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Elisha Scott Loomis DOCUMENT RESUME ED 037 335 SE 007 848 AUTHOR Loomis, Flisha Scott TITLE The Pythagorean Proposition, Classics in Mathematics Education Series. INSTITUTION National Council of Teachers of pthematics, Inc., Washington, D.C. PUB DATE 68 NOTE 310p. AVAILABLE FROM National Council of Teachers of Mathematics, 1201 Sixteenth Steet, N.W., Washington, D.C. 20036 EDRS PRICE EDRS Price MF-$1.25 HC-Not Available from EDRS. DESCRIPTORS Algebra, Geometric Concepts, *Geometry, *History, Mathematical Models, *Mathematicians, Mathematics, *Secondary School Mathematics ABSTRACT This book is a reissue of the second edition Which- appeared in 19410. It has the distinCtion of being the first vintage mathematical work published in the NCTM series "Classics in Mathematics Education." The text includes a biography of Pythagoras and an account of historical data pertaining to his proposition. The remainder of the book *shows 370 different proofs, whose origins range from 900 B.C. to 1940 A.D. They are grouped into the four categories of possible proofs: Algebraic (109 proofs)_ jGeometric (255)i Quaternionic (4) ;and those based on mass and velocity, Dynamic (2) . Al§o-included are five Pythagorean magic squares; the formulas of Pythagoras; Plato, Euclid, Maseres, Dickson, and Martin for producing Pythagorean triples; and a biblibgraphy with 123 entries; (RS) t; Y. T ,-,:;.;41-it: ,t -.2i MICROFICHEREPRODUCTION .1'.7. t ie I , .8 4121. gttilittt r:gZ 8 fst,r..'. i r.:44fl l'IwfwFt10*, ..... :. F 4. "-5.7:.;:t. .F-.--,-, ---,:-:,;::.'e: :IP, i /,:i .f. Lti:';",ttriz::,.......:i .,......: ::3 .::I,, U S. DEPARTMENT Of HEALTH, EDUCATION & WELFARE OFFICE OF EDUCATION PROCESSWITH AND MICROFICHE PUBLISHER'S MICROFICHE PRICES, ONLY. REPRODUCTION PABP0811AN11E1N ELISOA S. LOOMIS Photograph-taken 1935 01. Pt% IiI188 kmto reproduce this copyrightedwork has been granted to theEducationalResourct;s (EllIci andto the organization InformationCenter operating under with the Office01 Education to contract _cluded_in-the-Efi-le-systeni reproduce documental:1 by Meansof but this rightis not conferred microfiche only fiche received to any usersof the micro- from theERIC Document Service.Furtherreproduction Reproduction of any partrequires mission of thecopyrightowner. per- HE PYTHAGOREAN PROPOSITION 1,4 Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the FoUr Kinds. of "Proofs" Ensile Scott Loomis-- NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS 1201,Sixteenth Street, N.W., Washington, D.C. 20036 j About the Author Elisha Scott Loomis,Ph.D., LL.B.,was professor of mathematics at Baldwin Universityfor the period 1885-95and head of the mathematics departmentat West High School,Cleveland, Ohio, for the period 1895-1923.At the time whenthis second edition was published, in 1940, hewas professor emeritus of at Baldwin-Wallace College. mathematics About the- Book The second edition ofthis book (published igan, in 1940) is in Ann Arbor, Mich- here reissuedas the first title in *a seriesof "Classics in MathematicsEducation." N C Classics M Copyright ,ri") 1940 byElisha S. Loomis Copyright 1968byThe National Council of Teachers ofMathematics, Inc. "PERM I SS IOX N TO REPRODUCE THISCOPYRIGHTED M R AMaEENGRANTED t k,ounc. Teach. h TO Elk AND ORGANIZATIONS OPERATINGUNDER AGREEMENTS WITH THE U. S. OFFICE OF EDUCATION. Printed in the U.S.A. FURTHER REPRODUCTION OUTSIDE THE ER IC SYSTEM REQUIRES PERMISSION OF THE COPYRIGHT OWNER." <. PREFACE Some Mathematical works of considerable vintage have a timeless quality about them: Like classics in any field, they still bring joy and guidance t.o the reader. Substantial works of this kind, when they concern fundamental principles and properties of school mathematics, ;are being sought out by the Supplementary Publications Committee. Those.that are no logger readily- available will be. reissued by the National Council of Teachers of Mathematics. This book is the first such classic deemed worthy of once again being made avail- able to the mathematics education community. The initial manuscript for The Pythagorean Proposition was prepared in 1907 and first published in 1927. With per- mission of the Loomis family, it is presented here exactly asthe second---edition appe-a-redi-n 1940.. Ex-ce-ptfo-rstrch necessary changes as providing new title and copyright pages, and adding' this Preface by way of explanation, no attempt has been, made to modernize the book in any way. To do so would surely detract from, rather than add to, its value. "In Mathematicsthe man who is ig norant of what Pythagorassaid in Croton in .500 B.C. about thesquire on the longest side of a rightangledtriangle, or who for gets what someone in Czechoslovakiaproved last week aboutintqualities, is likelyto be lost. The whole terrificmass of well established Mathematics,from the ancient Babylonians to the modernJapanese, isas good today as itever was." 1.4e11, 1931 FOREWORD According to Hume, (England's thinker who interrupted Kant's "dogmatic slumbers"), arguments may be divided into: (a) demonstrations; (b) proofs; (c) probabilities. By a demonstration, (demonstro, to cause to' see), we mean a reasoning consisting of one or more catagorical propositions "by'which some proposition, brought into question is shbwn to be contained in some other prOposition'assumedlyhose truth and cer- tainty being evident and acknowledged, the proposi- tion in question must also be admitted certain. The result is science, knowledge, certainty."'The knowl- edge which demonstration gives is fixed and unalter- able. It denotes necessary consequence, and is synonymous with proof from first principles. By proof, (probo, to make credible, to demon- strate), we mean 'such an argument from experience as leaves no room for doubt or opposition'; that is, evidence confirmatory of &proposition, and'adequate to establish it. The object of this work is to present to the future investigator, simply and concisely, what is known relative to the so-called Pythagorean Proposi- tion, (known as the 47th proposition of Euclid and as the "Carpenter's Theorem"), and to set forth certain established facts concerning the algebraic and geo- metric proofs and the geometric figures pertaining thereto. It establishes that: First, that there are but four kinds of demon- strations for the Pythagorean propobition, viz.: I. Those based upon Linear Relations. (im- plying the Time Concept)Aithe Algebraic Proofs. vii THE PYTHAGOREAN PROPOSITION II. Those-based-upon Comparison of Areas (implying the Space Concept)--theGeometric Proofs. III. Those based upon Vector Operation (im- plying the Direction Concept)--the Quaternionic Proofs. IV. Those based upon Mass and Velocity (im- plying the Force Concept)--the DynamicProofs. Second, that the number of Algebraic proofs is limitless. Third, that thereare only ten types of geo- metric figures from whicha Geometric Proof can be deduced. This third fact is not mentionednor implied by any work consulted by theauthor of this treatise, but which, once established, becomesthe basis for the classification of allpossible geometric proofs. Fourth, that the number of geometricproofs is limitless. Fifth, that no trigonometricproof is possi- By consulting the Table of Contentsany in- vestigator can determine iii-what fieldhis proof, falls, and then, by reverence tothe text, he can find out wherein it differsfrom what has already been established. With the hope that this simpleexposition of this historically renowned andmathematically funda- mental proposition, without whichthe science of Trig- onometry and all that it implies wouldbe impossible,- may interest many minds and prove helpful andsugges- -tive to the student, the teacherand the future orig- inal investigator; to eachand to all who are seeking more light, the author, sends it forth. ry, CONTENTS Figures Page Foreword vii Portraits xi Acknowledgments xiii Abbreviations and Contractions xv The Pythagorean Proposition 3 Brief Biographical Information 7 Supplementary Historical Data 11 An Arithmetico-Algebraic Point of View .. 17 Rules for Finding,Integral Values for a, b and h 19 Methods of Proof - -k Method6 22 I. Algebraic Proof:. Through Linear Relations . 23 A. Similar Right Triangles--several thousand proofs possible 1- 35 23 B. The Mean Proportional Principle 36- 53 51 C. Through the Use of the Circle 54- 85 60 (I) Through the Use of One Circle 54- 85 60 (1) 'The Method of Chords 54- 68 61 (2) The Method by Secants 69- 76 68 (3) The Method by Tangents 77- 85 74 (II) ThroUgh the Use of Two Circles 86- 87 80 D. Through the Ratio of Areas 88- 92 83 E.,ThroAgh the Theory of Limits 93- 94 86 F. Algebraic-Geometric Complex 95- 99 88 G. Algebraic-Geometric Proofs Through Similar Polygons, Not Squares -1, 100-103 91 II. Geometric Proofs--10 Types i 97 -Type. All three sq's const'd exterior . 104-171 98 B-Type. The h-square const'd interior . 172 -193 144 C-Type. The b-square const'd interior . 194-209 158 D-Type. The a-square const'd interior . 210-216 165 E-Type. The h- and b-sq's const'd interior 217-224 169 F-Type. The h- and a-sq's const'd interior 225-228 174 G-Type. The a- and b-sq's const'd interior 229-247 176 H-Type.All three sq's const'd interior 248-255 185 ix x THE PYTHAGOREAN PROPOSITION FiguresPage I-Type. One or more squares translated,giv- ing 7 classes covering 19 differentcases 189 (1) Four cases 256-260 191 (2) Your cases 261-276 193 (3) Four cases 277-288 201 (4) Two cases 289-290 206 (5) Two cases 291-293 208 (6) Two cases 294-305 209 (7) One case 306 215 J-Type. One or more of thesquares not graphically represented--Two sub-types 216 (A) Proof derived througha square, giv- ing 7 classes covering 45 distinct cases 216 (1) Twelve cases, but 2 givsg . 307-309 218 (2) Twelve cases, but 1 given . 310-311 219 (3) Twelve .cases, none given . 0 (4) Threecases, 312-313 22.1 (5) Three cases,none given 0 (6) Three cases, all 3 given .
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