## Philosophy 167: Class 8 - Part 3 - Discourse on Method- Descartes' Theory of Lenses, and the Reduction of Symbolic Methods to Geometry.

Smith, George E. (George Edwin), 1938-

2014-10-21

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Discourse on the method of rightly conducting one's reason, and seeking the truth in the sciences, and in addition, the optics, the meteorology, and the geometry, which are essays in this method. The originally proposed title is closer to his spirit.The plan of a universal science which is capable of raising our nature to its highest degree of perfection.

In addition, the optics, the meteorology, and the geometry in which the author, in order to give proof of his universal science, explains the most abstruse topics he could choose. And does so in such a way that even persons who have never studied can understand. That's his title, proposed title.

I think it was Mercer who talked him into a shorter title, but I can't be sure. The scope is rather large, of course the part we call discourse on method, is again published now as a little pamphlet its in ergo cogito ergo sum. The ergo doesn't occur in the meditations by the way, it's only in the discourse.

But that's at least taught. All this is in French. The optics as you see here is fairly extensive. I'm gonna look at it in just a little. The geometry I'll talk about more. It's a very important work in the history of mathematics. Though not quite in the form here.

And then the meteorology, you can see again how much is doing. Salt, winds, clouds, snow, rain, hail, storms, lightning. It's explaining everything there is that's in the meteorological world. The most famous part now is probably his account of the rainbow. Which is where I have Gal Kroger I think open to.

Because I was trying to look up something about it. While preparing this class. I'm not going to talk much about it. I will pass, we now have a reasonably good English translation of the whole thing, adapted from a prior translation. That Hackett has put out. Hackett can do some very nice thing.

But this is complete, and at the back is the original index for it. Which runs some, I think, 20 some pages. And it's not really an index. It's a table of contents on topic by topic, okay. And I've put that in supplementary material. So you could get a sense of just how many things are covered in these three essays.

It's mind blowing how much he tried to cover. The optics is the first place, as far as I know, that Snell's law of refraction was actually published. Snell discovered it and we think Descartes rediscovered it independently of Snell but we have no way of knowing that. Descartes didn't credit Snell, but Descartes wasn't prone to giving anybody credit for anything.

So you can't draw much in the way of conclusions from it. He, Descartes claimed to have derived Snell's Law or the Law of Refraction let's call it. Derived it from the physics of light, a claim that Newton, you'll see in the Principia disputes, in fact, ridicules the very thought of it.

The point though is he really did understand the principle underlying lenses. Maybe that the sine of the angle of refraction. Excuse me, the sine of the angle of incidence, and the sine on the other side of this. Is a constant for any material. The ratio of those two is a constant for any material.

And that lets him construct the first real theory of lenses. In which he shows why the phenomenon of spherical aberration occurs. It is a spherical lens using Snell's law does not focus all the rays in the same point. I've already mention I didn't emphasize this. The way they started fixing that in practice was not use the whole lens, cut the aperture down.

So you're getting less and less of an effect from the more curved part of the lens. But of course then you have less lights. So you see, you get less well defined images, that's the price you pay. But he did account for that, discovered that a hyperbolic lens would give you proper focus.

And even try to invent, try may be unfair. He certainly worked an inventing a means for grinding hyperbolic lenses. I hope you can all easily picture why it’s not hard to grind a spherical lens. Because you just turn it on a lathe, and move the cutter appropriately. Hyperbolic lenses don’t turn on lathes, circular lathes quite so easily.

So you need a different kind of machine, and he definitely worked on it. He also thought wrongly that he had explained what we now call Chromatic Abohration. Namely the circle that's almost like a rainbow on the very edge of an image in a telescope. It turns out he did not.

Newton did that in late 1660s. You'll see that in a few classes down from here, three or four classes after this one, of Newton doing it. But this is the first place where we have a complete mathematical theory of lenses that you can actually work with. It's a real accomplishment.

Was in French, and I do not know that Newton ever read it. He certainly knew what was in it, okay? But he could've learned that in any number of ways. Because it was a very, very widely referred to book. Geometry's more complicated. I could easily do a whole course on the geometry, I'll pass that around as well.

Again, we have the fortune of somebody doing facing page French and English. Which is always so nice. The Lamone I'm passing around is from my late good friend, Michael Mahoney. You'll encounter him again. He was a professor at Princeton, and a Dibner fellow with me. This book goes back.

The geometry goes back a little further. Geometry was published in 1637 in French. Then in 1649 a Dutch professor at Leiden reissued it in Latin. And then again 10 years later, a second edition in Latin. It's those two editions in Latin that Newton owned. But the picture of them as Descartes' geometry is misleading in one respect.

The first Latin edition ran about three times longer then Descartes' geometry. Because what Van Shuten did was have his students solve problems with the methods and the geometry. And then added them in, problem after problem with the solution in the book. By the second edition, the second edition is like this thick.

And it has problem after problem, by student after student including Hoigans. And that's the second edition with all those problems solved using the methods that Descartes was promoting. That's the one that had the huge effect on turning Newton into a mathematician. What happened, just quick aside, Newton started studying math as a sophomore in college.

And the easiest way to describe it is, 18 months later he was the world's leading mathematician. Because he was in front of everybody else 18 months later. And no single book did more to move him forward than the Von Chuten's edition of Descartes geometry. So that's an important book.

Unfortunately, it's never been translated. I keep trying to find somebody who will translate it, cuz I think it'll do a wonderful service. I read portions of it. We had copies of it when I was at Dibner Institute. And I certainly looked at it a great deal. And, it's extraordinary to look at the individual problems solved, and start appreciating how powerful the method really was.

For Decart himself, he solved one problem in the book. They made it, you know, convince people the method was very powerful. It's called Pappas's problem. You'll see it solved in a different way in Newton's Principia, so I'm not gonna worry about it right now. It's an old, classical problem.

The general idea, there are two points to make about Descartes geometry. The first is that it's usually said that it's creating analytic geometry that has an absolute upside down. What it's doing is legitimating algebraic methods, by showing they can be reduced to geometric methods. Geometry had the standards of rigor.

The issue was whether algebraic methods had comparable standards of rigor. And Descartes' argument, and you're seeing the principal argument right here on this page. For example, let AB be taken as unity, and let it be required to multiply BD by BC. He does the geometric construction. If the square root of GH is desired I add, and you've already seen this construction.

This is the mean proportional. I describe the circle FIH about K in center and draw from G a perpendicular, etc. So that much you've seen. Now the part stands out. Often it is not necessary thus to draw the lines on paper, but it is sufficient to designate each by a single letter.

Thus to add the lines bd and gh I, I call one little a the other little b and write a plus b. Then a minus b. Then you see the squared root, the cubed root, etc. He's just popping these all out. And his point is said very explicitly.

It's not actually necessary to draw the lines. You can just do it symbolically. Now the trouble is, they can't all be reduced to geometric methods. Cube roots are not so easy to do as mean proportionals. So what Descartes then does, the other page I'm showing you from this, is introduce motion.

Now this motion is a very complicated one, so I'll read it out loud. Now s the angle XYZ is increased, the point B describes the curve AB, which is a circle. While the intersections of the other rulers namely the points D, F, and H describe other curves. Which the latter are more complex than the first.

And this complex in the circle. Why did they do that? Because the points E, G, and C, are moving as you open up the angle. Those points are moving horizontally down. And so the dotted line is what you describe, what D describes, on this ruler. A different dotted line is what f describes.

And a still different one h describes. As you open the ruler up and the footnote gives you the different curves that are being described. And what Descartes is proposing is this is just as legitimate as a construction with ruler and compass. There's no reason whatsoever not to include certain kinds of motion in to geometry.

Even to the point that he remarks near the end of this, or any other That can be thus described cannot be as clearly. I should read the whole thing. Nevertheless, I see no reason why the description of the first, the circle, cannot be conceived as clearly and distinctly as that of the first, the second one.

As clearly and distinctly as that of the circle, or at least as that of the conic section. or why that of the second, third, or any other that can be thus described, cannot be as clearly conceived of as the first. And therefore, I see no reason why they should not be used in the same way in the solution of geometric problems.

So this is a real expansion of geometry, simultaneous with showing algebra can be reduced to geometry. So that we can thereby reduce algebraic problems that don't sit with compass and ruler, to the extended geometry. That's what he's doing. And he ends up imposing a criterion for distinguishing geometric curves from mechanical curves.

Geometric curves are ones that are constructed out of a single continuous motion, or two motions in which there is a relationship between the two. So that the first motion totally dictates the second. To say it differently. Curves that are generated by two separate motions, that are not forcibly linked to one another.

Those are not geometry. Those are mechanical curves. And he takes those not to be exact in the way that his geometric curves are. That's the point he's making. And Newton railed against that. Newton saw no reason whatsoever why geometry should stop with one kind of motion, instead of going beyond that.

That's one of the reasons I'm showing this to you because it's gonna make a significant difference later on. But at any rate, this is the new geometry that had a very, very substantial effect towards making purely symbolic methods. That is symbolic methods without an accompanying figure, making them fully legitimate.

And, of course the fruition of that came when Newton, using symbolic methods, discovered the Calculus. And 12, 14 years after that, some such number, Leibniz independently discovered the calculus too. But it's the legitimation of symbolic methods that's going on here. It's really producing a major transformation. And it's hard not to credit Descartes with the push for doing it, because Because geometry did.

The only thing one has to be careful about is giving him credit for everything. Other people picked it up and did things. He did not come back to it. He did not continue to work in this framework. He published the geometry, and did virtually nothing else in math.

And by Newton's standards he was a very, how to phrase this is always an interesting, mathematicians being rather sensitive about statements like this. Newton regarded him as a very not deep or insightful mathematician. He didn't really see things at the depth he should have the way Newton, of course, did.

Now you can judge that for what it's worth, but I just gave you an example of the kind of thing Newton couldn't see the reason for. Why, if you're gonna allow motion, doesn't any determinant motion suffice? You can prove theorems about any determinate motion, why does it have to be linked in this way?