Euclid’s Elements, from Hilbert’s Axioms THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Peter James Ward Graduate Program in Mathematics The Ohio State University 2012 Master's Examination Committee: Professor Rodica Costin, Advisor Professor Ronald Solomon Copyright by Peter James Ward 2012 Abstract This project is an exposition of Book I of Euclid’s Elements consistent with modern mathematical rigor. This text is designed to serve as a first introduction to geometry, building from Hilbert’s axioms the tools necessary for a thorough investigation of planar geometry. ii Vita May 2006 .......................................................Christian Brothers Academy 2010................................................................B.A. Mathematics, Rutgers University 2010 to present ..............................................Graduate Teaching Associate, Department of Mathematics, The Ohio State University Fields of Study Major Field: Mathematics iii Table of Contents Abstract ................................................................................................................................... ii Vita ......................................................................................................................................... iii Table of Contents ................................................................................................................... iv List of Figures .......................................................................................................................... vi 0. Introduction ......................................................................................................................... 1 I. Axioms of Incidence ............................................................................................................ 10 II. Axioms of Order ................................................................................................................. 13 III. Axioms of Congruence of Segments .................................................................................. 34 IV. Axioms of Congruence of Angles ....................................................................................... 41 V. Triangles ............................................................................................................................ 58 VI. Axiom of Circles ................................................................................................................ 80 VII. Axiom of Parallels ............................................................................................................ 84 VIII. Archimedes Axiom and Area ........................................................................................... 94 IX. Axiom of Completeness .................................................................................................. 116 References ........................................................................................................................... 117 iv Appendix A: List of Axioms .................................................................................................. 119 Appendix B: A Note to the Instructor ................................................................................... 121 v List of Figures FIGURE 1: FINITE GEOMETRIES ............................................................................................................................ 12 FIGURE 2: O IS BETWEEN A AND B ........................................................................................................................ 13 FIGURE 3: O IS BETWEEN .................................................................................................................................... 21 FIGURE 4: LINE AND SEGMENT ............................................................................................................................. 14 FIGURE 5: DIAGRAMS WHICH SATISFY AXIOM II.4 .................................................................................................... 14 FIGURE 6: DIAGRAMS WHICH DON’T SATISFY AXIOM II.4 .......................................................................................... 15 FIGURE 7: THEOREM 2, CASE 1 ........................................................................................................................... 18 FIGURE 8: THEOREM 2, CASE 2 ........................................................................................................................... 19 FIGURE 9: THEOREM 2, CASE 1 ........................................................................................................................... 20 FIGURE 10: THEOREM 2, CASE 2 ......................................................................................................................... 20 FIGURE 11: ON THE LEFT, A AND B ARE ON THE SAME SIDE OF A; ON THE RIGHT, A AND B ARE ON DIFFERENT SIDES OF A ...... 21 FIGURE 12: ON THE LEFT, A AND B ARE ON THE SAME SIDE OF O; ON THE RIGHT, A AND B ARE ON DIFFERENT SIDES OF O ..... 21 FIGURE 13: RAY ................................................................................................................................................ 21 FIGURE 14: INTERIOR AND EXTERIOR OF AN ANGLE .................................................................................................. 22 FIGURE 15: THEOREM 3 ..................................................................................................................................... 25 FIGURE 16: COROLLARY TO THEOREM 3 ................................................................................................................ 25 FIGURE 17: THEOREM 4 ..................................................................................................................................... 26 FIGURE 18: THEOREM 5 ..................................................................................................................................... 27 FIGURE 19: THREE BROKEN LINES ......................................................................................................................... 28 FIGURE 20: THREE POLYGONS. THE TWO POLYGONS ON THE RIGHT ARE SIMPLE POLYGONS. ............................................. 29 FIGURE 21: INTERIOR AND EXTERIOR OF A POLYGON ................................................................................................ 29 FIGURE 22: CROSSBAR THEOREM ......................................................................................................................... 30 vi FIGURE 23: THEOREM 7 ..................................................................................................................................... 31 FIGURE 24: EXERCISE 5 ...................................................................................................................................... 33 FIGURE 25: AB ≡ CD ≡ EF, AND GH ≡ JK. ....................................................................................................... 35 FIGURE 26: AXIOM III.3 ..................................................................................................................................... 35 FIGURE 27: AB + CD ≡ AE ............................................................................................................................... 36 FIGURE 28: AB – CD ≡ AE ............................................................................................................................... 37 FIGURE 29: AB < CD < EF < GH < JK < LM .......................................................................................................... 38 FIGURE 30: COROLLARY TO THEOREM 8 ................................................................................................................ 40 FIGURE 31: ∡ABC ≡ ∡DEF AND ∡GHJ ≡ ∡JHK ≡ ∡KHL .................................................................................. 42 FIGURE 32: CONGRUENT TRIANGLES ..................................................................................................................... 43 FIGURE 33: THEOREM 9 ..................................................................................................................................... 44 FIGURE 34: SUPPLEMENTARY ANGLES ................................................................................................................... 45 FIGURE 35: THEOREM 10 ................................................................................................................................... 46 FIGURE 36: VERTICAL ANGLES ............................................................................................................................. 47 FIGURE 37: COROLLARY TO THEOREM 10 .............................................................................................................. 47 FIGURE 38: THEOREM 11 ................................................................................................................................... 49 FIGURE 39: ∡ABC + ∡DEF ≡ ∡ABG ................................................................................................................. 50 FIGURE 40: ∡ABG - ∡DEF ≡ ∡ABC ................................................................................................................. 51 FIGURE 41: ∡ABC < ∡DEF ................................................................................................................................ 52 FIGURE 42: THEOREM