Smith, George E. (George Edwin), 1938-

2014-12-09

Next is a long paragraph added to the scholium after theorem four. Theorem four is the three halves power rule for ellipses. This is a single paragraph and the only change in it is Newton twice modify scratching out the word hypothesis and replacing it by law at the two places.

But otherwise it's beautifully, neatly written, and it is stunning. This has come to be known as the Copernican scholium. It's a single paragraph, I break it into four parts to treat each one separately. Each of the four parts is remarkable. I'm going to go through them, I'm going to read the whole thing out to you.

Then I'm going to jump to the third part, fourth part, then come back to the second. So let's start with the top. Moreover, the whole space of the planetary heavens is either at rest, as is commonly believed, or moves uniformly in a straight line. And hence the common center of gravity of the planets by law four, is either at rest or moves along with it.

In either case, the motions, the planets, among themselves. Of course, now we're working with my translations. I didn't just lift Harivel's out. Among themselves, by law three, are the same. And their common center of gravity is at rest with respect to the whole space. And thus can be taken for the immobile center of the whole planetary system.

So we've got an answer to the point to which we refer all motions. Next, hence indeed the Copernican system is proved a priori. He does not mean a priori as philosophers mean it. He means we don't need any direct observations like annual stellar parallax. We're gonna be able to conclude it without anything like that.

For if in any position in the planets their common center of gravity is computed this either falls in the body of the sun or will always be close to it. Pause a moment on that before I go on. What both Curtis Wilson and Tom Whiteside say about this is Newton already had, first of all, they say, Newton has to mean near center of mass.

Okay, which is wonderfully interesting cuz he hadn't invented the word mass for this yet, as you'll see later. So therefore he really has to have the law of gravity to be safe. I guess I can take credit. The article that I put on reserve on how did Newton discover the law of gravity, it actually happened in this course in 1991 that I realized he doesn't need the law of gravity, how to prove this.

And now it's become sort of standard in the community that he got the law of gravity much later. But it does, offhand if you're reading from the perspective of the Principia, this looks like he's got the law of gravity. You have to turn around and read your way up, and ask how could he conclude this.

I'll come back to that and show you how he can conclude this. I want to do the next two paragraphs first because they're more important. The little short paragraph is important because it's probably why Newton decided to write a whole book. Namely to prove the Copernican system, really do it right.

And so this is a driving mechanism, but the next two paragraphs totally alter the state of evidence that he's gonna have to deal with. But again these are my translations. By reason of the deviation of the sun from the center of gravity, this centripetal force does not always tend to that immobile center.

And hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. Hear that? They don't even repeat the same orbit ever. Each time a planet revolves it traces a fresh orbit as in the motion of the moon, and each orbit depends on the combined motions of all the planets.

Not to mention the actions of all these on each others. Here's the picture. The sun is moving in tandem, let's say with Jupiter, and there are only two bodies. It has to move always in the same ratio between the center of gravity for the two, for the center of gravity not to be affected.

That's the balance principle, okay? But we've got six bodies going around the Sun, and they're going at totally different speeds. Totally different periods. Saturn takes 29 years to go around. The Earth takes one year. The sun has to have a motion balancing those six such that the center of gravity is not affected by them.

And he looks at that and says even without the planets interacting. That's a hell of a hard problem. Where's the Sun, what's its motion have to be like for the center of gravity not to be moving? Well, it's going to be very complicated. If it's very complicated, so too are the motions of all those planets because they're being controlled by centripetal forces directed not to the center of gravity but to the sun.

So the sun's doing this crazy complicated dance that's making all the planets deviate from elliptical orbits. In fact from even normal orbits. In fact read this next statement. But to consider simultaneously all these causes of motion and to define these motions by exact laws, admitting a VC calculation, exceeds, if I am not mistaken, the force of any human mind.

So here he is concluding before he really even starts the Principia that it's impossible to calculate the motions of the planets. That's a far removed from okay? Just think for a moment what it's saying. In the case, if these were exact, then you'd say okay, we have a deviation from our expected orbit.

Surely that's gotta be an observation error. Our atmospheric correction, refraction corrections, are wrong. Or something like that. We're gonna be looking at something wrong, or there's an external force of some sort acting on the body. But otherwise, it would be exact. Okay. Now we're saying no, no, it's not exact.

We can't even calculate, it's a total mess what the motions are. So here's a deviation from your calculation. Question, what's it telling you? Bad observations? Or you just haven't gotten near the actual motion yet. Well, that's gonna be ambiguous. It's worse than that. Remember the point Descartes made.

Anything like Keplerian motion is a parochialism. Apochal pariochalism, I did a Skinarian move. And as a result the actual motions are so complicated it's silly thinking who can compute them exactly. What's in the the same damn thing. Except Newton wants to use the actual motions as evidence. And of course Descartes' telling you forget it, you can use those sevens.

They're way too ill behaved because the neighboring vortices are screwing everything up. Now that's the ambiguity that's going to bother Newton most. Once you grant that the actual motions are inordinately complicated, by the way we still can't calculate them exactly, I trust you all know that. His statement was apocryphal in more ways than one.

Once you grant that, then you've got a constant ambiguity between the possibility of external systems affecting this one and throwing them into motion, or their interactions with one another creating the complex motions. And how are you supposed to sort that out? Okay, you really can't draw very strong conclusions from the fact that Kepler gets reasonably good results for ellipse with area rule.

Last way to say this, this immediately opens up all the other ways to do things, beside the area rule. If the actual motions are very complicated then most you can do is approximate them. What's the grounds for saying the area rules the thing at work? Okay. Well it's a nice, it's an explanation of everything.

See you can have a nice explanatory theory inverse squares intrepidal force. But when it comes to try and to give evidence for it. You're going to be constantly in the situation of under determination with respect of very different causes. Now the next paragraph makes that worse. And it's an important paragraph, because it's historically important.

The point made is historically important. Omit these minutiae and the simple orbit, and mean among all the deviations, will be the ellipse that I have already discussed. What's he assuming? He's assuming that whatsever's going on is periodic. Remember the Saros cycle of the moon. Every 18 years, the Moon, Sun, and Earth, realign.

During that time, the Moon's gone through over 200 revolutions. And the Earth's gone through 18, but they repeat. It's a periodic pattern to the, okay. And that's what he's assuming here. Whatever's going on in the motions of the Sun etc, it's periodic and somewhere down the road you'll return to the same angular positions, etc.

When you can almost figure it out by looking at the 29 years For Saturn and figuring out what the great cycle has to be when all of them finally get back to the same one. Continuing. By the way, he's wrong about that. That would be true if all the deviations from Keplarian motion were periodic.

It turns out, some of them are not. The one's that are not are called secular. They're not periodic, that's why they're called secular. And we will get to those. And Newton comes to realise them not to long after this, realize they have to exist. But that's okay. Nothing like this paragraph appears in the Principia, and I should have said on the one that says the orbits are beyond calculation.

Nothing like that occurs anywhere in the Principia. People figured it out by 1730. When Fontanelle does his eulogium for Newton, one of his remarks is if Newton's right, the motions of the planets are incredibly complicated, surely beyond human reckoning. But nothing like the Principia totally suppresses that paragraph.

I'll come back to why that's important. Let me continue the next one. If anyone tries to determine this ellipse by trigonometrical computation from three observations As is customary. He will have proceeded without due caution. For those observations will share in the minute, irregular motions here neglected and so make the ellipse deviate a little from its just magnitude and position which ought to be the mean of all the deviations.

And so will yield as many as ellipses differing from one another. As there are trios of observations to be employed. Therefore there are to be joined together and compared with one another in a single operation, a great number of observations which temper each other mutually. And yield the mean ellipse in both position and magnitude.

Well, Saturn takes 29 years per orbit. So we're talking about collecting observations over many, many years. And now doing a mean for each of the orbits. Okay. Notice what this is challenging. It's one thing to say, I've got deviations from my Keplerian orbit. I don't know if they're due to observational error or due to the complexity of the motion.

But now he's telling us, you don't really know how to define the Keplerian orbit. Of Mars, Jupiter, Saturn, etc., unless you collect observations over a much, much longer time and do a mean of all of them. To give you just an off-hand feel for this, when Newcomb does it, Simon Newcomb does it for the four inner planets, He started this in 1878, he finished it in 1900.

But he publishes his memorandum, his monograph summarizing the work in 1895. What he describes is he used 62,000 meridian anno observations of Mercury, Venus, Mars and Earth. And did, of course, a huge statistical analysis to get the elements of the orbits. That's the sort of thing you need to do, 62,000 observations, okay?

And he knew that. Okay, but he didn't know Newton said this because this too never appears anywhere in the Principia. But the nice little solution for orbits that's given in following problem four, it's not worth much anymore unless you do lots and lots and lots of observations. So my point, the way I usually put it, I rarely show this whole Copernican scholium when I talk.

But the third paragraph isn't in virtually anything I publish. It's very rare, it's not there. And I'll make the comment for all of you. It's in the context of that third paragraph that you'd need to read the Principia. The Principia is a response to this complexity. And the oddity is that was not made clear till 1893 when this paragraph first emerged.

There's nothing in the Principia that tells you, you must understand what this book is a response to is the inordinate complexity of the actual motions. And the problem of extracting evidence if they're that messy. Okay? So we will come back to this. The opening class next semester will feature that paragraph.

I'll drop the rest of the Copernican Scholium but that paragraph sets it all up. The next paragraph's really very important too. For the reason I gave. You're not handed the right, Keplerian orbits. Remember, Kepler was struggling with them. You're gonna see later tonight, Flamsteed was struggling with them.

And you can see reasons why now. Okay, cuz it's not automatic just from the observations you've done that you've picked out an elliptic orbit at all. Fair enough? Changes the game, right everybody? De Motu Corporum is wonderful but now you have to ask what's it good for if the actual motions are like that and I don't mean that rhetorically.

Because it's gonna be good for a lot. But it's a different thing than if all you're doing now is showing. Well, there is such an inverse square force therefore, the motions are Keplerian. The motions are not Keplerian, and Newton knew that from very, very early on in the development of the Principia.

And why, you know, well, the easiest way to say it is once you start reading the Principia through the lens of that paragraph it becomes a very, very different book for everybody.