McGILL UNIVERSITY FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS MATH TOPICS IN GEOMETRY

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1 McGILL UNIVERSITY FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS MATH TOPICS IN GEOMETRY Information for Students Fall Term Pages 1 through 7 of these notes may be considered the Course Outline for this course this semester. W. G. Brown November 27, 2003

2 Information for Students in Math Contents 1 General Information Instructor and Times Calendar Description Class Test Homework Term Mark Calculators Final Grade Text-Book Other Published Materials References to other textbooks, and to papers in the mathematical literature Website Printed Notes Examination information Plagiarism Non-prerequisites Synthetic geometry Background from Logic Background from Set Theory Tentative Timetable 8 3 Notes to accompany a brief discussion of Euclid s Elements Objectives Definitions in Book I of Euclid s Elements [9, Volume 1, pp ] Postulates of Euclid s Elements [4, 1.2, pp. 4-5] Common Notions (Axioms) of Euclid s Elements [9, Volume 1, pp. 155] Some conventions Line segments vs. lines Line segments and their lengths Angles Unit of angular measure Triangles The real numbers Proposition I.1: Enunciation of the proposition in general terms Detailed restatement of the enunciation: Construction and Proof Q. E. F. and Q. E. D Proposition I Enunciation of Proposition I.2: Detailed restatement of the enunciation: Proof Proposition I Enunciation of Proposition I Detailed restatement of the enunciation Construction and sketch of proof Proposition I A problematic proposition Enunciation of Proposition I Restatement of the enunciation in more modern terms i

3 Information for Students in Math ii Further comments on Proposition I A detailed look at Proposition I.5, sometimes called Pons Asinorum Enunciation of Proposition I.5 in general terms Detailed restatement of the enunciation A proof of the Pons Asinorum proposition by Pappus of Alexandria What about equilateral triangles? Proposition I The converse to the first part of Proposition I Enunciation of Proposition I Detailed restatement of the enunciation Proof Difficulties with this proof Proposition I Enunciation in general terms (slightly modernized from Euclid s language) Enunciation in detailed terms Proof Difficulties with Proposition I Proposition I Enunciation Detailed restatement of the enunciation Idea of Euclid s Proof Philo s proof A congruence theorem Proposition I Enunciation in general terms Detailed restatement of the enunciation Construction Proof Proposition I Enunciation in general terms Detailed restatement of the enunciation Construction Proof Proposition I Enunciation in general terms Detailed restatement of the enunciation Construction, based on Euclid s but with some ambiguity removed Proof Proposition I Enunciation in general terms Detailed restatement of the enunciation Construction Proof Proposition I Enunciation in general terms

4 Information for Students in Math iii Detailed restatement of the enunciation Proof Proposition I Enunciation in general terms Detailed restatement of the enunciation Proof Proposition I Enunciation in general terms Detailed restatement of the enunciation Proof Proposition I Enunciation in general terms Detailed restatement of the enunciation Proof Proposition I Enunciation in general terms Detailed restatement of the enunciation Proof Proposition I Enunciation in general terms Detailed restatement of the enunciation Proof Proposition I Enunciation in general terms Detailed restatement of the enunciation Proof Proposition I.20 The Triangle Inequality Enunciation in general terms Detailed restatement of the enunciation Proof Proclus s commentary on I Proposition I Enunciation in general terms Detailed restatement of the enunciation Proof Proposition I Enunciation in general terms Detailed restatement of the enunciation Construction Proof Proposition I Enunciation in general terms Detailed restatement of the enunciation Construction Proposition I Enunciation in general terms Detailed restatement of the enunciation Proof Proposition I Enunciation in general terms Proposition I

5 Information for Students in Math iv Enunciation in general terms Detailed restatement of the enunciation Proof of Case Proof of Case Proposition I Enunciation in general terms Detailed restatement of the enunciation Proof Proposition I Enunciation in general terms Detailed restatement of the enunciation Proof of Case Proof of Case Some other propositions of Euclid cited in the textbook 50 4 First Problem Assignment 53 5 Symmetries in the Real Plane Sets and Functions Basic definitions Textbook convention: functions compose on the right Associativity of function composition Inverses Vectors in R Example of the use of geometric vectors : Concurrence of the medians of a triangle Isometries Introductory Examples Basic Definitions Further investigation of symmetries of the square Isometries of the Euclidean plane The product of two reflections in the Euclidean plane Half-turns The composition of two half-turns The composition of two translations Composition of three reflections in the plane A simplification: commuting reflections 73 6 Second Problem Assignment 74 7 Symmetries of R 2 (continued) Isometries (continued) Miscellaneous results on half-turns and translations Groups of isometries The periods of isometries of the plane Abstract Groups; Isometry Groups The Cayley Table or Composition Table of a group The Direct Product of Groups Types of symmetry groups for configurations in R

6 Information for Students in Math v 8 Solutions to Problems on Assignment Third Problem Assignment The symmetries of friezes (strip patterns) Comments on the table of symmetry groups of friezes What would have to be proved? How can two different friezes both have group C? How can three different friezes all have group D? How can we differentiate the cases of... DDDDD... and... HHHHH... from the others, and between themselves? Solutions, Second Problem Assignment Two results on plane lattices Sylvester s Problem of Collinear Points The Crystallographic Restriction Class Tests Version Version Version Version Brief discussion of (symbolic) logic Statements Truth values of sentences Tautologies and contradictions Logical Implication Rules of Logic Rules of inference Reductio ad absurdum = Proof by Contradiction The converse and contrapositive of an implication The Predicate Calculus The axioms, postulates, and theorems of a logical system Solutions, Third Problem Assignment Solutions to Problems on the Class Tests Version Version Version Version Fourth Problem Assignment Ordered Geometry The structure of our geometry Excluding the trivial geometries with 0 or 1 point Axioms concerning points on a line Ordered Geometry (concluded) 174

7 Information for Students in Math vi We require that not all the points lie on the same line Pasch s Axiom; Points are dense in any line Projective Geometries The Real Projective Plane Homogeneous coordinates; models of the Real Projective Plane The Euclidean plane as a subgeometry of the Real Projective Plane The points and lines which are adjoined to the Euclidean Plane to create the Real Projective Plane Some incidence properties of the real projective plane Projective planes over other fields than the reals What is a field? Fields of residues modulo a prime The projective plane over F p The Fano Geometry Projective Geometries (continued) Axiomatic Definition of Projective Planes What can we infer from the 3 given axioms only? The Duality Principle Some simple properties of finite projective planes Solutions, Fourth Problem Assignment 186 A Solved Assignments from Previous Years 1001 A / A.1.1 First 2002/2003 Problem Assignment 1001 A.1.2 Second 2002/2003 Problem Assignment 1004 A.1.3 Third 2002/2003 Problem Assignment 1007 A.1.4 Fourth 2002/2003 Problem Assignment 1009 B Tests and Examinations from Previous Years 1015 B.1 Tests from Fall, B.1.1 Version B.1.2 Version B.1.3 Version B.1.4 Version B.2 Solutions to Problems on the 2002 Class Tests B.2.1 Problems on all four versions of the test B.2.2 The problems that were each on only one test B.3 Final Examination, December, B.4 Supplemental/Deferred Examination, May,

8 Information for Students in Math General Information Distribution Date: Wednesday, September 3rd, 2003 (all information is subject to change) 1.1 Instructor and Times 1.2 Calendar Description INSTRUCTOR Professor W. G. Brown OFFICE: BURN 1224 OFFICE HOURS W 14:35 15:25; (subject to change): F 10:00 11:00 or by appointment OFFICE PHONE: BROWN@MATH.MCGILL.CA CLASSROOM: BURN 1B24 CLASS HOURS: MWF 12:35 13:35 CRN: 660 (3 credits) (Fall and Summer) (Prerequisite: Previous course in Mathematics) Selected topics - the particular selection may vary from year to year. Topics include: isometries in the plane, symmetry groups of frieze and ornamental patterns, equidecomposability, non-euclidean geometry, and problems in discrete geometry. 1.3 Class Test A test will be administered during the regular class hour on Wednesday, November 5th, No provision is planned for a make-up test for a student absent during the test. Any change in this date will be announced in the lectures. 2 In your instructor s eyes the main purpose of the test is as a dry run for the final examination. 3 1 Date changed as of 8 October, Note that the date of the test is after the deadline for withdrawal from the course. 3 Notwithstanding the minimal contribution of the test grade to the student s final grade (cf. 1.5 below), the test is to be considered an examination in the sense of the Handbook of Student Rights and Responsibilities, to be found at the following URL: UPDATED TO November 27, 2003

9 Information for Students in Math Homework There will be approximately 4 or 5 assignments. While students are not discouraged from discussing assignment problems with their colleagues, written solutions that are handed in should be each student s own work. 4 Submitted homework should be stapled with a cover page that contains your NAME, STUDENT NUMBER, the COURSE NUMBER, and the ASSIGNMENT NUMBER. Other pages should always include your student number. You can minimize the possibility that your assignment is lost or fragmented. 1.5 Term Mark Graded out of 30, the TERM MARK will be the sum of the HOMEWORK GRADE (out of 10) and the CLASS TEST GRADE (out of 20). 1.6 Calculators The use of calculators, computers, notes, or other aids will not be permitted at the test or examination. 1.7 Final Grade The final grade will be a letter grade, computed from the maximum of the Examination Mark (out of 100); and the sum of the Term Mark (out of 30) and 0.7 times the examination mark (out of 100). 1.8 Text-Book The primary textbook for the course will be: [4] H. S. M. Coxeter, Introduction to Geometry, Second Edition. John Wiley and Sons (1969). ISBN hardbound; or the paperback edition, published 1989, ISBN (The paperbound edition is the one to be stocked by the Bookstore.) 4 From the Handbook on Student Rights and Responsibilities: No student shall, with intent to deceive, represent the work of another person as his or her own in any...assignment submitted in a course or program of study or represent as his or her own an entire essay or work of another, whether the material so represented constitutes a part or the entirety of the work submitted. cf. 15(a) 5 UPDATED TO November 27, 2003

10 Information for Students in Math Most of the material needed for the course is supported by material to be found in this textbook, mainly from the following sections of the following chapters; this list is tentative, and subject to revision: Part I 1. Triangles: 6 1.2, 1.3, 1.4 This part of the course will be supported substantially by notes on the Web. (Added September 19th, 2003: Propositions I.1 through I.1 through I.12 of Euclid s Elements were discussed in the lectures; the First Assignment includes problems on other propositions in the range I.1 I.28; students are expected to read Propositions I.13 I.28 privately. 2. Regular Polygons: 2.3, 2.4, 2.5, 2.6, 2.7, Isometry in the Euclidean plane: 3.1, 3.2, 3.3, 3.4, 3.5, Two-dimensional crystallography: 4.1, 4.2, 4.3; (omit 4.4); 4.5; (omit 4.6); Similarity in the Euclidean plane: omit 6. Circles and spheres: omit 7. Isometry and similarity in Euclidean Space: omit Part II 8. Coordinates: omit 9. Complex numbers: omit 10. The five Platonic solids: 10.1, 10.2, 10.3, The golden section and phyllotaxis: omit Part III 12. Ordered geometry: 12.1, 12.2, 12.3, 12.4(part), 12.5, Affine geometry: 13.1, 13.2, to be completed 14. Projective geometry: 14.1, 14.2, to be completed 15. Absolute geometry: 15.1, 15.2, 15.3, 15.4, 15.5 to be completed 16. Hyperbolic geometry: omit Part IV Omit all chapters, except possibly 6 In this section of these notes, refers to sections of the textbook; elsewhere in the notes, unless otherwise indicated, refers to sections of these notes.

11 Information for Students in Math Topology of surfaces: part Not all topics in these chapters will be studied in depth. Some topics will be studied through a careful development of properties as the axiomatic systems are gradually expanded. Other topics will be studied only descriptively, since there will not be time to cover all chapters in depth. 1.9 Other Published Materials References to other textbooks, and to papers in the mathematical literature The following book, which has been used as a reference for some topics during recent years, will be kept on reserve in the Schulich Library: [8] D. W. Farmer, Groups and Symmetry. A Guide to Discovering Mathematics. Mathematical World, Volume 5. American Mathematical Society, Providence, R. I. (1995), ISBN Website These notes, and other materials distributed or posted to students in this course, will be accessible at the following URL: (1) The notes will be in pdf (.pdf) form, and can be read using the Adobe Acrobat reader, which many users have on their computers. It is expected that most computers in campus labs should have the necessary software to read the posted materials. Where revisions are made to distributed printed materials for example these information sheets it is expected that the last version will be posted on the Web. The notes will also be available via a link from the WebCT URL: but not all features of WebCT will be implemented. It is, however, intended to mount grades on the WebCT site: students should verify that their grades have been properly recorded. 7 7 Always keep graded materials homework and test to substantiate any claim that a grade has not been properly recorded. Of course, these materials are also useful in preparing for the final examination.

12 Information for Students in Math Printed Notes Typeset notes will be made available from time to time to supplement material in the text-book or lectures; these notes will be available through WebCT or the URL above (1); possibly some of the notes will be distributed in hard copy. Normally, when there are revisions and corrections to printed notes, these will not be distributed in hard copy, but will be posted on the Web. Unless you are explicitly told otherwise, you should assume that all printed notes whether distributed in hard copy or mounted on the Web or both are as much part of the required materials in the course as if they had been written on the chalkboard during a lecture Examination information 1. Will there be a supplemental examination in this course. Yes. 2. Will students with marks of D, F, or J have the option of doing additional work to upgrade their mark? No. 3. Will the final examination be machine scored? No Plagiarism McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures. (See for more information). L université McGill attache une haute importance à l honnêteté académique. Il incombe par conséquent à tous les étudiants de comprendre ce que l on entend par tricherie, plagiat et autres infractions académiques, ainsi que les conséquences que peuvent avoir de telles actions, selon le Code de conduite de l étudiant et des procédures disciplinaires. (Pour de plus amples renseignements, veuillez consulter le site Non-prerequisites Some parts of this course will be purely descriptive; but the intention is that some parts of the course will contain real mathematics, i.e., theorems and proofs. Since the course description requires only a previous course in mathematics, students can expect to be 8 Sometimes the notes may be nothing more than an edited version of material that was written on the chalkboard, or a preliminary version of materials to be used in a subsequent lecture. 9 Inserted at the request of the Provost of the University.

13 Information for Students in Math provided with the background needed to understand the topics under discussion. Some parts of this background material could be review for students who have studied MATH 318, MATH 240, MATH 235, and certain other courses, but the only material that will be formally assumed is the pre-calculus contents of a course like MATH 112 or CÉGEP Synthetic geometry [5] Synthetic geometry is ultimately based on certain primitive concepts and axioms, appropriate to the particular kind of geometry under consideration (e.g., projective or affine, real, complex, or finite). Each problem belongs to one kind (or to a few kinds), and I would call a solution synthetic 10 if it remains in that kind, analytic if it goes outside. The use of coordinates is one way of going outside; the use of trigonometry is another. In part of the course we will try to develop several types of geometry synthetically, and will use analytic methods only for the provision of examples and models. Elsewhere we may shift to analytic methods in order to be able to cover more territory in limited time. Even though we will usually accompany our geometric proofs with a sketch, the proof must be able to stand without any reference to the drawing; the sketch only serves to help us develop, understand, and remember the proof. For example, in several theorems we will be concerned with proving that certain lines are concurrent that they all pass through a common point. Our proofs will never be based on a drawing in which the lines appear to have the property, although we may make such a drawing in the course of trying to understand the proof Background from Logic The textbook is not written in the notation of symbolic logic, and we don t plan to write most of the proofs we study symbolically either; however, we need to have some of the notation of symbolic logic available to streamline proofs when it is convenient; and also we need to be comfortable with the notion of a proof based on inference from axioms and hypotheses. The ancient Greeks did not have symbolic logic available when they developed Euclidean geometry; nor did they have modern notation or modern writing paraphernalia, nor the theory of sets. However, by using these tools judiciously we will be able to reduce the study of geometry from a Herculean task requiring the genius of a Euclid or a Pythagoras to an interesting challenge. 10 italics added

14 Information for Students in Math Background from Set Theory The language and notation of set theory are standard in most mathematical writing today. We will review the simplest notions as we require them, so that we have them available when we need them. Students are not expected to have been exposed to these ideas before, but those for whom they are new should devote the time necessary to become comfortable with the concepts and notations.

15 Information for Students in Math Tentative Timetable Distribution Date: (0th version) Wednesday, September 3rd, 2003 (All information is subject to change.) MONDAY WEDNESDAY FRIDAY SEPTEMBER 1 LABOUR DAY 3 Introduction R ; 1.2, Euclid axioms, etc , Euclid I.1 10 Euclid I.1 I.4 12 Euclid I.5 Course changes must be completed by midnight, September Euclid I.5 I.6 17 Euclid I.6 I.9 19 Euclid I.10 I.28 Deadline for withdrawal with fee refund = September Notes, 5.1, Notes 5.1, Notes 5.2, Notes 5.3 OCTOBER 1 Notes 5.3.2, Notes NO LECTURE IN MATH , , 3.7 Deadline for withdrawal (with W) from course = Oct THANKSGIVING , Notes 7.2 DAY (Canada) 20 Notes Notes Notes Notes 10.1; text , Logic 4.5, 4.7 Notation: n = Assignment #n due R = Read Only X = reserved for expansion or review Section numbers refer to the text-book unless Notes are specified. The next page will not be distributed until the syllabus has been revised.

16 Information for Students in Math MONDAY WEDNESDAY FRIDAY NOVEMBER 3 Logic + Axiomatic 5 CLASS TEST 7 Systems DECEMBER 1 3 Notation: n = Assignment #n due at midnight on Monday this week R = Read Only X = reserved for expansion or review Section numbers refer to the text-book. UPDATED TO November 27, 2003

17 Information for Students in Math Notes to accompany a brief discussion of Euclid s Elements 3.1 Objectives We shall explore some of the simpler geometric results in Euclid 11 s Elements. We make no serious attempt to modernize the treatment, beyond the introduction of some modern terminology; however, we will indicate from time to time features of this remarkable work that do not meet current standards of scholarship. These comments should not cloud the profound debt that all modern mathematics owes to Euclid and his predecessors. (We will omit the last part of Chapter 1 of the textbook, [4, pp ].) 3.2 Definitions in Book I of Euclid s Elements [9, Volume 1, pp ] The following definitions would need some polishing if we were to use them today. However, the comments provided after some of them should not detract from an appreciation of the power and coherence of the theory that Euclid and his predecessors constructed. In some instances in the sequel we will, when presenting results from Euclid, update the terminology in whole or in part to current usage. 1. A point is that which has no part. 2. A line is breadthless length The extremities of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 5. A surface is that which has length and breadth only. 6. The extremities of a surface are lines. 7. A plane surface is a surface which lies evenly with the straight lines on itself. 11 Heath [9] places Euclid intermediate between the first pupils of Plato (d. 347/346 B.C.) and Archimedes ( B.C.) ; educated in Athens from pupils of Plato; not to be confused with the philosopher, Euclid of Megara, Euclid taught and founded a school at Alexandria. 12 Modern terminology would reserve the word line for one that has no extremities, and does not admit further extension. Where there is one extremity, modern terminology would use the word ray ; where there are two extremities, the word line segment would be used. 13 It will be clear immediately that the intention is that the extremities of a line segment must be different.

18 Information for Students in Math 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. 9. And when the lines containing the angle are straight, the angle is called rectilineal. 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. 13. A boundary is that which is an extremity of anything. 14. A figure is that which is contained by any boundary or boundaries. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another; 16. And the point is called the centre of the circle. 17. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. 18. A semicircle is the figure contained by a diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle. 19. Rectilinear figures are those which are contained by straight lines, trilateral figures are those which are contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle is that which has two of its sides alone 14 equal, and a scalene triangle that which has its three sides unequal. 21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acuteangled triangle that which has its three angles acute. 14 Modern terminology would not require that an isosceles triangle not be equilateral. In the sequel we shall normally use the modern definition.

19 Information for Students in Math Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled 15 ; and a rhomboid 16 that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction Postulates of Euclid s Elements [4, 1.2, pp. 4-5] (We shall not be concerned with Euclid s distinction between Postulates and Axioms.) 1.21 A straight line may be drawn from any point to any other point A finite straight line may be extended continuously in a straight line A circle may be described with any centre and any radius All right angles are equal to one another If a straight line meets two other straight lines so as to make the two interior angles on one side of it together less than two right angles, the other straight lines, if extended indefinitely, will meet on that side on which the angles are less than two right angles. 15 Modern terminology would permit the right-angled case i.e. the square as being a special type of rhombus 16 We would call such a figure a parallelogram, and would include the rectangle and rhombus as special cases. 17 Modern usage would usually reserve the word trapezium for a figure in which a pair of opposite sides are parallel, and would include all parallelograms. 18 Modern terminology would change this definition in several ways. 19 Note that this translation is ambiguous, as it uses the term straight line with two somewhat different meanings. Modern usage would reserve the term straight line for the doubly infinite object; a line beginning at a point and extending out indefinitely would be called a ray, and a finite straight line would be called a line segment. Note that in the use of the word finite here it is assumed that the length is non-zero, which would not be the current interpretation of that word.

20 Information for Students in Math Common Notions (Axioms) of Euclid s Elements [9, Volume 1, pp. 155] (Some manuscripts have variations and extensions of this list. 20 ) 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. 3.5 Some conventions Line segments vs. lines In preparation for future refinements, let us distinguish between the line segment joining points A and B (which we can denote by AB), and the length of that line segment, which we can denote by AB. Keep in mind that length will be unsigned, that is, where we wish to talk about positive and negative distances, we shall use other terminology and another notation. For the present, AB and BA represent the same line segment. Where a proposition is not concerned with the length of a line segment, we may be casual about replacing the segment by the line or a ray in the line. This can be justified by appealing to axiom 1.21 of Line segments and their lengths In some cases Euclid has not distinguished verbally between a line (segment) and its length; in the versions of the propositions and their proofs given below we have tried to sharpen the distinction Angles When we wish to talk about the angle at a vertex A between a ray 21 BA and a ray BC, we may write BAC or CAB. We can think of the angle intuitively as the collection of rays between these two bounding rays; but, by modern standards, the concept is not adequately defined, and additional postulates or definitions are required. 20 For example, If equals be added to unequals, the wholes are unequal. 21 We haven t defined this term formally.

21 Information for Students in Math Eventually after we leave the study of Euclid s Elements we may wish to introduce a sign convention. One could argue that we should be using a symbol like A to distinguish the name of A from its length. We will not do this Unit of angular measure Nowadays we usually measure angles in degrees, or in radians, where the former measure may be linked to ancient Babylonian practice, and the later measure is linked to the length of an arc on a circle of unit radius. Euclid s angles are expressed in terms of right angles Triangles We will denote the triangle with vertices A, B, C by ABC; no orientation is associated with the vertices in this notation, so ABC = BCA = CAB = ACB = CBA = BAC The real numbers Euclid s axioms could form the basis of an axiomatization of a number system: the usual definitions employed today would include axioms equivalent to these, and more. But, even with the limited machinery that he created, Euclid was able to prove (in later books of the Elements) non-trivial results about numbers although he expressed these ideas in geometric language.

22 Information for Students in Math Proposition I.1: Distribution Date: Monday, September 8th, 2003 In this first result of his Elements, Euclid proves the validity of a constructive procedure. The procedure uses Postulate 1.23 of 3.3, which asserts that A circle may be described with any centre and any radius. If we wish to interpret this postulate in terms of the use of a modern compass, we have to understand that Euclid does not claim that you can first set your compass at a particular radius, and then place the point of the compass on paper and draw a circle. Rather, you are to first place the point on the paper, then adjust the compass so that the circle will pass through a particular point, and then describe the circle; think of a compass that has no memory the moment the point is lifted from the paper, the radius reverts to zero. In Proposition I.3 he will show that one can, on the strength of his propositions, indeed set the compass at any desired radius. Euclid s economy in his use of postulates is very much in the spirit of modern mathematics: mathematicians always try to weaken their assumptions to obtain best possible results, while assuming the least. Euclid s construction involves the drawing of two circles: the apex of the triangle he is constructing will be the intersection of these circles. There is a gap in his axioms: he did not see the need to ensure that the circles actually meet in a point. To see that there is a problem here, think of the circles in the Cartesian plane with radius 1 and centres (±1, 0). Suppose that the only points that we can see on these circles are those with rational coordinates. Then the circles will both miss the point ( 0, 3 ) where we would like them to intersect. The property we need here is called continuity and is related to the concept of continuity that one meets in calculus. It is where, in the following proof, he says...the point C, in which the circles cut one another... that he has made an inference not justified by his logical system. To modernize this part of his geometry, we would need to supply some additional axioms; this can be done in various ways, and is beyond this course. You should make a sketch and follow every step on your sketch; of course, the sketch is not part of the proof, but it helps you understand what Euclid is claiming. The proof given is based on the translation in [9, pp ] Enunciation of the proposition in general terms On a given finite straight line, to construct an equilateral triangle.

23 Information for Students in Math Detailed restatement of the enunciation: Let AB be the given finite straight line. To construct an equilateral triangle ABC, i.e. to find a point C such that the line segments AB, BC, CA are of equal length Construction and Proof By Postulate 1.23 of 3.3 we may draw a circle with centre A, passing through B; and a circle with centre B, passing through A. Let C denote a 22 point of intersection of these circles. By Postulate 1.21 of 3.3 line segments may be drawn from C to each of A, B. By Definition 15, the lengths of the line segments joining the centre of a circle to any of the points on the circle are equal: in particular, for the circle with centre A, AC = AB ; and, for the circle with centre B, BC = BA. By Axiom 1, AC = AB = BC. By Definition 20, the triangle with vertices A, B, C is equilateral, and, as required, it has been constructed on the base AB. Q.E.F. (see next below) Q. E. F. and Q. E. D. The proofs of the first three propositions end with abbreviation of words meaning what it was required to do 23 ; the next three end with words meaning what it was required to prove 24, etc. The Latin abbreviations, especially Q.E.D., are still in current use by mathematicians; however, some mathematicians now use some sort of a box symbol instead of writing Q.E.D., e.g Proposition I This second proposition is, like its predecessor and its successor, a construction. Euclid is proceeding to prove develop results that will enable him to compare lengths. Euclid did not have the real number system available to him. (Even in later books of the Elements, where Euclid proved theorems about the integers e.g. that there exist an infinity of primes 25 his proofs were geometric.) Remember that Euclid s axioms do not permit a circle to be drawn with a compasscarried distance ; however, if one foot of the compass is placed at the proposed centre, the axioms permit the placing of the other foot at a point at any desired distance, and to 22 There are two such points of intersection unless the given points coincide. 23 Greek oπɛρ ɛδɛι πoιησαι; Latin Q.E.F. = quod oportebat fieri. 24 Greek oπɛρ ɛδɛι δɛιξαι; Latin Q.E.D. = quod erat demonstrandum. 25 Proposition IX.20, [9, Vol 2, p. 412]

24 Information for Students in Math then draw the circle through that point. It has been suggested that one should suppose the compasses to close of themselves the moment they cease to touch the paper Enunciation of Proposition I.2: To place at a given point (as an extremity) a (straight) line segment whose length is equal to that of a given (straight) line segment. The proof below is paraphrased from the version in [9, p. 244] Detailed restatement of the enunciation: Let A be the given point, and BC the given line segment. Thus it is required to place at the point A, as an extremity, a line segment whose length is BC Proof On the line segment AB (whose existence is postulated in Postulate 1.21 of 3.3), construct an equilateral triangle (using Proposition I.1 3.6) whose third vertex will be called D. Produce line segments DA and DB respectively beyond A and B sufficiently far 26 to points E and F. By Postulate 1.23 of 3.3 there exists a circle with centre B passing through the point C. Assuming that segment DB was extended sufficiently far, this circle will meet segment BF. Call the point of intersection G. We again appeal to Postulate This time the circle with centre D and passing through the point G will meet DE, the extension of DA; call the point of intersection L. Now 27 DA + AL = DL (2) = DG (3) = DB + BG (4) = DB + BC (5) = DA + BC ; (6) by Axiom 3, this implies that AL = BC, so we have constructed a line segment with one end at A, whose length is equal to BC. Exercise 3.1 Explain why each of equations (2) (6) holds. Solution: 26 The proof depends on these extended line meeting circles, so the line segments have to be long enough. 27 Can you explain why each of the equalities stated should hold?

25 Information for Students in Math (2): since L lies on the extension of DA beyond A (3): since L lies on the circle with centre D and radius DG (4): since G lies on the extension of DB beyond B (5): since G lies on the circle with centre B and radius BC (6): since ABC is an equilateral triangle Exercise 3.2 In the proof of the preceding proposition, care has not been taken to accommodate special cases where certain pairs of points coincide. What would happen if 1. A = B? (Here ABD would degenerate to a single point D = A = B.) 2. A = C? There are two questions here: Is the proof, as stated, correct? Could one give another proof, different from the general one stated, which could accommodate the situation.? Can you see any other possible degeneracies that would require adjustments in the proof? 3.8 Proposition I This constructive proposition is an application of its immediate predecessor. Students are again reminded that Euclid s postulates do not include the drawing of a circle with a compass-carried distance. Once the present proposition has been established, however, that operation will have been proved legitimate: if you wish to construct a circle with centre A and radius equal to BC, you may first construct any line segment through A, for example one of AB or AC 28 (by Postulate 1.21 of 3.3), then extend that line segment (by Postulate 1.22), then apply the present proposition to find a point on the line at a distance equal to BC from A, and finally apply Postulate Enunciation of Proposition I.3 Given two unequal line segments, to cut off from the greater a line segment whose length is equal to the less. 28 Note that one of these might be degenerate, if the ends of the line segment coincide.

26 Information for Students in Math Detailed restatement of the enunciation Let AB and CD be the two given line segments, of which AB is the greater length; it is required to cut off from AB a part equal in length to CD Construction and sketch of proof Use Proposition I.2 to construct a segment AE such that AE = CD. Then the circle with centre A, passing through E, will meet AB in a point whose distance from A is equal to AE, hence equal to CD by Axiom Therefore, given the two straight lines, AB, CD, of which AB is greater in length, a line segment AE has been cut off, equal in length to CD. 29 Here again one can quibble that there is an assumption of continuity that was not explicit; an axiom of continuity would state, in technical language, that there could not be holes in either the circle or the line. One can raise another type of objection also. The axioms have not been precise about what would be meant by greater length remember that Euclid did not have a formalized real number system available.

27 Information for Students in Math Proposition I.4 Distribution Date: Sunday, September 14th, A problematic proposition By current standards the proof presented for this proposition is inadequate. Coxeter observes [4, p. 5] that Euclid s principle of superposition, used in proving Proposition I.4, raises the question whether a figure can be moved without changing its internal structure. This principle is nowadays replaced by further explicit assumptions. If the principle of superposition is a legitimate tool, could it not have been used in the proof of Proposition I.3? Some explanation is needed for the language of the enunciation below. Euclid s custom was to designate one side of a triangle as the base, and the other two as the sides. 30 When he says that the triangle will be equal to the triangle, he appears to be referring to a relation that we would be more likely to call congruence today asserting that the two triangles are equal in both the lengths of corresponding sides and the magnitudes of corresponding of angles; triangles related under the relation of congruence are said to be congruent Enunciation of Proposition I.4 If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal line segments 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 Restatement of the enunciation in more modern terms If two triangles have the lengths of two sides of the one equal to the lengths of two sides respectively, and have the angles contained by the equal line segments 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 Further comments on Proposition I.4 Some of the ambiguities in the enunciation could be resolved by more careful use of the terms; but the intention of the proposition becomes clear when the points and angles are 30 Heath [9, I, p. 249] suggests that the term base must have been suggested by the practice of drawing the particular side horizontally...and the rest of the figure above it.

28 Information for Students in Math named. Consider the first sections of the proof 31 : Let ABC, DEF be two triangles having the two sides AB, AC equal in length to the two sides DE, DF respectively, namely AB to DE and AC to DF, and the angle BAC equal to the angle EDF. I say that the base BC is also equal to the base EF, the triangle ABC will be equal to the triangle DEF, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend, that is, the angle ABC to the angle DEF, and the angle ACB to the angle DF E. The removal of ambiguities does not remove objections to the basing of a proof on the questionable concept of superposition. One solution could be to designate the enunciated statement above as a new axiom, or to introduce new axioms from which the proposition would be a logical consequence A detailed look at Proposition I.5, sometimes called Pons Asinorum Euclid s proof of this proposition repeatedly uses Proposition I.4 and the constructive propositions I.1, I.2, I.3. We will discuss his proof in detail. The textbook gives a newer proof, due to Pappus of Alexandria [4, 1.3], which we will also discuss. The textbook discusses another approach to the theorem, in which the apex here the vertex between the two sides of equal length is joined to the midpoint of the base. Such a proof would require substantial reworking of other proofs in Euclid, since it is not shown until I.10 that the midpoint can be found, and the congruence of the two triangles formed would require not I.4, but I.8. However, the proposal to use the concept of reflection to prove theorems in geometry is an important one, and will be seen again in the textbook. 32 The name, Pons Asinorum. While the name was used for a proposition in Euclid over an extended period of time, it has not always been applied to the same proposition! (cf ). 31 called, by the ancient Greek logicians, the ɛκθɛσις = ekthesis = setting out and the διoρισµoς = diorismos = specification. In the sequel we will call this Detailed restatement of the enunciation, and may paraphrase Euclid s version. 32 It is possible to reconstruct Euclidean geometry by taking reflection as one of the building blocks. This is done, for example, in the book [2]. This reference is given only for academic completeness whenever possible, a reference to a theorem should be substantiated by a citation; students in MATH 348 are not expected to chase down these references, or even to remember that there was a reference.

29 Information for Students in Math Enunciation of Proposition I.5 in general terms In isosceles 33 triangles the angles at the base are equal to one another; and, if the equal sides be extended below the base, the angles under the base will also be equal to one another Detailed restatement of the enunciation Let ABC be an isosceles triangle, having AB = AC ; and let sides AB and AC be produced further (away from A) to form line segments ABD and ACE (by Postulate 1.22 of 3.3). I say that ABC = ACB, and that CBD = BCE. A B C F G D E 1. Let F be any 34 point in line segment BD (the extension of AB). Construct a circle 33 Euclid did not include equilateral triangles among those he called isosceles. His proof is valid for equilateral triangles as well. 34 We shall see from the construction that there could be some restriction on the choice of F, since