Now the overall system, this is a schematic, totally non to scale 100% not to scale. To begin with other than the moon Ptolemy had no idea how far any object was away. So, he simply decides and gives a plausible argument for it. The longer it takes to go around the zodiac, the further away it is, okay.
So, the sequence goes from the moon, being the nearest, to Mercury, then Venus you have the periods up there, I give here the correct period for Venus. Followed by the Sun, followed by Mars, Jupiter, and Saturn and each of them is shown as having an epicycle. The motion on the epicycle locking down from the North is counter-clockwise, just as the motion on the primary circle is.
So you get retrograde motion when any of the objects on an epicycle is nearest the Earth, and that fits observation because they're brightest when they're in a retrograde loop, that's when they're nearest. The other odd thing here istwo oddities, if you look at Venus and Mercury, their, the center of their epicycle is in line with the sun, it happens to be the mean sun.
Where the sun would be in the apparent sky if it moved uniformly and everything in Ptolemaic astronomy is set off of the mean sun, which is a fiction okay. Where the sun would appear to be if it moved uniformly. Venus and Mercury are aligned with that and then there epicycle is smaller moving around the center aligned with the Sun.
By contrast, the epicycles of the three outer planets, require the planet, align to the center of the epicycle to the planet, to be parallel to the line from the Earth to the mean Sun. Those are requisite features of the model. So, in other words, everything is all five planets are geared to the mean Sun, in one way or another, either on the epicycle itself or on what's called the difference the primary circle.
All right, so there are oddities here, and a natural question that would be asked is why this common feature of tie to the sun. It's a perfectly good question. It's a question others are going to ask, in particular, is going to ask it because as I say, it's a fairly natural question.
I repeat, this is not remotely to scale, and you'll see that in a moment. And this is a woefully oversimplified picture, as we're about to see because the actual orbits are significantly more complicated than these. This is the kind of picture you see at Galileo's Two Chief World Systems, where everything is grossly simplified.
All right. Now the models. Just briefly I'm gonna come back at length, talk about the model for Venus, Mars, Jupiter and Saturn. I already said that what Hipparchus did was to put the observer off the center, and yet, have the epicycle go around the center of the circle in an equiangular motion and it did not work.
Ptolemy did the following, and you'll have to follow me for a second here. First of all, how do you figure out where that line is from A to Pi? Pi is the perigee, the point where it's closest to the observer, A is the apogee where it's furthest from the observer.
Answer, where the planet or object is moving, it's this case planet, slowest, that's the apogee, where it's nearest, that's the perigee and that gives you the line. Next how much eccentricity is there? That's dictated by the ratio of the two speeds. That is if you're looking off center and the motion is simply a parent speeding up and slowing down, then that ratio of the fastest to the slowest will give you how far away the center is.
Okay? The point about which it's doing equiangular motion, but that didn't work for Hipparchus. So what did Ptolemy do? He did what's called bisecting the eccentricity. He actually measured where the point C should be that describes the circle on which the center of the epicycle moves. And to the accuracy of his measurements, it's midway between the point of equiangular motion and the observer and that's his invention that made him legendary.
Okay, it's that simple and we'll come back to that, so don't expect to get it the first time. It's what most of the rest of the night is going to be about. But that's his great solution, is in effect to move the circle itself on which the epicycle center is moving, move it off the point of equiangular motion.
And now just add, since Patrick asked before, real versus apparent motion. Half the motion is real, because you're moving equiangularly around a point that's not at the center. Half the motion is apparent, because the observer is off center. That's what I meant by half and half. That same half and half is in Kepler.
The exact same, almost the same numbers. Okay? So, what Ptolemy did was totally sensible. The solar theory is Hipparchus's theory. You have the vernal equinox and the autumnal equinox passing through the observer. You have the, again, the line of Apsides where apogee and perigee, perigee being the Greek letter pi.
And summer solstice and winter solstice are shown relative to the observer O, and you can start seeing why given the location of the solstices and the equinoxes why the time from a solstice to an equinox is not the same both ways because of the off center character of this.
So, it's Hipparchus' theory, he takes it over almost entirely from Hipparchus, it has faults. One fault is the year is not quite, the measure of the year is not quite right. Another is the rate of the procession of the equinox is wrong. We'll come to those at the end but, it's still the theory.
That essentially you end up with a version of this very theory in Copernicus, but now of course for the Earth instead of the Sun. It'll be slightly different. There's a further complication. The Moon is the striking one, and I've already said why. When he started on the Moon, he, instead of treating it the way the Sun is treated, he decided to use a small epicycle.
And he worked that model out, so it's just an eccentric according to Apollonius' theorem, being produced be two circular motions. He did that because he had to accommodate this motion, this eighteen year motion of the line of nodes and the nine year motion of the apogee. The apogee going around.
So he decided he didn't want it just to do eccentricity, he wanted a epicycle. Then he discovered in the quadrants, now I have to start introducing notion to you. Picture the Earth, Sun, and Moon. The moon is in the quadrant when it's 90 degrees away from the Earth's Sun line.
In the quadrants, the motion is significantly different from what the simple model said it was. That had never been picked up by the Babylonians because they looked only at eclipses and it doesn't show up When you're 90 degrees away from an eclipse. So he's the first to discover this feature.
It's fairly striking, it's one of the most complicated things in our moon. It came to be known as the eviction. I'll describe it just briefly. The apogee moves three degrees on average per cycle of the moon 28th a cycle of the moon. But as part of that it swings back and forth between plus 12 degrees and minus nine degrees.
So the apogee is doing this swing with a bias toward a gradual motion and at the same time the eccentricity of the orbit is changing. It's called an evection because a 17th century astronomer named Bayou said, it's like throwing the Moon out of one orbit into another and throwing it back.
Okay? Ptolemy discovered that and had to model it, so he put that inner crank in there. Which gives you variable eccentricity and allows you to get all three of the inequalities at one time. And then discovered there's a fourth inequality in the octants midway between the quadrants and the line of the Earth's sun, there's a speeding up that nobody had noticed before and that you wouldn't see until you got the other one in there.
So he discovered two new inequalities that nobody had found before. And he fixed it, the second one he fixed it by referencing everything to the mean location of the apogee, a bar in that circle rather than A. Now this is to scale. Okay. This is what the orbit is supposed to be to scale, it is fiercely wrong in one respect.
As you can see immediately it brings the moon much closer and further away front he earth than the moon's apparent diameters can possibly allow. This has the Moon coming much closer to the Earth and then much farther away when in fact if you just look at the size of the Moon, which on average is 30 minutes of arc.
By the way you can't see that when it's low in the horizon because of atmospheric refraction. Look at the Moon over head, it's half a degree It deviates from half a degree very, very little. So this is a glaring empirical failure of Ptolemaic astronomy. The most obvious failure that everyone picked up.
Why Ptolemy tolerated it? The only answer I can give is he couldn't find any other way to get all four of the all four of the anomalies that had been discovered. Now the last thing about this the Moon is so complicated, in fact the easiest way to describe how complicated the Moon is.
We got the Moon to the accuracy that we had the planets in 1800 for the first time in 1919. Not too much before I was born was the first time we had a successful theory of the Moon. The Moon is fiercely complicated. That theory has 1,405 correction terms, and that'll give you a feel for how complicated.
So I'm not gonna talk much more about the Moon till we get to Newton. I'll occasionally bring it up but nobody was able to get a decent theory of the Moon. Many of them had multiple theories, none of them worked and they all knew it perfectly well. The Moon is remarkably complicated notion.
I bring it up here, partly of course, because it's a major error of Ptolemy, but there's a more important thing. This was the one case where he used his initial models, they're called theories. As soon as you get fixed values for the parameters, it becomes the theory of the Moon, instead of the model.
He used his initial theory of the Moon, compared it to observation and made new discoveries with it and that's a remarkable step forward. And he tells you that in the Almagest and he suggests the same thing should be done with other objects. And for reasons that are perfectly mysterious, nobody began doing it till around 1600.
Nobody made a serious effort to learn from discrepancies till around 1600, and it isn't Ptolemy's fault. He knew it perfectly well. It, we need another explanation. I once asked an Islamic astronomer, why nobody picked it up, and he smiled at me, as if patting me on the head, he's fifteen years older than I am, no longer alive.he died last spring.
Patted me on the head and said isn't that an interesting question. Meeting one of the foremost experts on Islamic astronomy, had no answer to why. We just don't know why people didn't pick up on it but it's a striking feature of Ptolemy.