I'm introducing a word here from Philosophy of Science, but it's not the word normally used. The word normally used is nomological or lawlike, nomological being an introduction from German speaking immigrants to the United States the logical positive is, I don't know who first coined it you think it was?
It's quite possible. Llamos of course is the Greek word for law, so nomological simply is law like. Goodman doesn't like the word so he uses the English law like. The way the distinction has normally gone, I'm just gonna be very brief about this.
There is an accidental generalization, all the coins in my pocket happen to be silver. And what that doesn't entail in any way whatsoever is that I put another coin in my pocket it would be silver. Okay. A nomological generalization would. That is if it were a nomological, it was due to the physics that all coins in my pocket, pockets magical, makes them silver then if I put a coin, a penny in there, it's gonna turn silver.
That's the distinction, the way it's usually drawn and I'm not gonna dwell on it because there's a really important distinction it's encapsulating and I just caught it. Can we project from the cases at hand to other cases? Okay, and we'll come back to this in the second after break.
But here I'm asking a simple question. What grounds did Kepler have for extrapolating, projecting his result for Mars which held for 24 years? Indefinitely into the past and indefinitely into the future. Now I hope you realize nothing in the data justifies that. Now you can get some if you got enough data from the ancient past, and that's one of the reasons he keeps bringing Ptolemy up because he wants to invoke that to say things are not changing.
But what you really need is to say, there's a physics there and the physics is not changing. But of course, he'd, just like Ptolemy, extrapolated. Projected is the right word. Project beyond the known data to unknown data. There's a similar thing. What grounds could Kepler have offered for projecting?
The ellipse area rule and diametral distance rule to Jupiter, Saturn, Venus and Mercury. And that's what he did. He did not try to rederive the ellipse etc for those planets, and you'll see why in just a moment. He simply said what's good for Mars is good for all the others.
Now again, in his case it really looks like he's comfortable with that because he's identified a physics for Mars and the physics is planetary. Its tied to the Sun so it would automatically apply to all the others. But it's a worthwhile question and it's the first of these questions I'm gonna ask about projectability, it's a worthwhile question to sit back and say, data for 24 years?
And you're now talking about Mars a thousand years in the past. What just advised that? And you're talking about Saturn, way out there while Mars is 1.5 astronomical years from the Sun, Saturn is ten. That's a totally different world. Why should the same thing etc, okay. And this is a table I did and you can see what it's doing.
The table itself is using the area rule and comparing how much difference in heliocentric longitude you have between and ellipse and a circle. For different eccentricities. And the top eccentricity I use is O2, which is little more than the eccentricity of the Earth. And as you can see the discrepancies, the largest discrepancy is 20 arc seconds difference from the Sun.
Way below the precision of Tycho's observations. The next one I calculate is 05. Jupiter's a little less, and Saturn's a little more than 05. It may be the flip, we'll see it in a couple of moments. There you actually get up as high as 128 arc seconds, which is two minutes of arc, but two minutes of arc is right on the edge.
Of what Tycho claimed for his accuracy, and that by then Kepler probably realized was an exaggeration in the sense that many of the observations exceeded it. You get up to point .09, which corresponds to Mars. And now you've got ones as much as nine minutes, or 6.9 minutes of arc, those are clearly recognizable from the Earth, so you really have to get up to more or less the level of Mars once you have the area role, to distinguish between a circle and an ellipse and in a foot note, I point out that I wanna see how I said that.
If you go just to circles and equant, you could not have picked out even the Mars area in ellipse versus equant inner circle. With the level of accuracy of the observations of ten minutes of arc from the past. You needed better than that to be able to distinguish them.
And that's the point I made about Tycho's data. It's not just the amount of data. It's having them to a uniform standard that's high enough that you can actually distinguish between the two. So if people had tried to do this on Jupiter and Saturn, or for that matter Venus, they would have fallen flat on their face.
There's no way to distinguish circle and ellipse. Somebody, David asked me, and I'll now repeat the point, .00429. That's the magic number, 429. That's the ratio. One minus .00429 is gives you the minor axis in ratio to the major axis. It is 4.429% less than the major axis.
That's what that number represents and I threw out constantly, it's .4, that's the rounding of that number. So this is meant to convince you he was perfectly sensible for not trying to repeat Astronomia Nova and do four or five years of calculation. As it were, each planet took a couple of years work, even after he was committed to the ellipse, etc.