So, look, there are questions of evidence here. There are really five principles, and in the notes I go through all five. Actual sun versus mean sun. And there the evidence is pretty strong. We've got the latitude result, we've got the experimentum crucis, and we've got the line of apsides going through the actual sun rather than mean sun for triangulation.
So the shift to the actual sun is very strong and almost everybody seems to have switched to that very quickly, for whatever reason. I've never been able to trace the reason why everybody shifted to the actual Sun. The second thing is the latitudes. And again, centuries of trouble with latitudes and crazy multiple tiltings disappear into a very simple story.
Again, people have almost no resistance to it. And the evidence for it comes down to repeated measurements of the inclination constantly showing more or less the same thing, one degree, 50 minutes and a few seconds. So that tends to have reasonably strong evidence. The bi-section of the eccentricity of the earth is a more complicated problem.
He did do things beyond what I described, for example, he did careful measurements of the apparent diameter of the Earth. Apparent diameter of the sun and how much it should vary around the orbit and if it's the total eccentricity it should vary twice as much as if it's half the eccentricity and it's 30 minutes of arc subtended versus 31 minutes of arc subtended.
And he concluded that was supported. So the really hard one. The one to ask about what kind of evidence did he have, is for the combination of the area rule and the ellipse. Which is of course is precisely because A, they're both extremely radical. They're totally new principles never before in astronomy.
You might just notice, there are three ways that uniform motion on a circle could be described. Equal arcs at equal times, equal angles at equal times. Nobody ever thought equal areas at equal times, but that's the right one, so to speak. It is a generalization of uniform circular motion, so that's a beast of one thing going for it.
But it ends up that his argument comes down to something like he's got these three principles. The diametral distance reel, he has direct observational evidence for, direct observational evidence of course a total misnomer here, cuz it's presupposing all those angles. And all those angles are coming out of theories.
The theories are not theories of motion but they are Earth's sun theories of theory of motion but he's not assuming a motion of Mars but nevertheless that's fairly strong. The fact that the timing along the trajectory by means of the aerial rule in a way the best argument for that and probably what convince him.
The area rule is to be preferred is that when you use it you get this match with the truth lying halfway between. So in a way the area rule's forcing an answer on you. And the one over r rule being less precise isn't forcing an answer midway between to you.
So the area rule has a virtue. But I, in my mind, the strongest feature supporting the motion of Mars is the fact that these three interlock with one another. Given any two, the third automatically follows,and you can develop independent evidence for at least it being an oval close to his ellipse.
And the diametral distance rule, holding to very high accuracy. That then gives you the area rule. You can instead justify the area rule cuz it gives you a definite answer and that combined with the diametral distance rule then forces the ellipse on you etc. Okay but what we have here, the question I wanna raise, is the evidence extracted from the observed geocentric longitudes confirms that all three principles hold to within bounds of uncertainty.
So let's grant that these three hold to within bounds of uncertainty. That's what the Table of 28 is really showing. Fine. Here's a question. Do these three principals hold or should be taken to hold exactly based on this evidence? Essentially exactly that's a phrase I've introduced. Would hold exactly in the absence of external perturbing factors.
And that Cameron has already led me to point out that I think it was Cameron. When it comes to the Rudolphine. No, no it was Michael. When it comes to the Rudolphine tables what we have is evidence that it's not exact. He's ready to say it's essentially exact.
There's an external perturbation on it or merely approximately. Without precluding, are alternative principles different from them. You should ask yourself. Can he really say anything stronger than approximate to a higher degree? And if so what justifies it? If not, why not. We know why not. Because too many sources of imprecision are entering into all of the reasoning, including the observations themselves.
Okay, so that's one question, and I think it's an important question. The question I really want to ask at this juncture is what kind of evidence would it take to show that the three rules hold either exactly or essentially exactly? And it's not obvious. Newton is going to claim the following.
The three rules would hold exactly were it not for inter-gravitational interactions of the planets with one another. And they hold essentially exactly in spite of that. That's gonna be the Newton conclusion at the end of this. And he thinks he's actually taking on the problem. Remember what he said?
Kepler only guessed that the orbit was an ellipse. I proved it. That's what he means. He's got evidence to talk about it exactly, and he never read Astronomia Nova, so he had no idea what this argument was. I don't think he would've concluded otherwise from this argument, that it's still only approximate.
Here's a different question. Generic versus specific principles for Mars. Here's the problem. We had some discrepancies up there. When does a discrepancy tale, count as evidence against the theory of the orbit and when does it count only as you don't have accurate enough values for the elements of the orbit.
Okay, and does the present evidence begin to allow you to discriminate between those two? Now in Kepler's mind it didn't. In Kepler's mind at this point any discrepancy is because either the observations are not good enough, or the elements need to be refined. And the elements do get refined in the Rudolphine tables.
He comes back to this and he has many more observations by then and does accordingly. Let me give you a different issue. These are all related. They're all philosophic questions about what sort of evidence do we need, and they show up. The reason I'm stressing them, they're not exactly out of philosophy of science, standard philosophy of science.
But when you do actual science, these are the sort of questions that hit you. What have I got here, a very good approximation or do I have claim to it being exact? Hell, there are discrepancies. What do the discrepancies do to us? Is it because I am fundamentally wrong, or something else is going on that is minor, all I need to do is clean things up?
That shows up all the time, practically. But the issue I throw up there in parenthesis is in many respects the more interesting one. The phrase take to hold exactly is Newton's. It is Newton's view that that's what we do when we reach a conclusion in science. We take it to be true or very nearly true.
Is the phrase he uses. And he says that's okay. That's different from saying it just is exact. So the issue now, and the issue I want to put before you, and I'm going to offer my own view of that in just a moment. Let's look at the question not whether Kepler proved that the area rule, diametral distance rule and ellipse are exact.
Let's look at it from the point of view did he provide adequate evidence to take it to be exact? Now, what's the point in taking something to be exact is the right question you should ask me. And the idea is well, we're gonna do further research. We're gonna do research on four other planets, okay?
Should we continue to mess around with Mars? Or should we conclude we've gone as far as we can go with Mars? Now, let's start looking at the other planets and see what happens with them? That's an example of taking it to be exact. Another example is more observations are going to be developed.
Little did he know that a telescope was going to appear a year after it showed up and the possibility of much more precise observation. But the idea here is taking it to be, taking it to hold exactly, or essentially exactly, has virtues that taking it to be hold merely approximately do not.
What's the virtue? If you take it to hold exactly or essentially exactly any discrepancy is to be taken as telling you something. If you take it merely to hold approximately, it's not necessarily telling you anything at all about the world, it may be telling you something about the math you're using, the curve that you're using.
So, by taking it to be exact or essentially exact you're in effect saying, okay, I'm now ready to take any discrepancy that's systematic that's not just a local, discrepancy that may be due to a bad observation. Any pattern discrepancy ought to be telling me something and I can pursue it accordingly.
And that's the point I want to make and I make it in the notes. I take Kepler's reasoning in here and again Curtis Wilson summarized it far less than I did. I'd only summarized it. I'd shown you maybe half of the cross checks he actually did. Just because, finite time he's just constantly finding another way to do a cross check.
The point I'm making, I think he had adequate evidence to take this to be exact pending further information, and to pursue the matter accordingly. And you should be disturbed by that only if you have the following view of science. When people take things to be true, they should have compelling evidence that they are actually true, period.
And what I say in response to that. That's asking for something almost impossible at the early stages of development of a science. You might be able to ask for that very later on. But initially all you wanna do is get on a path that allows you to develop further evidence.
And in all fairness to Kepler, while his physics is all wrong, he put us on a path that very much did open the way to further evidence. Though it took the better of a century to get to the point, and that's what we're gonna do the rest of this course, get to the point where Newton suddenly discovers that he can show both the area rule and the ellipse hold under very specific circumstances.
Okay, so I'm ready to defend Kepler, just the way Charles Sanders Peirce did, but not to defend him as saying okay, he's now established the ellipse in the area rule. He simply hasn't, and nobody else considered them established at the time of Newton's Cornucopia. They were very much open, particularly, the area of rule was totally open to question.