Just continuing for a moment now, this is jumping ahead to 1676. Nicholas Mercator, in 1669, published a paper in field transactions, which again is coming up. The philosophical transactions of the royal society. The sequence goes as follows, Cassini published those values for the motion of the orbits of Jupiter's satellites.

And for the outermost satellite, where the eccentricity is not negligible, so it's not a circle, he used an equant at the empty focus. And Mercator publishes a fairly nasty, there's a lot of nastiness at this time. If those who want to see how bad it is, look in my select sources at the end, the exchange between Wing and Street over whether or not parahelia and apelia precess, where Street said no and Wing said yes.

And just the titles, when you look in the notes at the end, they will amaze you. I didn't bring those today cuz I couldn't carry everything. But they really, in the titles, they're calling one another stupid, etc. Correcting his stupid remarks. It's a better word than stupid. I think it's ignorant, but regardless.

So Boulleau, when this work of Cassini's comes out, he publishes a short article simply announcing the equant doesn't work. It's always going to give you significant errors. The larger the eccentricity, the greater the error. And he concludes any construction that does not come very close to the area of rule is always gonna fail.

And Wilson took this to almost be, well he actually, in the paper I assigned you, what he wrote in the 1660s I don't think he had looked at this book. Wilson took that to be an endorsement of the area rule by Mercator. The thing is, just a few years later, 1676, he publishes this book which I have two copies of.

I have a copy I use all the time because it's pocket book. This is a quite beautiful book published in 1676. It is the book Newton recommended for serious mathematical astronomy to prepare for the Principia and very definitely read it. He even wrote, at one point, a slightly nasty letter to somebody else complaining that Mercator had written him a letter asking him a question.

He had given him the answer. Mercator had adopted it in the book without giving him credit and he was a little peeved at that. But neither here nor there. What Mercator does in this book on the area rule is he has a whole section of the book. He starts with the equant and shows you the equant is inadequate.

Then he takes the area rule, shows you it's adequate but it doesn't appear to be exact. Then he takes Boulleau's construction and shows it's just as good as the area one. The choice between them, observation can't choose between them. And then what does he do? He adds another construction himself.

So his construction is actually to take the circumscribed circle and literally displace the whole ellipse out of the circle to give you an alternative to the area rule. So this is the fifth alternative way of locating planets on their orbits. We have two from Weng, one from Boulleau, one from Mercator, and of course, the area rule.

And they're all in competition with one another. And to show you how comparable they are, I'm just going to go back. Here's something you saw in the third week, Kepler's table for the 28 oppositions of Mars, in which he, if you remember, he got the errors down, the discrepancies down to no worse than five minutes of arc and the latitude's down.

Well, he doesn't show error in latitude here, but it's an enormous achievement he was very proud of. So what does Boulleau publish in 1657? The exact same 28 showing he agrees with him as well as Kepler does. Actually, his are slightly better than Kepler's when you look at the differences here.

But it's the same 28 observations. Quick question. Why are they using only these observations? Because none of the rest of Tico's data had been published. So, here sits Kepler. He's bragged about this. They're all gonna take exactly the same observations as Kepler and show they do just as well.

Here's Street doing it. Street's largest error is four minutes in the same place that Kepler has a five minute. But again, it's clear that Street does just as well as Kepler. And remember, he's using two different principles from Kepler. He's using the three halves power rule to get the mean distance, and he's using the Boulleau construction.

Then we get the Wing. I didn't bother to show the earlier one, just the 1669 one. Wing doesn't do it as a table, he has a summary table. But then he gets in there and comments on each observation, discussing the observation and the calculation and showing how good he is.

So he's really quite bragging about the whole thing. And finally you have Mercator. And you notice the largest Mercator error is less than, oh no it isn't, there's a four minute. So what we now have is the same 28 observations being used to judge the quality of mathematical astronomy at the time.

We have completely different ways of locating planets on the orbit, five different ways of locating planets on the orbit, four of them represented here and they're all comparably accurate. And of course, the one that is most computationally extreme is Kepler's. That is, all of them are simpler than Kepler's, so if you're gonna to do it by a simplicity criteria, Kepler's is the computationally most complicated.

Doesn't mean it's complicated in other ways. That's as of 1676. Now, the one other crucial thing here is Newton knew all these tables I just displayed except the Kepler one. But the Kepler one is shown in Mercator's book anyway to reproduce that. Okay? So Newton knew there was the serious question about which of the alternative ways of doing mathematical astronomy is to be preferred.

And as you will see, that's actually the primary question the Principia answers is which of these is to be preferred. How good were they? Well, the situation's much more complicated than I've let on. I've only shown you the different theories and where they stood. All sorts of people were putting out ephemerides telling you what you were gonna observe in the next year, and nobody was recording how good or bad they were.

So we don't actually have records of people doing observations and comparing if they're ephimerides. All we really have is that wonderful remark by Flamsteed saying he judged Streets esteemed, Streets to be the most exactest. But you can see here, these are Wing and Agroly and constant. The errors are quite substantial.

In the case of Jupiter and Saturn, those are real errors, by the way. Those are Jupiter and Saturn. They're starting to realize they're wandering all over the place versus any calculation. But the point I'm making here, good. Let me stop with that. The point I'm making is if you're somebody wanting to know which of these is the best, either of the five I just put on display, or any of the others that are put forward, there's no good clearing house at the time.

Okay? All you can do is look at the comparison of the 28, and they're more or less the same. But then you have the question, well that's good for 28 spots. What about Mars all the time, Mercury all the time, Venus all the time, all of that? There's no place to judge it.

Okay, at this time in the 1660s, there's no real means of finding out. Now by 1670s, we have two observatories, one sitting in Paris with Cassini and one in Greenwich. But they're not doing much to publish on this. They're too busy trying to reconstruct astronomy from the ground up using telescopes.

So Newton was not in a position to judge how good any of these approaches actually turned out to be mathematically at the time, 1679, when as you'll see in two weeks, he first gets started on the path to the Principia. And nor was anybody else. And of course when many different people publish ephemerides and claim they're just as accurate as anybody else's and nobody's doing careful observation to evaluate them, the situation can be pretty confusing.

Some people in the know, knew. So for example, when Newton was desperate after 1679, he would write Flamsteed a letter asking who can I trust? And Flamsteed and Cassini respected one another and they had very little regard for anybody else on observations, but that's the state we arrive at.