So 1638 is when two new sciences came out. And I want to compare and contrast, as of 1638 where things stood in celestial motion versus motion near the surface of the Earth. So, what I'm saying here is that in both cases what is desired, and I'm trying to say this in as neuter a form as I possibly can, in both cases, we want a mathematical representation specifying the location of a body versus time along a trajectory known or unknown.

When it's vertical fall, we know the trajectory. When it's on an incline plane, we know the trajectory. When it's a pendulum, we know the trajectory. But when we, for the projectile motion, the trajectory was unknown. Of course, the trajectory in the celestial cases was very much unknown. But they're sort of doing the same thing.

Trying to get a geometric representational motion that locates the body versus time. But look at this difference. If we take 1638, and the point I made maybe be too quickly last time, it's one of the reasons I decided to add this slide this time. And I'll just read this out.

Dop Kepler's proposed horizontal parallax of one minute of arc which he had proposed, but not himself adopted. That led Horox to reduce the eccentricity of the Earth Sun orbit from 0.018 to 0.0173. Still an error should be 0.0168, but that's because our atmospheric refraction corrections were wrong. That, in turn, had led to an increase in eccentricity of Venus's orbit from 0.00692 to 0.00750.

Notice that's an increase really in the fourth significant figure, virtually so. And in the length of the semi major axis from 0.74413 to 0.7233. That change in the major axis, that's the astronomical units, mean distance of the sun from the Earth. That revision Then eliminated a 0.11, a tenth of a percent error in the three halves power relation between mean, distance, and period.

The second largest of any of the discrepancies. Only Saturn was larger. And Saturn was very much, as Cory pointed out to me on Friday, it had not gone around once during Tico's period of observation. So Saturn could be handled separately. That, of course, is why Horrocks turned to Saturn right after he did Venus.

Okay? That was the natural place to turn next if he's focusing on the three halves power rule. But based on the removal of that error, he decided that Kepler's Three S Power Rule should be taken to be exact. And that allowed him to tack on one more significant figure to the distance, 0.72333.

Three threes, that did not come from observation. That came from the Three Halfs Power Rule. And when he put all those revisions in, he reduced discrepancies in the Rudolphine tables from five minutes of arc down to two minutes of arc. That's the state at that time of what was being done on celestial motion.

Where fussing over reducing five minute of arc errors, down to two minute of arc errors by doing refinements in the fourth and fifth significant figure, okay? Now, people didn't realize Horrocks had done that till the 1660s. But the point is, that was an appropriate thing for him to be doing at that juncture.

The Rudolphine tables were there. The crucial issue is, what are the discrepancies that they're exhibiting versus observation telling us? And the natural first step is not to assume there's something fundamentally wrong. But instead, start looking at the parameters and see if that doesn't remove them. And that's still what we do in astronomy to a very great extent.

Now, contrast that to the situation Galileo was in. The sole natural local motion that is near the surface of the Earth. That's what the word local means here, is vertical fall. Galileo had originally concluded that bodies have a characteristic natural constant speed of dissent that depends on their density and the density of the medium.

We call that now, the terminal velocity of a body falling in a medium. You never really quite reach the terminal velocity, but any heavy object gets within a smigeon of less than 1% of it. Very, very quickly. And then, continues to fall that way. So you know some of these things.

If you fall out of an airplane and you spread your arms and you don't have a parachute, you'll hit the ground at about 140 miles per hour. And that is the terminal velocity for a shape of a human being going down, spreading his arms, etc. Why do I know those?

That's cuz I work on airplanes. But now Galileo, by the way, that describes nature, that idea. There's a terminal velocity, and objects seem to fall more or less at that characteristic velocity. And it depends primarily on their density and the density of the medium. And it's very Archimedean if you think about it.

Archimedes says, on bodies floating in water that if the density is greater than water, the body falls. He doesn't say how it falls. But it becomes very natural to conclude that it falls at a terminal velocity. Now, he was instead proposing that in the absence of any resisting medium all bodies are uniformly accelerated as they descend.

The rate at which they gain speed is the same for all bodies regardless of their weight and shape. And any observed departure from these results, from this, results from an affect induced by the motion. And hence a second-order consequence of it, namely a resistance that's caused by the velocity relative to the medium as it moves through.

So if the medium is moving with the body, there is no resistance at all. That's a totally different picture. And he's at a point, he's worked this out earlier, but he's publishing for the first time, a view of what nature tells us happens. We should not think that way.

We should instead consider what happens in the absence of air resistance and make claims about that. With air resistance or water resistance, if it's falling in water, being a second order effect that should be considered separately. Now, these are essentially qualitative claims. They're quasi quantitative, but we're not talking about differences in agreement that are running in the third and fourth significant figures.

It's a different world. To go back to Leon's challenge to me two or three weeks ago, in this case, we actually have some grounds for saying, he's at an early stage of a new science, starting a new science. Okay. And even though there was a rich tradition beforehand, nobody had ever worked out the theory he's proposing here.

They'd never really worked out the first theory either, because they didn't have exponentials. And that's what you need to do terminal velocities, etc. You will see it worked out. It's in the Principia quite straightforwardly. In Newton's Principia. All right so there, we're talking about very, very different states of affairs in the two fields.

And what we're going to watch in the next five weeks and we'll culminate in the mid 1670's 16 73 primarily. In publishing a book that would have been the most important book published in the 18, in the 17th century on the subject of physics had it not been for Newton's Principia 14 years later.

So that's where we're going. We're building up to in the mid 1670's. And finally getting to, more or less, the same point that Corox is doing. By the time we get to Huygens, he's working to four significant figures on what we would call the acceleration of gravity. He's got that measured to four significant figures.

So, there's gonna be a similar culmination by the mid 1670's, 40 years later. Building off of Galileo, but with various people intervening.