Analytic Geometry

CHAPTER 2 Analytic Geometry Euclid talks about geometry in Elements as if there is only one geometry. Today, some people think of there being several, and others think of there being infinitely many. Hopefully, after you get through this course, you will be in the second group. The people in the first group generally think of geometry as running from Euclid to Hilbert and then branching into Euclidean geometry, hyperbolic geometry and elliptic geometry. People in the second group understand that perfectly well, but include another, earlier brach starting with Rene Descartes (usually pronounced like “day KART”). This branch continues through people like Gauss and Riemann, and even people like Albert Einstein. In my mind, two of the major advances in the understanding of geometry were known to Descartes. One of these, was only known to Descartes, but Gauss, it seems, figured it out on his own, and we all followed him. The other you know very well, but you probably don’t know much about where it came from. This was the development of analytic geometry, geometry using coordinates. The other is not known very well at all, but Descartes noticed something that tells us that geometry should be built upon a general concept of curvature. What is generally called Modern geometry begins with Euclid and ends with Hilbert. The alternate path, the truly contemporary geometry, begins with Descartes and blossoms with Riemann. Note that, as is typical, modern usually means a long time ago. We’ll look at analytic geometry first, and Descartes’ other piece of insight will come later. You’ll often hear people say that Descartes invented analytic geometry, but they usually don’t go into much more detail than that. Descartes’ Discours de la Methode was first published in French, I believe, in 1637, and it is an important book in philosophy. An appendix to this book is known as La Geometrie [Descartes], or in English The Geometry. This, apparently, is where analytic geometry was invented. Many people associate axioms and proofs with the word geometry, and you may think that your only exposure to geometry was in your high school geometry class. On the 7 1. CONSTRUCTIONS WITH STRAIGHTEDGE AND COMPASS 8 other hand, most of what you know about geometry probably came from your high school algebra and college calculus classes. Think about that. This is a manifestation of the power of Descartes’ approach. I’ll be referring to the Dover publication of a translation of La Geometrie. A lot of the “classics” are available through Dover and other publishers, and they’re a lot cheaper than most of our textbooks. I paid $8.95 for my copy a few years ago. The book I have has copies of the pages from the originally published version along with a translation into English. There should be a copy in the library, and it wouldn’t be a bad idea to buy one for yourself. When you’re teaching calculus or algebra, and you want to say that analytic geometry and cartesian coordinates are due to Descartes, it would be nice to be able to wave the book around in front of class. 1. Constructions with straightedge and compass Traditional geometry often deals with constructions with straightedge and compass. You can see this in Euclid’s First Postulate, “To draw a straight line from any point to any point,” and his third, “To describe a circle with any centre and distance.” With this, we allow ourselves only the ability to do the following. Given two points A and B, we can draw a straight line through the two points with the straightedge, and given a third point C, we can draw a circle with center C and radius equal to the distance between A and B with the compass. It should be emphasized that the straightedge is not a ruler, and so measuring lengths with it is against the rules. I don’t see this as a practical approach, but like driving with your left foot, it is exciting, and it will instill a greater appreciation for life. In some sense, Euclid’s Elements is as a very methodical description of the things you can do with straightedge and compass constructions. It actually makes more sense to think of it this way, as opposed to thinking of it as an axiom system. The first three postulates tell you how you can construct geometric figures, draw a line through any two points with the straightedge, extend a line you already have with the straight edge, and draw circles with the compass. The last two postulates give some basis for interpreting what the figures represent and that the things you see in a figure always behave the same way. Euclid’s Elements consist of thirteen Books, and these contain 432 Proposi- tions. The propositions are basically theorems that tell you that a certain kind of 1. CONSTRUCTIONS WITH STRAIGHTEDGE AND COMPASS 9 figure can be constructed (the proof tells you how) or some fact about a particular kind of figure. We’ll look at some of these propositions to get some sort of feeling for Euclid and to help motivate Descartes’ work in analytic geometry. Euclid’s Proposition I (from Book I) states [Euclid, p 241] On a given finite straight line to construct an equilateral triangle. Here Euclid is saying that if you have a line segment (finite straight line), then you can construct an equilateral triangle (a triangle with three equal-length sides) with this segment as one of the sides. Euclid’s proof goes something like this. Let’s say our segment has endpoints A and B, and we’ll call the segment AB. We then draw two circles each with radius AB, one with center at A and one with center at B. The circles will have two points of intersection, C and C0. Both 4ABC and 4ABC0 are equilateral triangles, since AC, AC0, BC, and BC0 are all radii of one of these two circles. See Figure 1. C AB Figure 1. Given segment AB, we can construct an equilateral triangle 4ABC. For some reason, Euclid used a collapsing compass. He could put one end at a point (the center) and the drawing end at another point, and then he could draw the circle. Once he picked it up, however, the length of the radius was lost. We’re obviously not talking about a real-world compass, and the motivations here are probably that Euclid was trying to start with the most basic assumptions possible. In his second proposition, Euclid shows that a collapsing compass is equivalent to a non-collapsing one. This is of no concern to me. I’m interested in how Euclid’s big geometric ideas got us to where we are today. 1. CONSTRUCTIONS WITH STRAIGHTEDGE AND COMPASS 10 Let’s skip up to Proposition 4 [Euclid, p 247] If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend. This proposition illustrates one of Euclid’s logical flaws as an axiom system. Eu- clid starts the Elements with some basic assumptions, and then seems to prove the propositions from these assumptions. The proof he gives for Proposition 4, however, says little more than “it’s true, because it’s obviously true.” If you read the statement carefully, you may recognize this as the side-angle-side criterion for the congruence of triangles or SAS. One of Hilbert’s fixes is to assume SAS as an axiom. Ge- ometrically, SAS tells us a couple of things. One is that a triangle only has three degrees of freedom. In other words, designating a angle and the two adjacent sides completely determines the triangle (the lengths of all its sides, the measures of its angles, and its area). It also expresses the uniformity of the Euclidean plane: the geometry of a triangle is the same no matter where it is. Before we move on, let’s look at two of Euclid’s early propositions and some of the constructions we’ll use in exploring Descartes’ work. Proposition 11 tells us how we can construct a perpendicular. For example, suppose that we have a line l and a point P on it. To construct a line through P that is perpendicular to l, we would do the following. Draw a circle (with any radius) centered at P . This would give us two points A and B. See Figure 2. ABP Figure 2. Given a point P on a line, we draw a circle centered at P to find points A and B. Now we do what we did in Proposition 1, and draw circles centered at A and B, both with radii AB. Then we draw the perpendicular through the two points where 1. CONSTRUCTIONS WITH STRAIGHTEDGE AND COMPASS 11 the two circles intersect. Let’s call the two points C and D. See Figure 3. The triangle 4ABC is an equilateral triangle, which means that the three sides have the same length. It’s probably obvious to you that the three angles must be equal also (i.e., that the triangle is also equiangular), but it’s not terribly easy to prove this (actually it’s technically impossible) within Euclid’s postulate system. That’s much of what the previous ten propositions are trying to establish.

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