Today my class reached the pinnacle of Ancient Greek astronomy: the Mathematical Syntaxis of Ptolemy (later referred to as the Almagest because of its Arabic name, meaning the "the greatest"). We had already looked at the basic epicycle/deferent model developed by Apollonius, and we saw that it could produce retrograde motion and link that motion with increased brightness in a natural way. We also saw that retrograde/brightness could be linked to opposition (for superior planets) or conjunction (for inferior) by synchronizing the motion of the epicycle (for superior planets) or the deferent (for inferior) to the motion of the Sun. We examined all of these things using my Superior Ptolemaic and Inferior Ptolemaic simulations.
Today we looked at some modification to the basic deferent/epicycle model. First we examined Hipparchus' idea of using an eccentric circular orbit for the Sun. The Sun moves uniformly along a circular path, but the center of this circle is located away from Earth. This causes an apparent change in the speed of the Sun as seen against the background stars: faster when the Sun is closer to Earth, slower when the Sun is farther away. This is particularly interesting because we had learned earlier that the Sun moves fastest relative to the stars in WINTER. So this implies that the Sun is closest to Earth in winter, which contradicts a commonly held misconception about the seasons (that they are related to the distance of the Sun from Earth - they aren't, they are determined by the inclination of Earth's rotational axis relative to its orbital plane).
One thing I had my students do was derive the formula that gives the eccentricity of the Sun's orbit from the maximum and minimum angular speeds of the Sun. The algebra is not bad at all, but it is entirely symbolic. Many of my students just got wide-eyed and didn't want to try it. In the end I got them all through with a few hints, and hopefully they realized it was not as scary as they were making it out to be. Math phobia is a terrible thing. I wish I could banish it from the world.
After the eccentric we moved on to the equant. This is the hardest part of Ptolemy to understand. The equant is a point that is off from the center of a circular orbit. A body moving along an orbit with an equant changes speed in such a way that the angular speeds as seen from the equant point is constant. So ti moves faster when it is far from the equant, slower when it is close to the equant. I have (unpublished) simulations to illustrate the eccentric and equant mechanisms by themselves, but they are also built into the Superior Ptolemaic and Inferior Ptolemaic simulations mentioned above.
Finally, we see how Ptolemy determined the periods of motion for the deferents and epicycles. The period of the epicycle, measured relative to the deferent, is just the synodic period of the planet (the time between retrogrades). This measuring relative to the deferent seems weird to modern eyes - we want to measure everything relative to "absolute space" (ie, relative to the background stars). But the Ancient Greeks thought of these circles as being the equators of great transparent spheres that rotated around. The epicycle gets dragged around, and actually rotates, because of the motion of the deferent - but the Ancient Greeks didn't count that as a rotation of the epicycle itself. Only the extra bit of rotation that didn't come from being dragged around the deferent was considered as a true rotation of the epicycle. It is the period of this rotation which is equal to the planet's synodic period. The period of the deferent motion is pretty obviously (once you look at it) equal to the planet's zodiacal period (the time for the planet to go all the way around the Celestial Sphere, on average).
Next up we will work on the sizes of the epicycles relative to the deferents so that we can actually try to sketch out the scale model of the whole system of planets. In doing so, we will see just how poorly the whole Ptolemaic scheme fits with Aristotle's cosmology. In some ways the merger of Aristotelian cosmology and Ptolemaic astronomy represents one of the great compromises in the history of science. This compromise was probably only possible because of strong skepticism about whether or not astronomy could tell you what was really going on in the heavens. Some astronomers, at least, took the attitude that Aristotle had the basic structure right. Ptolemy didn't fit that structure, but Ptolemy was just giving us formulae for predicting planetary locations, not telling us what was really going on in the heavens. This attitude still existed (perhaps even more strongly) in the Renaissance. It was the opponents of this brand of skepticism who would ultimately revolutionize astronomy.
Thursday, September 27, 2012
Wednesday, September 26, 2012
Eudoxus and Apollonius: Spheres and Epicycles
On Tuesday my class began to move beyond the two sphere model of the universe and started focusing on explaining the motions of the planets. We examined two theories regarding planetary motions that were devised in antiquity.
The first theory we looked at is due to Eudoxus of Cnidus. Eudoxus' basic idea was that each planet had associated with it a set of nested spheres. All of these spheres were centered on Earth. The planet itself lay on the surface of the innermost sphere (which we will call sphere 1). That innermost sphere spins about an axis. The ends of this axis are fixed to the inner surface of the next sphere out (we will call it sphere 2). Sphere 2 also spins about an axis, and it turns out that it MUST spin at the same rate as sphere 1, but in the opposite sense (we'll talk about why in a bit). If the axis of sphere 1 is aligned with the axis of sphere 2 then the planet would just stay motionless, because these two spheres would cancel each other. However, if you tilt the axis of sphere 1 relative to the axis of sphere 2, the planet executes a little figure-8 shaped jaunt. It doesn't really get anywhere, it just does figure-eights (actually a shape known as a hippopede) in one part of the sky. But for now we don't want it to get anywhere. That's why the two spheres spin at the same rate in opposite senses, so that the planet wiggles around a bit but doesn't actually go to a new part of the sky.
Now we connect the axis of sphere 2 to the equator of sphere 3 (the next one out). Sphere 3 spins around so that in addition to the figure-8 motion we get a motion of the whole assembly on a big circle around the sky. You set this up so that this motion is eastward, and the big circle is the Ecliptic. The result is pretty amazing: the planet has a general eastward drift, but occasionally it stops and goes west for a bit before returning to its eastward motion. In other words, the first three spheres qualitatively reproduce the retrograde motion of the planets. Sphere 3 then attaches to sphere 4 at a 23.5 degree tilt, so that sphere 4 has its axis aligned with the axis of the Celestial Sphere, so that sphere 4 can produce the daily motion of the stars around our sky.
The whole thing is pretty amazing. It is hard to visualize how it works from a description. You really have to see it in action, so my student spend some time playing around with my Spheres of Eudoxus simulation. How Eudoxus figured out that these motions would produce the retrograde behavior is just beyond me.
But then I lead my students to see one of the big problems with this model. The planet lies on a sphere that is centered on Earth. So its distance from Earth never changes. So there is no explanation for the changes in brightness that are observed in the planets (particularly Mars). It also turns out that Eudoxus theory fails at the level of quantitative prediction - you just can't build a version of this theory that will accurately determine the future locations of the planets in the sky. But it gets a major qualitative feature right, and it fits in beautifully with the Aristotelian cosmos.
Moving on from Eudoxus we investigate the deferent-epicycle model originally formulated by Apollonius of Perga. The pure version of the model consists of a large circle, known as the deferent, centered on Earth. Another smaller circle, known as the epicycle, has its center on the deferent circle. The deferent spins around Earth, and the epicycle spins around its (now moving) center. This combination of two circular motions produces a planetary orbit that is akin to a spirograph, but the Ancient Greeks didn't really think in terms of the planets moving in orbits through space. They thought of rotating circles (which may or may not be the equators of rotating spheres). So the spirograph picture wasn't really discussed until much later (by Kepler).
Again, this idea is hard to picture. My students explore the idea with my SuperiorPtolemaic simulation. Again, this model is pretty amazing, although for me it seems easier to figure out than Eudoxus' model. The cool thing is that it not only reproduces the retrograde motion of a superior planet, it also brings the planet closest to Earth during retrograde. That explains why superior planets, like Mars, are brightest when they retrograde. It is an automatic feature of the model. It occurs naturally from the basic premises of the deferent and epicycle. Pretty neat.
Not only that, but you can set things up so that the planet does its retrograde/brightness thing when it is in opposition to the Sun, as happens with real (superior) planets. In order to do this you must synchronize the epicycle motion with the motion of the Sun around Earth (we are still geocentric so far). This is not at all a natural thing to do in this model, although there is also no reason why you shouldn't do it. It's a bit ad hoc, but it makes it all work.
So on a qualitative level, the deferent-epicycle model was a huge success and it quickly supplanted Eudoxus' homocentric spheres (although homocentric spheres made a comeback among Arabic speaking astronomers during the Middle Ages, and again among Europeans during the Renaissance). But the basic model was not quite sufficient to give accurate quantitative predictions. It still needed to be tweaked, and it was Ptolemy (following some important work by Hipparchus) who would do the necessary tweaking.
This activity really makes me think about the underdetermination thesis (sometimes called the Duhem-Quine thesis), which states that for any set of observational facts there are many theories (perhaps an infinite number) that will fit those facts. From a Logical Point of View* that is certainly true, but most of those theories would be immediately dismissed as unreasonably complicated or arbitrary. In practice, it is really hard to come up with EVEN ONE reasonable theory that fits even just the qualitative features of the observations. Eudoxus got part of it right with his theory, but Apollonius really nailed it. To some extent, all subsequent theories about planetary motions were variations on the theme of the deferent-epicycle model. Yes, there were many different versions (which would seem to fit the Duhem-Quine thesis) but they employ almost the same geometry. I guess my point is just that coming up with even one good theory is hard, and we should all be very impressed by the ideas of Eudoxus and Apollonius even if we no longer think they were quite right.
*inside joke for logicians (see the works of WVO Quine if you want to get the joke)
The first theory we looked at is due to Eudoxus of Cnidus. Eudoxus' basic idea was that each planet had associated with it a set of nested spheres. All of these spheres were centered on Earth. The planet itself lay on the surface of the innermost sphere (which we will call sphere 1). That innermost sphere spins about an axis. The ends of this axis are fixed to the inner surface of the next sphere out (we will call it sphere 2). Sphere 2 also spins about an axis, and it turns out that it MUST spin at the same rate as sphere 1, but in the opposite sense (we'll talk about why in a bit). If the axis of sphere 1 is aligned with the axis of sphere 2 then the planet would just stay motionless, because these two spheres would cancel each other. However, if you tilt the axis of sphere 1 relative to the axis of sphere 2, the planet executes a little figure-8 shaped jaunt. It doesn't really get anywhere, it just does figure-eights (actually a shape known as a hippopede) in one part of the sky. But for now we don't want it to get anywhere. That's why the two spheres spin at the same rate in opposite senses, so that the planet wiggles around a bit but doesn't actually go to a new part of the sky.
Now we connect the axis of sphere 2 to the equator of sphere 3 (the next one out). Sphere 3 spins around so that in addition to the figure-8 motion we get a motion of the whole assembly on a big circle around the sky. You set this up so that this motion is eastward, and the big circle is the Ecliptic. The result is pretty amazing: the planet has a general eastward drift, but occasionally it stops and goes west for a bit before returning to its eastward motion. In other words, the first three spheres qualitatively reproduce the retrograde motion of the planets. Sphere 3 then attaches to sphere 4 at a 23.5 degree tilt, so that sphere 4 has its axis aligned with the axis of the Celestial Sphere, so that sphere 4 can produce the daily motion of the stars around our sky.
The whole thing is pretty amazing. It is hard to visualize how it works from a description. You really have to see it in action, so my student spend some time playing around with my Spheres of Eudoxus simulation. How Eudoxus figured out that these motions would produce the retrograde behavior is just beyond me.
But then I lead my students to see one of the big problems with this model. The planet lies on a sphere that is centered on Earth. So its distance from Earth never changes. So there is no explanation for the changes in brightness that are observed in the planets (particularly Mars). It also turns out that Eudoxus theory fails at the level of quantitative prediction - you just can't build a version of this theory that will accurately determine the future locations of the planets in the sky. But it gets a major qualitative feature right, and it fits in beautifully with the Aristotelian cosmos.
Moving on from Eudoxus we investigate the deferent-epicycle model originally formulated by Apollonius of Perga. The pure version of the model consists of a large circle, known as the deferent, centered on Earth. Another smaller circle, known as the epicycle, has its center on the deferent circle. The deferent spins around Earth, and the epicycle spins around its (now moving) center. This combination of two circular motions produces a planetary orbit that is akin to a spirograph, but the Ancient Greeks didn't really think in terms of the planets moving in orbits through space. They thought of rotating circles (which may or may not be the equators of rotating spheres). So the spirograph picture wasn't really discussed until much later (by Kepler).
Again, this idea is hard to picture. My students explore the idea with my SuperiorPtolemaic simulation. Again, this model is pretty amazing, although for me it seems easier to figure out than Eudoxus' model. The cool thing is that it not only reproduces the retrograde motion of a superior planet, it also brings the planet closest to Earth during retrograde. That explains why superior planets, like Mars, are brightest when they retrograde. It is an automatic feature of the model. It occurs naturally from the basic premises of the deferent and epicycle. Pretty neat.
Not only that, but you can set things up so that the planet does its retrograde/brightness thing when it is in opposition to the Sun, as happens with real (superior) planets. In order to do this you must synchronize the epicycle motion with the motion of the Sun around Earth (we are still geocentric so far). This is not at all a natural thing to do in this model, although there is also no reason why you shouldn't do it. It's a bit ad hoc, but it makes it all work.
So on a qualitative level, the deferent-epicycle model was a huge success and it quickly supplanted Eudoxus' homocentric spheres (although homocentric spheres made a comeback among Arabic speaking astronomers during the Middle Ages, and again among Europeans during the Renaissance). But the basic model was not quite sufficient to give accurate quantitative predictions. It still needed to be tweaked, and it was Ptolemy (following some important work by Hipparchus) who would do the necessary tweaking.
This activity really makes me think about the underdetermination thesis (sometimes called the Duhem-Quine thesis), which states that for any set of observational facts there are many theories (perhaps an infinite number) that will fit those facts. From a Logical Point of View* that is certainly true, but most of those theories would be immediately dismissed as unreasonably complicated or arbitrary. In practice, it is really hard to come up with EVEN ONE reasonable theory that fits even just the qualitative features of the observations. Eudoxus got part of it right with his theory, but Apollonius really nailed it. To some extent, all subsequent theories about planetary motions were variations on the theme of the deferent-epicycle model. Yes, there were many different versions (which would seem to fit the Duhem-Quine thesis) but they employ almost the same geometry. I guess my point is just that coming up with even one good theory is hard, and we should all be very impressed by the ideas of Eudoxus and Apollonius even if we no longer think they were quite right.
*inside joke for logicians (see the works of WVO Quine if you want to get the joke)
Tuesday, September 25, 2012
Measuring the Earth
On Monday my class used Eratosthenes method, and some simulated observations in Stellarium, to measure the diameter of Earth. The result is amazingly accurate considering the technology available to Eratosthenes (which consisted of, basically, a stick and somebody who was willing to walk long distances and count his steps).
Here's the idea: Eratosthenes had heard that in the city of Syene (now Aswan, Egypt) on the summer solstice, sunlight would reach to the bottom of a very deep well. This implies that the Sun gets directly overhead at Syene on the summer solstice (and therefore that Syene is on the Tropic of Cancer). So a gnomon (a vertical stick) would cast no shadow at all in Syene on that day. However, Eratosthenes lived in Alexandria, Egypt. He could measure for himself the shadow cast by a gnomon in Alexandria on the summer solstice. His measurement, plus some trigonometry (really the method of chords, because the Greeks didn't have what we now call trigonometry) told him that the sunlight at Alexandria was coming in at about 7 degrees off of vertical.
Eratosthenes assumed that the sunrays hitting Alexandria and those hitting Syene were parallel (and we investigate why this is a reasonable assumption, although there are some problems with the finite angular size of the Sun). In that case, the 7 degree difference in the direction of the Sun rays relative to vertical must come from a difference in the latitude of the two locations: Alexandria must be 7 degrees North of Syene. Eratosthenes believed that Alexandria was DUE NORTH of Syene, and he knew that the distance between the two cities was about 500 miles (in modern units) because someone had paced it. So he decided that 7 degrees around Earth corresponds to a distance of 500 miles. Proportionality implies that 360 degrees corresponds to a distance of just under 26,000 miles, which would be the circumference of Earth. Divide by pi to get the diameter, a bit more than 8000 miles.
The activity guides the students through this procedure, and lets them make measurements along the way using Stellarium. They can set their location to Alexandria or Aswan, and set the time to 200 BC, and measure the altitude of the Sun at local noon on the summer solstice in both locations. But in the activity we also investigate Eratosthenes assumptions. He assumed Alexandria and Aswan had the same longitude, but we find that this is not true because local noon occurs at a later time in Alexandria than in Aswan, which implies that Alexandria is a bit to the west of Aswan. He also thought that the Sun's altitude in Syene on the solstice would be 90 degrees, thus giving no shadow. In fact, it is 89.5 degrees which would give a 0.5 degree shadow (hard to detect, to be sure, but 0.5 degrees is not 0).
The really remarkable thing is that these two errors nearly cancel each other out. I don't have my students work through this, but I probably should. It's not that hard to see how it works. Local noon in Alexandria is roughly 10 minutes later than in Aswan, which implies a 2.5 degree difference in longitude. The real difference in latitude is 6.5 degrees, not 7 degrees. But if we now combine the 6.5 degrees north with the 2.5 degrees west using a simple Pythagorean theorem approach (which isn't really correct, but it is a good approximation because the angles are small and we are close to the equator) we find that the angular distance over Earth's surface from Syene to Alexandria is the square root of (6.5 squared plus 2.5 squared), which works out to 6.96 degrees, or essentially the same as the 7 degrees that Eratosthenes thought it was.
Sometimes it is better to be lucky than good, although I would argue that Eratosthenes was both. It's such an elegant solution to the problem of measuring Earth's diameter. I think my students enjoyed this one. It was a bit short, but maybe next time I'll have them think their way through the impact of the two errors mentioned above so that they can see why they basically cancel each other out. It will be challenging for them, but if I can get them to do it I will be very happy. That kind of analysis of error is a pretty high-level scientific thinking skill.
Here's the idea: Eratosthenes had heard that in the city of Syene (now Aswan, Egypt) on the summer solstice, sunlight would reach to the bottom of a very deep well. This implies that the Sun gets directly overhead at Syene on the summer solstice (and therefore that Syene is on the Tropic of Cancer). So a gnomon (a vertical stick) would cast no shadow at all in Syene on that day. However, Eratosthenes lived in Alexandria, Egypt. He could measure for himself the shadow cast by a gnomon in Alexandria on the summer solstice. His measurement, plus some trigonometry (really the method of chords, because the Greeks didn't have what we now call trigonometry) told him that the sunlight at Alexandria was coming in at about 7 degrees off of vertical.
Eratosthenes assumed that the sunrays hitting Alexandria and those hitting Syene were parallel (and we investigate why this is a reasonable assumption, although there are some problems with the finite angular size of the Sun). In that case, the 7 degree difference in the direction of the Sun rays relative to vertical must come from a difference in the latitude of the two locations: Alexandria must be 7 degrees North of Syene. Eratosthenes believed that Alexandria was DUE NORTH of Syene, and he knew that the distance between the two cities was about 500 miles (in modern units) because someone had paced it. So he decided that 7 degrees around Earth corresponds to a distance of 500 miles. Proportionality implies that 360 degrees corresponds to a distance of just under 26,000 miles, which would be the circumference of Earth. Divide by pi to get the diameter, a bit more than 8000 miles.
The activity guides the students through this procedure, and lets them make measurements along the way using Stellarium. They can set their location to Alexandria or Aswan, and set the time to 200 BC, and measure the altitude of the Sun at local noon on the summer solstice in both locations. But in the activity we also investigate Eratosthenes assumptions. He assumed Alexandria and Aswan had the same longitude, but we find that this is not true because local noon occurs at a later time in Alexandria than in Aswan, which implies that Alexandria is a bit to the west of Aswan. He also thought that the Sun's altitude in Syene on the solstice would be 90 degrees, thus giving no shadow. In fact, it is 89.5 degrees which would give a 0.5 degree shadow (hard to detect, to be sure, but 0.5 degrees is not 0).
The really remarkable thing is that these two errors nearly cancel each other out. I don't have my students work through this, but I probably should. It's not that hard to see how it works. Local noon in Alexandria is roughly 10 minutes later than in Aswan, which implies a 2.5 degree difference in longitude. The real difference in latitude is 6.5 degrees, not 7 degrees. But if we now combine the 6.5 degrees north with the 2.5 degrees west using a simple Pythagorean theorem approach (which isn't really correct, but it is a good approximation because the angles are small and we are close to the equator) we find that the angular distance over Earth's surface from Syene to Alexandria is the square root of (6.5 squared plus 2.5 squared), which works out to 6.96 degrees, or essentially the same as the 7 degrees that Eratosthenes thought it was.
Sometimes it is better to be lucky than good, although I would argue that Eratosthenes was both. It's such an elegant solution to the problem of measuring Earth's diameter. I think my students enjoyed this one. It was a bit short, but maybe next time I'll have them think their way through the impact of the two errors mentioned above so that they can see why they basically cancel each other out. It will be challenging for them, but if I can get them to do it I will be very happy. That kind of analysis of error is a pretty high-level scientific thinking skill.
Saturday, September 22, 2012
Aristotle's Cosmos
In Thursday's class we did an activity on Aristotle's physics. My goal in this activity is for them to become familiar with Aristotle's ideas about matter and motion, and also to see how reasonable those ideas are if you are describing things qualitatively rather than quantitatively. I want them to understand this because Aristotle's physics and cosmology were the main reason people initially rejected the Copernican theory. I want my students to see that this rejection was not unreasonable, given what was known at the time. In fact, it may have been the most rational thing to do at the time. Hindsight, as they say, is 20/20.
I have read bits and pieces of Aristotle's Physics and On the Heavens, as well as small parts of his Meteorologica. I am very much not an expert on Aristotle. But what I have read seems very qualitative. I know that later commentators tried to interpret Aristotle's words in a more quantitative way, but from what I have read it is not clear if Aristotle was really making quantitative statements. This is particularly true when you consider that our "modern" mathematical definitions for quantities like velocity or acceleration simply did not exist at that time.
So Aristotle seems to be saying that objects fall with a constant (instantaneous) speed. But he almost certainly didn't have a good mathematical definition for instantaneous speed. So you can read some of his statements as just saying that when you drop a certain object from a certain height, it always takes the same amount of time to fall to the ground, and therefore its average speed during its fall is always the same. But I perhaps I just haven't read enough of Aristotle, or read him closely enough, to see that he is actually saying more than this.
Likewise with the idea that heavier bodies fall faster. Later "Aristotelians" read Aristotle as saying that the speed of a body's fall is proportional to the body's weight. But it is not clear to me that he was stating such a definite quantitative relationship. And there is no doubt that, in general, heavier objects fall faster than lighter ones. I have my students drop a metal ball and wad of paper towel. The two objects are about the same size, but the metal ball is heavier. When you drop them from the same height, the metal balls hits first. The greatest challenge in doing this activity is getting the students to look at what really happens. They have all heard in school that objects fall at the same rate, so they want to claim that the metal ball and the paper wad hit at the same time. But they don't. I usually have to make them repeat the experiment and watch closely to see what really happens.
Of course, this experiment completely refutes the more quantitative notion that the speed is proportional to weight. The metal ball weighs several times what the paper wad weighs, yet it hits the ground only a fraction of a second sooner. But qualitatively, it is a fact that the metal balls hits before the paper wad.
Lest anyone worry that I'm going to convince my students that Aristotle's physics is "correct", let me reassure you that we will get to Newtonian physics by the end of the course and show that Newton does a better job of explaining things in a quantitative way. Even at the qualitative level we discuss some problems with Aristotle's ideas. He does pretty clearly indicate that no object can move without a force causing it to move (excepting natural motions like falling, which he treats differently). The classic case against this is the flight of an arrow. The bowstring pushes the arrow initially, but once the arrow leaves the bow the bowstring can't continue pushing it. So what keeps the arrow moving?
Well, it turns out that Aristotle's theory of motion implies that no vacuum can exist in Nature. Because without the resistance of some medium, an object could move with infinite speed. Aristotle found this notion ridiculous (and modern physicists would agree), so he claimed that there could be no vacuum. Well, when the arrow moves from one place to another it no longer occupies the space it used to be in. To avoid a vacuum, air (or whatever the medium is) must rush in to fill that vacated space. Aristotle felt that the air rushing in might push on the back of the arrow and keep it moving along. Not a very satisfying answer, perhaps, but it does illustrate the coherence of Aristotle's ideas. He is able to fix an apparent flaw in his theory of motion (the moving arrow) by appealing to a conclusion (Nature abhors a vacuum) that derives from that very same theory of motion.
Aristotle's cosmology has a similar coherence. His notion of the five elements (four terrestrial, one celestial) ties in very closely with his ideas about the spatial structure of the universe. And both tie in to his ideas about natural motions. With a handful of ideas he built up a pretty comprehensive picture of the world. These ideas were each subject to criticisms, but the problem was that if you tossed out one of the ideas the whole thing would start to fall apart. People don't want to throw the baby out with the bathwater, and because Aristotle's ideas were so reasonable it took a long time before people started to realize that there was no baby in this bathwater. Serious criticism of Aristotle, and new proposals about motion, began to surface during the Middle Ages and by the 14th century there was a robust new theory of motion (the impetus theory). But it was not until Galileo that someone came along with was willing to not only discard Aristotle's theory of motion (and replace it with a better, more quantitative theory) but also Aristotle's cosmology.
I have read bits and pieces of Aristotle's Physics and On the Heavens, as well as small parts of his Meteorologica. I am very much not an expert on Aristotle. But what I have read seems very qualitative. I know that later commentators tried to interpret Aristotle's words in a more quantitative way, but from what I have read it is not clear if Aristotle was really making quantitative statements. This is particularly true when you consider that our "modern" mathematical definitions for quantities like velocity or acceleration simply did not exist at that time.
So Aristotle seems to be saying that objects fall with a constant (instantaneous) speed. But he almost certainly didn't have a good mathematical definition for instantaneous speed. So you can read some of his statements as just saying that when you drop a certain object from a certain height, it always takes the same amount of time to fall to the ground, and therefore its average speed during its fall is always the same. But I perhaps I just haven't read enough of Aristotle, or read him closely enough, to see that he is actually saying more than this.
Likewise with the idea that heavier bodies fall faster. Later "Aristotelians" read Aristotle as saying that the speed of a body's fall is proportional to the body's weight. But it is not clear to me that he was stating such a definite quantitative relationship. And there is no doubt that, in general, heavier objects fall faster than lighter ones. I have my students drop a metal ball and wad of paper towel. The two objects are about the same size, but the metal ball is heavier. When you drop them from the same height, the metal balls hits first. The greatest challenge in doing this activity is getting the students to look at what really happens. They have all heard in school that objects fall at the same rate, so they want to claim that the metal ball and the paper wad hit at the same time. But they don't. I usually have to make them repeat the experiment and watch closely to see what really happens.
Of course, this experiment completely refutes the more quantitative notion that the speed is proportional to weight. The metal ball weighs several times what the paper wad weighs, yet it hits the ground only a fraction of a second sooner. But qualitatively, it is a fact that the metal balls hits before the paper wad.
Lest anyone worry that I'm going to convince my students that Aristotle's physics is "correct", let me reassure you that we will get to Newtonian physics by the end of the course and show that Newton does a better job of explaining things in a quantitative way. Even at the qualitative level we discuss some problems with Aristotle's ideas. He does pretty clearly indicate that no object can move without a force causing it to move (excepting natural motions like falling, which he treats differently). The classic case against this is the flight of an arrow. The bowstring pushes the arrow initially, but once the arrow leaves the bow the bowstring can't continue pushing it. So what keeps the arrow moving?
Well, it turns out that Aristotle's theory of motion implies that no vacuum can exist in Nature. Because without the resistance of some medium, an object could move with infinite speed. Aristotle found this notion ridiculous (and modern physicists would agree), so he claimed that there could be no vacuum. Well, when the arrow moves from one place to another it no longer occupies the space it used to be in. To avoid a vacuum, air (or whatever the medium is) must rush in to fill that vacated space. Aristotle felt that the air rushing in might push on the back of the arrow and keep it moving along. Not a very satisfying answer, perhaps, but it does illustrate the coherence of Aristotle's ideas. He is able to fix an apparent flaw in his theory of motion (the moving arrow) by appealing to a conclusion (Nature abhors a vacuum) that derives from that very same theory of motion.
Aristotle's cosmology has a similar coherence. His notion of the five elements (four terrestrial, one celestial) ties in very closely with his ideas about the spatial structure of the universe. And both tie in to his ideas about natural motions. With a handful of ideas he built up a pretty comprehensive picture of the world. These ideas were each subject to criticisms, but the problem was that if you tossed out one of the ideas the whole thing would start to fall apart. People don't want to throw the baby out with the bathwater, and because Aristotle's ideas were so reasonable it took a long time before people started to realize that there was no baby in this bathwater. Serious criticism of Aristotle, and new proposals about motion, began to surface during the Middle Ages and by the 14th century there was a robust new theory of motion (the impetus theory). But it was not until Galileo that someone came along with was willing to not only discard Aristotle's theory of motion (and replace it with a better, more quantitative theory) but also Aristotle's cosmology.
Thursday, September 20, 2012
The Round Earth
On Tuesday we did a short activity in which students used the Two Sphere model (small spherical Earth at the center, big Celestial Sphere spinning around it) to predict the motions of stars as seen from various locations on Earth. We had already done a fair bit of this using the Celestial Globes, but this gave them a chance to make predictions without having the globes in their hands and then test the predictions by watching the night sky (via Stellarium) from three different spots: North Pole, South Pole, and Equator.
These astronomical observations provide convincing evidence for a curved Earth, although technically they only indicate a North-South curvature (since we are only changing latitude and longitude doesn't affect what you see, just when you see it). So we discuss some of the other evidence for a spherical Earth: ships disappearing hull first as they sail away, the Earth's circular shadow on the moon during a lunar eclipse, the fact that lunar eclipses occur at the same (universal) time for all observers but happen later in the day (local time) for those in the east, etc.
Part of the point of this is about scientific inquiry: we don't just believe that Earth is spherical because we were told so, or because we have always seen those spherical globes. There are lots of things that we can observe which make sense if Earth is a sphere, but would be very surprising coincidences if Earth had some other shape.
But the other point is historical. The Ancient Greeks knew that Earth was spherical. Heck, they measured its diameter (we'll do that in class next week). Sometimes you hear the myth that ancient people all thought the Earth was flat, and that it was Christopher Columbus or someone like that who "discovered" that Earth was round. But Columbus got the whole idea from a medieval encyclopedia which in turn got it from the Ancient Greeks. Even Plato and Aristotle agreed that Earth was a sphere. Ptolemy gives a wonderful case for a spherical Earth at the beginning of his Almagest. So (educated) people have known the Earth was a sphere for well over two millenia now. Of course, the "educated" epithet used to apply to only a very small portion of the population. I like to think that is no longer true - although if students only think the Earth is round because they are told so, then maybe that's not much different from thinking it's flat because it looks flat. In fact, it might be worse.
Next up, physics. Aristotelian physics. If you don't understand Aristotle's physics, you won't be able to understand why the Copernican theory was rejected so strongly at first.
These astronomical observations provide convincing evidence for a curved Earth, although technically they only indicate a North-South curvature (since we are only changing latitude and longitude doesn't affect what you see, just when you see it). So we discuss some of the other evidence for a spherical Earth: ships disappearing hull first as they sail away, the Earth's circular shadow on the moon during a lunar eclipse, the fact that lunar eclipses occur at the same (universal) time for all observers but happen later in the day (local time) for those in the east, etc.
Part of the point of this is about scientific inquiry: we don't just believe that Earth is spherical because we were told so, or because we have always seen those spherical globes. There are lots of things that we can observe which make sense if Earth is a sphere, but would be very surprising coincidences if Earth had some other shape.
But the other point is historical. The Ancient Greeks knew that Earth was spherical. Heck, they measured its diameter (we'll do that in class next week). Sometimes you hear the myth that ancient people all thought the Earth was flat, and that it was Christopher Columbus or someone like that who "discovered" that Earth was round. But Columbus got the whole idea from a medieval encyclopedia which in turn got it from the Ancient Greeks. Even Plato and Aristotle agreed that Earth was a sphere. Ptolemy gives a wonderful case for a spherical Earth at the beginning of his Almagest. So (educated) people have known the Earth was a sphere for well over two millenia now. Of course, the "educated" epithet used to apply to only a very small portion of the population. I like to think that is no longer true - although if students only think the Earth is round because they are told so, then maybe that's not much different from thinking it's flat because it looks flat. In fact, it might be worse.
Next up, physics. Aristotelian physics. If you don't understand Aristotle's physics, you won't be able to understand why the Copernican theory was rejected so strongly at first.
Monday, September 17, 2012
Shadows and Gnomons
In today's lab we used my EJS Gnomon simulation to track the motion of the Sun using shadows. The simulation can display the shadow cast by a vertical stick (or a sundial gnomon that is aligned with Earth's axis, but we didn't use that feature) at any time of day, on any day of the year, from any location on Earth. My students explore how the shadow can be used to track the direction (azimuth, using the direction of the shadow) and height (altitude, using the length of the shadow and some trigonometry) of the Sun in the sky over the course of a day, and how these directions vary throughout the year. They make observations from different locations on Earth to see how this affects the patterns that they find.
They pretty quickly discover some interesting regularities. Some are fairly obvious. The Sun rises due East and sets due West only on the equinoxes, while in summer it rises northeast and sets northwest and in winter it rises southeast and sets southwest. They find that in the northern hemisphere the shadow at local noon points North. Then they find that in the Southern hemisphere the noon shadow points South. On the equator they find that it sometimes points north and sometimes points south. Once they have narrowed down the range of latitudes for which the direction can changes (23.5 degrees S to 23.5 degrees N) I try to help them make the connection to our notion of the "tropics" and specifically to the Tropic of Cancer and the Tropic of Capricorn.
Moving on they discover that the median noon altitude for the Sun over a year is just 90 degrees minus the observer's latitude. The variation in altitude is plus or minus 23.5 degrees (there it is again). Once they have the pattern down they can predict the noon altitude as a function of time of year for any latitude. They notice that the curve gets flipped in the southern hemisphere (so that highest altitudes are near the winters solstice - the southern summer). They also discover that for some latitudes on some days the noon altitude is negative, so the sun never rises on those days in those places. By exploring the latitudes for which this occurs (above 66.5 degrees N and below 66.5 degrees S) they can define the Arctic and Antarctic circles.
It's pretty cool to see students finally understand these features on the globe that they have heard about for years. They know the tropics are near the Equator where it is hot, and the Arctic/Antarctic circles are near the poles where its cold. But they have never known why those lines should be drawn at particular latitudes rather than other latitudes. Now they know. And hopefully they have 23.5 degrees burned into their brains by this point.
Everything mentioned above can be explained fully with a simple Celestial Globe model. But the best part of the lab is when they take a closer look at the noon shadows and find that even in the northern hemisphere (above the tropics) the noon shadow doesn't always point exactly north. This is because the simulation keeps time by the mean sun, so local noon occurs when the sun is on the meridian ON AVERAGE. The true sun does not move uniformly across the Celestial Sphere, so sometimes it runs ahead of the mean sun (meaning it is east of the mean sun, and thus produces a noon shadow that point just west of north) and sometimes it runs behind (it is west of the mean sun and produces a noon shadow slightly east of north). By examining the variations in the shadow direction over the course of a year, my students find that the Sun moves faster relative to the stars in winter and slower in summer. This variation in the Sun's speed across the Celestial Sphere will be important when we discuss why Hipparchus displaced the center of the Sun's orbit away from Earth (and why Copernicus displaced the center of Earth's orbit away from the Sun).
Overall I really like this lab. There is so much more that could be done with it. In the future I want to add a project in which students must set up their own gnomon and make their own shadow observations in the real world. They can't do as much as they can do with my simulation, but a handful of observations in the real world may be enough to make the virtual world seem more realistic.
I'm not the only one who teaches this stuff to college students. Joe Heafner at Catawba Valley CC has built a substantial curriculum around observing shadows, and some of his materials are available at SticksAndShadows.com. He may even be using my Gnomon simulation. Anyway, his stuff is worth checking out. He and I share a vision of teaching astronomy in a radically different way, a way that emphasizes scientific inquiry and critical thinking rather than surveying vast amounts of factual information. We go about things somewhat differently, but I think the spirit of our efforts is quite similar.
They pretty quickly discover some interesting regularities. Some are fairly obvious. The Sun rises due East and sets due West only on the equinoxes, while in summer it rises northeast and sets northwest and in winter it rises southeast and sets southwest. They find that in the northern hemisphere the shadow at local noon points North. Then they find that in the Southern hemisphere the noon shadow points South. On the equator they find that it sometimes points north and sometimes points south. Once they have narrowed down the range of latitudes for which the direction can changes (23.5 degrees S to 23.5 degrees N) I try to help them make the connection to our notion of the "tropics" and specifically to the Tropic of Cancer and the Tropic of Capricorn.
Moving on they discover that the median noon altitude for the Sun over a year is just 90 degrees minus the observer's latitude. The variation in altitude is plus or minus 23.5 degrees (there it is again). Once they have the pattern down they can predict the noon altitude as a function of time of year for any latitude. They notice that the curve gets flipped in the southern hemisphere (so that highest altitudes are near the winters solstice - the southern summer). They also discover that for some latitudes on some days the noon altitude is negative, so the sun never rises on those days in those places. By exploring the latitudes for which this occurs (above 66.5 degrees N and below 66.5 degrees S) they can define the Arctic and Antarctic circles.
It's pretty cool to see students finally understand these features on the globe that they have heard about for years. They know the tropics are near the Equator where it is hot, and the Arctic/Antarctic circles are near the poles where its cold. But they have never known why those lines should be drawn at particular latitudes rather than other latitudes. Now they know. And hopefully they have 23.5 degrees burned into their brains by this point.
Everything mentioned above can be explained fully with a simple Celestial Globe model. But the best part of the lab is when they take a closer look at the noon shadows and find that even in the northern hemisphere (above the tropics) the noon shadow doesn't always point exactly north. This is because the simulation keeps time by the mean sun, so local noon occurs when the sun is on the meridian ON AVERAGE. The true sun does not move uniformly across the Celestial Sphere, so sometimes it runs ahead of the mean sun (meaning it is east of the mean sun, and thus produces a noon shadow that point just west of north) and sometimes it runs behind (it is west of the mean sun and produces a noon shadow slightly east of north). By examining the variations in the shadow direction over the course of a year, my students find that the Sun moves faster relative to the stars in winter and slower in summer. This variation in the Sun's speed across the Celestial Sphere will be important when we discuss why Hipparchus displaced the center of the Sun's orbit away from Earth (and why Copernicus displaced the center of Earth's orbit away from the Sun).
Overall I really like this lab. There is so much more that could be done with it. In the future I want to add a project in which students must set up their own gnomon and make their own shadow observations in the real world. They can't do as much as they can do with my simulation, but a handful of observations in the real world may be enough to make the virtual world seem more realistic.
I'm not the only one who teaches this stuff to college students. Joe Heafner at Catawba Valley CC has built a substantial curriculum around observing shadows, and some of his materials are available at SticksAndShadows.com. He may even be using my Gnomon simulation. Anyway, his stuff is worth checking out. He and I share a vision of teaching astronomy in a radically different way, a way that emphasizes scientific inquiry and critical thinking rather than surveying vast amounts of factual information. We go about things somewhat differently, but I think the spirit of our efforts is quite similar.
Saturday, September 15, 2012
Night Lab
A few nights ago we had our first night lab for the course. I've always struggled a little bit about what to do for a night lab. My colleague used to have students sketch a constellation, trying to be accurate about orientation and angular separation, and then they spent the rest of the time looking through the telescope or just looking at the overall night sky. I have abandoned the constellation sketch. I think his motivation was to get them doing SOME real observations. But now I have them observing lots of things (via computer simulation) so I don't feel a need to force the constellation sketch. I also have them do real observations of the Moon as part of a project, which I will describe later.
So now the night labs consist of a tour of the night sky with special emphasis on features discussed in the course: finding Polaris, the connection between Polaris' altitude and our latitude on Earth, the apparent rotation of the night sky, the Ecliptic and the constellations of the zodiac (for this lab we mainly looked at Scorpius/Ophiuchus and Sagittarius), and I usually get to point out that any visible planets or the Moon are right near the Ecliptic. We discuss some other prominent constellations (this time Pegasus and Andromeda, Cygnus, Cassiopeia, and of course Ursa Major) as well as particularly bright stars (for this lab: Arcturus, Antares, Vega).
The rest of our time is spent looking through the telescope. I try to make sure they get to see a wide variety of things. We were able to look at Saturn and Mars just before they set. The view was pretty hazy, but good enough to clearly see Saturn's rings through the 14 inch SCT. After that we looked at two globular clusters (M 13 and M 5), two planetary nebulae (the Ring and the Dumbbell), a double star (Albireo), and a galaxy (good old Andromeda). Not a bad mix. They were all pretty impressed with the views of M 13 and the Ring Nebula. We had some trouble with the alignment of our 155 mm apochromatic refractor (RA and Dec locks kept coming loose) so we didn't get to see as much as I hoped. But we saw enough that, along with the roll-off roof and my green laser pointer, they were impressed.
And that, I think, is the real purpose of the night lab. Get them out where the sky is dark and show them how beautiful it is. Many of them had never really seen the Milky Way before. Then show them the additional beauty that is accessible via telescope. Once you start talking about what we are seeing (that galaxy is 2.5 million light years away, that cluster has half a million stars in it, that glowing gas cloud is the outer layer of a star that has been blown off, etc) it is hard for them not to be fascinated. Hearing a student say "that's the coolest thing I have ever seen" and then hearing her basically say the same thing again later in the night makes all the effort of a night lab worth while.
So now the night labs consist of a tour of the night sky with special emphasis on features discussed in the course: finding Polaris, the connection between Polaris' altitude and our latitude on Earth, the apparent rotation of the night sky, the Ecliptic and the constellations of the zodiac (for this lab we mainly looked at Scorpius/Ophiuchus and Sagittarius), and I usually get to point out that any visible planets or the Moon are right near the Ecliptic. We discuss some other prominent constellations (this time Pegasus and Andromeda, Cygnus, Cassiopeia, and of course Ursa Major) as well as particularly bright stars (for this lab: Arcturus, Antares, Vega).
The rest of our time is spent looking through the telescope. I try to make sure they get to see a wide variety of things. We were able to look at Saturn and Mars just before they set. The view was pretty hazy, but good enough to clearly see Saturn's rings through the 14 inch SCT. After that we looked at two globular clusters (M 13 and M 5), two planetary nebulae (the Ring and the Dumbbell), a double star (Albireo), and a galaxy (good old Andromeda). Not a bad mix. They were all pretty impressed with the views of M 13 and the Ring Nebula. We had some trouble with the alignment of our 155 mm apochromatic refractor (RA and Dec locks kept coming loose) so we didn't get to see as much as I hoped. But we saw enough that, along with the roll-off roof and my green laser pointer, they were impressed.
And that, I think, is the real purpose of the night lab. Get them out where the sky is dark and show them how beautiful it is. Many of them had never really seen the Milky Way before. Then show them the additional beauty that is accessible via telescope. Once you start talking about what we are seeing (that galaxy is 2.5 million light years away, that cluster has half a million stars in it, that glowing gas cloud is the outer layer of a star that has been blown off, etc) it is hard for them not to be fascinated. Hearing a student say "that's the coolest thing I have ever seen" and then hearing her basically say the same thing again later in the night makes all the effort of a night lab worth while.
Thursday, September 13, 2012
Precession of the Equinoxes
Today we tracked the Sun through the constellations, finding which constellations it goes through (the zodiac) and when it enters and leaves each one using Stellarium. We compared these to astrological dates and found that they were about a month off. Stellarium lets you go back in time, so we were able to go back to 200 BC and see that back then the dates for the Sun to be in each constellation matched up well with the astrological dates, and specifically that the vernal equinox was in Aries back then (now it is in Pisces).
We also found out that there is a 13th zodiacal constellation, Ophiuchus. In the traditional ecliptic coordinate system, Ophiuchus is considered to be part of Scorpius - but the truth is that Ophiuchus has a better claim to be a zodiacal constellation than does Scorpius. The Sun barely spends any time in Scorpius, and spends much more in Ophiuchus. But I have to admit that Scorpius is cooler than Ophiuchus, and much easier to spot in the sky. (Besides, if you go by where the Sun was on the date of my birth, I'm actually an Ophiuchus. But I won't admit to that publicly.)
Finding out that the vernal equinox used to be in Aries led us to investigate the motion of the equinoxes relative to the stars. My students found that the Celestial Equator moves relative to the stars (but the Ecliptic doesn't), and they found that it takes about 26,000 years for a full cycle of this motion (which we call precession of the equinoxes). They determined this period using both Stellarium and my Celestial Globe program. Once they had determined that the equinoxes move westward relative to the stars, they were able to figure out that the tropical year (vernal equinox to vernal equinox) is slightly shorter than the sidereal year.
I don't know the history of the discovery of precession very well. I have read that Hipparcos was the first person to notice the effect. Apparently one important factor in the discovery is that Hipparchus had access to Babylonian data from centuries before his own time. A comparison of data over such a long time scale is essential for measuring precession because precession is so slow. If you wait a century the equinoxes will move only 1/260th of the way around, or about 1.4 degrees. So without a long time scale, the changes are so small that they are hopelessly difficult to measure with naked eye instrumentation.
But just because the data were available doesn't mean that the discovery will be made. I assume that Hipparchus was very concerned about determining the length of the tropical year. The tropical year is much harder to measure directly than the sidereal year, but it is the tropical year that we really care about (since it is the year that gives us the full cycle of seasons). Hipparchus had every reason to assume that the two years were the same, but perhaps he had noticed that calendars based on the sidereal year failed to match up with the seasons after centuries of use. I really don't know the history, but something must have motivated him to look for this tiny effect.
In any case, it is precession that explains the difference between the dates for the Sun entering/exiting the constellations and the horoscope/astrological dates. The astrological "signs" were established at least by the time of Ptolemy's Tetrabiblos (which became the authoritative source on astrology, just as Ptolemy's Almagest became the authoritative source on astronomy) around AD 150. I suspect they date back further, maybe to the time of Hipparchus in about 130 BC, but I don't know that history very well either. In the intervening two millennia since then, precession has moved the equinoxes around a bit.
In my class we won't spend much time talking about precession, except when we discuss theoretical explanations for it (a rotation of the ninth sphere, Copernicus' imperfectly synchronized motions of Earth). But in practice it was an important issue for astronomers because one had to correct for precession to make high precision astrometrical measurements. This issue plays a more important role in my galaxies course than it does in my Copernican Revolution course, so maybe I'll write more about it in the Spring.
We also found out that there is a 13th zodiacal constellation, Ophiuchus. In the traditional ecliptic coordinate system, Ophiuchus is considered to be part of Scorpius - but the truth is that Ophiuchus has a better claim to be a zodiacal constellation than does Scorpius. The Sun barely spends any time in Scorpius, and spends much more in Ophiuchus. But I have to admit that Scorpius is cooler than Ophiuchus, and much easier to spot in the sky. (Besides, if you go by where the Sun was on the date of my birth, I'm actually an Ophiuchus. But I won't admit to that publicly.)
Finding out that the vernal equinox used to be in Aries led us to investigate the motion of the equinoxes relative to the stars. My students found that the Celestial Equator moves relative to the stars (but the Ecliptic doesn't), and they found that it takes about 26,000 years for a full cycle of this motion (which we call precession of the equinoxes). They determined this period using both Stellarium and my Celestial Globe program. Once they had determined that the equinoxes move westward relative to the stars, they were able to figure out that the tropical year (vernal equinox to vernal equinox) is slightly shorter than the sidereal year.
I don't know the history of the discovery of precession very well. I have read that Hipparcos was the first person to notice the effect. Apparently one important factor in the discovery is that Hipparchus had access to Babylonian data from centuries before his own time. A comparison of data over such a long time scale is essential for measuring precession because precession is so slow. If you wait a century the equinoxes will move only 1/260th of the way around, or about 1.4 degrees. So without a long time scale, the changes are so small that they are hopelessly difficult to measure with naked eye instrumentation.
But just because the data were available doesn't mean that the discovery will be made. I assume that Hipparchus was very concerned about determining the length of the tropical year. The tropical year is much harder to measure directly than the sidereal year, but it is the tropical year that we really care about (since it is the year that gives us the full cycle of seasons). Hipparchus had every reason to assume that the two years were the same, but perhaps he had noticed that calendars based on the sidereal year failed to match up with the seasons after centuries of use. I really don't know the history, but something must have motivated him to look for this tiny effect.
In any case, it is precession that explains the difference between the dates for the Sun entering/exiting the constellations and the horoscope/astrological dates. The astrological "signs" were established at least by the time of Ptolemy's Tetrabiblos (which became the authoritative source on astrology, just as Ptolemy's Almagest became the authoritative source on astronomy) around AD 150. I suspect they date back further, maybe to the time of Hipparchus in about 130 BC, but I don't know that history very well either. In the intervening two millennia since then, precession has moved the equinoxes around a bit.
In my class we won't spend much time talking about precession, except when we discuss theoretical explanations for it (a rotation of the ninth sphere, Copernicus' imperfectly synchronized motions of Earth). But in practice it was an important issue for astronomers because one had to correct for precession to make high precision astrometrical measurements. This issue plays a more important role in my galaxies course than it does in my Copernican Revolution course, so maybe I'll write more about it in the Spring.
Wednesday, September 12, 2012
The Mystery of the Planets
Yesterday my class observed the motion of the planets against the starry background using Stellarium. In part this was an exercise in quantitative astronomy. They learned how to measure the synodic and zodiacal periods of the planets, and they will need to know how to make such measurements when they begin examining their personalized solar system later in the semester.
But the first part of the activity was qualitative, and this is where they were in for a big shock. As they follow, say, Mars eastward across the stars they find that it suddenly stops, turns around and goes west for a bit, and then resumes its usual eastward motion. This retrograde motion must have perplexed the ancients to no end, and my even my students with their modern educations ask me "why does it do that??!!" It is one of the most satisfying moments of the course, especially when I get to answer their questions with "we will spend the rest of the semester answering the question you just asked."
These qualitative observations also help us to establish the first major classification of planets. Mercury and Venus exhibit similar qualitative behavior. They both remain relatively close to the Sun at all times. During their retrograde motion they pass the Sun, so that at the center of their retrograde they are in conjunction (near the Sun in the sky). In contrast, Mars, Jupiter, and Saturn have a different type of motion. They can be far away from the Sun, even on the opposite side of the sky (in opposition). In fact, it is when they are in opposition that they do their retrograde thing. Curiously, they are brightest at this time as well (although the effect is only dramatic for Mars). These two different types of motion lead to two different classifications: Mercury and Venus are "inferior planets" and Mars, Jupiter, and Saturn are "superior planets".
I'm actually very interested in the origin of these terms. I know that eventually ancient astronomers settled on a system in which Mercury and Venus were below the Sun, while Mars, Jupiter, and Saturn were above the Sun. But not all ancient astronomers used this ordering. Eventually, Copernicus would shift the meaning of inferior and superior, but I'm very curious as to how these terms first came to be used, long before Copernicus.
One thing I have found in teaching this material is that modern students jump easily to what was known as the Capellan system: with Mercury and Venus orbiting the Sun, but no such claim about the other planets. This may be partly because the high-speed view you can get from Stellarium is very suggestive of this arrangement, but I think it is also a product of their schooling. They have been told all their lives that planets orbit the Sun, so when they see Mercury and Venus running back and forth across the Sun in the sky it is very easy for them to connect this to what they "know." They are just seeing what they expected to see. But it is harder with the superior planets. The motions of the superior planets are tied to the Sun in a much subtler way, and even that subtle connection does not really suggest that these planets are orbiting the Sun.
For now, I want the motion of the planets to remain a mystery for my students. We have established some empirical facts (qualitative and quantitative), and now we must come up with some rules that will help to explain these facts. As I said above, we'll spend the rest of the course doing just that, with history as our guide.
But the first part of the activity was qualitative, and this is where they were in for a big shock. As they follow, say, Mars eastward across the stars they find that it suddenly stops, turns around and goes west for a bit, and then resumes its usual eastward motion. This retrograde motion must have perplexed the ancients to no end, and my even my students with their modern educations ask me "why does it do that??!!" It is one of the most satisfying moments of the course, especially when I get to answer their questions with "we will spend the rest of the semester answering the question you just asked."
These qualitative observations also help us to establish the first major classification of planets. Mercury and Venus exhibit similar qualitative behavior. They both remain relatively close to the Sun at all times. During their retrograde motion they pass the Sun, so that at the center of their retrograde they are in conjunction (near the Sun in the sky). In contrast, Mars, Jupiter, and Saturn have a different type of motion. They can be far away from the Sun, even on the opposite side of the sky (in opposition). In fact, it is when they are in opposition that they do their retrograde thing. Curiously, they are brightest at this time as well (although the effect is only dramatic for Mars). These two different types of motion lead to two different classifications: Mercury and Venus are "inferior planets" and Mars, Jupiter, and Saturn are "superior planets".
I'm actually very interested in the origin of these terms. I know that eventually ancient astronomers settled on a system in which Mercury and Venus were below the Sun, while Mars, Jupiter, and Saturn were above the Sun. But not all ancient astronomers used this ordering. Eventually, Copernicus would shift the meaning of inferior and superior, but I'm very curious as to how these terms first came to be used, long before Copernicus.
One thing I have found in teaching this material is that modern students jump easily to what was known as the Capellan system: with Mercury and Venus orbiting the Sun, but no such claim about the other planets. This may be partly because the high-speed view you can get from Stellarium is very suggestive of this arrangement, but I think it is also a product of their schooling. They have been told all their lives that planets orbit the Sun, so when they see Mercury and Venus running back and forth across the Sun in the sky it is very easy for them to connect this to what they "know." They are just seeing what they expected to see. But it is harder with the superior planets. The motions of the superior planets are tied to the Sun in a much subtler way, and even that subtle connection does not really suggest that these planets are orbiting the Sun.
For now, I want the motion of the planets to remain a mystery for my students. We have established some empirical facts (qualitative and quantitative), and now we must come up with some rules that will help to explain these facts. As I said above, we'll spend the rest of the course doing just that, with history as our guide.
Monday, September 10, 2012
The Moon
Today in class we studied the Moon. The Moon plays an odd role in the history of astronomy. The regular, repeating progression of the Moon's phases must have been one of the first signs of order that ancient humans found in the night sky. This "law of nature" was likely discovered even before the seasonal variations of the Sun, although perhaps after the regularity of the daily motion of the sky. In any case, it is surely something that the first humans to pay attention to the sky would have picked up on quickly.
The explanation for the phases, in terms of the Moon reflecting Sunlight, may have taken longer since there is some basic geometry involved. But even this is not a great challenge and certainly the Ancient Greeks had a complete handle on how the Moon's phases worked (even though they were not entirely sure that the Moon shone by reflecting sunlight). My students learn how the phases work by playing with my EJS Phases of the Moon program.
But the Moon's motion relative to the stars is tricky. At first glance it seems simple: the Moon drifts eastward relative to the stars just like the Sun does, only faster (13 degrees per day, compared to the Sun's one degree per day). My students find this for themselves using Stellarium. But whereas the Sun follows a repetitive circular path through the Celestial Sphere (the Ecliptic), the Moon never quite repeats its motion. It moves roughly along the Ecliptic, but not exactly. Sometimes it is above, sometimes below. So while the rough outline of the Moon's motion is pretty easy, the details of its motion are hard.
Really hard. Ptolemy was heavily criticized for his theory of the Moon, because in order to explain the quirky motion of the Moon across the Celestial Sphere Ptolemy had used a model in which the distance of the Moon from Earth changed by a factor of 2 from minimum to maximum. This would mean that the Moon's apparent diameter should sometimes be twice what it is at other times - and this just doesn't happen (the apparent diameter does vary, but only by a tiny bit). Copernicus came up with a new and improved theory of the Moon (with not one, but TWO epicycles) and in some ways his theory of the Moon was the most popular part of his De Revolutionibus. Tycho developed a new lunar theory, etc, etc.
With most aspects of solar system astronomy, this trend of improving theories ends with Newton. But not so with the Moon. Newton got the physics right, but the problem of the Moon being simultaneously attracted by Earth and Sun was too difficult for even the great Newton to solve. He could only give a rough approximation to the solution. This approximation was later refined by Euler, Clairault, and d'Alembert. Eventually all of this culminated in the Hill-Brown theory of the early 20th century. Even that has been superseded in recent times by calculations using computers. So in some sense, the quest to predict the Moon's motion is still ongoing.
For my class, we content ourselves with the rough view: understanding the phases, defining sidereal and synodic months, determining the Moon's average rate of motion relative to the stars and to the Sun, determining approximate rising/setting times for various Moon phases, and such. But for ancient and early modern astronomers this was not enough, for one very particular reason: they wanted to be able to predict lunar and solar eclipses. To do this one needed an accurate theory of the Moon's motion, and as we have seen this was a long time coming. The problem was that even a small error could make the difference between an eclipse occurring or not occurring. Still, those astronomers were able to do well enough to get it right most of the time. And they certainly understood why we don't get a lunar eclipse at every full moon (or a solar eclipse at every new moon), which is something my class is learning too! (To learn about this we use Mario Belloni's EJS Solar and Lunar Eclipse program.)
The explanation for the phases, in terms of the Moon reflecting Sunlight, may have taken longer since there is some basic geometry involved. But even this is not a great challenge and certainly the Ancient Greeks had a complete handle on how the Moon's phases worked (even though they were not entirely sure that the Moon shone by reflecting sunlight). My students learn how the phases work by playing with my EJS Phases of the Moon program.
But the Moon's motion relative to the stars is tricky. At first glance it seems simple: the Moon drifts eastward relative to the stars just like the Sun does, only faster (13 degrees per day, compared to the Sun's one degree per day). My students find this for themselves using Stellarium. But whereas the Sun follows a repetitive circular path through the Celestial Sphere (the Ecliptic), the Moon never quite repeats its motion. It moves roughly along the Ecliptic, but not exactly. Sometimes it is above, sometimes below. So while the rough outline of the Moon's motion is pretty easy, the details of its motion are hard.
Really hard. Ptolemy was heavily criticized for his theory of the Moon, because in order to explain the quirky motion of the Moon across the Celestial Sphere Ptolemy had used a model in which the distance of the Moon from Earth changed by a factor of 2 from minimum to maximum. This would mean that the Moon's apparent diameter should sometimes be twice what it is at other times - and this just doesn't happen (the apparent diameter does vary, but only by a tiny bit). Copernicus came up with a new and improved theory of the Moon (with not one, but TWO epicycles) and in some ways his theory of the Moon was the most popular part of his De Revolutionibus. Tycho developed a new lunar theory, etc, etc.
With most aspects of solar system astronomy, this trend of improving theories ends with Newton. But not so with the Moon. Newton got the physics right, but the problem of the Moon being simultaneously attracted by Earth and Sun was too difficult for even the great Newton to solve. He could only give a rough approximation to the solution. This approximation was later refined by Euler, Clairault, and d'Alembert. Eventually all of this culminated in the Hill-Brown theory of the early 20th century. Even that has been superseded in recent times by calculations using computers. So in some sense, the quest to predict the Moon's motion is still ongoing.
For my class, we content ourselves with the rough view: understanding the phases, defining sidereal and synodic months, determining the Moon's average rate of motion relative to the stars and to the Sun, determining approximate rising/setting times for various Moon phases, and such. But for ancient and early modern astronomers this was not enough, for one very particular reason: they wanted to be able to predict lunar and solar eclipses. To do this one needed an accurate theory of the Moon's motion, and as we have seen this was a long time coming. The problem was that even a small error could make the difference between an eclipse occurring or not occurring. Still, those astronomers were able to do well enough to get it right most of the time. And they certainly understood why we don't get a lunar eclipse at every full moon (or a solar eclipse at every new moon), which is something my class is learning too! (To learn about this we use Mario Belloni's EJS Solar and Lunar Eclipse program.)
Saturday, September 8, 2012
Westman's The Copernican Question
About a week ago I finished reading Bob Westman's monumental book The Copernican Question. The book is jam-packed with scholarship and was very thought-provoking. I could write a lot about it, but I don't have time. I just want to jot down the two major takeaways I got from reading Westman's book.
The first has to do with how important astrology was in the development of Copernican astronomy. These days we tend to think of astronomy as the supremely UNpractical science. Astronomy is the poster child of "fundamental science". Modern astronomy is all about understanding the cosmos, not building a better mousetrap (although NASA will always claim that the space program helps us develop new technologies, and some folks still dream of human colonies on Mars, etc). But back in the 16th century there was a practical side to astronomy. Partly this had to do with basic issues of seasonal change, which has to do with the apparent motions of the Sun and can be well understood with the good old Celestial Sphere model. But another practical aspect to astronomy in that age was astrological prognostication, which intimately involved the motion and ORDER of the planets.
Westman points out that Copernicus' work came at a time when astrology was under attack, particularly by Pico de Mirandola. Pico's criticisms were many, but among them he noted that astronomical predictions were not of sufficient accuracy to allow for accurate astrological prognostication. He also noted that astronomers weren't even certain of the ordering of the planets, or how far they lay from Earth. Westman suggests that Copernicus' De Revolutionibus can be seen as a response to these ASTRONOMICAL aspects of Pico's attack on astrology. So it may be the case that Copernican astronomy was an attempt to put astrology on surer footing.
The subsequent development of Copernican astronomy was also closely tied to astrology. Some astronomers who favored Copernicus' ideas disavowed astrology (Maestlin), some practiced astrology but did not really write much about it (Galileo), and some saw Copernican astronomy as the foundation for a reform of astrological practice (Kepler). But any attempt to reform astronomy could not, in that day, completely ignore the practical applications of astronomy in the form of astrology.
The other major thing I learned from Westman's book is just how divergent the early "Copernicans" were in their views. So much so that Westman claims the term "Copernican" doesn't even make sense if applied to these people as a group. Although these people may have shared a commitment to a model of the solar system in which the Sun was stationary near the center, they diverged on nearly every other question about astronomy. Some believed in a finite universe (Kepler) some believed in an infinite universe filled with populated worlds (Bruno). Some thought physics should form the basis of a reformed astronomy (Kepler), some thought astronomy could be used to motivate a reformation of physics (Galileo), and some thought physics and astronomy should be kept entirely separate (Maestlin). Some never even clearly stated whether they thought the Earth orbited the Sun or not (Gilbert).
This second lesson speaks to me because of some recent experiences I have had in trying to reform something. The opponents of reform have a unity of vision that is lacking in the proponents of reform. Those who oppose reform are armed not only with criticisms of the reform effort, but also with a solidarity in support of the existing situation. What we have is good, and here is why your suggested change is bad. On the other hand, the proponents of reform often want reform for a host of divergent reasons. They may have completely different criticisms of the status quo. They may be excited about totally different parts of the proposal for reform. In short, they are much less unified than the supporters of the status quo. I think this is a big reason why change is so hard. And maybe that's the way it should be.
Thankfully, though, it is possible for divergent views to coalesce behind a a really strong proposal for reform. And so Copernican astronomy (as reconceived by Kepler and again by Newton) ultimately prevailed.
The first has to do with how important astrology was in the development of Copernican astronomy. These days we tend to think of astronomy as the supremely UNpractical science. Astronomy is the poster child of "fundamental science". Modern astronomy is all about understanding the cosmos, not building a better mousetrap (although NASA will always claim that the space program helps us develop new technologies, and some folks still dream of human colonies on Mars, etc). But back in the 16th century there was a practical side to astronomy. Partly this had to do with basic issues of seasonal change, which has to do with the apparent motions of the Sun and can be well understood with the good old Celestial Sphere model. But another practical aspect to astronomy in that age was astrological prognostication, which intimately involved the motion and ORDER of the planets.
Westman points out that Copernicus' work came at a time when astrology was under attack, particularly by Pico de Mirandola. Pico's criticisms were many, but among them he noted that astronomical predictions were not of sufficient accuracy to allow for accurate astrological prognostication. He also noted that astronomers weren't even certain of the ordering of the planets, or how far they lay from Earth. Westman suggests that Copernicus' De Revolutionibus can be seen as a response to these ASTRONOMICAL aspects of Pico's attack on astrology. So it may be the case that Copernican astronomy was an attempt to put astrology on surer footing.
The subsequent development of Copernican astronomy was also closely tied to astrology. Some astronomers who favored Copernicus' ideas disavowed astrology (Maestlin), some practiced astrology but did not really write much about it (Galileo), and some saw Copernican astronomy as the foundation for a reform of astrological practice (Kepler). But any attempt to reform astronomy could not, in that day, completely ignore the practical applications of astronomy in the form of astrology.
The other major thing I learned from Westman's book is just how divergent the early "Copernicans" were in their views. So much so that Westman claims the term "Copernican" doesn't even make sense if applied to these people as a group. Although these people may have shared a commitment to a model of the solar system in which the Sun was stationary near the center, they diverged on nearly every other question about astronomy. Some believed in a finite universe (Kepler) some believed in an infinite universe filled with populated worlds (Bruno). Some thought physics should form the basis of a reformed astronomy (Kepler), some thought astronomy could be used to motivate a reformation of physics (Galileo), and some thought physics and astronomy should be kept entirely separate (Maestlin). Some never even clearly stated whether they thought the Earth orbited the Sun or not (Gilbert).
This second lesson speaks to me because of some recent experiences I have had in trying to reform something. The opponents of reform have a unity of vision that is lacking in the proponents of reform. Those who oppose reform are armed not only with criticisms of the reform effort, but also with a solidarity in support of the existing situation. What we have is good, and here is why your suggested change is bad. On the other hand, the proponents of reform often want reform for a host of divergent reasons. They may have completely different criticisms of the status quo. They may be excited about totally different parts of the proposal for reform. In short, they are much less unified than the supporters of the status quo. I think this is a big reason why change is so hard. And maybe that's the way it should be.
Thankfully, though, it is possible for divergent views to coalesce behind a a really strong proposal for reform. And so Copernican astronomy (as reconceived by Kepler and again by Newton) ultimately prevailed.
Tuesday, September 4, 2012
Celestial Globetrotting
Today my students got their first chance to work with a celestial globe. A celestial globe is basically the embodiment of the Celestial Sphere theory, along with a theory of the Sun's motion that has the Sun moving along the ecliptic at a uniform rate over the course of a year. Given a particular date you can set the position of the Sun on the globe so that it is in the right place among the stars (it has the correct right ascension and declination). You can tilt the axis of the globe to match the latitude of an observer on Earth. Then you can rotate the globe around its axis to orient it for a particular time of day. In this way you can determine the position of the Sun and brighter stars for any observer on Earth on any day of the year at any time of day. It's pretty incredible that such a simple device can tell you so much - but that is exactly the power of a good theory.
My students mostly use the globe to gain more familiarity with the Celestial Sphere theory and the motions of the Sun (what used to be called "spherics"). But my interest in the globe is in the fact that it is a device that embodies theoretical knowledge. I heard a talk a few years ago about embodied knowledge and I was intrigued. There are certainly many astronomical devices that seem to embody theoretical knowledge. The Celestial Globe, or its ancient cousin the armillary sphere, fit the bill. So do sundials, orreries, astrolabes, and the famous Antikythera mechanism.
My response to this idea of embodied knowledge is to wonder to what extent the device itself really embodies knowledge, or to what extent it just serves as an aid to recall knowledge already possessed by the user (and even more so by the maker). I think in most cases the answer lies somewhere between. Without a celestial globe or something like it, I certainly couldn't tell you at what time Aldebaran will rise on May 4 as seen from Rome, Georgia. But with the device I can tell you quite quickly. So there is some knowledge embedded in the device itself that is not already in my mind. At the same time, if I gave a celestial globe to the "man on the street" and said tell me when Aldebaran rises, etc, I doubt he would have much success. It is not enough to the device to embody the knowledge - the user must also understand HOW the device embodies the knowledge, and thus how to extract useful information from the device.
With the celestial globe some of this information is pretty obvious. There are stars and constellations etched onto the plastic surface of the globe, and that embodies knowledge about the arrangement of stars in our sky in a way that almost anyone could interpret without any training. But the other parts of the globe are harder to use. Could an untrained person figure out how to use the thing without guidance? What if the globe was improperly assembled, or if it was broken? Could the untrained user identify the problem? If not, then clearly there are critical parts of the knowledge that is supposedly embodies in the globe which really lie in the mind of the user.
(As a side note: I ordered some replica armillary spheres to show my class. They were ALL improperly assembled. I notified the company, and they seemed unaware of the problem which meant that everyone who had bought one of their armillary spheres in the past didn't really know how to use one. It was just a decoration. Did the armillary sphere embody any astronomical knowledge for these folks? Probably not.)
So I'm still intrigues by the question, but I'm convinced that it presents a false dichotomy. Devices can surely embody knowledge in some way, as shown by the celestial globe. But any device can only embody knowledge in a meaningful way if there are users who know how to extract useful knowledge from the device. Makes me wonder about all the computer simulations I have created. They could be said to embody theoretical knowledge. But if I burn the programs onto a CD and then wait 20 years and find that no computer can run the programs any more (or read the CD!) then can they still be said to embody knowledge? Not in any way that I find worth thinking about.
Speaking of computer simulations, if you want to check out a virtual version of the Celestial Sphere model (with the drifting Sun), see my EJS Celestial Globe Model.
My students mostly use the globe to gain more familiarity with the Celestial Sphere theory and the motions of the Sun (what used to be called "spherics"). But my interest in the globe is in the fact that it is a device that embodies theoretical knowledge. I heard a talk a few years ago about embodied knowledge and I was intrigued. There are certainly many astronomical devices that seem to embody theoretical knowledge. The Celestial Globe, or its ancient cousin the armillary sphere, fit the bill. So do sundials, orreries, astrolabes, and the famous Antikythera mechanism.
My response to this idea of embodied knowledge is to wonder to what extent the device itself really embodies knowledge, or to what extent it just serves as an aid to recall knowledge already possessed by the user (and even more so by the maker). I think in most cases the answer lies somewhere between. Without a celestial globe or something like it, I certainly couldn't tell you at what time Aldebaran will rise on May 4 as seen from Rome, Georgia. But with the device I can tell you quite quickly. So there is some knowledge embedded in the device itself that is not already in my mind. At the same time, if I gave a celestial globe to the "man on the street" and said tell me when Aldebaran rises, etc, I doubt he would have much success. It is not enough to the device to embody the knowledge - the user must also understand HOW the device embodies the knowledge, and thus how to extract useful information from the device.
With the celestial globe some of this information is pretty obvious. There are stars and constellations etched onto the plastic surface of the globe, and that embodies knowledge about the arrangement of stars in our sky in a way that almost anyone could interpret without any training. But the other parts of the globe are harder to use. Could an untrained person figure out how to use the thing without guidance? What if the globe was improperly assembled, or if it was broken? Could the untrained user identify the problem? If not, then clearly there are critical parts of the knowledge that is supposedly embodies in the globe which really lie in the mind of the user.
(As a side note: I ordered some replica armillary spheres to show my class. They were ALL improperly assembled. I notified the company, and they seemed unaware of the problem which meant that everyone who had bought one of their armillary spheres in the past didn't really know how to use one. It was just a decoration. Did the armillary sphere embody any astronomical knowledge for these folks? Probably not.)
So I'm still intrigues by the question, but I'm convinced that it presents a false dichotomy. Devices can surely embody knowledge in some way, as shown by the celestial globe. But any device can only embody knowledge in a meaningful way if there are users who know how to extract useful knowledge from the device. Makes me wonder about all the computer simulations I have created. They could be said to embody theoretical knowledge. But if I burn the programs onto a CD and then wait 20 years and find that no computer can run the programs any more (or read the CD!) then can they still be said to embody knowledge? Not in any way that I find worth thinking about.
Speaking of computer simulations, if you want to check out a virtual version of the Celestial Sphere model (with the drifting Sun), see my EJS Celestial Globe Model.
Monday, September 3, 2012
Is there a scientific method?
So on my first day of class this semester I told my students that there is no such thing as the scientific method. We talked about the 5-step process they learned in high school (and which, coincidentally, my 9-year old son is learning this year in his 4th grade class), and how science isn't really like that. I told them they would get to learn about real scientists and how they did what they did, and they would see that science is much too complicated and messy (and exciting!) to be contained within some recipe with a handful of steps.
More recently my son's 4th grade teacher (who is awesome, by the way) asked me to look over her unit review for science, and one of the things on the review was the 5 step scientific method. My response to her was that while I believe there IS no scientific method, this 5 step process is a good place to start for 4th graders. And I really think she is right to use it with those kids. My college students need to learn that reality is more complicated, just as they must do in their history, government, English, etc, courses.
So while I endorse the use of this 5-step so-called "scientific method" in grade school, I think there are a few problems with labeling this process with that title. First of all, it is really just evidence-based reasoning, a combination of empiricism and deductive logic. This process is, surely, of great importance in science. But it is by no means exclusive to the things we think of as science. Historians use the same methods, and so do police detectives and auto mechanics. We all use it in our day-to-day lives, like when we are trying to figure out where we left the car keys. If using this method makes one a scientist then we are all scientists (which is a conclusion I would be happy to embrace).
Second, it is simply a fact that much that is done in professional science (astronomy, physics, chemistry, biology, etc) does not follow this method at all. I've probably never followed it myself, and while I'm no great scientist I have published in respected journals, etc. Theoretical physics does not lend itself to description by this method, and even experimental physics often violates the norms of this simple process. Often we already have the data and we just try to come up with some way, ANY way, to make sense of it.
Third, the recipe makes the "formulate a hypothesis" step seem way too easy. Most of the time it is extremely difficult to formulate a hypothesis to explain some phenomenon without that hypothesis being obviously wrong. It takes inspiration to come up with good ideas that are even worth pursuing. And who knows where these ideas come from? It takes creativity to come up with a good hypothesis. Trying to encapsulate this in a recipe is like having a recipe for a birthday cake that goes: "Step 1: bake a delicious cake. Step 2: coat with icing."
For that matter, even the first step, which is usually stated as "ask a question" or "recognize a problem" or something along those lines, is really hard. Often it takes creativity and incredible insight just to be able to see that a problem exists, or that there is an interesting question to be asked. Of course, the 4th graders aren't dealing with questions of this sort and so this method works for them. But Copernicus, Kepler, Galileo, et al WERE dealing with these tough questions and so my students need to know that the 5-step method won't work for the Copernican Revolution.
Finally, my biggest problem with the 5-step method is that it makes science seem boring. This isn't a problem for 4th graders. They LOVE science. They haven't been conditioned by boring textbooks full of authoritative facts and no discussion of how those facts were found. But by the time students get to college their love for science has been beaten out of them. They might be perfectly willing to believe that science can be done by robots mindlessly carrying out the steps of The Method. They need to see the human side of science, and part of that is finding out that science is not all that methodical.
So, no, I don't believe there is a scientific method. Scientists solve their problems using the approach coined by Jean-Paul Sartre and advocated by Malcolm X: by any means necessary. College students need to know this. It may actually give them a greater appreciation for science and scientists. But for 4th graders, the good ole' "scientific method" is a great place to start.
More recently my son's 4th grade teacher (who is awesome, by the way) asked me to look over her unit review for science, and one of the things on the review was the 5 step scientific method. My response to her was that while I believe there IS no scientific method, this 5 step process is a good place to start for 4th graders. And I really think she is right to use it with those kids. My college students need to learn that reality is more complicated, just as they must do in their history, government, English, etc, courses.
So while I endorse the use of this 5-step so-called "scientific method" in grade school, I think there are a few problems with labeling this process with that title. First of all, it is really just evidence-based reasoning, a combination of empiricism and deductive logic. This process is, surely, of great importance in science. But it is by no means exclusive to the things we think of as science. Historians use the same methods, and so do police detectives and auto mechanics. We all use it in our day-to-day lives, like when we are trying to figure out where we left the car keys. If using this method makes one a scientist then we are all scientists (which is a conclusion I would be happy to embrace).
Second, it is simply a fact that much that is done in professional science (astronomy, physics, chemistry, biology, etc) does not follow this method at all. I've probably never followed it myself, and while I'm no great scientist I have published in respected journals, etc. Theoretical physics does not lend itself to description by this method, and even experimental physics often violates the norms of this simple process. Often we already have the data and we just try to come up with some way, ANY way, to make sense of it.
Third, the recipe makes the "formulate a hypothesis" step seem way too easy. Most of the time it is extremely difficult to formulate a hypothesis to explain some phenomenon without that hypothesis being obviously wrong. It takes inspiration to come up with good ideas that are even worth pursuing. And who knows where these ideas come from? It takes creativity to come up with a good hypothesis. Trying to encapsulate this in a recipe is like having a recipe for a birthday cake that goes: "Step 1: bake a delicious cake. Step 2: coat with icing."
For that matter, even the first step, which is usually stated as "ask a question" or "recognize a problem" or something along those lines, is really hard. Often it takes creativity and incredible insight just to be able to see that a problem exists, or that there is an interesting question to be asked. Of course, the 4th graders aren't dealing with questions of this sort and so this method works for them. But Copernicus, Kepler, Galileo, et al WERE dealing with these tough questions and so my students need to know that the 5-step method won't work for the Copernican Revolution.
Finally, my biggest problem with the 5-step method is that it makes science seem boring. This isn't a problem for 4th graders. They LOVE science. They haven't been conditioned by boring textbooks full of authoritative facts and no discussion of how those facts were found. But by the time students get to college their love for science has been beaten out of them. They might be perfectly willing to believe that science can be done by robots mindlessly carrying out the steps of The Method. They need to see the human side of science, and part of that is finding out that science is not all that methodical.
So, no, I don't believe there is a scientific method. Scientists solve their problems using the approach coined by Jean-Paul Sartre and advocated by Malcolm X: by any means necessary. College students need to know this. It may actually give them a greater appreciation for science and scientists. But for 4th graders, the good ole' "scientific method" is a great place to start.
Saturday, September 1, 2012
Motions of the Sun
So in my last class we started learning about the motion of the Sun. I have them measure the transit time for a star (23 hours, 56 minutes) and then the transit time for the Sun (the expected 24 hours). We think a bit about relative motion and realize that both the stars and Sun are moving Westward across our sky, but the stars are moving faster (since they go all the way around in a shorter time). Therefore the Sun is moving Eastward relative to the stars. The concept of relative motion is challenging if you haven't thought about it before, but I usually talk about driving down the highway heading West and passing another car. To you, that car appears to be moving backward (Eastward). So the motion of that car relative to you is Eastward. They usually get that.
But the Sun now throws a wrench into our wonderful Celestial Sphere theory. ALL of the stars move as though they are attached to our Celestial Sphere - but the Sun does not. Therefore we must posit a new motion for the Sun. Does this represent an ad hoc modification of our Celestial Sphere theory? According to some philosophers of science, like Karl Popper, we shouldn't make ad hoc adjustments to our theory when we encounter an anomaly like the Sun. Should we, instead, just give up on our Celestial Sphere?
There are a few reasons why we shouldn't. The first involves the idea of classification. The stars all look very similar: tiny points of light that are only visible in the night sky. But the Sun appears to be a fundamentally different critter (forget what you've been told for the time being). It is much brighter, is noticeably disk-shaped rather than point-like, and it lights up the day so much that we can't even see the stars when the Sun is in the sky. Why should we expect the Sun to follow the same rules as the stars when it is obviously a very different thing.
This classification helps us to limit the scope of the Celestial Sphere theory (to the stars, but not the Sun) without having to toss out the whole thing. This is really what classification is all about. We classify things because certain groups of objects follow certain rules that others don't follow. To extend our Game of Science analogy, think about chess. All the pawns move the same way, but the thing that looks like a horse moves in a very different way. It looks different from the pawns, so maybe we shouldn't expect the knight to move the way the pawns do. Some deal with the stars and Sun.
The second reason we wouldn't want to throw out our Celestial Sphere theory is that it actually helps to explain the Sun's motion, even though it doesn't fully explain that motion. The Sun's actual motion across the sky is not quite circular like that of the stars. Instead, the Sun spirals (albeit very slowly). This spiraling motion, viewed in full, would appear quite complicated. But if we start off with the spinning Celestial Sphere, we find that we can describe the Sun's spiraling motion by simply having the Sun move slowly along the Celestial Sphere on a circular path. It is the combination of the circular motion of the Celestial Sphere (which takes 23 hours, 56 minutes to complete) and the circular motion of the Sun along the Celestial Sphere (which takes a year to complete) that produces the observed spiral motion of the Sun. If we don't start with the Celestial Sphere, the Sun's motion would be much harder to model.
My class will continue exploring this model (Celestial Sphere plus moving Sun) next week. This simple model explains much that is of great importance to humans, which is why the study of the "The Sphere" was the focus of the medieval and early modern astronomy curriculum in the universities. The classic work was the De Sphaera of Johannes de Sacrobosco (John of Holywood), but many students learned "spherics" from later commentaries which elaborated on Sacrobosco's book. We will be finished with spherics in another week or so!
But the Sun now throws a wrench into our wonderful Celestial Sphere theory. ALL of the stars move as though they are attached to our Celestial Sphere - but the Sun does not. Therefore we must posit a new motion for the Sun. Does this represent an ad hoc modification of our Celestial Sphere theory? According to some philosophers of science, like Karl Popper, we shouldn't make ad hoc adjustments to our theory when we encounter an anomaly like the Sun. Should we, instead, just give up on our Celestial Sphere?
There are a few reasons why we shouldn't. The first involves the idea of classification. The stars all look very similar: tiny points of light that are only visible in the night sky. But the Sun appears to be a fundamentally different critter (forget what you've been told for the time being). It is much brighter, is noticeably disk-shaped rather than point-like, and it lights up the day so much that we can't even see the stars when the Sun is in the sky. Why should we expect the Sun to follow the same rules as the stars when it is obviously a very different thing.
This classification helps us to limit the scope of the Celestial Sphere theory (to the stars, but not the Sun) without having to toss out the whole thing. This is really what classification is all about. We classify things because certain groups of objects follow certain rules that others don't follow. To extend our Game of Science analogy, think about chess. All the pawns move the same way, but the thing that looks like a horse moves in a very different way. It looks different from the pawns, so maybe we shouldn't expect the knight to move the way the pawns do. Some deal with the stars and Sun.
The second reason we wouldn't want to throw out our Celestial Sphere theory is that it actually helps to explain the Sun's motion, even though it doesn't fully explain that motion. The Sun's actual motion across the sky is not quite circular like that of the stars. Instead, the Sun spirals (albeit very slowly). This spiraling motion, viewed in full, would appear quite complicated. But if we start off with the spinning Celestial Sphere, we find that we can describe the Sun's spiraling motion by simply having the Sun move slowly along the Celestial Sphere on a circular path. It is the combination of the circular motion of the Celestial Sphere (which takes 23 hours, 56 minutes to complete) and the circular motion of the Sun along the Celestial Sphere (which takes a year to complete) that produces the observed spiral motion of the Sun. If we don't start with the Celestial Sphere, the Sun's motion would be much harder to model.
My class will continue exploring this model (Celestial Sphere plus moving Sun) next week. This simple model explains much that is of great importance to humans, which is why the study of the "The Sphere" was the focus of the medieval and early modern astronomy curriculum in the universities. The classic work was the De Sphaera of Johannes de Sacrobosco (John of Holywood), but many students learned "spherics" from later commentaries which elaborated on Sacrobosco's book. We will be finished with spherics in another week or so!
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