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Studies in Mathematics, Volume II. Euclidean Geometry Based on Ruler and Protractor Axioms

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Studies in Mathematics, Volume II. Euclidean Geometry Based on Ruler and Protractor Axioms DOCUMENT RESUME' ED 143 544 SE' 023 028 AUTHOR Curtis, Charles W.; And Others ' TIrLE Studies in Mathematics, Volume II. Euclidean Geometry Based on Ruler and Protractor Axioms. Second Reviged Edition.---, INSTITUTION Stanford Univ., Calif. School Mathematics Study , Group. SPONS AGENCY Rational Science Foundation, Washington, D. C. PUB DATE 61 -NOTE 185p.; For related documents, see SE 023 029-041 EDRS PRICE MF-$0.83 HC-$10.03 Plus Postage. DESCRIPTORS *Geometry; Inservice Education; *Instructional Materials; *Resource Materials; Secondary Eddcation; *Secondary School Mathematics;, Teacher Eddcation; *Teaching,Guides ; IDENTIFIERS *School Mathematics Study Group ABSTRACT These materials were developed to help high school teachers to become'fatiliar with the approach to tenth-grade Eu :lidean geometry which was adopted by the SchoolMathematics Study Group (SMSG). It is emphasized that the materialg are unsuitable as a high school textbook. Each document contains material too difficult for most high school students. It is assumed that teachers whostUdr- 0:he notes, have good backgrounds in axiomati'c geometry. In particular, some familiarity with Euclid's Elements is presuppose d4.Chapters include:(1) Historical 'Introduction; (2) Logic;(3) Toints,',Lines, and Plahes; (4) Real Numbers and the Ruler Axidm; 0) Separation in Planes, and in Space; (6) Angles 'and the PrOtractor Fostplates; (7) Congruence; (8) Parallelism; (9) Area; and (10) Circles and Spheres. (Ahthor/RH), *******************************************************,*I ************" Docume's acquired by ERIC include many informal-unp bliShed , * * materials'n6t available from other ,sources. ERIC Mak$esyery effort * * togobtain-the bept copy availableNevertheless, items ofmarginal. *, * reprod'ucibiiity are often encountered and this affec the quality * * of the microfiche and ha'rdcopy reprOTuctions,ERiaakes eiaifable, * ,* via the ERIC Document ReproductionSeruice,NEDES).:EDRS is not * responsible for the quality of the original document. Repro actions * * supplied'by EDRS ,are the best that can be made .from the ori§:nal. *************************************************************** ****** SCHOOL J S OEPARTMENT OF HEALTH MATHEMATICS EDUCATION & WELFARE NATIONAL INSTITUTE OF STUPY--GROU P EOUCAtiON BEEN AEPPe E N E,", RFCF ROM -E A 2P.' ON C.,,G, N , 1/' N C F h OP OP,N ONS :N, PE PPE ". ),. NS'Tt 'EOP En .. : `N a h1i ^v w STUDIES INMATHEMATICS' VOLUME II EuclideanGeolnetry Bcisklon Ruler and ProtractorAxioms (second revised edition) By CHARLES W. CURTIS, Universityof Wisconsin- PAUL H. DAUS, Univeriily Of California,Los Angeles ROBERT J: WALKER, NrrieLlUniversity _ PE R'.s1/ J",If) R-PPO'H.JrE SIISG . j ^,''( ''' I R EWE Ar1 "iv 1( 2 SCHOOL. MATHEMATICS STUDY GROUP 'STUDIES IN MATHEMATICS VOLUME SII ' r Euclidian Geometry Based on Miler 4 41, s and Protractor Axioms (second revised edition), By CHAkES -.-W. CURTIS,Ilitiverlity of Wisconsin PAUL H. DAUS; University of California., Los Angels , . ROBERT J. WALKER, Cornell University. e Written for tSCHOOL MATHEMATICS STUD'.CRoUP Under' a gra t from the NATIONAL SCIENCE FOUNDATION ti y . Copyright 1961 by Yale U iversity. 4.) 55 . ii PHA TOLITHOPRINT EU f1,Y CUSHING - MALLOY INC AN ARBOR MICHIGAN NITEU S YATES OF AMERICA **. Preface', These noteshave'been prepared tohelp high school teachers to become familiar with the, approach to tenth grade Euclidean geometry which has been adopted, by the School Mathematics Study 4 Group (SMSG).They are intended specificalto be used/during the summer of 1959,1 courses on geometry' for high school teachers. °A The 4M,SC1-,133preparing a tenth grade geometry text book anda 1 M teachers' manual, and these notes follow the preliminary outline of the `text book. It should be emphasized,- that these notes are , quite unsuitable as a text book for high school stUdents,' nor can they be used as a teachers manual. -They contain twin material ...which is too difficult to bt"presented to most tenth graders, but ,which we believe it is important for tenth grade t eachers to know. -,The'notes probe deeply into the beginni f thesubject,:brt do not'cover much of the material .of tandard tenth gradg-ieometry 'CR=26. In particular, the notes do nocover such topics as a parallels, circles, areas, Pythagorean theoreM, lytic geometry, etc., whereas all of these will be treatedin the SMSG text book. 0, ., - It.-.4.8 assumed that teachers who study these notes have good . \--/ . ' . I 'backgroundsin axiomatic geometry. In particular some familiarity ..k.... '. , with tuclidts Elements is presupposed, and the teachers should have access to these. (Heath's translation i Everyman's Lit'rary. (E.p. Dutton) is convenient.) .1rhenotee, oontai onfg occasional 1, . .t ). i '71° 4 r" - .4 references to the common high schooltreatments ofgeometry, but Ot the treaders should continually, make comparisons ofthe two types of treatment, especially the proofs of the More familiar theorems. Although we have tried to anticipate the content, order, notation, etc. of the proposed text book there is bound to be .ome divergent However, we are confidet that we are presenting the spirit of the new course; and that anyone who understands the . material these notes will be able to use the text book in an: intelligent and interesting manner. A instructor .in a summer ,course forteachers is urged tb spend most of theti me,onChapters4r 7. These contain the material on the Ruler and PrOtractor Axioms, and the theoryof separation or order Of 'points on a line, lines in a plane, and , planes in space.'Mtch of this, material is unfamiliar to riliky teacher*. Chapters 1 -'3 are introductory in nature, and can be covered . swiftly at the beginning, and referred to'fromtime to time as,the I'. course pr greSses...' / 0.4 ,These notes,are by no means a polished work, and it is: . ,- .:/ expebted that the . instructor will use good judgement indekding tr. 11 which'parts toam;Ify,which to light, what extramate*to flik : :i , 'put in, etc. Some exercises have been included but many additional *- k ,, 0 . ' : . ones will have to be Supplied. The starred theorems * ,(e.g. Theorem 3.1) can be 'used as exercises, and it is intended 1. - . P , that asman -,.aspossible .ofthese10 proved. Most of the early , - , , r "1? onep are used in later proofs, and their ommissionwould leave , 0 sriouB.,ga.rilit,,in the pryent.ation. 4 . i ne of our'proofs are accompaniedby diagrams. This was 'don partly to urge the reader to draw his owz diagram aid partly to emphasize that afOgical proof should be indepeindnt of any diagram.'Ieommission doed not mean that we \ wishtominimizetheimportanceof drawing -a figure to fix the" to ideas in the mind. The readers should make copatantme of diagrams as an aidto understanding tie theorems and discovering' proofs. 1 Future work byithe'SMSG on the text book, the teachers' . 'manual, and the teacher training, manual oan profit, greatly from comments,and,criticisms of these notes. You are 'urged to send 44: yo4r sftsgestions to School Mathematics Study Group::. ',Drawer 2502A Yale Station New Haven, Connecticut C- .7t t Tiii . 7 Prefa6eto Second Revision Several changevhave been made iii.this'edition to impiove the , material andbring it more into line with the text book., . 1. The notationsAtfor a line andABfor a distance have beeri adopted, On the other hand it did not seem 'worthwhile to change m( LATi.L)al ,mZABC and arcAB tO tN 2.'' The distance and separation postulates haVe been reworded to conforin to the text boat. However, tochaege the order of / 1 . presentation of these postulates to that used in theteXtbook . .-. would have necesgitated extensive alterations inChapters 3 and 4.. , ,-_____ The text bdokorder was dictated by pedagogical considerations, .1 p#inciPally the desire to avdict indirect Tiroof's intheifiist _ .). , .. , . theorems, which do not apply to this boor. It was the efore , . decided to leave the order as in the earlier editions. The . pottulate's thus differ in numbering as follows: These Notes. Textbook Incidence postulated 1, 2:3, 4, 5\ 1, .5,6", 7, 8 Distance ppsiulates 8. 2, 3, 4. e . I " '3. Chapters on Parallels ,and on Ares.', lave been', inserted to- f ':. clarify the position of these topics in oti.r presentation.. , _ _ _ __._ , .' r 4., Numerous minor'thanges, insertiond, corrections, etc,. have-been made, and one-significant error has been corrected (Section 4 of Chapter 5). 6 'iv -8 The user of this book', .71 find much additional material in the text book, including,si exercises'and expository material in the text book proper and lementary reading in-the Teachers, Commentary, the Appendices, an articularly the,Talks to Teachers In this book we present Eu can gedmetry as a mathematical theory. We havelett aside all, a ications of geometry to questions in physics-and to other wiles of mathematics. This is not to say that we regard these licatiorts as unimportant for the teacher to know and to use in the ,lassroom; in tact, it would be a mistake to teach geometry"to high' chool students without 1 bringing in some of the significant aPp cations, especially to elementary physics.The reader of this k ill find abundant_ supplementary material on applications of entary geometry in the following books: e G. Polya, Mathematics and kauiible/R Princeton University Press, vol. 1, especially the chapter on "Physical Mathematics1' R. Courant and H. Robbins, What is Mathematics ?, Oxford University Press. -N. H. Rademacher and 0. Toepljtz, The Enjoymentof Mathematics, Princeton University Press. II CONTENTS * * * * I Chapter Page 1. Historical Introduction 1 2. Logic 37 f 3. Points, Lines, and Planes . .. 63 4. Real Numbers and the Ruler Axiom 69 . 5. Separation in Planes and.in Space 107 6. Angles'and'the Protractor Postulates 115 7. Congruence .31. 8. Parallelism 1k5 9. Area 153 I . - 10. Circles and Spheres 169* 10 .4 "ot - -;,----- a Chapter 1 Historical Introduction 1. The Glory' that is Euclid's.
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