All right, the other thing that's added, there's a diagram changed in problem five. It's interesting, the diagram changed but the words did not. So I don't know what he had in mind when he changed the diagram. He just put the triangle in there, as you can see, the dotted triangle.

I don't know what he's doing. And then following problem five, down here there's a new scholium that goes on three long paragraphs. I call it the resistance scholium, but it runs, you see, this is the third paragraph, second paragraph. And down here, I can't always see where it breaks.

I think it's there, first paragraph. It replaces the scholium that led into the last two problems from last time. And you can see now there's a new title leading into those problems. So let's look at those three paragraphs there. Thus far I have explained the motions of bodies in non-resisting mediums, in order that I might determine the motions of celestial bodies in the ether.

We haven't got empty space here, folks. For I think that the resistance of pure ether is either nonexistent or extremely small. Quicksilver resists strongly, water far less, and air still less. These mediums resist according to their density, which is almost proportional to their weights and hence, I may almost say, according to the quantity of their solid matter.

That slightly strange quantity of solid matter is, of course, a Cartesian expression. So it looks like he has a Cartesian picture of how gravity works here. Ether is filling the space all around it, but it's just the quantity of solid matter that's controlling the weight. Therefore, as the solid matter of air is diminished, so it is the resistance of the medium in about the same proportion to the point, in about the same proportion.

Notice that up to the point that it obtains the tenuity of air. Now why's he saying about the same? Because the ether's in there too, and he's got to allow the ether to have some effect. If air were to penetrate the parts of bodies freely and so were to act not only on the external surface of the whole but also on the surfaces of its internal individual parts.

That's the Descartes picture of how ether works. Its resistance will be far greater. Ether penetrates very freely and does not resist. All those sounder astronomers think that comets descend below the orbit of Saturn, I misread that. All those sounder astronomers think that comets descend below the orbit of Saturn, who know how to compute their distances from the parallax of the Earth's orbit more or less.

Of course, that sort of leaves Galileo out. We'll worry about that later. These, therefore, are indifferently carried through all parts of our heaven with an immense velocity, and yet they do not lose their tails nor the vapor surrounding their heads, which the resistance of the ether would resist and tear away.

Planets have, in fact, now persevered in their motion for thousands of years, so far are they from experiencing any impediment. I singled that out because that's going to become the basis for saying there's no ether in the Principia. In particular, the fact that comet trajectories can be done and the tails remain intact are screaming at you there's no resisting medium out there.

Motion in the heavens, therefore, is ruled by the laws demonstrated. Okay, so he's still holding onto that he's got the laws that are going on but the motions are remarkably complicated. But if the resistance of our air is not taken into account, the motion of projectiles in it are known from problem four and the motions of bodies falling perpendicularly from problem five, assuming indeed the gravity is reciprocally proportional to the square of the distance from the center of the Earth.

For one kind of centripetal force is gravity, and from my computations it appears that the centripetal force by which our moon is kept in its monthly motion about the Earth is to the force of gravity on the surface of the Earth reciprocally as the squares of the distances from the center of the Earth, more or less.

So it looks like he's redone the moon test and now concluded it's showing the terrestrial gravity is inverse square. I've got to include the last one because it doesn't appear in the Principia. From the slower motion of pendulum clocks on the summits of high mountains than in valleys, it's clear that gravity diminishes with increase of distance from the center of the Earth, but in what proportion has not been determined.

What he's referring to is Halley's report that when he checked gravity on the top of the Mount at Mount Saint Helena, which is about 2,700 feet high, about half the height of Denver, the pendulum was slower. And he attributed it to altitude. Hooke told him that in one of the letters to Newton in 1679.

Newton is using that to conclude that what Halley observed was a variation due to altitude. You only have to calculate how high 2,700 feet is, versus the center of the Earth, to realize that can't be. So this never appears in the Principia. Somewhere along the way, he and Halley figure out Halley got the result he got because Saint Helena's near the equator.

It's another take on the Rochet example. Halley already knew about the Rochet example. Newton did not know about the Rochet example at this point because we have a memo where actually Newton, in his own hand, reports what Rochet found as Halley told him on the occasion of the memo.

We can't date that memo but we can be confident it's later than this. All right, the new moon test. This is a slightly revised version of the one I gave you before, and I did the ratio slightly different, but it's a number over 40. The ratio of the tendencies at the surface of the Earth versus the moon's tendency to fall toward the Earth is 4,302 if you use as the radius of the Earth 17,500,000 Italian feet.

Meanwhile, however, Picard and Cassini have both gotten very careful measurements of one degree of longitude, and you'll notice five significant, into the sixth significant figure, the two of them differ by 6 Paris feet out of 342,000. Those are reasonably decent measurements. If you do that, then the Earth's radii is 19 million, not 17 million.

If you run through the same calculation totally in parallel, it's now 3,617, only 0.5% difference between the two. Not exact, but certainly near enough to be evidence that terrestrial gravity is what's holding the moon in orbit and terrestrial gravity is inverse square. So I take this to represent the corrected form of the moon test.

There is one in the Principia itself. The numbers are slightly different from this, but when you get to the Principia you'll see it. But this is just my working it out from knowing Newton knew Picard's and Cassini's values. They were published, so he could have used either one.

I use Picard's cuz that's what he does in the Principia, uses Picard's. All right, so the moon test, we now have conclusion the terrestrial gravity is inverse squared. He says he has the evidence for it. He can't have evidence for it. Last paragraph of this new scholium. The motions of projectiles in our air, moreover, are to be referred to the immense and indeed motionless space of the heavens, not to the moving space which is revolved along with our Earth and our air and is naively regarded as immobile.

This is part of my evidence to say that what provoked him into version three of De Motu was the question of to what center to refer the motions, because he's meticulously now gonna tell us how to do that properly. The ellipse which the projectile describes in that motionless space is to be found, and this its motion in a moving space is to be determined.

With this agreed, it will be gathered that the heavy body which is let fall from the top of a tall building will be deflected a little from the perpendicular in falling so that the amount of its deflection and direction thereof may be determined. And conversely the motion of the Earth may be gathered from the deflection as established by experiments.

When I myself formerly communicated this deflection to the celebrated Hooke, he confirmed that it was so by an experiment three times repeated the heavy body always, etc. Okay, that's the last place I know of where Newton speaks nicely of Hooke. But there's no complaint here. Again, the last part of this does not occur in the Principia because by then he decides you'd have to be very, very high to detect the effect, that there was no way they could do it.

So he just hasn't done the calculation on either the height at the mountain on Saint Helena or what you need in dropping it. But we've now got all the changes made in the augmented version of De Motu. All the propositions are the same, but we're now getting meticulous about multiple centers and what we do about the multiple centers of attraction.

And we're getting careful about when you drop an object on the surface of the Earth and ask what trajectory it describes, you need to be careful whether you're talking about the surface of the Earth as your reference point, the bottom of the point where you dropped it, or you're talking about a space in which the Earth is rotating, because those are gonna give you different answers.

He's getting careful about the reference. Fair enough? Any questions? That’s all of the augmented De Motu. The shocking part of it is 1893 that it first appears. And I really think the whole history of reading the Principia would have been different had it been available all along. People would have seen what he saw as the problem that made the book 500 pages long.

That he can't just run off of Keplerian motion and draw conclusions. He's gotta be very, very careful, using Keplerian motion as at most an approximation to the actual motions. Maybe a very proper approximation, but that's a different thing between it and the actual motions.