## Philosophy 167: Class 2 - Part 7 - Heliocentrism: Copernicus' Mathematical Transform (of Ibn al-Shatir's Mathematical Transform) of Ptolemy's epicyclic system.

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

2014-09-09

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How did he get to heliocentricity? Well here's a passage from Almagest. Notice it's tucked way back in book 12. Book nine is where the theory of planets begins. Book 12 is where you're doing calculations. In the definition of this kind of problem, calculating stationary points, there is a preliminary limit demonstrated for a single anomaly by Apollonius.

If this synodic anomaly is represented by the epicyclic hypothesis, then you get one way of calculating it. Now here's the key, if the anomaly related to the Sun is represented by the eccentric hypothesis which is a viable hypothesis only for the three planets which can read any elongations from the Sun.

Mars, Jupiter, Saturn. Come back to that, in which the center of the e-center moves about the center of the ecliptic backwards along the signs with the speed of the mean Sun. While the planet moves on the e-center forwards along the signs of this speed with respect to the center of the e-center, equal to the mean motions in anomaly, dot, dot, dot.

What this amounts to, retrograde motion of the outer planets does not have to be represented by any epicycle. Okay, I'm gonna show you how to represent it in just a moment. Reggio Montanus commenting on this passage in epitome to the Almagest shows how Ptolemy is wrong. It's not true, of just the three outer planets.

You can do the same thing with Mercury and Venus. So Copernicus knew that there was a transform possible. To transform, this is Noel Swerdlow's diagram for it, you're seeing two ways of doing this. One way has the defferant centered at Z, you see the circle going through, diameter AG going through Z, with an epicycle on top of it.

And in both cases, motion is going counterclockwise. Now here's the alternative you can do, and this is the way Owen presents it as well. Though it's slightly more sophisticated when you see this diagram. What you can do is flip those. Let's move the epicycle on the inside. Then, we get an e-center whose center is at N, but that center is revolving in the circle in H.

And at all times, the line between Z and N points to, of course, the mean Sun. Because when you put the epicycle on the, and you always point to the mean Sun, you've now had that happening on the inside, okay? Now it doesn't give you much, but it gets rid of having to work at all with the epicyclic model of retrograde motion.

We can get retrograde motion this way. Come back. I'm just going to go on for a minute and then I'll answer questions. But notice something now, that's fairly neat. Mainly, this works for all four planets! In the case of Venus you already have what is here, the epicycle being moved on the inside.

It's already on the inside. Kay, that's number one. Number two, what's to stop you from saying, look, the Sun could be anywhere here. Why not put the Sun at end? We have no idea how far is the Sun is away. What happens if you put the Sun it in.

Will you get couple of things right away for free. Oh, that's one orbit common to all of these planets. We've unified, by putting the Sun at end. First thing. Second thing, if we put the Sun at end, then we know two angles. Well, do it in terms of longitude.

We know the position of the planet, which is at O. You notice these two are equivalent. We get o both ways. We know the position of the planet at O. Helioc-, excuse me geocentric longitude. But if the Sun is at end, we know it's heliocentric longitude, and if we know both of those longitudes, and we take Z in to be unity.

Call it the mean distance of the Sun to the Earth and what's now called the astronomical unit, we set that to some fixed number. We can use the triangle ZON where we know two angles plus the side NZ by stipulation to get the distance of O from Z in units of NZ, astronomical units in other words, we've solved the distance problem, okay if we move this on to Z.

I'm not quite done, let me just go one last step. There's a problem however. When you do this, the orbit of Mars intersects the orbit of the Earth's sun, and you can't do that with crystalline spheres. So the general view is, and we don't know this. We have notebooks but it's even a surmise from the notebooks.

Copernicus got this far, saw that the crystalline spheres intersect with one another and solved the problem by simply doing a relative motion argument, have the Earth go around the Sun rather than the Sun around the Earth, and that's how he got to heliocentrism. That's the Noel Swerdlow version of how he got to heliocentrism, but the important point is what's putting him there is the very fact that different representations are possible here.

You don't have to do the epicycle at all in the standard way. You can do the same thing geometrically in another way. Anyway that's pretty much the standard story. And what happens now is those radius ratios. I'll come right back. Those radius ratios take on a whole new meaning because now all five of the orbits have a suborbit in common.

It's still two compound circles, but they all have one orbit in common, the Earth Sun orbit. So, now the radius ratio is giving you the mean distance of each of the other objects from the Sun, under the assumption that the mean Earth Sun distance constitutes the basic unit.

And you see what they are using the radius ratios in the Almagest, but flipping them. They're remarkably close to the modern, remarkably close to De Rev. Other than, I guess the one that most departs is Saturn. Saturn has a long period. What is it? 20 years, as I recall.

So the amount of observations you get on Saturn are low compared to each of the others. But the point is, that radius, and this is a point I'm gonna make in various ways and then take a break. The point is, in fact I'll take a break at eight.

Camera, will that be tolerable? Got a little more to do before eight. The radius ratios that Ptolemy had.

Are fundamental. Fundamental parameters. They weren't what he thought they were. They are instead, the magnitudes of the orbits of all the planets relative to the magnitude of the orbit of Earth's Sun.

Whether you do it as the Sun going around the Earth, or the Earth going around the Sun. And of course, the eccentricities carry over. That's not surprising, because the motions remain the same variability, etc. So those Ptolemaic elements carry right over into modern astronomy. Why? Because all the did, was a mathematical transform.

Of Ptolemaic astronomy. And all Copernicus really did was a mathematical transform of. Okay, there's no new observations, no new accuracy here, it's just three different ways of representing the same set of appearances.