At any rate, though, it's very good science. How good are his? Well, I have four more slides. We'll be out of here. What you need to remember from tonight is the model at the top. That's the basic Ptolemy model for longitudes. And it's my schematic drawings. It's purely schematic, but it's shows the midpoint.

It shows what we now think of as eccentricity of an ellipse. The full eccentricity that he's talking about, in elliptical terms, is from one focus to the next. What we now call the eccentricity of the ellipse, half of that. But for Ptolemy, it's the two Es together that constituted the eccentricity.

And you see the equant there, the observer, the mean sun. Mercury is a different story. Mercury is near the moon and partly there are several factors in play here. First, Mercury is by far the most elliptical orbit and maybe Ptolemy had good enough observations to pick up on that.

I'm dubious. Mercury's visible at best for about 20 minutes when it's at full elongation either in the morning or at night. I once took a whole group of students to Wesley to use their 12 foot refracting telescope to look at Mercury. We could dial it in and then we couldn't find it.

We had a computer to dial it in and we couldn't find it in the 15 minutes. Last year when I taught this course at Stanford and one of the students bought a telescope and was proud to show us all Mercury. Saying to me see George you actually can see Mercury but it's very hard to see.

It's very faint. It's up there low in the sky which means atmospheric refraction is fierce, and it's there for a very short time. So it could be nothing but bad data that led him into this. But, his actual model, it employees the same parameters, but notice what happens.

It has a variable, it has an inner crank just like the moon with the center of the different, moving around that crank, and the middle of that center having lying between the two eccentricities, but now the equant is sitting in the edge of that circle. So, it's a very different model from the other four.

And there are two things he can say to justify it, his data forced it. Number one and number two, since it's between the Moon and Venus, it's not surprising it has features of the Moon in common with the Moon, unlike the other planets. That's his justification. How good were the models?

Well, this is just Mars and it's Neugebauer's numbers compared to observation. Real observation, well, the real observation. I brought this in. This is a book by Bryant Tuckerman, Planetary, Lunar, and Solar Positions, 601 BC to AD 1. Every five nights, and there's a second volume twice as large and heavier in volume than I care to carry in this humidity that goes from AD 2 to 1650.

So we have calculated using modern gravitation theory. Where everyone of these objects should have been from 600 BC down to 1650. That's the comparison that's being shown. Now you look at that comparison and hopefully you can see it better in your handout than you can see up here.

We're talking several degrees difference between the dotted line and the solid line. The moon width is 30 minutes of arc. One degree is two moon widths. Two degrees is four moon widths. These are very, very substantial discrepancies. The largest discrepancy in the case of Mars occurs every 15 years.

It's four degrees, eight moon widths. Now the way to do the width of the moon is hold your thumb up about that far in front of you. That's the width of the moon, when you go out and just thumb up. You can see how far you have to hold it away to capture the moon.

Eight of those is trivial to see versus the stars. Yet I have not seen, and people I ask who knows Islamic astronomy had not, as I gave you the story with Bocchi Sobra before, say they don't know of anyone to have pointed these errors out. Anywhere before the 15th century.

They were there to be seen at all times. They're not cumulative, but what it's saying is nobody picked up on Ptolemy's idea that those discrepancies are telling us something. And the only way I can discuss, there are different ways of discussing this, one way to say it, what they wanted out of science was not agreement in every detail.

They didn't care that much about daily longitudes and latitudes. Enough so, they didn't record them. We have not tables of them from Ptolemy forward until the 16th century. So, they weren't recorded. Nobody did any comparisons. They just sat there. They're not cumulative. They don't affect predictions of stationary points.

They don't affect predictions of what the loops look like. Those are pretty good. So it looks like Ptolemy astronomy, even though you can calculate everything every day, every longitude and latitude of every object, that was not its goal. There are comparable things in modern science and physics today where there are phenomenon that are just ignored because we can't calculate them.

I'll give you one. The strength of steel. We have to measure it for each steel at each temperature because we can't derive it from first principals. I give you a lot more. So it's not unique to Ptolemy that we have all the quantum theory but we don't know how to do calculations for things simple.

I'll come right back to your question. As a matter of fact for what it's worth, the strength of steel is one order of magnitude less than the chemical bonding strength of the crystals. That's the mystery for quantum mechanics, why steel is so much weaker than quantum mechanics says it should be.

They have answers, but they can't do it quantitatively, they're qualitative answers.