## Philosophy 167: Class 11 - Part 2 - Alternatives to the Area Rule: Ismal Bullialdus, Vincent Wing, and Three New Methods for Calculating Orbits.

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

2014-11-18

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All right. The first person to come along and offer a full system alternative to Kepler's was Ismael Boulleau. He was clergy in France, needless to say, and I struggle to do his French name, Boulliau. He deeply disliked Kepler's invocation of physics, he thought astronomy should be purely geometric.

But he fully recognized, in fact he was one of the first to say, I quoted him earlier, in a slide back when we were looking at Kepler. He was one of the first to say that before Kepler proposed the ellipse, everybody was just kidding themselves, especially about Mercury.

So, he's totally sold on the ellipse, and outspokenly sold on it. But, he doesn't like the physics, and he doesn't like the area rule. So, he gets this idea that the ellipse is happening, because you're actually doing uniform circular motion, but on a cone, and you slide up and down the cone.

And as you slide up and down the cone, the actual trajectory becomes an ellipse even though at every moment, you're on a trajectory, an instantaneous trajectory that's uniform circular motion. He did this in 1645 and thought he had worked out a solution to do tables, and then at the last moment realized that for Mars he was not matching Kepler.

So he actually threw in the area rule in the first publication of this, without telling the reader that's what he had done. And a few years later, Seth Ward, I threw that name up before at Oxford, wrote an attack on this book, saying that in fact, what it is equivalent to is having an equant at the empty focus.

And it's funny, Ward was arguing that against the expressed denial by Boulliau that, that was the case. He was arguing that, because he wanted to use equant of the empty focus. And he wanted Boulliau's support for it. Boulliau knew an equant at the empty focus doesn't work. So what he did was devise an alternative to the area rule.

That's the first serious alternative. F is the focus, and if you looked at this, it's a little like equal angular motion around that isn't equant at the empty focus, except then a correction is made. So that the correction becomes one. I'm not going to go into detail on this, but if this is the equal angular motion, this is where it would be if you were at the equant.

You now draw perpendicular through that. Draw this line to the point on the circumscribed circle, and where that intersects the ellipse is where the planet should actually be. Okay. And, of course, being at aphelion, furthest away, it's going slower, and therefore it lags behind the equal angular motion, just as it should.

This is as accurate as the area rule within the bounds of the Tycho observations. And it's the first real alternative to the area rule. And we'll come back to it. It's a non-trivial step. In 1651, this is Boulliau's book. If people want, I can pass Boulliau's book around.

I don't have it with covers. I just have a xerox. When I was at Dibner, of course, Dibner Institute with 50,000 rare books in the vault below me, I had all of these in first editions, and many of them I copied, so that I had them. But the ones I do have, Bernard Cohen, had reproduced that way and bound himself.

And I'll pass some of those around. But this is the Boulliau book that finally came out in 1657, and you notice the reference to Seth Ward in there. Professor at Oxford, etc. It's a complete astronomy. As complete as the Rudolphian tables. Coupled to the theory behind the Rudolphian tables presented in the last parts of the epitome of Copernican astronomy.

So it's the first total alternative that's of fully comparable accuracy to Kepler for all the planets, okay? Not a small thing. Somewhat in this same time period, 1647 to 1660, Vincent Wing, in England, comes along. And this one is in English. It's always nice to have something in English.

It's not the only thing that's gonna be in English, but I'll pass this around. Bernard did wonderful things as far as binding books, but this is a Vincent Wing first book, the Celestial Harmony book. And as you can see, it's very much in English all the way through.

This Book did something that Kepler expressly rejected, yet it works. First of all, look at the little epicycle on the edge of the circle. That epicycle has a radius of one-fourth the eccentricity. And it generates an ellipse by moving along and rotating, it's a way of drawing an ellipse, a circle within a circle, done properly.

So that's the outer one. The inner circle, at the empty focus, is doing the very thing that Kepler said would be possible. We read it when we looked at Astronomia Nova, an equant point sliding up and down around the empty focus would give relatively high accuracy. And that's exactly what he's doing there.

If you see the point, the crosshairs are at the focus. But at that juncture, the equnt point is a little bit below the focus. And it just rotates, moves up and down, driven by that little circle, that little epicycle on the inside. So that's one way, and Wing produced in 1651 a complete, a second complete system.

Actually it's the first published. But Boulliau I said is the first then because, it's more thorough than this one of Wing's. But Wing came out with this in '51, and it probably has more claim to being the first, even though it was in English and did not sweep across the world in the same way.

Five years later, he came to realize that he was getting about five minute errors. As much as five minute errors with this one. So he then came up with a different way. So the left side shows you how he gets, that's the 1651 approach. The subsequent approach in 1656, that became part of his Astronomia Brittanica of 1669, gives you an alternative geometric construction to Boulliau's.

It's, again, a correction to the equant. So you have the equant there, x, m is the mean motion. So you're doing equiangular motion around x. That gives you a .h corresponding to the eccentricity e, and then you put in this correction term. And that's described in here, I'm not gonna run through it.

But both of these are described by Curtis Wilson in the text below, and you can read it for yourselves. So here are two alternatives to the area rule, both roughly comparably accurate. Boulliau's, of course, is the third, and the area rule is the fourth. So by 1657, we have three alternatives to the area rule, and are rule all producing essentially the same level of accuracy.

Okay. Things aren't gonna stop there, but it's fairly dramatically that you got that far. This is Astronomica Britannica. And I will pass the whole thing around. These are heavy volumes. But it's the two volume work, and in Latin. It's not in English like the earlier work. I'll pass these.

I'll start them separately, and then you can see what's going in them. But again, complete tables by 1669, where the theory was all in place in the 1650s. Now what's important about all the ones I just listed to you, with the possible exception of Boulliau, Newton read them.

Newton read both of the Vincent Wings. And we think he read Boulliau, because he either got Boulliau second hand or first hand. He certainly knew what was in Boulliau, because when he starts the Principia he actually just says Boulliau and Kepler have comparably accurate systems. And as you'll see at the end tonight, the first issue of the Principia is having to decide is which of the competing systems in preferable.