all answered in the Principia 19 of the 20 are our current answers.

So one way to think why the Principia is 500 pages long is it answers a whole lot of questions besides the obvious ones. To understand those questions, that's why I'm taking you through 14 weeks of two hour and 45 minute classes to do. Because when we read the Principia, you should say, oh I know why he's doing that, that's that question.

And know where it came up, who it came up with, and why it became so important, and why it was hard to answer. Okay? So, you can go through those 20 questions on your own. Final thought for the day and for the course, I picture here three revolutions occurring.

The third one, is just starting to show up in the material for tonight. So the first is this idea of a commitment to the principle that the empirical world is the ultimate arbiter. I saw that strongly in Tycho, I saw it, it continues and becomes most strong in Newton.

The second, the discovery that extended mathematical theory can open the way to more telling evidence than can be achieved through testing hypotheses in isolation. You see that in Galileo, you see it carried more forward by Huygens, you see it in. Mathematical theory is very, very effective in allowing things to be done.

Then the third one, and this is one we just saw at the beginnings of tonight, the emergence of a new conception of exact science in which every systematic discrepancy between theory and observation is taken as telling us something about the world that we have to pursue. Remember the extent to which that's not true of Galileo, discrepancies are to be explained away, dropped, even really for Ptolemy, Copernicus and Kepler.

Kepler was somewhat wanting every discrepancy to be informative, but it's Newton who's gonna push that to the extreme. Every systematic discrepancy with theory is telling us something fundamental about the world, we have to figure out what, okay? That's the revolution that is occurring during this 17th Century, that culminates in the Principia.