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!
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