[19847386_16_0] In lecture 15 I talked about Leonhard Euler, the greatest of all mathematicians. But if there is one person who might challenge him for that title, it's Carl Fredrich Gauss, about whom I'm going to talk in this particular lecture. Gauss is responsible for referring to mathematics as "the queen of the sciences". And largely because of that, later mathematicians and historians of mathematics would refer to Gauss himself as "the prince of mathematics". Gauss was born in 1777. He was born in Brunswick, the son of a labor. And eventually the Duke of Brunswick would become his patron and support him through his early work in mathematics and the sciences. The Duke of Brunswick unfortunately was killed during the Napoleonic Wars and so Gauss would need to seek employment. And he would become director of the astronomical observatory in Gottingen. And eventually he would take a position as chair of mathematics at University of Gottingen.

He did absolutely groundbreaking work in a number of areas of mathematics including algebra, geometry and in number theory. He also did very important work in astronomy. And one of the important results that he did, one of the things that really made his mark as an astronomer, was when the asteroid Ceres was spotted in the heavens in 1801. It was spotted just before a path behind the Sun. And only a few observations had been made and astronomers tried to figure out where the asteroid would reappear as it came out from the other side of the Sun. And Gauss was among those who did those calculations. And Gauss' calculations were the most accurate. He was able to just from a few observations before the asteroid disappeared. He was able to predict the exact location and the exact time when Ceres would reappear.

Gauss' also noted for his work in electricity and magnetism. The unit the gauss that is common in this field this name directly for him. One of his important books early in his career was a book on number theory, called Investigations of Arithmetic, published in 1801 and this is arithmetic in the sense of Diophantus' Arithmetica. So he calls it arithmetic what he really means is what today we refer to as number theory, the study of the integers. And this received a lot of attention throughout Europe. And one of the people who became fascinated by this book and began to study it was a woman by the name of Sophie Germain. Sophie Germain had been born in 1776. She was interested in questions in algebra and questions in number theory. And she corresponded with a number of the great mathematicians of the time including Lagrange and also Gauss. And she went under the pseudonym of Monsieur Leblanc. She was afraid if she admitted the fact that she was a woman and her correspondence, people would not take her seriously. Eventually both Lagrange and Gauss realized that they were corresponding with a woman and we know that Gauss was quite delighted to discover that there was a woman who was doing such important work in mathematics. We're going to return to Sophie Germain later on when I talk about Fermat's last theorem. Because she will be one of the many players in the story of the eventual proof of Fermat's last theorem.

But in this lecture I want to focus on Gauss' work in geometry. And I'm going to back up a bit right back to Euclid's Elements. Euclid's Elements include five postulates, five axioms, five things that he assumed in order to be able to proceed with the development of geometry. The first of these is that if you have two points in a plane they determine a line. And there is only one line that goes through these two points. If you've got a point and a length that denotes the radius that uniquely determines a circle. All right angles are equivalent. And the fifth postulate is a rather strange sounding one. It says that if you've got two lines and you've got another line that cuts across it. And you take [19847386_16_1] a look at the two interior angles of the line that cuts across. If both of those angles are less than right angles, then the original two lines must meet some place on that side of the transversal. This is known as the parallel postulate. And it's a little bit easier to state what's going on if it's put in an equivalent form people fairly quickly realize that this fifth postulate is equivalent to saying that if we've got a line and if we've got a point that is not on that line, we're working in a plane, then there exists exactly one line that passes through that point and never intersects the given line. So there is one parallel line through this other point. And this was a postulate that a lot of people wondered it was really needed. Is it possible simply to take these natural assumptions about geometry and show that the parallel postulate must follow from those. There seem something arbitrary about the parallel postulate. And a number of people tried to do this, to find a way of proving the parallel postulate or at least reducing it to something that's seemed obvious where was clear that there have to be one line through this point and only one line through this point. One of the people who made progress on this was Girolamo Saccheri. Born in 1667, he died in 1733. And he looked at what would happen if there was no line through this point that did not intersect the given line. In other words there's no parallel line. Every line through the point must intersect at some point. Or what would happen if there's more than one line through this point that never intersects the given line. And what Saccheri was able to show is that if there is no parallel line, then the sum of the angles in any triangle must be greater than 180 degrees. And if there is more than one parallel line through this point, then the sum of the angles in any triangle must be strictly less than 180 degrees. Now we all know that the sum of the angles in any triangle is 180 degrees. So that seems to settle it. But in fact all that shows us is that the parallel postulate and this result about the sum of the angles in any triangle are actually equivalent statements. If you know the sum of the angles in any triangle you can derive the parallel postulate from that. If you know the parallel postulate you can derive the formula for the sum of the angles in any triangle. But you can't really say that one is more obvious than the other. They simply are equivalent statements.

The next person who really made progress is Johann Lambert. Born in 1728, he would die in 1777. And he fairly quickly dismissed the possibility that you might have no parallel line through that point. But he was really concerned about this question of more than one line through the point that never intersects the given line. And how could you show that that could not happen? And he explored the kind of geometry that would follow if you could have more than one parallel line. And he came up with something that's seemed totally impossible. He was able to show that if you've got more than one parallel line through the point then if you look at the sum of the angles in a triangle not only is that less than 180 degrees, but the sum of the angles in a triangle is going to depend on the area of the triangle. And that seems totally counter intuitive. Why should the sum of the angles in a triangle depend on the size of the triangle? We're used to Euclidean geometry where you can scale things up and that does not change the sum of the angles in a triangle. But what Lambert showed is that if you've got this strange kind of geometry with multiple parallels when you try to scale things up that necessarily forces the sum of the angles to change. What real progress on this question really now begins to happen with Gauss, who study geometry. He did not set out to study this problem of whether or not the parallel postulate is correct. He was interested in geometry on various kinds of surfaces. So considering the geometry on the surface of the sphere, or considering geometry on the surface of a cone, or some other two dimensional surface. And he explored the idea of distance, so curves that trace the shortest distance between two points, or areas of pieces of these surfaces and how to do that. And he realized that the key to finding distances or finding areas was really to use the key ideas of calculus, to go back to Leibniz's idea of the differential, this infinitesimal little change and build up your distance by piecing together little straight lines, even though you might have a curving surface. Build up your distance by these little straight lines, the infinitesimals, and use the technique of calculus in order to define distance in the same sort of way you can use the techniques of calculus to define the idea of areas.

Now when we're working on a surface, say the surface of a sphere, there is something that corresponds to a straight line but it doesn't look like a straight line in a normal sense. I live in Minneapolis St. Paul, and there is non stop service from there to Amsterdam. When I fly from Minneapolis to Amsterdam though, I do not fly mostly east. Amsterdam is a little bit north of Minneapolis St. Paul, but only a little bit, it's mostly east. Yet when my plane takes off from Minneapolis I start out going almost to do north. What I'm doing is traveling on a great circle. If we're on the surface of a sphere, the shortest distance between two points is given by what's called the great circle. And the great circle is defined by taking the plane that goes through the two points that I want to travel between, and that goes through the center of the Earth. So I take a plane that includes my starting point, my ending point, and the center of the Earth. That plane is going to slice the Earth along an arc, and that arc is the great circle. That arc gives you the shortest distance between the two points that you want to work with. And so straight lines now correspond to, on a surface, straight lines become these great circles. And you can begin to build triangles on a surface out of these great circles. So for example one of the triangles that you can get is the triangle that goes from the North pole down along a line of longitude to the equator, and then you take another great circle that goes from the North pole, along another line of longitude down to the equator, and then as the third great circle you simply follow along the equator. And this gives you a triangle on the surface of the Earth, a triangle on the sphere. And if you think about it, each of these lines of longitude hit the equator at a right angle. So what I've got is a triangle that has two right angles, plus another angle at the top. I've got a triangle where the sum of the angles is strictly greater 180 degrees. And actually if you think about it if your lines are great circles, then if I take any great circle on the surface of the Earth [19847386_16_2] a complete circle and I take any point that's not on it. And I look at it at any great circle that goes through that point. It must intersect the other great circle. There is no such thing as a pair of parallel lines when we are working on the surface of the Earth. And so this exactly agrees with Saccheri's prediction that if you don't have parallel lines then the sum of the angles has got to be strictly greater than 180 degrees.

Now Gauss was also interested in something that is called curvature. So he looked at these surfaces and a surface that is flat is said to have curvature zero. If I take a sphere whose radius is one, Gauss said that that has curvature equal to one. And as the radius gets smaller, as the surface turns more abruptly, the curvature is going to get larger. And so as I take smaller spheres, the curvature is actually going to be the reciprocal of the radius. So the curvature is going to get greater as the surface bends more sharply; the curvature gets closer to zero as the surface begins to flaten out. Now not every surface is the surface of a sphere, and so the general way that you have of trying to determine the curvature is to look at a piece of the surface and try to find that sphere that the most closely approximates the way that surface is bending at that point. So a surface might have different curvatures at different points. And you can also have a surface that is bending into different ways, consider a saddle where if you travel along it from the front to the back of the saddle it curves upward. But if you travel over the saddle from side to side, it bends downward. And so what Gauss said in this case we have something that he called negative curvature where in one direction it curves down while at right angles to that it's curving upward. And so every surface has a curvature. If it's a flat surface the curvature it's zero. And otherwise it has either a positive curvature if it looks like a piece of the surface of a sphere, or a negative curvature if it looks like a piece of the surface of a saddle. And a nice example of a surface that has negative curvature every place is the Enneper surface. And I have a nice picture of a snow sculpture that was created by a colleague of mine at Macalester Stan Wagon and a team that he put together for a snow sculpting competition at Breckenridge. This Ennerper surface stands about eight feet tall, eh stood about eight feet tall. That of course no longer exists. It did win second place in the Breckenridge snow competition and they called it Rhapsody in White. And one of the things that Stan Wagon has pointed out is that surfaces of negative curvature are ideal for trying to do snow sculptures because if you've got two different ways in which the surface's bending [19847386_16_3] that really tends to strength in the surface and so this is an idea that is often used in architecture. If you want interesting curved surfaces, do surfaces of negative curvature. They not only are aesthetically appealing, but they also tend to have greater strength than a surface that has positive curvature.

Now as I said on the surface of a sphere we've got positive curvature, and we also have the fact that the sum of the angles is greater than 180 degrees. And if you remember what Lambert said was that the sum of the angles is going to depend on the area. And Gauss actually found a relationship between the curvature, the area of a triangle and the sum of the angles. This today is known as Gauss's Theorem. And what he proved was that if you take a look at your triangle, and you figure out the total curvature of the triangle. So what you do is at each point on that triangle you look at the curvature of that point. And you find a small area around that particular point and multiply the curvature times the area. So we subdivide this triangle into little pieces. We take curvature times area on each of this little pieces. Add those up. And then let the little pieces become smaller and smaller. It's exactly the idea behind calculus. And we look at the limit of curvature times area as we sum these up over the entire triangle. And the value that we're approaching then becomes the total curvature of the triangle. And what Gauss was able to prove is that the sum of the angles in any triangle on any surface, is 180 degrees plus the total curvature. In the case of a sphere of radius one, the curvature never changes. That's always equal to one. And so the sum of the angles of any triangle on a sphere is going to be 180 degrees plus the area of that triangle times the curvature which is one. In other words the sum of the angles is always 180 degrees plus the area of the triangle.

Now Gauss was working with surfaces and he clearly saw that surfaces could be non-Euclidean. But all of the surfaces that he looked at exist within Euclidean space. We're looking at the surface of the sphere but it still rests inside normal three dimensional Euclidean space. And so a question that was still out there was could space itself be non-Euclidean? And there are two mathematicians in the early nineteenth century who really explored this and I want to say a little bit about them. The first is Janos Bolyai, born in 1802. His father Farkas Bolyai has actually been a schoolmate of Gauss. And they maintained a close friendship throughout their lives. And the son Janos began working on this possibility that space itself might not be Euclidean, and began to explore what that would mean. And in 1832 he wrote up what he had done on this. And he included it in a mathematics book that his father was publishing. It was included as an appendix of that book. And the father Farkas sent a copy to Gauss, thinking that Gauss might be interested in this. And Gauss wrote back with an interesting quote. "If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it, would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last 30 or 35 years. ... So far as my own work is concerned, of which up till now I have put little on paper, my intention was not to let it be published during my lifetime." This was a real putdown. And Janos actually would not publish anything more or very little more about his idea of non-Euclidean geometry. And it points to one of the real flaws of Gauss. Gauss was brilliant. Gauss' model though was few but right. He didn't want to publish anything until he felt that he had perfected it and the time was right to put it out there. And while he understood a lot to what was going on with non-Euclidean geometry and almost certainly his investigations of what happens on surfaces led him into exploring non-Euclidean geometry in general. He felt that the time was not right to publish. And because Gauss had already done all of these things that really put it damper on what other people were interested in doing. And unfortunately this did not just happen with geometry. It happened with many of the areas in which Gauss worked. It was not until after Gauss died and people were able to go through his manuscripts that they found that he had indeed discovered many of the things that other people discovered in the early nineteenth century, but often decades before the others had and often in a much better and more complete form. And mathematics certainly would have benefited if Gauss had been more aggressive in getting his results out there and distributing what he knew because he really was incredibly talented.

[19847386_16_4] Another person who worked on the question of non-Euclidean geometries was Nikolai Ivanovich Lobachevsky. Tom Lehrer has made this mathematician famous in his song about plagiarism. And there really is no connection between Lobachevsky and plagiarism. Lehrer simply like the name and it fit into the song very well. Lobachevsky was a Russian from Kazan, born in 1792. He also had a connection to Gauss. His teacher was Martin Bartels, who earlier had taught in Germany and had been Gauss' teacher of mathematics. And so when Lobachevsky came up with his ideas on non-Euclidean geometry he also corresponded with Gauss. Gauss would eventually share these with Bolyai. And there was some thought especially on the part of Bolyai perhaps pulling together what he and Lobachevsky had accomplished. But nothing ever really came of this. And it would not be until geometers much later in the nineteenth century that non-Euclidean geometry would finally be accepted and fully worked out.

I want to close this lecture, talking about something else that Gauss did. That's going to be important for later work in later lectures. And that's his study of elliptic functions. If we go back to the trigonometric functions, the sine function. I talked about how Euler looked at the sine function and thought about this. As a function of a variable which is the arc length, the arc length of a circle of radius one and so we can easily define the sine for any number from 0 up to 2 pi. But there's no reason that we need to stop at 2 pi. Once we get beyond 2 pi we just start repeating the values of the sine function. And we repeat the values up to 4 pi and then repeat them again up to 6 pi and so on. And we can also continue in the negative direction. And so we can build up the sine function as a periodic function, a function that keeps repeating. And the same is true of the cosine function. We can do the same thing with the exponential function in the imaginary direction because recall that the exponential function e to i times x is the cosine of x plus i times the sine of x. So once we get up to e to the i times 2 pi, we are back to one. And so we can continue it beyond that. And if I think of the exponential function as I take imaginary values as I travel up along the imaginary axis, taking the exponential function that also is going to continually repeat its values. And one of the questions that then comes up once you start working with functions of complex numbers and Gauss was the one who really push this idea of working with functions of complex numbers. You realize that you've got the trigonometric functions that repeat as you move in the real direction. And you've got the exponential function that repeat as you move in the imaginary direction. Can you get a doubly periodic function? A function that repeats both in the real direction and also in the imaginary direction. This is a kind of function that actually had been studied earlier by the French mathematician Adrien-Marie Legendre. Gauss became fascinated by this doubly periodic functions, what today we call elliptic functions and he studied them in a great deal of detail. They're called elliptic functions for a strange reason. There is a connection with the ellipse but it's rather obscure. One of the ways of building elliptic functions is out of certain definite integrals. So they come up in working with the integral calculus. And one of the definite integrals that gives rise to the elliptic functions is an integral that is related to the problem of finding the length of the arc of an ellipse. And so all of these type of definite integral got the name elliptic integrals because one of them was useful in finding an arc length on an ellipse. And then these elliptic integrals became one of the ways of finding this doubly periodic functions and so the doubly periodic functions became known as elliptic functions. We will also see them sometimes referred to as Abelian functions. It's Niels Henrik Abel who I will be talking about in the next lecture who did a lot of important work on these functions. And so his name especially in nineteenth century was often associated with them.

Now if we think about a doubly periodic function, we've got a function that's going to be repeating as we move horizontally and repeating as we move vertically. And so we can look at a fundamental domain for this function. We can look at a rectangular region and as we translate that rectangular region the function is taking on exactly the same values. We can translate it horizontally. We can translate it vertically. If I know all of the values of the function within this fundamental domain I know the values of the function every place else in the complex plane. Now if I think about this fundamental domain, this rectangle that determines the value of the function everywhere, and I think about what happens as I move off the right side of that fundamental domain. I'm really coming back in on the left side. I'm moving into the next copy of that fundamental domain. And the values I'm taking on are exactly the same as the values that I would get coming in on the left side of that fundamental domain. So one of the ways of thinking that fundamental domain is to wrap it around in a cylinder. So I take this rectangle and I simply wrap it around to reflect this fact, that is I move off one side I'm coming back around onto the opposite side. But the same kind of thing happens if I go off the top of this fundamental domain I come back into the bottom of the fundamental domain. So I don't just want to wrap the left and the right side around into a cylinder I want to take the top and bottom and wrap them around and what I'm going to get is a donut, or a torus. And so this elliptic functions naturally live on a torus, on a donut. And this is the beginning of the realization that interesting functions, especially functions defined on the complex values, often really exist on some other kind of surface. And we begin here to get a connection between geometry and interesting sorts of surfaces, and problems in analysis and calculus, and the study of functions.

For the next lecture we're going to be turning to algebra and seeing how this idea of transformations will come to play a fundamental role in the development of algebra in the nineteenth century. [End]

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av19847386 听写_Episode 17

Episode 17 Gauss-Invention of Differential Geometry

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