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DOCUMENT ED 037 335

RESUME

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007

848

AUTHORTITLE

INSTITUTION PUBDATE

NOTE AVliiLABLE FFOM

. National Council of Teachers of Mathematics, 1201 Sixteenth Stteet, N., Washington, D.C. 2003.

Washington, 68 310p.

Loomis , Elisha Scott The Pythagorean Proposition, Classics in Mathematics Education Serieso National Council of ---Teachers of athematics, Inc", ' -I> D.. c.

EDRS PFICE DESC RIPTURS

Price MF-$1.25 HC Not Available fro m EDRSu Algebra, Geomet ric Concepts, *Geometry, *History, Mat matical Models, *Mathematicians, Mathematics, *Secondary School Mathematics

EDRS

-ABSTRACT

This book is a reissue of the second edi tion fiitK appear-ed in 194 .. It has the dis tinction of being t he first v intage mathematical work published in the NCTM series "Classics in Ma:thematics Educ.tion." The text includes a biography of Pythagoras

and an a ccount of historical data pertaining to his proposition. he remainder of the book shows 370 different proofs, whose origins rang_e from 900 B .. C. to 1940 A. D.. They are grouped into th, e four cat egqies r=--- o .Po.s.sihle -troo-fs;__ _P,._l_gQra;i,_(.; _(jj)_9 J?-roofs)_; Geometric (255)_; _ _ _ _ Quaternionic (4 ) ; and those base d on mass an-d velocity --Dynamic (2) _ Aloi-I)cluded are five Pythagorean magic squares;- the formulas of Pythagoras; P lato, Euclid, Maseres, Dikson, and Martin for pro_ucing Pythagorean triples; an d a biblibgraph y ith 123 en tries: (RS)_ __

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--IC!HWJCHEONLY.

PHOC:J or ganization onti ng under contract with Uw Otfice 01 Educ

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HE PYTHI&.OREAN PROrOSITIOIi. __ ---

Its Deqmnstrations Analyzed and Classified . and Bibliography of Sources for D t Qf the _l!_JL.

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Four Kinds. of "Proofs"

Elisha scon Lomls--

. NATIO!'!'A COUNCIL OF TEACHERS OF MAT -IE MATIQS 1 l20t.S1xteenth Stree t, N W Washington, D.C. 20036

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A bout the A.uthorElisha Scott Loomis, Ph.D., LL.B., was professor of mathematics at Baldwin Unive.rsity for the period 1885-95 and head of the mathematics department at West High School, Cleveland, Ohio, for the priod 1895-1923. At the time when this second edition was published, in J.940, h was professor emeritus of mathematics at Baldwin-Wallace College. ,.

About the BookThe second edition of this book (published in Ann Arbor, Mich igan, in 1940) is here reissued as the first title in a series of "Classics in Mathematics Educati on."

DDDllClassics

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Coptf1'ight '0 J.fJltO by Elisha. S . .Loomis Cop 1ight "PERMISSION TO REPRODUCE THIS COPYRIGHTED MAIER IAL BY MICROFICHE ONLY. HAS_BEEN. GRANTED BY t. ounc. Teach Math TO ERk AND o'RGANIZAHONS OPERATING UNDER AGREEMENTS WITH THE U.S. OFFICE OF EDUCATION . FURTHER REPRODUCTION OUTS IDE nfE ERIC SYSTEM REQUIRES PERMISSION OF THE COPYRIGHT OWNER."

Council of Teache1s of Mathematics, Inc.

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1968

by

The

National

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Printed in the U.S.A.

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PREFACE..

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Some inathematical workH of considerable vintage have a _ _ tjmeless quality about them. Like c.Iassics in any field, they stilfbring joy and guidance t.o th reader. Substantial works of this kind, when they concern fundamental principles and . properties of chool mathematics, are being sought out by the Supplementary Publications Committee. Those ..that are no lopger readily. available will be. reissued by the National Council of Teachers of MJheinatics. This book is the first such classic deemed -worthy of once again being made avail able to the mathematics education community. The fnitial manus!_!---c!

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B-YIJe.C-Ty:pe. D-Ty:pe. E-Ty:pe. F-Ty:pe. G-Type.

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xte:r:'ior

The h-square const'd interior

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The b-square const'd interior The a-square const'd interior

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Te h- and b-sq's const'd interior The h- and a-sq's const'd interior The a- and b-sq's const'd interior All three sq's const'd interior

H-Ty:pe.

104-171 172193 194209 210-216 217-224 225-228 229-247 248-255

98 144 158 165 169 174 176 185

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ix

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THE PYTHAGOREAN PROPOSITION FiguresI-Type. One or more squares translated, giv

Page

ing 7 olasses covering 19 different cases (1) P'our cases (2) Four cases . ! (3) Four cases (4) Two cases -

(6) (7)J-Type.

(5)

Two cases Two cases One case

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256-26o 261-276 277-288 289-'290 291-293 294-305 306

189 191 193 201 ?06 208 209 215 216

One or more of the squares not

graphically represented--Two sub-types

(A)

Proof derived through a square, giving

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classes covering .

45. .

distinct . .

cases

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(B)

Proof based upon a triangle through calculations and comparisons of equivalent areas

(6)

(1) (2) (3) (4) (5)

Twelve cases, but Twelve cases, but Three cses,

2 1

Twelve .cases, none given

given

giv

Three cases, none given. Three cases, all

3

given

307-309 .310-311 0 312-313 0 314-328

216 218 219 221 222 230 230 244 246' 248 252 254 258 271 277 281

329-346 c

Why No Trignometric, Analytic Geomety Nor Calculus Proof Possible III. IV. Quaternionic Proofs Dynamic Proofs.

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347-350..

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351-353 354 355-359 66-366..

A Pythagorean curiosity Pythagorean Magic Squares Addenda.-

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Bibliography Testimonials Index . . .

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PO.R TR Al TS

1. Loomi s, Eli sha 2. C opernicus 4. Euc lid

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. . . . . Fronti spie-ce . . . . . . .. .

3.

Des c art e s .'

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5 . Gali lee . . . . . . . . . .'

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. fac ing pageII II II II II II II II II II. II II II II II II II II II II

244 118 188 -.-,'58 l-44 168 8 210 16

88

8. Lobachevsky . .9 . Napier

7 . Leibniz . . .

6. Gau s s . . .

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10 . Newt on

12.. Sylve s t er .

1 1 . Pythagora s

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266

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ACKNOWL E DGM ENTS-

E v e r y man bu t l d s upo n h t s p r e d e c e s s o r s .

My prede c e s s or s made thi s work po s s ib l e , and may tho e who make further investigation relative to. thi s - renowned prop o s i tion do b e t ter than their predec e s s or s have done . The author herewith expre s s e s hi s obliga t ions: To the many who have re c e ded him in thi s fie ld, and who s e t ext and proof he ha s acknowledged here in o the page Where such proof i s found; To tho s e who, . upon reque s t ,. c ourte ou s ly gr ant ed him permi s s i on t o make u s e. of such pro of, or refer t o the same; To the following Journal s and Magaz ine s who se owners .so kindly extended o him permi s s ion to u se . proofs found the rein, viz. . . :I ,

The American Mathematical Month1y; Heath ' s Mathematic al Monographs; Te Journa of Educ ation; 'The Mathemat i cal Magaz ine ; The School Vi sitor ; The Scient ific American Suppl ement{ Sc ienc e and Mathema t ic s ; and Scienc e . -To. The odore H . Johns t on , Ph . D . , formerly Principal of the We s t High S choo l , C leveland , Ohio , for hi s valuab l e s tyli stic s ugge s tions after reading the origial manu script in 1 907 . T o P rofe s s or O sc ar Le e Dustheiiner Prof.e, s s or of Mathema t.ic s a_nd Astronomy , Baldwin-Walac\ Pol lege , Berea, .O[li o , for hi s profe s s i onal a s s i s tanc e and ad vide ; and t o David P . Simpson, 33 , former Princ ipal of We s t High School , C leve land; Ohio , for hi s br other! ly enco uragement , and helpful suggst ions , 1926.,

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To Dr . Jehutnfe l Ginsburg , ..p ubli sher of Sc ripta Mathemat ica , New York C i ty, for the right to reproduc e the photo _plte s of t en of hi s "Fortrai t s of Emineht Mat hemat i c i ans . " To Elatu s G . Loomi s for hi s a s s i s t ance in drawing the' 366 figure s which appe ar. in thi s Second Edi tion . Apd to "The Ma s t e r s ;and Ward.ens