Okay, here's where we've gotten to. We've gotten to where the Earth-Sun orbit has got bisected eccentricity, and therefore, Kepler can go ahead and say velocity in the Mars orbit is inversely proportional to the distance of Mars from the Sun. And the same can be said of the Earth going around the Sun.
We can now say inversely proportional. So he's replaced the equant, he's got a new rule. That's the rule that sets the timing on the orbit, it doesn't give you the trajectory. So you've got two separate problems, timing and trajectory. He says, fine, let's go back to a circle, but now use my new rule, the rule that the velocity is everywhere proportional to one over the distance from the off-center Sun.
Fair enough? Is that clear? So he starts doing it and he finds it's a very difficult problem mathematically. He really needed the calculus. So what he then did was to divide the circle into 360 one degree arcs, and then tried to get the time over each equal arc by in effect, summing over the number of the distances in there.
Okay, he's trying to do what we call integration. Didn't exist yet. Okay, but he's trying hard to do it and he's approximating it. And then he notices something. That too was computationally very much a nuisance. And then he notices something. By the way, that idea of breaking it into units and adding the segments of it, that's from Archimedes.
Okay, so he's consciously borrowing a trick from Archimedes there. He notices that it's very close, if you look at the distances of each of these little triangles, it's very close to just having the area of the triangle. So he then decides he's going to approximate his one over r rule for velocity with instead, each equal area takes the same amount of time.
Now, this is nothing but a computational device to approximate the rule he really wants when he introduces it. Okay? He has no reason at all to take it to be correct at the time. It's just gonn save him a lot of labor. To double check it, he goes back to his new Earth-Sun orbit, that he's revised from Tycho, and he sees if there's any difference between the two rules on it, and concludes the difference is within observational accuracy.
Which is not surprising, because the Earth-Sun orbit is much nearer a circle than the Mars orbit is, okay. So what he proceeds to do for a while is to do both rules, okay, the area rule and the one over r rule. One over r is my description, of course.
He didn't have one over r, but inverse with distance. Okay. And he takes a circle and does that. This happens to be with the area rule. And he compares it to the longitudes from the vicarious hypothesis. And what he discovers is, and you've seen this in the Curtis Wilson, at the octants, the octants nearest aphelion, farthest away from the actual Sun, you are ahead of yourself by eight minutes and 21 seconds, while at the other two octants, you're behind yourself by eight minutes and one second, okay.
And what's that mean? You're going too fast when you're at the extremes of the line of apsides and too slow in the middle. Well, that means you've got to remove some area in the middle. Which means you've gotta go to an oval. Okay, so he concludes, this looks like it has to be an oval.
Great, which oval? Well, the first thing he does is, he's very uncomfortable with the idea of an oval. But he knows something. He knows that one, there are two ways. This is Apollonius' theorem again. There are two ways to achieve the effect of eccentricity. One is to have an e center.
The other is to have a minor epicycle that goes around, just what Apollonius said. So he decides he's gonna put a minor epicycle around and let that describe an oval. And that minor epicycle, I now have to be careful to look at what I'm doing. It ends up generating an egg shape oval with the epicycle moving nonuniformly in order to keep the SM distance the right amount, okay.
And while he is doing that and struggling with the calculation, he comes up with another very clever idea. This, am I doing this right? Yes. He decides he can approximate this calculation of the egg shaped if he introduces just an ellipse that approximates it. So at the bottom here you see the eccentric circle.
You see the true ellipse. You see this oval that he's generating. And you see how the approximating ellipse, the dotted line, comes very close to the oval he generates with the epicycle. Okay? So he runs the calculation that way to see what he gets. And again, you've seen this result.
The slide you can read for yourself to get the full explanation. That's from Owen Gingerich. What he gets is an overcorrection by almost the same amount as the excess was before, he's now gone to too much of an oval. Okay. Now, there's a natural conclusion to draw at this point.
And Curtis, excuse me for using his first name, he was very close and enormous influence on me and he's no longer alive. Wilson presents this in, especially in the Scientific American article, where there are limits to how much detail we can go in, he presents this as, isn't it obvious now that what you want is the ellipse midway between the ellipse that is generating this and the circle?
Because they balance one another out almost exactly. And that's a way to get to the ellipse. And that's exactly what Kepler did not do.
Okay, it's even more striking when you look at what Kepler presents. He presents a table of comparisons done several different ways, ending up with the Vicarious Hypothesis, the physical hypothesis of a perfect circle.
And now the physical hypothesis with a so-called perfect ellipse. This is not his final ellipse. This is what Owen Gingerich calls the auxiliary ellipse. And he remarks down here, you will note that the truth is exactly in the middle between these two. That's the point that Wilson makes.
But does it lead him to conclude, therefore it's the ellipse midway between the two? No. In fact, he goes almost two years before he finally concludes it's an exact ellipse. Now, why is he hesitating? Well, think about it for a moment. The area rule itself is just an approximation to another rule.
The auxiliary ellipse is just an approximation to another calculation. The observations are not exact. Why in the world would he take that seriously the fact that midway between an ellipse and a circle you get what you want? Okay? Maybe this is totally accidental that he's coming up with that nice pattern of numbers.