Philosophy 167: Class 3 - Part 12 - Astronomia Nova, Part 4: the Diametral Distance Rule, and Settling on the Ellipse.
Smith, George E. (George Edwin), 1938-
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Okay, then he notices something that really strikes him and I'll just read it. The breadth of the lunule of Chapter 46 above. Borne to us out of the opinion of Chapter 45, which instructed us to cut it off from the semicircle. This breadth I say was found to be 858 units, of which the semidiamater of the circle is 100,000.
But then, by two arguments by no means obscure, which I have already presented in chapters 49, 50 and 55. I concluded that the breadth of the lunule is to be taken as only half that, namely 429, or more correctly, 432. And in units, which the semidiameter of Mars is 152,350, nearly 660.
What he's talking about is the distance between the circle and the oval. And units where the radius of the circle is 100,000. And it's 429. While I was anxiously turning this thought over in my mind, reflecting that absolutely nothing was accomplished by Chapter 45.
And consequently my triumph over Mars was futile.
Quite by chance, I hit upon the secant of the angle, 5 degrees, 18 seconds, which is the measure of the greatest optical equation. That's the measure of where it is furthest from the Sun angularly, at the two quadrant points. And when I saw that this was the secant of that angle, 100,429, it is as if I were awakened from sleep to see a new light, and I began to reason thus.
At the middle longitudes, the lunule, or shortening of the distances is greatest. And it has the same magnitude as the excess of the secant over the greatest optical equation, etc. He then generalizes that into a rule. He calls it the Diametral Distance Rule. And what he says is Mars Sun distance is always ca times 1 plus the eccentricity times the cosine of x.
Where x is gonna be very significant next week. It's the angle he's gonna use to measure things in the ellipse. And he says various things about it. This is the way nature works, a nice little sinusoid, okay. But the obvious question is, okay, what justifies going to that as a Mars subdistance?
Well he can test it with triangulation. And he proceeds to test it with triangulation. So on the left, from the observations of Chapter 51, we have different observations at different times. Triangulated distances at several different points in the orbit, ranging from very near aphelion 166,000 to very near perihelion 139,000.
And we then compute it by the diametral distance rule. The distance of Mars from the Sun by this rule and you will see how spectacularly good those numbers are. Now again, they're not exact, but it looks like the diametral distance rule is accurate to within the precision of the observations.
I think the largest error is in the fourth significant figure, but don't hold me to that. Largest discrepancy, it's not error here, it's discrepancy. I'm just looking through. I think that's right and it sure looks that way, the largest error's in the fourth, maybe even the fifth significant figure.
So now he accepts the diametral distance rule. That's the way to figure out what the trajectory is, is to use the diametral distance rule, okay. So now the question is how do you lay it off? At any point, I've got the distance but I don't know what the angle is.
So he first lays it off to the point where a radius coming off the center in it. It barely touches a radius. Now everybody's gotta picture this. There are a whole lot of radii, only one of them will that distance exactly touch. And he thinks that's it. That's the oval.
And he check it, it's slightly egg shaped. He called it puff cheeked. And he checked it and what he concluded is, it doesn't satisfy the area rule. And by this time he wants to stick to the area rule. So then he lays it off again and has it intersect the perpendicular from the line of apsides which in this case is point e.
ge is the perpendicular from the apsides. se is the diametral distance. Where those two intersect, that's the trajectory. And that is, A, it satisfies the area rule, B, it is an exact ellipse. It is a perfect ellipse, okay? If you can't see why that's a perfect ellipse, you'll see it next week when we learn how to work with this.
That's what finally convinced him that it's an ellipse. Okay, so it's really a combination of three rules. The area rule, the ellipse, and the diametral distance rule. Given any two of the three, they entail the third. In any combination. Fair enough? And the diametral distance rule probably has the best evidence for it, of any of the three.
Because the triangulations can measure distances. They have trouble measuring exact trajectories. They have trouble measuring time. Fair enough? That's how he got there. Having got there, he now goes back and compares to 28 observations from Tycho. They're all near opposition from 1582 to 1595. There are multiple observations in each case.
This is from the Latin. I'm gonna switch to the English, the same thing. It's a little easier to read cuz it doesn't use signs, it uses names of the Zodiac. If you look through these, 10 of the 12 signs are covered. This is a pretty good coverage of the entire orbit.
If you look over on the right, first of all notice how complicated this table is. You're given the Sun's position from the Earth at a specific time on a specific day. At quite specific time to within, I guess it's within six minutes generally speaking, in time. You're given the Sun's position, you're given the Sun Earth distance and since people asked about it before, notice how it varies from around 98,000 up to over 101,000.
It is variable, but it's a very near circular orbit. You're then given the Sun Mars distances, as calculated with the diametral distance rule. You're given the Mars eccentricity. And now you've got its computed longitude. This is observed geocentric longitude. And you've got its observed geocentric longitude. And finally, you have the difference between the two.
And all but four of them are less than, I think it, I know this. Less than three and a half minutes of, are different discrepancy, there are less than three and a half. Three of those four are bad Tycho observations. Bill Harper and I, running an Owen Gingerich calculation, simply redid it and decided that the Tycho, I'll come right back, Tycho observation was wrong.
The only one we weren't able to dismiss was the 5 minute 39 second discrepancy. Now this is an interesting table because what you're gonna see in the rest of this course is I think it's five further people offering alternatives to Keplerian theory. They have only way to show that they meet Kepler's standard, namely the same table with their calculated versus observed.
So this table became the standard for everybody in astronomy. If you have a theory of any orbit, you better have a theory of the Mars orbit, and it has to be at least this good. And there end up being, we'll worry later whether it's four or five. I can just do it.
Wing, it's four others, two by Wing, no. Yeah, two by Wing, one by Bouillaud, and one by Mercator. And always they have to show they're no worse than Kepler. The point I'm making, this table set the standard. Most of the others didn't have access to Tycho's data, so all they could use was this table for comparison.
They had to take things for granted. Now it's not within what he claimed, Tycho's observations were good to two minutes of arc. In fact, if we go back a moment, point to make. I'll jump right back up here. Go back to this first figure. If Tycho's observations were good to ten minutes of arc, Kepler would have stopped at this point.
He had an orbit within ten minutes of arc. It was only because he believed Tycho's observations were much better than eight minutes of arc, that he couldn't accept this. Finally, when he gets up here where we're looking, and he's got it down to worst discrepancy's around five minutes of arc.
In the notes, I actually have added them up and done a mean. I think, on the mean, it's around two minutes of arc. And keep in mind, he knows he can't trust Tycho's corrections for atmospheric refraction and parallax, and those are the order of one minute of arc.
He's at a point where he's ready to stop. He really can't do much more. He's gonna do something more and I'm gonna show it in just a moment, but that's not the key point here. The key point is he's now got a whole new standard for astronomy. And notice the complexity of the standard.
The old standard was reproduce salient events to reasonable time. The new standard is to agree with observation everywhere on the orbit at all times, at least to this level of precision. Okay, and that's an enormous step forward. How much? It's 20 times more accurate. It's more than an order of magnitude.
It's between one in order of magnitudes improvement in a single work, single publication. That's how big of a jump the new astronomy is. You went from errors in the order of four to five degrees to errors in the order of four to five minutes of arc. And those are the worst.
You're generally better than that.