1. Euclid s postulate (axiom) system

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1 Math 3329-Uniform Geometries Lecture Euclid s postulate (axiom) system Book I of Euclid s Elements (I ll always refer to the Dover edition of Heath s translation) starts on page 153 with some definitions. Some of the twenty-three that he gives are 1. A point is that which has no part. 2. A line is breadthless length. 3. The extremeties of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. 10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. 11. An obtuse angle is an angle greater than a right angle. 12. An acute angle is an angle less than a right angle. Note that we would probably say curve instead of line in definition 2 and later. Keep that in mind when you re reading. This is somewhat normal usage in older books like Heath s translation from about a hundred years ago. Also keep in mind that Euclid did not say line. I believe the Greek word that Heath translated as line was γραµµή. The letters in this word are gamma-rho-alpha-mu-mu-eta, and if you sound out the initial letters in the names of the Greek letters, it sometimes makes sense. I would guess (probably badly) that Euclid s word for curve sounds like gramme, or maybe even gramma. If you don t believe me (and there is good reason not to), consider the word παραβoλὴ (pi-alpha-rho-alpha-beta-omicron-lambda-eta), which we ll run into later. This word translates as parabola. Euclid then gives his five postulates. These are [Euclid, p 154] 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Euclid s postulates are criticized for having logical flaws as an axiom system. To be fair, of course, we should understand that this was not his intention. In particular, if you look at the first three postulates, he is really describing the things you could construct with a straight-edge and compass. This is a scientific theory for straight-edge and compass constructions. At the same time, however, the things you can draw on an infinite piece of paper characterizes properties of the paper. Bump this up one dimension, and we re talking about the underlying nature of space. Geometry, therefore, studies scientific theories of space Quiz. 1 The second and third definitions indicate that the use of the word line is different from what we would normally call a line. What would we call Euclid s/heath s lines? 2 One rule for a construction with compass and straightedge is that you can draw a line through two points you already have. Which postulate is this? 3 If you have a line, you can use your straightedge to make it longer. Which postulate is this? 1

2 2 4 Given two points, you can put the pointy end of a compass on one, and draw a circle through the other. Which postulate is this? 5 What word do we use for the word distance in the third postulate? 2. Straight lines Before we go on with Euclid s postulates, let s talk about what a straight line is. In modern terminology, we would just say line. Now, there s no way Euclid could have known this, but we know now, thanks to Einstein, that our intuitive notion of what a straight line is can t exist in the real world. Since geometry studies the underlying structure of space, we have to be open-minded about what a line could be. Now, Euclid says a line is straight if it lies evenly with the points on itself. I m not quite sure what that s supposed to mean, but it s not particularly useful, in any case. Let s have you think about it a bit Quiz. 1 Suppose I handed you something, a yardstick, a rifle, a stretched-out piece of string, and asked you to determine whether the thing was straight or not. What would you do? 2 Another way to approach this problem is to list out characteristic qualities. In terms of measuring distances, what s the most important property that we associate with straight lines? For the first problem in the quiz, I would expect that most of the procedures you would have come with depended on the assumption that light travels in straight lines. If not, then your procedure probably depended on some notion of symmetry. For the second and third problems, I was looking for the shortest distance between two points is along a straight line, but being symmetric might have also come up. We ll expect, therefore, that if we were walking along a straight line, it shouldn t turn left or right, and it should not turn left or right equally. In general, curves with these properties are called geodesics. Definition 1. A curve l in a space is called a geodesic, if for any pair of points A and B on l, sufficiently close together, the shortest distance (within that space) between A and B is measured along l. In Euclidean spaces, like the xy-plane or xyz-space, the geodesics are called (straight) lines. That s really all lines are. We ll use the terms (straight) line and geodesic interchangeably. On the sphere, the great circles are the geodesics. We ll sometimes refer to these as lines also. Let s look at some other examples. We could take a piece of paper, roll it up, and tape two edges together to form a cylinder. At the end of these lecture notes is a rectangle with some lines drawn on it. You can cut this out and make a cylinder to see what geodesics will look like. Notice that when you roll the paper up, the length of a line segment measured along the paper will not change. When the paper is flat, the shortest distance between A and B is along the straight line, not the dotted curved one. Since the distances don t change, the same must be true on the cylinder. The straight lines on the flat paper become the geodesics/lines on the cylinder. In particular the geodesics on the cylinder are straight lines running up and down the cylinder, horizontal circles, and otherwise, helixes. Note that the distance between A and C on the helix on our paper model is shortest along the helix geodesic. That s why we require the phrase sufficiently close together in the definition. The shortest distance between A and C is measured along another geodesic, however. We can also make a cone out of a piece of paper, so just like with the cylinder, straight lines on the flat paper will become geodesics/lines on the cone everywhere except maybe at the point. There are also plans for a paper model of a cone at the end of these lecture notes. We remove a quarter-wedge, and tape the edges together. We have to be a little bit careful about how we have our geodesics pass over the cut, but since we could have put the cut anywhere, there really isn t anything special going on here. The point/vertex is different, however. The line that passes through the points A, O, and B when the paper is flat, should look a little funny on the cone. The geodesicness is messed up at the vertex. Notice that the dotted line from A to B across the cut is shorter than the solid line through O. If you try to play around with this a bit, you ll come to the conclusion that the angle AOB must equal 180 on both sides. The angle on the cut side is

3 3 only 90. One last observation. Note that the points C, D, and E are the vertices of a triangle with three right angles. There were no geodesics through the vertex of the cone, because if the angle AOB < 180 c irc on either side, then there are shortcuts on that side. We wouldn t have that problem, if AOB 180 on both sides. We have that on a flat piece of paper. Can we have that on a cone? Yes. All we have to do is to add a wedge instead of taking a wedge away. The third model shows how to do this. Here, the line through A, O, and B is a geodesic. There are 180 on one side, and 270 on the other. There is also a geodesic through A, O, and F. There are infinitely many others in between Homework. 1 Make a paper model on a cone that has a pentagon with five right angles.

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8 8 References [Bonola] Roberto Bonola (1955). Non-Euclidean Geometry (H.S. Carslaw, Trans.). Dover Publications, New York. (Original translation, 1912, and original work published in 1906.) [Descartes] Rene Descartes (1954). The Geometry of Rene Descartes (D.E. Smith and M.L. Latham, Trans.). Dover Publications, New York. (Original translation, 1925, and original work published in 1637.) [Euclid] Euclid (1956). The Thirteen Books of Euclid s Elements (2nd Ed., Vol. 1, T.L. Heath, Trans.). Dover Publications, New York. (Original work published n.d.) [Eves] Howard Eves (1990). An Introduction to the History of Mathematics (6th Ed.). Harcourt Brace Jovanovich, Orlando, FL. [Federico] P.J. Federico (1982). Descartes on Polyhdra: A study of the De Solidorum Elementis. Springer-Verlag, New York. [Henderson] David W. Henderson (2001). Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces 2nd Ed. Prentice Hall, Upper Saddle River, NJ. [Henle] Michael Henle (2001). Modern Geometries: Non-Euclidean, Projective, and Discrete 2nd Ed. Prentice Hall, Upper Saddle River, NJ. [Hilbert] David Hilbert (1971). Foundations of Geometry (2nd Ed., L. Unger, Trans.). Open Court, La Salle, IL. (10th German edition published in 1968.) [Hilbert2] D. Hilbert and S. Cohn-Vossen (1956). Geometry and the Imagination (P. Nemenyi, Trans.). Chelsea, New York. (Original work, Anschauliche Geometrie, published in 1932.) [Motz] Lloyd Motz and Jefferson Hane Weaver (1993). The Story of Mathematics. Avon Books, New York. [Weeks] Jeffrey R. Weeks (1985). The Shape of Space. Marcel Dekker, New York.