All right, cuz what I wanna look at now is where astronomy stood, I'll finish tonight with this, where astronomy stood in 1679. The reason 1679 is so important is, it's the year Hooke is desperate to get articles for the now defunct Phil Transactions. And he starts writing everybody asking, do you have an article.
And he writes Newton, and Newton replies with something, and then they have an exchange, and then it breaks off because Newton gets angry at Hooke. But in the meantime, Newton has probably made the key discoveries that led to the Principia that sat moribund for five years till Halley comes and asks him the same question Hooke had asked him in 1679.
And he gives Halley the answer and Halley gets, as you'll see, gets more than a little bit interested. And that will be the end of this course. Namely, Newton writing a nine handwritten page document, sending it to Halley, in effect tying Kepler's three laws to the principle of inertia, free fall, and projection.
The six are all tied together in a single nine-page handwritten document. Which when Halley reads, he gets on the stagecoach and goes back to Cambridge to wanna talk to Newton more about it. And by the way, that's an overnight. You go halfway to Cambridge, you stay in an inn for the night, and you go the other half.
And this is in December that he did that so you'd get about seven or eight hours of really good daytime in England at that time. But what your last paper's gonna be about is to try to substitute for Halley his report on returning from going and visit Newton.
What he saw in all of this. But it's the seed from which all of the Principia grows. And every one of the propositions in that eight-page document appears in the Principia with essentially the same proof as in that eight-page document. So it's the start of everything in 84, but it's the letter from Hooke in 1679 that starts things, and you'll see why I am emphasizing that in just a moment.
So what's the state of astronomy? Well, the obvious question is how good were they, how successful were they in mathematical astronomy at matching observations? And the answer is, not that good. You look at the Saturn and Jupiter, and we now know it's not surprising, they're wandering all over the place.
But even Mars is not that good. You're seeing errors down there not, from Street as much as 20 minutes of arc. The other one's consistently bad. And then from Argoli and Wing and Condstat, you're seeing errors, substantially more than ten minutes of arc. In fact these are all, by the way, calculations made in the 1980s by Owen Gingerich.
Okay, so nobody published this kind of display of how inaccurate things were. But the point is they weren't that good. The next figure, striking because this is Street doing very, very well in the case of Venus, far better than the other two. But remember Street's using Horrocks for Venus and that's where Horrocks had his great success.
He got Venus down very fine, and always keep in mind, Venus is the nearest to circular orbit among all the planets known at the time. I can't remember whether it or Neptune is the most circular. But it's eccentricity is 0.005, so it's almost a circular orbit. So it's not that surprising you can do well with Venus.
It's the easiest of the orbits, provided you get other things correct in it. But did anybody know that Street had done that? Well, we know that Flamsteed said he had done better than anybody else. And you notice Curtis Wilson here does single out the Horrocks, and the fact that Street was inheriting Horrocks.
But far more useful now, this is a table that I've published several different places now. This is the state of theoretical mathematical astronomy as of 1679, 1680. We had seven different ways of doing, seven different versions of orbital theory. The one thing that they all agreed on is the ellipse.
They had four different geometric, one, two, three, four different geometric constructions plus the area rule. So there were five different ways of locating a planet on an ellipse. And two people, Horrocks and Street, got the mean distance from the sun by means of the three-halves power rule. Everybody else was doing it from observations.
Now the important thing is Newton knew all seven of these ways, as of 1679. He may have known the Kepler way best from reading Horrocks. In 1672 and 73 all of Horrocks' remaining papers got published posthumously. Newton owned those volumes. And I brought them in when Horrock, I brought one of those volumes in when we were looking at Horrocks earlier.
It's not just Venus' transit of the Sun, it's everything Horrocks did, his theory of the Moon and many other things. So Newton knew all of these and the obvious question, very natural to ask, is which of these seven is to be preferred? All we know is that the group and Paris were going to proceed with Kepler.
They might have actually meant Horrocks because Huygens was a big fan of Horrocks'. So they may have actually been taking the Horrocks version of Kepler and pushing it for all it was worth to see where it broke down. As I said, that's still a long way off of bearing fruition as of 1680, okay?
I keep, when I first published this slide the whole point was to tell people, this is the question The Principia primarily answers. It selects among these seven. It selects Horrocks, by the way, not Kepler. And you'll see when we get in The Principia, it's very carefully choosing the Horrocks line there.
It would be an ellipse were it not for action of other planets. The area rule would hold were it not for action of other planets. And the three halves power rule is the proper way to get the mean distance. All of those come out of the Principia, and from the Principia forward, they become standard.
From roughly 1740 on, that's what everybody does. And that's not because of Cassini's observations, that's because of the Principia itself. Okay, as of 1679 this is a collection of unanswered questions. They're all very important, so I'm gonna run through them one by one. Which of the several more or less comparably accurate yet still discrepant orbital calculation procedures is to be preferred?
That's the one I just did. Second, does Kepler's or any other of these procedures amount to anything more than just a transient approximation to the true motions, as Descartes would have them? We don't know that. We know that the Moon has not been successfully described by anybody. And we know that Jupiter and Saturn are doing something weird.
Kepler was the first to say that. The elements, the values of the elements of those two orbits were changing over time. They don't know what's going on with it. Third, what's the nature and source, I've just done that one, comparatively large discrepancies exhibited by Jupiter and Saturn? Fourth, what are the proper corrections to observations for parallax in atmospheric refraction?
That was the project that Cassini considered primary. Fifth, is the speed of light really finite and, if so, what corrections to observations are needed to adjust for it? This is, oh it's 1679, it's only months after Roemer publishes. What's the motion of the Moon and why is it so much more complicated than the satellites of Jupiter?
Satellites of Jupiter are remarkably well behaved side-by-side with our moon. Why is that? Seems very incongruous. Seventh, what are comets, and what trajectories do they describe as they pass through the planetary system? It's slightly more complicated than I showed you. I showed you the Christopher Wren straight line, which seemed perfectly reasonable, because it seemed to fit the observations reasonably well.
In 1678 Hooke proposes, know the magnetic effect of the Sun slightly displaces comets, but not the way you do planets, okay? And in effect he's saying that comets are made of different material, they're being drawn curvilinear much less than the planets are. This is 1678 from Hooke, in a book called Cometa that I've put on supplementary material.
Next, does the strength of service gravity really vary from place to another and if so, according to what rule? All we have at this time is Rochet's claim which Huygens doesn't trust. Newton doesn't even know of Rochet's claim and doesn't find out about it til the middle of 1686, after Book One of the Principia's in press.
And he goes back and makes changes to it after learning of it. Next, are the planets being carried around by vortices and if not, then what retains them in orbits that are at least roughly elliptical? That was the question that Borelli launched into more than anyone else. But it's very much a live question as you're about to see in a couple minutes in the last slide for the night.
And finally, what, if anything, should be made of the seeming fact that the centrifugal of the planets varies in an inverse square ratio with the mean distance from the Sun? I should explain that because it's fairly significant. Huygens published the v square over r solution for uniform circular motion in 1673, without a proof.
So it appears the only two people who knew how to prove that result were Newton independently of Huygens, and Huygens until the Principia published. The first published proof of v square over r is in Newton's Principia. You'll see it in this document you're gonna be reading it two weeks.
It's, If you know the v square over r result, and you know the three halves power rule, it is child's play to say that varies from planet to planet by the inverse square of the mean distance. It's a very trivial thing to notice. Huygens didn't notice it because he paid no attention to orbital astronomy, because he had Descartes' vortices quote blinding him, obscuring him.
Newton had noticed it already in the 1660s but didn't tell anybody. Christopher Wren noticed it in the 1670s and started telling a lot of people. So there was a lot of talk about about inverse square suddenly showing up in the 1670s. Enough that Wren actually visited Newton in the 1670s and talked to him about inverse square, in Cambridge.