9 Ptolemy Day 2 the Basics of The

DAY 2

DIAGRAM.

The Exercises in Day 1 contain some of rs ta S the most basic observations at the d e ix F foundation of Ptolemaic astronomy. f o re There was no diagram to accompany e h p Polaris (NCP) them because it is good to begin from S raw data. That way, one can distinguish pure observation from the model You produced for the sake of understanding N it. We will now begin to introduce W K E diagrams and to try to account for the Your Horizon S facts, beginning with just the daily Earth rotation of the stars, and working our R way to the Sun’s “backward” motion South Celestial Pole along the “ecliptic”, and eventually to the planetary motions, introducing new observations and phenomena along the way, as needed. The first diagram depicts the Earth, and a very large you (for clarity) standing upon it. Suppose you are in the northern hemisphere, and you are looking at Polaris in the night sky, the tail end of the “Little Dipper” (or Ursa Minor). We draw an imaginary plane under your feet where you stand, tangent to the surface of the Earth (which we depict as a sphere, although we will argue for that shortly), and that plane is extended till it meets the “sphere of fixed stars”, at whose center the Earth sits, immobile. This celestial sphere has all the “fixed stars” fixed upon it, and as it rotates, we see them all make counter-clockwise concentric circles about P, Polaris (or, more precisely, about the North Celestial Pole, which is very near Polaris). We have not yet argued for any of these ideas, but these are the basic elements of the Geocentric Model. The circular plane under your feet, tangent to the Earth, is called your “horizon.” It extends out as far as the stars. If the stars lie on a sphere, then the line which forms your horizon is a circle on that sphere. More than that, if the Earth is of insignificant size compared to the sphere of fixed stars (its size is exaggerated in the diagram just for the sake of clarity), and if the Earth sits at the center of the sphere (as Ptolemy will propose and defend), then your horizon is a “great circle” on the celestial sphere. A “great circle of a sphere” is a circle whose center is the same as the center of the sphere on which it lies, and so it is a circle of the largest possible size on that spherical surface. Not every circle on a

9 spherical surface is a great circle. For instance, the circles which the stars in the Little Dipper make around Polaris are not great circles, although they are circles on the sphere of fixed stars. Their centers lie closer to Polaris than to the center of the sphere itself. Now let’s familiarize you with some of the properties of your horizon. First of all, you can see everything above that plane, but nothing beneath it. Choose any star R on the sphere of fixed stars which lies below that plane, and join R to you by a straight line. Then that line has to cut the sphere of the earth at some point K. That is a basic fact of spherical geometry. rs ta S d e The line joining you to K must lie entirely inside the ix F f o Earth, another fact of spherical geometry. Hence there re e h p Polaris (NCP) is no way for you to see R, since a big chunk of solid S earth lies between it and your eye. You The diagram might make it seem as though you N are lucky to be seeing fully half of the sphere of fixed W K E Your Horizon S stars, since you happen to be standing at the center of Earth the celestial sphere, whereas other people on Earth R would not be, but would be standing some terrestrial South Celestial Pole distance away from you, and hence away from the center of the sphere of fixed stars. But since the Earth (as we shall see) has no significant size compared to the celestial sphere, those terrestrial distances make no real difference. It is as if everyone were standing at the center of the celestial sphere. Imagine growing the Earth inside the diagram (but put its center at the center of the celestial sphere) so that it becomes almost as big as the celestial sphere itself: Then you will get scrunched up near the top of the celestial sphere, and your horizon will cut off a very tiny portion of the celestial sphere, and so you will see very little of it; much, much less than half. Conversely, if you imagine shrinking the Earth till it is a microscopic speck in the diagram, sitting right at the center of the celestial sphere, your horizon will cut off pretty much exactly half the celestial sphere. And whether we think there is a celestial sphere or not, we can show that the size of the Earth is insignificant compared to the distances out to the stars (as we shall see). Now consider the point on your horizon directly below Polaris, and call that N. The line joining you to N points to terrestrial North. If you were to walk in that direction long enough, Polaris would keep moving up in your sky until eventually it was directly overhead and you were standing on Earth’s North Pole. (Don’t try that.) Extend the same straight line from N behind you, now, and that will point due South on your horizon. Directly to your right is East, and to your left is West, if you are facing Polaris. As we noted in Day 1, all the stars, as well as the Sun and the Moon, rise somewhere on the Eastern portion of your horizon, and move together in apparently concentric circles around Polaris, and set somewhere in the West. If we extend the line “from Polaris to you” below your horizon, through the Earth, and out to the celestial sphere, that point is the South Celestial Pole. This straight line joining the North Celestial Pole (near Polaris) to the South Celestial Pole is the “axis of the daily motion,” the axis around which the whole celestial sphere spins once every 24 hours (roughly). Strictly speaking, that axis goes not through you, but through the center of the Earth. But since you are an insignificant distance away from the center of the Earth,

10 compared to the distances out to the stars, it makes no difference which we say, so far as the stars are concerned. That is our first diagram, and our first taste of the Ptolemaic geocentric model of the universe. Soon we can begin to bring forward some of Ptolemy’s arguments in favor of it.

SCHOLIUM ON THE FOUR POINTS OF THE COMPASS

How are North, South, East, and West defined? “Celestial North” means the spot in the heavens that does not move, near the pole star. On earth, the place called “Terrestrial North” means the spot where Celestial North is directly overhead. (Do not confuse this spot with magnetic north, which is not exactly the same!) On earth, if you can see Celestial North (Polaris), but it is not directly overhead, the direction called “North” means the direction toward the place on your horizon directly below Celestial North in the sky. If you cannot see Celestial North, then you can see Celestial South, in which case the spot directly beneath it on your horizon is the direction called terrestrial “South,” and the opposite direction is “North.” “West” = directly to your left as you face north. “East” = directly to your right as you face north.

PROPOSITION 1: THE HEAVENS MOVE SPHERICALLY.

In Book 1, Chapter 3 of his Almagest, Ptolemy argues that the heavens move spherically and uniformly. In these arguments, one is presuming that it is the heavens, not the Earth, that is moving. The arguments do not prove this, but assume this, and go on to prove that the motion of the heavens must be spherical and of uniform speed (so far as the unaided senses can determine).

[A] If you pointed a perfectly circular tube directly at Celestial North, through the tube you would see the stars on the edge of the tube trace out the circular shape of the tube. They would not stray from it. And this is true no matter how big a piece of sky you can see through the tube. Also, if you use a watch (Ptolemy would have had to make do with an hour glass or a water clock), and time how long it takes the star to go through any eighth-arc of its full circle, it will take exactly that long for it to go through any other eighth-arc of the circle. So the motion is uniform.

[B] Other ideas about how they move A B A B make no sense. For example, if we 2 supposed that the stars move along straight 1 lines, away to infinity, then the same stars would never return. But the same stars do return. So that cannot be how they are moving. Earth

11 [C] In fact, if the fixed stars moved around us by being carried along any surface other than a spherical one, they could not possibly keep the same angular distances from each other from our point of view unless they changed their actual linear distances from each other, which would be very complicated and bizarre—as though the universe were playing a trick on us, making it look as though the stars kept the same linear distances from each other, but really they didn’t. To see this, imagine yourself observing two stars, A and B, at two different times. Suppose A and B are carried along on some surface other than a spherical one—say a flat surface, moving by like a conveyor-belt. At one time, the angle between A and B and your eye at the vertex (which you can measure with a graduated protractor whose arms are sighted against the stars with your eye at its vertex) is angle 1. Later, if A and B stay the same distance apart from each other, the angle between them will be angle 2, a much smaller angle as A and B get further away from you. The only way to keep the apparent angle between A and B the same, then, would be to have the distance between them increase as they got further from you. And that makes it seem as though the universe is playing a trick, just especially on you. Since that is ridiculous, we have to say instead that A and B keep the same distance from each other, as well as the same apparent angle from each other, which happens only if they are moving on a sphere. And, Ptolemy points out, if you think the fact that the stars appear larger near the horizon (like the moon) proves they are closer to us there, you are wrong. He attributed this fact (which is at least to some extent an illusion) to the atmosphere. We are looking out through more atmosphere, and more moisture, when we see a star near the horizon than when we see it nearly overhead, and that affects its appearance.

Ptolemy gives other arguments, but these are the clearest and best of them. Note that no argument here establishes that the fixed stars are all the same distance away from us. They could be stuck in a giant crystalline sphere at different distances from us, so far as these arguments are concerned. But they maintain their distances from each other (hence they are called “fixed” stars), and always appear to have the same angle between them when we observe them with a sighting instrument.

PROPOSITION 2: THE EARTH IS SENSIBLY SPHERICAL

Ptolemy proves this in Book 1, Chapter 4 of his Almagest. Note again that he knew this long before Christopher Columbus—and it was known long before Ptolemy, too. But why does he say “sensibly” spherical? I think for two reasons. One, because you can actually see the curvature of the earth, e.g. when a ship approaches from over the horizon, first you see the top of the mast, and later the ship itself, so that the bulge of the Earth between is visible. Two, because the earth is not a perfect mathematical sphere, as we can see from its mountains and valleys, but when one gets back far enough, the whole impression is very nearly a sphere. It is truer to say that the Earth is an “oblate spheroid,” a bit fatter about the equator and squashed in at the poles, but not much.

ARGUMENTS FOR THIS BASED ON LONGITUDE:

Positive Arguments

(a) Even without watches, we can know the sun rises and sets sooner in the eastern parts of the earth than in the western parts (e.g. in New England vs. California). And an eclipse of the sun is a more or less simultaneous event that can be seen from regions fairly far apart on earth (from very different longitudes). But the sun is higher in the sky for eastern folk (when an eclipse occurs) than it is for western folk. So the earth is roundish in an east-west direction. And since the further east you go, the higher the sun is in the sky proportionally for a given moment (like an eclipse), the earth is circular in an east-west direction.

When an eclipse occurs, for instance, the sun is right overhead for Europeans, but still rising for midwestern Americans.

East

USA Moon SUN West

(b) Mountains on land appear to rise out of the sea as we sail in from sea toward the shore, and so the ocean is convex. This argument is independent of direction, unlike (a) which argues that earth is round in the east-west direction, or longitudinally.

Negative Arguments

(1) A concave earth (from E-W) would mean that the people in the West would see the sun rise before people in the East. That doesn’t happen.

Sun West East (California) (Maine)

(2) A flat earth (from E-W) would mean that stars would rise and set at the same time for people in the East and in the West. That doesn’t happen.

West East Sun (California) (Maine)

(3) A polyhedral solid earth (from E-W) would mean that stars would rise and set at the same time for any people at the eastern and western edges of a flat face on that solid. But again, that does not happen. Even at relatively small distances, e.g. if A lives 200 miles west of B, there will be observable differences in the time of the rising of the sun. Today, we can easily verify this. If you live in New Hampshire, call your friend in California just as the sun is rising for you. He will be angry with you, because it will still be completely dark where he is—the sun is above your horizon, but still below his for another three hours. Ptolemy did not have telephones, nor could he bring a watch with him and travel 200 miles or so west and just record the time the sun rose on his unadjusted-to-local-time watch. He would rely on things like eclipses, which could be observed simultaneously by people 200 miles apart, and who could observe how high up in the sky the sun was for them at the time.

E W

ARGUMENT BASED ON LATITUDE

(4) A cylindrical earth (standing N-S) would mean there are no always-visible stars for inhabitants of the curved surface. All stars that could be seen from the curved surface would move from East to West and eventually lie behind the earth, and hence be invisible until they rose again. But, with the exception of earth’s equator N (where all stars move from east to west with the exception of Polaris, which sits still right on the northern horizon, or nearly so), from anywhere on earth there will be stars that are always visible, i.e. that never set (although they will be hard to see during the day, W E thanks to the Sun). Also, a cylindrical earth (standing N-S) would mean there would have to be some stars which are visible from all latitudes along that cylindrical surface. S

14 For example, Polaris would be visible from all latitudes along that surface. But in fact, it is possible to go south far enough that Polaris disappears below one’s horizon (just as we cannot see the south celestial pole if we live in the northern hemisphere). According to a cylindrical model of the earth, there would be no change in which stars were visible as we went north or south. As it is, when we travel north, certain stars in the south dip below the horizon and are never to be seen from that latitude, and new stars become visible above the northern horizon that were not visible where we were before. As Ptolemy puts it: “the more we advance towards the north pole, the more the southern stars are hidden and the northern stars appear.”