Philosophy 167: Class 4 - Part 6 - Celestial Harmonies: Nested Platonic Solids, Physical Theorizing, and the 3/2 Power Rule.
Smith, George E. (George Edwin), 1938-
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This is the picture, I'll show you a more dramatic form, from Mysterium Cosmographicum. The idea, of course, is there are five regular solids. I don't know if I can do this. The cube, the pyramids, the three-sided pyramids, the octahedra, the isocothedron, is that right? I'm mispronouncing it, I think.
And the duoadeconhedron. And, Theaetetus approved those are the only regular figures. Figures with, think of it for the equilateral triangle. All the sides are the same length, all the faces are congruent with one another. Those are the five. And they nest inside of one another, in a way that take any one of them.
You have a sphere on it's outside, and a sphere on it's inside. The one on the outside hits the corners. The one on the inside hits the tangency points. And, of course, now we want the one on the inside to be on the outside of the next sphere.
And that's what he thought he had. These are the ratios of outer to inner sphere, going around inside or outside each of the five regular solids, is what is from Gingrich's article. This is a blow up from the actual book. It's that full page scale. I did the best I could from working.
I don't have, it's very frustrating to me. When I was at Diptner Institute, I had first editions of all of his books. An awful lot of books I scanned. I did not scan Kepler because they're all available. In his complete works. I wish now I had scanned the originals, and had the original figures.
The original figures are probably much better than the ones issued in the beginning of the 20th century. But anyway, you can see the idea for what it's worth. And I'm not gonna push it further. It shows up at the beginning of the part four of the epitome. So we'll get to that in a moment.
This is the original Latin title page, I had a choice, English or Latin and I decided to do the LAtin. I'll pass the book around of Harmonices Mudni. This is in English that I'm passing around. It has five parts, the first part is the geometry of regular figures.
The second part, the architectonic describes how they fit inside one another. The third part on psychology, astrology, and metaphysics. Talks about how the celestial realm, and the rays, and the making from it affect things on Earth. The fourth part, I skipped a part, I'm sorry. I knew something was wrong.
The third part just is the classic theory of harmonies. Going all the way back to the Greeks, but now being advanced at this time fairly much, musical harmonies. So the fourth part that I've already told you about. Then the fifth part said, the harmonies God designed the whole system so he could listen to these beautiful harmonies in effect.
That's a little bit sacrilegious. Kepler would never have said it that way. But the point is, the beauty of those harmonies explains the design of the whole system. And that's the book, and as you can see now why I might be irritated with Owen saying this is his favorite book.
Because I like to emphasize a Kepler who does things different from this, namely labors with numbers for four years. But the same man did both okay, all the time. And he was quite seriously religious. And really wanted to be a theologian all along. And this is his way of being a theologian, finally.
The nice thing about Harmonices Mundi, from our point of view, it's gonna be very important to the rest of the course. You see that's in a 19. I thought my data of the prior slide is simply wrong. That should be 1619, not 1618. I thought I was wrong.
Late in the book, while he's looking for patterns between numbers that would give you some sort of harmony, he stumbles on something. And it's going to be extremely important. The way it's normally said, the cube of the ratio's of the distances of the planets from the sun, are proportional to the square of the periods of the planets going around.
I will henceforth usually refer to it as the three has power rule. The period is proportional to three halves of power of the mean distance. Where proportional means take two periods, take two three halves power of mean distances, you have the same ratio. That's what proportional to always means.
Because for Kepler and people like, that you can't compare anything but like things. So you're gonna compare two times, you're gonna compare two, three half powers of distances. You don't write a formula down the way we would with a constant in front of it. That's just not being done.
By the way the reason for that is Eudocia ratios in Euclid, you'll see them in two weeks. Anyway if you look at it. The numbers agree pretty well. One at the top I simply take. These are his values by the way. I take the periods, I take his mean distances, I square the periods.
I cube the distances and compare. Down below I do a comparison with percent differences.The percent difference on Mercury is easy to explain. Because we observe it so, we have so much trouble observing it. The percent difference on Saturn is easy to explain. The data simply werent there. We didn't have enough good data on Saturn going around.
Because Tycho didn't live for 100 years. Even as he took data for a relatively short time, so there's less for Saturn than anything else. The curious one will turn out to be Venus. Why is Venus that far off? And we'll see next week, somebody looking at that and concluding that there was something wrong in Kepler.
And working out much of what was wrong. One person before 1640 picked up Kepler's ideas, and pursued them in a Keplerian fashion. And that person died in his 20s, unfortunately. Jeremiah Horrocks, but it was one of the things that caught his attention. Was the error being larger in Venus, than it is on either Mars or Jupiter.
Because it'd make no sense. Venus is in almost perfect circular orbit as you'll see in a few minutes. All right, that's the three has power rule. Any questions on that? That's gonna be a very big deal later in the course. Because that's gonna turn out for circular motion to entail inverse square.
That's where Newton gets the inverse square from.