I've been away from the blog for two weeks, but I'm going to try to get caught up. So hopefully many posts in the next few days. I want to record some thoughts about each of my classes from these past two weeks.
The first was a discussion about different perspectives on the purpose of science. The discussion was motivated by the anonymous ad lectorum at the beginning of the De Revolutionibus. This preface was written by Andreas Osiander, a Lutheran theologian who was in charge of supervising the printing of Copernicus' great work. Osiander added the preface without Copernicus' authorization (at that seems to be the case), most likely in an attempt to protect Copernicus' ideas from hasty criticism. But what he basically says is that the notion of a moving Earth may be ridiculous, but that doesn't matter. Who cares if what Copernicus says is true? That's not the point of astronomy. The point of astronomy is to give us a way to calculate the apparent positions of the planets, and that is all. This is a view of science known as phenomenalism.
Copernicus himself was not a phenomenalist. He was a realist, who believed that astronomy could reveal the truth about the heavens. In the first book of De Revolutionibus he goes to great lengths to argue for the truth of his system (these are exactly the bits that would be censored by the Catholic Church much later). Copernicus thinks not only that the goal of science should be to reveal the truth, but he is convinced that he has achieved that goal.
Our class discussions focused on the different criteria that one might use to evaluate a scientific theory depending on whether one was a phenomenalist or a realist. Both, of course, want their theories to match the existing data (observations, measurements, whatever). Both value the successful prediction of future data (or past but previously unknown data). But for a phenomenalist that is really about all that matters. Phenomenalists might prefer simple theories to observationally equivalent but more complicated ones, but that is just for convenience. Otherwise it is all about fitting the data for a phenomenalist.
Realists, by contrasts, can bring in a host of other criteria to judge theories. For example, a realist may be concerned about whether or not a theory is compatible with other accepted theories. For a phenemenalist this is not an issue: each theory is designed to match certain types of data, and so what if it happens to be logically incompatible with a different theory that matches different data. So we have a theory for Venus that indicates the the distance from Earth to Venus changes dramatically. And theories of optics and planetary luminance that state that if the distance to a planet varies by a large amount then the planet should get much brighter and dimmer. But the brightness of Venus doesn't change so much. A realist would be bothered by this, but for a phenomenalist the theory of Venus' motion is just designed to predict where it will be in the sky. It isn't designed to predict Venus' brightness, so that issue is just irrelevant.
Likewise, realists (at least in the past) might be concerned about their theories being compatible with religious doctrine. In 16th century Europe you had to be cautious in promoting a scientific theory that might seem to contradict the Bible. Phenomenalists didn't have to worry about this issue, since they never claimed that their theories were true.
Finally, realists can bring to bear a host of aesthetic criteria. My students were pretty receptive to the idea that aesthetics are important in science, but maybe that's because I talk about it all the time. But it seems to me that a phenomenalist shouldn't worry about whether or not a theory is ugly or elegant. They might value ease of use, but not beauty. But for a realist, beauty is a sign of truth. Some explanations just seem more natural than others, and that sense of naturalness is a guide to truth for a realist.
What we discovered through this discussion is that, in the 16th century, it made a lot of sense to embrace the Copernican system as a phenomenalist tool. But if you were a realist then the cards were stacked against Copernicus. He scored major points on aesthetics, but his incompatibility with accepted theories (Aristotle's physics), with Scripture (interpreted in the most literal way), and with observational data (ie annual parallax) would lead most most realists to reject the Copernican system. And that's just what most of them did. Thankfully there were a few for whom the aesthetic properties trumped all else.
Tuesday, October 30, 2012
Tuesday, October 16, 2012
Perspective in Astronomy
The parallels between changes in astronomy and changes in art discussed in Hallyn's The Poetic Structure of the World has gotten me thinking about the introduction of perspective into art and astronomy. Hallyn talks about this a little bit, but I want to engage in some wild speculation. This is a blog, after all.
Here's my thought: the Aristotelian cosmos is like pre-perspective art, while Copernicus effectively introduced perspective into astronomy. I can't back this up with any research, but I'll at least try a little bit to justify this idea (which I'm sure has been suggested by others, so it's not even new). The Aristotelian cosmos is a hierarchical structure. It has a geometrical arrangement to it, but the ordering is really a moral/philosophical one. Perfection exists at the outer boundary. Corruption at the center. Christians would later adopt this moral structure by placing heaven outside of the sphere of stars, and hell at the center of Earth. The universe has a moral order with bad at the center, increasing goodness as you make your way outward (so that man has a somewhat ambiguous location between heaven and hell, but much closer to hell), and perfect goodness at (or beyond) the outermost limit.
Likewise, much pre-perspective art was arranged hierarchically. The most important figures in the scene were often placed at the top of the composition and they were usually much larger than the other figures. Figures of less importance were at the bottom and were smaller. There was little, if any, attempt to depict a realistic view of the 3D setting. The only way to tell if one object was in front or behind another is if the images of the objects overlapped so that one image blocked a part of the other image. But the spatial arrangement wasn't really what was important. What was important was the hierarchy, the ordering by importance. One has this sense of the Aristotelian cosmos: that the fundamental order was a philosophical one, and the geometric arrangement was just there to illustrate the philosophical hierarchy. This becomes very explicit in Dante's Divine Comedy.
Ptolemaic astronomy was an attempt to construct geometrical motions that would fit into this philosophical space. It never really quite succeeded. There was always a tension between Ptolemy and Aristotle (not personally, since they didn't live at the same time, but you know what I mean). Copernicus abandoned the philosophical space of Aristotle and tried to develop a geometrical theory that employs the ideas of perspective - the recognition that what we see is dependent upon our point of view. In addition to making this fundamental step, Copernicus went further by introducing a new perspective which he felt was more fundamental than ours: the perspective of the Sun.
You can argue that Copernicus was really just introducing a new Neo-Platonic or Hermetic or some such philosophical hierarchy, to replace the Aristotelian one. That may be true. If so, he certainly wasn't the last to try that. Kepler very explicitly introduced a new hierarchical structure to the world, but his hierarchy is a hierarchy of harmonies. It has a relation to the spatial structure of the world, but in a much subtler way than did Aristotle's. And long after Kepler others tried to impose a philosophical/moral ordering on the universe that dictated a particular geometric structure: Stukeley and Wright both proposed explanations of the Milky Way that made use of such a moral-geometric ordering.
So even after astronomers recognized that the world could look very different depending on your perspective (the classic exposition of this is Kepler's Somnium, in which he describes what astronomy would be like if done form the Moon), they still continued to look for one particular perspective that was somehow better than all others. The main thing that seems to have guided them in this search for the ideal perspective is a sense of harmony and order. Apparently the same thing happened in art. The artist, of course, must choose a perspective from which to depict his scene. Not all choices were considered equal, and artists sought the choose the perspective that would produce the most harmonious composition.
Often this ideal perspective was couched in religious terms. Copernicus did that a little, Kepler a lot. Stukely and Wright were explicitly religious in their suggestions, giving Heaven and Hell specific locations in their models of the universe. The view these men sought might be called the "God's Eye View". But eventually the idea of God's omnipresence trumped this notion of a God's Eye View. If God was everywhere, then there was no single location or point of view that was particularly God's. In effect, this made all perspectives equally valid from a moral/philosophical point of view. You see this with Bruno and again with Newton. But Newton showed that some perspectives were better from a physical point of view (what we would now call inertial reference frames).
Anyway, I find it interesting that the search for a harmonious and well-ordered model of the world was, at least in some cases, motivated by the desire to find the ideal perspective, God's perspective. We still make use of harmony and order in guiding our search for scientific theories, but few scientists would claim that the search for harmony or symmetry is religiously motivated. What, then, convinces us that scientific theories that are harmonious, well-ordered, symmetric, or beautiful are better than empirically equivalent theories which are not? This is a question that I find deeply fascinating.
Here's my thought: the Aristotelian cosmos is like pre-perspective art, while Copernicus effectively introduced perspective into astronomy. I can't back this up with any research, but I'll at least try a little bit to justify this idea (which I'm sure has been suggested by others, so it's not even new). The Aristotelian cosmos is a hierarchical structure. It has a geometrical arrangement to it, but the ordering is really a moral/philosophical one. Perfection exists at the outer boundary. Corruption at the center. Christians would later adopt this moral structure by placing heaven outside of the sphere of stars, and hell at the center of Earth. The universe has a moral order with bad at the center, increasing goodness as you make your way outward (so that man has a somewhat ambiguous location between heaven and hell, but much closer to hell), and perfect goodness at (or beyond) the outermost limit.
Likewise, much pre-perspective art was arranged hierarchically. The most important figures in the scene were often placed at the top of the composition and they were usually much larger than the other figures. Figures of less importance were at the bottom and were smaller. There was little, if any, attempt to depict a realistic view of the 3D setting. The only way to tell if one object was in front or behind another is if the images of the objects overlapped so that one image blocked a part of the other image. But the spatial arrangement wasn't really what was important. What was important was the hierarchy, the ordering by importance. One has this sense of the Aristotelian cosmos: that the fundamental order was a philosophical one, and the geometric arrangement was just there to illustrate the philosophical hierarchy. This becomes very explicit in Dante's Divine Comedy.
Ptolemaic astronomy was an attempt to construct geometrical motions that would fit into this philosophical space. It never really quite succeeded. There was always a tension between Ptolemy and Aristotle (not personally, since they didn't live at the same time, but you know what I mean). Copernicus abandoned the philosophical space of Aristotle and tried to develop a geometrical theory that employs the ideas of perspective - the recognition that what we see is dependent upon our point of view. In addition to making this fundamental step, Copernicus went further by introducing a new perspective which he felt was more fundamental than ours: the perspective of the Sun.
You can argue that Copernicus was really just introducing a new Neo-Platonic or Hermetic or some such philosophical hierarchy, to replace the Aristotelian one. That may be true. If so, he certainly wasn't the last to try that. Kepler very explicitly introduced a new hierarchical structure to the world, but his hierarchy is a hierarchy of harmonies. It has a relation to the spatial structure of the world, but in a much subtler way than did Aristotle's. And long after Kepler others tried to impose a philosophical/moral ordering on the universe that dictated a particular geometric structure: Stukeley and Wright both proposed explanations of the Milky Way that made use of such a moral-geometric ordering.
So even after astronomers recognized that the world could look very different depending on your perspective (the classic exposition of this is Kepler's Somnium, in which he describes what astronomy would be like if done form the Moon), they still continued to look for one particular perspective that was somehow better than all others. The main thing that seems to have guided them in this search for the ideal perspective is a sense of harmony and order. Apparently the same thing happened in art. The artist, of course, must choose a perspective from which to depict his scene. Not all choices were considered equal, and artists sought the choose the perspective that would produce the most harmonious composition.
Often this ideal perspective was couched in religious terms. Copernicus did that a little, Kepler a lot. Stukely and Wright were explicitly religious in their suggestions, giving Heaven and Hell specific locations in their models of the universe. The view these men sought might be called the "God's Eye View". But eventually the idea of God's omnipresence trumped this notion of a God's Eye View. If God was everywhere, then there was no single location or point of view that was particularly God's. In effect, this made all perspectives equally valid from a moral/philosophical point of view. You see this with Bruno and again with Newton. But Newton showed that some perspectives were better from a physical point of view (what we would now call inertial reference frames).
Anyway, I find it interesting that the search for a harmonious and well-ordered model of the world was, at least in some cases, motivated by the desire to find the ideal perspective, God's perspective. We still make use of harmony and order in guiding our search for scientific theories, but few scientists would claim that the search for harmony or symmetry is religiously motivated. What, then, convinces us that scientific theories that are harmonious, well-ordered, symmetric, or beautiful are better than empirically equivalent theories which are not? This is a question that I find deeply fascinating.
Sunday, October 14, 2012
Poetic Structure of the World
Today I finished reading Fernand Hallyn's The Poetic Structure of the World. Hallyn analyzes similarities between the work of Copernicus and Kepler and contemporary movements in art, music and literature. Although he does not really claim that Copernicus and Kepler were specifically influenced by these movements, he does claim that there was a set of common aesthetic concerns that underlie both the science and the art.
In discussing Copernicus he particularly focuses on the notion of symmetry and order. Copernicus makes his case for the truth (as opposed to the possibility or utility) of this theory almost entirely by pointing out the symmetry and beauty of his theory in contrast to the monstrosity of the Ptolemaic theory. Copernicus' arguments about the planetary orbits being in proper proportion, with the Earth's orbit as a common measure, are mirrored by Renaissance artists concerns about properly displaying the proportions of the human figure. Hallyn argues that Copernicus views the cosmos as an organism, thus imbuing the cosmos with a coherence and integration that it did not have for Ptolemy.
Hallyn also points out the ways Copernicus uses synecdoche: giving to a part (the Earth) a property (gravity) normally attributed only to the whole (the Universe). Copernicus viewed gravity as a local attraction among like bodies (Earth is attracted to Earth, etc) rather than as a universal attraction to an absolute center. Hallyn also notes Copernicus' use of metaphor in arguing why the Sun ought to be located at the center, and how this parallels similar discussions about the placement of altars in churches.
Hallyn associates Kepler's work with Mannerism. He stresses Kepler's desire to uncover the signs that reveal God's plan for Creation. Kepler's obsession with the Platonic solids was apparently shared by some Mannerist painters. Likewise, Kepler's ellipse was, for Kepler, a conjunction of the divine curve and the physical line - thus representing the plan of the divine Creator executed within the physical world, rather than simply a deformed circle.
An important aspects of Kepler's work is his belief that God created the universe according to a specific plan that can be known by man. He justifies this belief through the idea that man is made in God's image, and that the world was made for man. Therefore we have an natural ability to recognize the divine signs and see the truth of the cosmos (which, of course, for Kepler, was heliocentric). This divine order must be found intellectually because it is hidden within the appearances we see. For example, Kepler thought the motions of the planets represented a musical harmony, but this harmony could only be directly perceived by viewing the apparent motions of the planets as seen from the Sun. This idea of viewing the world from another point is one I want to return to.
For now I just want to comment on how ideas from outside science can influence science in some very important, and positive, ways. Hallyn makes the case that there were important aesthetic concerns that were prevalent in the cultural milieu of Copernicus and Kepler that played an important role in their scientific thinking. There were also important religious considerations, such as that man should be capable of discerning the true structure of the cosmos. This idea follows much more easily from a belief in a personal God who created the world for the benefit of man than it does from belief in an Aristotelian Prime Mover.
I think this is one of the dangers of scientism. If science is held up as the only possible source of rational belief, then what can inform the development of science itself. How can the criteria by which science judges itself evolve and grow? From whence will come the new aesthetic notions and new philosophical perspectives that have, in the past, led to important revolutions in scientific thinking? If scientists listen to no one but other scientists, science will suffer.
In discussing Copernicus he particularly focuses on the notion of symmetry and order. Copernicus makes his case for the truth (as opposed to the possibility or utility) of this theory almost entirely by pointing out the symmetry and beauty of his theory in contrast to the monstrosity of the Ptolemaic theory. Copernicus' arguments about the planetary orbits being in proper proportion, with the Earth's orbit as a common measure, are mirrored by Renaissance artists concerns about properly displaying the proportions of the human figure. Hallyn argues that Copernicus views the cosmos as an organism, thus imbuing the cosmos with a coherence and integration that it did not have for Ptolemy.
Hallyn also points out the ways Copernicus uses synecdoche: giving to a part (the Earth) a property (gravity) normally attributed only to the whole (the Universe). Copernicus viewed gravity as a local attraction among like bodies (Earth is attracted to Earth, etc) rather than as a universal attraction to an absolute center. Hallyn also notes Copernicus' use of metaphor in arguing why the Sun ought to be located at the center, and how this parallels similar discussions about the placement of altars in churches.
Hallyn associates Kepler's work with Mannerism. He stresses Kepler's desire to uncover the signs that reveal God's plan for Creation. Kepler's obsession with the Platonic solids was apparently shared by some Mannerist painters. Likewise, Kepler's ellipse was, for Kepler, a conjunction of the divine curve and the physical line - thus representing the plan of the divine Creator executed within the physical world, rather than simply a deformed circle.
An important aspects of Kepler's work is his belief that God created the universe according to a specific plan that can be known by man. He justifies this belief through the idea that man is made in God's image, and that the world was made for man. Therefore we have an natural ability to recognize the divine signs and see the truth of the cosmos (which, of course, for Kepler, was heliocentric). This divine order must be found intellectually because it is hidden within the appearances we see. For example, Kepler thought the motions of the planets represented a musical harmony, but this harmony could only be directly perceived by viewing the apparent motions of the planets as seen from the Sun. This idea of viewing the world from another point is one I want to return to.
For now I just want to comment on how ideas from outside science can influence science in some very important, and positive, ways. Hallyn makes the case that there were important aesthetic concerns that were prevalent in the cultural milieu of Copernicus and Kepler that played an important role in their scientific thinking. There were also important religious considerations, such as that man should be capable of discerning the true structure of the cosmos. This idea follows much more easily from a belief in a personal God who created the world for the benefit of man than it does from belief in an Aristotelian Prime Mover.
I think this is one of the dangers of scientism. If science is held up as the only possible source of rational belief, then what can inform the development of science itself. How can the criteria by which science judges itself evolve and grow? From whence will come the new aesthetic notions and new philosophical perspectives that have, in the past, led to important revolutions in scientific thinking? If scientists listen to no one but other scientists, science will suffer.
Monday, October 8, 2012
The Scale of the Universe
Today's activity involved determining the radii of the planetary orbits (in Astronomical Units) from observational data. It is pretty straightforward for inferior planets. You measure the maximum elongation of the planet from the Sun, and the sine of that angle gives you the radius of that planet's orbit in AU (assuming all orbits are circular and centered on the Sun, which is a decent approximation). For superior planets it is more difficult. It involves measuring the time from opposition to quadrature and then doing a few calculations before finally getting to the trigonometry that lets you find the radius of the planet's orbit.
The main point of this exercise is to show the students that in the Copernican system you can actually find the sizes of the planetary orbits, relative to the size of Earth's orbit (with a radius of 1AU), from observational data. This was not the case for Ptolemy. In the Ptolemaic system we can only determine the size of a planet's epicycle relative to the size of that same planet's deferent from observational data. There is no COMMON distance measure for the planets, nothing that they share that can allow us to connect the orbit of one planet to the orbit of another. But in the Copernican system all of our observations of planetary motion are a mixture of the motion of the planet and the motion of Earth. Once we learn how to disentangle those motions, the orbit of Earth becomes our common unit of measure. All other planetary orbits can be related, via observations, to the orbit of Earth. Everything hangs together.
This is yet another illustration of the remarkable coherence of the Copernican system. As Copernicus says, Ptolemy's theory is like taking a hand, a leg, a head, a torso, all from different individuals and sticking them together. The result is a monster, not a man. In the Copernican system everything is in just the right proportion to everything else. All the pieces fit neatly. And there is no freedom to have it any other way.
Perhaps this is the strongest aspect of the Copernican theory: necessity. Copernicus establishes lots of NECESSARY connections between things. There are necessary connections between the sizes of the planetary orbits. There is a necessary connection between whether a planet's orbit is smaller or larger than Earth's and a variety of observational characteristics for that planet. There is a necessary connection between the retrograde motion of a planet and that planet being in conjunction (for an inferior planet) or opposition (for a superior planet). Ptolemy has some necessary connections, particularly between retrograde and brightness (which Copernicus also has) - but Ptolemy doesn't have nearly as much necessity as Copernicus does.
Of course, necessity can be a weakness. If a certain connection is a necessary outcome of a theory, and then that connection is found to not exist, then the theory will be refuted or modified (or some auxiliary theory must be modified). But when the necessary outcomes of a theory are found to, in fact, exist then it speaks powerfully in favor of that theory. Saying that something MUST be true, and then finding it is true is much stronger than saying that it CAN be true and finding it is true. The latter is good, but the former is much better.
In today's activity we also discovered another beautiful feature of the Copernican theory. It is not a necessary outcome of the theory, but it is a feature of the theory that Copernicus would say is "pleasing to the mind." If we assume the Copernican theory, then we can show that the more distant a planet is from the Sun the longer its orbital period. Moreover, we can even show that the more distant the planet from the Sun the slower it moves along its orbit. There is a nice harmony here that Copernicus appreciated, even though he had no explanation for why it should be that way. Later, Kepler would seek to make this harmony a necessary outcome of his celestial dynamics. And even later, Newton would succeed in this venture.
I'll say more about the aesthetics of the Copernican theory in a later post. I'm reading Hallyn's The Poetic Structure of the World, and it is worth commenting on. But for now I'll just point out that these aesthetic features are really the main thing going for the Copernican theory in the 16th century. Without these features Copernicus offers nothing more than a theory that is observationally equivalent to Ptolemy, but which conflicts with Aristotelian physics and requires a gigantic Celestial Sphere with a tremendous void between Saturn and the stars. Most people didn't find the aesthetic features sufficiently appealing to overcome all of the negatives, but thankfully a few did.
The main point of this exercise is to show the students that in the Copernican system you can actually find the sizes of the planetary orbits, relative to the size of Earth's orbit (with a radius of 1AU), from observational data. This was not the case for Ptolemy. In the Ptolemaic system we can only determine the size of a planet's epicycle relative to the size of that same planet's deferent from observational data. There is no COMMON distance measure for the planets, nothing that they share that can allow us to connect the orbit of one planet to the orbit of another. But in the Copernican system all of our observations of planetary motion are a mixture of the motion of the planet and the motion of Earth. Once we learn how to disentangle those motions, the orbit of Earth becomes our common unit of measure. All other planetary orbits can be related, via observations, to the orbit of Earth. Everything hangs together.
This is yet another illustration of the remarkable coherence of the Copernican system. As Copernicus says, Ptolemy's theory is like taking a hand, a leg, a head, a torso, all from different individuals and sticking them together. The result is a monster, not a man. In the Copernican system everything is in just the right proportion to everything else. All the pieces fit neatly. And there is no freedom to have it any other way.
Perhaps this is the strongest aspect of the Copernican theory: necessity. Copernicus establishes lots of NECESSARY connections between things. There are necessary connections between the sizes of the planetary orbits. There is a necessary connection between whether a planet's orbit is smaller or larger than Earth's and a variety of observational characteristics for that planet. There is a necessary connection between the retrograde motion of a planet and that planet being in conjunction (for an inferior planet) or opposition (for a superior planet). Ptolemy has some necessary connections, particularly between retrograde and brightness (which Copernicus also has) - but Ptolemy doesn't have nearly as much necessity as Copernicus does.
Of course, necessity can be a weakness. If a certain connection is a necessary outcome of a theory, and then that connection is found to not exist, then the theory will be refuted or modified (or some auxiliary theory must be modified). But when the necessary outcomes of a theory are found to, in fact, exist then it speaks powerfully in favor of that theory. Saying that something MUST be true, and then finding it is true is much stronger than saying that it CAN be true and finding it is true. The latter is good, but the former is much better.
In today's activity we also discovered another beautiful feature of the Copernican theory. It is not a necessary outcome of the theory, but it is a feature of the theory that Copernicus would say is "pleasing to the mind." If we assume the Copernican theory, then we can show that the more distant a planet is from the Sun the longer its orbital period. Moreover, we can even show that the more distant the planet from the Sun the slower it moves along its orbit. There is a nice harmony here that Copernicus appreciated, even though he had no explanation for why it should be that way. Later, Kepler would seek to make this harmony a necessary outcome of his celestial dynamics. And even later, Newton would succeed in this venture.
I'll say more about the aesthetics of the Copernican theory in a later post. I'm reading Hallyn's The Poetic Structure of the World, and it is worth commenting on. But for now I'll just point out that these aesthetic features are really the main thing going for the Copernican theory in the 16th century. Without these features Copernicus offers nothing more than a theory that is observationally equivalent to Ptolemy, but which conflicts with Aristotelian physics and requires a gigantic Celestial Sphere with a tremendous void between Saturn and the stars. Most people didn't find the aesthetic features sufficiently appealing to overcome all of the negatives, but thankfully a few did.
Thursday, October 4, 2012
Copernicus' On Planets
Copernicus' theory of a rotating and orbiting Earth manages to explain the daily motion of the sky and the annual motion of the Sun just as well as Ptolemy, but no better. If that was all he had, his theory probably would have died right there. But Copernicus had an ace up his sleeve: epicycle-free retrograde motion for the planets.
The first thing to note about Copernicus' theory of the planets is that by making Earth and the other planets orbit in circles (roughly) about the Sun, Copernicus automatically creates two different categories of planets. First there are planets whose orbits are enclosed within Earth's orbit (ie, planets that are closer to the Sun than is Earth). These planets will always be seen near the Sun from Earth's perspective, so clearly these must be the inferior planets (Mercury and Venus). The other category is the planets whose orbits enclose Earth's orbit (those farther from the Sun than is Earth). These planets can at times be opposite the Sun from Earth's perspective, so clearly these are the superior planets (Mars, Jupiter, Saturn).
This is nice - the division between inferior and superior has a very natural explanation in the Copernican system. But you ain't seen nuthin' yet. In order for an inferior planet to undergo retrograde motion as seen from Earth, that planet must move along its orbit faster than does Earth. In that case, the planet will appear to retrograde (as seen from Earth) when it passes Earth on the inside. Of course, if it is passing Earth on the inside then it must be between the Earth and the Sun. In other words, it just be in conjunction as seen from Earth. And we already know that the inferior planets do, in fact, retrograde only when in conjunction.
For superior planets to retrograde they must move slower than Earth. In that case, they will appear to retrograde when Earth passes them on the inside. If Earth is passing them on the inside, then Earth must be between the planet and the Sun. So as seen from Earth the planet and Sun will be in opposition. And we know that the superior planets do, in fact, retrograde only when in opposition. What is more, when Earth is passing the planet it will be as close as possible to that planet. This explains why superior planets are brightest during retrograde/opposition - that is when they are closest to Earth.
So Copernicus gets retrograde motion without having to use epicycles (although he does, in fact, use epicycles for something else as we will see later). Moreover, his simple theory automatically achieves all of the links we find observationally: retrograde is linked to brightness and also to conjunction/opposition. No special tweaking necessary. In the Copernican theory it CAN'T BE ANY OTHER WAY. This is a severe constraint on the Copernican theory and it makes it much more likely that the Copernican theory would conflict with observations if the theory were not, in fact, true. But it doesn't conflict with observations. Pretty cool.
But there is a complication. In the Copernican theory we cannot directly observe the periods of the planetary orbits. This is because while the planets are moving, WE are also moving. So the cycles we observe (synodic periods, zodiacal periods) are really a mixture of the motion of the planet we are observing and our own motion. My class spent much of today's class period working to disentangle those motions. We succeeded in determining the orbital periods of all five visible planets, but in order to do so we had to do a bit of geometry and some calculation.
Next week we will finish up our discussion of Copernicus by determining the sizes of the planetary orbits. This will help us to see the overall coherence of the Copernican system, and why Copernicus felt that Ptolemy's system was a monster.
The first thing to note about Copernicus' theory of the planets is that by making Earth and the other planets orbit in circles (roughly) about the Sun, Copernicus automatically creates two different categories of planets. First there are planets whose orbits are enclosed within Earth's orbit (ie, planets that are closer to the Sun than is Earth). These planets will always be seen near the Sun from Earth's perspective, so clearly these must be the inferior planets (Mercury and Venus). The other category is the planets whose orbits enclose Earth's orbit (those farther from the Sun than is Earth). These planets can at times be opposite the Sun from Earth's perspective, so clearly these are the superior planets (Mars, Jupiter, Saturn).
This is nice - the division between inferior and superior has a very natural explanation in the Copernican system. But you ain't seen nuthin' yet. In order for an inferior planet to undergo retrograde motion as seen from Earth, that planet must move along its orbit faster than does Earth. In that case, the planet will appear to retrograde (as seen from Earth) when it passes Earth on the inside. Of course, if it is passing Earth on the inside then it must be between the Earth and the Sun. In other words, it just be in conjunction as seen from Earth. And we already know that the inferior planets do, in fact, retrograde only when in conjunction.
For superior planets to retrograde they must move slower than Earth. In that case, they will appear to retrograde when Earth passes them on the inside. If Earth is passing them on the inside, then Earth must be between the planet and the Sun. So as seen from Earth the planet and Sun will be in opposition. And we know that the superior planets do, in fact, retrograde only when in opposition. What is more, when Earth is passing the planet it will be as close as possible to that planet. This explains why superior planets are brightest during retrograde/opposition - that is when they are closest to Earth.
So Copernicus gets retrograde motion without having to use epicycles (although he does, in fact, use epicycles for something else as we will see later). Moreover, his simple theory automatically achieves all of the links we find observationally: retrograde is linked to brightness and also to conjunction/opposition. No special tweaking necessary. In the Copernican theory it CAN'T BE ANY OTHER WAY. This is a severe constraint on the Copernican theory and it makes it much more likely that the Copernican theory would conflict with observations if the theory were not, in fact, true. But it doesn't conflict with observations. Pretty cool.
But there is a complication. In the Copernican theory we cannot directly observe the periods of the planetary orbits. This is because while the planets are moving, WE are also moving. So the cycles we observe (synodic periods, zodiacal periods) are really a mixture of the motion of the planet we are observing and our own motion. My class spent much of today's class period working to disentangle those motions. We succeeded in determining the orbital periods of all five visible planets, but in order to do so we had to do a bit of geometry and some calculation.
Next week we will finish up our discussion of Copernicus by determining the sizes of the planetary orbits. This will help us to see the overall coherence of the Copernican system, and why Copernicus felt that Ptolemy's system was a monster.
Tuesday, October 2, 2012
Motions of Earth
Today we began our study of Copernicus. The focus of today's activity was the three (really four) motions that Copernicus assigns to Earth. We also discover, right away, the obvious way to refute Copernicus' idea that Earth orbits the Sun (and also Copernicus' seemingly ad hoc solution to this problem).
The first two motions of Earth are pretty straightforward, and are easily demonstrated with my Daily Rotation simulation (not yet available on OSP). The Ancient Greeks thought the daily motion of the sky could be accounted for by a Celestial Sphere that rotates with a period of 23 hours, 56 minutes. Copernicus achieved the same effects by holding the Celestial Sphere still and letting Earth spin about the same axis, but in the opposite sense, with the same period.
The second motion accounts for the annual motion of the Sun around the Celestial Sphere and is demonstrated by my Earth Orbit simulation. The Greeks just had the Sun orbit the Earth in the Ecliptic plane over the course of a (sidereal) year. Copernicus instead has Earth orbit the Sun, also in the Ecliptic plane and in the same sense (counterclockwise as viewed from the North Ecliptic Pole). The rotational axis of Earth (see the first motion above) is not perpendicular to the Ecliptic plane (the plane of Earth's orbit). Rather, it is inclined away from perpendicular by 23.5 degrees. This gives rise to all of the seasonal variations in sunlight (both in the altitude of the Sun and in hours of daylight) that the Ancient Greeks accomplished by simply tilting the Ecliptic circle relative to the Celestial Equator.
So far, so good. Copernicus matches up with Ptolemy perfectly, although one might argue that he has done nothing new yet so what's the point? There is a point, but you don't really get to see it until you look at the planets. As far as Earth is concerned, its all downhill from here.
The first problem Copernicus runs into can be easily demonstrated. Grab a stick and hold it in your hand. Extend your arm. Don't point the stick straight up, but tilt it at an angle (say, 23.5 degrees from straight up?). Now spin your body, keeping your arm extended. Your hand orbits around you body just like Earth orbits the Sun in the Copernican system. The stick plays the role of Earth's rotational axis. You will find that the stick does not point in a fixed direction. Instead its direction changes as you spin around. This was a problem because the sky appears to spin about the same points (around a point near Polaris in the Northern Hemisphere) all year long.
These days we wouldn't even see this as a problem. We think of Earth as a freely spinning body that will automatically maintain the orientation of its rotational axis unless something messes with it. But for Copernicus the Earth was a heavenly body that had to be carried around by heavenly spheres. So in his mind it was much more like a hand holding a stick than like a freely spinning ball. But he solved the problem in a an ingenious way. While Earth orbits the Sun, its axis or rotation rotates around the Ecliptic poles in the opposite sense and at ALMOST the same rate. If he made the rate exactly the same (one rotation per sidereal year), then the rotational axis would maintain the same orientation forever. By making the rates SlIGHTLY different, Copernicus found an explanation for the precession of the equinoxes (see my Equinox Precession simulation). Brilliant! He turned what could have been a major weakness of his theory into a strength.
But a problem still remains. Even if the axis maintains a fixed orientation, that doesn't mean it will intersect the Celestial Sphere always in the same point. As Earth orbits the Sun, the rotational axis gets dragged around with it. So technically the point where the axis hits the Celestial Sphere (what we call the Celestial Pole) moves in a circle over the course of a year. This is NOT the same as the circle the poles make due to precession. For one thing, the poles complete this circle once per year, not once every 26,000 years as with precession. For another thing, the precession circle is centered on the Ecliptic pole, while this new effect has no relation to the Ecliptic poles.
Again, we simply do not see this effect. The North Celestial Pole stays right next to Polaris all year long. It doesn't move about the sky in any noticeable way. So it would seem that Copernicus' theory of Earth's orbit around the Sun has been refuted. It makes a prediction, that predictions fails the observational test. Throw out the theory, right? Of course, that's not what we do. Because Copernicus has an answer: the Celestial Sphere is just REALLY BIG compared to the size of Earth's orbit. If you push the Celestial Sphere far enough out, then the little circle traced out by the Celestial Pole becomes a tiny dot - so small that it looks like a point, even though it is really a very tiny circle.
For Copernicus himself, this was a satisfying explanation of the failure to observe the effect that his theory predicted. For most of his contemporaries it seemed like an ad hoc defense of the theory, exactly the kind of thing that Karl Popper would later say should never be done. Of course, in this case Copernicus' ad hoc defense would turn out to be completely correct. The Ancient Greeks thought the Celestial Sphere could hold more than a trillion Earths, but as it turns out they had vastly underestimated the distances to the stars.
Before I close, I should mention Copernicus' fourth motion for Earth. He introduced this little wobble to account for an effect known as "trepidation". This was an variation in the rate of precession that had been originally proposed, I think, by Medieval astronomers in the Middle East. However, this effect is not real and so this fourth motion is not needed. We barely even mention it in class, although I am quite interested to know the history of how astronomers eventually discovered that trepidation was not a real effect. Once you have measured something it is hard to make it go away, even if it is not really there. This issue comes up in my other astronomy class with regard to the rotation of spiral galaxies as measured by Adrian van Maanen - but that's another story.
The first two motions of Earth are pretty straightforward, and are easily demonstrated with my Daily Rotation simulation (not yet available on OSP). The Ancient Greeks thought the daily motion of the sky could be accounted for by a Celestial Sphere that rotates with a period of 23 hours, 56 minutes. Copernicus achieved the same effects by holding the Celestial Sphere still and letting Earth spin about the same axis, but in the opposite sense, with the same period.
The second motion accounts for the annual motion of the Sun around the Celestial Sphere and is demonstrated by my Earth Orbit simulation. The Greeks just had the Sun orbit the Earth in the Ecliptic plane over the course of a (sidereal) year. Copernicus instead has Earth orbit the Sun, also in the Ecliptic plane and in the same sense (counterclockwise as viewed from the North Ecliptic Pole). The rotational axis of Earth (see the first motion above) is not perpendicular to the Ecliptic plane (the plane of Earth's orbit). Rather, it is inclined away from perpendicular by 23.5 degrees. This gives rise to all of the seasonal variations in sunlight (both in the altitude of the Sun and in hours of daylight) that the Ancient Greeks accomplished by simply tilting the Ecliptic circle relative to the Celestial Equator.
So far, so good. Copernicus matches up with Ptolemy perfectly, although one might argue that he has done nothing new yet so what's the point? There is a point, but you don't really get to see it until you look at the planets. As far as Earth is concerned, its all downhill from here.
The first problem Copernicus runs into can be easily demonstrated. Grab a stick and hold it in your hand. Extend your arm. Don't point the stick straight up, but tilt it at an angle (say, 23.5 degrees from straight up?). Now spin your body, keeping your arm extended. Your hand orbits around you body just like Earth orbits the Sun in the Copernican system. The stick plays the role of Earth's rotational axis. You will find that the stick does not point in a fixed direction. Instead its direction changes as you spin around. This was a problem because the sky appears to spin about the same points (around a point near Polaris in the Northern Hemisphere) all year long.
These days we wouldn't even see this as a problem. We think of Earth as a freely spinning body that will automatically maintain the orientation of its rotational axis unless something messes with it. But for Copernicus the Earth was a heavenly body that had to be carried around by heavenly spheres. So in his mind it was much more like a hand holding a stick than like a freely spinning ball. But he solved the problem in a an ingenious way. While Earth orbits the Sun, its axis or rotation rotates around the Ecliptic poles in the opposite sense and at ALMOST the same rate. If he made the rate exactly the same (one rotation per sidereal year), then the rotational axis would maintain the same orientation forever. By making the rates SlIGHTLY different, Copernicus found an explanation for the precession of the equinoxes (see my Equinox Precession simulation). Brilliant! He turned what could have been a major weakness of his theory into a strength.
But a problem still remains. Even if the axis maintains a fixed orientation, that doesn't mean it will intersect the Celestial Sphere always in the same point. As Earth orbits the Sun, the rotational axis gets dragged around with it. So technically the point where the axis hits the Celestial Sphere (what we call the Celestial Pole) moves in a circle over the course of a year. This is NOT the same as the circle the poles make due to precession. For one thing, the poles complete this circle once per year, not once every 26,000 years as with precession. For another thing, the precession circle is centered on the Ecliptic pole, while this new effect has no relation to the Ecliptic poles.
Again, we simply do not see this effect. The North Celestial Pole stays right next to Polaris all year long. It doesn't move about the sky in any noticeable way. So it would seem that Copernicus' theory of Earth's orbit around the Sun has been refuted. It makes a prediction, that predictions fails the observational test. Throw out the theory, right? Of course, that's not what we do. Because Copernicus has an answer: the Celestial Sphere is just REALLY BIG compared to the size of Earth's orbit. If you push the Celestial Sphere far enough out, then the little circle traced out by the Celestial Pole becomes a tiny dot - so small that it looks like a point, even though it is really a very tiny circle.
For Copernicus himself, this was a satisfying explanation of the failure to observe the effect that his theory predicted. For most of his contemporaries it seemed like an ad hoc defense of the theory, exactly the kind of thing that Karl Popper would later say should never be done. Of course, in this case Copernicus' ad hoc defense would turn out to be completely correct. The Ancient Greeks thought the Celestial Sphere could hold more than a trillion Earths, but as it turns out they had vastly underestimated the distances to the stars.
Before I close, I should mention Copernicus' fourth motion for Earth. He introduced this little wobble to account for an effect known as "trepidation". This was an variation in the rate of precession that had been originally proposed, I think, by Medieval astronomers in the Middle East. However, this effect is not real and so this fourth motion is not needed. We barely even mention it in class, although I am quite interested to know the history of how astronomers eventually discovered that trepidation was not a real effect. Once you have measured something it is hard to make it go away, even if it is not really there. This issue comes up in my other astronomy class with regard to the rotation of spiral galaxies as measured by Adrian van Maanen - but that's another story.
Monday, October 1, 2012
The Ptolemaic Universe
Today we continued our exploration of Ptolemaic astronomy by focusing on size. In particular, we figured out the ratio of the radius of the epicycle to the radius of the deferent for each planet's orbit. We did this using a simplified Ptolemaic model without eccentrics or equants, which makes the geometry doable.
What I hope my students see from this is the interplay between what we can observe and what we can calculate. Essentially, to find the ratio of the epicycle radius to the deferent radius we need to set up a right triangle that has the radius of the deferent as one side and the radius of the epicycle as another. Then, if we can determine one of the non-right angles in this triangle, we can use trigonometry to find the ratio we want. So we are constrained to use right triangles because it is only to these triangles that we can apply our trigonometry (or chords, as the Ancient Greeks would have done).
For an inferior planet this turns out to be surprisingly easy. We wait until the planet has reached its maximum elongation from the Sun (East or West). At that point we can show, geometrically, that the Earth-Planet line must be perpendicular to the line from the center of the epicycle to the planet. Now draw a line from Earth to the center of the epicycle and we have our right triangle. Not only that, but one of the non-right angles is simply the elongation of the planet!
This situation occurs because, as we figured out earlier, the center of the epicycle must lie along the Earth-Sun line in order to keep the inferior planet close to the Sun. So all we have to do is measure the maximum elongation of the planet, and trig will give us the ratio we desire. It's really quite pretty the way the need for a right triangle is met at just this moment of maximum elongation, and how the requirements of the model (epicycle center of Earth-Sun line) allow us to connect this geometry to something we can actually observe. My students are able to figure all of this out with some guided exploration of my Inferior Ptolemaic simulation.
For superior planets it turns out to be much harder. We can set up a right triangle again, no problem. We just wait until the planet is in quadrature (90 degrees from the Sun, as seen from Earth). Again, our theoretical principles help us out. In order for Ptolemy's model to work we know that the Earth-Sun line and the line from the epicycle center to the superior planet must remain always parallel. When the planet is in quadrature the Earth-planet line is perpendicular to the Earth-Sun line, by definition. But this also tells us that the Earth-planet line is perpendicular to the line from the epicycle center to the planet. This gives us two sides of a triangle with a right angle between them. We close the triangle with a line from Earth to the center of the epicycle, and voila! A right triangle. Again, my students can get to this point with some guidance and my Superior Ptolemaic simulation.
Here's the problem: we cannot directly measure any of the non-right angles in this triangle. BUT we can figure one of them out. If we measure the time from when the planet is in opposition to when it is in quadrature, then we can determine the angles through which the Sun and the center of the epicycle move during this time. We can do this because we know both of these things move at a uniform rate (remember we are disregarding eccentrics and equants here), and we know the period for each motion: one year for the Sun, and the planet's zodiacal period for the motion of the epicycle center around the deferent.
Once we have found the angles through which the Sun and epicycle center have moved, we can find a simple relation between these two angles and one of the angles in our right triangle. So although we cannot measure any of the non-right angles in the triangle directly, we can calculate one of them by making use of something we can observe (time from opposition to quadrature) and our theoretical principles (uniform motion on circles with known periods).
OK, so once we have that angle we can do some more trig and find the ratio we want. So it works out, in the end, for both inferior and superior planets.
Armed with the ratios of epicycle to deferent, we can then determine the ratio of the maximum distance of the planet from Earth to the minimum distance. Then, armed with one more principle and one more observation, we can determine the size of the Ancient Greek universe. The principle we need is that there should be no empty space. So the maximum distance to Mercury, for example, should be equal to the minimum distance to Venus, and so on. That way there is no overlap between the planets, but there is also no wasted space.
The additional piece of data we need is the daily parallax of the Moon. More on this later, but the Ancient Greeks were able to use this measurement to determine that the Moon was about 30 Earth diameters away from Earth's center. This agrees well with modern measurements. From there, we can assume Mercury's closest approach is just beyond the Moon's orbit. Venus grazes the maximum distance of Mercury. The Sun orbits just outside Venus' reach, and Mars comes in to just graze the Sun's orbit on the outside. The minimum distance of Jupiter is the maximum distance of Mars, and then we do the same thing with Saturn and Jupiter. Finally, Saturn's maximum distance will equal the distance to the Celestial Sphere on which the fixed stars sit.
It works out that the radius of the Celestial Sphere is just over 5000 Earth diameters. So the diameter of the Celestial Sphere is 10,000 times that of the Earth. If you compare volumes, we find that you could fit one trillion Earths inside the Celestial Sphere if you could pack them in with no empty space. To the Ancient Greeks, this universe seemed quite large. One trillion is a big number, and the Earth is a big thing, so a trillion times the volume of Earth is pretty darn big. Of course, they didn't know that the actual size of the solar system (much less the universe) makes their size for the universe seem tiny.
This semester was the first time I tried this activity. Overall I think it went fairly well. Some groups were already a bit behind, and I don't think they caught up. It was a time consuming activity because there is a lot of math (geometrical diagrams, trigonometry, proportionalities, as well as lots of basic arithmetic). My students are all capable of doing this math, but it doesn't come quickly to some of them because they have not had to think mathematically very much, at least not for some time. But I think they all got through the hardest parts, with a bit of guidance.
Now they are all set to the take their observations of their fictitious solar system and build a Ptolemaic model. And now they know why we cared about measuring the time from opposition to quadrature for a superior planet.
Tomorrow we jump forward in time by more than a millenium and start talking about Copernicus!
What I hope my students see from this is the interplay between what we can observe and what we can calculate. Essentially, to find the ratio of the epicycle radius to the deferent radius we need to set up a right triangle that has the radius of the deferent as one side and the radius of the epicycle as another. Then, if we can determine one of the non-right angles in this triangle, we can use trigonometry to find the ratio we want. So we are constrained to use right triangles because it is only to these triangles that we can apply our trigonometry (or chords, as the Ancient Greeks would have done).
For an inferior planet this turns out to be surprisingly easy. We wait until the planet has reached its maximum elongation from the Sun (East or West). At that point we can show, geometrically, that the Earth-Planet line must be perpendicular to the line from the center of the epicycle to the planet. Now draw a line from Earth to the center of the epicycle and we have our right triangle. Not only that, but one of the non-right angles is simply the elongation of the planet!
This situation occurs because, as we figured out earlier, the center of the epicycle must lie along the Earth-Sun line in order to keep the inferior planet close to the Sun. So all we have to do is measure the maximum elongation of the planet, and trig will give us the ratio we desire. It's really quite pretty the way the need for a right triangle is met at just this moment of maximum elongation, and how the requirements of the model (epicycle center of Earth-Sun line) allow us to connect this geometry to something we can actually observe. My students are able to figure all of this out with some guided exploration of my Inferior Ptolemaic simulation.
For superior planets it turns out to be much harder. We can set up a right triangle again, no problem. We just wait until the planet is in quadrature (90 degrees from the Sun, as seen from Earth). Again, our theoretical principles help us out. In order for Ptolemy's model to work we know that the Earth-Sun line and the line from the epicycle center to the superior planet must remain always parallel. When the planet is in quadrature the Earth-planet line is perpendicular to the Earth-Sun line, by definition. But this also tells us that the Earth-planet line is perpendicular to the line from the epicycle center to the planet. This gives us two sides of a triangle with a right angle between them. We close the triangle with a line from Earth to the center of the epicycle, and voila! A right triangle. Again, my students can get to this point with some guidance and my Superior Ptolemaic simulation.
Here's the problem: we cannot directly measure any of the non-right angles in this triangle. BUT we can figure one of them out. If we measure the time from when the planet is in opposition to when it is in quadrature, then we can determine the angles through which the Sun and the center of the epicycle move during this time. We can do this because we know both of these things move at a uniform rate (remember we are disregarding eccentrics and equants here), and we know the period for each motion: one year for the Sun, and the planet's zodiacal period for the motion of the epicycle center around the deferent.
Once we have found the angles through which the Sun and epicycle center have moved, we can find a simple relation between these two angles and one of the angles in our right triangle. So although we cannot measure any of the non-right angles in the triangle directly, we can calculate one of them by making use of something we can observe (time from opposition to quadrature) and our theoretical principles (uniform motion on circles with known periods).
OK, so once we have that angle we can do some more trig and find the ratio we want. So it works out, in the end, for both inferior and superior planets.
Armed with the ratios of epicycle to deferent, we can then determine the ratio of the maximum distance of the planet from Earth to the minimum distance. Then, armed with one more principle and one more observation, we can determine the size of the Ancient Greek universe. The principle we need is that there should be no empty space. So the maximum distance to Mercury, for example, should be equal to the minimum distance to Venus, and so on. That way there is no overlap between the planets, but there is also no wasted space.
The additional piece of data we need is the daily parallax of the Moon. More on this later, but the Ancient Greeks were able to use this measurement to determine that the Moon was about 30 Earth diameters away from Earth's center. This agrees well with modern measurements. From there, we can assume Mercury's closest approach is just beyond the Moon's orbit. Venus grazes the maximum distance of Mercury. The Sun orbits just outside Venus' reach, and Mars comes in to just graze the Sun's orbit on the outside. The minimum distance of Jupiter is the maximum distance of Mars, and then we do the same thing with Saturn and Jupiter. Finally, Saturn's maximum distance will equal the distance to the Celestial Sphere on which the fixed stars sit.
It works out that the radius of the Celestial Sphere is just over 5000 Earth diameters. So the diameter of the Celestial Sphere is 10,000 times that of the Earth. If you compare volumes, we find that you could fit one trillion Earths inside the Celestial Sphere if you could pack them in with no empty space. To the Ancient Greeks, this universe seemed quite large. One trillion is a big number, and the Earth is a big thing, so a trillion times the volume of Earth is pretty darn big. Of course, they didn't know that the actual size of the solar system (much less the universe) makes their size for the universe seem tiny.
This semester was the first time I tried this activity. Overall I think it went fairly well. Some groups were already a bit behind, and I don't think they caught up. It was a time consuming activity because there is a lot of math (geometrical diagrams, trigonometry, proportionalities, as well as lots of basic arithmetic). My students are all capable of doing this math, but it doesn't come quickly to some of them because they have not had to think mathematically very much, at least not for some time. But I think they all got through the hardest parts, with a bit of guidance.
Now they are all set to the take their observations of their fictitious solar system and build a Ptolemaic model. And now they know why we cared about measuring the time from opposition to quadrature for a superior planet.
Tomorrow we jump forward in time by more than a millenium and start talking about Copernicus!
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