All right let me go back now to Ptolemy and the bisection of the eccentricity. What I've given you here is what he actually says in the book on the bisection of eccentricity. So it starts. But since it is not clear whether the uniform motion of the epicycle takes place about point D, here to, we tuck two greatest elongations.

This is of Venus, one is Eveningstar at 48, and whatever it is. 48 and a third. That's degrees away from the sun. And the other is Morning star 43 and seven twelfths degrees. In each of which the mean motion of the sun was a quadrant from the apogee, 90 degrees from the apogee, and now he goes through a calculation, that I give you just dot, dot, dot, dot.

He takes those two observations, works out a calculation. You can look at the book and see what the calculation is. From which he concludes, BE is 60.3. Now let me explain that number for a moment. This is the Babylonian sexadecimal number system. Everything is based on 60. Instead of decimals, which were invented in the 17th century, we have parts of 60.

So that's 60, semicolon, three sixtieths. Next semicolon, would be three. 36 hundredths. But it's like the decimal system except everything's done with 60. 60 has wonderful virtues. Think of all the prime numbers divisible into it. It has a wonderful fault. Think of memorizing the multiplication tables for 60.

All of astronomical math, until at least the 17th century is done in this number system. Okay, he goes through and the radius of the epicycle is 43 ten. That's given that the radius of the is 60. Therefore, BD as we already know is two and a half, but we show that the distance between B and the center of the ecliptic, and the center of e-center on which the epicycle is always scary, is one and a quarter in the same units, that is half BD.

In other words, he is taking measurements of Venus and concluding the point around which the circle is centered. The circle on which the epicycle center is moving. Is not the point of equal angular motion but midway between the point of equal angular motion and the observer and this is something he obtained by simply saying, let the center of that circle, the circle you see up there, be arbitrary.

Let's figure out where it is given that we know where the center of equal angular motion is. And off it pops and it turns out to be more or less midway between. Then he turns to the other three planets. For the other three, Mars, Jupiter, and Saturn, the hypothesis which we find for the motion is the same, and like that established for Venus.

Namely one in which the e-center on which the epicycle center is always carried is described on a center which is the point bisecting the line joining the center of the ecliptic and the point about which the epicycle has its uniform motion. For in the case of each of these planets too, using rough estimation, the eccentricity turns out to be about twice that derived from the size of the retrograde arcs, had greatest and least distances of the epicycle.

However, the demonstrations by which we calculate the amounts of both anomalies, and the apogees, cannot proceed along the same lines for these planets, as for Mercury and Venus, since these reach every possible elongation from the sun. And it is not possible as it was for the greatest elongations for Mercury and Venus, when the planet is at the point where our line of sight is tangent to the epicycle.

So since that approach is not available, we have used the observations of their oppositions to demean position of the sun to demonstrate first of all, the ratios of their eccentricities and the positions of their apogees for only in such positions do we find the ecliptic anomaly isolated with no effect from the anomaly related to the sun.

That anomaly is the number of degrees on a circle. So the one moving around the Zodiac versus the one in the epicycle. Now the main point here is he does not give us a similar calculation procedure. And in fact he wouldn't be able to almost certainly for Jupiter or for Saturn.

Cory mentioned before I think it was Cory on the ellipses versus circles. What you should know is these planets, we think of them as on ellipses. The ellipses are incredibly near to being circles. The most elliptical is Mercury. Its minor access is 2% shorter than its major access.

If I drew it on the board you would have to have a very fine measure to realize it's not a circle. Mars, it's four tenths of 1% difference between the two axes. And all the others are much less than that. So these are almost perfect circles. That's one of the things that's remarkable about being able to get the eccentricities, etc.

You're getting them not geometrically, you're getting them from speed differences. All right. That said, people don't believe modern scholars, don't believe, that Ptolemy didn't try to work out what the so called bisection of eccentricity, whether it was true for the outer planets or not. So two different people, Jim Evans whom you're reading for next week, the paper, and I'll show you that now, cuz I'm introducing you to the reading, and separate from that, Noel Swerdlow, and his paper is being put on supplementary material, have both tried to work out what Ptolemy had to do.

Jim Evans's approach is not very Ptolemaic, cuz it's essentially trial and error, and that's not the way Ptolemy presents himself. Swerdlow identifies a series of measurements for Mars that would give you the bisection of eccentricity, and then of course claims this must be what, Ptolemy did. Since he's an expert on Ptolemy and he knows the amalgest backwards and forwards.

We really don't know. This is the total information Ptolemy gives us. I've given you everything he says about the bisection of eccentricity. The point by the way of equal angular motion. He never calls it this, but it became known as the equant. And of course it's a violation of a uniform circular motion, and therefore a violation of Aristotle.

So, Ptolemy is twice violating Aristotle. The earth is not in the center and the motions are not uniformed. Double violation.