Swerdlow, in an attempt to explain that, and it's in this paper, gives you a way as he says at the bottom here where I marked it out. Gives you a way of having the radius ratio. The ratio of the epicycle to the deferent radius. Determining that from observations, but then he adds, right away at the bottom, Ptolemy's actual way is more complicated.

So let me just start with what the parameter is. They had no way of knowing distances. So for every orbit, you sit the main circle's radius at 60. That's the unit, and then everything else is based on that 60. So all the orbits have the same radius for there main circle, mainly 60.

Sun, moon, everything. And now what we want is the size of the epicycle relative to that. And what you have here is a count of the calculation using the ratio of the velocity of the center of the epicycle, that's v sub c, versus the velocity of the planet.

And he does it. This is Noah Swerdlow doing his Ptolemaic type calculation simpler than Ptolemy. What he says at the end. Surely Ptolemy did this first and then got it more complicated. Typical Noah Swerdlow remark that. He's a very good friend and I trust him on remarks like this one, let me say.

But he takes the average size of half the retrograde arc. What you see is Saturn's three and a half, Jupiter five degrees, Mars eight, Venus seven and a half, Mercury six. He takes the time of, average time of the loops, he takes the ratio of the velocities. He does a calculation, sketched out here, that you can read more about in the paper, and you get radius ratios.

So for Saturn the epicycle is six and twenty four-sixieths of sixty. That is 10% the size of the main circle for Saturn. The next one over is 1135. The next one after that is 3924. So, the Mars epicycle is just under two-thirds the size of 60, 39 out of 60.

You saw that when Swerdlow drew it to scale, it's very dramatic. Then the next two 43.6, 43;6, 23:26. Those are his calculations. The next row give you Ptolemy's actual values. And the next row gives you the modern values. What do we have for the modern values? Where are they coming from?

It's essentially the size of the Earth's sun orbit. Versus the size of the planetary ruins. Okay, that's what the radius ratios become in modern form. So again the radius ratios are being determined from observation. Everything is come observation driven as Swerdlow says on the paper I'm putting on supplementary material, the remarkable thing about Ptolemaic astronomy is the extent to which it's empirically driven.

It is not a bunch of hypothesis he makes a few, three basically, initial plausible simplifying assumptions, from that, he constructs systems of measurements and the very stability of them over time says something has to be right about this model. Okay, and those measurements kept repeating essentially the same values again and again for centuries.

And they're not significantly different from our modern values. Okay, once you adjust what it is you're looking at. So that's the essence of Ptolemaic astronomy. And when Swerdlow spoke here in 1991, this fits my view of evidence about as well as anything possibly could so I was exceedingly pleased.

And when he said, this is the way science really works, not the way a philosopher says it works. Swerdlow's a historian, a typical historian, snide attitude toward philosophers. I made a remark to the effect of yes, Naoh, it's at least as good as modern economics. And he blew up saying, modern economics would give a hand and a fist to be remotely this good!

And, of course, he's right. Okay? Modern microeconomics and macroeconomics can't do anything like this. Okay? They, too, make simplifying assumptions, rationality of human beings, and microeconomics, etc., but their actual predictive success is very, very low. So from lots of points of view this is very, very good science.

In fact, it's such good science, I'll come to that in a moment, that one of the interesting historical questions is, how did it ever become a laughingstock? The answers very simple, Galileo. Galileo and a polemic that you will be reading later, but. And of course the Galileo trial, which Protestants then turned into a wonderful anti-Catholicism weapon.

So Galileo's laughing at Ptolemaic astronomy got enormously augmented when he's tried and Protestants turned on the Church with that.