This dissertation has been microfilmed exactly as received 69-22,104 BYHAM, Frederick Charles, 1935- INDIRECT PROOF IN GEOMETRY FROM EUCLID TO THE PRESENT. The Ohio State University, Ph.D., 1969 Education, theory and practice University Microfilms, Inc., Ann Arbor, Michigan INDIRECT PROOF IN GEOMETRY FROM EUCLID TO THE PRESENT DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University <• By Frederick Charles Byham, B.S. in Ed., M.S. The Ohio State University 1969 Approved by Adviser Department of ] PLEASE NOTE: Not original copy. Several pages have indistinct print. Filmed as received. UNIVERSITY MICROFILMS • ACKNOWLEDGMENT The author gratefully acknowledges the assistance and counsel given so generously by Professor Nathan Lazar. He is also indebted to Harold P. Fawcett, Professor Emeritus, for his guidance during the writer's earlier graduate work in mathematics education. To his wife, Patricia, the author expresses sincere appreciation for her encouragement and patience. 11 VITA April 2, 1935............ Born - Meadville, Pennsylvania 1957•••••••••••••••••••••# B.S. in Ed« , Edinboro St^t6 College, Edinboro, Pennsylvania 1957-1962................ Teacher, Dunkirk High School Dunkirk, New York 1963«#•••••••••••••♦•••••• M.S., Clarkson College of Technology, Potsdam, New York 1963-196 4................ Instructor, Department of Mathematics, The Ohio State University, Columbus, Ohio 1964-196 9...... *.......... Assistant Professor, Department of Mathematics, State University College at Fredonia, Fredonia, New York FIELDS OF STUDY Major Field: Education Studies in Mathematics Education. Professors Harold P. Fawcett and Nathan Lazar Studies in Higher Education. Professors Earl W. Anderson and Everett Kircher Studies in Mathematics. Professors Robert J. Bumcrot, Norman Levine, and Earl J. Mickle iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS..................................... ii VITA............... '.................................iii Chapter I. INTRODUCTION.............................. 1 Statement of the Problem Importance of the Problem Scope and Limitations of the Study Outline of the Remainder of the Study II. THE NATURE OF IMPLICATION................ 10 Introduction Material Implication Strict Implication Entailment Summary and Recommendations III. ON THE CONTRADICTORY OF AN IMPLICATIVE STATEMENT......................41 Introduction Contradictory Statements Contraries Contradictory of an Implicative Statement Summary IV. DEVELOPMENT OF INDIRECT PROOF IN GEOMETRY TEXTBOOKS.................... 51 Introduction Giving a Definition or. Analysis of Direct Proof Showing Examples of Indirect Proofs Stating Schemes for Indirect Proofs Contrasting Indirect Proof with Direct Proof Giving a Precise Definition of Indirect Proof Summary . V. LITERATURE EXCLUSIVE OF GEOMETRY TEXTBOOKS...................... 109 Introduction Definitions of Indirect Proof Importance of Indirect Proof Dissatisfaction with Indirect Proof Suggestions on Teaching Indirect Proof Summary VI. PROPOSALS FOR INDIRECT PROOF.................. 138 Introduction Inconsistent Statements Method of Inconsistency Method of Contradiction Method of Contraposition Summary VII. INDIRECT PROOF IN EUCLID'S ELEMENTS........... 1 84 Introduction Illustrations of the Method of Inconsistency Illustrations of the Method of Contraposition Summary VIII. SUMMARY AND RECOMMENDATIONS................... 194 Summary of the Study Recommendations for Teaching Indirect Proof Recommendations for Further Study APPENDIX A.......... 202 B.......................... 204 C.......................................... 207 D......................... 208 E........... 209 F........................ 210 v 6 211 BIBLIOGRAPHY................... 213 vi CHAPTER I INTRODUCTION Statement of the Problem In recent years the mathematics curriculum has undergone such rapid changes that the situation has been described as "The Revolution In School Mathematics." Modern mathematics has placed a new emphasis on the postu- lational structure of mathematical subject matter which "lends emphasis to the need for clearer understanding of the basic techniques used in valid patterns of deductive thinking."1 It is to this need as it pertains to Indirect proof that this study is directed. The usual manner in which students learn how to prove a theorem is by imitation, they study model proofs and then try to arrange their proofs to follow the patterns exhibited by the model proofs.2 The following statement from the Report of the Cambridge Conference points out the inadequacy of'this approach: "it is hardly possible to do ^Charles H. Butler and P. Lynwood Wren, The Teaching of Secondary Mathematics (*J-th ed.; New York: McGraw-Hill Book Co., 1965), p. 70-. 2Myron F. Rosskopf and Robert M. Exner, "Modern Emphasis in the Teaching of Geometry," The Mathematics Teacher, L (April 1957) t P» 27^. 1 anything in the direction of mathematical proofs without the vocabulary of logic and explicit recognition of the 3 inference schemes."^ A survey of the literature indioates that indireot proofs frequently present difficulty to the student and the teacher. This difficulty with indireot proof by students and teachers has resulted in a continuous decrease in the number of indirect proofs used in plane geometry. The indirect method was used by Euclid to prove eleven of the ip forty-eight theorems in Book 1. In current geometry programs teachers, too often, use indirect proof for only one or two theorems.^ The authors of a recently published calculus text list four points which have proved troublesome to their students: In our experience they often have difficulty with (a) proof by contradiction (b) implication (c) finding convenient equivalent forms for the denial of complicated statements, and (d) reasoning from false statements.° ^Cambridge Conference on School Mathematics, Goals for School Mathematics (Boston: Houghton Mifflin Co.. t&tttpryr.----------- Clifford B. Upton, "The Use of Indirect Proof in Geometry and in Life," Fifth Yearbook (Washington, D. C.: National Council of Teachers of Mathematics, 1930), p. 10^. ^Donovan A. Johnson and Gerald R. Rising, Guidelines for Teaching Mathematics (Belmont, Calif.: Wadsworth Publishing Co., 1967), p. 79. ^Stoughton Bell et al., Modern University Calculus (San Francisco: Holden-Day, Inc., 196^), p. 1. 3 It is important to note that all four of these arise in the normal course of constructing indireot proofs of the reductio ad absurdum type. The problem of this study is as follows: 1. To establish the logic required in the methods of indirect proof. 2. To seek methods of teaching indirect proof which are logically correct, yet simple enough that they may be readily understood by students of high school plane geometry. Importance of the Problem The development of logical thinking has been an objective of the teaching of mathematics from the Pythagorean School (500 B.C.) to the present.? The Com­ mission on Mathematics in its report published in 1959> stated that one of the principal objectives of mathematics in general education is to develop "an understanding of the deductive method as a method of thought. This includes the ideas of axioms, rules of inference, and methods of proof."® i ?Myron P. Rosskopf, "Nongeometric Exercises in Geometry", Twenty-second Yearbook (Washington, D.C.: Nation­ al Council of teachers of Mathematics, 195*0» P. 283. ^Report of the Commission on Mathematics (New York: College Entrance 'Examination Board, 1959)* P* H • Today, as in years past, a major objective of the high school geometry oourse is to develop an understanding of the nature of proof. The National Committee on Mathematical Requirements in its 1923 report stated that a principal objeotive of instruction in demonstrative geome­ try is "to develop understanding and appreciation of a deductive proof, and the ability to use this method where it is applicable."9 Upton, in 1930, stated that "our great aim in the tenth year is to teach the nature of deductive proof and to furnish pupils with a model for all their life thinking."10 Butler and Wren in their current text­ book on the teaching of mathematics state that "the prime objective in the geometry of the senior high school is to build a feeling for a deductive system in which the valid­ ity of conclusions is accepted only when these have been established by formal reasoning from bases already accepted or established."11 Can a complete understanding of the nature of deductive proof be achieved without an understanding of the nature of indirect proof? There are two forms of deductive ^National Committee on Mathematical Requirements, The Reorganization of Mathematics In Secondary Education (Boston: Houghton toifflin Co., 1923)* p. 3^. 10Upton, op. clt.t p. 132. 11Butler and Wren, op. clt.. p. ^58. 5 reasoning, direct reasoning and indirect reasoning. Indirect reasoning is a form of deductive reasoning which is extensively used by nearly everyone. In fact, Jevons has estimated that about one half of all our reasoned conclusions are arrived at through indirect reasoning.12 It would then seem to necessarily follow that if students are to gain an understanding of the nature of deductive proof, they must achieve a basic understanding of the nature of indirect proof. H. C. Christofferson expressed this need for emphasis to be placed on indirect proof as follows: If geometry is to be taught largely because of its inherent possibilities to pro­ vide experiences in the science