Opening segment: "Has everybody who wants to be on trunk given me their information? All right, it'll be up here after class. Please do so, cuz I can put you on, but I can't do it if I don't know your email. All right, Ptolemy."
Has everybody who wants to be on trunk given me their information? All right, it'll be up here after class. Please do so, cuz I can put you on, but I can't do it if I don't know your email. All right, Ptolemy. Complex person. I'll tell you about him in a moment.
Let's look at his book first. The title Almagest, is a Arabic modification of Latin. The original title appears to be Mathematical Syntaxis or Mathematical System. To become the majestic one is something the Arabs did to it. Though picking it up from, it's a trade off on the word which is Latin.
It has 13 books. I'm just gonna run through them in order. Book one starts out be giving his reasons for various assumptions. One of which is the Earth does not revolve daily. The Earth does not move, etc. He does give reasons, the reasons are sensible reasons. And that part of book one, I'm putting on supplementary material so that you can read it and see that his claims are not coming out of left field.
Following that, he gives you spherical trigonometry, if you notice. A table of chords that the Islamic people or the Islamic astronomers are the ones who introduces sine, cosine and tangent. The classic trigonometric function is called the chord. It's the chord of a circle. And the chord of an angle is the same thing as twice the sine of a half-angle.
If you can picture that, just take the chord, draw a line down the middle. So, you can get all the other trigonometric relations from the chord. The chord happens to be the he's functioning with. And he started showing you how to do things with it. Preliminaries for spherical proofs, etc.
Menelaus Theorem, there are six elements to it. If you have any five of them you can get the sixth. That's what the theorem let's you do. And it let's you transfer from one coordinate system to another very easily. If you take a course in Orbital Astronomy, that's almost the first thing you learn in the first week.
How to go from equatorial system to an ecliptical longitude latitude system because you gonna have to do it all through the course to go back and forth with stars. Books two and three talk about the tropical year which is his choice for the year, its unfortunate choice, for a reason.
The tropical year is the number of days it takes to return to the vernal equinox. Unfortunately the vernal equinox is not sitting stationary on the Zodiac. You all know it. We are entering the Age of Aquarius. That's the motion of the vernal equinox through the stars. Ptolemy adopted the lowest bound that Haparcus proposed, which is one degree per century, which would be 36,000 years.
That's wrong, it's 26,000 years. It's significantly faster and it's one of the basic errors in Ptolemaic astronomy that's exceedingly easy to fix. Because you don't, all you need is to get the location of the vernal equinox correctly among the stars. So you can adjust that in any way.
And then you proceed, in this case, always proceed all measurements from the vernal equinox. As I say, the tropical year, you'd be better off using sidereal year. We will get to that later. Books four, so that's books one through three, and it gives you the theory of the sun.
The theory of the sun is essentially Hipparchus's theory. I'll come to that in a moment. Books four through six are about the moon. And it's striking. I will come back to this almost immediately. He gives you a basic theory of the moon and then uses it to discover two inequalities that no one has ever discovered before.
And it's a striking example of what you can do with a reasonably good model, when you start comparing it to observation. You can make discoveries and just discover new patterns. It's the only place Ptolemy did that, and it's the only place anybody did it for several centuries. As you'll see, it's a striking feature of Ptolemaic astronomy.
But books four through six lay out the theory of the moon. And you will see it ends with how to calculate eclipses. Books seven and eight give you a star chart. Over a thousand stars in their relative positions, with respect to one another. That's a star chart that survived until the 1590s.
It was the reference for everybody. Nobody redid it. And it's quite decent, on the whole. And you understand why that's important. What you're going to constantly measure is the position of a planet relative to a star. And the location of the star, on the overall system, is then going to give you the location of the planet.
So, you won't be going directly to getting angles of the planet. You will instead be going directly, measuring distances using an instrument opp no lenses, but using a measuring instrument to get distances to various stars in order to locate a planet at any time. That's why the stars are in here before the planets.
Books 9 through 12 are on longitudes of the planets. He splits the problem of longitudes and latitudes. What is new in the history of astronomy, is his theory of planetary longitudes. Books 9 through 12, that's the principal thing I'll be focusing on much of the rest of the night, and then book 13 is on the latitudes.
Notice something here while I have it up. What's it set up to calculate? Now in fact his system lets you calculate longitude and latitude of every one of the key objects at any moment anywhere on the surface of the earth at any time, in principle. But he doesn't tell you how to do that.
He tells you how to calculate retrogradations of Saturn, Jupiter, Mars, Venus, Mercury. A table for the stationary points, etc. He's not interested in where planets are every night. He's interested in the principal salient phenomenon. If he can reproduce that he's perfectly happy. So while there's a general calculational scheme, that's not what it's really about.
I've passed two copies of the Almagest around in English. I don't have a Greek copy, and I've held Arabic copies, but I don't read Arabic, so it's quite useless to me. So I'm depending on Toomer's translation. Toomer is one of Drugabauer's colleagues at Brown. He spent years on this translation.
It's his authoritative, probably the most authoritative translation we will have for a century or two, but certainly for a long while. I rely on that. There are two editions, cuz there's a correction to it. So, the paperback is more recent and a more valuable book. The hardbound I keep in my office, the paperback I keep at home where I work.
So, but, I passed them around so you see both. Ptolemy did a lot of other things besides the Almagest. Let me stop a moment. I just skipped one thing. I want to say something about the latitudes. He needs the latitudes to get the shape of the loops, right, because the loops don't just move in longitude, they dip down in various ways.
How is he supposed to get them? Well, he's got epicycles. So what he does is have the epicycle and the main circle tilty. And it's a system of tilts. And how to make it work is one of the principal complaints the Islamic astronomers had against them. It seemed there was no sensible way to make the mechanism work.
And he himself is very guarded in his claims about the latitude. But he does do well in producing the loops. It's for that reason I'm not gonna talk about the latitudes but let me come back to that. Now about Ptolemy. The Almagest is what he's most famous for.
To quote Toomer, it's a complete exposition of mathematical astronomy as the Greeks understood the term at the time. That's 150 AD. The Roman Empire was of course occupying Alexandria. Alexandria had the great library at, for all intents and purposes, universities in the modern sense of the word, and for all intents and purposes, Ptolemy was a professor at a modern university.
The Almagest he's most famous for. There's another work called the Planetary Hypothesis. Where he tries to figure out the minimal size that all the visible objects have to fit within with all of the epicycles, etc., and not intersecting with one another. It's an exercise is one of the reasons to take him as doing something more than just recovering the phenomena, because he's very serious about the possibility of this model being realized.
It's funny though, one of the Islamic complaints against him is he is inconsistent in the planetary hypothesis versus the Amalgest in various places. He simplifies to make the planetary hypotheses more possible than they would be under the Almagest. You don't have to worry about that. I just wanted to mention.
Other things he wrote. I passed it around. The Tetrabiblios, that's the work I have in both Greek and English besides, the only work I have is Greek and English. It's a textbook on how to do astrology, through and through. And how to use the material and the Almagest to do astrology.
And that's what people wanted this for primarily. Was to conduct astrology. I'm not saying that's what he wanted it for primarily. But those were the people who were paying for things like this. He wrote a book on optics. He wrote the first real geography which I'm passing around.
I'm not passing around the whole geography. Alex Jones did the translation and reproduction only a part of it. But it's the first half way decent geography of the part of the world they had access to that we had. And remain fairly significant as the prevailing geography for several centuries.
He produced a work on harmonics, musical harmonics. On logic, on sun dials, on stereographic projection. Some of these we have, some these we do not, the Almagest we of course have, goes without saying. The reason I passed the book around is when you look at it you're going to see almost immediately, this is not an easy book to read.
It is a mathematically difficult, demanding book. Why? Because what it is a textbook. Textbook to do what? First, how to set up a epicycle model. Second, how to use observations to get the values of all the constants in each of the individual models. Now having got that, how to calculate all the salient phenomena.
So it's a how to book, through and through, and it's exactly for that reason that it remained pertinent, right down to the 15th century, because this is where everybody learned how to work from observations to models, and how to do calculations. Even Copernicus, when he was departing from Ptolemy is working very heavily from Ptolemy.
The best way to say this, had Ptolemy not supplied that book, astronomy as a science would have been delayed for a long, long time. Because all we would have had is qualitative models with no systematic way to put numbers to things. You'll see that later when I get into this.