## Philosophy 167: Class 3 - Part 7 - Astronomia Nova, Part 2: the Vicarious Hypothesis, and Mars' Heliocentric Longitudes.

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

2014-09-16

Part 2. He now decides to take a first shot at the Mars orbit. And what he wanted was heliocentric longitude to the Mars orbit, so that he could do triangulation of the sort I described. Now the problem with getting heliocentric longitudes is the problem I mentioned last time where I got questioned on the phrase, question baking.
We can't go to the Sun and observe Mars from the Sun. We have to infer it. And the way Copernicus did it for triangulation was to adopt a full orbit for Mars, and then he's got heliocentric longitudes and then he can do triangulation. But that is seriously question begging.
What Kepler wanted was a way of having heliocentric longitudes under the assumption that Mars is going around the Sun, that are independent of any theory of the motion of Mars. So, the question is how do you get that? Here's what he thought. A clever idea. At opposition, particularly opposition relative to the actual sun.
That means the actual sun is on one side of the Earth. The Earth is here and the moon and they're in a straight line. Guess what? You know the heliocentric longitude of Mars. It's the same as the geocentric longitude of Mars. Right, everybody see that? It's the one time you know the heliocentric longitude had opposition.
And since he is doing it relative to the actual sun, he has to take Tycho's observations and rework them to be inline with the actual sun so that he can get the oppositions. But he ends up with 12 oppositions scattered very nicely around the zodiac. Okay? Now if Mars is going around theSun, he can in effect take an orbit and ask what are the features of this orbit to give me those 12 locations of heliocentric longitude on time?
So, what he does it set up a Ptolemaic orbit. It has a circle. It has an equate at C. It has a center at B, and it has a, I think that's an O or an A. I can't quite read it. This is the original figure, not from the translation.
And O is us observing. So, the fact that the line runs through to the equinoxes shows that's where it is. Now he's got four unknowns in this. The first unknown is the distance from us to the center of the circle. The second unknown is the distance to the equant.
He is not assuming bisected eccentricity. He's letting the data fix that. The third unknown is the line of apsides, O to H. And the fourth is the time at which the planet is of age. He's got four unknowns. He's got 12 observations. Four unknowns means he only needs four of the observations to calculate the four unknowns.
And he can then check them against the other eight. Okay. That's his idea. Unfortunately, it's a rather tedious long calculation because there's no way to solve for it algebraically or directly. It's all by trial and error. He's got four unknowns. He goes through something like 100 different iterations until he gets it.
While doing this, he writes a letter to someone expressing that he, and this is translated, I wish I had a computer. What he meant was a human assistant doing the six-digit calculations, okay. But he was doing them all himself. The net effect is, he finds, and this is the differences down here, he gets a fit where the largest discrepancy is 2 minutes and 12 seconds of arc.
All of the others, I think the next largest, well all the rest are less than two. You can see which ones he used for the solution, because they're the ones that have very low error to them when you back fit. But again since he's doing it iteratively and not exactly, they're not perfectly zero, cuz what he does is good enough approximation and stop.
So at this point he concludes he's got if Mars is going around the Sun, he's got heliocentric longitudes everywhere around the zodiac, independently of any theory of the motion of Mars, because he's got these fitting so well. Fair enough? Impressive achievement. Now, they're still assumption-based, namely Mars is going around, but it's no longer committed to any trajectory whatsoever.
What about on a Ptolemaic view? What does he just measure? Answer, the location of the center of the ignix. I'm gonna say this better the location of the attachment point of the epicycle on the deferent. We'll come back to that. Okay, cuz we'll show he uses that too.
Okay, this is what he ends up calling his vicarious hypothesis. In the notes, I tend to call it vicarious theory most of the time, but he really does, he uses the word vicarious hypothesis. What he finds almost immediately is that it's disturbed by is the equant. The center b is not midway between the equant and the observer, but it's 0.6 of the way over.
In other words, this does not have bisected eccentricity. Yet, there is a 1400-year tradition of Mars having to have bisected eccentricity. So could, could that 1400-year tradition be wrong? Typical of him. What he then does is to go out and use first geocentric longitude, excuse me, first using arguments from latitudes, then using arguments from longitudes to verify, that the ratio of a, I guess it's a here, ab to ac is very near 0.5.
He can't get it exactly 0.5, but all the evidence shows that it's in between, very near the middle, and therefore this theory is false. Okay, period. It does not have bisected eccentricity, the real ones have bisected eccentricity. So of what good is this theory. Well it's a curve fit and it can be used for triangulation.
Cuz what we need to know, is we need to know the geocentric position of Mars, geocentric longitude of Mars. The geocentric longitude of the Sun, and the heliocentric longitude of Mars. If we know those three, we have all three angles. We have two angles of the triangle. If we know the Earth-Sun distance, then, we get the distance to Mars.
That's what he wants. So, these give him that. Under the assumption Mars is going around the Sun. Okay? And it's a much weaker assumption than assuming the whole motion. Okay? People buy that, that's a less question baking in what Copernicus did. You'll see how much less question begging in just a few minutes but I want to go past here to get there.
So the way he proceeds to do, is he's now not gonna worry about what the heliocentric longitudes of Mars are. He's got them, he can interpolate in this table. He can use the vicarious theory for anything, to get him anywhere else. He can just calculate heliocentric longitudes at all times with confidence under that one assumption.
Okay, that puts him in a position to do triangulation constantly. Any questions on this though? Because this is a big deal. He's in effect done it in the manner of the ancients and he's concluded he can fit the data but it's false. In fact, what he says about it is a wonderful ending.
The end of this whole portion section two. The blame for this discrepancy among the different ways of finding the eccentricity. I am repeating this over and over so that it will be remembered. He has, it's not really a flipping style, but it's close to a flippant style at places as you'll see.
Repeating it over and over. So, it's not Donahugh, that's Kepler. Falls entirely upon the faulty assumptions studiously entertained by me in common with Tycho and all who have ever devised hypothesis. For the necessary consequence of this inquiry is that there is no single fixed point on the planet's eccentric, about which the planet always sweeps out equal angles at equal times.
The equant is dead. We would instead have to make such a point reciprocate up and down along the line of apsides, if indeed we can keep the other assumption of a circular orbit. And how such a reciprocation could be reconciled with natural principles I do not see. This is not the last time we will hear of an equal point sliding up and down.
Okay. It will come back again and again. But in fact, the other assumption will be demolished in chapter 44 below. That is, the orbit of the star is not a perfect circle, but an oval. And it's greatest diameter is the line of apsides while it's least is that passing through the center to the middle longitudes.
No wonder, then, that the other observations at points not at opposition to the Sun, do not accord with a hypothesis constructed in chapter 16, since in it we have made two false assumptions. So in effect he's announcing here at the end of part two that we're gonna throw out two of the classic points of pieces of astronomy.
Equant or any point of equal angular motion and the reliance on circles.