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

Washington, D .. c. 68 310p. . National Council of Teachers of Mathematics, 1201 Sixteenth Stteet, N��., Washington, D.C. 2003�

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EDRS Price MF-$1.25 HC· Not Available from EDRSu Algebra, Geometric Concepts, *Geometry, *History, Mat��matical Models, *Mathematicians, Mathematics, *Secondary School Mathematics

This book is a reissue of the second edition �fiitK appear-ed in 194� .. It has the distinction of being t he first vintage mathematical work published in the NCTM series "Classics in Ma:thematics Educ.�tion." The text includes a biography of Pythagoras and an account 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 categq�ies

r=--- o£ .Po.s.sihle �-t�roo-fs;__ _P,._l_g��Qra_;i,_(.;� _ _(jj)_9 __ J?-roofs)_; Geometric (255)_; _ __ _____ __:

Quaternionic (4) ; and those based on mas_s an-d velocity� --Dynamic (2) ��­Al�o�i-I)cluded are five Pythagorean magic squares;- the formulas of Pythagoras; Plato, Euclid, Maseres, Di�kson, and Martin for pro_�ucing Pythagorean triples; an d a biblibgraph y �ith 123 entries: ·(RS)

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PROei<:ss WITH MICRO!o'ICHE AND PUBUSHER•s PRICES. MICUOJ.'ICUE REPRODUCTION ONLY.

HE PYIHIBOREIN PROPOSITION

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ELISHA S. LOOMIS i

Photograph taken 1935 I . I I r

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h·: IIIJijsion to repl'odu�:c this (!Opya·ighted work has boon gnmted to the 1-;ducationai Reaou1·ccs JnformaUon Center (1-:lHc, und to iht> organization o�nting under contract with Uw Otfice 01 EducatiOn to reproduce d�u��nts.Jn..._ - .. · _____ - · --=clwJ�.Jn_the-ElH<:-systenrt>y·inoans· of-microfiche only, but this rigl\t Js not conferred to any usore of the micro­fiche recet'ved f.rom the EHJC Document Reproduction Service. Jo'urther reproduct�on of any part r6Qufrea per­mission of the copyright owner.

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HE PYTHI&.OREAN · PROrOSITIOI

Deqmnstrations Analyzed a.nd Classified

and ·

D t f the Bibliography of Sources for __ _l!_JL.Q --�-

·�--- � Four Kinds. of "Proofs"

Elisha scon Lo�mls--

. . UNCIL OF TEACHERS OF MATi-IE.MATIQS NATIO!'!'A� CO

S t ·N W Washington, D.C. 20036 l20t.S1xteenth tree , 1 • ••

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A bout the A.utho·r Elisha 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 p�riod 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· Book The second edition of this book (published in Ann Arbor, Mich­igan, in· 1940) is here reissued as the first title in ·a series of "Classics in Mathematics Education."

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Classics Dll

Coptf1'ight '0 J.fJltO by Elisha. S . .Loomis Cop�1·ight «:> 1968 by The National Council of Teache1·s of Mathematics, Inc.

"PERMISSION TO REPRODUCE THIS COPYRIGHTED MAIER IAL BY MICROFICHE ONLY.hHAS_BEEN. GRANTED

BY � t. �ounc. Teac • Math TO ERk AND o'RGANIZAHONS OPERATING UNDER AGREEMENTS WITH THE U.S. OFFICE OF EDUCATION. - Printed in the U.S.A. FURTHER REPRODUCTION OUTS IDE nfE ERIC SYSTEM REQUIRES PERM ISS ION OF THE COPYRIGHT OWNER."

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PREFACE ..

Some inathematic�al workH of co_nsid_erable 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 M�Jheinatics. This book is the first such classic deemed -worthy of once again being made avail­able to the mathematics education community.

The fnitial manus<;ript for The Pythago1·ean P1·oposition was prepared in 1907 and first published in 1927. With per-

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mission of the Loomis family, it is presented here exactly ·

:as· the-second·,-edition-app·eare·d-in-t940::-Exc-ept-for-su·ch--- --- - -----------�---- -- --i ·.necessary ch�nges as pr�viding new title and copyright _

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pages and· adding� t�is Preface by way of expl�nation,. no -attempf·has been made to moderp.ize the book in ·any way. To do so would surely d�tract from, rather .than add to, its -value. .

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"In Mathematics the man who is ig-norant of ��a� Pythagoras said in Croton in -500 B.C. �bout_the squ�re on t�e lo�gest side of· a right-angled triangle� �r who for­gets what someone in Czechoslov&kia proved last week about inrequaliti-es, is likely to be 1ost. The whole terrific mass of well­established Mathematics, from the ancient B�bylonians t6 the modern Japane·se, is as good today as it ever was."

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FOREWORD

According to Hume, (Eng�and ' s thiP�er who interrupted Kant ' s " dogmatic slumber s " ) , argument s may be divided int o: ( a ) demonstrations ; (b ) proofs ; ( c) probabilities .

By a demonstration, ( demonstro, to cause to· see ) , we mean a reasoning consisting of one or more catagorical proposit ions "by ·which some proposition:. brought int o que st ion is shown to be contained in s ome other proposition·as sumed, .whose truth and cer­tainty being evident and acknowledged , the propos i ­tion i n que stion must also be admitted certain . The re sult i s sc:t,ence, knowledge, cert·ainty . u· The knowl­edge which demonstration give s is fixed and ��alter­able . It denote s ne_ces sar_y consequence , and i s synonymous with proof from first principle s .

--------------·----------------�--- By-proof, (probo, to make credible, t6 demon-

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strate.) , we mean 1 such an argument from eXPerience . as leave s no room for doubt or opposition ' j that i s ,. evidence confirmat ory of a. proposition, ana adequate to e stablish it .

The ob ject of thi s work is to pre sent to the future inve stigator , simply and concisely , what i s known relative to the �o-called Pythagorean Propo si­tion, (known a s the 47th proposition of Euc lid and as the " Carpenter ' s Theorem" ) , and to set forth certain e�tablished fact s concerning the algebraic and geo­metric proofs and the geometric figures pertaining thereto .

It e stablishe s that : Ffr st, that there are but four kinds of demon­

str�tions for the Pythagorean proposition, viz. : \

I. Those based upon Linear Relations. ( im­plying the Time Concept ) .... the Algebraic Proofs.

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viii THE PYTHAGOREAN PROPOSITION - -- -- -- --�----- -- ------- II �-·Tnose ·oas.eu upo-ri--Comparfscm -of Areas ·

( implying the Space Concept ) ·- -the Geometric Proofs . III. Those .based upon Vector Operation ( im­

plying the Direc tton Co�cept ) - -the Quaternionic Proofs.

IV . Tho se based upon Mas s and Velocity ( im­plying the Force Concept ) - -the Dynamic Proofs.

Second, t�at the number of Algebraic proofs is limitle s s .

-%hird, that there are only ten type s qf geo­me tric figur•e s from which a Geometric Proof can be 'd.educ e d .

Thi s-third fac t i s not mentioned nor implied by any work_consultea by the author of this treatise, but which, onc e e stablished, become s the bas·i s for the c las sification of all pos sible geometric proofs .

Fourth, that _the number of geometric proofs is limitle s s .

ble . Fifth, that no trigonome tric proof �s po s s i -

By consulting the· Tabie o f Content s any in­ve stigator can determine in-what field hi s proof

.._ __ _ , __ _., ,. falls , and then, by reterence to the text, he can find out wherein it differs from what has already­been e stabli shed. ..

With the hope that thi s simple expo�ition of thi s historically renowned and mathematically funda­mental proposition, without which the science of Tnig­onometry and all that it i�plie s would be impo2-srbre,-·may intere st many minds and prQv-e helpful and sugge s ­

·��·t�ve to the studen�� the t�acher and the future orig­inal inve stigator, t� each and to all who are seeking more light, the author , s ends it forth .

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CONTENTS

Figl_!res Page

Foreword Portra�ts Acknowledgme�ts . . . , . . . . . . Abbreviations and Contractions . . • .

The Pythagorean Proposition . . . . . . . . . Brief Biographical Information . . . . . . Supplementary Historical Data . . . . . . An Ari thmetico-Algebraic Point of View . . .. . . . Rules for Finding.Integral Values for a, b and h . Methods of Proof--4 Methods . . . . . . .

I. Algebraic Proof� Through Linear Relations A. Similar Right Triangles--several thousand proofs

possible . • • • • • • • • • • • • • . 1- 35 B. The Mean ProporUonal Principle • • • • . 36- 53 C. Through :the Use of the Circle • • • • • • 54- 85

(I) Through the Use of One Circle • • . 54- 85 ( 1) 'The Method of Chords • • • 5.4- 68 (2) The Method by Secants • . • 69- 76 ( 3) The Method by Tangents • • • • • 77- 85

(II) ThroUgh the Use of Two Circles • • • 86- 87 D. Through the Ratio of Areas • • • • • 88- 92 E • . Through the Theory of Limits • • • • 93- 94· F. Algebraic-Geometric Complex • • • . 95- 99 G. AJ..sebraic-Geomet:ric Proofs Through Similar

Polygons, Not Squares • • . • • -. • • • 100-103

IL- Geometric Proofs--10 Types • . . • •

�-Type. �11 -���13 __ �9,' _s_ C_s>!_!�-�-�-c! __ �xte:r:'ior B-�YIJe. C-Ty:pe. _

D-Ty:pe. E-Ty:pe. F-Ty:pe. G-Type. H-Ty:pe.

The h-square const'd interior . . The b -square const 'd interior • •

The a-square const'd interior . •

T�e 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

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104-171 172�193 194�209 210-216 217-224 225-228 229-247 248-255

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23 51 60 60 61 68 74 8o 83 86 88

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144 158 165 169 174 176 185

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Figures Page I-Type. One or more squares translated, giv-

ing 7 olasses covering 19 different cases • • • •

(1) P'our cases • • • • • • 256-26o (2) Four cases .- • • • ! • • • • • 261-276 ( 3) Four cases • • • • • • • 277-288 (4) Two cases • • • • • • • • • • • • 289-'290 (5) Two cases • • • • • • • • • • 291-293 (6) Two cases • • • • • • • • • • 294-305 ( 7) One case • • • • • • • co • • • • • 306

J-Type. One or more of the squares not graphically represented--Two sub-types • • • • •

(A) Proof derived through a square, giv-ing 7 classes covering 45 distinct cases . . . . . . . . . . . . . . . . (1) Twelve cases, but 2 giv� • • 307-309 (2) Twelve cases, but 1 given • • .310-311 (3) Twelve .cases, none given • • • • 0 (4) Three c�ses, 312-313 (5) Three cases, none given. • • • • 0 (6) Three cases, all 3 given • • • • 314-328

(B) Proof based upon a triangle through calculations and comparisons of equivalent areas • • • • • • • •

Why No Trig�nometric, Analytic Geomet�y Nor Calculus Proof Possible • • • • • •

• 329-346

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III. Quaternionic Proofs . . . .

IV. Dynamic Proofs . . . . . . . . .

A Pythagorean ·curiosity Pythagorean Magic Squares· . . .

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Addenda . . . . . . . . . . , . . .

Bibliography • • . . . . . . Testimonials . • . . . . . . Index . . . . . . . . .

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347-350

351-353

354 355-359 �66-366

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189 191 193 201 ?06 208 209 215

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252 254 258 271 277 281

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1. Loomis, Elisha s. . . . . . Frontispie-ce 2. C opernicus . . . . . . fac ing page 88 3. De·scart e s II II 244 . . . . . . . 4. Euclid II II 118 . . . . . . . . 5 . Galilee '

II II 188 . � .. . ' . . . . . . . . . 6. Gauss II II 16 . . . . . . . 7 . Leibniz II II -.-,'58 . . . . . . . . . . 8. Lobachevsky II II 210 . . . . . . . . . •

9 . Napier II II l-44 . . . . . . . . . . 10 . Newton II II 168 . . . . . • . . . .

11 . Pythagoras II. II 8 . • . . . .

12 .. Sylve ster II II 266 . . . . . . . . . .

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ACKNOWLEDGMENTS-

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

My predece s sor s made thi s work po s sible , and may tho�e who make further investigation� relative to. thi s - renowned proposition do better than their predece s sor s have done .

The author herewith expre s s e s hi s obliga-tions:

To the many who have �receded him in thi s field, and whose text and proof he has acknowledged herein o� the page Where such proof i s found;

To those who, . upon reque st ,. c ourteously grant­ed him permi ssion to make use. of such proof, or refer to the

owners proofs

I , same; To the following Journals and Magazine s who se

.so kindly extended �o him permi s sion to u se found therein, viz . . .. :

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The American Mathematical Month1y; Heath ' s Mathematical Monographs; T�e Journa� of Education; 'The Mathemati cal Magazine ; The School Vi sitor ; The Scientific American Supplement{ Science and Mathematic s ; and Science .

-To. Theodore H . Johnston, Ph . D . , formerly Principal of the We st High School , C leveland , Ohio , for his valuable s tyli stic s ugge s tions after reading· the origi�al manuscript in 1907 .

·To Profe s sor O scar Lee Dustheiiner , · Prof.e, s sor of Mathema t.ic s a_nd Astronomy , Baldwin-Wal�ac�\Pollege , Berea, .O[li o , for hi s profe s s i onal assi stance and ad­vide ; and to David P . Simpson, 33°, former Principal of We st High School,· C leveland; Ohio , for his brother-! ly encouragement , and helpful sugg�stions , 1926.

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• • ·::...- >J ... , .... � .... _. - �--;:_� ;. • :�·,4'; . To Dr . Jehutnfel Ginsburg ,

-.. publi sher of

Scripta Mathematica , New York C i ty, for the right to reproduce� the photo ·_pl�te s of ten of his "Fortrai t s of Emineht Mathematicians . "

To Elatus G . Loomi s for hi s assistance in drawing· the' 366 figure s which appear. in thi s Second Edition .

_ · . Apd to "The Masters ;and Ward.ens As sociation of The· 22nd Masonic Di strict of the Most Worshipful Grand Lodge of Free and Accepted Masons of Ohio, " owner of the C opyright granted to i t in 1927 , for .i�s gen�z:oous permi s sion to>publi sh thi s. Sec::ond Eqit_ion of The Pythagorean Proposition, the author agreeing that a complimentary copy oi.'· i t shall be sent to ·the known Mathematical Libraries of the World, for pri­vate re search work, .and al so to such Masonic Bodie s a s i t shall selec t . (Aprcil 27 , 1940 )

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A�BR EVIATIONS.AND CONT R A C110NS

Am.' Math . Mo . = The American Mathematical Monthly, 100 proofs , 1894.

a-square = square upon the shorter leg . b-squaf.e = 11 ) ' " longer leg. Colbrun = Arthur R: Oolbrun ,-- LL . M., Dlst . of Columbia

Bar. -·'const . = co"nstruct .

const ' d = constructed . cos = cosine. Dem . = demonstrated, or demonstration . Edw. Geom . = Edward ' s Element s of Geometry, 1895 . e q . = equation . e q ' s = equations. Fig. pr fig . = fig�re. Fourrey = E. Fourrey ' s Curiosities Geometriques . H'eath = Heath ' s Mathematical Monographs , 1900 ,

Parts I and II- -26 proofs . . . h-square = square upon the �ypotenu se . Jo.ur . Ed ' n = Journal of Education. Legendre · = Davie s Legendre , Ge'ometry, 1858 . -Ma t:n:-- = rna thematic s Math. Mo· . = Mathematical Monthly, 1858-9 . Mo. = Monthly . . No. or no. = number . Olney ' s Geom. = -Olney ' s Element s of Geometry, Uni-

versity Edition . outw ' ly = outwardly. par. = parallel. varal. = parallelogram . perp. = perpendicular. p. = page. p t! = point. qua� . = .quadrilateral. re sp ' y = re spec tively.

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xvi THE PYTHAGOREAN PROPOSITION

Richardson = John M . Richardson- -28 proofs . rt . = right . rt . tri . = right triangle . rect . = recta�gle . Sci. Am. Supt . = Scientific American Supplement ,

1910,· Vol . 10 . sec = secant . s in = sine . s q . = square . sq 1 s_ :::; squares . tang = tangent . :. = therefore . tri . = _t _riangle . tr11s =�triangles . trap . � trapezoid . V . or v. = vol�e . Versluys = Zes en Negentic ( 96 ) Beweijzen Voor Het

Theorema Van J>ythagoras , by J . Versl.uys , 1914 . Wipper = Jury Wipper 1 s " 46 Beweise der Pythagor-_

aischen Lehr�atze�, 1 1 1880 . ·

�HE2, or any like S)�bol. = the square of, or upon, the line HE , or like s�bol .

ACIAF, or-like symbol = AC + AF , AC or - · AF See proof 17.

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GOD ·GEOMETRIZES - CONTINUALLY -PlAl'o.

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PYTHAGORAS From a Fresco by Raphael

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CARL FRIEDRICH GAUSS 1777-1855

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NICHOLAS COPERNICUS 1'7�-1543

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J'AMES · J OSEPH SYLVE�TER 1814- 1897

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THE PYTHAGOREAN PROPOSITION

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This celebrated proposition �s one of the most important theorems in the whole realm of geome­

\ try and is known in history as the 47th proposition, that being it s nUmber in the first book of Euclid ' s Elements .

It is also (erroneously) sometimes c�lled the Pons Asinorum . Although the practical application of this theorem was known lohg before the time. of· Pythagoras he, doubtle s s, generalized it from an Egyp­tian rule of thumb (32 + 42 = 52) and firs t demon­s tr��ed it about 540 B . C . , from which fact it is gen­erally known as t�e Pythagorean Proposition . This famous th�orem has always been a favorite with geo­�etricians .

(�he statement that Pythagoras was t� in­ventor of the- 47th proposition of Euclid has bee�_de­nied by many s tud�nt s of the subject . )

Many purely geometric demons trations of this famous theorem are· acqes sible to. the teacher,. as well as an unlimited .. number of prqofs .'Qased upon the al-

. gebraic method of geometric investigation . Also _quaternions and dynamics fu_rnish a few proofs .

-No ¢loubt many other proofs-"lhan __ these now known will be res olved by future investigators, for the possibilities of the algebraic and g�o�etric re-­lations impliea· in�""the theorem are· limitles s .

·Th;I;s-�t-heo�em--with its many proofs is a strik­ing illustration of the fact that there 1-s more than one way of establishing the - same truth .

But before proce�ding to the methods .. of dem­onstration, the following his tories� account trans ­lated from a monograph by Jury Wipper, published·_ in 1880, and entitled "46 Beweise des.�ythagoraischen� Lehrsat�es," may ppove ·both interesting and profits-· ble.

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I THE PYTHAGORijAN PROPOSITION 4

Wipper acknowledge s hi s indebt edne s s to F. Graap who translated i t out of the Rus s ian . It i s as follows : "One of the weightie st propositipns in geometry if not the weightiest with reference to i t s deduct ions and applications i s doubtle s s the so­called Pythagorean propos ition . "

;. The G�eek text i s as follows : 'Ev �o(� opeoywvtot� �o ano ��� ��v 6pe�v

ywv(av uno�EtVOUcr�� nAEupa� �E�paywvov taov £a�( �Ol� &na· �@v ��v 6pe�v ywv(av neplExouaQv nAEup@v �E�paywvo t �.

The �atin reads : In rectangulis trianguli s quadratum, quod a latere rec tum angulum subtendente de scribitur , aequale e st eis , guae a lateribus rectum angulum continentibus describuntur .

German : In den rechtwinkeligen Dreiecken 1st das �uadrat, welches .von der· dem rechten Winkel gegenuber liegenden Seite be schrieben Wird, den Quad­raten, welche von· den ihn umschlie s senden Seite.n ·be schrieben werden , gleicp.

i .According to the te stimony of Proklos the

demonstration of thi s proposition i s due to Eucl id who _adopted i t in hi s elemen�s ( I , 47 ) . The method of the Pythagorean demo�1stration remains unknown to us . It i s undecided whether Pythagoras himself di s ­covered t]:li s characteristic of the righ�. triangle , or learned it from Egyptian priest s, or took i t from.' Babylon : regarding thi s opinions vary .

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Ac'?ording to that one mos t widely di's seminat-ed Pythagoras'learned from the Egyptian pri e s t s the characteri s.tic s of a triangle in which one leg = 3 ( de signating O siri s ) , the second = 4 ( de signating Isis ) , and the hypotenuse = 5 ( designating Horus. ) : f9r which reason the triangle itself i s also named the Egyptian or Pythagorean . *.

*(Note. r The Grand Lodge Bulletin, A.F. and A.M., of Iowa, Vol. 301 No. 21 Feb. 1929, p. 42, has: In an old Egyptian manu­script� recently discovered at Kahan, and supposed to belong

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THE PYTHAGOREAN PROPOSITION 5

The characteristics of such a triangle, how­ever , were known not to the Egyptian priests alone, , II the Chinese scholars also knew them . In Chine�e hist ory , " says Mr. Skat schkow, " great honors are awarded to t.he brother of the ruler Uwan, T schou-Gun, who lived 1100 B . C . : �e knew the characteristic s of the right triangle, (perfected) made a map of the star s , discovered the compas s and determined the length of the meridian and the equator:

Another schoiar'(cantor) says : this emperor wrote or shared in the composition of a mathematical treatise in which were discovered the fundamental feature s , ground lines , base line s , of mathemati c s_, in the form of a dialogue betwe_en 'l'sqhou-Gun and ··, Schau�Gao . The title of the book i s: Tschaou pi� i . e . , the high of T schao . Here too are the sides of a triangle already named legs as in the Greek; Lat in, German and �us sian languages .

Here are some paragraphs of the 1 s t chapter 'of the work . Tschou-Gun once said t o Schau -Gao : " I

learned, - s ir , that you know numbers and their appli­cations , for which re�son I would like to ask how old Fo-cpi determined the. degrees of the c�lestial sphere. There are no s teps on which one can c limb up to the sky, the chain and the bulk of the earth are also in­applicable; I would like for thrs reason, .to know how he determined the numbers . "

Schau-Gao replied : " The art.of counting goes back to the c ·ircle and square . "

If one divides· a right triangle into i t s part s the line_which unites the ends o f the sides

(Footnote continued) to the time of the Twelfth Dynasty, we find the fol�owing equations: 12 + (�)2 = (lt)2; 82 + 62 ·

= 102; 22 + (1�)2 = (2-k)2; 162 + 122 = 202; all of which are forms of the 3-4-5 tr:J.angle. • • • . We also find that this tri­angle wa� to them the symbol �f universal nature .. Th� base 4

. � ' · represented Osiris; -the perpend.icular 3, Isis; and the hypote-

nuse rep�esent�d Horus, their £Jon, being the p�.·oduct. of the two principles, male and female.)

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when the base = 3, the altitude = 4 i s 5 .. T schou-Gun cried out : "That i s indeed ex­

cellent." It is to be ob se�ved that the relations be­

twe�n China and.Babylon more than probably led to the as sUJpptiqn that this characterlstic was already known to the · Chalde�ns. As to. the geometrical demonstra-

� tion it�comes doubtles s from Pythagoras himself . In busying with the addition of the series he c ould very naturally g6 from the triangle with sides 3, 4 and 5, as a sin.gle instance to the general characteristic s of the right triangle .

After he observed th�t addit�on of the series of odd number ( l + 3 = 4 , l + 3 + 5 = 9, etc.) gave

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a series of squares, Pythagoras formulated the rule fdr finding , logically, the s ides o� a right triangl� Take an odd number ( say 7) which forms the shorter side, square i t (72 = 49) , subtrac t one _Li9- l = 48), halve the remainder ( 48- 2 = 24); thi s half i s the longer s ide, and this increased by one ( 24 + l = 25), is the hypotenu se . �

The ancient s recognized already the signifi ­cance of the Pythagorean propo siti on for which fact may serve among o thers -as proof the acc ount of D�og�­nes Laertius and Plutarch concerning Pythagoras . The latter is said to have offered (�acrificed) the Gods an ox in gratitude after he learned the notable char� acteri stic s of the right triangle . Thi s story i s without doubt a fiction, a s sacrifice of animal s , i . e., blood- shedding, antagonizes the Pythagorean teaching .

During the middle ages thi s proposition which was also named tnventua necatombe dt�num ( in-as-much as it was even believed that a sacrifice of a heca­tomb--100 oxen--was offered) won the honor-designa­tion N a� tste r •a�heseos, and the knowledge thereof was some decade s ago s till the prooT of a solid mathe­matical trainirig ( or education) . In examinations t o obtain the mas ter ' s degree this propo�ition was often given; there was indeed a time , as i s maintained,

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when from every one who submitted himself to the test a.s mas ter of mathema tic·s a new ( o·riginal ) demonstra­tion was required .

This latter circumstance, or rather the great ' �-�---"·

significance of the proposition under consideration w.as the reaqon why numerous demonstrations of it were thought out.

The collec tion of demonstrations which we * bring in what follows , m-qst , in our opinion, not

merely satisfy the s imple thirst for knowledge , but als o as important aids irr the teaching of geometry . The variety of demonstrations , even when some of tnem are finical , must demand in the learners the develop­ment of rigidly logical thinking, mus t show· them how many sidedly an 9bjec t can be considered, and spur

.them on to test their abilities in the disc overy of like demons trations for J;he one or the other propo si­tion. "

Brief .13iographi,ca1 lnf9rmation

Concern! �� �ytha�oras

"The birthplace of Pythagoras was the i sland of Samo s ; there the father of Pythagoras , Mnes sarch, obtained citizenship for services whi ch he had ren-

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dered the inhabitants of' Samos during a time of fam-ine . Ac companied by his wife Pithay, Mnessarch fre-

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quently traveled in busines s interest s ; during the year 569 A.C. he came to Tyre; here Pythagoras was born. At eighteen Pythagoras , secretly, by night , went · from ( left ) Samos ,- which was in the power of the tyrant P olycrate.s , to the i sland Lesbos t o hi s. ·unc le who welc omed him very hospitably . There for two years_ h� received instruction from Ferekid who with Anak­simander and Thales had the reputation of a phi loso-pher . .

*Note. There were'but 46 different demonstrations in the mo�o­graph by Jury'Wipper, which 46 are among' the c!assified collec­tion found in this work.

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8 THE PYTHAGOREAN.PROPOSITION

_ _ _ _ ___ .A.f.ter---P.ythagoras .hacl Jllade the . .r.eligi.o.us ideas. of his teacher hi s own, he went to Anaksimander and Thales in Miletus (549 A. C . ) . ·The latter was- then already 90 years old. With these men Pythagoras stud­ied chi�fly c osmography, i . e . , Physic s and Mathemat-ics. --

Of Thale s it is known that he borrowed the solar year from Egypt ; he knew how t o calculate sun and moon eclip se s , and determine the elevation of a pyramid from it s shadow; t o him al s o are attributed the_discov.ery of geometrical project ions of great im-port ; e. g . � the char�cteri stic.of the angle which i s inscribed and res t s with i t s s ides o n the diameter,_ as well as the characteris tic s of the angle at the base of an (equilateral) i sosceles triangle .

Of-Anaks imander it i s known that he knew the use of the dial in the determination of the sun ' s ele­vation ; he was the ·firs t who taught geography and drew geographical maps on c oppe� . I t must be observed t oo , that An"aksimander was the first pr.ose wri ter, as

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down to hi s day all learned works were written in , verse, a procedure which continued longest among the East Indians.

Thales di-rec ted the eager youth·to Egypt as the land where he could satisfy hi s thirst for knowl­edge . The Phoenic ian priest college in Sidon mus t in s om� degree ·serve as preparation ·for this journey-. Pythagoras spent an entire year there and arrived in EgY.?t 547, .

Although P olikrates who h�d fo��ven Pytha­goras ' nocturnal flight addre s se s to Amasi s a letter in which he c ommended the young scholar , i t c o st Pythagoras as a foreigner,_as one unclean, the mo st incredible toil to gain admis sion to the pries t'caste which only·unwillingly initiated even their own peo­ple into their mys teries or knowledge .

' The priests in the temple Heliopoli s to whom the king· in person brought Pythagoras dec-lared it im­possib le t o'receiv� him into .their midst , and direct -

ft ed him tq the oldest pries t c ollege at Memphi s , thi s

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THE PYT�GOR EAN _P_R.OE.OSI.T-ION-- -------�------9---�--� � �-�----�---

commended him.to Thebes. Here somewhat severe condi­tions were laid upon Pythagoras for his reception into the priest caste; but nothing could deter him. Pythagoras performed all the rites, and a�l tests, and his study began under the guip.ance of the c·hief :priest Sonchis.

During �is 21 years stay in Egypt Pythago�as succeeded not only�n fathoming and absorbing all tne Egyptian but also became sharer in �he highest honors of the prie-st caste.

In 527 Amasis died; in th� following (526) year in the reign bf Psammenit, son-of Amasis, the Persian king Kambis- invaded Egypt �rid loosed all his fury aga�nst the priest �aste. .

Nearly all mem�ers thereof fell into captivi­ty, among them Pythagoras, to whom as abode Babylon was assigned. Here -in the center of tHe world com-- \ merce where Bactrians, Indians, Qhinese}. Jews and

,, other folk came tog-;ther, Pythagoras had during 12 years stay opportunity to acquire those learnings in which the Chaldea.ns were so rich.

oA singular accident secured Pythagoras liber­ty in consequence of which he returned to his native land in his 56th year. After a brief stay on the island Delos where he found his teacher Ferekid still alive, he spent a half ye.ar in a visit to .Greece for the purpose of making himself familiar��ith the re-

, ligious, scientific_ and social condition thereof. The opening· of the teaching activity of Pytha­

goras, �n th� island of S�mos, was extraordinarily sad; in order not to remain wholly" witho�t 'pupils he was forced even to pay his sole pupil, who was also named Pythagoras, a son of E�atokles. This led him

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to abandon his thankless land and seek a "new home in .

the highly cultivated cities of Magna Graeci� (Italy) .. In 510 Pythagoras came to Krqton. As is

known it was a turbulent year. Ta�qu!n was forced to .flee from R ome, Hippias from Athens; in �he ne�ghbor­hood of Kroton, in Sibaris, ,insurrection broke out.

The first ap��ar�nc� o� Pyt�agoras before the· people of Kroton began ·wfth an oration to the youth

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wherein he rigorously but at the same time so con­vincingly s·et forth the duties of toung men that the elders of the city entreated him not t·o leave them without guidance (counsel). In his secon� oration he called attention to law abiding and purity of mor­als as the butresses of the family. In the two fol­lowing orations he turned to the matrons and chil­dren. The result of the last oration in which he specially condemned luxury was that thousands of·· ---

- costly garments were brought to the temple of Hera, bec-a-use no matron could make up her mind to appear

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� in them on the street. P-y.thagoras spoke captivatingly, and it is for

this reason not to be -wondered at that his orations brought about a change in the morals of Kroton's in­habitants; crowds of listeners streamed to him. Be-. I sides the youth who listened all day long to.his teaching some 600 ·of- the .worthiest men of the city, .. matrons and maidens, came toget�er at his evening entertai�ents; among them was the young, gifted and beautiful Theana, who thought it happines·s· to become the wife of the 60 year old teacher.

The listeners divided accordingly .into d·isci-, ---- '

c ples, who formed a school· in the narrower sense of the word, and into auditors, a school in the oroade.r sense. · The former, the so-called rna thema ticians. ·were giyen the rigorous teaching of Pythagoras as a scien­tific whole in logical success�on from the prime con� cepts of mathematics up to the highest abstraction of

·philosophy; at the same time they learned to-regard everything fragmentary in knowledge as more harmful than ignorance even. ' ··

From. the mathematicians must be distinguished the auditors (university extensioners) 'out. of whom subsequently were formed the Pythagorea·ns! Th� �e took part in the evening. lectures oniy in which noth­:ing rigorously scientific was taught. The chief themes of-these lectures were: ,ethics, immortality of the soul, and transmigration--metempsy�hology.

About the year. 49u when the ;pythagorean schqol l'eached its highest �plendor�-brilliancy--·a ·

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certain Hypasos who had been expelled from the school a s unworthy put himself at the head of the democratic party in �oton and appeared as ac cuser of his former colleague s . The school was broken up, the property of Pythagoras was confiscat�d and he hlmself exiled . ·

-·The �ubsequent�l6 years Pythagoras lived in TarentUm, . but even here the dem0cratic party gained the upper hand in 474 and Pythagoras a 95-year old man. must flee again to Metapontus where he dragged out his poverty-stricken exi s tence 4 �ears�mdre . Fi­nally democracy t riumphed there al so; the house in which was the school was burned, m�ny di sc ·iple s died a death of torture and Pyt�agora� himself with dif­ficulty having e scaped the flame s died soon after in his 99th year . "*

Supplementar y Historical Data

Tq.the following ( Graap ' s) translation, out of the Russian, relative to the gre·at master ,Py;tha­goras, the se ln�ere sting s tatement s are due .

"Fifte en hundred years before the time of Pythagoras, ( 549�470 B.C.),** the Egyptians construct ­ed right angle s by so plaqing thre e pegs that a rope" measured off into 3, 4 and 5 unit s would just reach around them, and for thi�purpose profe s sional ' rope fasteners ' -were employed .

"Today carpenters and masons make right an­gle s by measuring off 6 and 8 feet in such a manner that a ' ten-foot pole ' comple t e s the ·triangle .

" Out of thi s simple Nile--c ompelling problem.

of the se early Egyptian rqpe-fastener s P ythagor�s i s said t o have generalized and proved thi s important and famous theorem,- -the square upon the hjpotenuse

* Note. The above tran�lation ie that of Dr. Theodore H. John-ston, Principal (1907) of the Weet.High School, Cleveland, 0.

**Note . From recent accredited biographical data ae to Pytha­goras, the record reads: ".Born at Samoa, c; 582 B.C. Died probably at Metapontum., c. 501, B.C •. "

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THE PYTHAGOREAN PROPOSITIOij

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squ!i.res upon its two leg s , - -of which- the right tri-angle whose side s are 3, 4 and 5 is a simple and par-ticul-ar case; and for having proved the universal truth implied in the 3-4-5 triangl� he made his name immortal - -written indelibly acros s the age s .

In speaking of him and hi s philosoph� , the Journal of the Royal Society of Canada , Section II ,

0 Vo_l . 10, 1904 , 'p . 239, says: "He was the Newton, the Galileo , perhaps tl)e Edi son and Marco�i 'or his Epoch . . . . . ' Schola� now go to Oxford, then to Egypt, for fundamentals of the past . . . . . . The philos ophy of P�thagoras is Asiatic - - the best of India- -in origin, in which lore he became proficient; but he committed none of his view� tq1writing'and forbifrhis f�llowers t o do·so , insi sting_�hat they listen and hold their tongue s . '" :t

' He w�s inde�d the Sarvonarola of hi s epoch; he excelled in philbsophy, mystic i sm, geoMet�y, a wr·i ter upon music , �nd· in the field of astrop.omy he

1 anticipated Copernicus by making the sun the center . ) - -

bf the" cosmos . · "Hi s mo st original m'athematical work however, was probably in the Greek Arithmetica, or theory of number s , hi s teachings being followed by all subse,quent Greek writers on the sub j ect . "

Whether hi s proof of the famous theorem was wholly original no one �nows; but we now know that geomet�rs of Hindustan knew thi s theorem centurie s b�fore hi s time; whether he knew what they knew i s al so unknown . But he , o f all the masters of antiqui� ty, carries the honor of it s place and importance in our Euclidian Geometry .· ·

On'account of its extensive application in \

the rie�d of trigonometry, �urveying , navigation and a s tronomy , it i s one of the mos t , if not the most , interesting propositions in elementary plane geometr�

It ha� b�en variously denominated as , th& Pythagorean ·Theqrem , The Hecatomb Propo s:tti�o..rjl , ' -.�h�· ; , · · 1 ,: Carpenter ' s Theorem, and the Pons Asinorum because of it� supposed difficulty. · But the term "Pons Asinorum"

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THE PYTHAGOREAN PROPOffiTION 13

also at tache s to Theorem V , properly, and to Theorem XX .erroneously. , of'' ·BooJ<: I of Euclid 1 s E'lements of Geometry .

It i s regarded as the most f�scinating Theo­rem of al� Euc.lid, so much s o , that thinkers_ from all c lasses and national�ties, from the aged philosopher in his a.rmchair to the young soldier in tpe trenches v'

next to no-man1s-land, 1917 , have whiled away hours seeking a new proof of i t s truth .

Camerer.,* in hi s note s on the Firs't Six Books of Eu.c lid 1 s Element s give s a c ollection of 17 differ­ent demgnstrations of thi s theorem, and from time to time others have·made collec tions , - -one of 28, qn­other of 33, Wipper of 46 , Vers luys of 96, t.he Ameri­·can Mathematical Monthly-has 100 , others of li st s ranging fr8m a few t o over 100 , all of which proofs , with credit, appears in thi s (now, 1940 ) c ollection 'of ·over 360 dLfferent proofs, reaching in time, from goo. B.C. , to 1940 A . D .

Some of the se 36 7 proofs,- - supposed t o be new- -are very o.ld ; some are short and simple ; others are long and c omplex ; but each i s a way of proving the same t·:r�uth .

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Re�d and take your choice; or better , find a new, a different proof,· for there are many more proofs pos sible.,_ whose fi-gure will be different from any one found herein .

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*Note.1 Perhaps J.G. 41, p. 41.

See Notes and Queries, 1879, Vol. v, No.

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Come and take choice of all my Library. -Titus Andronicur.

'Behold!" l'iam lnfJtniam aut Faciam •

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"Mathematics is queen of the sci-ences and arith•etic i�.qQeen of Mathe­matics. She often COJldes cen ds to render �ervice to astronomy and.other natural . sci�nces, but under all circumstances the first ptace is her due."

Gauss -(1777-:-1855)

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THE PYTHAGOREAN THEOREM '

From an Arlthmetico-Algebraic Point of View

Dr . J . W . L. G£asnier in his addres s before Sec tion A of the �ritish As soc iation for the Advance ­ment ·of Sc i'ence , 1890, said: "Many of the greates t masters of the Mathematical Scienc e s were first at­tracted to mathematical inquiry by problems' concern­ing numbers , and no one can glance at the periodicals of the pre sent day which contains questions for- solu­tion without' noticing how s ingular a charm such prob­lems c ontinue bo exert . �

One of these·charming problems was the deter­mination of "Triads of Arithmetical Integers " such that the sum.of the· square s of the two les s e� shall equal the square of the greater number .

The�e_triads , groups of three, represent the three side s of a right triangle , and are infinite in number .

Many ancient master mathematicifins $ought _

general formulas for finding such groups , among whom worthy of mention were Pythagoras ( c . 582-c . 501 B.C� P lato ( 429-348 B . C . ) , and Eucl id ( living 300 B . C . ) , because of their rul e s for finding such triads� �- -

In our public'librarie s may be found many publications c ontaining data relating to the sum of two square numbers whos e sum i s a square number among which the follow:t.ng two mathematical magazines are e specially worthy of notice , the first being "The Mathematical Magaz ine , " 1891 , Vol . I I , No . · 5 , in which, p . 69. , appears an article by that master Mathe­matical Analyst , Dr . Artemas Martin, · of Washington, D . C . ; the second being, "The American Mathematical Monthly, " 1894 , Vol . I , Ng_. ___ l, in which, p . 6, ap-pears an·articie by Leonard E . Dickson, B . Sc . , then Fellow in Pure Mathematic s , Univer sity of Texas .

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18 THE PYTHAGOREAN PROPOSITION

Those who are interes ted and de sire more data relative to such numbers than here culled therefrom, the same may be obtained from these two Journals .

From the artic le by· Dr. Martin . "Any number of square numbers whose s�m i s a square nwnb�r can be found by various rigorous methods of solution."

Case I . Let · i t be required to find two square numbers · whose sum i s a square number.

First Method.· Take the well -known identity (x + y ) 2 = x2 + 2xy + y2 = (x - y ) 2 + 4xy. - - - ( 1 )

Now if. we can transform 4xy into a square we shall have expre s sions for two square numbers whose sum is a square number. _

A s sume x = mp2 and y = mq2 , and we have 4xy = 4m2p2q2 , wh�ch i s a square number for all val­

----- ----�-tias_£_f m, p and q ; and ( 1 ) become s , by sub s titution, . (mp2 +m�� = (mp2 - mq2 r2 + ( 2mpq ) 2 , or striking

� . 2 ( 2 2 ) 2 out . the common squ�re-factcr m , we have p + q := ( p 2 -· q 2 )2 + ( 2pq ) 2 . _-.::::.-n�-,--- -...: .. ----Dr. Martin follows thi s by a �Gond and a third method , and discovers that both�d� ·third) m8thods reduce, oy simplification, tb. formul�----( 2 ) .

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Dr . Martin dec lare s , ( and support s hi s decla­ration by tlie .inve stigat-ion of Mat.thew Collins t "Tract on the Possible and Impos sible Cas e s of Quad­ratic Duplicate Equalitie s in the Diophantin� Aqaly­si s , " publi shed at Dublin in 1858 ) , that no ·expre s ­sion for two square numbers whose s� i s a square can be found which are not deducib le from thi s , or re ­ducible to this f�rmula, - - that ( 2pq ) 2 + (p2 - q� ) 2 · is always equal to (p2 + q2 ) 2 •

Hi s numerical illus trations are : Example 1 . Let p = 2 , and q = 1 ; then

p2 ·+ q2 = 5 , p2 - q2 · = 3 , 2pq = 4 , and we have 32 + 42 = 52.

Example 2 . Let p = 3 , q = 2 ; then p2 + q2 = 13 , p2 - q 2 = ··s ' 2:i;q = 12. :. 52 + 122 = 132 , etc. ' ad inftni tum. � ·

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THE PYTHAGOREAN THEOREM 19

From the article by Mr . Dickson: ' Let the three integers used to expre s s the three side s of a right triangle be prime t o each other , and be symbol­ized by a , b and h . ' Then the se fact s follow : 1 . �hey can not all be �g numbers , otherwise they

would stili be divi sible by the c ommon divi sor 2 . 2 . They can not all be odd numbers. For a2 + b2 = h2 .

And if a and b are odd, their square s are odd, &nd the sum of their s��re s i s even; i . e . , h2 i s even. But if h2 Ls even h must be even .

3 . h must always be odd; and., of the remaining two , one mus t be even and the other odd . So two of the three integers , a , b and h, must always be odd . (For proof, see p . 7 , Vol . I, of said Am . Math . Monthly . )

4 . When the side s of a right triangle are integers , the perimeter of the triangle i s always an even number, and i t s area · i s al so an even number .

and h . Rules for finding integral value s for a, b

1 . Rule of P:[thagoras : Le t n be odd; then n, n2 - 1 2 n2 + 1 and 2 are three such numbers . For . ( 2 1) 2 4n2 + n4 - 2n2 + 1 = ( n2

2+ 1) ? n2 + .n . -

4 2

2 . even divi sible m2 m2 �y 4 ; then m, lf - l , and Jf + 1 are three such

numbers . For

= m4 m2 6 + - + 1 1 . 2 .

( m2 ) 2 m2 + 4 - 1 ' ( m2 ) 2 = T + 1 .

m2 - + 1 2

3 . Euclid ' s Rule : Let x and y be any two even or odd numbers, such that x and y contain no common fac ­tor greater than 2 , · and xy i s a square . Then � X - y X + y

2 and 2 are three such number s . For

20 THE PYTHAGOREAN PROPOSITION

(x - y )2 x2 - 2xy + y2 - ( x + Y )2 (t/XY) 2 + 2 = xy + 4 - 2 .

4 . Rule of Mas ere s ('1721-1824 ) : Let m and n be any m2 + n2 two even or odd, m > n, and -

2n an integeit . - - 2 2 - 2 +-· 2 2 m - n m n - -Then m , 2 n and

-· 2n are three such ntunbers. m2 - n2 4m2n2 + m4 - 2m2 + n� + n4 For m2 + = ���--�--��--��--�

2n 4n2 = ( m2

2: n2 )2 .

5 . D�ckson's �ule : Let m and n be any two prime in­teger s , one even and the other. odd, � > n and· 2mn a square . !Ilhen m + v'2mn, n + V2nin and m + n + v'2riiii are three such number s : For (m + v'2riiD.) 2-+ (11 + v'2riiii ) 2 + m2 + n2 + 4mn + 2m V2nin + 2n v'2ffii1 = ( m + n + V 2mn ) 2 .

- 6 . By inspection i t i s evident that the se five rul e s , ----�-- - the formulas of Pythagoras , P lato , Euclid, �-� Masere s and Dickson, - -each r�duce s to _ the�ormula

• 1 of Dr . Martin . . --- ··-

-------------� In the Rule of Py..trr-ragoras: multiply by 4 and

- squar.e -and there _ _ . .re sulls (2.n ) 2 + (n2 - 1 ) 2 = (.n2+ 1 ) 2, __ ...--- -in �cn-p�n and q = 1 .

__ ___-- In .the Rule of: P lato: multiply by 4 and L----------- square and the;re results' ( 2m) 2 + (m2 - 22 ) 2

= _('m2 + 22 ) 2 , · in which p = m and q = 2 .

'· ,,

i'

I. - - . - .. . --

In the Rule of Euclid: multiply �y 2 and square there re sult s ( 2xy) 2 + (� . - y) 2 = (x + y ) 2 , in which p = x and q = y . . In . the Rule of Masere s : multiply by 2n ·and ·square and re sul t s are ( 2mn) 2 + (m2 - n2 ) 2 = (�2 + n2 ) 2 , in which p = m and q = n .

, In Rule o f Dickson : equating and sblving

P =1m + n + 2 � + Vm - n and

I

_j I /

I , I

, t

q = Jm

THE PYTHAGOREAN THEOREM

+ n + 2 v'2nii1 - lm - n 2

21

Or if de sired, the formulas of /�artin, Pythq­gora s , Plato , Euclid and Ma seres ma"y;- -"be reduced to that of Dickson .

The advantage of Dic�un ' s Rule i s thi s : It . give s every pos sible set o£/valu�s for a, b and h in their lowe st terms , and/gives this set but once . ·

/

; To appiy h).�" rul-e , proceed as foll·ows : Let m be any odd s�ware what soever , and n be the double ,of any squap��umber what soever not divi sible by m .

/-///Example s.. If m = 9 , . n may be the double of / � 4 , 16, 25 , 49 , etc . ; thus when m = 9 , and n = 2, then . . m + ./2mn = 1§-, -n + -V2i!ID = 8, m + n + v'2nii1 = 17 . So a = 8 , b = 15 and · h = 17 . .

If m � = l , and n � 2 , we get a = 3 ; b = 4 ,

If m = 25, and n = 8 , · we get a = 25 , b = 45, h = 53 , etc . , etc .

Tables of integers for values of a ; b a .. 1d h have been calculated .

Hal sted ' s Table { in hi s "Mensuration" ) � s ab ­solutely complete as far as the 59th set of value s .

I

l

I �--

" M E T H O D S O F P R O O F

He t h o d t s t h e fo l i o w t n & o f o n e t h t n g llE:.QJf'-R- ano t h e r . Qc.�g,c. t s t h e fo l l o w t n g o f o n e t h t n g g[!g,c. a n o t h e r .

The tyPe and �orm of a figure nec e s sarily de ­termine . the pos s ible argument of a derived proof ; hence , as an aid for reference , an order of arrange­ment of the proofs i s of great importance .

_ In thi s exposition of some. proofs of the Pythagorean theorem the aim has been t o olassify and arrange them as �o method of proof and type of fig­ure u�ed ; t o . give the name , in case i� has one , by w�.fch the demons tration is known; to give the name arid page of the � journal , magazine or text wherein the P.±'oof may be found, if known; and occasionally t o give other intere sting data relative t o certain proofs . .

·

Th� order of arrangement _ herein i s , only in part , my own, .being formu�ated after a �tudy of the order found in the several groups of proofs examined, but more e specially of the order of arrangement g�ven in The American Mathematical Monthly, Vol s . III and

\ '

IV , 1896-1899 . ·

It i s as sumed that the per s on using this work will know the fundam�ntal s of plane ge ometry, and that , having the figure before him, ·he will readily supply the "reasons why" for the steps t�ken a s , often from the figure , the proo·f i s obvious ; there­fore only such statements of construction and demon­s•tration are set forth in the text as i s nece8 sary t o e stabli'sh the agrument of the particular· proqf .

'

/

' 22 / '

ALGEBRAIC PROOFS

Th e M e t h·o d s ·O f P r o o f A r e :

I. . ALGEBRAIC PROOFS THROUGH LINEAR RELATIONS

A . S i m i l a r R t g h t ·Tr i an g l e s

23

From linear relations of similar right trian­gle s it may be proven that , Th e s qu a re o f t h e hyp o t e ­n u s e o f a r t g h t t r i a n g l e i s e qu a l to t h e s u m o f t h e

. s qu a r e s o f t h e o t h e r t wo s i d e s � !;::::,.

And since the algebraic square i s · the mea sure of the geometric square , the truth of the proposLtion

--as just stated invol_va s th.e -truth of the propo sition a s s tated under Geometric Proofs through compari s on . of areas . Some algebraic proofs are the following :

Fig . 1

In rt . tri . ABH, draw RC perp . t o AB . The tri ' s ABH, ACH and HCB are similar . For ·convenience , · denote BH, HA, AB , HC , CB· . and AC by a , b , h, x, y and h-y resp ' y . Since , . from · three similar and related tri- c-··

angle s ,, there are po s sible nine sim-' pie propo�tions , the se proport ions

and their re sulting equations are : ( 1 ) a x = b h - y :� ah - ay· = bx . ( 2) a y = b x :. ax = by. ( 3 ) X Y = h y X :. X = hy - y . 2 - 2 (4 ) a ,x = h b :. ab = ·hx . ( 5 ) a y = h a :. a 2 = hy . ( 6 ) x � y = · b a � ax = by. ( 7 ) b & y = h : � .� b2 = h2 - hy . ( 8 ) b x = h : a :. ab = hx . ( 9 ) h - y : x = b : a � ah - at = bx . See Versluys ,

p . 86 , fig . 97 , Wm . W· . Rupert . . Since equations ( 1 ) and ( 9 ) are identical ,

al s o (2) and ( q ) , and ( 4 ) and ( 8 ) , there remain but s ix different equations , and th� problem be�ome s ,

\

24 THE PYTHAGOREAN PROPOSITION

how may these six equations be c ombined so �s to give the de sired relation h2 = a2 + b2 , which - geometrical­ly interprested is AB2 = BH2 + HA2 � ·' In t;his proof On e , and i.n every case here-after , a s in proof S i x t e e n , p . 41, the symbol AB2 , or

-2 a like symbol , signifie s AB . Every rational solution of h2 = a2 + b2 af­

fords a Pyth�gorean triangle : See "Mathematical Mon­ograph , No . 16 , Diophe tine Analys i s , " ( 1915 ) , by R . D . Carmichael .

l s t . - - L e g e n d r e ' s So l u t i o n a . From no single equation of the above nine

can the de sired relation be determined� an� there i s but 'one combination o f two equations which will give it ; viz . , ( 5 ) a2 = hy; ( 7 ) b2 = h2 - hy ; adding the se give s h2 = a2 + b2 •

TP,i s i s the shortest proof poss'ible of the Pythagorean Proposition .

b . Since equations ( 5 ·) and ( 7 ) are implied in the principle that homologous side s of similar tri­angle s are proportronal it follows that the truth· of thi s important proposition i s but a c_orollary to the more gen�ral truth--the law of similaritx .

c . See Davis �egendre , 1858, p . 112 ,

Journal of Education, '1 888 , V . XXV, p . 404, fig . v . .:

Heath ' s MatQ . Monograph, 1900 , No . 1 , p . 19 , proof III , or any late text on geometry .

d . W . W . Rouse Ball , o f Trinity College , Cam­bridge , England seems to think Pythagoras knew of this proof .

2 n d . - - O t h e r So l u t i o n s a . By the law of c ombinations there are po s ­

s ible 2 0 set s o f three equations out o f the s ix dif­ferent equatioris . Re jec t ing all sets containing ( 5) and· ( 7 ) , at;:d. all set s containing dependerit equat±ons , there are remaining 1) set s from which the elimina-

. t ion of x and y may be acc ompli shed in ·44 different

I

\ \

' \

, ,

j

ALGEBRAIC PROOFS 25

ways , each giving a di stinct proof for the relation h2 = a2 + b2 .

'

b . See the American Math . Monthly, 1896 ,

V . III , p . 66 or Edward ' s Geometry, p . 157 , fig . 15 .

' Fig . 2

( 1 ) 1J h = a ( 2 ) b :: a = a ( 3 ) h a = x ( 4 ) b h = h (5 ) b a = h

Produce AH t o C so that CB will be perpendic ular to AB at B .

Denote BH , - HA, AB , BC and CH by a , b , h , x and y re sp ' y .

j

The triangle s ABH, CAB and BCH are s-imilar .

From the continued propor­tion b : h : a = a : x : y = h : b + y : x, nine different s.tmple pro­porti6ns are possible , viz . :

x . ( 7 ) a x = h b + y . .y . ' ( 8 ) a : y = h : x .-y . ( 9 ) x : b + y = y : x , from b + y . which six different X .

(6 ) h a == b + y : x . equations are pos sible as in O n e above .

! s t . - - So l u t i o n s From Se t s o f Two E q u a t i o n s a . As in On e , there is but one set of two

equations , which wi ll give the relation h2 = a2 + b 2 . b . See Am. Math . Mo . , V . I I I , p . 66 .

2n d. -- So l u t i o n From S e t s o f Th r e e Equ a t i o n s a . As in 2nd under proof O n e , fig . 1 , there

are 13 set s �f three eq ' s , giving 44 di stinct proofs that give h2 = a2 + b2 •

b . See Am. Math . Mo . , V . II I , p . 66 .

c . Th�refore from three similar , rt . atri ' s so related· that any two have one s-ide in common there are 90 ways of proving that hP = s2 + b2 �

I ,

i I 1

"

\

\ \

, ,

26 · - .� THE PYTHAGOREAN PROPOSITION

Fia. 3

Take BD = BH and at D draw CD perp . to AB forming tha two simi­lar tri ' s ABH and CAD .

a . From- the cont inued propor­tion a : x = b : h ' = h : b - x the simpl� proportions and their re sult­ing eq ' s are :

( 1 ) a x = b h - a :. ah - a 2 = bx . ( 2 ) a x =

- h b - .x :. ab - ax = hx . ( 3 ) b h - a = h : b - x :. b2 - bx = h2 - ah .

.. As there are but three equations and as each eq uation contains the unknown x in the l st degree , there are pos s ible but three solutions giving h2 = a.2 + b2 . ,)

--- ·

b . See Am . Math . Mo . , V . III , p . 66 , and Math . "Mo . , 185�, V . II , No . 2 , Dem . Fig . 3 i on p . 45 I

by Richardson . '

I 'l r

f.Q!H.

In Fig . 4 extend AB to C making BC = BH, and draw CD perp . to AC . Produce AH t o D , forming the �wo similar tri ' s ABH and ADO .

, From the continued pro-��--+--......_ - � - 1 C portion b : h + a = a : x

= h : b + x three eq�ations are Fig . 4 po s sible giving, as in fig . 3,

three proofs . , a . See Am . Math . Mo . , V . III , p . 67 .

H

AA ' J) Fig . 5

Draw AC the bi s�ctor of the angle HAB , and CD perp . to AB , · form­ing the similar tri ' s ABH and BCD . Then CB = a - x and DB = h - b .

/

. .

I '\

/

J I

' /

. I

I I

•

I

_ALGEBRAIC PROOFS 2T

��om the c ontinued proportion h : a - x = a : h - b = b ' : x three equations are pos sible :;i v ­ing , as in . fig . 3 , three proofs for h2 - = a2 + b 2 •

a-.:: ..Prigirial with the author , Feb . . 23 , i926 .

Throush D , any pt . in either leg of the rt . triangle 1 ABH , draw DC perp . to AB an,d extend it t o E a pt . in the other leG produced, thus fo�ming the four similar rt . tri ' s

"'b ABH, BEC , ACD and EHD . · From .A ...,.,. __ .,..IJJ the c ontinued proportion (AB = h )

: . ('BE = a: + x ) : (ED = v ) Fig . 6 : (DA = b - y) = (BH = a )

(BC = h - z ) � : (DH ·= y ) : (DC = w) = (: HA = b ) : ( CE = v + w) : (HE = x) : ( CA = z ) ,

. eighteen simple proportions and eighteen different equations are po s sible .

· From · no single equation nor from any se t of two eq ' s cart the relat ion h2 � a2 + b2 be found but from combination_ of eq ' s involving three , four or five of the . unknown elemen1t s u, w, x ,. y, z , solutions may be obtained .

1 s t . ;;.. .;.·pro o fs From Se t �. I n uo z u·t n g Th r·e·e ·uirk n o wn E l e ­/ m e·n t s

a . �t has been shown that there i s pos sible but one combinat:l,on of equations invol:ving but three - of the unknown e],ement s , viz . , x, y and z which will g:tve· h2 = a2 4- b2

• .

� b . See Am . Math . Mo . , V . III , p . 1 11 .

2n d . - - P ro o fs F rom Se t s I n vq l u t n g Fo u r U n k n o wn E l e­m e n t s

a . There are po s sible 114 combinations in­volving but four of the unknown element s each of which will give h2 = a2 + b2 •

b . See Am . Math . Mo . , V . III , p . 111 . i

I I I

I ! I

I

I

I I

/

f

� , , , , .

I . I I I

28 THE PYTHAGOREAN PROPOSITION

3rd. -� P ro o fs Fro • S�t s l n uo l v t ri j A l l F t u e U n k n o wn E l e • e n t s

a . �imilarly, there are 4749 combinations in­volving all five of the unknowns , from each of which h2 = a2 + b2 can be obtained .

b . See Am: Math . Mo . , V . I I I , p . 1 J,. 2 . c . Therefore the total no . of proofs from .

the rel�tions involved in fig� 6 i s 4864 .

A..._ __ .......liH - - .. �E \

. Fig . 7 7 i s 9728 .

\ \

' 1 '

' �])

�roduce AB to E , fig . 7, and through E draw, perp . to AE , the line OED meet'ing AH pro­

. duced in C. and HB produced in D , forming the four similar rt . tri ' s ABH , DBE , CAE and CDH .

a . As in fig . 6 , eigh­�een different .equgtions are pos sible from which ther� are aiso 4864 proofs .

b . Therefore the total 2 no . of ways of proving that h ,

· z a2 + 'b2 from 4 similar rt . tri ' s related as in fig ' s 6 apd

c . As the pt . E approache s the pt . � , fig . 7 approached fig . 2, above , and become s fig . 2 , whan· E fail s on B . .

d . Suppo se E fall s on AB so that CE c ut � HB between H and B;-then we will have 4 similar rt . tri 's involving 6 unknowns . How many proofs will re sul t ?

In fig . 8 produce BH to D , making BD =' BA , and E , the middle pt . of AD , draw EC · parallel to AH , · and join BE , forming the 7 stmilar rt . triangle s AHD , ECD , BED, BEA , BCE , BHF' and AEF , but six of which

/

r !

•

I

Fig . 8

ALGEBRAIC PROOFS 29

need c onsideration, since tri t s BED and BEA are congruent and, in sym­bolization , identical .

See Ver sluys , p . 87 , fig . 98, · Hoffmann, l818 .

From these 6 different rt . triangle s , set s of 2 tri ' s may be s�lected in 15 different ways ,. s e t s of 3 tri ' s may b e selec ted in 2 0 different ways , sets of 4 tri ' s may

be selected 1-n 15 different ways , . s ets of 5 tri ' s may be selected in 6 different ways� and set s of 6 ' tri 1 s may be selected in 1 way, giiing , in all , 57 differ­ent ways in which the 6 triangle s may be., c ombined .

But ? S all the proofs derivable from the sets of 2 , 3, 4 , or 5 tri ' s are al so Tound among the proofs from the set of 6 triangle s , an inve stigation of thi s set will suffice for all .

In the 6 simi lar rt . tri ' s , let AB = h, BH = a , HA = b , DE = E� = x , BE = y, FH = z and BF = v , whence EC - � ' DH = h - a , DC = h - a , EF = y - v ,

� 2 BE = � ; a, AD = 2x and AF = b - z , and from the se ·dat� tqe cont inued proportion i s

b . b/2 . y . (h + a )/2 . a . X . . . . . = h - a . (h a ) /2 : . X b/2 z . y - v . .

= 2x . X . h . y . v . b z . . . . . . -From thi s continued proportion there re sult

45 simple proportions which give 28 different ·equa­. tions , · and , a·s groundwork fr,r determining the number

of proofs pos sible , they are here tabulated . . ( 1 ) b b/2 = h - a. . (h - a ) /2 , where 1 = 1 . Eq . 1 . . ( 2 ) · b b/2 = 2x . x , whenc e 1 = 1 . Eq . 1 . . (3 ) h - a . (h - a ) /2 == 2x . x , whence 1 = l : Eq . 13. . . {4 ) b y = h - a. . x, whence bx = (h - a ) y . Eq . 2 . . ( 5 ) b y = 2x . h, whence 2xy = bh . Eq·. 3 . . (6 ) h - a X = 2x . h ,, whence 2x2 = h2 - ah . Eq . 4 . .

I I) i

oo THE PY1HAGOREAN. P-ROPOSITION

1 '

( 7 ) b . . Bq .

( a'· + h ) /2 = h - a : b/2 , whence b2 = h2 - a2. 5 . (h + a ) /2 = 2x : y, whence (a + a ) x := by . · 6 .

( 8 ) b . . Eq .

a : b/2 = 2x : y, whence bx = (h - a ) y . Eq . 2 . ( 9 ) h -

( 10 ) b a = h - a : z , whence bz = (h - a )� . Eq . 7 . ( 11 ) b a = 2x : v , whence ?ax = bv . Eq . 8 . ( 12 ) h - a : z = 2x : v , when·ce 2xz = (h - a ) v . Eq . 9. ( 13 ) b X ::: h - a : Y - V , .. �hence (h - a )x = b ( y _: v) . Eq . 10 . ( 14 ) b : ·x = 2x : b - z , whence· 2x2 = b2 - bz . Eq . 1 1 . ( 15 ) h - a : y · - v = 2x : · b - z , whence 2 ( y - v ) z = (h - a ) (b - z ) . Eq � 12 . ( 16 ) b/2 : y = (h - a ) /2 : x , whence bx = (h - a ) y . Eq . 2 . . . . . ..

( 17 ) b/2 : y = x : h , whence 2xy = bh . Eq . 3 . ( 18 ) · (h - a ) /2 : x ·= x : h, whence 2x2 = h2 - ah . Eq . 42 • . . ( 19 ) h/2 : (h + a ) /2 = (h - a ) /2 : b/2 , whence b2 = h2 - a2 . . Eq . 52 . ( 2 0 ) b/2 : (h + a ) /2 = x y, whenc� (h · + � ) x = by . Eq . 6 . ( 21 ) (h - a ) /2 : b/2 = x y, whence bx = (h - a ) y . "4 Eq . 2 .

- ( 22 ) b/2 : a = (h - a ) /2 z , whence bz =. (hv - a ) a . Eq,. 72 • ( 23 ) b/2 : � = x �' whence 2ax = bv . Eq . 82 . ( 24 ) (h - a ) /2 z = x : v , whence 2xz = (h a ) v . Eq . 92 •

--�25 ) b/2_ : x . = (h - a') /22

: Y . - v , whence (h a ) x = �( y - v1 . Eq . 10 �

( 26 ) b/2 : x = x : b - z , whence 2x2 = b2 - bz . Eq . 112 • ( 27 ) (h - 'a ) /2 ·: y - v = x : b - . z , whence 2 ( y - v ) x = '{h - a ) ( b - z ) • Eq . 12.2 • . , ( 2 8 ) y :· ( h + a ) I 2 = X : b I 2 ' whence ( h + a ) X = by. 63

. ' Eq . · .

. . ( 29 ) y : (h + a ) 2 = h : y, whence 2y2 = h2 + ah . Eq . 13 . '\. \ \

/

I /

' I

/

(30 ) X b/2 = h , .)l ) y a = X ( 3 2 ) y a = h ( 33 ) X z = h ( 34 ) y .x = ' x (35 ) y X = h ( 36 ) X y - v =

= h ( y - v ) . (37.) (h + a ) /2

Eq . 20 . (38 ) (h + a ) /2

Eq . . 21 .

ALGEBRAIC PROOFS

3 : ¥, whence 2xy = bh . Eq . 3 . z , whence ax = yz . Eq . 14 . v , whence vy = ah . Eq . 15 . v , whenc e vx = hz . Eq . 16 .

31

y - v, wh�nce x2 = y ( y - v ) . Eq . 17 . b - z , whence hx = y (.b - z ) . Eq . 18 . h : b - z , whence . (b - z ) x

Eq . . 19 . a = b/2 : z , whence (h + e ) z = ab .

x =· y : v , when( e 2ay = ' (h + a ) v .

(39 ) b/2 : z = y : v , whence 2yz = bv . Eq . 22 . ( 40 ) (h + a ) /2 : x = b/2 y - v , whence bx = (h + a )

(-�/ - v ) . Eq . 23 . ( 41 ) (h + a ) /2 : - x = y · : b - z , whence 2xy =· (h + a )

. (b - z ) • Eq . 24 . ( 42 ) b/2 : y - v = y : b - z , whenc e 2y ( y - v ) = b

2 . " = o - ·bz . ( 43 ) a ( 44 .) a

X = Z X = v :

( 45 ) z y - v = = (b - z ) z .

Eq . 25 . y v , whence xz = a ( y - v ) . o - z , whence vx = a (b - z ) . v : b - z ; whence v ( y - v ) E q . 28 . .

Eq . 26 . Eq . 2'1 .

The symbol 241, see ( 21 ) , means ·. that 0equation 2 may be derive!]. from, 4 different proportions ." Simi­larly for 63 , etc .

Si'nce a; definite nb. of s e t s of dependent equa­tions , three equati ons iri each set , is derivable from a given c ontinued proportion and since the se sets mus t be known and dealt with in e stabli shing the no·. of pos sible proofs for h2 = a2 + b2 , �t become s : ne·c ­e s sary .. to determine the no . . of such set s -. In any .,. continued proportion the �ymboli zation 'for the no . of ,. n2 (n + 1 ) s�ch s et s , three equations in eaph set , i s . 2

' . in whi-ch n signifies the no • . of s1mpl� ratios in a member of the continued prop ' n . Hence for the above c ontinued proportion there are deriv�ble 75 such s e t s o f dependent equations . They are :

..

'

I

32 THE PYTHAGOREAN PROP.OSI�'ION ' - .

( 1 ) , ( 2 ) , ( 3 ) ; ( 4 ) , ( 5 ) , ( 6 ) ; ( 7 ) , ( 8 ) , ( 9 ) ; ( 10 ) ,

( 11 ) , ( 12 ) ; ( 13 ) , ( 14 ) , ( 15 ) ; ( 1 6 ) , ( 17 ) , ( 18 ) ; ( 19 ) ,

( 20 ) , � ( 21 ) ; ( 22 ) , ( 23 ) , ( 24 ) ; ( 25 ) , ( 2.6 ) , ( 27 ) ; ( 28 ) ,.

( 29 ) , '( 30 ) ; ( 31 ) , ( 32 ) , ( 33 ) ; ( 34 ) , ( 35 ) , ( 36 ) ; ( 37 ) ,

( 38 ) ' ( 39 ) ; ( 40 ) ' ( 41 ) ·, ( 42 ) ; ( 4 3 ) ' ( 44 ) ' ( 45 ) ; (1 ) ' ( l� ) ' ( 1 6 ) ; ( 1 ) ' ( 7 ) ' ( 19 ) ; ( 1 ), ( 1 0 } , ( 2 2 ) ; ( 1 ) ' ( 13 ) '

( 25 ) ; ( 4 ) , ( 7 ) , ( 28 ) ; ( 4 ) , ( 10 ) , ' ( 31 ) ; ( 4 }, ( 13 ) ,

( 34 ) ; ( 7 ) , ( 10 ) , ( 37 ) ; ( 7 ) , ( 13 ) , ( �0 ) ; ( 1 0 ) , ( 13 ) ,

� ( 43 ) ; ( 16.) , ( 19 ) , . ( 20 ) ; ( 16 ) , ( 22 ) , ( 31 ) ; ( 1 6 ) , ( 25 ) ,

( 34 ) ;' ( 19 ) , ' ( 22 ) , ( 37 ) ; ( 19 ) , ( 25 ) , ( 40 ) ; . · ( 22 ) , ( 25 ) ,

( 4 3 ) ; ( 28 ) ' (.31 ) ' ( 37 ) ; ( 28 ) ' ( 34 ) ' ( 40 ) ; . ( 31 ) ' ( 34 ) '

( lf3 ) ; ( 37 ) ' . , ( 4.0 ) ' ( 4 3 ) ; ( 2) ' ( 5 ) ' ( 1,7 ) ; ( 2 ) ' ( 8 ) ' ' .

( 20 ) ; ( 2 ) , ( 11 ) , ( 23 ) ; ( 2 ) , ( 14 ) , ( 26 ) ; , ( 5 ) , ( 8 ), ( 29 );

( 5 ) , ( 1 1 ) , ( 32 ) ; ( 5 ) , ( 14 ) , ( 35 ) ; ( 8 ) , ( 1 1 ) , ( 38 ); ( 8 ),

( 14 ) , ( 41 ) ; '( 1 1 ) , ( 14 ) , - ( 44 ) ; ( 17 ) , ( 20 ) , ( 29 ) ; (17 ) ,

. ( 23 ) , ( 32 ) ; ( 17 ) , ( 26 ) , ( 35 ) ; ( 20 ) , ( 23 ) , ( 38 ) ; ( 20 ) ,

( 26 ) , ( 41 )-;� ( 23 ) , ( 26 ) , ( 44 ) ; (29 ) , ( 32 ) , ( 38 ) ; ( 29 ) ,

( 35 ) , ( 41 ) ; ( 32 ) , ( 35 ) , ( 44 ) ; ( 38.) , ! 41 ) , ( 44 ) ; ( 3 ) ,

.( 6 ) , ( 18 ) ; (3 ) , ( 9 ) , ( 21 )-;· - {3 ) ; ( 12 ) , ( 2_4 ) ; ( 3 ) , ( 15 ) ,

(27 ) ; ( 6.)_. ( 9 ) , ( 30 ) ; ( 6 ) , ( 12 ) , ( 33.} ; ( 6 ) , ( 15 ) ,

·( 36 ) ; ( 9 ) , ( 12 ) , , ( 36 ) ; ( 9 ) , ( 15 ) , ( 42 ) ; ('1 2 ) , ( 15 ) ,

( 4 5 ) ;: ( 18· L ( 21 ) �� ( 3 o ) ; \ 18 ) , ( 2 4 ) , ( 3 3 ) ; C 18 ) , ( 87 ) ,

( 36 ) ; ( 2i ) , ( 24 ) , ( 39 ) / ( 21 ) :, C27 -) , ( 42 ) ; ( 24 ) , ( 27 ) , ( 45 ) ; ( 30 ) ' (.33 ) ' ( 39 ) ; ( 30 ). , ( 36 ) ' ( 42 ) ; ( 33 ) ' ( 36 ) '

( 45 ) ; ( 39 ) ,; ( 42 ) ' ( 45 ) . Thes e 75 s et s expre s s ed - in the strnboltzation

. of the 28 equations give but 49 s et s as follows : 1 , 1 , 1 ; 2 , 3 , 4 ; 2 , 5 , 6 ; 7 , 8 , 9 ; 10 , 1 1 , 12 ; 6,

13 , 3; 14, 15 , 16; 17 , 18 , 19; 20, •2 1 , 22 ; 23, 24,

25 ; 26, 27 , 28 ; 1 , 2 , 2 ; 1 , 5, 5 ; 1 , 7 , 7; l , '1 0 ; 1 0 ;

1 , 6 , 6 ; 2 , 7 , 14 ; 2 , 1 0 , 17 ; 5 , 7 , ao ; 5 , 10 , . 2? ; 7 ,

1 0 , 26; 6 , 1 4 , 20 ; 6 , 17 , 23 ; 11� ; 17 , 26; 20, 23, 2'6 ;

' 1 ' 3' 3 ; 1 ' 8 ; 8 ; 1 ' 1 1 ' 11 ; 3 ' 8 ' 15 ; 3' 11 ·, 18 ; 6'

8, 2 1 ; 6 , 11 , 24 ; 8 , 1 1 , 27 ; 13 � 15 , 21 ; _ 1}, 18, 24 ;

15 , 18, 27 ; 21 , 24, 27 ; 1 , 4 , 4 ; 1 , 9, 9 ; 1 , 12 , 12 ;

4 , 9, 16; 4 � 12 , �9 ; 2 , � 9) 22} 2 i · 12 , 25; 9� - 12 , 28 ;

3, 1 6, ,22 ; 3J' 19 ,· 25 ; 16, · ·19 ,-: 28 ; 2'2, .25 , 28·. I . '• Sinc e eq . � is an identity and eq . 5 gives ,

at onc e , h2 = a2 + b 2, there are remaining 26 equa­tlons involving the 4 unknowns x , y , z and v , and

( . I . I

(

. .

ALGEBRAIC PROOFS

proofo may be po ssible rrom s e t s of equations involv­ing x and y, x and z , x and v, y and z , y and v, z and v , x , y and z , x , Y tand v , x , z and v , y, z and v , . i and x , y, z and v .

l s t . - - P ro o fs Fro m· Se t s I n u o l u t n � Two Unk n o w n s 'k

> a . The two unknuwns , x and y, oc c ur in the follow·ing five equations , viz . .. , 2 , 3 , 4 , 6 and 13 , from which but one set of two , viz . , 2 and 6 , will give h� + a2 = b2 , and a s �q . 2 may be derived from

· 4 dif:ferent proportions and equation 6 .from 3 differ­ent proport ions , the no . of proofs from thi s set are 12 . .

Arranged in sets of three we get , 24 , 33 , 13 giving 12 other proofs ; ( 2 , 3 A 4 ) a depe�dent s et - -no prriof ; 24 , 42 , 13· giving 8 other proofs ; (3 , 6� 13 ). a dependent set :-no �roof ; 33 , 42 , 63 giving 18 other pro0fs ; 42 , 63 , 13 giv�ng 6 other proofs ; .

33 , 42 , 13 giv±ng 6 other proofs . ' i

' ' '

·Therefore there are 62 proofs· from s e t s in-vo�ving x . and y . .

b . Sim.1.,.larty, from ,.set s i·rivolving x and z there, are B. proofs , the - equations for which are 4 , 7 , 1 1 ; and 20 .

I ' ; • " J

C • s·et S involVing X a.nd V giVe no additional proofs .

d . �e t s involving y and z give 2 proofs , but the equatiops w.ere used in a and b , hence · cannot be c ounted again, th�y ar� 7 , 13- and 20 .

· e . $et s involving y and v give no proofs . ! I f . S'et·s--t-nv·o-Tv-ing z and v give same re sult�

a s d . Therefore the ng. o f proofs from set s involv­

ing two unknowns i s 70, makiryg, iri all 72 proofs s·o far, since P-2 '=_. a2 + b2 i s obtai;q�q. direc t ly frqm two different 1prop "s. ·;

••

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·'

/,

34 ·THE PYTHAGOREAN PROPOSI_TION

2nd.--Proofs From Sets Involving Three Unknowns a. The three unknowns X-, y and z occur in

the following 11 equations, viz., 2, 3, 4, 6, 7, 11, 13, 14, 18, 20 and 24� and from these 11 equations

sets of four can be selected in 1� · .LK · 9 • 8 = 330

ways,, each of which will give one or more proofs for h2 = a2 + b2• But as the 330 �ets, of. four equa­tions each, include certain sub-sets heretofore used,

• • certain dependent set§ of three equations each found among those in '\he above 75--sets, �nd certa�n sets of four dependent equations, all these must be qeter­mined and rejected;. the proofs from the remaining sets· will be proofl·s additional to the 72 a+ready de­termined.

Now, of 11 consecutive things arranged in

4 10 . 9 • 8 sjts of �ach, any o�e will .occur_

�n -[i

120 of the 330 sets, �ny _two in·

_ 9[i8 :or 36

_ of

$30, and a�y three in �� or � of the 33'0 se�·s.·

or

the

There-

_ ________ ,.tore any sub-set of two equatJgns will be found in 36, and any of thre�- equat�ons �n 8, of the 330 sets.

_But some orie or more. of the 8 may l;>e some\ one or more of the 36 sets; hence a sub-set ·of two and � sub-set of three will not nece�sarily cause a rejec­tion of 36 + 8 = 44 of the 330 sets.

' /

• I /

I I I

The sub-sets �h±ch gave the 70 proofs are: --2, 6, f.or which 36 sets must be rejected; 7, 20, for which 35 sets must be rejected·, sinc·e 7, 20, is found in �ne of the .36 sets abovej 2, 3, 13, fo1� which 7 other sets must qe i>e jected,

since 2, 3, 13, is found· in one of the 36. sets above; 2, 4, 13, t:.or which 6 otl:le.:r sets must be rejected; 3., :4,- _ q,. for rhich 7 other sets must be rejec-ted; 4, _6, 13, for which 6 other sets must be .rejected; 3, 4, 13, for which 6 other sets m�st be rejected; 4, 7, 11, for which 7 other sets must be rejected;

�nd ' � . . { ' I

\

/

• i I I ' '._. ' ')o ., � ;>

/

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ALGEBRAIC PROOFS 35

4, 11, 20, for which 7 other sets must be rejected; for ·all of which 117 se.ts must be reJected.

Similarly the dependent sets of three, which are 2, 3, 4; 3, 6, 13; 2, 7, 14; 6, 14, 20; 3, 11, l�; 6� 11, 24; and 13, 18, 24; cause a rejection of. 6 + 6 + 6 + 6 + 8 + 7 + 8, or 47 more sets.

,/ · Alse the �epen�ent sets of four, and not al­ready reje·cted, which are, 2, 4, 11, 18; 3, 4, 7, i4; 3, 6, 18, 24; 3, 13, 14, 20; 3, 11; 13, 24; 6, 11, 13, �8; and 11, 14, 20, 24, cause a rejection of 7 more s�ts. The dependent sets of (ours 'are di�cove�ed

· as follows:· take any two dependent sets of threes having a common term as 2, 3, 4, and 3, 11, 18; drop the common term 3, �nd write the set 2, 4, 11, 1$; a little 'study will qi:sclose the 7 sets named, as well ·

�s other sets already rejected; e.g., 2, 4, 6, 13. J • '

Rejecting the 117 + �9 + 7 = 171 sets there remain 159 sets, each of' whigh 'will give one or more proo\s, determi-ned as foll,ows. Write .down the 330 sets, � ' thing ·easily done, strike out the 171 sets which must be rejected, and, taking_ the r�maining sets one �by one, det-ermine how many proofs each w.ill give; ·e.g.', take -the set 2, .3, 7, 11; 'W·rite·it thus 24; 33, 72, 112, the exponents denoting the different proportions 'f'rom which the r.espective equations may be derived; the product of the exponents, 4 x 3 x 2 x 2 = -48, is the number of proofs possible for that set. The set 63, 112, 181, 201 gives 6 proof�, the set 141, 181, 201, 2lf1 giVes but· 1 pz:aoof ;' etc. -

-

The 159 sets,:by investigation, give 1231 proofs.

b. The three unknowns x, y and v occur in the following twelve equations,,--2, 3, 4:, 6, 8, 10, ll, 13, 15, 17, 21 and 23, 'Which give 495 different sets of 4 equations each, many of which.must be reject€1-

·. for same reasons as in -a. · Having �stabl�shed a met�od - in a, ·we leave details t·o ��e C?ne .�ntere�t.ed. • ' I c. Similarly for proofs from the eight 'bqua-

tions containing x, z and v, and the seven eq's con­taining y, 'z and v.

I ' I

·'

36 THE PYTHAGOREAN PROPOSITION

3rd.--Proofs From Sets Involvtn� the Four Unknowns x, y, z and u. ·\

a. The four unknowns occur in 26 equations;

th 26 . 25 . 24 . 23 . 22 65780 t hence ere are . . [i .

. , = differen ·

sets of 5 equations each. Rejecting all set� con­taining sets �eretofore used and also all remaining sets of five dependent equations of whic.h 2, 3, 9, 19·, 28, is a type, the remaining sets will give us many addit�onal proof�, the determination of which involves ,a vast amount of time and labor if thetmeth­od given in the prece�ing pages is follow�d. If there

I be a shoPter .method, .I·am linable, as yet, to discover it; neither ani. I able to find anything by any other investigator.

,4th. --Spec t· a l Sol u t tons a. By an inspection of the 45 simple propor­

tions given above 1 it is· found that certain pr.opor­tions, are worthy of special considerati.on as they ·

' give equ�tions from which v�ry simple solutions fol-low.

From proportions (7 r·aiicf (19) h2 =: a2 + b2 __ .follows immediately. •-Also from the pairs (4.) and . (18), ·and (10) _ and (37.),_ so�utions are readily ob-, tained. .

\

Q. Hor"fmann's solutiqn . .F Joh. �os. 'Ign. Hoffmann made a cpllection of

'32 proofs, publishing the same in "Der Pythagora�sche Lehrsatz ," 2nd edition. Mainz, 1821, of which th� so­lutioh ·f.rom (7') is one. He selects the two triangles 1 (see fig. 8"), AHD and BCE, from which b : (h +· a)/2 ·. -·

� h - a : b/2 �o��ows, ·giving at once h2 = a�+ b2• .

'· . See Jury Wipper's 46_pr9ofs, 1880, p. 40, fig . . . 41. Also see: ·versl.uys·, 'p. 871 fig. 98, credited to

HofrinE.nn, 1818: Also see Math. Mo., Vol. II, No. II, p·. 45, as, given �n Notes ae.nd. Queries, Vol. 5, No. '43,, p. 41. > ' '

c.' Similarly frore the·two·triangles BCE and ECD b/2 : (h + a)/2 = (h -- a)/2 : b/2, h2 = a2 + b2•

I

I

. ' )

..

.. '

---

..

_ALGEBRAIQ PROOFS 37

Also from the three triangles AHD , BEA and BCE pro­portions (4) and (8) follow, and from the three tri­�ngles AHD , BHE and BCE proportions , (10) and, (37) �ive at once h2 = a2 + b2•

See Am. Math. Mo. , V. III, pp. 169-70.

Pro.duce AB to any pt.

� E D. From D _draw ·bE,. perp. to , ,, AH produced, anq. from E drop

H. , "fl • \ the perp. EC, thus forming � t, the 4 similar rt. tri's ABH,

A :c \J). AED, ECD and ACE. .

/'I��-..... -.... �- - � -, . From the homologous sides o.f. t_n,es� .. §J:.milar tri-

Fig. 9 angl�s -the. following con­tinued propo�tion results:

(AH = b ) : (AE = 'b + v ) : (EC = w ) : (AC ·= h + X ) = (BH = a ) : , (DE = y) : (CD = z ) : , (EO = w ) -

= (AB = h ) : (AD = h + X + z ) : (DE .,;, y) : (AE . = b + v ) • Note- -B and C do not coincide. · ' ·

. a. From this continued prop'n �8 simple pro­,pQrtions are possible, giving, as. i� fig. 6, several ''t:tio.usand proofs.

b. See Am. ·Math. Mo., V. III, p. 171.

In fi·g. 10 are three S1.m1- . .

ltJ £ l�r rt. tri 1 s. ;.'"ABH, EAC and DEF, ·J� :, . from w�_ich the continued 1propor-

, ". 1t\;)) tion

�. A �---� v:._C .

Fig. 10

(HA = b ) : . .(AC\ = 11 + v ) . (DF1 = DC = x ) ;

= (HB = a ) : ( CE =· . y).

: (FE = � ) = (AB . = h ) (AE = h + v + z ) : (DE = y - X ) •

' \

/ '

•,

' -

/ •

. . \' .

\

\

38 THE PYTHAGOREAN PROPOSITION

follows giving 9 simple proportions from which many more proofs for h2 = a2 + b2 may be obtained.

a. See Am. Math. Mo. , V. III, p. 171.

- Fig. 11

From D in HH, so that DH = DC, draw DC par. to HB and DE perp. to AB, forming the 4 similar rt. tri's ABH, ACD, ODE and DAE, from whi.ch the continued proportion

(BH = a) : ( CD = DH = v) : (E C = y) - : (DE = X ) = (HA = b ) : (D A = b . - v )

: (DE = X ) :: (AE = z ) = (AB = h ) : (A C = z + y ) : ( CD = v ) : (AD = b v )

. follows;' 18 _simple proportions ,are possible from which many more proofs. for h2 = a2 + b2 result.

By an inspection of the 18 r�oportions it is evident tha� they g!ve no simple equations from'which easy solutions follow, as· was found"in the investiga­tion of fig. 8, as ·in a under proof E.t'tht.

a. See Am. Math. Mo. , V. III, p. 17i.

The construction of fig. 12 gives five similar rt. triangles, which are: ABH, ARD, HBD, A C&:"-'S.+l-9-BCH, from which the continued

'! prop'n A� ...... �.. (BH =·· a) : (HD = x) : (BD = y)

Fig. 12 : ( CB = a 2 ) : ( CH � � ,) =.

(HA = b ) I X . X ' -- . ---

: · (D A = � - y) :. '(DH =7 x y : (BA = h )· : (HB = a Y ' = ·(m = � ') (AH = b ') : CHB = 'a ) :\ (AC =, b· + . 7)

(:Be = S: )'

,. �

/·

·.

ALGEBRAIC PROOFS 39

follows:, giving 30- simple pr•oportions from which O!llY 12 different �guations result. From these 12 equa­tions several proofs for h2 = a2 + b2 are obtainable.

a. In fig. 9, when C falls,on B it is obvious that the graph become that of fig. 12. Therefore, the sol uti on of fig. 12 ,, · is only a .•. .Particular case of fig. 9; also· note that several of the pr·oofs of �case 12 are identical with those of case l, proof One.

b. __ The above La_an ori�inal method of proof by the author of this work.

..

•.

Fig. 13

Ih.lr.itto.

Complete the paral. and draw H F perp. to, and EF;par. with AB

resp 1 ly·; fornung th� 6 similar tri 1 s, B ijA, HCA, BC H, AEB, DGB and DFE, from which 45 simple proportions are ob­tainable, resulting in several thou­sand more possible· proof for h� = a2 + b2, only one of �hich we mention. (1) From tri 1's DBH and B HA,

2 DB : (BH = a; ) = ·(BH = a ) : . (HA = b ) ; :. DB = ab

and ( 2 ) HD (AB = h ) = . (B H � a ) (HA = b ) ; · ah

:. HD = b (3) From tri1s DFE �nQ B HA,

or DF

(4) Tri.

DF : (EB - DB ) = (BH = a ) : (AB = h ) ,

b2 - ab

2 a : h;:.DF = a(-22 �

ha2) . 1 1 1 -

AB H = 2 par� HE = ·2 AB.x HC = 2 ab

:;r l (AB( A� + CF)]. ·= l�(� .+ DF )] 2 . ' 2 ,.. 2

. ? 1 [ (ah - b 2 - a 2 \\ · J = 4 h b +(a bh );

:. (5) �- ab = ,

I I

. I j

I .! I ; .

/

\ ., ' ' l

\

40 THE PYTHAGOREkN PROPOSITION

vhence (6) h2 = a2 + b2• a. This particular proof was produced by

Prof. D. A. Lehman, Prof. of Math. at Baldwin Uni­. versity, �erea, 0., pee. 1899.

' p. 228. b. Also see Am. Math. Mo., V. VII, No. 10,

/

f.QY.ti!!rl

Ta�e AC and AD = AH and draw: HC and RD.

Prqof .. Tri 1 s CAH and r � ]· � are isosceles. Angle CHD �---- • -\il.._ ........ ._ is a rt. ahgle, since A is

I Fig. 14

equidist�nt from C, D �nd H. Angle HDB ,;, angle CHD

+ angle DCH. = angle AHD + 2 angle CHA = angie CHB. :. tri' s HOB and CH13 are: similar, having an­

gle; DBH in common and angle DHB·�-= angle ACH.

; .

:. c:e<- : .:B� = B� : DB, or h +- b : a = a : h - b. Whence h2. = a2 + b2•

/ /

/. /

- a. See. Math. Teacher·, Dec., 1925. -Oredited to Alvin Knoer, a Milwaukee High School pupil; also Versluys, p. 85, fig··. 95; aiso Encyclopadie der Ele­mentar Mathematik, von H. Weber und J. Wellstein, Vol yi�, p. · 242, ·where � _ (19.05), it ts credited. to · ·

C .//Q. Sterkenburg. ·

In fig. 15 the canst's is obvious giving four similar right triangles ABH, AHE; HBE ·and HCD, from whic� the continuad proportion

A · (BH = a ) : (HE = x ) : (BE = y) : (CD = y I 2 ) = (HA = b ) ': (EA = h - y )

Fig . 15 : (EH = x ) : (DH = x/ 2 ; = (AB = h ) 1 1: . (¥ ;�· b ) : (HB i:: a ) : (HC = a/ 2 )

tollo�s, giyfng·-18 s�mple proportions,

I - ! -- - - .

\

'\

'>¥ \ \. -I .

(1) a (2 ) b

ALGEBRAIC PROOFS 41

a. From the two_.sfmple proportions

y = h : h - y =

b. This

a and , li. : b we 1�et easily h2 = a2 + b2•

,J solution11 is original with the author, but, like cases 1.

11 ��d �2, _it is subordinate to case /\ I I

c. As the numb�� of' ways in which three or more similar right triar�gles may be constructed so as to �on�a�n �elated lf�ear relations with but few

�unknowns involved is ��limited, so the number of pos­sible proofs therefrom �ust be unlimited.

� A �·· The two following proofs,

differing so 'much, in metb,od, . from those prec�ding, are certainly worthy of a place among selected proofs.

Fig. 16 1st. --This --proof rests on the axiom, "The whole is equal to the

sum of its parts. " Let'AB � h, BH = a and HA = b, in the rt. trL

ABH, and let H C , C be�ng the pt. where the perp. from .� ' �

H intersects the line AB, be perp. to AB. Suppose h2 = a2 + b2• If h2 =�a2 + b2, then a2 = x2 + y2 and b2=x2+ (h-y )2, or h2= x2+y2+x2+( h -y)2

= y2 + 2x2 + (h - y )2 = y2 +\2y (h - y ) + (h - y )2 = y + [ (h - y ) ] 2

i� true.

• I

:. h = y + (h - y ) , 1. e. , AB == BC + CA , which

:. the supposition is t1•ue, or h2 a2 + b2• ·a. This proof' is one of' Joh. Hoffmann's 32

proofs� · See· jury Wipper , 1880, p; 38, fig. 37 .

2nd. --This proof is the "Reductio ad Absurdum" proof'-.__ ,

-�-

h2 <, =, or )- -(a� + b2 ). Suppqse it is l�s�. ·--

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' .

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r

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I

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l

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.. T.ll�n, �ince .h2 = [ (h _f> y) + y ]2 + · [(h - y) + x2 + (h - y ) ] 2 andl b 2 =' [ ax + (h - y ) ] 2, the� [ (h - y ) + x2 J. . (h - y ]2 < [ax + (li - y) ]'2 + a 2 •

:. [ x2 + (h - '1) 2 ] 2 < . a 2 [ x2_ t (h -- '1) 2] • :. ·a2 > x2 + (h - - y )2 , whic� "is absurd . For,

if the rsupposi tion be true , we must have a 2 < x2 + (h - y }2, . as is easily. shown_.

�imilarly, the supposition that n2 >. a2 + b2 , will be proven false .

· Therefore it follows that h2 = a2 + b2 • a . See Am. Math. Mo . , V . ·III, p . l'TO •.

. ·• ,__;...,·

�ake AE = 1, and draw EF · . H perp . to AH, and HC perp. to AB.

� A HC' = (AC X FE )/FE' Be-.-.�-� (HC- � FE )/ PfF .A � = (Ac x -FE)/AF x FE/AF �-J\c x FE2/AF�

- · and :AB · = ·Ac x CB =· AC · + AC x FE2/AF2 1 } .. Fig-: 17 = AC (I + FE2)/AF2 � .Ac (AF2:__+. FE)2 /A� . .. (l )· .

' . But AB : AH = 1 : Pu�, whence AB = AH/AF, and

AH = AC/AF. Hence AB � AC/AF2• (2 ) . ·�" AC (AF2 + EF2 )/AF2 = AC/AF2.• : •. AF2 + FE2 = ·1. : •. AB : 1 = � : AF . � :. AH = AB X AF. (3 ) • and BH. = AB x FE. ( 4 ) (3 )2 �+ (4 )2 = (5 ).a' or; AH2. + BH2 = AB2 x AF2·: + AB2 X-FE2 = .I\B2 (AF2 + FE2) = AB2. :. AB2 = HB2 + HA2' 0�.

h2 = a2 + "b2. a. See Math . Mo . , '(1859), Vol . II, No. 2,

·nem. 23, fig_. 3. . b . An indirect proof foll.ovs . It is : If AB2. # (HB2- + HA2 ) , let x2 = )IB2 + HA2 then

x = (HB2 + HA 2)1 /2 ·= HA (1 + HB2 /HA2 .)l / 2 =

HA .. ,-(1 + FE2/FA2 )1· 2- = _HA[ (FA2 + FE2 )/FA2]112 = HA/FA

= AB, since AB :�AH = 1 : AF. :. if i = AB, ·-x2 = 'AB2 = HB2 + HA2• Q . E . D .

\c . See said Math . Mo .• , (1859), ·val . II, No . 2, Dem. 24� fig . 3 .

' ·.

'

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i I. I l ' I

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(?

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..

f

.,,

CA .

l

Fig. 18

.:• (<

:· ALGEBRAIC PROOPS 43

flab.!ttn

. _From sim. tr11 s ABC and BCH, HC = a 2 /b . An81e ABC = angle CDA = rt • · ahgle·. From ·sim. tri .' s AHD·-and DHC, C:Q = ah/b ; CB = CD . Area of tri . ABO on bas _e __ AO = i (b + -a 2/b )a. Area o�Acn on base AD = i (ah/b )h . :. (b ·+. a2/b )a = ah2/b = (b2 + a2)/b-

ab2 + a3· " x a = b

a .• See _-.VersluJ:s -! ·p . 72, fig . 79 •

I

---- I '

. j I

t!lnt1ttn

!�tD.!l_ �

! I ' .--. \ l

Draw HC perp . to AB and = AB, Join CB and Draw ·en· and .CE perp. resp '� to· HB and HA.

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44 I I

�HE PYTHAGO� PROPOSITION

Fig. 20

- -Area BHAC = area ·ABH

-:;_ area AB6 = lh2 • But �rea tri . CBH.-= la2 , and ·or tri . CHA = lb2 •

. :. j-h2 = la2 + lb2 . . . .. h2 = a2 "+ b2 . P,. See Versluys , p . 75 ,·­

fig . -�2 ,f whe�e credited to P. Arm�nd Me�·er, £876 • . I

I

J) If--. .... It HO = HB = DE; HD = HA. Jgin : .J .',. r �� .;,-6-__E.A__and..JO. .. __ Dr�w. EF and HG perp . to ;

1 , 1". · ' 1 . . ·Area of trap . ABCD = area.

I I .

I I. I

I

: (;;,·.! 1 AB and EX pe:rp�- to DC . · -·----------�-�-----� 1 ,� . • .: . (ABH + HBO + CHD + AHD )-= .ab + l-a2 ·

I 1 . • c ·

v J4. I • • 'B. + ib2 . (1 -) .l � ·i Fig. 21··

= area (EDA + EBC + ABE +. ODE ) = t·a;b- + "*sb--+ (iAB-,r"EF-= fAB' X .KG' as tri ' s BEF ·_and HAG are congruent )

= ab + t (AB = CD ) (AG + GB ) = ab + th� • . (2 ) :. ab + lh2. = ab + ia2 + tb.2 • :. 112 =, a2 + b2 • Q . E . D .

. : ------+---·-·-···-'·-----! '

a . See Versluys , p . 74 , fig . 81 .

I l

I I I I t\' \ . I , I II •'' I .

I I L:'l>l ' \ • . I � \ . ,, , , . , .

''l f '"'• ,, f!OI ,..,

Ct..�-- .C t.. - jl) Fig. 22

In fig. 22. , it is·.obvious that : (1) T�i . ECD = th2 , (2 ) Tr� . DBE

= ta2 • (3 ) Tri . HAC· = ib2 � :. (1:�- � . (2 ) + · (3 J = (4 ) ih2. = ta2 + j-b . :. h2 = a + b2 • Q .E •. D .

·

a . See Versluys , p . 76, fig . '83 , credited to Meyer, (1876 ) ; also this· work, p . 181 , .fig . �238 :for a similar geome.tric pr'oof . · ·

________ .....

0 .

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I •

·.

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I \ ALCJBBRAIC PROOPS "

I�ta.1t:.!b.ttt \ 0 I

I . -Por figure; use fig. 22 above, omitting lines

EC and /Ial. · Area of sq·. AD ·• (2 area of tri . ·n.m • rect . BP) + {2 a��a of tri . HAC = �_ect . !3Y • 2 x ia2_. + 2 x tb2 = a2 :+- b2 = h2• :. h2 :.:: a2 + 'b2 • �

Or use similar parts of fig:. 315, in geometr-ic proofs . a. See�Versluys; p . 76, proof_66, credited·

to Meyer ' s, 1876, collection.. ·

!lt!l!I:.f2Y.t

. In fig . 22, 4enote HE py x � Area of.tri . ABH -+--+-area-ot--sqo--AD�hx::_+-h·�-area-of-(tr·i"��-:-�oH�t-rl.:-­

CDH + tri . DBH) = . j-b2 + j-h (h + x) + j-a2 :=:__._._ip2 + j-h2 - - 2 • . 2 2 .: . ' 2.. ' . .. . + l� + ia . . . h ·• a + b ..• . a . See Versluys,. ·p . 76 , proof 67, and there

• I I . ·

cr�dited to P . Armand �eyer ' s collection mad� in --J---l-876-;-_: ___ ! - ----

-� - - --- ----- - - -- ---- - . -· -

·

b. ·PPoofs TwentT-Two, Twenty-�ee �d Twenty-Four are on�7 variations of -the Mean Proportional Principle, --se ·e p . 51, thi's boo� .

!�t!l!I�fl�!

At A. erect AC = to, and perp . to AB: :·.and from -C drop (en· = AH) perp . ·to All. Join CH, CB and DB . Then AD'• HB = a� Tri . CDB = tri .. CDH = rtQD x DH �

Tri . CAB = tri. C AD + tri . DAB + (tri • . BDC = -tri . CDH = tri .

Fig. 231 : CAH + tri . DAB ) . :. ih2 = ia2 + fb2 • · • -� h2 = a2 + b2. a . See Vers�uys , p . 11: fig . 84 , one of

Meyer 1 s , 1876 ,_ collection. ·

,,.. ... _ �---�-" IltD.!l:.ll!

. . From A draw AC perp. to , and = to!AB . Join CB,� and draw.BF parallel and = to�, and CD parallel to AH and = to �.: Join CF ana· BD .

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C·tr' :rig. �4

.

- f ', ' t 'l'HI PY'l'BAOORJWI PROP08IT.IQI

�. '

�ri� CBA ; tri . BAP + tri� :PAC + tr.1.- · - C.BF •

- tri . BAP . ../.! tr{. FAC � t�i •. . PDB (a�n�e �ri. BO�· • �·� BD�) • tr1 . :PAC: + tri •. ADB. • • . th .

.1. 2 .1.·a.· . a · 2 a • Ia _ +. -2'� • • • _h I • • a + b • ) .

·a . See Versluys, ·p. 77, · fig. 85,·_ ·b.ei:ng .. one ot Me7er •·il£o-llectiQn.

... . ..: • -/ • � • ' .. .,! •

. -, > .., "" ... ""� ;r'"'t $5'

. From. A.. draw· ·AO · perp . - to, . - H.. · . �d = -tp �. From 0 draw OF e)qu�l ·.

A ·

to HB and par$lle'l tQ ·AH. _�-Join OBi A' B . � and HF and 4r�w �- parsl_.�el to ., ...

-

_ , _ _ _, _ ,, /HA� OF = ·EB = BlJ = a·� AOF and _:AB� I �Ei?, . are . congruent� so ar6· CFD and �.

• • • • •• •

ABB � tr" • � + tri � B)' A + tri . .OF I . ,, . � Cl �,, . . _ + (· tri. FCD + otri . l)lm.- :. tha + tab

.,... · · ·

� ia• + tb2 + tab. :. 1;12: =:: -�2 + -b2 • . ·Fig. 2"- c .... Q.E.D� . .

.

1 • .a . · ��e Versluys , p·.Q 78,_ fig . . 8�; a_lso see . "Vriend de Wi�Itlmde , " 1.898, by F. J. ' '

Vaes • , . .

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i

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1\ . i I .

ljj ••

Draw PHlt perp :· to AB ·,4-;J' ·

and· mpke PH;· -:.·.AB;. ·,Join PA, ·

,_F ' ·•-, -PB , AD an4 GB. · f � , � · ' , 1 1 , . $ -��1:-J;s. :BDA �d BHP are

D'/ 1.( · I, ' , .. contntuent ,•L so are· ti'i ' s GAB ::... '. f � - ':- . .., • \ ·'"'06 \' ,, ' ' ,. �' and·�. ·�uad· . AHBP = tri. •

•,_ ' --- ....; � BHP + tri . AliP • . :. th2 = ia2 · r- · '.u· · · a 2 2 2 ' .... - . , , ' -- + tQ . • :. h. = a + b • Q.E . D .

"

a . See - Versluys, p. 79, fig . &e. Also the Sqientif�que

. r1s. '26 .-· Re!ue{, Pf;)b . -. �6·� �889, B. Re.nanj

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ALGEBRAIC PROOFS 47

· al�o Fourrey, p . 77 and p . 99, --Jal de-Vuibert, 1879�

: ' . ! +

!

Be • .. • I

.I�t!l!I:.!ln.!

Through H draw PK perp . to

e.Jl. .·.! . AB! ,m��ip� �H = AB,_ and ij�i� PA and

'·· . I.,_. P.B-.;-·\ . ... \ \,-lo. l •• ----..:. . .• : ' : . ' . . I' . I . - ' • / 1 • · · Since area AHBP = [area PHA 1 I '\· + area PHB = �h x AK ·+ -ih � BK

1 : · = ih (AX + BK)� · ih x fh = !h2J = (area

,' AHP -:+" area ·BHP · = !b2{ + -ka2 )·. :. !n2 -= -ba2 + lb2. . :. h2 -;,1 a2 + b2 . .

a . See Versluys , p . 79, fig . 89 , being qne of Meyer ' s , 1876 , col-

Fig. ·27 lection .

Draw PH perp . -to AB, making . �H = CD = AB. Join PA, PB, CA and CB .

Tri . ABC =. (tri . ABH .+ quad . AHBC ) = (quad . AHBC + qu�d . --ACBP ) , . since PC = HD. In tr! . DHP, angle BHP = 180° ·- (angle BHD.; 90.9 + . angle . HBD ) . So the alt • . of tri . BHP from the verte� P = a, and its area � !a2 ; likewise tri . AHP· = ib2. But as in

) . F!g. 28 - fig . 2 7 above , are� AHBP = �h2. :." h2

', . 2 2 Q E : ·

= a + b • • .D .•

a. See Versluys , p . -80 , fig . 90 , as one of Meyer ' s , '1876 , col�ections .

and DB

!b.lt!I:lD. e

Tri ' s ABH · and BDH are 2imilar ,_ so DH = a2/b = ab/h . Tri . ACD = .2 tri •. ABH + 2 tri . DBH.

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48

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.. -··

THE_PYTHAOORBAN PROPOSITION . -

�' •' �.,.. ' I . • • AI I .

I

. Area of tri. ACIJ =· �/b· =!= are.a-7of 2' tri. . ABH + 2. tri . DBH

,.. 3/'"' . h2 2 - b2 Q E D c av + a U'o • • - . ==. a + • • . • - •

91 . a. See Versluys- , p . 87,. fig ., ,. ..... ,,.; ;; - · � "

!b.lt!r.:!�Q --�r-.

. !fother Reductio ad - Absurdum ·• proof -•sep pro·or Sixte�n -above.

· sluppose a2· + b2 > h2• · 'Then AC2 + p2) b2; and-CB2 + p2 > a2 •

. ··:;.-AC-2 + CB2 + �p2 > a2 + .b2 > h2• As F�g. 30 -2p� = - 2 {AC x BC) then AC2 + CB2- + 2AC

�, _ x CB > a2 + b�, or {AC + CB)2 > a2 --'""'-b2 > h2 r/r h2 > a.2 +. p2 > h2 , or h2 > . h2, an ab:

-.

Similarly, if ·a2 + b2 < h2• ..� h2 = a2 + b2 . . .

surdity. Q.E·.�.

a . See. Ve�sluys , p . 601, fig: 64.

.. \ , . - §" • • •

' . . ,.. • IN • r I I'-\ ttl' ' I , � \ I , \ ·� C� .. .• - . • �J.u

Fig. 31 .

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i ALGEBRAIC PROOFS / 49 ---�·---�--��--------��-------7--�------��

0 ! I

Let BH := x, and HF = y; then AH = x + --.y; sq . AC = 4 tri • .

ABH + sq . HE = 4[x(x ; y)J.+.y2 = 2xn 2�y + y2 = x2 + 2xy + y? + x2 � y ) 2 + x2 .. · :. sq. · on AB = sq. of + sq. of BH . :. h

-� 2 • 2

= a + b Q . E . D . I This proof is due to . ' ' l -;. Fig. ·32 R�v . J. G. Excell , Lakewood,"O . , !.:J- "i July:�. 1928; ai=so �iven by R . A .

Bell , . Cleveland,/ 0 . , Dec;. 28, 1931 . And it app'ears in "Der.Pythago.reisch Lehrsatz" - (1930 ) , by Dr . W. Lei tzmann, in Germany.

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·. .

T.h lr.tv� f-l:v•- -.----- -�-----------�--,.,--�----- - -- ------------------1 �------------------- • -J; ·- -.,_--. - - - �----�-�---------- ---H

. . -- -- -- - . ' · -- I In fig . 33a, sq . CG

A 4 2 -·.

· �,_ ___ ...., = sq. AF + x tri . ABH = h C �-., 1 + 2ap . --- (1 ) (' I , .. - .- In fig� '3b , sq . KD

' 1. ,·_ 'I" � sq . ,KH + sq . ilm + 4 x tri . ,. ' >" 2 2 .. r .

� �H = a + b + 2ab . --- (2 ) \ I ' 1 , , ·

-n� _ ._---A But-sq . CG = sq . KD, by ., \ , � F' const ' n • .' :. (1 ) = (2 ) or }?.2

�g � Fig. 33a - + ·2ab = a2 + b2 + 2ab . :. \h2 = -a 2 + b 2 • Q . E • D •

Eig. 33b

a . See Math . Mo . , 1809, dem. 9, · and there , p;. 159, Vol . I, credited _to Rev· .. A . D. . Wheeler, of Bruhswipk,

· Me . ; als.o see Fourrey, p . .So, fig ' s a and�; also see "�er rythagorei!sch Lehrsatz" (1930 ) , by Dr . W_; Leit�mann . ·

b . Using fig . 33�, a second proof is : Place 4 rt . triangles BHA,· ACD� DEF and FGB so that their legs form a

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TijB PYTHAGOREAN PROPOSITION •.

square vhofte aide ia HC. Then.it is pla1� that: } ! <

1. Area of aq •

..

HE • a2 + 2ab + b2 • 2. Area ot tri. BRA • ab/2 . '· A�ea ot the 4 tri ' s • 2ab. 4. ,Area ot 'j·· AP • area of �q. HE '- area

· tri' a � a + 2ab + b2 ·- 2ab = -a:e + b2 •

5. But �ee; ···o·r aQ. AP • h2• ·

of t�e 4

6 • ,. a . • :.·. h \,II\" a• +,_ b"'""' Q.E.D. · This. proo� vas dev.�sed by -Maurice Laisne-z � a·

higb _aoh_ool bo� '·in th� Junior -Senior High School of - . ···'-;"SouthBencl,-Ind., and sen't to�Jne, May- _16, 19�9, .. J>Y

�is cl�ss teacher, W1lson Thornton, o •

\ \ ' , c·...,

lig.--·�

\

_ !b.lt!l:IIX!D.

. • b -� a, Drav�OD.

Since tri '-s ABH and CDH are sim1lar, and- CH then .CD-• h (b - a )/b , an·d ni(� a·(bl - a ) /b •

Nov area of tr1 .- CDH = i(b-. a ) x·a (�- a )/b • ia{b

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.ALGEBRAIC .PROOFS 51 ------�----��------�.1 ----�--�--�-

·Area-of tri . ' DG* = �GA x AD = �b F [ 2 a (b - a.) J �' 1 [ 2 c·:' )] A •· � 1J - b ·= 2 b - . a_. b - a ' , . \

�--,'' ',-c. (2)

�:----: � •' Area of' tri . . GDO = �h(b·� 8)h

\ ' . . � ·) = th�cb - a ) . ;_.:._ (3) ��- -..,........ b '

-Fig. 35 " I

• I

:. area of sq,. AF :;: (1),. + {2 ) + (3) + tri. _9QF = ta (b' -�a )2/b

+ t[l:>2 - _a_{b --: a. ) ] + !h2_(b .- a )/b + lab = b2, which reduced: and c ollected _give s h2 (b ... �n - �b :.. a )a2 = .(b --a )b2• :. }?.2 = a2 + b2 • Q.E.D.

fl· See Vepsluys,· p . 7�-4 , solution 62. -b . An Arabic work of Annairizo; 90�N.C. has

a-- siinilar proof . c . A:s last 5 prOofs- .show., figure s for geo- l

--metrtq--proof'-a-re--rtgur·e-s··---rq:r- alge·bra.1:c·--p�e:>rrr·s-a1.-so-. ----- --'-- ---;------t---------- --

Probably foroeach-geometric proof there�§ an aige• · i

1 I ! t I I '

• • I braic p'roof. . -- -'

B. --Th e Kean P ro p o r t t onal Pr t nctple

The ,mean ·proportional principle le�ding to equivalene� qf areas 9f triangles �nd parallelogr�ms , is very prolific in prool's !!> • • •

1

, :By rejecting-all, similar right triangles . � other than those obtained by droppi� a perpendicular - � . , . . from the vertex of the right angle to the hypqte�use I of � right triangle and omitting a.ll equations re -

-sulting from the thre,e similar right triangles thus : fQrl!led, save only equa-tions (3) , (5) and (7 ) , as given in proof ·on e, we will have l�mited our field

.greatly • . But in this limited field.--the--proofs po ssi­�le are many, -of which a few very interesting ones �ill now be1given . I I • . ' In every figure under B we will let h = the

" - hy:pote�use, a-= the short�r leg, and b = the longer leg of the given right triangle ABH •

.. )

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.

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·�

Fig. 36

I ! 2 I Since AC : - AH = AH : AB, AH 1

= AC x AB, and BH2 = BC x BA . Theh··�·. '

BH2. + HA 2 = (AC + CB )HB = AB 2 • :. h 2 1 = a2 + b2 . !-

a . See Versluys , p • . 82 , fig. · ! 92 , as given by Leonardo Pisano , l

1220, in _prac tic a Geometriae; Wall1 s, Oxford, 1655 ; j· Math� Mo . 1859, De:g�. . 4 , and,-credited to Legendre ' s 1

l -1

·-

--- �- . -' Geom. ; Went�orth ' s. New Plane Geom. ' , p . 158 (1895 ) ; ·

I '

-�!�o��:�::=:\:�m�f;����· P:-�;�rf��o��:-�i!!so----�J · _---

"Mathematics ·for the Million, "- (1937) , p . · 155, fJ-g . 51 (i ) , by Lancelot Hogben� F .R . S .

· ·

. Fig. ,38-

Extend AH and KB to L I thro�--c �aw.cn paif . to AL,

'AG. · -T . ·

perp . to CD, and LD. par . to HB, · ! and, PXtelld HB to· · F. / · i BJ£2 = AH � HL = FH X HL = FDLH = I a2 • Sq . AX = ·pa�al . HCEL = ' paral . AGDL = a2 + b2 • :. h2 _.

= a2 + b2 • Q.J.D. a . See Versluys , p . 84 ,

fig . 94, as�given by Jules ca.m;;s; 1889 in S . Revue

···' ( f2t1r.

Draw �c.. Through C. draw CD par·. -sto BA�· &ld the perp 1 s AD, HE a_nd. BF . :' '

Tri . ABC = i sq. BG = i rect . BD . :. sq . BG = a2

. �"� = rect . BD = sq. EF + rect . ED = sq. EF + (� x ED= EH2 ) = sq. EF + ��2.;_ · · But tri ' s -ABH and BHE

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2 '2 2 are similar . :._, if in tri . BHE, BH = BE + EH , then in it s similar, the tri . ABH, AB2 = BH� + HA2 • -:. h� = a2 + b2 • - Q. E .D •

. a: See Sci . Am. Sup . ,. Vol . 70 , · p . 382 , Dec . 10-, 1910, - fig . 7--one of_ the 108 proots of 'Arthur E . Colbu�n,· LL .M. , of Dist . of Columbia Bar .

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Const ' n obvious . Rect . LF = 2 tri . FBH + 2 tr1 . ADB = sq . HD = sq . LG + (rect . KF = KC X OF = AL X LB = HL2 ) = sq . LG + HL 2 •

j . I<�� 19 I - . - I

But tri ' s ABH and BHL are similar . Then as in fig. 36 , h2 = a2 + b2 • .

I • \ I t 1 - - � -1 --n- -

a . See Sci . Am. S�!J_V_ . _- ______________ ___ _

10, p . 359, one of -co!b�n '-s -108 .

I

I .__ CJ:-- J F --- .... ..,. ___ ,.....,...._._ �w_, --·-

Fig. 39

' E�u:.!t:I�2 "-

Const_ru_ction as in fig . 38 . Paral . BDKA = rect-. AG = AB x BG = AB · x ac· = BH2 • And. .AB X AC = AH2 • :Adding BH2- + AH2 = AB X BC. + AB X AC = AB (AC + CB ) = AB2 • :. h2 = a2_ + b2 • Q .E.D .

Fig· •. 4o a . See Wipper, 1880, p .

39, fig . ·· 38 and there credi.ted to Oscar Werner, as recorded in

"Archiv .• d . Math. und Phys .. , " Grunert , 1855; also see Ver.sluys, p . 64, fig. 61, and.Fourrey, p . 76 .

0

----- - - -�--,----- --�---- -� - --:

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Two �quares , one on AH const'd outwardly, the other on JIB overl�pping the given ,triangle .

Take HD = HB and cons 1 t rt . tri . CDG . The� tr.i 1 s CDH and ABH are equal. Draw� GE par . to - -

'

AB meeting GKA produced at E·.--- _·--

- Rect . GK = rect . GA + sq. HK = (HA _= HC )HG + sq •. HK. � HD� + sq . HK. ·

Fig. 41 Now· GC: DC =· DC : (HC = GE)

" :. DC2 = GC x GE = rect . GK = · sq . HK + sq . DB = .AB2 • :. }?.2 � a2 + b2 • : , , :

j a • . See Sci . Am. Sup .; V . 70, p . 382 , Dec . 10 , ,.

---·----1.

I 1910 . Credited to A. E . Colburn . . -----�---------- � --� ------ -�------- ---- ------ -- _---_---�-- ---------------------------�--- 1 - ---u- ---

f2t!l:fsuu:

I

I

'

AK = sq! on AB . Through (! draw . GD _par . to HL .: . , and meeting FL produ�ed at n· .

and· draw EG. Tri . AGE is common to

sq. " AK and rect . AD. ,·. tri . AGE = i sq. AK = i rec·t. AD. �·. sq�� ·a-=-- :rect . AD; Rect . AD = sq . HF + .(rect . HD· = sq. HO ,, see .argumen� 'in proof 39 ) . :.

Fig. 42'· sq •. BE = sq . HQ + HF, or h2 !'

= a2 + b2 • . -a. See Sci •. Am. Sup . , v. 70, p . 382, - nee . 10,

1910 . Oredi ted to· ·A .- E . Colburn . b . I regard this proof, wanting ratio, as a

geometric , rathe� than an algebraic . proof .. E-.-- S· . Loomis . 1 '·

- - - -�.. . .... --- - -- -� --- ---- - ------...... • �.,. _ _ ,._...,. __ , -..,.. -- _ • w • -\

---- ----

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' ALGEBRAIC PROOFS 55

<'

,-

·HG -= sq . on Ali. Extend · KB to .M and thi'ough M draw ML pa� . to HB' meeting GF extended at L and draw CM.

T�i . ACG = t�i . ABH . . _l 1 T�i . MAC =· � �ect . AL = 2 -sq . AK. :. sq- . AK = rect . AL = sq . HG + ·

. (�ect� HL = ML . x MH).� HA x HM = ,HB2 � sq . HD� = sq . HG + sq . HP. :. h2 = a 2 + b2 .

a . -See Am. Sci . -Sup . , V . 10, · p . 383 , Dec . 10, 1910 . C�ed­fted to A . E. Co1b�n.

f�!!I:.!l!. -

·Extend-:J{B t� 0 in HE. Through 0 , and pa� . to HB .�aw

·NM,_ making OM and ON e�ch · ·=. to HA. Extend GF to N,. GA to L, making AL = to AG and draw CM.

'

·T�i . ACL = t�i . OP� - tri . ABH, and t�i . CKP = tri� ABO;

. :. �eC"t . OL =:= sq: · AK, hav�ng po1ygon-�B in common. :. sq . AK = rect . ·AM = sq . HG + �ect . HN = sq. �G + sq . HD; see p�oof. Fo�ty-Fo� al:fove .

·:. h2 = a2 + b2 -o' Q.E.D_.� "l'l.' .

a . Se_e Am. Sci. -Sup .• , V. 101 p . 383 . Credited to A . E. Colburn .

- - ............ ,

- ---- ------- - -- --- ------:

L

-- - ________ __,

' . .

-- --- ------ �--- --------�-- - - ---�--- --- ------ - --- - - -- ---- --- ---------- ----�--��--------- ----�------------------- -- -- - - -THE PYTHAGO�- PROPOSITION

fi!ll:!IXIfl

Transposed sq . LE = sq . on AB.

Draw through.H, perp . to AB, ·GH and produpe it to meet MC produced at F . Take ·HX = GB; and t.�ough K draw LN par . and equal -t9 AB. Complete the tran�posed sq . LB • . Sq: �LE = rect . nw·+ rect . DL •. (DI: )( XN = LN )( KN = AB - X AG K �2) + (rect . LD = paral. AF

· Fig. 45 • sq. AC ) for tl'i . FCH = tri . � . and tri . CPR = tri . SLA . :. sq . LE

= HB2 + sq . AC, or h� • a2 + b2 a . Original with the author or this work,

· Fe]? . _

__ ·?_! __ ��g§. _ _ __ _ __ __ ____ _ -- _ _ _ _

i '\..._

f2t1x.:.tistb.1

Construct ·

tri . BHE = tri .• BHC and· .tri . .. AHF = trt .

-� . AHC, and through p.ts. ��.... · F . H, and E draw the

/ . ..... ............. f....... H lin� �� �:(cing FG

1 -'-·- _.... and EL each .. =· AB, L _ ___ _. ..,... ' R ..... \ 1 -' � · .... and aomplete the

· .... .... .... < .... ....... ..:.J reQt 1 s 'FK and ED, ·-·-- � ....

', '

and draw· the 1 ... _ 1es ... ' ........ HD- and HK.

• F.ig. 46 , .. Tri . HKA . = ' AK X AF = ' ·AB

x AC - t AH2• Tri . HBD � i BD x BE = i AB x BC = i . HB2 • Whence AB x AC • AH2 and AB x BC = HB2• Add­ing, .we g�t AB x AC + � �..x BC = AB (AC + BC ) = AB2 ,

.

or AB2 = J3H2 + HA2• :. h2· = a2 + b2 • ai. Original with the author, discovered Jan .

31, 1926 ..

.-.

- ' .

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. .,

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•

' • -----

l.

' : ' "' ' - -

'• ' �7·� .. '

' ' y ' . �;; � .

,,;,; ,. �-- � .�:� ..... � ' ,..,;)-�·'

fi • I I

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"-

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_

_____ 57---�----:--- ----,-� � a • & l

I ' : . " I ' I_ \1 • I . I ' \,. I

fRtil:.l!!!!! .�

t'ons.truction. Draw HC , AE r- and Bl!. each perp . to � 1 maldng each

equsl to AB. . Draw EC and FCD . .Tri·' s "

ABH.and.HCD a�e equ81�and similar . _

· Figure FCBBHA = paral . CB + par�l • . CA • QH · ;t'rJB· + CH x c;tA = AB x GB + AB x AG • HB2 ·+ HA2

I �'C '\, I �I ;�' \ f t-

,� �: ., 0. "r · = AB·(GB + .A:G}. = AB x �· = AB2•

a . See Math. Teacher, V. XVI,. 1915. Credited t9·Geo. G. Evans, C�rlestqn High �chool, Bpston,. Mass.; also Verslu7s , · p-. 64, fig. 68, and

p . 65, .,ttg. 69; al�o Journal ·14-e Mathe1n, 1888, -

·p. Fabre; . �d f'ound in "De Vriend der Wirk, 1889, " by .A . E . B. Du�fer .

. Eif�l

t - am giving this figure. ot Cecil Hawkins as it -appears in Vers.lilys I work, --not 'reduc- � ing it to 11!7 scale of h.=·l".

Let BB' = HB = a, and HA 1 • vA '=- b and draw A 1B1 ·to � <

,

,. . D in AB.

' �'18· 48· <>

to baae.an�altitu�e

Then angle BDA 1 . is a rt . .

ang�e ,. -since �ri 1 __ s 'BHA and E I HA.1 �e congruent-h&ving pase and. alt�tude ot the .one . res ' ly perp . of. the other .• -

Now .tri. BHB' + tri . AHA' = tri . BA'B' + tri . AB'A' ·• :tri� ·BAAt· � tri . BB·1A. .�. t �2 + t b2 ·

_, t (411 X ·A ' D - t (AB )( B 1D') - t[��A 'B I '+ B '' D )] - t (AB. )( 'B 'l) ) .. t AB )(, A 1-B ' + i At!. )( B I D - t AB )( B I D

···tAB xA'B' �-th Xh•th2• :.h2 • a2 +b2 • Q.B.D:: .

. a., �ee VerslU7B, .. p . 71, fig. 76, as given by Cecil �wktna·, 1909, of England.

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El!1t:.2nt

Tri . ACG = tri . ABH . ' 2 . :. sq . HG = quad . ABFC = b �.

Since ang�e BAC = rt . angle .

'

i \

:. tri . cAB = -ih2 • :. b2 = __ qua.d •. · . ]._ 2 ' .1. 2 ' ABFC � . 2h + tri . BFC = 2h + l (b + · a ) (b - a ) • - - - (1 ) Sq . HD = sq. HD 1 • Tri . OD 1 B = tri . · RHB . :. sq . HD ' = qu�d . BRE ' O = a2 + tri . ABL - tri . AEL . :. a2 = ih - t (b + a )

Fig. 49 (b -· ·a ) . - ·:..- (2 ) · (1 ) + (2 ) = (3 ) . a2 + b2 = ih2 + th2 = h2 .

2 2 2 -:. h = a + b • Q. E • D . ·

· Or .from . (1 ) thus : . - lh2 + ! (b .+ a ) (b - a ) = b2 = ·lh2. + ih - la -. ·Whence 112 · = a2 + b2 •

"' ·a .• See Versluys , p . 67, fig . 71 , as one of Meyer ' s collection, of 18?6 .

" ... _,

fl!!t:.!�Q

Given the rt . tri . ABH . T�ough B draw BD = 2BH ·and �ar . to AH·. From D ·draw perp . DE to AB . Find �an prop ' l between AB and AE ·

A . :-a which 'is BF . From A, on AH, lay off AT = BF . Draw TE and TB, for��

· ing th� two si�lar tri � s AET and l)�'.... • • ..

ATB, from which AT : TB = AE : AT , ,.· · ·

• ' • F or (b • a ),2 = h (h - EB ) , whence ·

· · · • • �·· EB = [ h - (b - a ) ?! ] /h·. --- ( 1 ) · A.. 0 1<. ,.. I . Al s 0 EB ' : AH � BD : AB .

_., · .-:- EB = 2ab/h . --- (2 ) Equating (1 ) Fig. 50 and (2 ) · gives · [,h '!" (b - a )2]/h

·

I 2 2 2 · · = 2ab h1 whence· h[ = · a + b • a . Devised by ·the author, �eb . 28 , 1926 .

. b . Here we introd�ce th� circle. in finding the mean pr_oportional .

- - �� - - - f_ ... _ - � - - - ---- ------- -�- --

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j /! / : /

,,

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------------· -------

--·

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.... -l. ' ' ])''• \ ·. '...... ' . . .... \ .

•• • ' . ... .\.£ · . ..-

· . - . . . . . . .

flf1I:!b.t!l

An indirect algebraic proof, said. to be due to the great Leibniz (1646.-1716 ) .

If . (1 ) HA 2 + HB 2 = AB·2 ,

then (? ) ·HA 2 = .AB2 - HB2,, whence ·(3 ) HA 2 = (AB + HB ) (AB - ·HB ) :

' \ IP1g. 51

Take BE and BC each equal to AB, and from B as center describe the semicircle CA 1E . Join AE and AC·, and draw· BD perp . to AE . 'Now (4 ) HE = AB + He, .and '(5.) HC = AB· < j

- HB . (.4 ) x (5 ) gives HE x HC ;-' ' 2 ' .

. · -= HA- , which· j:s true· only ·when tri�ngles . .A.HC and ·EHA Q

ar�t s_�l�!' .__ . - - -- - - -- - - - · ·

:. (6 ) ·angle CAH = angle AEH, and so (7 ) HC . · : HA = HA : ·HE ; since angle· HAC = ' arigle E, then angle

CAH· • angle EAH . ,. :. angle AEl{ . + angle EAH � _90° and ·angle CAH + angle .EAH = . 90° . :. angle EAC = 90° . :.

· vertex A lies on the semicircl�, or A coinctd�s with A I • :. EAO is inscribed in a semici:rcle and is a rt •

ansle . �Since equ�tion (1 ) leads through the data de­rived from it to a rt • . triangle , then startill8 with such a triangle and reversing the argument we · arrive at h2 =:= a2 + b2 • __

a . See Ver.sluys , p . 61 , fig . 65, as given by von Leibniz .

. flf1I:fQII.t

J'ig. 52

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a . This proof was .s·ent to me by . J. Adams of T:q.e . Hagu� , Holland . �ec�ived it March 2 , 193.Jf, but the author was not given . ·

f1!1I=.fl�!

. Assume (1 ) HB2 1+ HA2 :;t AB2 • · . Draw HC ·perp . ·to AB . Then . (2 ) AC�

+ CH� I: HA2 • (3 ) CB2 + CH2 =·-�� I ( 4·) . Now AB = AC + CB-� so - (5 � AB2 = -AC2 + 2AC X CB .+ CB� - AC + 2HC2

Fig . 53· + CB2·• _' But (6·) HC2 = AC x CB . : . . ·(7 ) AB2' =_AC2 + 2AC x CB +. CB2 and

(8 ) AB . =: AC + CB . . :. (9 ). �-= AC2 -t: 2Ac x· CB + CB2 • (2.) + (3 ) = . (10 ) HB2 '+ HA:2{ • AO� · + �C2 . +. CB2-, or .

(11 ) �-2 = HB2 ? + HA2-. :. __ ('12 ) h2 = a2 + b2 • Q.E .D • . - -- ·a . ·se·e ·ver sluys , ·-·p\ ·62 ,. fig; ·66 .

·

b_.�Th1.s_p_r_oo.f..J .. s�o.I\e_o�offms:�s,-181_8_,;. __ col.--- .. ___ _ _ . ; \ lect�on , '. ,. ' ' . \

c_. -:.. the �t rc l e t n Co!Zn�c t t o� w t t h. t h e Rt l h t rrt anl l e . .

(! ) . -�Through the Use ol On� Circle · From certain Linear 'elations or the ·chord,

Seca�t :and T_ange��

_in · conjun�tion with a right tr1-.;,- --

... angle , · or with si�J,ar. ... ·r.elat�d i'ight triangles , it may also be p:roven that : The_ ! square .o f the hvpo t enuse 'o f a r

_t l h t ·t r t anl l e t a e qu al ) to t he aua ol ·th·e . s qu are s

o f the_ o t h�r:.

t wo . . s t-de�. 1 , .

' ·

And . since the· �lgebrria is the .�easure or · translite�ation of the geom�tric square t�e trut� by �� proof .through , the alg��aic method involves �he truth of t�e geometric methpd .

Furthermore these proofs thro� tqe use of circle ele�epts are true ,/�ot because of straight-line properties of the -circ�e • . but because of the law of similarity, as each proof may ·be reduced to the propo��

,��onality of t�e homologous sides of similar

triangl)es , "the circle -being a factor only · in this, . that the homologous angles are measured by _ equal arc s.

. . -(

I --1,

I i .

1-------- - - - - - -- -- - - - - - -�-�-��-- - - ----- - - -�� -·�--� --- --- -- - -------

'•""---

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I

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----�....,...--------�----- ·--- ----- --- ---··--:-·-- --- - --------···· -- -- -.--- - - --- -- - -- � - ---- - - . - --- � -- � · - : · ·· - - - - -- - - - -4 61

' -

-

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+ ! ! • '

I . I ! I

·-

1 · l

ALGEBRAIC PROOFS

(1 ) The Method by Chords .

flf!l:ll!

H is any pt . on the semi-Q·irc le BHA . :; the tri • AB:ft is a rt . triangle . Complete ·the sq. AF and

" draw the perp . EHC . BH2 = AB x BC (mean propor­

tional ) AH2 . = AB x AC (mean propor­

tional ) Sq . AF = rec_t . BE + re_ct . AE = AB x BC

Fig. 54 + AB x AC = BH2 + · AH2 • :. h 2 =

a 2 2 + :b •

a . See Sci . Am-. Sup . � V . 70 1 p . 383 � Dec . 101 1910 .- Credited to A. E . Colburn.

b . Also by Rtchard A . Bell1 - -given to me Feb . 281 1938� He says_ he produced it on Nov . 181 1933 .

f.lf!I:.lt�l!! -

AJ). · - P<� ' . "' ' . ,� '

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. \ . . --�...,--- ­'-- �,. -r -· ,. 1 -

' N L · - -- - - -' Pig . 55

3 � 1910 . · Credited to A .

• 0 Take ER = ED and ...-------

B�sect HE . With Q as cen-ter desc�ibe · semicircle AGR . Co�plete sq . EP • Rect·. m{'= HC X HE = HA

2 x HE = HB = sq . HF.. EG i s a mean proportional be­.. tween EA and (ER =· ED ) • :. sq . EP . = rect . AD = sq . AC + sq . HF . But AB is a

...

mean prop ' l between, EA and (ER f1 � ) . :. EG = AB . sq. BL = sq. AC + sq . HF • . :. h2 + 82 + b2 .

a : See Sci . Am. Sup . � V. 70� p . 359� Dec .

E . Colburn.

a-·

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f 1 l . I

l

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l

- - - - - --· -- - - -------------------·-----------------;-"��

J

62 THE PYTHAGOREAN PROPOSITION

_ _ _ _ .,... ____ _

-------

------ ------ -- -- - - . ---·-

·----------"::"""""'�------M---""'"'-----'"''""-=== . -- - - ---· · - .. . .

. . ' tJ r � -.,- -

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fl!!l:.(ltb.1 • • ., l

In any circle �pon any diameter, EC in fig . 56; take any distance from i the center less than the r�ditis ,

, -�s B.li. At H dra:w a chord � AD per�. to the diameter,

and join !J3 fornrl.ng the rt . tri . ABH.'

�a·. Now HA. x HD = HC x. HE, or b 2 • (h -(· a ) (h - a ). :. h 2 ;; a 2 + b � •

·, "' b . By joining A and

C , and E and D, two similar ___ rt . tri·' s are formed� giv-

, ing HC : HA = HD : HE, or·, ,. again., b·2 � (h + a ) (�. - a) . .�. h2 ;a a2 + b2 • .

- - - - ---.--------·But·-·by--j·oil11llg -·c- ·and 1ll. the tri . DHC • t:ri . . AHC, apd since ·the · tri . DEC is a ,particular Qase of

One; f1.,g . · 1 , · as. is obvious, the above proof is sub-. ordinate to, being but &�culaP c.ase of�:f!

of, One . c . See Edwa�s ' Geometry, p . 156, fig , 9,

and. Journal or. Education, 1687, V. XXV, p . 404, fig . VII .

.J f1!1r.:.!1D.l .,. .

' . f .. - - - -- . ... .. ¢ . ... , . .. � ' ' I \ • '· ' •

,._.._ ... __ � - - - - - ·- �E I • , I . , ' ,

' , \ , . �

' , .... , ... _ ,

�'· · - - - -- ""'

Fig. �� •.

With· B as center,� and radiu� = A1:l' describe circle �c • .

. Since ·CD is a ,mean proportional bBtween AD and DE; and as CD • AH, b2 '

2 2 ' = (h - a L(h + a ) = h a •

• ·h2 2 ba .. . ·- = a + • a . See . Jo�nal of

Education, 1888, Vol . XXVI� p . 327, 21st proof; also Heath ' s Math. Monograph,

- -

' ..

�----�-------'---------- --� � - -·-- --. ' . . � . .

i . !

,.

I .

ALGEBRAIC PROOFS

No . 2, p � 30, . 17th of the 26 proofs there given . b . By analysis and compaJ;-i_son it is obvious ,

by sub'stituting for ABll its equal , tri . CBD, that . above solution is subordinate to that of F�fty-Six .

' In any circle draw any chord as AC -perp . to any diam-.-· eter as BD, and join A and B, B and C, and C and D, forming the three similar rt . tri ' s �; CBH and DBC .

Whence AB : DB = BH : Be , giving AB x Be = DB x BH

= (DH + HB )BH = DH. � BH + BH� = AH· x HC + BH2 ; or h2 = a2 + b2.

Fig. 58 · a . Fig . 58 is closely k , re�ated tp Fig . 56 .

b . }'or solutions s·ee Edwards 1 Geom.· , p . 156., ��g . 10, Journal of Education, 1887, V . XXVI, p . 21, fig . 14, Heath ' s Math . Monographs , No . 1 , p. 26 and Am. Math . Mo o , v. III, p . 300 , solution XXI .

•

L�t H · be the center of a -· ----�-�-�.--.-·-· -· • ...-.-,-rl-. =- --ci�c-l:'e·;-andaC- anei---aDtwo�aiameters-.-----�

," , · f� perp . to each other . - - Since HA = HB, I ' , ' , · ., ,' . ' we have the ·case -�rt_!qular, same I . ' . U ·' ' I ' �, • • as in fig . , under Geometric Solu-' • tions • . , · ' Pr.oof 1 . AB x BC = BH2 .A + AH X CH. :. AB2 = HB2 +. HA 2 • :.

Fig . 59

• HA ·x HC ) = BH2

h2 = a2 t b2 . · Pro.of· 2 ..- AB x BC = BD x BH

= (BH + HD ) X BH = BH2 + (HD X HB + AH2 • :. h 2 = a 2 + b 2 •

\

I I

I

I I I

" I i

. - ----1

. i

I I

64 THE PYTHAGOREAN PROPOSITION

a . These . two proofs .are from Math: .. Mo . , 1859, Vol . 2, No . 2, Dem. 20 and Dem. 21 , and are applies-. tiona· of Prop . JCXXI , Book IV,. Davies Legend:re, (1858 ), p . 119 ;. or Book III, p . ··173 , Exercise 7, Schuyler ' s Geom. , (1876 ) , or Book III, p . 165 ,· Prop . XXIII, Wentworth ' s -New Plan& · Geom. , (1895 ) .

b . But it does not follow· that being true when HA = �� it will b� true when HA > or < HB . The .author .

Fig. 6o

§.l!.!I:.!�2

At B erect a perp . to AB and > • prolong AH to C , and BH to · n . BH

= HQ. Now AB2 = AH x AC = AH (AH + HC ) = Ali� + (AH X HC = HB2 ·) =· 'AH2 + HB2 •

h2 = 2 + b2 . Q E·--D

----:. a • • • • a . See Versluys , p . 92 , fig .

105 .

. !l!!I:.!b.tl!.

1\ From the fi,gure it E .. .. .. . .. ;,�._ . i s· evident that AH x HD �� ,. ' , 2 . ' , / ' = HO . x HE, or b = (h + a )

/' ' (h - a ) = h 2 - a.2 • :. h 2

....--· --+---------- ----- - - - "/ -·-- · '·� - · · .c · ; \� = a" +_ bZ: s��:E��si_u_y_s�--------: \ -':..J- p_._· 92�, . . fig . 106 , and credit-

I . '

' ' 1 ed to Wm. W. Rupert , 1900 . ' \ I ' ' I ' ' / ', �,.,

.. ... -· '-' - - - - · -

...

. ,,

l L 'I I I t

. ....

• j

_..-; •

j '

Fig. 62

I T HA. = (b - a )/b � )( fb = ' (b - a ) .

ALGEBRAIC PROOF�-

With CB as radius describe semicircle BHA cut-­ting HL at. K and: AL at M . Arc BH = arc KM • . >. BN = NQ = AO = MR and 1m = KA..; also ' 0 arc BHK = arc AMR = MKH = 90 . So ·tri ' s BRK and KLA are cbn­grue_nt � . HK = HL - KL = HA - OA . Now HL : - KL = HA : OA . ' \

.So HL - XL : 1JL = HA - OA : HA, -or . � - KlJ ) + HL = (HA - OA )·

:. KQ = (HK -: NL )LP = [ (b - a ) + b]

Now tri . KLA = tri . HLA.- - tri . AHK = :!-b2 - tb x !-('� � � ) =, tba_ = t tri . ABH, or tri . ABH = tr.i . BKR ··+ tr;1 . KLA, whence , .trap .. LABR - tri. . ABH . . __ _ __ .... · ·-- �--

= · trap • . LABR -� ( tri . BKR + tri . KLA.) = trap . LABR ·

- (tl'i . HBR + tri . HAL ) = trap . LABR - tri . ABK. :. tri . ABK = t:r-i . ··HBR + tri . HAL; -or. 4 tri r ABK = 4 , tri.

- 2 2 2 ' . HBR + 4 tri . HAL.. :. h = a - + b •· Q.E . D . . a . See Versluys , p . 93, fig . 10�; and found

in Journal , de Mathe in,. :J-897, credited to Brand . (10/�3, ' 33, 9 p . m. E . ·s·. L . ) .

I '·

" ------- - -- ----------- -- - - � - r� ---- � - - --. · The construction is obvious. . I From the similar tri$ngles HD� [ and

HBO , we have HD : liB = AD . : OB, or HD · x CB = HB x AD . _:_ (1 )

_ In like manner, from the similar triangles DHB and AHC, :H:P · x AC = AH x DB . - - - (2 ) Adding (1 ) and (2 ) , ·HD -x AB -�-- HB x AD + AH

. ' ( • 2 2 2 1 x DB . - - - 3 ) . . . h = a + b •

a . See ' Ha�sted ' s Elementary Oeom. , 6th Ed ' n, i895 for Eq . (3 ) , p . 202; Edwards ' Geom. , . p . 158, fig . l7 ; Am. Math. Mo . , V . IV, p . 11 .

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66 THE PYTHAGOREAN PROPOSITION •

b . Its first appearance in print, it seems, was in Runkle-' s- Math . Mo . , 1859� and by Runkle cred­ited to C . M. Raub, of Allentown, Pa .

c . May not a different solution be obtained � ..... ' from other proportions· from these same triangles'?

"

�l!.!l:.�l!.

Ptolemy ' s Theorem (A . • D . 87-168 ) . If ABCD is any cyclic '(in­scribed ) qua¢r�lateral, _ then AD x BC +" AB X CD ' = AC X BD i

As appears· in Wentworth ' s Geometry, revised edltion (1895 ) , ·p . 176, Theorem 238. Draw DE making ' ' Fig . 64 LODE . = lADB . · Th�n the tri 1 s ABD and ODE are similar..; ... also the tri ' s BCD

and ADE a�e similar . From·. these pairs of si¢-lar tr�angles ·. it follows that AC x BD = AD x BC '+ DC x AB. (For full demonstration, see Teacher ' s Edition of Plane and Solid Geometry (1912 ) , by Geo . · ·Wentworth _and Dav+d E . Smith, p . l90, Proof 11 . )

In ca�e th� , quad • . ABC� be­comes . a rectangle then AC = BD, BC ' 2 2 = AD and AB = CD . So AC = BO t 2 � 2 2 2 • + AD , or c = a + b •

· o • • a special

case . Qf Ptolemy ' s Theorem gives a ,' proof of the Pyth . Theorem.

, ' I ' - a . As - formulated by the _ : . " .. ' -- - -- --- - - - -F.J:g .•. -6�---------a-uthor-.---A-1-so- --see- --A-Gompan±on-·to- -- -

� -- � -. i . --

' • . Elementary School Mathematics (1924 ) � -. by F . C . Boon, B .A . , p . 107, proof 10 .

\ \,.

.. ._ _____ -----'-·---

Circumscribe aqout tri . ABH circ.;Le � . Draw AD = · nB . Join HD . Drliw_ CG perp_. to HD at H, and AC and :Bd each perp . to 'CG; als.o AE and BF perp . to HD.

Qu�d ' s CE and FAG are squar-e.s... ·TI,'i- ' ·�·. HDE and

-- - ---t

1 - 1

-·� -=-�

- .

� \' l I

---- ----------'-'---- - -� - - - - - -

<·

! . � . { � i ! � -I

"( -

- I l1g • . 66

b • . See

- --

- - --- - - - --- �--------

ALGBBRAIC PROOFS "' �.. . 67

DBF are congruent . - :. AE = DF = EH • AC . BD • HIP + P'll = BG + AC • Quad . ADBH • tHD (BP' + AE ) -= iHD x CG . Quad. ABOO • j- (AO ·-+ ·BG ) · � OG = tHD x CG� :. trl . ADB = tri . AHC + tri .

-HBG. :. 4 tri . ADB = 4 tri . AHC + 4 tri . HBG! :. h2 • -�2 + b2 • Q .E . D .

a . Se� E . Fourrey ' s C . Geom,, 1907; credit�4 to Pit9n-Bre�sant ; -see. Versluys, p . 90,- fig. l�3 .

fig . 333 tor Geom. Proof--s_o-oalle.d .

lll1l:l!lb.1

Oonstruotioh same as· in fig. 66, for points· C , D and G. Join _DO. Prom H draw HE perp . ' to AB � and join EG and ED . From G ��-rt--¥¥'f.i·:.::.:..·· -�- -=--- draw ox· pe-ri>� to HE-anCf(].p--peFp :

J'ig. 67

to AB, and ·exte nd .AR . to F . KF is a -sq�e, :with diag. GE. . .. •• angle ' 0 . ·BEG • angle EBD • 45 • :. GE . and BD are parallel � Tri . BOG = tri . BDB . --� (1 ) Tri . BGH -= tri . BGD. --- (2 ) :. · (1 ) • (2 ) , or tri . BGH ·

• tri . BDE . - Also tri . BOA • tri . ADE • :. tri . - BGH + tri . HOA • tri . :' ADB . So 4 tri . - ADB • 4 tri . BHG + ·4 tri_. -BOA. :. h2 • a� + b2 • Q.B . D·.

. -

. a . See VerslU781 p . 91 , fig. 104 , and credit�

ed also .to Pi�on.:.Bresaant; as_ · round 1n B . Fourrey •·s Geom. , 1907, p ; 79, IX .

b' . -see fig·. 3�4 -of .Geom. Proofs . "

•

-+--------------------����dl�tn�.�- -------

In fig. 63 above '-·t is obvious th�t AB · x BH • AH )( DB + AD . x BH . · :. AB2 • HA2 _+ 'BB_2_. - �_.: •. h2 =!= a2

a· --

. � - - - - . .

+ b . a . S�e Math. Mo .· , 1859, by Rurikle, Vol . II,

No . -2, Dem. 22.,· fig. 11 .

1

......

I I 1-1

-, . I 1 -

: 0

l :.

--'

•

-·------------ ----- --

' . . .

' ,,

' '

:!>

' . . ' .

68 THE PYTHAGO� PROPOSITION ' ' '

b . This is a particular case of Prop . XXXIII , Book IV, p • . 121 ; Davies Legendre (1858 ) which is. Ex­ercise 10 , in Sc;huyler ' _s Geo�. · (1876 ) , Book III, p . 113 , or ExerciAe 238, Wentworth ' s New Plane Geom . (1895 )., Book III--, p . 176 .

-

r" - ... .. ... C1 • ... -· ·( . . .. ,�E

' \ , .. ,,

, ' , ' ' , , �

. . ' '

• • • ' I ' • • • - , . · - -

,A'.--��� . ' ��� ., '� - ... -, , '

On. any diameter as · AE = aAH, const . rt . tri . ABH, and ·prod\lce the side i-to chords . Draw ED • . From the sim. tri ' s ABH and _AED , AB : AE = AH : AD, or h : b-· + HE = b : h + BD . :. h (h + BD ) =-b

'(b + HE ·=· 1;>2 + b X HE = b2 + HF X HC = b2 + HC2 •

. "'! - - (1 ) . Now c onceiv� AD to ·re-· volve on A . as a .center until D . coincides with q , when AB = AD = AC = h, BD = 0 , and � = HC

= a • . . Substi cuting --in (1 ) · we have h2 = -a2 + b2 •

. a . This . . i s the solution . of G. I . Hopkins of Manches·ter, ,N-�H . . · 1See his . Plane Geom. , p ; 9�, ·ar-t . "427 ; also see Jol:Ir ." �f' :Ed.� :: 1lS88, v. xXvti, '. p . 327 , 16th prob . Also Heath ' s Math. Mo�og�aphs , No . 2, p ; 28,. proof XV •

. · b . Special case . . When H coincides with 0 we get (1 ) BC· = ·(b + c ) (b - a.)/h, and0 (2 ) BC '= 2b2 /h - h·. ' . 2 2 2 Equating, .�. h· = a + b •

c . Se·e Am. Math . Mo·. , V . III , p . 3.00 .

(2 ) The Method by Secants . ' ..

�!Y.ll!.!I:Jll!.!

. ' With H a� center �nd HB as �ad�us des�ribe ·

the circle EBD . 0

The secants and their external segment s bring :rec�prooally proportional, · .we have , AD : AB = AF : AE,

0

()

'

I \ \

. I

,,

•

·"·

- I I . j I . j '-1

' � - - - - - --- - ------ ---- - �- -- - - - - -

..

' . ALGBaRAIO PROOFS·

I . 2� Ol' b + a : h = (h - 2CB ==-h - h)

· .... · · · · - : b - a, whence 'h2 = �2 + b'2 • , -- . -· ·�· =....

, � 0 . a . In case b = �, " the . ! , "" \ points A, E and F coincide and the ; . ! proot: still hoids .; for sub.sti tuting \ : . r b for a the-abov� pro� ' n reduces to ... ha · 2 a -- a a � i . - a = 0 ; .

·. h = 2a as it

1 shoU,ld. Fig . ··�69 b • B7 Joining E and B , and.

F and n·, the similar triangles upon which the above rests are �ormed. 0

• • • • .• Wi t}1., H as .center and HB as 0 • • •

• •• · ·,')) radius �ascribe circfe �D, and .. draw

,' _ , ' • · . � an� HO to middle of BB . '

: • ILE · )( AB = AF � .AD,. or · , -� f · (AD · � 2BO }AB. = (AH ... : HB ) (AH -+ HB } .

- 4. • �' · �·; ·.AB�- '"'i ·2BO· ·)(· :AB = .:AHa - HB2·� And ,. .__ • l •

as BC : JH • �H : AB, then BC � AB . .

• HB2 , or 2Bc )( · AB • 2BH2 • . .So. �·a

Fig. 7o - 2BH2 • AH• -.-.·BH2 • · :. ABa = HBa · · a· · · a 2 a ·

· . + HA, .. :. h = · a +. u • Q . E . D . a . Math. Mo . _, Vol . II·, No � 2 , Dem. ·,25, fig . , 2. >t

Deri·ved. from: .. Prop . XXIX, Book IV, 'p . · 118, Davies Legendre (l�58.) ; Pro:p .• �III , Book· Iri , p . 171 , �chu7,l�r • s · .aeometrr , · (l876 ) ; P·rop . XXI, Book III, p . 163 , Wentworth.' s . New Plane Geom. (1895 ) . ·

' • .... fl. � •'

• ' c

• 'i/ I ,

AE : AH' = AH : AD • :. ;

� AHa - AE )( AD .. AB (AB )( BH ) ' . a -,------ - ·=·--AE·--·)(· --AB- + A! x mi. So AI! � � - 2 '

"���-� - -· - - .,u + BH = AE )( AB · + ·AE )( HB � \ '7J ! . + HB 2 = AE )( AB + HB (AE + BH )

'·· ./ = AB (AE + BH ) • ABa . .:� .ha · · · � . • -a 4 a '

· • • •• � = a . + b • Q . E • D-. ·

Fig. 71 �� · See Math • .. Mo ; .. , . \

t

' t I i

. I

. l I · l '

0

"'-------- - - - - - - - --- -- - - · ·

'

. ·

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----·----

------- - - - - ----

�

--

• • f. •

10 THE· PYTHAGOREAN PROPOSITION

(1859 ) ; Vol .- II, No . 2, Dem. 26, p • . 13; derived ·rx-om . . .

Prop . XXX, p . 119, pavis Legendre ; Schuyler ' s �Geom. , Book III , Prop . XXXII, Cor . p . 172 (1876 ) ; Went- �

worth' s Geom.. , B ook III,- ·Prop . XXII, p . 1'64 . It. is· credited to· C . J. Xemper, Harrisonburg, V$ . , and Pro� Charles A-. Young (1859 )., at Hudson; 0 . Also to\llfd in Pourrey' s collection, p . 93, �s given by J . J . I . · Hotf'marin, 1821 • .

1'18. 72

llllll!l:f21t

In fig . 72, E vil.l tall· be­t_vee�.A-.and F at P, or between P and .�, as BB is less than, equal to, or greater than. HE . Hence there are three cases ; but. inv�stigation ot one �ase--when it ralls at middle ·point or- AB--is sufficient . - ,

. J_Qin L and B , an� P and C , ' making the two stmilar triangles

· APO and ALB; whence h .: b + a • b-. b2 2

- a : .A.P ; :. Ail • . - a -·:.._ (1 ) -. h

· : - J'qin � and 0, and B _-and: D making the two sim-- ilar tri ' s PGB and BDB, whence th : a - th � � + th -

I - • 0 a a - th 2 . , 'c )

'' ( ) : PB, whe�ce PB • ih

, --- 2 • Adding 1 - and

( - - 1 a2 + b·2 -- !h21 . - a 2 . 2 2 ) gives 2'h • h f'rhence . h • a + b . -a o The- &b0V8. -S01Ut·ion .t·� gi Veil by ·Jtrueger 1

in "Aumerkungen uber Hrn. geh. R . Wolf ' s Auszug aus der .Oeometrie , " 1746.. Also see Jury Wipper, p . 41 , r�g. 42, and Am. Math. Mo·� , v . IV, p . 11 •

b ; When 0 falls . in1:dwa7 between P and B , then �

are ·closely rel�ed. ·-�

\. . I

I

0

I l

· · - - --- -- 1 . - ---------

r

' .

,.

• I '

-

- -- -----· �1 ---- - � -------- ·--- ------- --- ---- ----------............... �--... - ---------·- - ·�----� -- ----- ---- --------·- - - -- - - ----------

•

, .

... .

..

-, ,

Cl .

, .

71g. 73a

; r '

"-- ����...--.. -- - _.,

71 .

. l!�l!l!f:.fl�!

. _ _In �fig . 73a, . take HP

• BB . With B as center, and BF as radius describe �emi­circ�e DBO, o being· the pt. where the circie in�er�ects AB . Produce AB. ·to D , and dra� PO, P'B, BE to· AH pro­duced, and DB, forming the similar tri ' s AOP and ABD, from 'which (/.0 • x ) · : . (AF • y) = (AE • "'1 + �PH,) : (AD • x __ _

+ 2BO ) � . "'f" _+ 2z : x + 2r whence x2 + 2rx • 72 + 2yz . - - - (1 ) • .

But if·, see figQ. 73b, HA··· HB ,. (sq: OE = h2 ) -. (.q . ;HB • a2 ) + (4·- tri . AHO ·= sq .

HA • b2 ) , whence h2 Ill! a2 + b2; then, '(see fig . 73 a ) when· BP • BG, we vil_l have B(J2 · .• Hs2 · + HP2, or r2

--"=-==+-'---'·�-;.�� .z 2 . + z 2·, . ·(-�e-sr· -- PH); �--fe-r• - -· ·:"�="""�=-;:.- : - - --

,.

0

I i .. I

.

· (1 ) + (2 ) • (3 ) x2 + 2rx + r2 • 12 - - + 2Iz .+ .z·2 _ . + z2 or (4!). (x + r}2, � ("'1 + z ')2 + z2 • :. (5-) �- • a2 + b·a, since x + r · = AB • h, "'1 + ... z • AH • b, ·and z • BB • a� .

. . a . See-· Jury W:ipper., p . 36, where Wipper also credit-s · it to Joh. Hoffmann . See al$0 Wipper, p . 31 , tig . 34, f�r another- statement ot s�e proof; and Po�re"'f; p . 94·'- tor Rottmann ·-�·- ·pr"Oof .

llltD.!I:.l-J;*'! --�

In tig. 74 - in the circle whose center is Q� and· WhC?Se diamet·er is ·.e�� - erect t�e perp . DO, joirf D· to A and B, produc� - DA to P; making AP � AH, and pro-duC?e' JiB--to G Mlcing- BO � BD, -thus· torua!pg the �-t_�-� _ _ _ _

isosce�es tri ' a P.RA and DGB; also the two isosceles ·

tri ' � .ARI) and � . As -angle 'DAR • 2· angle at P, and �le HBD ·� ·2 angle at G, and as angl_e DAB and_ angle·

. -. - �

I

'I

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�

i I

_..;:; .. � _ _ .., __

1-------

---

·-·--

--·

--

-

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-

-

-

-

�:-:·-.·

;,... . -· ··-· •' " - --·

r -- -

j '

r

•

72

-

-

-

--

-

-

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-

�

-

--

-

-

-

-

-

-

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--

�

-

--

-

-

--

\ \

..

THE PYTHAGOREAN PROPOSITION

HBD are measured by same arc � .. H S HD , then angle a_t · F = angle.

. , ... ·.�� at G . :. arc AP = arc QB . · (/ ; . '-��� , And as angles ·ADR �,:,; 0 -·�;� and BHS have same measure , t

. --��'\ · /'�� . . of arNPQ, and i of arc BQP , t_,r " .. " � .. - - .... ·

\.(1 respectively, then tri ' s ARD ' . and BHS ar� similar, R is the

Fig. 74 . intersection of AH and DG, .

and S the intersection of BD and HF . Now since tri ' s FSD and GHR · are . similar, be� ing equianSular, we have , DS <: DF = HR : HG. :. DS : (DA + AF ) = .HR : (HB + BG ) .

. :. DS : (DA +, AH ) = HR : (HB + BD ) I :. DS : (2BR + RH ) = HR : (2BS J SD ) , .

:. (1 ) DS2 + 2DS X BS = HR2 + 2HR X BR . A�(2 ) HA2 = (HR + RA )2· =· HR2 + 2HR X RA. + "RA2 = HR2 ' + 2HR X RA" + AD2

. .

. '

(3 ) HB2 -= BS2 = (BD - DS.) 2 ·= . BD2 - 2BD x DS + DS2 - . = AD2 - ·( 2Bl? x DS - DS 2 ) = AD2 - 2 (BS q + · SD )DS + DS2 = AD2 - 2Bs x sn - 2ns2 + ns2 = AD2 - 2BS x ns - _DS2 � AD2 - (2BS X. DS - DS2 )

(2 ) + (3J = (4 ) HB2 + HA2 = 2AD2 • But as in proof, fig . 73b , we foUnd, {eq . 2 ) , r2 · = z2 + z2 = 2z2 • :. 2AD2 (in fig . 74 ) - = AB2 • :. h2 = a2 + b2 •

a • . See Jury Wipper, p . · 44 , fig • . _43 , · and there · ·credited to Joh •. Hoff'manrt,_ one of his 32 solutions . ·

In fig . 151 let BOA be_ any tri�ngle , and let AD, BE and CF oe the three perpendiculars from the. -thre� verti.cle s , A, B .and C , . to the. three l:lides , BC� .

CA and AB', respectivel7. Upon AB, BC �nd CA as diam­eters describe circumferences , and since the angles . I

- ·--- - - - - - - ADO , - �EC and CFA are rt . angles , the circumferences

I - {

pass through t�e points D and E , F and ·E , and F and D, . re�pectively.

. . -- -�----1 -

I ! � '

.;

' . . �

I • •

. I I J,

1 I

ALOiBRAIC PROOFS

•

. . • .

Fig . 75a

f'

\

73

Fig . 75b

Sirice BC x BD = BA x· BF, CB x CD = · CA x CE , and AB x AF = AC .x AE, therefore

' ' '

[BC X BD. + CB X CD = BC{BD + CD ) = BC2 ] . = [BA X BF' + CA X CE = . BA 2 + AB X . AF + CA 2 + AC x, AE

=· AB2 + AC2 + 2AB x AF (or ·2AC x. AE ) ]-: When the · angl� A is . acute (fig . 75a ) or obtuse (fig . 75b ) the sign i' � or + respectively . And aa angle A approaches 90° , AF and AE ·approach o , · and at 90° they become 0 , rand we have BC2 = AB2 · + AC2 •· : •. whe-n A • a rt . angle h2 • a2 + b2 •

a . See - Olney' s Elements of Geometry, Univers­itY' Edition, ?a:rt III, p; 252, art . 671, ·and Heath ' s Math: Monographs , No . 2, p . 35, proof XXIV .

lt�lD.1l:.&lib.1

� -- - ·Pro·duo·e· ·-xc- and HA to M, complete the rec t • MB , �a11 BF p�r . _to ·

L AM, and . draw ON and AP r- • • • • • • • • • • perp to HM I . ..:,. I ' • • ,

1 �:,;j '

'. Draw the .semi-

l , _;� : ,� J \.. ciiCle � on th: diame-! . , fl. .• -� 1 , JJ � : ter AC . Let � x . a , , ', 'lit 'f · 1 Since the area of \�he � - .. - - - - .. t' :.;.ll.. -· - - - • • .J· · paral . MFBA • the area of ' . . c r I the s'q . AX� and since, by

Fig . 76 the Theorem for the

, . .

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c , .

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.

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� · ·----� --; - - � · -·· . , .. -· - · · .> 4 -� , . �·· '

I \

74 THE PYTHAGOREAN PROPOSITION

-:=---me-asiirement C1f a parallelogram, (see fig . 3...08, this text), we have· (1 ) sq. AX = (BF x AP • AM x AP ) = a (� + x ) . ·But , in tri . MCA, CN is a mean propor-tions:!. be�v€Jen AN and NM. :. {2 ) b2 -=- ax . (1 ) - . ·(2 ) • ' (3 ) 'h2 - b�- •· a2 '+ . � :.. ax = ·a2 • :. , h2 = a2 · + -b2 •

.. ·Q .E .D . a . This· proof is· No • . 99 of A . R . Colburn ' s

·· i08 ·solutions , being devised Nov .. 1 , 1922 � I

(3 ) The Met·hod- ·by Tangents 1 s t . -- 1he BuDo t enuse a� a Tan� en t

lltta.1r.:.llril

Draw HC perp . to .AB, and with H as· a center. and HC as a radi-- . . us describ� circle �EF �

From the siDJilar· tri 1 sr ACG . and AEC, AC : AE = AG : AC , or AC · : . b + r • b . - r : AC ; :. (1 ) AC2 • b2 - r' • . . From the similar �ri ' s ·

• I

CBD and BPC, we get·-f2-)�-o��-82- r2.--.--... , _ _ _ _

Jig. 77 From the �similar rt.. tri I s BCH and . . HC�, ve g�t (3J BC x AC = r2 • .

:. (4 ) 2BC x .AC • 2r� . .(1 J + (2 ) + (4 ) <:�gives (5) AC2 �+ ·2AC x .BC q+ BC2 • a2 + b = (AC + BC ) 2 = AB2 • :. h2 = ·a2 + b2 . I

'

a . See Am.-. 'Math. Mo . , V . III, p . 300 .

0 , the center of the circle , l�es on the bisector of angle B, and on' AH. �

With the construQtion com-A pleted, from the similar tri ' s ACD

and AHC, ve, get, calling OC = r , 718. 78 . -(AC • · _ h - � ) :-�(1JI • b ) �- .(AD • b - 2r J

· :. (AC � h - a ) • :. (1 ) (h ... a ) 2 • b - 2br . But (2J a� ·- a2 • (1 ) + (2 ) • (3 ) (h - a )2 + a2 • .a2 + b2 2br, or (h - a )2 + ' 2br + .:a2 = �2� b2 •

...

'

.- .-

- t

•

"

' � . '

I I j - ! l J ! l l

I

I I -l I

I �.

·--- - -- ----�-�--- -�-- --------- .---------.--. ......... . � 75

Al·so (AC = h -' a ) : (AH = b ) = ( OC = OH �. r ) : (!IB : a ) , whence

(4 ) (h - a )a = br . .. ·. (5 ) . (h a )2 + 2 (h - a )r + a2 = a2 + b2 :. (6 ) h2 :;:: a2 + b2 • Or, i� (3 ) .above , expand and factor gives

(7 ) h2 - 2a (h - 'a ) = a2 + b2 - 2br . Sub . for a (h - a ) its equal, __ �ee (4 ) above , and collect, we have ·-

(8 ) h2 � a2 + b-2-.-- - ------ -� 1 � -- - - --

a . See Am . Math . ' ;Mo . , ,V . IV, p. 81 .

Fig. 79

2nd. -- 1he _ Byo o t en u s e a Se can t �h t c h P as s­e s Th ro u l h t h e Ce n t e r o f t h e C t rc l e and O n e o r Bo t h L e� s Tan � en t s

• • <to--.

Having HB , the - shorter leg, · a tangent at 0 , �ny convenient p t . on HB, the construction ' i s evident . _

Fro� the similar tri ' s BCE and BDC , we get BC :. BD = BE : BO , whence BC2 = BD x BE' = (BO + OD )BE = (BO + 00 )BE � --- (1 ) From �imilar tri ' s OBC and ABH, . _we get OB ·: AB

OB r = OC .: AH·, whence h. = Q, ; :. BO

- .h! '(. 2 ) BC : BH = OC - b • - - - : AH, whence BC �· � . ---(3 ) b �ubstituting (2 ) and (3 ) in (1 ) , gives , a:��� = (� + r )BE = (hr � br ) (Bo - oc ) = (hr � br ) (hr + br ) (4 ) 2 2 2 b .

. • --� . whence h = a + b • Q.E . D . a . �pecial case i s : , when, in Fi� . 79, 0 co-

incides · with A, as in Fig . 80 . "

... .. v

,.

r '

' j , , } i 1 -,

_.___ __ , __ � - - - - -- - -

•

-----�-- -- --- - - - - -- - - ------ � -- �-� �- - - - � -------------'---------,; 76

I I •

, ,

THE PYTHAGOREAN PROPOSITION

Wt�h A as center and �AH as radius, describe the semicircle BHD .

• •

From the similar tri­angles BHC and BDH, we get, .

· h - b· : a = a : h + b, whence _ directly h2 = a2 + b2 •

a . This o case is found .. ixf:.,_;;.;;"";Hei"tlr•:s Math . Monographs, No . 1/ p. 22 , px-o_of VII.; Hop­kins ' Plane Geam. , p . 92 , fig . IX; Journal of Education,

1887 , , V. XXVI, p-. _ 21, fig . VIII ; Am; Math . Mo . , V . III, p . Q229; Jury Wipper, 18.80, · p . 39;, fig . 39, where he says it is foun� _ in Hub�rt ' s Elements of Algebra, Wurceb, 1792, also in Wipper, p . 40 , f�g . 4�� as one _of - ·Joh. Hof:tmann', s 32 proofs . Also by- Richardson in Runkle ' s Mathematical (Jom-nal ) Monthly� No . 11 , 1859·- ---one of Rich�rd�on' s 28 proofs ; Versluys , p . 89, � fig . 99 .

-·

b . llany persons , independent of above sources , hav� found this proof .

·

c . � When 0, �n fig . 80-,· is the middle pt . of AB, it becomes a spacial ca�e 9t fig. 79 .

' �. ���-.. - -.. . . - .

--·

- As-sume HB < HA, and eJDPloy tang . HC and 13e?ant �� �henc.e --HC2 = HE )( :HD = AD X AE = AG ')( AF = BF . x BG

·= BC 2 • Now employing like argu­

ment as in proof Eighty-One we g�t h2 = a2 + b2 . -

a . When 0 is the mid4le Fig. 81 _ point ot" AB, and HB = HA,, then HB

and HA are tangents , and AG = BF, secants , the arg�ent is same ' as (c ) , p�oof Eightz•. !!£, by applying theory of limit s .

· b . When 0 is any pt . in AB, and the two le�s i 1..

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---------77---�-------- --------------

------------------------------------------------�

are tangents . This is only another form of fig . 79 above , the general case . But as the general case -give s , see proof" case above , h2 = a2 + b2 , therefore the special 11u s t be true , whence in this case (c )

-h2 = a2 + b2 • Or if a p roo f by explicit argument is desired, proceed as in fig . 79 .

�By prov�ng the general case , as in fig . 79 , and then showing that

, some case i:s only a p-articular of the , ,,. ' J- general·, and therefore true immedi-

A�rJ�&:....,..�..,.'D · ately, is here' contrasted with t-he \

' " ·"

.. .. _ . .

Fig . 82

p . 80 :

following long. �nd ·complex solution ,_ . of the assumed _partic-ular case •

w,he following solution is gi1l.en +n The Am. Ma. th . Mo • , V . TV,

"Draw OD perp . to . AB . Then, AT2 = AE x AF- = A02 - E02 = A02 - TH2 . - - - (l ) . .

BP2 = BF x · BE = B02 - F02 = B02 - HP2 . - � • (2 ) Now, AO : -OT = AD : OD ; :. AO X OD =. OT X AD • And, since OD = OB , OT = TH = HP, and AD = AT + TD

= AT + BP •

:. AT X 'TH + HP .. x BP = AO X OB • --- (3 ) Adding (1 ) , (2 ) , and 2 x (3 ) ,-AT2 + BP2 + 2AT X TH + 2HP X _BP = A02 -

· -- '- . :. - . ' - HP 2 + 2AO X OB ; :: AT2 + 2AT X TH + 'FH2. . :+ ·BP�- - + .... 2BP X HP + HP 2 = A02

+ 2AO X OB + B02 • . ------ - ---- --:. (AT + TH )2 + (BP + CP )2 = (AO + OB )2 • ;·. AH2 -+ BH2 = AB2 • ·n Q .E . D .- �

.: . . h2 = a2 + b2 .

3rd. - - Th e Hvpo t en u s e a Se can t No t P as s­t n� Th ro u ' h t h e Cen te � o f th e C t r­c l e , and Bo t h L e� s Tan, e n t s

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78 THE PYTHAGOREAN PROPOSITION

fl1�1l:.flY.l

_ Through B draw BC parallel to HA, making BC. • 2BH; with- o , the middle point ot BC·, as center 1 de­scr�be a circumference; tangent at B and E , and draw CD; forming the two simil�r rt . tri ' s ABH and BDC , whence BD : . (AJI • b ) _ -=- (BC • 2a ) : (AB -= h ) from w��h, DB =

2:b. (1)

Now, b.y---the p:t':f.ncipal of t�. and sec . relations , (AB2 • [b - a f ) • (AB • h)- (AD • h - DB ) , .whence

· (b - a )2 · ( L- · ·

DB • h - . •· --- 2 )\ ' h ) ,.

�quating (1� �d - (2 ) give� h2 = a2 · + b2 • . .

a . If the legs HS and HA are: equal, by theory of limits same- result obtains . ·

b . S�e Am . Math . Mo . , V . IV, p . Si,. NQ. XXXII. - -c . See proof .Pii'.tr-'l'wo �bove·, ·and observe

that this proof 116htz-P-ive is s�p�ri·or. to it .

. .

4t h . --Hupo tenuse and Bo t� Lela ronten�a

' ' The ta�ent poi�ts of the three sides are C , D and· E .

Let OD = ·r • OE = OC, AB = h, BH = a and AH = b .

I (1 ) h + 2r = a + b .

Fig . 84 (2 ) h2 + 4hr + 4r2 = a2 + �a'b = b2 • (3 )' ·Now if' 4hr + L 2 � 2ab ; then

� 2 2 a-� ·; 1. -· h = a · +r -b , � · , · ;-- � · -:- ·

(4 .) Suppose 4hr +.4r2' = 2ab . ·

·

·',- J (5 ) 4rlh + r ) - . 2ab.; :.' 2r (h · + r ) =, tab ,

" n

(1 ) = (6 ) 2r • a + b -- h • . . (6 ) in (5 ) gives (7 .> (a + b � h ). (h + r ) • ab • �

I � I t

. .

..

- - -..

y • '

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I

I

---ii - -----�------� -- ---�-- ----------- ----�� · -�-��----;1 ALGEBRAIC -: ROOFS 79

I (8 ) h (a + b - · h -� -r ) + ar + br. = ab . , (1 ) = · (9 ) r = (a + b - h - r ) . (9 ) �n (8 ) gives

(10 ) hr + ar + br = ab . - - ·

(11 ) But hr + ar + br = 2 area tri . ABC . (12 ) And ab = 2 area tri . ABC . : . . (13 ) -hi'-� -ar--+--br = -ab - hn , + r (a +- b ) = hr

+ r (h + 2r ) :. -(14 ) 4hr + 4r2 = 2ab . _ . · .

:. the · S\lpposi tion in · (4_ )' is true . :. (15 ) h2 = a2 + b2 • Q . E .D .

a . This solution was devised by the author De� . 13, 1901, before· reQeiving Vol-. VIII , 1901, p . 258, Am . Math. Mo . , �ere a �ike solution i s given; also. see Fourrey, p . 94, where , cr�dited.

\ b . 'By drawing a line OC , �n .fig . 84, we. bave

the geom_.. .fig . .from which, May:, 1891 , Dr . L . A'. Bauer, o.f Carnegie Institute , W&.sh . , D . C . , deduce.d a· proof

· through the equations · -(1 ) Area o.f tri ABH· = ir (h + a +- p ) , and (2') HD + HE = a + b - h·. S�_e pamphlet : On

Rational Right-Angied Tr�les , A\lg . , '1912, by Artemu� Martin for the' Bauer pr9of . In same pp�phlet, is stiil another· proof attributed to Lucius Brown .o.f Hudson, Mass . _ . .

> c . See Olney' s Elements ot Geometry, Universi­. . ty ;Edition, p . 312 , .art . 971, or Schuyler ' s Elements .of Geometry, p .. 353, exercise 4.; also Am_ • . Matll. -Mo . ·,

- V . rv; p . �� proof XXVI; also Versluys , p . 90, fig . •102 ; also drunert ' s Archiv . der Mathein,,._and .Physik, 1851, credited to Mtlllmann. �

d. _Re•a rk. --By ingenious deyices , some if not

.,.

c..,J

I , . l I

.______,�-----�- - f--a-1-l-,-o.f-the-se--:!:n-wh1:ch- -t-he-c.t-re-le.-.has.-b..een employed . - can be. proved without · the use of the circl;::iiot _______ -----�---:----1----____ :__J i nearly so e�si�y perhaps , but p�oved . The figure , r without · the cir.cle, would suggest the �evice to be I employed. By so, d'oing new proofs· may be discovered . l• - \ I

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eo THE PYTHAPOREAN PROPOSITION

.. �!ab.1I:.lt�t!l

H Complete rect . HG. Produce '

l) .., DO to P and EO to X. Designate AC · ·

= AE by p, BD = BC QY' q and HE = HD A , ' 1 ·;) by: ' r . .

· . · j · ;;: \ . ', --�'!!hen a • q + r., --b-= p ... + r, ,.. " . .

F\ , ' and .h = . p + q. Tri. -FMA = _tri . OMC ' &v and tri . COL = tri . XLB . l"ig . 85 . :. tri�- AGB = rec t . P(UtO

= tri . ABH • i rect . HG. Rect . FGKO = rect . �oB--+· sq . ED + rect . OiCBD .

So pq = pr + r2 + qr . whence �pq = t!qr + 2r2 + 2pr . But p2 + a2 = P�- + q2 • . :. p'2 + 2pq + q2 ;a (q2 + 2qr + r2 ) + (p2 + 2pr + �2 )

or (p .. + q )2 = ( q + r ) 2 . + (p + r ) 2 - � h2 � a2 + b2�

a . Sent to me by J. Adams, f�om ·The Hague, and credited to J. F . -V ae s , XIi

_I 1 4 (i917 ) . .

,.

(II ) . -�Throujh the Use of Tvo Circle� . . .

l"tg. · 86

�!ab.1I:.[lab.1

. Qonstr.uction . · Upon the legs of the rt L tri . ABH, as diam- · eters 1 construct circles and draw HC 1 forming t�ee similar rt ._ tri ' s ABH1 HBC and HAC . Whence h : b = b .: AC . :. hAC

= b2 . --..; (l ) Also h: a. = a : BC • :. hBC

= a2 • - -- (2 ) (1 ) + (2 ) =

' a . Another form is : (1 ) JiA. 2 :: HC x AB • (2 ) BH2 = · BC .x AB • -- -: - A�ding, (3 � AH2 +. BH2 = AC x AB + BC _x AB . • AB (AC :._ BC ) • AB • :. h 2 = a 2 f. b 2 •

"

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------------+--------------------------- ------ --

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/ - -

ALGEBRAIC PRO,OFS

b . �ee Edwards ' Elements of Geom. , p . 161 , f-ig .. 34 · ·and Am·. Math.. Mo . , V . . IV, p . 11 ; Math . Mo .

81

(l859 ) , Vol . II, No . 2, Dem.· ·27, fig . 13 ; Dav-ies Legendre , · 1858, Book IV, Prop . ' XXX, p . 119; Schuyler's Geom. (1876 ) , �ook III, Prop � XXXIII, co� . , p . 172 ; Wentwort� ' s New Plane Geom. (1895 ) , Bopk III , · Prop . XX�I, p . 164, from e·ach of said Pfop�ositio�s, the· above proof Eighty-Eight �may be derived. · .

'

/ . Fig • . 87

�! ib.1t:lllll . '

With the legs .

-, of the ·rt . tri·. ABH as ' . .

radii describe circum-

an<;l HDH, AF : AH · =" Ali .AD ••• b2 = AF � .Ab . --- (1 )

F�om the similar tri 1 s CHB and HEB, .,� _ _,

. CB : HB = HB : BE . :. a2 = CB x BE . --- (2 ) (1 ) + (2 ) = (3 ) a2 + b2 = c� x BE. + � x AD

= (h + b ·) (h - b ) + (h + a ) (h - a ) = . ha - ba + ha - a·a ;

: . .(4 ) 2h2 = �a2 + 2b2 • :.· h� = a2 + b2 • a . Am. Matli. Mo . , V . IV, p . 12 ; also, on p . 12

is a proof by Richardson. But �t is much more dif: ficul t than the above method ..

llD.J!x

For proof. Ninety us·e fig . 87 • •

AH2 = AD (AB + BH ) . � = - (1 J · BH2 = BE (BA + AH ) . - � - (2 ) (1 ) +' (2 ) = ·(3 ) BH2 + AH = BH (BA + AH ) + .Ap (AB + BH ) ' = BH X BA· + BE x. AH + AD X HB + AD X BH a: ·HB (BE + AD ) + AD X BH + BE X AH + BE X AB . - BE X AB

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82 THE -PYTHAGOREAN.·PROPOSITION •'

,, � - I ,I J

= AB (BE + AD ) + AD x BH + BE (AH + AB ) - ·j BE x AB = AB (BE +. 'AD) + AD · X BH + BE (AH + AE + BE) - BE X ' AB'· = AB (BE +. AD ) + AD-...Jt BH + BE (BE + 2AH ) - BE X AB � :A:B (BE- -ft. fD ) + AD' X BH + DE2 + 2BE X . AH - BE X AB

' = AB (BE + AD ) +_,AD, X BH + BE2 + 2BE."' X AE - BE (AD + 'BD ) = � (BE + An ) + .AD X BH + BE2 + 2B� X AE,_ - BE X AD

J BE x BD = AB (BE +. AD ). + AD x BH + BE (BE + 2AE ) - BE (AD + BD ) = AB (BE o+ AD ) + AD x BH + BE (AB + AH ) � BE (AD '+ BD') =· AB (BE + AD ) + AD ·�"�-:BH + (BE x BC· = BH2 = BD2 )

- BE (AD + BD ) : . t = AB (BE + AD ) + (AD + BD ) (BD - BE.) = ·AB (BE + AD ) +. -AB x DE =. AB (BE + AD + DE ) = AB X AB = AB2 • :. h2 = a2 + b2 • · Q .. E .D .

a . See Math. Mo-. (1859 ) , Vql . +I , �9-·" 2, Dem. 28., f�g . _ 13--derived trom Prop . , XXX , Bi>ok �� p . 119, Davies Lege11dr.e , 1858; also Am. -Math. Mo . , Vol . IV, .

p ' .

p . 12, proof XXV.

' .

For proof Ninety-One use .fig . 87 . This proQf ls .known as the "Harmonic Proportion Proof . "

From the ' similar tri.' s AHF an4 ADH, . .

AH : AD = AF : AH1 .or AC : AD. = AF : AE whence AC + AP : . AF + AE = AD : AE or ·CD : · OF = AD : ·AE,

. and AC - :AD = AF - AE = AD. : AE, ' Ol,'

, DE : EF = AD : AE • :. OD : CF .� .DC : EF .

or (h + b a ) : (h + b +. a ) � (a - h. + b ) : (a + h + b ) :. by expanding and collect;1ng, we get - - -

h2 = a2. + b2 . ·a . See Olney ' s El�ments of Geom. , University

Ed ' n, p .� 312 , art . 971, or Schuy�er ' s Elements· ot Geom. , p . 353 , Exercise '4;; also Am. Ma�h. Mo . , V . IV, p . l2, proof XXVI .

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ALGEBRAIC PROOFS

D. -�Ra't t o · o f A reas: As in the three p�eceding divisions , so here

in D we must �est o� p�oofs on simila� �t . t�iangles .

lllD.t!i:.!�2

' '

:rig. 88 Since simila� s�faces a�e -p�opo�tional to the squaiPes of thei�

homologous dimensions·� thel'efo·�e_, , ' .

· [t (x + y )i, + j-yz = · h2 · + a2 ] . = [ tyz + ixz = .a2 + ·b2] ·

. . = [i (x + y)z + j-yz = _.(a2 + b2 )a2 ] ..� ' . :. ha + �2 =

· (a a. + b2 ) + a2 :. 'h.a = a2 + b2 .

·

a . See J�y Wippe�, 1880, p . 38, fig . 36 as • ' <> • , I

£oun� .in Elements- of Geom��y .of Bezout ;' Po��ey, p . 9.1, as in Wallis •· .T�eatise . of Algeb�a1 (Oxfo�d ) , 1685 ; ·p . 93 of Co�s ·de Mathematiques , Pa�is, i768 : Also· :Heath ' s Math.- Monographs, No . -2 , p . 29, p�oof•

· XVI; Journal of . Ed�cation,' 1888, V . XXVII, p . 327�· 19th p��of, whe�e it is c�edited to L. J. Bulla�d� o� Mancheste�, · N .H �

• "> . -.

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0

As the t�i ' � ACH, HCB and ABH are simila�, then t�i . HAC " ·: t�i. BHC : t�i . ABH - AH2 ,: BH2 : AB2, and so t�i . AHC · + t�i . BHC : t�i . ABH • AH2 + BH2 : AB2 • J(ow t�i . AHC

ll'1g • .' 89 + .t�.i . BHC ,: t�i . ABH = 1 • . :. , .AB2 . ·:}-- " �.

. = B:JI2. + AH2 • .�. h2 = a2 + b� : Q .E .D .

. . ,.... a . S�e Ve�sluys, ·p . 82_, .p�oof 77, w:he�e Q�ed­ited to Bezout, 1768; also Math. Mo . , 1859, Vol . II, Dem. 5, p . 45 ; also c�edited. to Olive�; the School

' '

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• Visitor , Vol . 201 p . 167 , says Pythagoras gave this proof--but no documen�ary ev�dence .

Also Stanley Jashemski a school boy, age 19, "' -

of So . High School � Yoimgstown, o·. , _in 1934·, sent me same proof, as an original discovery on hi s part .

b . Other proportions than the explicit one $s given above may be ' deduced, and so other sym-

� '

boliz'�d proofs , . fro.m same figure , -are derivable--see V�rsluys , p·. 83 , proof -78 .

I, ,

. .

· Tri ' s ABH and ABH ' are con­gruent ; also. tri ' s AHL &nd AHP : also tri ' s BKH and BPH . Tri • . ABH = tri . BHP + tri . HAP = · tri . BKH + tri . AHL ._ ' ' '

2 :. tri . ABH : tri . BKH : tri . AHL ::;, h : a2 : b2 , · and so tri . ABH : (tri .

� , ,,> 2 2 • 2 BKH- + tri . AHL) = h : a + b , or ' 1 = h2· + (a2 + b2 ) . :. h2 = a2 + b2 • ·Q .E .D .

t' a . See Versluys , p . 84 , ffg • . is attributed to Dr . H . A . Naber, ;1:908 . Leitzmarin ' s work, 1930 e�' n, p . 35 , fig .

l!lnt!I:.El�!

. CompletP, the paral . HC , and the rect . AE , thus forming ·the simi-/

I I

I . I

lar tri ' s BHE, HAD and BAG. Denote � the areas of these, - ltri ' s by x, y and ! z respectively. 1

/ ,

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'I'ftf;no Z { Y : X = 1}2 :. a2 : b2 • obvious that z

26, 1926, .

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�=---=-=-:=·-=-:::c=--�------------·-----------�---"--------------.t---------�-------- ---. --n.OIBRAI.C PROOFS �.t85

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Fig . 92

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D�aw HL pepp � �o AB . . Sine� the tri 1 s· ABH., .AffiJ, and HBL are ,simile:r , . so . al:so the squares AK, �E and HG, and since s�milar polygon� are to eaqh other a� the squares of their . homologpus di�ensions ,

· we 'have ·

tri . ABH : tri.. HBL : tri . AHL ·

= �2 : 82 : b2 = sq . ·AK .: sq . BE : sq. - HG. But tri . ABH = tri . HBL. + tri t .-'

AHL . ' :. sq . �. = sq . BE + sq . HG. � h2 . = a� + b2 : A

a . Dev:tsed by the author, J.u�y· 1 , , 1901 �- and afterwards, Jan . "{> 13 , 1934, found in Fpurrey·, s ·Curj_o Geom. , ·p . 91 , where credited ·to R . P . Lamy, 1685 .

' 0

use fig . 92 and •ta . 1 . �) Since , b7 equ&tt·on (5) , see fig . 1 , Proof

One , BH2 = BA x BL • rect . LX, and in like manner, .

AH� = AB x_ AL • rect • . AC, t�erefo�e sq . AK = rect . · LK + t�ct . AC • sq. BB + sq. HG.

h2 - a - ba Q E D . • ·., 1 .. . a + • • • .•

a . Devised by the author ·July ?, 1901 . b. TAis· principle · of " '1mean proportional'! can

be made use of in many of the . here�in-after figures among the Geometric P�oofs , · thus giving variations as to th� proof of sai� figures . · Also many oth�r fig­ures may be const�ucted based upon the use of the "mean proportional" relation; hence all -such proofs ,

. since they result from an algebraic relationship o� cor�e�ponding lines of s�lar triangles , must be classed as algebraic proofs . ,

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THE PYTHAGOREAN PROPOSITION �-... ' ... " 86

1. - - A l � e�b ra t c P ro 9 (, .Th ro u t h The o ry o ( . L t m t t s I I

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The so-called Pytha­go�ean Theo�em, in its simp­lest fo�m is that in which . the two l.egs a�e equal . The g�eat �oc�ates (b . SOO. B . C . ) 1 by �awing �eplie$ f�om a slave , using his staff as a pointe� and a figure on the

' pavement (see' fig . 93) as a model, made him (the slave ) see :that the equal t�iansles in the squa�es on 'HB---and ;IIA· · were just as many as like equal t�i ' s +n the sq. on AB, as i s evident by inspection.

(See Plato • � Dialogues , Meno , Vol . I ,_ pp . 256-�60 , Edition of 1883, Jowett ' s t�anslation, Chas . Sc�ibne� and So�s . )

0 a . Omit-ting the lines AX, CB; BE and FA, which eliminates the numbe�ed t�iangles, the�e �a­mains the fig�e. which, in F�ee Maso�y, i s called �he Classic Fo�m, · the fo�� usually found on the mas­te� ' S' carpet· . -

b . The fol1owi�. �ule is c�edited to Pytha­go�as . Let n be anz odd numbe� ,. the short side ; squa�e it, and f�om this square subt�act 1 ; divid� the .�emainde� by 2 , which gives the median side ; add 1 to this quotient , and this sum is, the hypotenuse ; e . g. , 5 = sho�t side ; 52 -· 1 = 24 ; 24 + 2 =,...12 , -" the median side ; 12 + 1 = 1� the hypotenuse . See said '>

Rule of Pythago�as, above , on p . 19 .

llo.t!l:ll!!.l

Sta�ting with fig .. 93 , and dec�easing the · 1 .� length of AH, which necessa�ily inc�ease s the length

..

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. -- - - -- -- - - - - -

I

ALGEBRAIC PROOFS . 87 •

� ... I \ '· / ' ... .... . ..g Jtt I \ '

of AH, which necessarily in­creases the length of HB, since AB remains constant , we

' decrease the sq . HD and in­. crease the sq. HC �(see fig . J \ ,.�,�1 \ I

, ,�, �� ' \ \ I ' \ \ I 94a ) .

I

' � Now we are to prove A '' , • ths.t the sum of the two vari-

J ' , J , • -1 , ,. • able squares, . sq .. HD and sq . 1 JJ )tf J.. I HC v.ill. equal tl'}.A constant I I , ' ' HF I , ·f · ' ' L. -'-�· _ --�

s ��---· _ . ·

E f.! _ _ --- -·� F" Fig . 94a

We haye, fig . 94a, ·

H l' L. l'

. . Fig . 94b

DA = AB = c DI = DK = a' = b

� DF = a -- x

· . FE =!= BK = x· IL = 1M =

.Y

EK = FL = h

2 2 2 ' ( .

h � a + b . --- 1 ) But let side AH, fig .

93, be diminished as by x, thus giving AH, fil . .. 94a, or bet ... ter, FD, fig,. 94b , ·and let DK be .

. increased by y, as determined by the hypotenuse h remaining con-

· . stant . Now,· fig . 94b ,· · when a = b,

82 +- b2 = 2 area of sq. DP . A.nd when a < b, �e have · (a. - x ) 2 · = area of sq . DN, and (b + y )2 = area of so . DR .

Als� c2· - ' ·(b +-- y )2 = (a - x )2� = area of �CLR, or

- (a - x )2 + (b + y )2 = c2 . -- - (2 ) Is �his true? Suppose it i s ; then, afte·r reducing (2 ) - (1 ) ,

, = (3 ) - 2ax + x� � 2by + y2 = o, · or (4 ) 2ax - x� =. 2'RY· + y2, which shows that the area by which (a2 • - -sq . J>P ) i llS. diminished = the

area by which b2 is increased. S�e graph ·94b . .·. the increase always equal·s the decrease .

· But a2 -- 2x (a - � ) - x2 = (a . - x )2 approaches ' # .

0 when x . �pproao}?.e·B a in ·.value . . .·. (5 ) (a ,� - � )2 = o, wl;}en x = a , _ which i s true· and (6 ) b2 + 2bt + l� • (b + y)2 = c21 ... when x =· a, for . when X bec9iner. a, (b + Y ), becomes c , and so, we

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88 THE PYTHAGOREAN PROPOSITION

have c2 = c2 which is true . :. equation (2 ) is true ; it rests on the eq ' s

(5 ) and (6 ) , both of which are tr�e . :. whether a < = or > b, h2 = -a 2 + b2 •

. a . Devlsed by the author, in Dec . 1925 . Also a like proof to the above is that of A . R . Colburn, devised Oct . 18, 1922, and is No . 96 in his collec­tion of 108 proofs .

F. -- A l t e b ra t c-Geo M e t r t c P ro o fs _. In determining the equivalency or· areas these

proofs· are algebraic ; . b�t ·in the fina1 . comparison of areas they are geometric .

Fig. 95

The construction, see fig . 95, being made, ve have �q. " FE = (a + b )2 •

But sq . FE = sq . AC + 4 tri . ABH

Let AD = AG = x, HG = HC = y, and BC = �E = z . Then AH = x- j-. ,y, arid BH , = y + z .

Wit}l. A as ·center and AH as radius de scrib-e arc HE ; with B a& center. and BH as �ad�tr� describe arc':HD; with B as center, BE as radius deecribe arc

"10 ; with A as center, radius AD , describe ·arc DG •

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. ALGEBRAIC PROOFS

Draw the parallel f lines as indicated . By in-- ;'\. / , specting the figure it be-

/ < 1� .. �, , ,, comes evident tha·t if y2 = 2xz , ( l �<" 4r \� ' : ,,1), _. t_:Q.en the theorem holds ." - Now,

· '<'" ' � • • � ,',..'i,J since .. . i�Al.J. j)s _ .a tapgent .and AR ' 1 : � '�� , \ is a chord· of same c ircle ,

1\ ' .. 14. , ' l

.A . • ,--� AH2 = · AR x .AD, o:r _ (x + y )2 �1-'-'-�!! - � = x (2y + 2z ) = x2 + 2xy + 2xz . J • t ' t I J .,, ' -�� •tJ ' f-.- - ., - ... -• --!.-'' I j _P-��/(

Fig . 96

Whence y2 = 2xz . :. sq. ·AK = [ (x2 + y2 + 2xy ) = .sq . AL ] + [ (z2 + 2yz + (2xz = y2 ) ] = sq . HP . :. h2

= 0.2 + b2 . a : See Sci . Am. Supt . , V . 84; p . 362 ,. Dec· . 8,

1917, and credited .to' A . R . Colburn . It is No . 79 in his (t�_en ) 91 proofs • .

b •. This proof is a fine tllustration of the fle�ibility of . geomet�y. It s value lies , not in a rep·eateq proof of . the manY, times established fact , but in the effective marshaling- and use of the ele- - .

--.......i.,ment·s of' a proof·, and· even mo;e also . 7n the better insight which it gives us to the interdependence of the various theo�ems of geometry •

Fig . 97

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Dra'W' the bisectors of 'angles A, B and H , and from t�eir common point C draw the perp ' s CR, � CX and GT ; take AN = AU ·= AP , and BZ = BPJ and draw lines UV par . tQ AH, � ­par; . to AB and SY par . to BH . Let AJ = AP = x, BZ = BP = y, and HZ � HJ = z = OJ = CP = oz .

X HA. =

+ XZ + . + rect .

Now 2 tri . ABH = HB ,

(x + ' zJ (Y + z ) = xy yz + z = rect . PM HW . + z:.e9t � .HQ + sq. SX •

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90 THE PYTHAGOREAN PROPOSITION

But 2 trl . 'ABH = 2AP x CP + 2BP x CP + (2 sq. HC � 2PC2 ) = 2xz + 2yz + 2z2

= 2.re¢t. HW + 2 rect . HQ + 2 sq . SX • :. rect . PM = · rect . HW + r�t . HQ -r s·q-. XX .

Now sq . AK = (sq. -· AO = sq . AW ) + (sq . OK = sq. BQ ) +_ (2 rect . PM = rect . HW + 2 rect . HQ . 2 2 2 + 2 sq. SX ) = sq . HG + sq· . HD . :. h = a + b •

a • . Th�s proof was produced by � · F . S . Smed­ley, a photographer, of Berea , 0 . , Juqe 10, 1901 •

. Also see �ury Wipper, 1880-, p ; 34, fig . 31, .credited to �t . M8llmann, as . given in "Arcb.ives ci • . Mathematik, u . Ph. Gr\lllert , " 1851 , for fundamentally the same -proof .

. ' , -

Let HR = HE = a = SG. Then rect . GT = rect . EP , and rec t . ·RA. =' rae t-., QB .

:. tri ' s 2 , ·3 , 4 and 5 are all equal . :. sq . AK = h2 = (area of 4 tri . ABH + area sq. OM ) = 2ba + (b - a �2 = 2ab" + b2· 0- 2ba

2 b2 2 • h2 2 + a = + a • . . . = a + b2 • Q .E .n :·

a .- See Math . Mo . , _ 1858-9, Vol ; I , - p . 36�� where above proof . is given by Dr . Hutton (tracts , Lon­don, 1812, 3 vol ' s , 820 ) in his History of Algebra .

- Take AN and AQ � .AH, KM and KR = -aH, and . through P and Q draw PM and QL parallel to AB ; also

"

draw OR ana · Ns. par . to · AC . - Then OR = h a, SIC = h - b arld ·· RS • a + b - h .

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ALGEBRAIC PROOFS ? 91

Now sq. AK = CK2 = CS2 + ·RK2 - RS2 + 2CR x ' sK, or h2 = b2 + a2 - (a + b - h ) 2 + 2 (h - a ) x (h - b ) ·

2 2 2·' � 2 2 = b + a- - a - b · - h - 2ab + 2ah + 2bh + 2h2 - ah � 2'bh + 2ab . :. 2CR

· x SK = RS 2 , or 2 (h - a ) (h - b ) = (a · + b - h )2 , or '2h2 + 2ab 2ah - 2bh = a2 + b2 + h2 + 2ab + 2ah - 2bh. - :. h2 = a2 + b2 •

a . Original with the author , April 23 , 1926 .

- - '

G. -- A l l e b ra t c- Geo • e t r t c P roo fs Th ro u � h S t m t l a r . Po l u-,o � s O t � e r Th an Squ a re s . 1 s t . - - S t m � l a r T r t an� l e s

Fig . 100

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Tri : s ACB, BDH and . -�

HEA are three similar tri ' s constructed upon AB, BH and

-HA , and AK, BM and HO are · thre� corresponding· rect ' s , . double in area to tri ' s ACB, _ BDH .and HEA respectively ;

·Tri . ACB : tri . BDH trr.-HEA = h2 : a2 : · b�

= 2 ' tri . ACB : 2 tri . BDA = · 2 ·tri . HEA = · r�ct . AK

: rect . BM : rect . HO . Produce LM and ON to their in­ter.section P , and draw PHG . I� �s perp . to AB� and by the Theoreni of Pappus , see fig . 143 , PH � QG . :. , by said theorem, rect . BM + rect . HO = rect . AK . :. : 2 ' - 2 2 tri . BDH + tri . HEA = tri . ACB . :. h = a + ·b

. a ' 1 Devised by the author Dec . 7 , 1933 .

., . ..

In fig . 100 extend KB to R, intersecting LM at S , and draw PR and HT par ; to AB • . Then rect . BLMH • paral . BSPH = 2 tri . BPH :;:: 2 tri (B:PH = PH x QB )

I = rect . QK, - In like manner, 2 tr·· . 'HEA = rec·t . AG .

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92 THE PYTHAGOREAN PROPOSITION . .

Now tri . ABH : tri . BHQ : tr� . HAQ = h2 : a2 : · ·b-2 = tri . ACB : tri . BDH � : tri . HEA .

But tri . ABH = tri . BHQ +. trl �- .. liAQ. :. · tri . 2 . 2 2 ACB = tri . BDH + tri . HEA . :. h = a + b •. Q .E .D .

a . Devised by aut�or �c. • 7 , _ 1933 . -

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�lnce in any tria�gle with · side s a, b an� c--c bei� the base , and h ' the altitude--the . formula for h ' · is :

h , 2 _ 2s x 2 (s - a 1 )2 (s - b 1 )2 (s - c ' ) -4c ' 2

ti · . · and having, a.s here , c 1 = 2a.� h ' = b, Fig . 101 a. ' = b ' = h, by substitu�ion in

. formula for h 1 2 , we get , after re-d i b2 h2 2 • h2 2 b 2 uc ng , . = - a. • . . = a + .

a . See Versluys , p . 86 , fi:g . · 96, where , taken from ·"De Vriend · des Wiskunde" it is attributed! to J. J. Posthumus .

2nd. '-.- St lll t l a r Po l v�·o n s o (- lfo re Th an l'o u r S t de s .

' R�gula.r Polygons l-- - - {' II!!L!!.I!.Il!l.r.!!U.b!r!

-, \ �� ... .... Q Any regular poly- · I I A, . ' \ , ' ... , L"' .

h .... gons .can be resolv�d into �·

•• ' , .... - - ,--""' 1 'i 1 , ,.,.,r as many equa so_sce e s ' , I I .

' 1 1 # ... �ri 1 s . as the polygo'\ has · A " �.. .. F . side s . As· the · tri 1 s. are

, ' · , , similar tri ' s so whatever ,' ' , · . / , • relations are established. , �\J, ' '. among these tri ' s AOB, ·BPH

C / 0. '• a�d HRA,· the same �ela�ion� . , ,')E , ,wi�l·}��!st amon� the· ·po�r:-_· : ___ ; , . . , gons 0 , ·P and R . ' ... ' � ,. . ' ""'ll , Fig • 102

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ALGEBRAIC PROOFS 93

As tri ' s AOB, BPH and HRA are similar isosce­+es tri 1 s , it follows that these tri ' s are a particu-iar case of proof One Hundred Six . .

And as tri . ABH : tri • . . BHQ : tri . HAQ = h2 : a 2 : b2 = tri.. AOB : tri . BPH : tri . HRA = penta­

:gon 0 : pezttagon P : pentagon R, since tri . ABH = tri . BHQ/ + tri . HAQ. :. pqlygon 0 = polygon P + polygon R . - :. h2 � a2 + b2 •

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a . Devised by the author Dec . 7 , 193? :

' Fi�� 103

er polygons = the area of the greater polygon . .

. · In general , an algebraic proof is i�ossible before �rans­formation . _ .But -granting that h2 = a2 + b2 , it i s easy to prove

that polygon (1 ) + polygon (2 ) ·= polygon (3 ) , as we knOlf that pol¥gon. (1 ) : polygon (2 ) : polygon (3'}

·· , :r::. . �ff.��:.: : b2· : h2 • But from .1thi s it do-es not follow ·

that' a� + b 2 = h 2 • ·

See Beman and Smith ' s New Plane and Solid Geometry (1899 ) , p . 211 , exerc��e 438 .

But an algebraic proof is . always possible by transforming the thfee similar -polygon� int9 equiva­lent stmilaF paral ' s and then proceed as in proof ·

One Hundred Six . _ _ ____ Knowing that tri . ABH : tri .• BHQ ·: tri . HAQ

• h2 : a2 : b2 • · :---· (L) �z:id _that P . (3 ) : P . �(l ) : P . ( 2 ) . ( P = polygon ] • h2 : a2 : b2 • ·- - - (2); by equating tri . ABH : tri . BHQ : tri . HAQ-

= P . (3 ) : P • (1 ) : P • .(2· ) . But

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94 THE PYTHAGOREAN PROPOSITION

tri . ABH = tri . BHQ + tri . HAQ . :. P : (3 ) = P . (1 ) + P • ( 2 ) • :. h 2 = a 2 + b 2 • Q. E • D • ·

,

a . Devised ;by the author Dec . 7, 1933 . b . Many more algebraic proofs are possible .

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II . GEOMETRIC PROOFS

All geometric , demonstrations must result from the comparison of areas--tbe foundation of which is superposition�

As the possible number of algebraic proofs has be.en shown to be limitless , so it will be conclu­sively shown that the possibl� number of geometric

· proofs through dissection and comparison of congru­ent or equivalent s.:reas i s also "absolutely 'unlimit­ed . "

The ·geometric proofs are classified under ten type forms , as determined by the figure , and only a limited number, from the indefinite many, will be ·

given; but among ' thase given w�ll be found all- here­tofore (to date , June - 1949 ) , re.corded proofs which have come �o �e , - to�ether with ail recently devised or new proofs . -

· The re·ferences to the authors in which the - ---- - ---- - -

proof, Qr figure , is ' found or suggested, are arranged chronologically so far a� poss.ible .

· - The idea_ of throwing - the suggested proof into the fprm of a single equation is my own; by means of it every essential element of the proQf is set forth, as well ·a s the comparison of the equivalen� or equal· ,, areas .

The wording of the · .theorem for the geometric proof is : Th e s qu a r e de s c r t b e d upo n t h e hypo t en u s e o f a r t � h t- anl l ed t r t anf l e t s e qu a l t o t h e sum o f t h e s q u a r e s d e s c r t b e � upori - t h e o t h e r t wo s t d e s . 6

TY P E S

It is obvious that the three squares con-• • • _! • • ; structed upon the three side_s of a right-angled tri-

- _ _ _ -���angle can have eight different posi ti6ns , . as per se-lections • . Let us designate the squ�re upon the ·

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THE PYTHAGOREAN PROPOSITION

by htpotenuse , by h , the square upon. the shorter side a , and_ tpe · square upon the other s·ide by b , and set forth the eight ar.rangement·s ; they. are :

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A . All squares h, a and b exterior . . B-. a : and b exterior and h interior . G . h and a exterior and b interior . D . h · and b exterior and a interior . \

inte:ttior . E . a exterior and h and b F . b exterior and h anQ. a interior . G . h exterio� and a and b interior . H . All squares· h, a

'

and b interior .

The arrangement designated above c onstitute _ the first eight of the following ten geometric type s , the other two · being :

' . . I . A transla.tioh · of one or more squa.re s-. . J . One or more square s omitted .

\ ' Also for some sel�cted fig�_e s · for provi-ng

Euc lid I., Proposi't!ion 47 , ·the re·ader i s referred to H . d 1 Andre , N . H . )\a.th.- (1846 ) , Vol . 5 , p . 324 .

,Note . · · By "exterior" i s ·meant constructed· · outwardly ..

. By · " interior " _is meant c onstructed overlapping the given r:tght tri�ngle .•

------. - --.·

Th�s type include s ail pr..oof? d�rived from

'

t�e figure determined by constructing ·square s upon each si�e of a -·.right -e.I?-gled triang�e , each square be- -­ing constructed outwardly �rom the 1 giveri- triangle .

·

The · pr<:>ofe· under· this type are c-laas-ified 'as . ·. . \ -- ' · follows : ·

·

(a ):C:Those · proofs in which'. pairs of the dis­

sected parts are congruent . -Congruency implies :iuperposition, the/ most . / fun�ental and self-evident truth found in plane

' . geometry.

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GEOMETRIC PROOFS 99

As the ways of di s section are so various , it foll·ows that the number of " dis section proofs" is un­limited.

(b ) Those proofs in which pair s of the di s­sected part.s are shown / t_9. be equivalent .

· As geometricians at large are not i� agree­ment as to the symbols denoting " congruency" and . " equivalency" (personally the a uthor prefers _ for cong�uency, and - for equivalency ) , the s-ymbol used herein shall 9e .= , tt � context de ciding it s import .

/ (a ) PROOFS IN WHICH ' PA. RS OF THE DISSECTED PARTS ARE

CONGRUENT .

Paper Folding "Proofs, " Only Illustrative

1 ' Cut out a square piece of F- ---._. - - • � � pap�r EF, and oh i t_s edge , using .

· ' �� 0 · : the edge of a seco:n,.d smal.;t square • / 1 }I of paper ; :El{, as a · measure , ma:rk

:z:> � ..1 - - off /EB·, ED, LK, LG, FC and QA. · t I e , , K (, - _ .JN £ 1 Fold on DA , BG, KN, KO ,

• �---·- 1 ' f CA_ , AB and BK . Open the ·sq . EF ' t . ,

. : ' �M· ' f and ob serve \thre·e sq ' s , EH, HF L , ., ' F

L - �GJ.. � C - - -' and BC , and t:qat sq . EH � s.q ; KG .-. 1 With sc+ sso�s cut off · } 'Fig . 104 �ri . CFA from s(L. HF, and lay it

I - on sq . 'BC in posi tiop BHA, ob-I

) serving th�t it covers tri . BHA or sq . BC ; next cut ' l

off KLC from s� · � NL and HF-. and lay it .on sq. BC ' in position of KN;B �6 that MG fal:t. s on PO . Now, ob se,rve that tri . I<Mri., i � part· of sq .

_ 'KG. and Sf! · BC and .th� t

the . part .HMC.A: :l s part of sq·. HF and sq . BG_, and that al;t of. sg,. · BC . . i.s nq.w , covered by .t�e .:two _I)art·s of

, sq·.

KG and the t-wo part s . of sq. HF . . ' Therefore the isq . EH = sq. KG ) + sq . HF

= �he sq . BC . Therefore the · .sq . upon the side BA which is sq . BC = the sq-. u�)on the side BH which is

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100 THE PYTHAGOREAN PROPOSITION

sq. BD + the sq . upon the side HA which is ·sq . 'HF . :. h2 = a2 + b2 , as shown with -paper and sci�sors , and ob servation .

· __ . a·. · See " Geometric 'Exercises. in Paper Fold­�ng, " (';!:' . Sundra Row ' s ) , 1905-, p . :1:4 , _fig . 13 , by Beman and S�th.; also School Visitor , . 1882 , Vol . III ,

· � --- - -

" . p . 209; al·so F . C. Boon, B . H . , 1-n A Companion to

Elementary School Mathematic s , " .(1924 ) , p . 162, proof 1 .

c

Cut cut three sq ' s EL who se edge �s HB , FA who se edge HA, - and BC whose e�gai.� AB, making AH = 2HB . . . - \ .

) Then fol� �� · FA along MN and OP , t and sepa­-rate

·:tnto 4 sq ' s MP , QA, ON

\ and FQ .each equal to sq . EL .

Next fold th� -4 pa­per 's.q ' s. (U, -R, S and T be­ing middle pt ' s ) , ·along HU, P.R, QS �nd MT, and cut , forming part s , 1 , 2 , 3 , 4 , .5 , 6 ; 7 and ·a .

Now place the 8 part s on sq . ac ��n posi­tions as inciicate'd, reserv-

Fig. 105 ing sq � 9 for last place . Ob serve that sq. FA and EL

exa.ctly cover sq . B.C . :. sq . upon BA = sq . :upon (HB = EL ) t.!'-''sq . upon AH . :. - h2· = -e.-2 + b.? . Q • . E •. F .

. a . Beman l;l.nd �ritith i' s Row ' s . (1;905 )., wo;r:k, , 1 . . , ,

p . - 15 , f ... g . 14.i al;so �chool Vi sitor_, 188� , Vol .• III , p :

• . 208; ·als;o ¥� c ! Boon, p . 102 , proof . • 1 .

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GEOMETRIC PROOFS 101

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, Cut. out three · sq ' s as in fig. 105 . Fold small �q . 9 (fig . 105 ) along middle �nd c�t , form-1ng ·2 rect ' s ; cut each re� t . along diagonal, forming 4 rt . tri 1 s , 1 , 2 , 3 and - 4. But from eaqh corner of sq. FA (fig . 105 ) , a rt . tri. each having a base HL = iHP (fig. 105; FT = iFM), giv-ing 4 rt . tri ' s 5, 6, 7 and 8

-· . (fig . '106 ) ; and a center par·t 9 (fig . 106 ) , _ and arrange the- p�-eqes as iJ?. fig . 106_, __ __.·

and observe that ·sq . HC = sq •. EL + sq . HG, as in fig . 105 . :. h2 . = a2 + b2 •

a . See �'School Vis_itor, " 1�82, Vql . III, p . '208 .

. ·'

. o . Proofs Two and Three are particular and i1�us�rs.tive--not. genel'�l--but useful as - � pap�l' .. -8,.� SCiS �Ol'S eXSl'Ci S� • _ .

•

. c . With paper· and scissors , many ot�er 'pl'oof�

-1;rue,· under· all conditions, \may be produced, using �igs . 110, lll , e�c . , as models of pl'ocedure .

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Part.icular -ca.'se--i:J-7-" lustrativ� rather than d�mdn­s·tl'a ti ve � __,.._ -

The sides a.l'e t'o each other as 3 ; 4, 5 unfts:. T:Qen sq . AK contains 25 sq . _unit s , H:b 9·- sq. units and· HG 1.6 sq. units . Now it-.:> is evident . that . \ .

the no . ·of- unit 1 squ�res in the sq . AX � the. s·um of the · unit squares in the squal'es HD and

· HG._ . :. squal'e AK .. = sq . HD +· sq. HG •.

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,. . 102 . . THE PYTHAGOREAN ·P-ROPOSITION ...

a . That by the use of the lengths 3 , 4, and 5, or length having the ratio .of 3 _ ; 4�: 5, a rlght­avgled triangle is formed was known. to the Egyptians i'-· • .... �

- ' e;s ,carly·· as 2000 B . c . ,· ror at -t�at· time there existed profe·s�ional "·rope-fasteners" ; . the.y: :were employed t;;o

7 con�truct right angle s which they did by plac�ng three pegs · ·so- tl:la:t a rope measuring off . . 3, 4 •and 5 units would just reach around them. This method is

, . ip. uf(e today by carpenters ·and .mason� ; sticks 6 and 8 feet · J:ohg form the two ·sides and a "tel\-foot·� stick

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forms the hypotenuse , thus completing a right-angled tri·angle , henc� establishing the rigb:t;; ,angle •.

f But granting that ' the early Egyp�ians formed rigbt ·angle's in the "l'ule or thumb" manner described �bove , /lit doe� not · f.ol-lov1 in ract. "rt- is not be-

. lieved that they knew tile a�ea of the square · -upon t�e hY,Potenus·e · to be equal to the eum of the ar.eas of

• .,. t�e ·squares. upon the othar tvo .eide e�· '"

/ �he d�scoyerr of this · fact . .is creQ.i ted to Pyth�gora�, a renowned philosopher and teacher, bo�� at S'amos about 570 B .• c . , after whom the theorem is caljieci . uThe P�hag-or'ean ·Theorem . ,,. . (See p . 3 ).

; b . S�e 'Hillr ' s. Geometry tor Beginners., p . 153 ; �all-'-s Histo�y. of Ma'th�matics , pp . -7-10 ; Heath ' s

� �th � Monographs·, ' No . 1 , pp . • · 15-17 i The School Visi-\ / to�_, ___ J!ol_. 20-, p . 16'7 .

--�--/-- -\ , .

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Another particular ·

· case is illu;strated· by ·fig.: : · 108, in whi�h BH = HA, show­

. . · i� ,16- equa:l. . .-�r.-is.ng1e s . . ' · , ·�. · _ since the sq . J\K. con-

tains 8 of .these triangies-, ,. ­

·�· sq. AK = sq . HD + sq. - -HP. ·,

:. h� = a2 + b2 • · ¢

\ a . . For this and many, other demonstrations by ais � sectfon, see - H . Perigal , in Messe�er of Mathematic s ,

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GEOMETRIC P�OOFS 103 --------------------------�--------------------

· 1873 ,. V � 2, p . 103 ; also see Fourrey, p .· 68 . b . See Beman and Smith ' s New P lane and Solid

Geometry, p . 103 , fig . 1 .· _CL, Also·- R . A . Bell, Olev'eland, 0 . , using sq.

AX and lines AK and BO only�

� ( -, In fig . 108, omit lines· AF, ·BE, � and NO ,

and draw 'line· FE ; this gives the fig . used in "Grand Lodge Bulletin, " qranq Lodge of Iowa, A . F .. and A .M. , Vol .� }0 , Feb-. 1929-,. No . 2 , p . 42 . -The proof is ob­vious , for the 4 �qual isosceles �t . tri ' s which

2 . . 2' 2 make up sq . FB = sq . AK. :. ,h = a + b • . ,. .�: a . This gives another form .for a folding pa-

per proof .

In fig . 108� omit lihes · as in .proof Six, and +t is obvious that .tri ' · s - 1 , 2 , 3 and 4 , in sq 1 s HG ·

. and HD will cover tri ' s 1 , · 2, 3 and 4 in sq . AK, or· · sq . AK = sq . ·HD .+ sq .• · 'HG. :. h2 ;:: a:2 +. q2 •

a . See Ve�sluys (1914 ) , fig . 1 , �p . 9 of his 96 · proofs .

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In fig . 109, let HAGF denote �he· �arger.·

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s·q . HG • . Cut the s�ller sq . EL into two equal rectangles -� and �' fig •

109 , and form with these

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. and the larger sq � th� · rect • . HD:.t;:F . Produce DH so .that HR = HF . On RD as a diameter describe a semicircle DOR . Produce

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' . 104 THE. PYTHAGOREAN PROPOSITION

't HF .. ·to C in the are . ·Join ·co, cutting FG in P , an9.· · .

AG in S . - ·complete the sq . HX. Now tri ' s CPF. and LBD· are ·congruent as are \ tri. 1 s CKL and PED. Hence sq . KH = ( &q . EL, 1:�g . · 1.05

• rect . AN + r�ct . ME, fig . 109 ) + ' (sq . �G, fig . 105

-. , ) 2 2 2 .• .quad. HASPF(. t tri . ·SGP , fig . 109 • :. h = a + b •

a. . See School ·Visito� , 1882 , Vol . III , p . 208 . b . This method, · embodi�d· in proof Eight, will

transform any rect . into a square . · · c : Proofs- -Two to Eight !�elusive are illus­

trative rather than demonstrat-ive .

Demonstrative Proofs

ti.l!ll ' /

I In ·fi"g . 110·) through

�)('F, ;Pth' Q, 1Rd and

f-St�h· the - ce

AKnterdrs of

1!. , J , � ' e s e s o e sq . aw V' � .. l J. 2.. ' /' ' PT. and .RV par . to AH, and Q�. \- V. - - S �'> and 811' par . to BH, and through , '\�-

I , "J) 0 , �he center of. the sq . HG,, �-..,.-�0� ·

draw XH par . to AB and IY ·

A, 1 � · J· . · �--_:_�--par . to AG , _ forming 8 _congru:-1 _.1 " ,; �r . .. . ent quadrilaterals] vi'z . , ,1 ,

-rl (...�· :G" \au , 2 , 3 and 4 in sq . AK, and 1 , 1-.: , ' ,.- � .

i ! � !

1 \ . · , j 2 , '3 and. 4 in sq. HG, and sq . C J l \V. � J J u 5 " AK - . (5· - HD·-\- '"�- - ·-1�--.:� .,, J·-:� ..J n ,-n sq. - sq. - . , . ·. , ··

- - - -Q - - The proof''. of their congrtienoy "' ·

i:s evident, _ since, in the .,. Fig� 110 par,al . OB, (SB. • · SA ) : (OH ·'

c OG • AP since .AP ... ��:" .AS ) • . (Sq . AX • - � qliad .· . APTS + sq. · TV: )· = (sq: HG �=: 4 quad . OIHZ.) + s,q . HD . :� .�q . ·on AB = · sq . on BH + sq • . on· AH . 2 . 2. . 2 ' . - . . \ :. h • a + b • · , · . .

· , f , ... to

.· ' fl . a� See :·Mess , ��h . � . v;ol:. �, 18'73� P � 104 ;, , by . Henry Fer:tgal·, P . R . �. S . , etc . , ·M&cmiilan and Co . , Lon�on -�d: Cambridg,e . j Here H . P·erigal"'" 'sJ:lows the great value of proof �7 dissection , and sugge�ts its application to other theorems also . Also �see· J�y

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1

GEOMETRIC -PROOFS 105

Wipper, 1880 , p . 50, fig . 46 ; Ebene Geometrie , Von G_. - -� �ler,. Leipzig, 1897 , p . 581 fig . 71 , _ and School . ·

. . Visitor, V . III, 1882 ,. p • . 208, fig . 1 , for a pa.rt·icu­lar application of the above d'emonstration; Versluys., 1914, P . 37 , fig . 37 taken fro� "Plane Geometry'_' of J . s . Mackay, as given by H . Perigal , 1830� Fourrey, p . 86 ; F . C . · Boon, pztoof ·.· , p . 105 ; Dr . Lei.tzmann,· p . 14, fig . 16 . .

b . See ' Todhunter ' s Euclid for a simple proof extracted from a p�per by De Morgan, in Vol . , I o� the Quarterly Journal · or .Math . , and reference i s also made there to the . . .:work "Der Pythagoraische- Lehrsatz , " Mainz , 1821 , by J ; J . - I . HorfniB.rin • .

c . By the above dis sectio� any two ·squares may be transfo�med into one square , ·a ·fine puzzle for p.upiis in plane geometry.

d . -Hence any case in which the three �q'll.a��� _

are eihibi ted, as set forth under 'the . fir.st . 9 types of J;I, Geometric Proofs, A to J :inclusive .C�ee Table of Contents for -said t-ypes ) may -be prg_y�-���_by. this method .

c . Proof Nine is unique in that the smaller sq . HD is not dissected. .., _ ·· .,

!!D. ,,

In ffg . 111 , · on CK const:ruc't tri . · O:J{L = tP-i-. ABH; produce OL ,.to P making LP = BH and take LN = :ml; . draw ·Nfti, AO and BP each perp . 'to QP i at . any angle of' the sq . GH, as F, cpnstruct _a 1 tr,1 . GSF = - tri . ABH, and fr9m �y_ angle of the �q . HD , ·as ¥,· �:!�� � �adi�� _

= KM, ' de't�rlD.ine :the pt •. · R and draw HR, thus di_s sec ting the sq ' � , as per f�gure .

It i s reaqily sho�

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106 ' - .

th�t sq . AK • (tri . CMN = tri . BTP ) + (trap . · NMKL -= trap, DRHB ) + . (tri . � =. tri . HRE ) + (quad . ' .AoTB ·

+• tl'i . B'l'P = trap . GABS ) + ( tri . ACO � tri . GSF ) • (trap . DRHB + tri � HRE = �q . BE ) + (trap . GAHS . + tr� . · GSF = sq . AP ) = sq. BE + Bq. ·� . :. sq . up_on . 2 . 2 2 AB • sq-. upon BH + �q . �pon AH. :. h = a + b .

a . THis dissection and proof were devised by - the author, on March 18, 1926, to -establish a Law of

Dissection, by which, no matter how the ·three squares are arranged, or p�aced, �heir resolutiqn in�o th� respective _ parts as - numbered in fig . 111, · can be read- �

ily obtaiiled� ·

:.... · · ·/ ' i ·

'b . In :marty of t�e geom�tric proofs herein the ' reader· will obser've that. -the above-- dis section, w�olly ·

-�r partia�+Y, ·li�s been employed . · Heuce these- ·proofs are but variation of this _ general proof .

Ip fig . 112, co�­ceive rect . - TS cut off. from sq-. }.F and J>l�.ced: in p<;>si­tion of re�t . QE,, AS co- . inciding with. �; the� DER ' is · a. st . line .. since these ' /

rect . were equal bi construe-. } ' '

tion. The rest of the · con-. struction and - dissection is �- evident .

sq._ AK = (tri . CKN = tri •.

P�D ) t· !�ri . KBO - = tri . BPQ ) ' Fig. 112 + (trfL . BAL · • tri:. TFQ)

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.. -.-. . , . , - + · (t;r! . AQM '� -'tr+ . FTG). · .

�: , _ , t (sq • . LN = sq. :RH ) = -sq·. ·BE + .rec·t·; QE · + ·rect'. ' GQ · • " , • • 1 � • \ \

.· +· sq. RH = �q . _BE + sq'•' \ GH . _: .. · sj·-· up�p. ';AB _= :sq. . up,on , �11 .� �q . \ ·upon AH. :� ·-h2

=� a� + , b • · . '� . · ' "' � � \ . ' .l a . · Original with the author after having care-" l -tully analyzed . the ' esoteric implications of B�askara's "Behold!." proof--see pro�f Two Hundr.ed Twenty-Four � · tjg . '25.

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GEO�TRIC PROOFS- 107 > ' ,•

b·: The reader�. wil_l notice that this dissec-. '

tion eontains some of the elements �of the preceding �is section, that it i s applicable to all three-square figures like the preceding, but· that it ·is not s'o simple ·or �unqamental , as it requires 's transposition or one �art . of the sq. GH, --the r.ect . TS- - , to the sq . HD, --tl;le rect . in position QE-- , so as tp form ·

the . two congruent r.ect 1 s GQ :and QD . . _c . · The student Vill note that all geometric

. . '

proofs hereafter , which make use of dissection and , congr�ency, are fundamentai�y only var�atiqns of the pro off! e'stablished by proofs Nine , Ten and Eleven and that all other geometric. proofs ·are based, · either par­tially or wholly on the equlvalengy of the correspond- . ing pai-rs of part s -�of the figures under consideration.

. This proof is a sim-r�

· · £' ple variation of the \proof _

� /. \ . "', Ten ab?ve -. In fig . 113 , ex-G,t' ,_ \f'l,."/ I ',"D . . tend . GA to .M, dr.aw c.� and BO t,---� ·- -- -� pe�p . to AM; take NP = BD ·

',, J : . 11 · /:D and draw PS par . to C�, and A' ·/ through H' drBJ{ QR par . to AB .

S 3 � Then since1 i t i_� . .. e.�sily

I ' , . �, I ..

, ... shq� tha-1#--:pa-rt�s ! ' a.nd. 4 of

J� � 2. t sq . ·n = parts 1 arid 4 of I .. / \' ! ·

sq . HD, and parts 2" and 3 of C�-'- ��..11( sq . . AK = 2 . and 3 of sq . HG,

:. sq. ·upon AB = sq . upon BH Fig . 113 + sq . up'on AH .

· · a . Original · with the author March 28, l926' :to' obt·airt � figure. more readily construc:te� than fig. 111 . · · · ' . · '

·. _ p� see s·chool Visitor·; · 18e2 , Vol . III , p . 208-9;, - Dr . Leitzm8nn, p . 15, fig . 17 , 4th Ed ' n.

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108

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.In fig . 114, prodUC$3 CA to 0, KB - to M-;-:-.GA-t-o-V� . _

l118.king AV = AG, DB to U, and draw KX and CW par . re sp . to· BH. and AH, GN and _Rr, p'ar . to AB, ·and OT par . to FB .

. - \____ . Sq . AK:.' = [ tri . ·(}KW = tri ... ,

� '

(HLA = trap . BDEJ.l .+ ·tri. NST ) ] + [tri . KBX . = tri . GNF

' ) '

= (trap . OQNF + tri . BMH )] + ( tri . JMU = , tri . OAT ) . ,

+ ( tri . KCV = tri .. AOG ) i Fig . 114 + (sq . VX ,= papal . SN ) - . 11 . = sq . BE + sq. HG . :. sq . . ', ' . 2 : 2 2 . upon AB = sq. upc;m BH + _ sq . upon AH. :. h = a + b • 'a .- Original 'with author March 28, 1926-, 9 : 30

..

p_.m . b . A variation of the proof Ele�en · above .

f.SUL!:!!!!l

·Produce· CJA -:to B , d�aw SP par . to F.B1 - take HT = HB, draw TR par . to HA,

;_ pi'oduce .GA to M., making AM = A(},- produce DB to L, dr� . . .

KO and CN par . resp . :to BH _ and AH, and dra;w .-QD . Rect .' mr-.-= rect . QB . Sq . AK = (tri • . CKN � tri-. -ASG ) + (tri . KBO = tri . SAQ ) + (�ri . BAL = . tri . DQP ) + (tri . ACM = tri . �E ) + (sq . LN = , sq . ST ) � rect .

ll'�g. 115 • - �E + �ect . GQ + sq . ST -= sq . \ 1 BE + rect . QB + rect . GQ

+ sq . ST = · sq . • BE + sq . GH . :. sq . upon AB = sq . upon BH + sq . upon Ali. :. h2"'"-= a2 + b2 •

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GEOMETRIC PROOFS 109

a . Original with author -March 28, 1926 , 10 ,_·

b . �Thi s is another variation of, fig . 112 . ·. '

',

-

Take HR = HE and FS = FR = EQ = DP �

Draw RU par . to •

AH, ST par·. to .. FH, QP par . to BH, and UP par . to AB . Extend--GA to M, making AM = AG, and DB to L and draw CN par . to AH and KO par . t·o BH .

Pla�e rect � GT in position of EP . Obvious

1 that : Sq . AK = part·s (1 + 2 + 3.) + (4 + 5 of rect. . - ____

HP ) ! :. S.q . upon AB = sq . upon BH + sq . upon AH . ·

:. h2 = a 2 + b2 • .

a . Math . · Mo .• , 1858-9,- Vol . I , p . 231, where this di'ssec�ion is credi t�d to David W. ·Hoyt , Prof .

··Math . ·and .Mechanic s , Po1.ytec�ic College , Phila., Pa . ; also to' Elin� Earle Chase , Phila., Pa .

b . The �th . Mo . was edited by J . D . Runkle , A .M . , Cambridge Eng . He · s.ays. this demonstration is e s sentially the same as .the Indian demonstration

1 found in "B:t·ja Gaui ta" and referred to as the figure of "The Brides Chair . "

_ c . · Also see .said Math . Mo . , p . 361 , for an­other proof ; and Dr . Hutton (tract·s , London � i812 , in his History of. �lgebra ) .

I 1

"

.In-fig . J J7_,-.t_Qe di ss·ection is ·"evident and . -- ' sliows that part s 1 , 2 and� ,in �q . AK are congruent·

to , parts l , � ·and '3 ih sq . " HG;-----..al.�o t'hat part s 4 and

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110 ' THE PY'l'HAGOREAN PROPOSI�ION

F - ��H � � l '

G, I ' \ t I I ; . ' -' \ .

.

"5 in sq. AK are , congruent to parts 4 and 5 in aq . HD .•

:. (sq � Ax: � parts ' 1 ·f- 2 + 3 + 4 + 5 ) -= (sq . HG = part s

. l + 2 + 3 ) + (sq . HD = parts 4 +. 5 ) • :. sq; on AB = · ·sq .

-

on .BH + sq . on ·AH. - :. h.2 / ��... 2 2 ' = a + -1:? .

a . See Jury Wipper , 1880, p . 21, fig . �4, as

· given by Dr .• - Rt!_dolf Wolf in "Handbook der Math�mat£ic,. e-tc - , ri 1869; _Journal of Ed-

. _ _ : ueation·, v . 1'.XVIII, ·1�88, p . -J:.7, 2�{th prso of, · by C � W . . Tyron� Loui·svilla 1 Ky. ; ·

1Se:man:::.and Smith '·s Plane _and 3oli4_ Geom. ; 1895 , p � 88, /fig •. 5; Am. Math . Mq� , V . IV, 1897;� · -p .• _. _ 169· proof i XXX�X; and Heath ' a Math.- Monographs , No � 2., p . 33 , · proof XXII . Also ·The 3�hool Visi tor, V . III, 1882 ,

P.• 209, for an applicati�Ii of it to ·a . pa.rticula.l .. case; Fourrey, p . .87, by 0 zanam; 1778',. R . Wplf, - 1869 .

_ .. b . See. also "Recreetions in Ma.th . i and Phys-

ics , " 'QY' · Ozanam; "ow.-iosities of Geometry, 11 1778, by Z1e E . Fourrey; �. �Bger, 1896 ; Vereluys, p • . 391. .

fig •.. 39, and p. '41 , fig. 41, an� a veriat1o:r.. �-s that qf 't/ers.±uys (1914 ) , p . 4o, · fig . 41 . ,. n 4 I

t

A f ,, l... . I ' ' .., /� I V- ' ·I 'I '?of J 1-. l/ ' I c - - �--'.J.J I<

:ria-· i1B

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Ext�nd CA t9 M and KB to Q, dl'aw ·MI'f par � to AB . Extend 'lA to· ·T ana. DB to · 0 r

Dr�� CP pa� . to -AB . � Take 0R = !tB and draw R� pe,r . to· .F.B . -

Obvious that sq � AK � sum of parts (4 + 5 ) i- (1 + 2 .+ 3 ) ·= sq . -!!D + sq , HG. ••• sq . upon AB = · sq:. _

upc�1 .Bli + sq . t1pon HA. · -:. h2 ,, 2 � �� + b • Q.E .D .

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GEOMETRIC PROOPS lll . .. J -_.;.··

.·) · -------- -- a·;-con��:tved by the � autkor, at Nashv:J:.l.le, o . ,\ i March "26, 1933,' for a high school _g_irl th�re, while \·

present for the funeral 91' his cousin; al�o see

I j I ! l . l -

SchoQl Visi:tor, VoL-.} 1.20, p . 167 . - . _ b . Proof an�- fig . 118, is practip·ally the

same· as proof Sixt��n, . fig . 117 . / •,

..

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.'On Dec . 17; 1939, there came· to me tnls : Der Pytbagoreische Lehrsats von Dr . W. Le�tzmann, 4th Edition, of 1930- (rst Ed 1 � 19lt, 2nd Ed 1 n, 1917, 3rd Ed 1 n, ).,. J .n which appears no �ess than 23 proof's of the Pythe.�orean P�oposition, of which- 21 were among m;y proof herein.

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This little book" or· 72 p�ges. i s an excellent treatise , and the bi�l�ography, nages 70, 7l ,o 7a, is valuable _ for ·investigators, listing 21 wo�ks. re .. Jthis t;heoremf.

· . . · t ' My �uscript, . for 2nd edition, ·credi-ts - this

work fdr all 23 prQ.of therein, and gives , as new proof 1 the twa no-t · lnpluded 1.n . . the. S$id ?1 . ,

- ,;.·

I Fig . 119

I I

�lib.!tl!l � ' ' I \

In fig . 119,. the , ,.. - 'dlss�c-<t�n is �v+:a��1& and

shows tha:� par� s 1 ,· · 2· ·and � in sq . aG ·are. congruent to parts 1, 2 and 3 in .rect . QC ; al so that p�rts 4"� 5 , 6 and 7 in sq . -HD- are congru­ent to p�rta 4 , 5 , 6 and 7 in rect . QR .

_ The-refore� sq. upon .

AB = sq. upon HB +· sq. upon HA. "' :. h2 = a2 + b2 • Q . E .D • .

. l

. -. · a . See dissection, Tafel II, in Dr .- . � · Leitz­mann1 s work, 1930. ed 1 n--on last leaf of s�id work . Not

credited to any one, but is based on H . Dobriner 1-s proq.f's • . '

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. -� PYTHAOORIWf . PROPOSITION

. . . .

In fig . _ 120 �araw GD, and from F and E �aw lines to GD par . to �C ; th�n extend DB and GA, forming t:he · rect . :AB ;

. tbrough C and � dra� li�es par. . r ':. .. respectivei7 �o AH .and BH� form-

. '•

ihg tri ' s equal to ·tri . � . - · - .

�ough points.-li and M .draw_ line par . -to GD . Take KP · ·= BD, and - draw MP , and through L. draw a li.ne par . to MP . . ·

. Number the parts �s �:n,. the ·:r!i.glire . - it :1& ·obvious that· - - l . . . --

��;."_i2o th� · �ssected .sq 1 � HG- $Ild HD., _ _ .. - -· gi vt� � tPiangle, s , .. c�- be �-�

ranged in sq . · AK ·1as nUlllbered; that is , the ;8;--tr� ' s in sq, AX can be �uper.imposed by th�ir � equivalent ·

tri ' s in sq 1 s HG and HD. .:. sq. AK = sq. BD + sq . HG. 2 - 2 2 -:. h = a + b • Q.E .D. ,

· a . See dissection, Tafel I, in Dr . W. Leitz­mann .work, 1930 ed•·n, on 2nd last leafi. Not credited to any one, but 1s . based on J. E . B6ttcher 1 s work.

, . 11g . ' '121

.. -;' , ..

1 I • ' .I

!!!tD:1l I ¢ I . -

In fi'g . _121 the- con­struction is_ readily seen, �� also the congruency ot

· the ·-corresponding dissected parts , from which sq. AX -. {quad. ··cPNA' ·= qua�. LAliT ) + (tri . CXP = tri . ALG}_ + (tri • . BOK � · quad • . DEHR + tri . TFL ). + · ( t�f' •. · NO�.

- = tri .- _ _ RBll}. · .:. .sq . upon AB • sq.

upon BH + sq. upon AH .

,,

' .

.. ,

<>

(_· .. / /"

)

' -; �l •

..

'

..

I . .I - · j

. .. . '

. . .,.

; . '

· � A • !..._:..

·?

· . (

,. -

.. - ' -

..

...... - .

.. ---- .

- -

,,

': :

·­.•

, " . :,.,.- I . ,

. l } i - - -

,_ .

- - - · ---- , _ - - . -- --· " - - - - - . . - I .. •.

.. . . .

I \ .

-� · ... OBOMBTRIC PR00JS - •

.

11,. . ·' . • . • I "">Q t

a. · - See · Math. Mo: , V. IV, - ).897, p . 169, pro or • r;.: ... -r-.. .... -..,._,....

XXXVIII. I . . .

/ , l . \ .• t .,

� -F Extend c� tc;; Q, · e - -. . /" "'\ . "'PrF� . .

.t_? P , draw RJ throUgh H, par.

9.(4 t, l _ $ ' /,(11 --� to AB , HS ���. tQ OK, Sl!

, _ -��-�-- - -� �- �d ZM P�·- �o �� � &)ld Z'l' ' 1.· . · _ · . 3; ! - , · par . �o AH �d· take . SV ;-=_ BP-,.

· , ,

' .

· · / · DN • PB, apd draw V'l par·. to -

; ·.Jt\ .7 / ;L-r' · · . .AIJ_ and NQ- par . to· B? •.

-U47' ·Vr �..,., sq. - � � parts (1 + 2 T""'- I, , · 'iM - · + ' 3 ·+ ' 4 · ·� -sq : · HD�) + p$rts· . ., ��--- ( · � A&.· (5 + 6 + 7 � · sq;. - H�) ; _s:o .dis-CL::__·..f.Jl1Jr :,ectted._

par-ts of sq . HD + dis-. .. · · · sec ted · parts · of sq. Hq (br ·

' : ·' t lis. �' superpo�i-�ion ) ,_ �q�a1s the dissected parts of ··sq . AK,

. . j -. - . ..

I ·

'D I� , .

. I :/

) .

r

t'

..

' � .';" , a / •

t

. r-·

•

�I - ��

' - ' � /,;#"" ... .-

' \ ' . . I

; , ,

}

0 i,

, ' \ ),

.,..

1 I ., ' • ·'

' . I . . . \ \ -

I

! . .

f. f,

I -

' · '

i /

1

. '

1: ' ' � ', !

·. ! ';< • ..... r -

I \

,.

114 .

' . .

THE P�OORBAN PR9POSITION .

. ..

.�. Sq: upon � • sq . upqn BH + sq: \l:Pon AH.• 1 :. h2 • a2 + b� . Q.E . D . ·· -

l - - - \ a . See Vers1UJ'S1 p . 43, rlg". 44.. l !

- b . Fig. and proof, of Twenty-Two 1� very much �ike that of Twent�-One .

# � -

. - . Afte� · showins l that � N

•• - •• - '- • •• each numb_er�d- part fotpt_d .+n

l ! I

r �-

, /! � . �� · i- · the sq 1 s HD E,md HG is -congru-

�t-h:�-- - ;!:fJ.' m ;:;!1:0 p�� ��r�:�����:n- . ,� 1 7 · : . lllt ·�,..:t is not. d1ff1c;ult ,: it fo�lovs . \ _ '-� · . that ·the s:uJJ, of the J>ar·�s in

, I .

A· , ' .t-, 1 ·1/ · sq·. AX· • the sum of the �arts . . ' r�Jl� JM : of t!le ljq, HD_+ the SUJn i Of L. w.· · j .. 1 i the parts _ -o� the sq. HG • .

· ¥ �- ,, - , :. the sq. · · upo.Q. AK

n :Y'3 . f ' i ll · = · the sq . upon HD + the sq .• .

� · - �'\ • ' 2 2. 2 . . . upon HA. :; h. • ,a + b �

�1'�;1 124 Q.E.D.. .. r -. . . a . See Geom. pf Dr . ·

H . Dobr1ner; 1898; also _Vers1uj"s, p . 45, fi·g . '46, ' / . ·rrom chi- • . Nie:tson; &;Lso Leitzmahn, p . 13 I fig-�. ·15 I

____,_-a-4th· Ed 1 n:-- : - - . - c. : - · · _ . . - _

· . • " .... >

\ . I TWIDft t v� F iut-r-- __ _

__ :,��,_ , _ _ .. ., _

I I ·,·-·

!

· Proceed as i.n fig. 124 and afte� . congr�ency is : estab1iened, / it is evident -that , since· the -eight dis-

· ,

. \.

0

'Jig. ·125

j

'·

.

, I

sected parts qt ):Jq. AKr ar�-. consrue�t -�o the .correspond-� ' i�. p.Umbered part·s . round in -. sq ' s HD and: HG:, par:t s -:>(1 + 2

+ 3 + 4 + 5· + 6 + 7 + 8 in sq . · AK ) � part� · (5 + 6. + 7 °

+ 8 ) + (� · +, 2 + 3- + 4 ) in � -�q' s -liB an4 HG-. .

. i

• N I

: ? '•

. '

-. I

• 0

i

-I

I ! I I

! ' '

r .

.J · J.

� -�-- - - -- .,._ _ --- - - - --! ,• >":, •

' . -

. '

. ·

- �

'• .

...

. .

·..;, . ·\ . -

. ..

.. - . �

,,

i : '

- ' I

i I f I I

\ ·-

• '

- w .

OBOMBTRIC PROOPS •115 \

•

. .. ., . ".�. sq. upon .Aa � sq_: ,1,_1_p.on � + \sq . upon HA . ·

· ••• h2 • a2 + ·b2 •

· · :1

a . See Paul Epstein' s_ (or S.tra:a�sbel'g ) , col­�ection �of proQfs ; a� so Versl uys , . p . 44 , _ fig . 4'5; also Dr •. L�i�zmann' s 4th · ed ' n, p . · �3, ·fig . �4 .

- ·-.. � '

·<>

I!!to.il:.fl�t

. ,..---. \ "--...... ...

I!!tD.1l:.ll!

-r; .F_ · ,. . _ _ �ince parts 1. and 2

_ /�, , _ � :� o� _ sq . HD a�e co�ruent to �/' I . \ · � -"It '\ l�e parts 1 an� 2 i� sq .• ' ' I Ll - · '� _ -:-- � AI:, apd parts 3, 4, 5 and 6

.

. · ''\- r;- 1 -� 1J. _Qf �q ._ -�� t_o like part� 3, . ,a _ , . · 4, 5 and ·6 ·in sq • . AX • • •• sq. I .. . JI. �,'--... _..� _upon AB. • sq.. tipon HB + �q •

I .,. I. I - �pon HA • ••• ha. •· &2fl+: ·p2

•

. · , . ' .... " " tt iN Q.E_ .D . : --- � · . . f L� --1 . £ � a . · a . · This dissection

. . . � / . ' 'f ' 6 C .k'�1- �-V( : · ' by- fhe author,

_ �r-ch 2 : ·

·. ·, ' 1933 . ' . J'ig. ' �

. . ' . · .. � -

- ----- _ ---- · ' .

· . . ) ' .

·.

\

v •

. ' .

- -- - - · _L ' ' - !

I I t I ,: � i l

� I ' ! I

, . t-··

\ ..

• • I •

.. - - - '

- - ,.. - - -- -- ..(._ ____ -----. I ,

/ ' f I I · \

. I I

. " . -. - . . . . . ·-t ' ' • .. ,

. :

.

, ' ',

" ! \' .. ' ., l \, ,

, . . - ·

. .

. ' I r

�

, ,

.. .

�

. � '

I � I !_

o,l· ... • .:t

. .

. ·.

.--

. �.

\ ' �

f {- -

' . •

\

., ·

�

<•

�

\.

-.

: --- --..

'•

/ /

,_....__- ,

, ,

I \ \

' �

\ \

/ ...... �

\

\ •

l:16 � P�GO�N P}tQPOSITION

.. -

!�!!l!l;!t�!!! - t · •

Take AU and CV = BH ·and draw ·uw pal' . to AB· a..q_� � par . "'to B:S::; from T draY. t

�L par . to AH ahd TS par� . to r . I . . BH, locating pts . _ L snd .S ; ' '

�omplete· the sq' s tN anQ ., $Q,· making si�e s 3� � and � p� � . t o AB . Draw S� par . te---HB and JCJ par . to _fui. ..The 10

. parts found· in. sq • s HlY anq H� are congruent to corre­sponding .Pal't3 ·in ag . AK. : .

. · _the sq . upon AB = � q . upon HB + ·sq • . upo� iiA � .-. h2

.:· a2+ b2 •

. Q.E .-D . . a . �is :pro·o.r, ·and dis sectio�� yas se�� t.o me by J! Ade.mp , C;tlass�s�reet 31, The Hag1,1e , l�ol:lap.d, April 1933 w i . . ·· · - f - ". b ., Aiil liz:res are· either perp·. or par . to the sides·. of the '1;;ri .. .. AJ3H-_-a unique· di ssectlon .

�- c ·. rt' l s a 'fine paper - and: scis sors exercise .

--·j · ��· D�av AF ana BE; pr6- :

J dUee GJ\ _to P making AP =,- AG; ""- fa , . - . .prod1:1ce ·DB to 0; draw CQ par .

�,._"' ,'t1 ' · · ,.·� to. AH and· . XR �a� . to BH; c on-' ' 1 ' ·' � \1{,"' 1 ', struct sq. LN = �q . OQ; draw

· '\,_ f, '- 1 - 1 � · � and FN.; take AT and KS : Lv,1. '� J ;; / JJ' = 'to. FM. Congruency of cor-:-A1,. - - � , . respopding �umbered pal't s hav• · r ',''"L�7 r ing. be�n

.e�t�bl�1shed, as _ _ i s

. 1 : .. o�>IJ , : ., ', easily done, it ·follows that : ..

1 .J � , 1 · sq . upon AB = sq. up� HB .,,, C.f $:.! I ',-' u + .sq . upon HA. :. h2

= a2 + b2• 'J'J_ .... :--z--iff. Q . E . D . · .

,

. Fig. 1a9

·•

• '

. ·.

/

! · l I ,

. ,-, ""'_.,.._ ,.......... _ __ ...-:

· ' \

r� �. � l -, 1 . l . . I

! . {

\ '·

1 '

•

Q

' .

' \

•'

J

. ... \

• I

,--·

. .. ,., __ , _ _ -- · - · ' - - - - -- - _,.

" ""� ..._ i'· �' I '- ""

', ! . ; - . ' I

� I - ......- -r ___ •

;:. l •

_GEOMETRIC ]l>ROOPS _

j . . I '

a . Be�jr v�n Guthe!l, oberleh.rer at Nurn- , -� bers, Oerill&ny ,_ produced th:e · ab�ve proof . He died in _

the trench�"'- in -Prance-, · 1914 . · so wr9te. J . A4Qms (•ee a, tlg.·- 128.) , August 19,3 • .

r J:,- b . · Let us ca·11 1 t th� B-. von Guth�+x-- World War Proof . · _ · · ·

c . Also see Dl' . Leitzmann1 _p . 15� �ig � iS, 1930 ed ' n � ;-:> ·

l •• : i

- i

J • j . l -_ _: IIi ·-fig : 130, ext&nd -

CA �o . 0, and drav ON ari.d KP -p�. to AB and BH' re spective., ly., - :an_d, �x·telld DB .�o . R . -T�e BM = AB ·and, drav DM •. ; Then we have

-sq . / � = (trap . �·ACKP

. -- ·� ....

= · trap . · OABN ='· pentago,n / 1 : - o�) +· • (tri . BRK = trap . Jq-���:¥

BDLH + ·tr;f: . -MHL = tri . OFN�- . +--:.( tri. PRB a tri: • �J �-- .�. . - ;

6 • sq . upon :AB • sq . upon :BH · :

.+ sq . · upon u. .·. h2 . • $2.+' 1?2 • a·, · See Ifath. Mo. :, V ., ·

·IV, 1897, p . 170, proof- XLIV • . / J � �

Th·lr t v : · ..... --�-/

, Fig . l31 · ob ject1f1e� the· ·lines tQ be dr�wn anq how they &l'e drawn is readily

• j

seen . - , . Sip.ee tri . ·oMN • . tl'i �

�a.,, tri . · MPL • tri . BRH1 . . , · ;tl'i . B� � tri . AOG, and · tri-.-- - ¥

- \ · OSA � -tr� . ·es (X i s t�e pt . _ �-..

v·-···- ·---:=-_, of intersection of the lines - ' . \ ' · - - . - 1.'

• "" . . •

t � ' .- MP and OS ) �hen sq . AX · • trap.

Cl

- . ',./(· ., AOXS. + tr1:. · KSB • tri . XOM

L- - - - ·- .J · - • trap . B!!JOS + tri . OSA

J'ig_, 131

" ·

== quad .. .AHPO + tri .- ABH

t" I, • -• I

0 . '

'- I""

. . i - I

I

. l !

" •

.::

...

r ,·

.._ I

. '

(-·--"_· I

' "

c �- \ .. , l:, _, I , I

. . . ' J

j. . .

I -'

. / - : ...-..� - -- --

•

/ I i-

. 1

-

� ! . � • ' • � :. "•. " 1

:. ..., __ - -:- ) ---

< 118

t":' c , I i . '

- -� ' '

r

THE ' P').'THAGOW:N P�OPOSITION .. .� ii

•!

•.

+ tr1 .• BMir + tr1 . MP� · = · qua.d. AliPo + trJ. . 1• OMN + tri . -AOQ- + tr1 . aRH • (pentagon. _-Al{PO(} ���tri :· O�·f ). . + (trap • PMNP l. .. �rap . .RBDE ) + . tri � �BH :. sq. H�. + sq . HD . 1· -

· sq . upon AB • sq.� upon HD + sq . upon Ali. :. ·f-2 = a�+�u�; --e�a . See Sc+ . Am. Sup . , v . 7�� p � 3.J33J, Dec . 10-,.

1910 . It i s No . 14 of A·.· R . Colburn ' s 108. /P!'C?���- ·-- .

\

,

/

J'1g. 132

Ext·end, QA mkirw -AP .

= AG; �xtend DB making BN ' : BD ·= CP . · Tri . CKP = 'tr± . _ .

ANB- �· t sq . HD = t �ect . LK. Tri-. APB . =·i sq•: .HG .= '.j. 'r�.ot . .AM·. Sq • .AK = rect· . AM. · � + rect . Lt.

'

:. sq . upon AB = sq • . ;, �pon liB + sq . up()n AlL. ; :. '' h2

2· 2 . ' .

= . a + b • Q, • E·;, D • ··

. a . This ia 1iuygens ' · proof

-

(1657 ) ; se� _ also Ver-. sluys, p . 25, · fig . 22 .

I!t!t1I:.!i'!2 ' ' ·

Extend GA making AD = · AO . -Ext eng .� to N, . draw

-1 CL end IOi.. Extend BF to s

E mS.ki� FS = -HB, compl-efte sq.-/�, SU, draw HP par .� to .AB , PR

" · � par � to All, and -dl';'aw SQ, . '

-� Then, abvious , sq . _ ,;.'J) -� :.: 4 · tri .; BAN of sq. NL

���-�.-.. ._ = tteot . AR + rect ._ TR +- sq··. ,. \ , I "r- �- ;'"· 1 GQ = reot •. AR + rect . _QF ' ' • f!r\ ( ) 1\" [; · 1 + sq.. � + sq . TF = sq . ND · I . tt � ...,, -1 / l) \ - ' : = sq .• HG + sq . HD. :·. sq . · .·

C• ..., · , ' u upon AB = sq . upon BH . . + s,q . t, � l A 2 2 � - - - - .. - upon AH. :. h = a + b" .

' J'ig .. 133 · Q .E .D .

..

- -�.7-�r -�- ., .! .Y,�-·�·/•'I.fo �� r� �·� -, .., __ .. -. f } .r

I

I

l I . I .

t

- ---- � --- - ----- �-_::..�- -- --

' \

. I

' .

,� .

1-----+------------------��-- -

..... ......_ _ __,

OEO�TRIC �ROOFS � ' ·· a . This proof !s " credited to Mis s E . A .

. 119 .

Coolidge , a blind girl . See . Journal of Education' · _ ·v. . XXV:� II., . _1888, p • 17 � �6th · propf :

· b. ·The r�a�er w!il nqte that this pr?o� em­ploy� ·exactly the same d�s secti"on and arrangement as round in -the solution by th� Hindu mathematician, Bhaskara . · .See . fig . . 32.4 , proof Two Hundred Twenty-'Five • ..

. .

"' (b) THO�E P�OOFS' �IN- 'WHICH PAIRS OF � DISSECTED . PARTS ARE SHOWN ., TO BE EQUIVALENT . .. ·

As · the triangle is fundamental in .th� dete'r-minatioh-• of .. the equlva.lency of two ar.eas ,. Eucl.id ' s proof will be give� first piace . 0.

!hlr.!I:.I!!.!:.!!

Praw HL perp . to OK, and draw HC., HX:,- AD· ·-an..-� BG • . '

Sq . AK = rect . AL ,+ rect . BL · = - 2 tri.. HAC + .·2 t'ri . HBK

, ,

= ·2 tri . G.Aa: + 2 -tri . DBA _

= sq. GH + · sq . HD . : .. . sq_. _ _ uporl AB = sq . upon BH + sq. upon AlL

-a . Euclid, about 300. 1

- -- -----

I -�+ - �------�----

i

/ /

/ ;

-B �C •.. discovered the_ above. · ; .,L __ ,

. ,

.proof_, and it has found a·place 1.�-/ - -��.ry : ���ndar� text on ge­o� try� . Logically no better . p· oof can be devised than Eu-

. . .

. _ Fig . 13'-'-�---�<Uid ' s . ' · _ ¢ �or �he old descrip�

1 • • ·- '+' ). . - - tive form of this proof se..e Elements· · of · Eti·clid by J C'\ / •

·- · ·Todhunter, 1887 , Prop . 47, l:)ook I . . For a modern mod-

..

'el proof, sec·ond to none·, see Beman and Smith 1 s New Plane a�d · So<!.id Geometr�,_ 1899, p . _102, Prop • .VIII , B9ok II . Also see Heath ' s Math. Monographs , No . 1 , 1900, p. 18, pro.of I.; V�rsl�ys ,· p·. J.O , fig . · 3 , and p .• 76, ·p�oqf 166 (aagebraic ) ; Four.rey, p . 70, fig . a;

I

-·-......oo;---� -� - ·-- �- -- - -- -·---·-·--7------I

·-· I

· • ·

120 THE PYTHAGOREAN PROPOSITION

R:-

--;-----t--------:;.;�==""""--� -�:.:.::...-=....::.::....:::.....:::.,;....:.__;:.....:-..;;._"""':"'"...;._,;.......:;...:;....;.;;........;...:;.,..,....,.....,...=== ==='-- -- - -- ---- - - -I ,,

-- - � - - . -·-

·-j

1

�lso The New South Wa�es Freemason,· Vol . �III, No . 4 , April l , 1938., p . 178, for a fine proof o� Wor . Bro . w. England, F . 8-. P • , · or· Auckland, New Zes.lafld-•

..... !Also- Dr . Leitzmann' s work - . (1930 ) , p . 29, fig '· s 29 and 30 .

. , . b , - ·r have noticed lately two or three .Ameri­

can te�s on geometry in which ·the above proof'" 'd.oes , not appear . I suppose the author -.wishe� to show his ' ., ori$i'nality or independence--possibly up-�o-�dateness . � ! He 1shows something else . The leaving out of Euclid<' s proof is like ·the pla� of Hamlet with Hamlet left out . . . -

c . About 870 th�re worked for a time , in Bag­dad·, Arabia, the · celebrated physician, philosopher and mathematician t a b i t i b n Qur r a i b n Me rvin ( 8 2 6-QO l ) '., /L.b u- ij a s a;t , a'+ - ij � r r ;nt·; a nat+v.e ot: H a� r an in -Mesopotamia . l_!e rev;f.sed 'I sh a q i b n H one i u ' s transla­tion of Euclid ' s Elements, as stated at foot of the " nhnto�tA. t ' -- --. . ., See David Eugene Smith ' s "History- of Mathe-

... ( ) matics , 1923 ; Yol . __ I, pp . 171-3 . · __ _

d. The figu�e of Euclid ' s proof , Fig . 134 - above, is known by the French as pon asiilorum, ·by the

· Arabs as the "·Figur� of the Brid:e . " - _

e . " The ma.the:Qlatical sQience ·c,r· modern Europe dates f-rom t'he thirteentb century, and received �ts fir.st stimulus from the Moorish ·Scho9_1s in Sp�in and Africa ,, - where the Arab works c;>f Euclid, Archimedes, Appollonius and P'tol�iny were · not uncommon • . . . . "

·

. . "First, fo·r the 'geo:q1etry . _'As. el:}.rly as ll20 . an· Erlglish -monk, named Adelhard ·(of Bath ) , had ob­·tained a copy _ of a Mporish e�ition of the Eleme�ts of Euclid-; ·anQ. another specimen waf3 secured by Gerard of Cremona i-n 1186 . The first of these was translated by Adelhard, and a copy of this fell into the hand� of G�ovaruii Campano .or ·Companus, who in 1260- 'repro-

· duced it as his o�n . The first printed edition was taken from it and was issued by Ratdolt �t Venice in

.. 1482 . " A History of Mathematics at Camb_ridge , by W. W. R : Ball, edition 1889:, pp . 3 and 4 .

" 1--- --:::;---p -. - � - - -- -- --� --- - --- -- - -- -- - - --- ---

--- - - . --- - - . . -- ·

-

-- - - ��--..- - � __,... __ __ ..........._ _ �- -- - - - , __ - ---

. - - ---=----- _.,l

GEOMETRIC PROOFS 121 •�--�--,---,-----J---�::=============--:-==::�==================��------+---- -------- ------. --�-- !

" ' '

. ' . '

P!'DIAGORKAN TliiCORKM m· TbiT IBN QORRA 'S ri:wmuTION 07 EUCLID

The translation vas made b7.

Is� !bn ionein (died 9l0) but was revised b7 ';t'ibit ibn Qorra1 c . 890 . This mariuscript was

·written- In �350 •

. '

1----------------- �-- --- - -- -�----, - · - -· --------- ---- ----- ----:-���---

l

..

·=--------. ==--===-;-': --------- --- - --

------------ ----- --- --�------ ---------r2·o·a , THE� PYTHAGOREAN PROPOSITION -- ------------ -

·�

_;_t _ .

· J

�-��------�-�������--�--���------------

-"'F'

!!!.ir.il:.E2Y.r.

\ \ \ ' \ I '

.. ,Extend HA to L making AL = HE, apd -'HB to N making BN ::;' HF, · di'aw -· the perp . HM,

- / ' '

6 / / . ' /'E. >

/' ' H , ' · · \����· and - join LC , HC , and KN . Ob-

--� - -- --­

.viously tzti 1 s ABH, CAL �pd 13KN azte equal . :. sq. upon ·::AK = ztect . AM + ztect . BM = 2 tzti . ·HAC + 2 .tzti . HBK = m\ X CL

, 1 - I \ f \ _. ., I I ' "

+ HB x KN = · sq • . HG + sq . HD . - -:. _sq . upon AB = sq . upon HB � I I 1 \ J \

\. ,. I I 1 ' \ ' \ I I · 1 I 't1 + sq. upon HA . :. h2 = a2 + b2• i

\ ' , ,� c�� - - JM _ _ U<:

a . See Edwai'ds I Gec;>m., I

p . 155, fig . (4 ) ; Veztsluys , "

Fig . 135 ·p .1 16;. fig . ·:12, credited to :be Gel__d_e� 11:80_6_)_

b . "To illumi�e and enlaztge the· field of con­sciousness , and to extend . the gztowing �elf, �s one I'eason why we study geometzty. " �

"One ·of the chief seztv·ices which J!lathematic s has · rendered the huma� ztace in the past centuzty i �

· to put ' ' common sense ' whe;re i t belongs , oil the top- .. most Sh�l.t• next to the dusty Canis tel' labeled I dis-CSI'ded nonsense . ' " Bezttztand Rus sell . '

c . "Pythagor�s . and his followezts found the ultimate explanation of things 1n their mathematical !'elations . "

Of Pyt��goztas , as of Omazt Khayyam :

"Myself when young did eageztly fztequent -boctoi' and S�int , and heaztd gzteat aztgument

About it and spout ; but· eveztmoi'e '

Caine out by the aame dOQI' whezte in I went . "

.;

l

r · I I

: "

l

. \ " _ ___ ,__� ---------�--------- ----- ------------------ -----H-----:------------ �----· -�---------- ,- -·:�- --- --� -�

"

·- ------

' - ·-r-7. -·

\.

-GEOMETRIC - P-R-OOFS 12la . , 1 lr -------�--- �----�----------------�-�-- --�----��----�----�- -�- -======------- ----�- ------ -------�

r ·

. -

'

.

< . • '

1 \ \

} ' f .... � .

'

� -• ' . .. '

HISTORY SAYS :

1 . "Pyt}lagoras ,. level-headed, wise man-j-- ·went quite mad ov�r seven .- He found seven sages , seven wonders' of the w�ld, seven gates to Thebes , sev-en heroes against Thebes, seven sleepers� of -

- Ephesus , seven dwarfs beyond ·.the mountains--and s.o on up to -sev�nty' times seven. " • . -

2 . .'1.Pyth�goJ?as wa_s inspired--a saint , prophet , found­-

·-�r or.: a f��tigaily reli_g�ous fiOCiety. " 3 : "-Pytfi�goras visited Ionia , Phonecia and Egypt ,. "

st·udie.d .. in _·Babylon,· . ·taught in Gre�ce·, committed nothing to �i ting and founded a philosophical ;- _ .

�ociety•. " ·- ' --4 . · "Pythagor�s declared the earth to be a sphere ,

and had · a- .movement in space . " 5·. �'·Pythagoras was one of the , nine saviors or· ci viii­

tat:tort. " 6 � "PY'tha2oras was oAe of .the four protagonis-cs of

modern ·sc�ence . " I 7 • . "Af·ter Pythagoras , because or the false .dicta of

P latp ' and Ar�stotle , it-�ook twenty centuries to .. �. prove tHat · thia··�earth is neither - fixed nor the

center or the univers� . " 8 . · "Pythagoras was someth�ng of, a na.tu.ralfst--he ·· was .2500 years ah�ad of the thoughts of · Darwin. "

· 9 . "Pythagoras was a belieyer ip. the Evol utio� of ·- " . . -�n.

10 . "The tea6hing of Pythagora� opposed: the ·teaching . "

- ' of Ptolemy-. _ ; . --t. .. ___ _ l� � "The SQl�r system �s·, we ·know it ·today . .. � the .one

PytQ.�g_ora�· knew 2500- years ago . " 12 . "What touc�ed Copernicus off? Pythagoras whb

taught tha\ the earth move� around the sun, · a � 8l;_'eat centra:J, ball of fire ·;" -

1} . "The cosmology .o.r PTt}iagoras contradicts t�a� of the ·-Book of ·Genesis·-�a barrfer to free thc;mght ·

. an<;l scientific progress . " . I , : 14 . "Pythagoras saw man- -not a cabbage , but an .ani-., . . . I -

· mal- �a bundle of poss!bilities--a rational ani-�1 . n · -

.. 15 . "�e . teaching "of Pythagoras rest,s upon the Soc-ial ,

Ethical and Aestheticai Laws of Natur.e . · " , . -

_____ ..

.

I ! I I

I ' I I . ! . .

) ' \ ...

,, -.

�, '.. . I

,,. ·- -

.· •-] • . � ' - · · --�

�

'

-

' . .

,., • '

--- -- ---- -- --- - --I I l

--� __ .,...,., . �� �-- --�-�----- --------------�-------------------.. -- --, -·::.....-

I

• \

122 • T;HE 'PYTHAGOREAN PROPOSITION - - -- ---·-·� -

Draw HN p�r . to AO, . ., '"'{ 1: KL par . to BF, ON par . ·to AH,

��, "' ' _ !.'-� ...... and e.xtend DB to M. It i s � - · - ' - u "' ,_ '

evldent. th_a t sq . AK = - hexa-, 'f� \

� � gon· jtQNKBH =

- par . AONH + par . ,. "' HNKB = AH X LN + . BH X m.,

•' '

� = sq. HG + sq . HD . ,, I "' .

, • "" � a :. ·sq . upon AB = sq.

! rt�lt': ·:. upon· Bit + sq . • · up�:m AH •

- t , ' a . See �dwards·' Geom., .

,.. • ;"" "iP , ., : I{ 1895, · p . 161, fig .- : (3g )';_ Ver- -

�,�- - - L _ J · aluys , p . 23 , fig . 21 , cred-

Fig. 136 "· ·· ·· '· " i ted to _ V�:p. Vieth (1805}-;..::....:... __ _.,; -�

·

also.,.- as an original proof, by Joseph ·zelson a s_ophomore in_ West Phfla . , Pa. . , --·

'High S�hoo_l , 1937 . · · . ,

b . In · each of th.e 3 9 figure s given by Edwards herein.

!b.lt!.x.:ll! ' ..

In fig . 136, produce, · liN to P . Then sq . AK - - (rec t . BP =' -paral . BHNK = sq . ,.HD ) + (r_ect .• AP = paral . HAON = sq. . HG'� . · "'

:. sq . upori AB = sq . . upon .BH + sq . upon AH . :. h2

= a2 + b2 . · _ . .

a . See Math . Mo . · (l:859 ) ; Vol . � ' Dem . . 17 ,� ._ fig . 1 . \

!f!.lr.-!.x.:.!tY!D. \

I '

-- ---------1 . .

..

-·

I .

..

I In fig . 137 , the -constry��:l,_Q:q._ i s evident .. · Sq . AK = r�ct •. BL :T rect . AL = ·pa�ai . BM + · paral . AM

'. I

= _paral . BN , + _ paral . AO · = sq . BE + sq . AF.. _ .

:. �q . upori AB - � sq_.� · upon BH + sq . upon AH . a . See· Edwards ' Geom. , 1895, p . 160 , fig .

(28 )'; _ Eb�ne qeometrie voh G. Mahler, _ L�ip�ig; 1897,

l�

. •.

� - .. - -- � - -'

GEOMETRIC· PROOFS

- _ __ _ _ .,._ __ _ �_ - --'

-.

I -- I

- - ---:-1 ' '

F .

-

- - - --

�

E--------·----+------�· � ------------p .. -80.,-f'-i-:S-.-:60-;---and.:.f4th�-- --------------------l , ' ---- --

' , , , . . . ,E __ Mo . , V . · IV·, 1897, p_. 168,. �_: - - - _.· ��.... ,

' ' proof XXXIV; Vers·l.uys, p . 57., · ', -�·- ..,.�:·- - - �� "'fig . 90, w�ere +-t · i_s credit -

' \ · · : . . , ., 'I) :_ eel to Hauff ' S· work, 1803 .

.,A. . I fB � .

_' .

.

j - .

. :. ' : . I I

-

�

1!. IM I . . • � . I . I , , , . .

·c_• .. ,." -· 11.... ' ·• ·K I� . • I• �... ' ' -.. ._ .- - .- � .J •'

fiS·� ·137'

. . ·�--

' . \ '

. . . ·· .. In fig . 138, the con- . .

7_/ struction _:t� �v�dent , as· · we,ll · ·: ,-ie� . · - $ as the pa:r-�21 ¢oP.tl,\inlng · l.ike

v . ' .

'

.. : I. 'o ...... · . -1( , _ , . ""' , . numerals . .. . ·

. . . . _ ...... - ..--�--..,....,...;- { , .IA3 - � - ,, � Sq--klf-=-tr±-BkJ:;----··----�- :--·-··--.....-:--.. --,; ' - \ "4\� -f -� � . . ·�� ,\ + ��i . Clqf. + sq . LN � (t�i . AC�_.. . - '

. · · ' t · ·_ :L:"' . ' , + tri . KBP ) + · tri . HQA; + .tri . , .

:l . '· : A.� , · . I . , ,IJ . QHS -+. sq • . RF + (r�ct . ·m., = sq. . i: •

'· .

•· '-l�.�, 1 . : Jn>_ ·+ rect � AP = sq . HD + rect . · ·

· I · · .t•;,,.. · N I - J GR) = sq . HD + sq. HG:.- -I· - ':S� I •

AB. _ 1 , M ' , _ 1 _ - �·; sq. upo� . . · = . sq .

· • , ..3 �I< upon BH + sq • . upoh AH . :"l!.. - - -_ � . "' � a . See Heath ' ·s· Math .

Fig. 138 Monogr_aphs , No . 2·, p . 33, proof

. ' \

!b.lt!I:.I!r!l .\ <., \ �--· -

. r.-Prod�ce CA to . P ,. cn:aw _ P�,. joi-n NE , -�raw HO :;P&rp . · to OR-, . OM p�r • · ·, to A:{{, · j o:rl} MK .and � . al1.d p_ro­d.uce DB to L . � 1Prom thls diasec.tion thezte results:.: Sq. AX·.•·

. r·ect . AO + ·:ec;t . ·BO

_=. (2 tl':J, • 1· �c . � · 2

-tri .

ACM = 2 t�i . HAM = 2 tr;L. AHP =· sq . 'Hq ) + �rec.t . �HMK • 2 tri . NHL = 2 .tri . · H:LN = 2 tri . NEH = SQ . HD ) . .,

' A

·.

. '

. .

f

\ ,_

I ·�

.

. -

/ • / / /

'

i

\, '

. .

,.

•

124 . . . THE PYTHAGO� P-ROPOSITION •

'fl.Af :. sq . .. upon AB. = sq. I ,., \ ' 2

c;._ , " 1 ' ,' �E upon HB + sq . upon HA . :. h

- ,'" I - � If , -' t \ = -- a2 + ·b2 • Q·.E·.D . r ' "' "' ' - • .

' � ' 1 • ' " · . a Devised by the

' \ ,· : " : , l> author No� . 16., i933 . . - ----:...--. \ I I

', jQ ' . . . . .

. :- . A I. ' t , "'� F o r t � I ' ' L, I . - - - -�

I 'Jt"' t .• I " I \ I I � I .

. c , ·� .. 1-

\ I I:J �- -- - .J.! - � ,,

.. _ ....

· Fig . 14Q - suggests its constructionr as all lines -dl.'awn are either perp •

Fig. 139 . , __

9� par . to a side of ��e giv­en tr� • � -ABH . Then we have

· / Q sq. AK = rect . BL + rect ; AL -� - ' ; -�--- -- - �JJ, "· : · = · paral . BHMK + :PE!raf. AHMO . '

. , ;1', . • = paral . BHNP + paral . AHNO f, \ I t . . •

a ' · . " 1 1 • ' '!"P · = sq. � + _.sq. HG . :. sq .

----- - - -- ---- - -- -----t--..j

• .. �--- --- -------...........,

. " - � - ---

, J ' � -t � . E -upon AB =· sq . upon BH + ·sq • .-------+----------·•.,.< ·�-' --: --' '��" i{ '

' -�pan-AH • .;,: This i S known

--a

-s

-_ ---+---

- ----- � - ·-:..:..-=- - ��f..:.. �:-·- - -- _ . ,· ·-- ---:� - -� �- �H�yne·s-•---p-rc:ro-r.;-:-.-'-S""e-e·-'-Ms;th·;--Ms:g-

r·�· - .

. j

I ! I , . !

•) i

1·------------ -�- ---- --

' . !

/

' 1 • ,, , ' · azine � .vo� . · r , 1882 , p .v 25., __

A 1 and-S�_no�ql Yi'sitor , V • . IX, · ·

1 , "" :r88'8�· ·. :P . 5, proof IV ; also ., , .,

·�� see _ Fourrey, p . 72 , fig . a, • , ' 1. , f in Edition arabe des Ele..:. I . , ' I

C u:� - J� �R ments. d 1Euclide s .

Fig . ·14o

Draw BQ perp . to AB meeti-ng ·GF ·extended, :tiN pa� .• "G_o BQ, NP _par.. to HF, thus forming OAR�; draw OL par . to AB-, OM par . to .Ali', AS and KT perp . to OM, and SU par . to AB, . th�s - dis secting sq. AK into parts

· 1 , 2 , 3., 4 and 5 . · -

Sq . AK = paral . AEQO, -tor sq . AK = [ {quad. ASMB ;: quad . AHLO ) + (tl'i . OSA = tri . :tfFH = .tri . OGH ) + (tri . SUT = tri . OLF ) = sq. HG ] + ( (trap . CKUS .

'·

' ..

f;

..

. " .. .

� �-- ,.... -

• 0

I

-

.

Fig . 141'

-

= trap . NHRP = tr1.. NVW + trap • . �, �inca tri • . EPR = tri . WNV = trap . BDER ) + · (tri • . NPQ = tri . HBR ) = s·q .

_ HD ] = sq . HG .+ sq • . mr. :. sq. upoz:1 AB = s·q .

upon BH + sq; upon HA. :. h2 . '"'2 ' 2 a + b •

a . "This proof and fig . was formulated by the author Dec . 12, 1�33', to _ahov j;hat , having ·given a paral_. , and a �q. _qf equal areas.,. arid dimensi-ons of ·

0 paral.. = t)lose . of the sq-. , · · the Pa1'�1 . can b� diss�cted •

into parts , _ e�ch equivalent tQ a _ like part in -the square .

<•

!--- .. �----- -.� - --- - - r--- ----· - - -�.- -- - - --- - - -,- -

0

.,-'rl'

S ., I ' · · o.·,\ r . 1 ' )M � '

r. , .. . . . --·::.Th.� cons.truc.t�on o_r f1g�.J.�2.�J.s- easil-y. S'een . - -.Sq . AX '=-� rect :" BL + rect . AL = paral . HBMlf + paral . AHNO . .

Q_ ., _, I . , \ I. ; � �

\ - '}K--·· > ,/ : \ . ' \ , 1- , . � - I � .

\ I ., t ' I .I' IJ ' . ., ....... I I..,_ •

ft I . ;B I I I f • I

I I I 1-. I .

I

c '- - .:.. - 'J-_J. R.

= s·q . 1ID + sq. . HG. :. sq � ,., upon AB = sq •. upon BH + sq . upon � · :. h2 = a� + b2 • -

a . This is Lecchio •·s "" . proof, 1753 . Also see Math . Mag . , 1859, Vol . 2, No . 2 ,

. :t•em. 3 , and cl!edi ted · to Chattles Young, .Hud�on, 0_. , ·

. (after�Brds P�or . Astronomy, P·r,i:nc.eton -O,ollege , N . J . ) ;

Fig . 142 Jury Wipper1 +880, p . 26, - fig •. 2!2 (Hist-oricai Note ) ;

Olney ' .s Geom. · , �872 , ·Part III, p . 251 , 5th methoq.; Jour . of Education, V. XXV , 1887 , p . 404, fig . III ; Hopkins�' Plane Geom . , 1891, - p . 91 , fig . II ; Edwards '

t) �

... ·• .

I . : !

---

v

-.- . . . -

.·

I .

/ /

t

I I

l-

.

, -

..

.. - - �----

'•

.. . ' . I

-

\ \ J •

'

' ·

-

�"--

THE PYTHAGOREAN PR�POSITION \ --·----�---------

126

----- Geom. �-�§.25�-P�_]59_,_�fig_!__jgs_)J_ � .Jtfath. _l{Q_._,__V .-:-lY·"-· - -�- _ __ - - � - ·-

---- --' ----- -- ---:r8971- -P�- -169-; - �I . 1Ieatli ,. $ -Math � Mono,sr·aphs , No�. 1 , 1900,_ p .• 22 , p:rioor· VI ; Vers1uys , 1914)- · p . 18, fig .

..

14 . . � .

-··

,. b . One ref'erenc·e says : "Thi :� proof :ts but a -ps.rt.ieular cas� of Pappus�' Theorem."" ·

c . P�pp�s was a Greek MathematiQian of Alex­andria, Egfl>t·., supposed to . have 11 ved ?etw�_�n 3�,0 �-_and 400 A .D � ' · . · .

' \ .

d . Tq.eorein of P.appus : " If up oil: f!-:rf.: 1;wo sides_ 9f any triang!J_e., paralle·l"'gi.'ams ar� constructed, (see .. fig-. · 143 ) , th�;ii• sum equals the pos �ible resulting · parallelog;ram det·errirl.ned upon the thir.d side of the t;riangle . "" " '

· e . See Chauvenet 1 s Eiem i y Geom. '(i890 ) , p . 147 ,. Theorem . l7 . -Also see F . e . Boon ' s proof, Ba,. p . · 106 .

. . 0 f . Therefore the - ·so-called Pythagorean Pro:ao-

sition is ' only a ��rti�ular �ase of _the - theorem 6f Pappus ; see fig . i44 herein .

. \ \ .;

- - - -- --·----l.

a.-------- --- ---- '-----�.--- ---- ---- - .'l'he.or_em o.f_ .P..app.us.---�- _ -- --�- - -

I

I :_. �� < '•'

\ � -\ - '

Let ABH Qe �ny tr+angle ; upon BH and AH co�-.struct any · two dis�ilnilatt lpar,aliel9grams BE and HG; . PfO<;lu�E, �F and DE to c·, their

po+nt ·or· intersectiop.; join C and · �C H J a�d j:>rodu�e- CH to L mak�ng KL .

' • '· E =1 OH; ·through A anQ. B drav MA' · to ,: {) N/, ·mak-�ng AN = CH, ·and OB to P :ma�-,1 -H� \ .

I . I \ :1Jng BP :f: .. o-H-:---· -.- . ' -

1 .1 \ · .:. / · •. · . . Since tri . GAM· :;:: tri . FHC ; · I I . f.L ·

�-1 . '

.' 0 . ·

· pei:qg equi�gul�Jt- and side GA · �F - -, - �- I= FH . :. · MA = CH 1= AN ; also BO 'I I \ I -

) ' ' ·.s I I I I 1)) ) = CH . = . BP = Jq, . lp aral . EHBD . - ·c.. B� + para:l . HFGA ::�. paral . CHBO

.A • 1· • + p�ral . HC� = paral . KLBP . f . I

1 . • • L ,...,..,. + paral�· � = para�. · AP, •

. N itt: .. - - !. - .J· I"" · Als·o paral . liD . + pa�al . HG J'ig . 143- = p�ral . -MB , as paral . _ MB = paral .

AP .

•.

. ..

· -·

i -

I •

)'"' ' . ' ,;: tlEOMETRIC PROOFS

' -·

..

'•"' ---�--1 ------- ------- --- - -�----------, . -

- -�- - ���_-x$-pilraT.- -HD e;;na· parar. --·H�·-are- not similar, . 1 . . it follows _that .BH�- + HA.2 l AB2 • ·

b . See· Math . Mo . ' (1858 j , Vol . i, p •. · 358·, De�.

8 ·, and Vol . · .II, pp . _ 45'!"52 , - in . whicl}.: this_' .theorem is _ given by Prof . Charles A . Young,· Hudson,· 0 . l no'W As-tronoJ;ne�, 'Princeton, N . J . Also D·av1d E . Smith ' s Hist . qf Math . !' Vol . ' I ,. pp . 136'-7 . . -

_- c . A7-so see Mas(?nic · qrand Lodge Bulle:t·in, of . Iowa·, Vol . 30 (19�9 ) , No . 2, p . 44, . fig � ; also"'"'Fotttt­

rey, p . 101 , Pappus , Collect�on, �� 4th century, A .D . ; also see p . 105, proof 8, 1:n . "A Companion tb Elemente,ry School �thematic� , " (lg2.4)..,_. . ..hy....F . C • Boon, A .B ; ; also Dr� Le�tzmann, p . 31 , fig . 32 , 4th Eqition;

·- "· , also Heath, Histpry, II, 355 . 0

J • d . See 11<rompa�on t·o Elementa;ry Schoo!l Mathe­matics , " by F. \0 . :Boon, � -�· . · (1924. ) , p . 1-4=;::-Pappus-�­lived at Alexandria _about A .D .• 300, though date is un­

·oertain. ·

I . i

e . This TheorE;tm a-f Pappus is a generalization �. ->+---"'·£.....the-P-y..thagor-eap----Theorem� .. -- �IDhe�e-.f!o.re-the--P-yt-hagorea.n--. Theorem ' i s only a corollary of ·-1:he Theorem of Pappus � · . ! � - __ __ _ _ _

-------t--- �----- � - - --- �- -- - -- -,--- ---------- --- · t ---� - · - - --:·---·- ---- ' I l

• t

'

f2t1r.:.!b.tlt . .

. .. , · By t_heorem of PappuS", '"f-L MN = LH . ·sin(3� apgle BHA is

E , " 1 ' , a rt . angle_, HD and HG are / � ' rectangu+�, and assumed

" ' ' E " '

·

� .squares (Euclid., �ook I , .Prop .

(J, " ' , 1 • , "-- \ 47- ) . But · by Th�orem of Pap-l, . H , ' ' ' pus , paral .' Hb + paral . HG

- . I

./ \

· · � I ' --

< '

. / . t '\ . �;�'i> . ·= parai . �AX. _ , : ·

' . . :. sq ..-·upon AB = sq . ' :

· ..

¢ •

A) .--'\ -! . upon BH + sq . upon HG. ��. h2�-'t � \ ·- I '2 ' ' I ·= . g2 + b • \ I I . I

I I I c :_ - :.. !N� 1/{ '

Fig . 144 -.

.a • . BY the .author·, O�t .• 26, 1933 .

..

..

. .. .

' . � .

' • • .;c.

.

.·

I

·.

I - __ , -

- . r - . - - - --� --

. !

. j . ·----­' l

· ·. I

...

I

1.1

128 THE P�GOREAN. PROPOSITION ' \' , f2t1I:.f21L!

L P�oduc�· DE to L mak-t' f 1 ' , 1� E:L = �-,t produce KB· to 0 , � � , and drav' LN perp . to CK. Sq . r / I ' )·0 · t:.. / / ' , .l - - ( � --AK = rect . MK + rect·. MC 1 - -

� . 1 4 ' _ 1 ,. , = [ rect . BL (as LH = - MN ) '' 1 H.' ./' : \ = _sq . HD l. + (simi}.arly, . sq.

' · I · � ' HG ) . , I , · u . . · ·

' . . __ _ !..,· sq . upon � =; sq . 2 A 1 ·

. ' ·

·o upon HB t sq. upon }fG:..,s:. h-t

-: = ' 8;� ·+-.-1-tg_-. -- --- _.::.._ ' -- �- . - . .. -- -

I_ , l . a . See Versluy.s , p �-•

' ,. I _ · • 19, fig -. 15, where cred.i ted ·.C L. _ _ _ 1.N. _J If ___; __ _ _ to Nasir-Ed:.D.1I?-

_

_ (1201-�.274 ) � ·

, Fig. '145-. .

· a_ls o Fourrey, , P. .• : 72 , flg . 9 •

., f2t!1:.El�-.

' .

t < !

I !

- - --- --1',., --�-

�---�-�--------�--- ----�---- - ----�--�-- - :,_ ____ . ----- ------ �----.

--- -�-- - - -

' :-

/.

. .

1

-�-

- \ _· ' --

-. �

'

... -

. �· ,, •,

• • ..

. . " ·

i

' i

- � --� . -� ·-- ---'---,{J!-��---�-�-- · -----·- -- ��!1 . . !;�-·�- -�-�6 ����-�� -·---1--.-,------• . ./' 1 , 'DE knd GF to ·P , CA and X& to

CL-f"' · • ' � : Q _ �d R respectively, draw crt . · ... 1

', f ' ··)>£ p�r . to_ � anq: ·dtt_av ·PL and

G ' 1 '

-, : • .!' 9t .'., KM perp . tJ� �- and ON respec-�, , ... :l -- , ·tively. . Take ES = HO an�

· ' ·1 . · t, · -� drav D�l. · t , � . . · Sq . AK = tri . KNM :w-..--•, -.rc + hexagqn ACKMNB-;tr.L.. -BOH

1 )1N. .+ pentagon ACNBH .= tri . • DSE t , .

, . t!t,' - t · � Pe�tagon �OFtP = :tr+ . ·DES , ' a ' t + . p��al . -AHPQ + .quad_. __ _f]IOR • - ; -�t c'' '

C 1..! ... _ _ -l� � 1< =· JJq. HG + tri . DEs,' + paral . · BP :- :tr1... BQH = sq.. HG + tri .

J'ig. l46- ------, ·· -.

. DES + tra� ; HBDS = sq . - HG + sq . KD . �·-"'

:. sq . upon AB = sq. upon BH' + sq. . upori AH •

. a . See Am. Math. 'l�o . , V. IV� ;t897, p ." 170; proof XLV.

( /

. . " ' .

\ '·

.= �

. . . , · .

. ,_ ..

.. ... ·

' I

' . .

. . l I .•

. .

. I

,.·

GEOMETRIC PROOFS l29

;t . / ' ..... 1

"' -

�J - . - \.\ ..... ::>"t '< . . , . / \ \ -- / - - \ \ . - i --:� /

The const�uction 1?-eeds no explanati-on; from it

· .�e get sq; AK + 2 tri. :ABH ·= hexagon AO�H = 2 quad. AOLH = 2 -quad. FEDG = hexagon

. ABDEFG = sq. HD + sq. HA -+ 2 tri. ABH. • A�. __ ...,._."" :. sq-;-upon AB . sq. I - . - ' 2 .1 . , upon BH + sq. upon AH. :. h 2 2.--

= a + b • , r I a. According �oF. c.

C: I :11 Boon, A.B. · (1924 ) , p. l07 ·or . '� -:-.r- �"7"� his ·"Miscellaneous MatJ:iemat-

.' .. ,1,// ics., -11 this proof ;ts that of � ��ona�do qa Vinci '(1452-1519).

Fig. 147 . b.'' s·ee Jury Wipper,

.: - 1880; p. 32, fig. 29, as 'rourid in �i 'Aufangsgrunden der· Geometrie11 von Tempel- . hoff, 1769 ;- Versl uy:s, p. 56; fig. -59, where Tempe).-� - -hoff, 1769, is mentioned; FourreY,, p. 74. - Alsq proof 9, - p. 107, in "A Oonipanion to Elementary_ School Mathe­matics, " by F. C. Boon, A.B·.; also Dr. Leitzmann, p. 18, fig. 22, 4th-Edition-.

I '

In fig. 148 take.BO · - - =--�afi-d-:Alr'·= BH; and' com::-------- �-1- _

____

__ , __ ----------- --

---Fig..-148

plete the figure; Sq. AK = rect. BL + rect. AL = paral. HMKB + paral. AOMH = para1. FODE + paral. GNEF = sq. DH + sq. GIL

· :. sq . upon AB ::;:: sq. ' 2 . upon BH + sq. upon AH. :. h '2 2 =:= a. + b • a. See Edwards' Geom.,

1895, · p. 158, fig. (21. ) , and

\

I I

, .. ...¥fll'--·

130 THE PYTHAGOREAN PROPOSITION

Am. Math. Mo., V. IV, 1897, p. 169 proof XLI. E2r.11:.�l9.b.! -----�------1� ---� ------ -

I �I

I .. . - '

,;- In fig. 149 extend OA to Q and complete sq. QB. Draw GM and DP each par. to AB, an� draw NO perp. to BF. This cbnstl."UCtion gives sq.' AB = sq. AN = rect. ·AL + rect. PN = paral. BDRA + (�ect . &� = paral. GABO} �-sq. HD + sq. HG·.

:. sq. upon AB = sq. · � ' 2 upon BH + sq. upon -AH. :. h = a2 + b2

O'L.:. _-- -IK . ·/� a: ·see Edwards ' · Geom.,

Fig •-· L4g- __ 1895, p. 158, fig. (29 )� �nd Am. Ma:th_._ Mo., V ··-IV, 1.897, p.�l68, �roof xxiv . · I

/" / / \ . r;-- s ')l (--:--r-1'

\ l I I

E.2r.!1:J!lnt In fig. 150 extend

;p .KB to · meet DE produced.at P,

9.� dra:W . QN par_. to DE, Nd par.

/ ' ' , 0 to BP, GR and HT par. to AB, - - -rr', .. "> ·extend OA to s, draw J{L par.

I Af.D to AO, -OV par. to AH, KV and

I / I l I I I

/ ' . -.--- -- -.�-r-· __ ,:. __ :x-�-......,iiiii-�--· -----MU--:�>B;r-.-to-BH-, --MX-Pal!-,--to�:AH., ._.-

1 '' 1 "'- I extend GA to W DB to· u· and

\ I I '\ /

\.' U.l.:'• I . . , ,

: , ' 1 , 1 0 draw AR and AV! Then we. will ·1 \'?\.V'.'\t have sq. AK = tri.- -ACW + tri .

. •- , , WL1 ��M OVLO + .quad. AWVY + tri. VKL CoJ�-- - '- �II{ + tri . KMX + trap. tiVXM F�g. 1'50

+ tri. MBU ·+.tri. BUY= (tri. GRF + tri. .AGS + quad-. AHRS ) +" (tri. BHT + tri. OND + trap.

·NOEQ, + tri. QDN + t.ri • . HQT). = sq . BE + sq. AF. ! r I

I ' "

I

.. 1

.. ,. ,

. ;

GEO METRIC PROOFS 131

:. sq. upon AB = sq" upon BH + sq. up on AH. :. h 2 = a 2 + b 2 •

-,, a . This is E . von Littrowis proof, 1839; see·

also Am. Math . Mo.,· V. IV, 1897 , p . 169, proof XXXV_II.

F1.g. 151

Elf.!l Extend GF and DE to

P, draw PL perp . to OK ; CN par . to AH meeting HB extenq­

' ed-, and KO perp . t.o AH. ·Then there results : sq . AK [ (trap . AC NH - tri . MNH = p'aral·�. AC MH ·= rect-, ALJ_ = (trap . J\HPG - tri . HPF = sq. AG ) ] · + · [ (trap . HO KB

tri . O MH = paral . HMKB = rect . �L) = (.trap . HBDP

tri . HEP = sq . HD )1] . :. sq. upon AB = sq.

upon BH + sq . upon AH-. :� h2 = a2 + b2 .

. a . See�. Math . Mo . , V.. IV, 1897, p . 16-9 , proof'··� XLII .

' AM = AH, complete sq. HM, ' � ' , -,ij/ , draw HL perp . to CK , draw CM ' � · par•. to AH, and KN par . · to BH; ' , 1 , � ",jJ this �onstruction gives : sq .

.ft. , . AK = rect' • BL�.+ rect . AL r' \ . I .. • 1 ' , '� ( /, '1'�----- -�--- -:=-�ara-1.- · HK + -pa�a-1-.- --HAGN·- ·-

1 , � 1 = sq. -BP + sq. HM· = sq. HD I )_ ·l� I + sq. HG . I / .,. I· ' I AB C1_, u :. -sq . upon · = sq .

·- 11::: - - - .tL...�"' upon BH + S<[. upon· AH. :. h2 � = a2 + b2 •

Fig. 152 '

I "' :· .

I

I

0

/

- --�-- -- - '---- - ------

.\

,,

,_ '·

... ,.

..

132 THE P�GOREAN PROPOSITION

�. Vieth' s � prQof--see Jury Wippel', 18801 p •.

24, fig. 19, as gi_ven QY Vieth, in "Aufangsgrundeii ·

der· M�t}lematiic," 1805; also .Am·. Math. Mo·. ,. V. Tv,· 1897,. p. 169,.proor XXXVI.

\\ 't\

].Pig • . 153 --. '

In fig� 153 co�s�ruct the sq. 1lT, �alf· GL,. HM, and PN pal'_ to. AB; :also KU -par. to BH, as par. ·�o AB, and join EP. �By anaiysi·s· we· rip.d that · sq. AX· � · ( tl'ap·. CTSQ + tri. I�RU) + ·[ t.r1:. CXU + quad., STRQ + . ( tri-� SON

=· tri. PRQ)- + rect. AQJ = (trap. EHBY + tri. EVD ) + [ tpi.. GLJi' + tri. HMA..· + (paral. SB· = paral.- �� ) ] ·

= sq_. HD + sq' � AF. :. sq. upo� :AB = sq.

upon BH +.sq . upon AH. � .. � h2 :::;: a.:Z·+b2• Q;E.D. -.

a. After three days of·analyzing an�'classi- J :rying solutions·based on the A type of. figure, the above dissection occurr.�d to· me, July 16-, · iB9o', :t:rom vhic.h I devised above proof . •

,.,f \ .

·.-/ \ - . In · fig .�154 through ;

r. �/ ' , ' $ "'( ... .,�,. , K draw:· NL par. t :' AH, extend \ , 'HB to L, GA to O, .DB. to M,

' � dr�w DL and·· MN par. to BK, y

l •

l I ) l ! I I

r '

. -·

, I

\ //1 u - , -· --· --�-- ------- ----T"'---an-d·'"C'N-pa�-;--t-o-A-6�. -- .----__::..·--+--.-------- ·1 ·

-'

A· , . / I Sq. AX .. = hexagon - . ·

· · • , " , ., t· AC�M = paral • . ' eM ·+ par�l.,· ·� t ,. /

· t \. I ' . 1 M ,..... ""h A' 1 · KM = sq. ·co + sq. ML .=·. sq.

· I r;,.. I . �- . 1 ,.1 , 1 ,.' HD + sq. HG.

·a' / 1

. ' lu/ , · :. sq. upon AB· =- sq. \"-- ,. - .j,. -II' .

. , 1 0>. upon BH + sq. upon AH •. t.t�., ""' Fig. 154

I I.

···'"' -. ... .. -

.. ... . .

,. . ' '

- ------'

(16).

�\ .

. ,-...

r'. ,.

a. -s,e·e- ·Edwards 1 .Geom. , . 1895, ·p. 157, fi'$ • p

,E. , ; ' \

\

fif.11:.f2JLt

- ..

In fig. .155 extend 1J;B t 9 M l!lSking BM ? AH, JIA to P ni_aking AP =_ BH, draw ON and KM eacn par. to AH, CP

'/I) and KO each �erp� to AH, and draw· HL :Perp. to AIL Sq. AK =.rect. " BL + rect. AL =para�. RKBH +· paral. CRHA =sq. RM

Fig. 155

+ sq. c·o· r= sq� HD + sq. 'HG . ·. �:. sq. upon AB = sq.

upon BH .f. sq. upon · AH . :. h2 � a2 + b2. . .

a. See Am. Math. Mo. , v • .,rv, 1897, · p. 169, pr.oo( xtfii.

f.lf!1:.f.l�l ._ _ _ ��tenQ. liA to N ltlaking AN = HB, DB and GA to M, draw, through 0, Nq ma�ing co = BH,: and .join MO and KO.

Sq. .AK = hexago� ACO�M =para. COMA, + paral.

�---. eRBM-=-�HD���HG-. - ---------.Af).._.__....ol!'..,. · � · · .. sq. upon. AB = sq. H"'" i ,. .

· !/ • _ -qp·on BH + sq. upon AlL :. h2 · \: ,, � l = a2 + b�.

\,. J, ;\ ·� ··1 a. This proof is. �I' I/( ·cred;tted to· C. Fr�ncn, Win- · -------- -- ------ �,-1-�J:t l chester; N. H. See Journal - of'

,. . Education, V. XXV;I:II, '1888,:

• " '

bt,"' · · · p. 17, 23d pro9f; �dw�rd� 1

Fig •. 156 · 'Geom. , 1895, p: .159., fig. (26); _ Heath 1 s· ftJa th. Monographs·, No •

2, p. 31; .proof XVIII.

I

� � , I I l !

;

I i I.!

l l .. !

�-� . .

. -�� �---

l J

..•

I'

' i '

, . •'

Q •

'\ \

--� -- �

------ - -

-

-

---

-

---

-

-

-

-

-

-

- --134. THE PYTHAGOREAN.PROPOSITION

� '

. '

t Complete the .sq's OP and HM,-.·which are equal.

Sq. AK = ·sq. LN -.--4 tri. ABH = sq. ·OP - 4 tri. ABH· = sq. ED + sq. HG. .:. sq� upo;n .AB = sq. upon :aH · + sq. _ .

HA . • h2 . 2 + b2 upon . • .. · = a . � . .. Q.E.D. ,

· · ·_,./ I '- .. ,_�--""- I· ,

. .a. See Versluys, p. � ·I v· ! ' 54, fig. 56, taken fro� Del-\ I -p . r' }N boeuf' s. vorlc, 1860; .Math. Mo;,

. I

\.! _ - -· · / . _- .1859, •Vol., .II, No • . 2, Dem .. l-8, -

\,i'(. _: ____ , fig.- 8;. Fo�rey:, Cilrios. · ·

'., / . Geom. , p. 82_; fig. e� 1683 • .

- �-tf<-Fig-._. 157

,.... tf

_f:' , , 1 � . . El!i'l:ltll!l _

_

k' I ,� / \ \ E. Complete rect. FE and ·t;.// \, I /,/', cons�ruct t-he tri's AIJJ' �nd

· -:'\- ·• ·

-

\ KMB , -each.= ·tr1. ABH-.---: '· _

, \ · � · I-t · is obvious that

' / _·..!'sq. � = .pentagon CKMHL - 3_ ·

.- ", '• t • I . , \ ·-\ I /.. ' I . /14 ht • ,

':-� --

-- �- -1( tlig·. 158

' lP

tri. ABH = penta�on �DNG

:. sq. upon AB = .sq. upon '2 2 HI;> + sq. upon :aA. :. h = a.

+ 'b2 .•

fig. 57.

r ! I

. '

• ., . ,... .. J • �-·- -:s,.

. ' " .,._ . ..

In fig. 159 complete the square.� AK,_ .. HI) and __ ·:. ______ _____ _ _ _____ _. ___ · ---·------"11 • I

HG, also the paral's FE, GO, AO, PK and BL. From ... ·' .. . '

" . . . ) ..

\ . I \, ' \ ' .

------- '" --�----� -� ----- - - --------- - - - ---- - - -- -------------

. ., . 1''

0

. . '

:

I '

GEOMETRIC PROOFS . 135

these we find ths.t sq. AK . , . /�,-

= he.xagop ACOKBP = paral. �,... \. OPGN -- paral. ' CAGN + paral .

/. I ,, E POLD - paral. BKLD = paral •... . G. � .

' \ ' ' A." -( ) , .. I / , ., LDMH _ - t��. MAE _ + tri . LDB

� . I ! · \ .: • + paral • ·GNHM �-:;..:· ( trl. GH'A

f ' . . . �--�'=��:,.t�:i HMF.) = sq. - HD +.1 sq. HG •

. · ' - ,/ 1 ' : .. sq . upon AB · = sq. upon BH · 0 ' -� .. 2 2 -2

�----- . . , + sq_ •. upon - AH. :. h = a + b • • ,. ' / "' I ' s 01 I Ge V' 1 \ ,/ ;t ' a. ee n�y ;S OJ:ll,,

• �. 1 -� • 1 '�L Un�ve.rs�ty -Edition, 187_2, p. \_\. t .f._.· t /.)- 251., ath �t�od; · Ectwar�� 1 ..

� ..J ____ L, �om. , ) .. 895, -�. 160·, t�g.

. -- .. ' . ----- -----....:

�\ I . /� · -(30�; Math-. �o. , Vol. II - � /' 1859, No, •. �� Dem� 16, fig. ,a, bt,,. ariP, W. ·· Rupert·; 1900.

Fig. 159·.

- E.l!-!l:.ll!!.! .(

-. �n��;!�g. -159, om+t lines GN·, LD, EM·, MF and MH, then tne dissect±o� come�. to: _ sq. iK = hexagon: A�P - 2 �r.i • . ANO = pa�al. PC + par�i. PK = sq. HD.' +' sa· HG. :. sq. upon AB = . sq. · upo� HD -+- sq. upon HA.: :. h • a 2 + b 2 • , Q. E. D • .

· .

a. See va·rsluys, ·p. - 66, t:t:g •. 70.

, .

t,

Sq.- AX_· • 2.(quad. AKBH . - .tri .• --ABH.) = 2(quad�· ABDG . . .

· - tri. ABH = t sq. EB + t sq. -FA) = aq. HD + sq � H;G •. :. sq. upon AB. = sq. upon HD + sq •. upon .HA. ;.·. ·h2 • a2 + b2•

.. f

-

t

.. •

....

. .

..

. �.

1·

• J

I ,. .

. r. - -

.� . '

. I

\ "

\ ' �

' ' 0'

• q

- ' --··

--

, · I>

I

:

• . '· •' �----"\-

--, - "136 THE PYTHAGOREAN PROPOSITION �----------�--����------��--------�-

..

fig. a .

-. ' ' .

. . . ../{'" · ·GL. and. DW are each

Q// \ j A perp . to .AB, LN par1• to 1m·,· �--- -·; y , _

QP. anq VK,par . to BD, .GR1 DB', , \ ___ 'X,'b MP, NO and. KW P�r. to AB and I \ ,?1 �'r* and RU perp . to AB.. _Trf .

� _ _ · _ , , :?!.Q DKV = tzii_ . BPQ . . :._ AN = ·MC.

I • Sq. AX • reqt . AP ; \ 'fbi �/.. ' . . (

- •

\ r- -:--"'T . t · + re�t . �o = paral . ·ABDs .. 'Hi----�_ ; 'v ·=:sq., HD ) + {rect . -�u = p�ral.

f I /1 , �ABR ·= sq . GH). /· sq. upon CL---���J.<L.� . ..JW AB = sq . upon BB + ,Sq� upon·

2 ' 2 . 2 - �

HA. :. 1:!- ::.,....a + 'b • Q .E· .D . · Fig: 16i: �a . s·ee 'Ver'sl:uys, P• �

· · . · 28, fig •. 2�--one of Wtrner' s ·· coll'n, cr�dited to Dobriher . · .

..

. ;:�· J'ig. 162

lll1l:.!!!2 · I .. ·' t ' I )

Construc-ted and num.:. bered as·her�-depicted, it foii_Qw� that sq.- AJt· = '[(trap •

'jim = trap . SBDT ) + (tri . oTQ; = t:ri . TVD ) + '(quad •. PWXQ ' -. = quad . USTE) =. sq . liD] - � .-+ [ (tri . ACN = tl'i . FMH) • · ·

+ . ( tri . CWO a= tri.-.� GLF) + (quad • · ANOX = quad . GAML ) =.sq.. H�.h.

:. sq . -upon AB = sq. ·upon BH + �q . upon �. · :. h2

2 . 2 . . = a + b • Q .E . D . ,

. �. Se� -Vel'sl uys, p . -33, fig . 32, as given by J�cob.de Geldel', -1806 .

. . -.

. '

., . I·

.. .

0

...

I,

-GEOMETRIC fROOFS 137

. ll!!r.:.!fi.t!!

· · Extend GF and DE to N, com­plete the square NQ, . and e�tend HA·to P , GA ·t·o R and HB to L.

From these ·

•

dissected. ·parts. of. ' the sq� NQ�e see that·· sq. AK + (4' tri� ABH + rect: HM ·

+ rect. GE + rect. OA_) ·= sq .- NQ = (rect: PR = sq. HD + 2 tri. ABH ) + (rect. AL

· = sqo HG + '2 tr.i. ABH}'.+ rect . JIM + �ect. GE + rect. .

F.iB.•- 163 AO • sq. AK. + (4 · tl'i .. ., ABH + rect. HM

. '

+ rect. GE + rect. OA - 2 tri. ABH : �} tri. ABH - rect • . HM �- ;rect. GE ':"' r�ct. OA = sq. lJD ·.:;.. sq. H(J..

-- ---- - ::sq. AK = sq. HD + set.· l!G. . · :. �q . upon ·Aa = sq . up�ni BH· + sq. up�n AH •. :. h 2 = 8,2 + b 2. . I

. .

· a. C.redited by ·Hoffmann, in "Der, Pytha- . gorai·sch� Lehrs�tz, " 1821, to Henry Boad�, of London,

·----:.�----·�--------"'=· _E=-hs . - . See ·Jury Vipper, 1880, p. ·la, · fig_J.g; 6Ve'!!,,r.....::.:.���-----+­sluys, --p.-- 53, fig . 55; also see Dr . Leitzmann, p. 20,

' ;

r:

fig. 23. . . ' b._ Fig. 163 employs 4 congruent triangules,

4 congr\Ient rectangles, 2 .congruent ·small squares, 2 congruent HG 1:1quares and sq. AK, if the line TB be ·�nser�ed� Several variations of p;roof Sixty-Three �t·b� p�o4uQe4 from it, if-difference is so�ght, �s­

� peci��ly· it. certain auxiiiary lines are drawn.

, ... �' - I

.� "'�"' It·

.. ( � ,J 'i

r " ,:;,

---------- .

•

' �- .,

•

-

· -

l J l ! � . , L

' .

' I I '

• I ! l I ,

� I

. (

,· .

/

138 . .

THE PYTHAGOUAN PROPOSITION·

n, - ll!.!I:.��II.t

;'>'

In fig. 164, produce HB to L, HA to R meeting CK · ' . . prolonged, DE a�d GF to b� CA to P,

,ED and FG to AB , prolope;ed • . Draw HN p�r. to, .and OH· perp. to AB. Ob-viously sq. AK-� tri.�RLH - (tri; RCA + tri. BXL .+ t1'i � ABH) .t tri. QMO.-""' . ( tri-.. QAF:� ---

. , Fig. '164 "�>:�) + t�'i. OHD · + tri. o <� • • ..1 •

•

•• ) <�· -�:g; • .·. · AB1L) = : (pal'al. PANO

= sq. HG )' + · (paral-._HBMN = .sq. HD ) • • I l · · · :. sq. upon· AB. = sq. upon,· HB + sq. -upon HA. 2 2 2' -•

•• ·h = a + ··b • . , ·

•. a. See Jl:J'y 'ffipper, 1880, P• 301. fig. �Sa·; Versluys, p . • 5.7, fig. 61; Fourrey, p. · 82, Fig. ·. c and d, . by H. �·o,nd, in Geometl'y, Londres, 1683 and :).733, also p·. 89 .

0

. ' 0

1-

In f1g.'.l65. extend--+----::-------�----. HB and CIC to L, AB and ED to. M·� DE and GF to ·O-;. CA and K8 to P an4 N· respe�- ·

/ . tively an� draw PN. Now observe ·'\ihat sq .• AI: =-= .(trap. 4CLB - tri. BLK) . = [quad. ·

�AMNP = hexagon AHBNOP� - ( tr1. NMB =: tri. BLX) = parai BO = sq. HD) + (pal'al.A9 �- s�. AF )J. . .

, :. sq. upon AB = sq. upon BH. + sq. upon AH •.

0

,- .. --

· '

. .. , '

. -f.

. ' -

-... . -

. I> I �t, r -��·

-� .. ,;,_. ····�-- ..... _41,..........,..__

" '

, .. r •

-----· --- ---

· ·GEOMB'l'RIQ .PROOPS·

• I

139. a. Devis��--b7 tl\e. author, cru�y 1, 1901., but

suggested qy f':f..g,�· 28b, in ·Jury'W:ip_Ref', 18�0, p� 31 • . · . ·b.· By om1tt:ir�, ·frbk .. the t,tg:., the sq. ·-AK, &nc:\ · the tr1 � s �LK an,.d 1BMD� an. algebraic proof through�

: 'the mean proportional is easily obtained. . · • • • • .; r., • •

' .

. ' '

In the· construction make CM - HA- =

·,PL , ':r,q ..

= FP, MK = DE =:= NQ • .• �. OL . .; -!.M . and· MN '

� = NO. Then. sq. AK - . = ·t�i . .-NLM�- (tri..

- . . LOA. + .tri. ·CMK .. + 'tri. m) = tri.

LNo· -. ( tri • . OPH +. .. tri. HAB .+ tt-1. QOH.) • . paral:. PLAH + paral. HBNQ_ = sq. - - -- -

. HG. + sq.' HD. :. sq. ' . .#

uppn AB . =:= sq. 1,1pon BH'+ sq. upon HA. :.-ha =a� +ob2 • .

·Q .E. D. , ., :. a. See .. versluys�. p. 22, f!'k"· 19; by J. D-: Kruit;;.. bosch • ••

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....... �-�-

- ..

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·.

· . �{. ·�..r · �. ?•, /i '\ :R. ·· . ud.x.�§.tuo..

-------

. - <('_,, t ,th . . . .I \ I 1 ."fl) · . · Ma�e AM = AH, BP .

: 1 / 1 • BH, complete,paral • . MC and l . . · PX; �xtend FG and NM to L,

-----

' ' �· ' i ., '

�� ... .:u_·_J-J4_-.�'tp. DE apd lCB to S, CA to T� OP

-.- to R, �d draw-�. . I · f· · J I ., . ·sq�· AX = paral. MC • ,_ · · � J + paral. PX = _paral. LA • ' -- . - t-

- · / � · . \ f +- parai� RB = sq. GH + sq. ·lJV. -j�. ·.i67 · \J Q . HD.

� ---------�--- - - �-�-------- -· - - • l

•

. . Q

fj

-- ----- ---------

' '

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----· ---

I I

=1�40�-·---------=T=lm=-�p���-�· ---oo_� __ -__ P_R _O _PO_ S_I_T _I _ON _______ ___

. h2 , .. :. _sq . '-Ip_on··AB ·;;: sq. ·u.pon HB + s�� upon HA.:k

= a2· + b2 . -- -

a. Math. Mo . (1859); Vol. II, No . 2, Dem. 19, .· fig . 9.

Gr·. <

l •

§.!!.1:i:1l!lb.!

· �om P, the �ddte · poin1;i .or' AB., · d.I'aw. PL, PM and

PN p�rp . respectively_ to 0�, DE a�d FG, dividing t�e sq's

. AK,, DH anfl. FA into equal :rect.' s .

Dr.aw EF, PE, OH to R, PF arid PC . .

Since· tr;t·t a· BHA and t.- - � -� ,...��Er·c_ong:t•�ent, Et = AB ';:-----AC· •. ----Since pi£·= Pt, the · -- -trl '"s PAC"" 11PE ·and PHF have

e-qual ba·ses . . J'�-- ·--· : Since �ri.' s having-- " ..

equal bases are ··'to each .. other a's ·-th�ir al·ti1;i'l.ldes; ·· tri . (HPE = ·EHP = ·aq. BD + -4 ) . ·

: tri . (Pf�F. = 'sq. HG + 4 ) = ER : <FR.'. :. t�i . HPE

, I

-f:- __ tri .(; P.HF :· tr.� . . PHF = (ER .+ FR = AC ) : FR. :. :k sq� -HD .:r· t �q·. HG : tri . PHF = Ac·_: FR. · .But ·(tri· . PAC

-- = t sc{:· AK) : tri.-. PHF � AC : .FR. ' :·. t . HD + t sq . t-�___:___::_··-�......:.....1---__,. __ �H·G�1-·�· -: �*�� s�q

·;.· --=

.AK4

=�=:;t� r:...::i=-i .

v-PHE:=.__'__:· -=:--::.

·t�r=:-ic:_� _P_...:HF....._..:_·�----'=!��-HD ____ '�--��- _____ -�-�----

{Jq. •. -�

;

- ...

. . ·

:. sq . \lPOn .AB = sg .• upon HB + sq. upQn HA. h2 2 b2 Q E"n ----· . :. . = a + . . • • • ,

_·a . Fig. 168 :1:s unique .. in ·t�at it is the fir�t ever.devised in whi�h all auxiliary ·lines and all '

-I ·triangles ·use�riginate at the middle �oint of the

'hyPotenuse· or. the· giv�n tr;t-,!!!18�8. . . · . . --"

b .• It was devised and �.proved by Miss Ann-Con�it·, a girl, aged 16 years, of Central Junior-· _

Senior Hi�-School, ·SQ�th Bend, In�., ·o�t . 1938. This 16-y.e�o�d gfrl ha�.4o�e· what no great ma�h�tician1 -Indian, Greek, ·or modern, is ever repprted to·have ·done . It :should be known as the Ann Condit Proo�. --: ' .

) '

' ' . ' '

I·

I " ,

/

. . Fig. 169 ..

GEOMETRIC PROOFS

• Pro+ong HB to 0 mak-

ing BO = HA; complete the :regt·. OL; ·on AC const. tri. ACM = tri. ABH; on OK const. · '. tri. CKN = tri. ABHo. Join AN, AK, AO, GB, GD, GE and FE.

It is obvious.that -tri. ACN = tri. ABO = tr1. ABG_= tri. EFG; and since t:oi. DEG = [ i (DE .) ' X (AE = AH + HE ) ] . = ·tri. DBG = .[ t (DB = DB). X. (BF = AE )·] =-t�i, .AXO

. = [! (KO = PE) X (HO .= AE ) ] = tri. AKN = [i (KN =DE) , -x· (AN· s AE) ] , . then hexagon -ACNKOB - (t.�i. . CNK + tri'.

BOK) = {tri. ACN =· tr� •. ABO = tri. ABG = tri. EFG ) + ( tri.. AKN =. tri. AKO == tri. GBD = tri. GED ) � ( tri. . .

'CNK + tr1. BOK) = 2 tri. AC.if + 2 tri. ABO - 2 tri. CNK = 2 trio. GAB + 2 tri. ABD • 2 tr1. ABH = sq. AK

• ' I = sq•. HG + sq. HD .

:. sq. upon AB = sq. upon HB + sq. upon HA. :. h2 -= a2 + b2 • . Q.E.D. .

a. Th:ts fig., and proof, is original; it vas devised by Joseph Zelson, a junior �n West Phila.,

, Pa., ijigh Sc�ool, and sent to me �y his'uncle, �ou�� ' \ � . . � - · G, Zelsoh, a teacher in a 6ollege near S�. Louis, Mo.,

on �'Y' 5, 1939 . It shows a high intell�ct �nd a fine menta�ity. _ ..

·

(9 _ , · b. The ;proof Sixty-Eight, by a girl �·of 16.,

and the proof Sixtr-Nine, by-a boy of 18, are evi­denc�s that deductiv� reasoning is not beyond our youth.

.·

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/A __ - _J�JI(. 'Fig. 170

� .L[�Q.r.�!.·­

If upon any con-venient lenRth; as .AB, three trt-, an�-l es are con-st'ructed, ·one havtn� the �ntle oppostte AB ob­tuse, the second havtnr that anrle rtrht, and ·the thtrd havtng that ofipostte an�Ze ·

acute, and upon the sides tnclud­tn� ·the obtuse, rt�h_t arid acute an�le squares are constructed, then 'the su11 of the th�ee square� are z ess t-hem, equa'l to, or lreater

? -

t'han_-the square constructed upon AB, accordtnt as· the an�ls ts ob-. tuse, rttht or acute.

o In fig. 170, upon AB as diameter describe the semicircumference BRA. Since all triangl�s. who�e ve.r­vertex H ' lie�·within the circumference BHA. is ob­t�se at.H1, all triangles whose vertex H li�s on tnat circumference is right at H, and all triangles whose vertex H2 lies without .s�i� circumference is acute at H2, let ABH', ABH and ABH2 be such. triangles, and on sides BH' and All' .�omplete �·t�e··- -squar.'es H'D ' and

. H'G': ;. ·

on sides BH and AH complete squar&s HD' and HG; on ·

. .. .

I.

\

..

.. �--

.....

GEOMETRIC PROOFS '"l.t - -

sides BH2 and AH2 complete squares_H2D2 and H2G2. De-termine the points P', P ·and. P2. and draw P 1H1 to L.' making N' L 1 = P '·R 1 , PH to L making NL = EH, and P 2H� tQ L2 making Ni�!l:-2 = P 2H2 � : �

ThroUgh A draw AC' , AO and AC 2; simil:arly ·

draw BK1, BK and BK2; complete the pa�allelograms AX', AK �nd AK2.

�.

The� the paral. AX' = sq. H 'D + sq. H 1A ·�. (Se� d unde:t' proof F'ortz-Two, and proo:f_under fig.

143) j tl!e paral. ·(sq.') -AX;. = aq • . JID· + sq. HG; and paral. AK2 = sq. H2D2 + sq. H2G2. 4' .......

· • Now the area o:f'AX1 is less than the area of Ax if (N•L.t :; P 'H') is less tha'n (NL = PH} and the area o:f AK2 is greater than the area of AX.if (N2L2 =· P.2H2) is grea t�r than (NL = PH).

_ � In :fig. 171 construct

. Fig •. 17l

· rect. FHEP = to the rect. ·

" FHEP in fig .. · 170·,. take -� 1 -� H1F in fig. 110, and com­ple�e F''H'E'P'; in like. man­na� con�truct F2�2E2P2 equal to same in ��g. 170 . Since a�le �·B is. always ob�use, �ngle E'1H1F1 is always acute,

�and the �more acute E'H ' F' be­come� the shorter P1H' be­comes. .�ikewise, s1.nee angle

·AH2B is always acute, angl� · E2H2F2...,.is obtuse, and the : I

. more obtuse��t becomes the longer P2H2 becomes. .

. So ',;f'irst: As the va;r.i$ble acute angle F 1 KiE·• ·approaches its superior limit, 90°., the length B:'P '·

r 1..

. il).creases and approaches the length .HP.L_as -��!c!. yar.i��----· ·

able angle app�oaches, in degrees, its inferipr limi� 0°, the- lensth of. HI p·t decreases and approache�) _as. its interior-limit, the length o:f the longer of the _

tv9 ·l:�ne� H:''4 or l;I'a, P' then. coi'nciding with either E 1 or F' 1 , an4 the distance of P 1 (now �' ·or F 1 ) f;rom a i�n� di-awn tl;lr.ough · H 1 par-allel tb AB� will be . the I . second-dimension o:f the parallelogram AK1 oh AB; as

i '

'

b

.. . '

I •

. .

' v 1 '

... �·

l 1

I • I i i i . 1' l

·•

said ang'ie F' H1E1 'continues to decrease, H1P1 passes , - ' through its inferior limit and increases continually and approaches its superior limit ro, and the distance of P 1 .from t.h�· par.allel 1:1:ne •.throu8h the ·cprre spend­ing point o.f' H' increases and_again approaches the. length :HP • .

:. __ said distanc-e is always less than HP and the p�allelogram AK1 is always less than the sq. AK.

.. ··And secondly: As the obt:use ·va-riable angle EaH2Fa appro�ches its inferior �imit, 90°, the length . -of HaP2 .decreases and-· approaeh�·s-�·-the length of HP;

"'as- said va:i'i�ble. angle approacheS' -its scuperior limit, '180°j the length of H2P2 inqreases and approaches oo in lengt�, and- �he .distance of P2-from a line ,through . ' , . the corresponding H� par.allel to � increases from the length HP to oo, which distance

r is -the second. · dimensio�:r;t of the parallelogram A2K2 on AB. _

.�. the said distance · is always greater ·than HP and the parallelqg�am AK2 is always greater than the sq. AK.

. _ .. � · :. the sq. upon AB = the sum of no ·other· two_

sq�ares exc.ept the two .squares upon HB and HA� _

:. the sq. upc,;>n AB = the sq; upon BH + the· sq. _ upon.AH. .

�:. h2 = a� + b2, and ne�!3r.., a 1 2 + b 1 2• . a. This proof· ,and f:te;ure was . .foi'niulateq by

the au�hor·, Dec. 16, .' ;1.933. ·

B .

• , 1 ' � "' -This type includes all proofs derived f·rora-'� the .figur.e ·;1n which the .. square .. constructed upoh the' . hypotenuse over�aps. the _ .. given triangle ana4 the squares constr'ijct�d upon the;).egs as in 'type A, :and the ··.·.

,

. -proofs are based ort the principle of �quivalenc�. . ;�· '�o

. . ..... - . ' '

·! ..

...

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.

1�

Fig. · 172 giv:es a par- . ticul�r�p�o9f. In rt. t�i • . ABH, legs AH and'.BH are equal. Complete sq. AC on AB, over­lapping. -the tri .• : ABH, and ex:­tend AH and BH to C and n; �nd there results 4 .equal equiva-· lent'tri1s.l, 2 , 3 an� 4. • G '

.' Fig. 172 ., The sq. AC :;: tri •.s . [(1··+ 2 + 3 + 4 ), of which �

tri.- 1·· + ·tri.-. ·(2 � 21 ) = sq. -Bq,-·an:d�tri.-'�} + tri-;

-· ---(4-- = -41 ) = sq .• AD}. .

·

:. sq. upon AB = sq. upon BH +. sq.' upon AH. . h2 - a2 + b2 ' -• • - • t::J

a. See fig. 73b and fig. 91 here�n. . · b Th.is ·proof (better, illustration ),_ by -Richard .. Bell, Feb: 22, 1938.. He used onlt ABCD of fig. 172 ; also creditea to Joseph Houston, a high school boy- of South· Bend,

.Ind., May 18,· 1939 . He

used the full fig.

Take AL = CP and �aw LM and ON perp. to AH.

v Since quad. CMNP � quad. KCOH, and quad . CNHP is commo� to both, then quad, PHOK.= tri. OMN, and we have: sq. AK = (�ri. ALM =.tr�. CPF of sq. HG) '+ (quad. LBHM

Fig. 173 .= quad. OBD� of sq. HD ) • 1 + (tri . OHB· common to. sq's AK

and HD) + (quad. PHQK;.= tri. CGA of -sq. · JiG ) + (qua�. CMI'ij? common to sq' s AK and HG ) = sq. HD + sq. HG. . . . 2 2 ;:. sq-. upon AB .= sq • upon ;BH + sq. upon HA . :. h = a + b2• 'Q.E.D.

' . I �-

a. This .pro.of-,-witn fig., discovered by the autho� March 26,· 1934, -1 p.m. . �

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THE . ..

PROPOSITION , . .

. ���!!l!r.:.!b.t!t, .

Assuming the . three - � . \

aquares construG�ed,�- -�s in fig. 174, draw Gb--�t must p_as s through H.

Sq. AK : 2 trap. ABz.n, . . . ' ' . = 2 tri. AHL + 2 tri. ABH .. + 2 tr�. HBM = � tri. AHL

'Fig. 174-

+.-2 (t:ri. -ACG = trt .. ALG +· �ri. GLC') + 2· tr� .'

HBM = (2' tri. AHL + 2 tri. �G) +. (2 tri. GLC .

. . = · sq_. !(F + sq; BE. j

· = 2 tri. D�-) + 2. tr'i. HBM .

:. �q. upon AB = s .q� upon BH + sq •. 1 URon AH. . • ... 2 '• % • h"' - a + b- '

.. 0 I I ' • • ., - R • • • j

R • ·

a. See Ani. ,Matl;t. Mo., . . . v. r:v, 1897, p. 250, proc!t .XLIX.

' .. _1_t�J!l!r.::f2Y. [.

' I

�

· _Take HM = HB, and ·

I• I I

·�aw KL par. :to AH and ".MN par·. to BH.

Fig. 175

. . Sq. AK = tri. ANM + trap. 'MNBH + tri; BKL + J���-· . KQL + quad •. AHQC = (tri. CQF +: tri. ACG + quad. AHQC ) . +-�trap. -RBD� + tr�. BRH ) = sq.· AF + sq. HD. .

. :. sq •. upo� AB �_sq. upon BH + sq, tipon-=-AH-.- .:. h2 =:= a2 + b2• . · .

a. See Ani. MB.th. Mo·., ·v·. IV, 1897, p._ 250, · · proof' L • . "J b. ' If OP-ls Cira� iri P.,lace of MN, (LO = HB )., ."

the proof is prettier,. but same in principle�

1938. c .• Also oredi ted· to R. A. B&ll� Feb. 28,.

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Fie. 177

- ' -',

I •

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GEOMETUT, 'DUI\1\'DO · --�-'�-

# �evtn·tv-Slx . �-x,__ ---'---- -

•

. In ·fig.: � 77, dra'! GN and DR par. to 'AB· and LM par. to�· R is the,pt. or inter­sect:ton of AG apd· DO:

Sq. AK = �ect. AQ _

9+ rect. ON·.+ rect. LK· = (p�ral. DA = sq·. BE ) + {paral. R!'f . . = pentagon �THMG +�tr+. CSF ) + (paral. ·. GMKC = ; til:ap •. GMSC + t�. TRA ) = sq. BE + sq. AF.

-- :. �q. upon AB = sq. upon BH + sq •. upon AH • • •• h2 = a2 + b2 . . . . " fJ a. See _ Am. Mat�. Mo., V. IV, 1897 , p. 2'50,

proof. XLVII; Vers)+lys, 191:4, p., 12, fig. 7 .

Fig. 3,78

I

,-·

� In fig. -i78, draw LM thr�ugh-H perp� to AB, and draw HK and HC •

. Sq . AK = rect. LB :+- rect. LA = 2 _tri. lOffi + 2 ttri. CAH = .s·q. _'A:b + sq.' AF. · ·

- .o. sq. upon AB = sq ... . 1,.1:ron BH + sq, upon AH.

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----- - --------:------- r s ,i

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...

:. h2 = �2 + b2. a. Versluys, 1914, p. 12 ,'fig. 7 ; Wipper,

1880, p. 12,. proof V; Edw. Geometry,· 1895, p. 159, fig. 23i Am. Math. Mo. � Vol. IV, 1897, p. 250, -proof i LXV-III ; E. - Fourrey, Curiosit�es of Geometry, 2nd Ed'n, p. 76,. fjg. ; e, ·c�edited to Peter Warins, 1762 !

+ tri.

"Fig. 179

' -!t�tn.1t::ttn1

Draw HL par. to BK, KM par. to HA, KH and EB.

Sq. AX = (tri. ABH = tri. ACG ) + qua�. -AHPC com­mon to sq_. .Ax ·and sq. � . + ( tri. HQM = . tri. CPF ) + ( tri • KPM = tri. ENP ) + [ paral. :QHOK = 2(tri. HOK = tri. KHB - tri. OHB ·= tri; EBB - tri�'OHB = tri. EOB)-= ·paral. OB� ]-

OHB conunon,_to sq. AK and sq. liD· . :. sq. o AK = sq. - HD + sq. AF. -'"; :. sq. upon AB = sq! upon BH + sq. upon All. 2 .2 ' \ a + b • · · ·

a. See�. Math. Mo., V. IV, 189,7,. p • . 250, proof'� LI .

b. See Sci. - Am. Sup. , V .• 70, 1910, p. 382, -£or a geometric proof, unlike the-above proof� but

. � -

based upon a similar figure of the B type.

Fig. 18o

·.

._ I

In fig. 180, extend DE to_K, dftd draw KM�erp. to F.B •

. ¥Sq. AK.= (tri. AP-� = tri. AyG)' + quad. ARLO com­mon to sq. AK and sq. AF + ( (tri .• KLM ::: tri. BNH ) + tri. BKM =.tri� KBD = tr��. BDEN + (t�l. KNE = tri. CLF ) ] •

.

�·. sq. AK = sq. BE + sq. AF ..

0'

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___ .....

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.....

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..

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•

GEOMETRIC PROOFS· . 149

. . <�- :. _sq. upon AB ·= sq. _upon BH + sq. upon AH. - • h2, = a2 . +. b 2 ·' . . .. . .

·a.' S�e E·dw�rds 1 Geom. , 1B95-, ·_ p. ).61·, :fig'. \; -

. · (36Jii·:Am·. Math. Mo. , y.- IV, 1897,:p. 2SI, proof LII; Vers·ltiys, · 1914, p. 36, fig-. ·35, credited to Jenny de I -Buck. \ . ' \ ·\

-_ In fig. ·181, e�te�d

GF to L makrng.,FL = HB and­draw KL an� KM1respectivelt par. to BH· and AH. \

. . Sq. AK.� (tPi. ABH

.= tri·. 0� " �rap � BDEN + tri. _ OOF ) + (tr.i • .

BXM �, tri. AOG) ·

. 1 + (tr� !. KOM·:: t�i��· .BNH) +quad . . r '�00 c!oimilon to Sq.: AK and sq._ " ' .f: -F • �81 HD'- +· s·q. H�·; · J , " :. ·sq •. upon AB = sq.

upon BH +.. sq. upon AH. · :. h2 = a2. + b2-. . . , • See Aln. Math. Mo· •.. , V. IV, 1897, p. 2S1_,

proof LV 1I.

' J · In. fig._ 1� , extend

.A:� DE to L maki,ng x;,. = HN, and ·

r - ' draw ML. I r �� ' I< . I= _ . -� � -�-�"-£· . Sq. AK = _ (���· · �H

� , iM\ ., 1� = tri. · AC'Cf) / ·( tri .... �MK :;: . i � I \ ,.-y \ 0 '[!. . BDE (

' ( , r -, rect. Bl;.. = � trap. ,N + tri •

\ ( \ · MKL.__ �ri( BNH) ] + quad. · AHMC ' t ,� :ommon_�o _sq. AK And _

_

sq. AF.

182

- sq. HD + _sq. HG. . ,_ ·

. . . / :. sq. upon ,AB = sq .• upofBH ! �q. upon AH. -� •. h2. = � + b • . I 1

a. See�Eawardsf oni�-,·189.5, p. 158, f'ig.r ' ·

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Fig. 183

:.

,, -�ltb.!l�!�st

In .fig. 183, extend GF ·

· and DE to L and draw LH •. .. ·s·q . AK = h�xagon AHBKLC

+ paral. ,HK + paral . HC = sq. HO· + sq. HG.

.. /. sq. upon AB = sq. upon BH + sq. up_on AH. :. h2 . = a2 + b2. ' '

a . -Origfnal -w1 th the author, July 7!:1901; but .old for it· appears i1;1 Olney' ·s Geom., university edit�on, ·1872, Pc

· 250; fig. ·3J4; Jury Wipper, _1880, 'P• 25, fig. 20b, as -given -by. M. -v-.- Ash, in ·"Ph:iios'ophical. Transac�i:ons, " 1683; Math. Mo.__, V. IV, 1897, p. 25�,. proof LV; Heath's kth. -Monographs, No�' l, 1900,_ p. 2_4, proof\' IX; -Versluys, 1914; p. S5,�fi�. 58, credited to Henry Bond.· -Based on the Theorem of Pappus. Also see"Dr. Leitzma.nn, p • . 21, fig. 25·, 4th ldi-tion-. , I,

. , : b.· By, e_xtending !Ji to AB, an algebraic pifoof can be

_ reaaiiy devised, thus

_inc�easing the r?· _or .

�imple proofs. _

-

' .

; - I

Fig. 184

.,

�l5l�1t=.itu:.tt

'

- ..,_ In i'ig. 184, extend GF an¢ DE to L, and. draw LH. .

: Sq. -AX = pentagon ABDLG - (3 tri. 'ABH • tri. ABli + rect. "· LH ) + s_q . HD + sq.-_ AF. *

:. sq . upon AB = sq . . _upon BH �+ sq. upon AH. :. h2 _

...

= a2 + b2. ---

_cat:\on, a. See Journal of Edu-1887, V. XXVI,· p. 211 Math. Mo-. , -1855, vo-1. 2, Dem. 12, fig. 2.

fig. ;t; II, No.

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. -... .. �ig. 185 .

•

! E.-itb.!I:.f.O.U.t I c. · -t ! In fi�. 185, extend H · draw LM perp � to AB, �d draw

HK and HC . ·

' a . . SQ.. AK = rect. LB

+ rect. LA = · 2 tri. HBK + 2 tzii. ARC = sq. HD + sq. HG.

, :. sq-. upon. AB = sq • . up.OJ:l BH + · sq. upon. AH . :. h2 = ·· a2 + b� •. ' .,..

. . I � a� See 8�;1 . Am.· Sup. , V. 70, -p�-3--=8::-.3-, ..

-D-e-c--.-_�;LJ-0,"' 1910, bei� No. 16 in A. R.

· · Colburn:• s 108 proofs; FourreY.:. p. 7 1, fig. e . ' .

! •

�ltb.!t:.f.l�t.

. . , 1'L. �- � \ ' ·C , ' k. . , 1- '"\- ' ·- -1 E.

In fig • . 186, extend GF anq. DE to L, and through H �aw LN,· .:N being-·the pt . of -i-nter- - ­

section - of NH · and AB.

·fi,� "'· , . 1 .)>, . Sq. AK • rect • . . .Ma \ + rect. MA = paral, HK + paral.

\ ·I · I $ H.C . = sq . HD + �q. HG .• j \ I J • AB'

• , 1 � �- - -_1 _ . -. . . 1 • • • sq. upon . , , = sq.

, A upon BHI T · sq. upon AH. :;. h2. = 82 + 02.

FJ.g . 186 a . See Jury Wlpper, . 1880, p • . 13, fig. 5b, and p. 25,

fig. 2 1, as given by Klagel 'in rrEncyclopaedie, ·" · � 1808; Edwards " Geom .• , 1895, p : 156·, fig • . (7 ); Ebene Geome­trie, von ·G� Mahler, 1897, p . 87, art. 11; Am. Math. ltto . , V � IVJ 1897,- p. 25l ,1 • LIII ; 'Math. Mo:· �--'+'859, Vol. II , N�� 2, fig. 2, Dem ... 2, pp. 4:5-52, wh��e credited to Charles A. Young, Hudson, . 0. , now Dr. Young, as­tr<;>�omer; . Princeton, N.J.. This proof is an .applioa- ·

t:f.on of Prop. txx�, Book IV, Davies. Legendre; also · ' � � ... • " I As�:, M. ·v. of D�bli!l; 1!'-lso Jpseph �elson, Phila., Pa.,

� ·a student in West �Chester Hign School, �939 . b . This figur� will give an a+gebraio proof.

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152 THE PYTHAGOREAN PROPOSITION

, In fig . 186 ·it is .evident. that sq . Ale' = hex­agon ABDKCG - 2 tri . BDK = hexagon AHBKLC = (paral . KH = ; rect . KN ) + paral . CH = rect . CN ) ::. sq. HD + sq .• HG. :. sq . upon AB = sq . upqn Bit. + sq. u:pon AH. :. h2 = a2 + b2 • Q .E . D . . .

a . See Math� Mo . , �858, Vol . I ,· p . 354; Dem. 8, where it is creditPd to David Trowbri4�e .

. b.. Thi-s proof is also based on the Theorem of Pappus . Also this geometric proof can easily be converted into · an algeb�aic proof �

' �lib.!l��!Y.�!l

r: In .fig . 187, _extend DE CA-.... .... H_ k to K, draw FE , and draw 'KM par .

(; / J-tJ; it1�£' to Ali. ( I /� ' Sq • . .AX = (tr·i . ABH . \ I I ' = tri . ACG }- + quad;; AHOC com-

' ' \ \ t ;/1J mon t o s q . AK and sq_. AK + tri .'

� BLH common to sq . ·AK ·and sq • .A . ·. ·HD + [ quad . Om,K· = 'pentagon

OHLP� ' + (tri . _P� = tri . PLE ) + (t.ri . MKN = tri . ONF ) · = tr: L

HEF = (tri . BDK = trap . BDEL + (tri . COF = tri . LEK ) ] . = 89.! liD + sq . HG .

...... - -- - -- J

:: sq. \ppon AB = sq . upon liD + sq. upon HG . :. h2 �- a2 + b2 • ·Q. E ! D .

-

a . See .1\m. Math . · Mo . , eJf . DL,. 1897·, . . p . 251, proof LVI .

\ -In. fig. 188, extend GF and BK to L; and

'through H . draw 'Mrj_'·p¥- . to BK, . ·and draw KM . ' , Sq ., AiC = -tJaral • :AOI;C .. = p�ral . HL +. paral . HC

= (paral . � . = �q� _ AD ) + ' sq� HG. . � �·· sq. upon AB = . sq • . upon BH + sq. upon AlL

:. h2 = 82 + b2· . . \ ' I

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GEOMETRIC PROOFS 153

a. See <Jury Wipper, t:J. .... 1 L 1880, p . 2·1, fig . 23 � where

� 1 it says' that this proof was r "" ' ' 8 ,, , , I given to Joh. Hof'fman11., 1 00,

t! .... � -� - � k by a friend; also Am . Math. , >T ' ' E 8 G...,." 1 . \ ' fJ.A Mo. , 1 97, V. IV, p. 251 , ' 1 � 1 .. � . ' , proof' LIV; Versluys, P·· 20,

\ _ , , , 1 � ·. f'ig. 16, a�d p. 21, fig . 18;

\ 1 .... 1' , Fourrey, p . 73, fig . b . # � �--.._� b. .From thi,s figure

� an algebraic proof is easily ·"?ig. :J-00 devised .

c . Omit line MN . and we have R. A. B-ell's f'ig . and a proof by congruency follows. He found it Jan . 31 , 1922 .

· Ext�nd GF to L making · FL = �H, draw XL, and�·draw CO par. to ·F.B and KM par. to AH.

· S� AX = (tri . ABH = tri. ACG) + tri . CAO common J to sq' s AK and HG +�\sq. MH com­mon to sq ' s AK. _and HG + ( penta­gon MNBKC = rect . � + (sq . NL = sq. HDj ] = sq. HD + sq . HG.

:. sq. upon AB =· sq. Fig . 189 upo:q. BH + .sq . upon HA. :. h2

\ = .a 2 + b 2 • Q. E • D. a. Devised by the authoP� July 30, 1900, and

afterw�rd·s .f'ound in .. Fotirre:r, p . 84, f'ig .. c . • . .

�n fig . 190 produce GF and DE to � � and GA I t �· ·and DB to M . Sq • . AK + ··4 tri. ABH = sq . GD = sq . HD + sq . nG .+ (rect � HM = 2 tri . ABH) + (rect. LH = 2 tri. ABH) w�enc� �q . AK = sq . HD + sq • . HG. � • :. sq . upon AB = sq. upon BH + sq. upon AIL :. h2 = 8.2 + b2 .

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15lf. THE PYTHAGOREAN PROrOSITION

Fig . 190

Society of Canada, lows :

a . See Jury Wipper, 1880, p . 17 , fig . 10, and is cred�ted to Henry Boad, as given py Johann Rpffmann; .in "Der P}-thagoraische Lehl'satz , " 1821 ; also see Edwards ' Geom. , 1895 , p . 157, fig . (12 ) . Heath ' s Math. Monographs , No . 1 , 1900, p . 18, fig . 11 ; al so attributed to Pythagp�as, by W. W. �ouse Ball . Also see Pythagoras and hi s Ph�losophy in Sect . II, Vol . 10, p . 239, �904; in procee4ing� ot Royal

wherein the figure appears a; � fol-

-L OOK Fig . 191

Tri 1 s BAG, MBK, EMC , ' AEF, . �H and DLC are each = to tri. ABH.

:. sq . AM =: (sq .. KF - 4 tri . :ABH.) = [ (sq . KH + sq. HF

M + 2 rect . GH ) � 4 tri . ABH ] = sq. KH + sq . HF . -

•

:. sq. · upon AB = sq . upon . F · 2 2 2 C,__...,;r;;;a.�---..a HB + sq . upon HA. :. h = a + � •

Fig. 192 I,

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GEOMETRI-C PROOFS' 155

"The Pythagorean .Theorem; or a New.Met�od of Demon­

�trating it . " Proof as above . Also Fourrey, p . 80 , . as the demonstration of Pythagoras according to

• • t Bretschenschneider ; see Simpson, and Elements of Geom-etry, Paris , 1766 . Joe . b. tn No . 2 , of Vol . I , of Scientia Bac-calaUiteus , p . '6l, Dr . Wm. B . Stn+th, of the Mis·souri State University, gave this method of proof as �· ­

. :But , see "School Visitor, " Vol . II, No . 4 , 1881, -for · same demo�istration ·by Wm. Ho.over, of Athens , o·. _, as

"adapted from. the French of Dalseme . " Also see "Math . Mo . , " 1859; VC;i;t_ . ) : , · No . 5, p . 159; also the same journal; 1859, Vol . II , No .• 2, pp . 45-52 , where

. Prof . John M . Richardson, Collegiate Insti-tute ,· Bou­don, Ga . , gives a collection of 28 proofs., among oWhich., p . 47 , :!,s the one above , ascribed to Young . · See als·o Q;rlando Blanchard 1 ·s Arithmetic , !852� published at Cazenovia:, N:Y. , pp . 239-240 ; also Thomas Simpso:h.! s "Elements of Geometry,.." 1760 , · p . 33 , and p . 31 of his 1821

-edition . '

s·. . . Pr<?f . Saradaran�a;n Ray of India gives it ,·on pp . 93-94 of Vol . I , of his Geometry, and says itr ·

" i's d.u� to the ·Persian Astronomer Na-sir-uddin who flourisned in the 13th century under Jengis Kha;nw �

Ball ,. in his· 11 Short History of Matherilatics , " giyes same m�thod 6f proof, p , 24, and thinks it is probably the one - originally offered by Pythagoras .

- Al�o see ".Math-. Magaziri� , " by: Arte�� Martin� LL.D. , 1892 , Vol . II , No . 6 , p . �-97 . ·, n:r·. Martin says : "Probabiy. /bo other tP,eor.em has r.eceived sQ much at­tention .from Mathematicians or been d�monstrated irC

<- -' s o �ny different ways as this celebrated p:roposi­tipri� which bears th� name of. its supposed discov·er-

.-er . " c . See T ; S�dra Row, 1905 , p • . 14, by paper

folding, 'fReade·r, take t:wo equal squares oJ' paper and a pair . of scissors, and quickly may you �ow

• that AB? ::= "BH2 + HA2. " · · ·

' . ' . Also see Ver.sluys·, 1914 , his 96 proofs, p . - 41,

fig �: 42 . The title page of Versluys . . is :

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A ·

' ,_ :' \ THE PYTHAGOREAN PROPOSITION

ZES EN 'N EGENTIG �EWIJ·ZEN

Vo o r Het . THEO REMA VAN PYTHAGO RAS

Verzameld en Gerangschikt

Doo r

J . VERSLUIS

Am sterdam-- 19 14

�n fig . 193 , draw KL par . and equal to BH, through H draw LM �par�. to B,K, and draw AD , LB anq CH . ·

Sq. AK = rect . MK + rect . MC = · .(paral . HK = � tri . BKL = 2 . tri . ABD = sq . BE ) .f. _(2 trf . . AHC = sq . AF ) .

:. sq . upon-AB = sq . · . . . 2 upon BH + sq . upon AH . :. h , - Fig . 193 .= .a2 +� b2 . .

a . This figure and , -proof is taken from the following work; now in my ii- , I • I brary; the title page of w�ich i s shown on the fol-

lowing page . · , � . ; ··

. The figures · of this book ar.e all grouped to­gether - �t the end of �he volume . �he aboye flg�e is n�bered · 62,· and is constructed for ''Proposi tio XLVII, " .in "Librum Primum, " which pr'qposition peads , '' I� recta_ngulis ··tr.iangul�s , quadr�tum.

qu�ci a latere rec�.tlll1 angul1lDl sub.tedente,. .. P,escribitur � . aequuale ·est ·

· eis , q'l!�e a lateribus_ rectum angulum continentibus I CJ,escribUn.tur quadratis .·"

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GEOMETRia PROOFS

"Eu cli des El e-mento rum Geom e t ri co rum "

Li b ro s - T�edecim. -

I si do rum e t � Hyp sicl e•

& Rec en ti o res de · Corp o ribu s Regulari bu s , '& P ro cli

P�o p o s i tiones Geome t ri c �s

- ·

Cl au dius 'Ri ch ards

-\

157

e Societate · Jesu, Sacerdos, _patria Ornecensis in libero Comitat� -- . I • . . -

,. &r�nda�, & Regius Ma.tlteroa.tica.rum . ' ' .,.

P ro fes�o r : dicanti que

Phi l i p p o III!. �i �p aniarum et Indic arum R�gi Ca.thJli co.

Antwe rp i ae ,.

ex O ffi cina Hi esonymi' Ve rdu s si i . M ;·n·c . XL V . ' Cum Grati a & P-ri vi l egi Q 11 .

Then ·co�es the . fol�owing sentence : "Pzaoclus in hunc libzaum,

'celebzaat Pythagozaam

Autho�em huius propositionis, pzao · cuius - demonstzaa­tione dici tur Diis Saczaificasse hecat.ombam TaUI'ozaum. "

' Following this , comes -the "Supposito , " th�n the "C�n-struc�io, " and the}1: the . "Demonstzaatio , :r whic}?. con-

. densed and translated is : (as peza fig . 193 ) tzaiangle BKL_ equals t�iangle ABD; squazae BE equals twice - tzai� ang�e ABM a!l:d: zaectangle MK equais twice tzaiangle BKL; thezaePozae zaectangle ,MK equals squa�e BE . Also ·aquazae - ' AG -equals twice tzaiangle AHC ; zaectangle HM equals-tw;LQe triangle CAH; thezaefozae squazae:AG · equal zaectan­gle HM. But squazae . Blt4��quals )W9tangle" kM plus zaec- . -tangle OM. Thezaef.oza� squazae BK equals squazae_ AG plus squazae BD .

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158 THE PYTHAGOREAN PROPOSITION ·

I � . The work from which the above is taken is a .

_ .,_ ...... , . · book -of 620 ._pages, 8 inches by 12 'inches, bound. in

vellum, · and, though printed ih 1645 A .D . , is well pres�rved. It once had a place in the S�derland Li­brary, Blenhei� Palace-, England, as. the book plate shows.:.-on the book plate· is printed--" Fr.om the Sunder­land Library', Blenheim Palace, Purchased, April, ·

1882 . " . .

The work h�s 408 diagrams, or g�ometric fig­ures, - is entirely in Latin� �nd highly embellished.

I found the book in · a: · se'c.ond:-hand bookstore· in Toronto, Canada, and on July 15, 1891, I purcha·sed

· it. · (E . S. Loomis. ) ·

0

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c

This 'type includes all proofs derived from · the figure in whi�h the · ·square constructed \lPOn the longer leg overlaps the given tri�ngle and 'the square -upo� the hY,P:otenuse.

_ Proofs by disseoti�and superpositio� are PC?S.sible, but nc;>ne were found·.

I

lll!l!!l:.!b.tp.t

· ,) ·J,.� In fig. 194 , extend KB � 'I , �to L, take qN. = BH and dl'EJ.W MN ' 1 , par:. to AH . . Sq. AK = quad. AGOB ,

� 2J c_onmion tc:> sq' s AK ap.d. AF + (tri. t · nilll--.--�u · �---:- ·coK = tri. ABH + tri • . BLH )

+ ( trapo �I}G� = trap·. BntL ) . j • + (tri. AMN = tri. BOF ) :: sq. Jill I " � f . " + S q .:M'lf(t. · ·- - --- � -- . . -- - - � -� .., - - -- . I � . . -I. (J. f. 1 :. sq. upon AB · = sq. upon , ' . 2 2 . ' 2

· Cl�;. _ _ . _ �•u __ _ _ _ _ __ _ _ _ BH _ _ i" sq. upon A){. :. h · = � + b • .... ._ - - - ,;;;;11, - . . / �

_ . · a • . S.ee Am. Math. }�c;> . , V .

· Fig . 1�4 . IV, 1897, p . 268, proof LIX. · , .b . In fig.�l94, ' omit -MN and draw KR perp. to OC; then ·take KS = BL �d draw ST pe�p. to oc . Then t�e fi�. 18 that of Ri�ha�d A .

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GEOMETRIC PROOFS 159

Bell, of Cleveland, 0 . , devised July 1 , 1918, �nd given .to me Feb ; 28, · 1938, along witJ:i. 4o ot�er proofs through disseqti·on, . and al.l derivation of proofs by Mr • . Bell (who knows practically nothing as to Eucli­dian Geometry ) are found therein and credited to him, on MarcH 2 , 1938 . He made no use of equivalenqy.

Fig . 195

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In fig : 195 , draw DL par . to AB, through G d.r·aw PQ par . to �

CK/' take GN =. BH, . draw ON par . AH and LM perp . to AB .

Sq . AK =· quad . AGRB com­mon to � sq ' s AK an4 AF + (tri . ANO = tri . BRF.) + · (quad. OPQN = quad. LMBS ) + (rect . PK = paral . ABDL = sq. B;E ) + (tri . GRQ ·= tri . AML ) = sq. BE + sq . AF.

:. sq . up·on AB = sq . upon BH + sq . upon AH . :. h2 = a2 + b2 •

a . · Devised -by the author , Ju.Iy 20 , 1900 .

...( '� -·

In fig. ·196 , through G and D draw MN and DL e�ch par . to ·

Afj, and dr�w GB . 'Sq . AK = rect . MK + rect .

MB = paral . AD + 2 tri • BAG :::: sq . __

BE + . sq . AF. :. sq·. upon AB = sq • . upon

,BH + sq . upon AH . · :. h2 = a2 -+: b2 • , · ; . ·a . See . Am . .Math . Mo • , · V . .

IV, 1897 , p . 268, proof LXII .

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THE PYTHAGOREAN PROPOSITION

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In fig . 197, extend FG E to G, draw EB, and t�_ugh C "� draw HN, and 'draw DL par . to · AB·. , 'LL�, - _ " ·sq . AK = 2 [ quad • . Acmt ,., � = (tri . CGN = tri . DBL ) + tri .

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Fig . · 197

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· : AGM conunon . . to sq. - AK- �nd AF + (tri . ACG = tri . ABH. = tri. AMH + tri . ELD ) ] = 2 t�i . AGH + 2 tri . BDE = sq� liD + sq. HG •

. :. sq . upon AB = sq. 1,1pon BH + �q . up_on AH.1 :. h2 = a2 + ·b2-.

·, a ._ See Am. Math. Mo . , V . IV,. - 1897, p . 268, proof �Il+ .

In fig . 198, extend FG, to C , draw HL. par . to AC , �nd draw AD and HK. Sq. AK = · rect . -BL + rec.t •. AL = (2 tri . KBH = . 2

-

tri . ABD + para1 . ACMH ) = sq . BE + sq. �· AF. · · :. sq . upon AB = sq .

· upon

BH + -sq • . upon AH. :. h2 = -a2 + b2 • ., ,·

. a •. See !J'l,lry Wippef , 18�0,

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p . 11, II ; Am. Math . Mo . , : V . IV, -i897, .P·· 267, proof LVIII ; Four�- ---- ---- ·· �ey, ·p ... . 10, ·� fig . 'b ; Dr . Leitz-

Fig_.�--- mann' 8 work (1920 ) , p . 30 , , f�g. ?1 . .

In fig . 199, through G draw MN par . to AB, draw HL perp • . to OK,. ·and draw· AD , HK and BG .

Sq . AK = rect-. MK + rect . AN = . · (rect . 1BL ·= 2 tri . .KBH = 2 tri •

. AB:O ) + 2 tri . AGB = sq • BE, + sq-. AF.

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I GEOMETRIC PROOFS 161

: . . sq . u,pon AB = sq. upon 'BH AH - ·- h2 2 2 .f .

+ sq . upon • . :. = a + b •

' , · ., _ .. _a . See Am .• . Math . �o • . , V. · · ., IV, 1897 , p . 26B, proof LXI .

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' . ' - · -

. In fig . 200 ,� ex�end FG

Fig. 199 ·-

- to c , . draw HL pa_� . to BK, and · draw EF end LK.. �q . AK ·= quad.

AGMB common to scf' s�'-AX ap.d AF -+ (tri . AOQ- � tri •. ABH ) + (tri . CKL · =

·· trapr. EHBN +:_ trt . BMF ) + (tri . ·. KML- = ·tri'." END ) =- f;Jq . HD ' -- I

· + sq. HG·. :. sq. upon AB - ::=: sq . up.oli

BH ' � • , 2 2. 2 . + sq . . upon AH. • • h =; a + .b •. . .:a . se·e .Am . Math; Mo . ,. v.

IV� 189�, p . 268, proof LXIV.

I

· Qnt_ttsuut't!sl / . Fig. 200

�ig. 201

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....., . ' . _ ;rn·· .r�g: 201, draw FL _par .

to AB, . eiK:t_enci FG to . C , and draw p

. EB and FK. Sq. AK = (re�t . LK" . = 2 tri . CKF = 2 tTi . ABE = ·2 tri . ABH + tri . HBE =· tri . AB� T tri . FMG + sq . HD,) + (rect . AN = par&l . MB ) . .

:. sq . upon AB = sq . upon BH ;+ sq. upon AH . :. � F ·a� + - b2 •

a . See Am. · Math . Mo . , V . :iv, 1897, . p. 269; proof LXvii ;.·

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162 THE Pl.'THAGO�- PROP'OSITION

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� 1 �-'• _, ' !u " ' L. c�-�- - -� , Pig . 202 , •.

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In fig . 2.02, extend FG. to C , Ha to L, draw KL par ; to AH, and .take NO = BH and draw OP &nd NK par . to BH .

Sq . AK =. _quad. ; AGMB _c o�­mon 'to ·sq ' s AK and AF + (tri . ACG = _tri . ABH} + (tri . QPO = tr;t . BMF)" + (trap . PKNO + tri : KMN = sq., NL = sq . HD ) = sq. HP + sq. AF.

:. sq . upon AB = �q-. upon 2 ' . 2 2 BH + sq . upon � . : .. h = ·a + b • .

a •. See Edwards 1 Geom. , 1895, P·· 157 I fig . (14· ) . .

In .f�g . 203 , extend HB to L mB.king FL = BH, draw HM perp . to OK and draw HC and HK.

Sq . _AK ::= rect . BM . + r�ct . AM = - � tri _. :tffiH + 2 tri �- HAC = sq. ·

HD + sq. HG .

_ __ : _ _ -. __ _ _ _ -·�· --·�<M· �P-�� AB -� �q! . upgn _ _

BH + sq . upon Ali. :� h2 = a2+ b2 • ' \ . .

. a . See ··Edwards 1 Geo¥1" ·, L . .:.of 1895,, p •. �.61 , fig .�. (�7. 1!�------t��-

Dr�w_HM, LB and EF par .

tq. ·BK. Joi;n CG, MB and FD . Sg_ . · Ax: = ·.paral . ACNL

:d para1 . !Q.'J. + para1 . HC =· (2 tri • BHM = 2 tl'i . DEF = _sq � HD ). + sq . HG = sq. HD + sq . HG. .

. :. sq:. upon AB = sq . tipo:p. 2 . 2 2 BH· + sq . upon AH. :. h . = a + b •

. .. a � . S�e �·· Math. Mo . , V .•

-

. . lY., 1897, p • 269, proo.f IXIX .•

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GEOMETRIC PROOFS

In fig . 205, extend FG A £ t o C , c;l!'aw, KN par . . to BH, take

,.. -,, ,.. 1 ' NM = BH, draw ML par . to HB , and 1

� � draw ·.MK, KF and BE c

• 1J Sq . AK ;;· quad . AGOB com-- mon to sq 1 s AK and AF + (t-ri . ACG

= tri . ABH·) + ( tr-1 . C.tM .= tri . BOF ) + ( (tri. LKM :: tri • . OKF )

·- + tr:l, . KON = tri • . BEH ] + (tPi . MKN · = tri . EBD ) = (tri . BEH + tri •

�ig. 20,5

• EBD ) + (quad. AGOB '+ tni . BOF + _triL-AB(J ) = :�q·. liD + sq . HG.

:. sq . upon AB = sq . upon - 2 . 2 . 2

.LXVIII.

BH + sq • • U}ton Ali. :. h =. a + b • a . See Math. Mo . , V . PI, 18971 P� .- 269, pl"oof

Fig. 206

. .

In fig . 206, extend FG to H,. draw HL p�r . to AC , - XL 'par . to HB, and draw KG, LB, FD and

�F .•. Sq . AK = quad . AGLB com� �

·mbn to sq ' s AK and AF + _ (t�i . AGG = tri . ABH ) + (tri . CKG = tr� . � EFD = � s·q . liD ) + (tri . GKL = tri'; BLF ) + (tri . aLK = � paral. -HE; = � sq. liD) = (� sq. HD + -k sq. liD ) + (quad . AGLB + tri . ABH + tri . BLFJ = sq . liD ·+ sq . AF .

:-. sq . upon AB · = s q . upon BH + sq. upon AH . :. h2 · = a2 · + b2 •

�

a . See· Am . Math . Mo . , V . IV, 1897, p . 268., proof r.i1:.v.

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164 TBE PYTHAGOREAN PROPOSIT;rQ�---

- . . - . . a .- - See Am . proof LXV'r. . '

• •• I

In fig . 207, extend FG-­·to C and N, making FN = BD, KB to 0., (K · being ··.the _vert�x opp . A .in the sq. · ·CB ) • draw FD, FE and FB , and ¢!raw HL par . to AC.. ,_.

Sq. , AK = para1 . �CMO := para1 . HM + parta1 . HC = [ (para1. EHLF =:= J�ect. EF. ) - (para1 . EOMF :;:: 2 tr:l . • EBF =- 2 ' tri . DBF_ =, rect . DF ) = sq. HD ] - = sq .• liD + sq . AF .

· . . :. sq . up_on AB � · sq. \.tppn r 2 2 2 BH + s_q . upon AH . :. h =· a + b • Math� M� . , v . IV, 1897, p . 26S,

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�n fig; 298, through c· and K .dJ:.aw & Y ·and -PM par . �espectivE;31y·· 1

" .N. " " I� ·.

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., ,_ I ' I "1. " ' . ' ) � -C�- --�_ T,-· ·� ·\ tl • ' � � I . . \""na " . . ' .;t'

.. Fig . 2o8 . " I .

�� . ' . ,_ to .BH ahd .JW�. and� ex��nd

ED - t9 ·M,. }IF," tq -L ,·: AG 'to Q;· HA to 'N .and F'G'•to a ·. -. �

�sq> .AK + rect . HM + 4 tri • . ABH = rect . ·NM

sq . HD ·-4: �q . · HG + (rect. �t • . HM.) + (x-ect . ·

2 tri . ABH ) + (rect . 2 tri . ABH ) •

:. sq. AK = sq. HD · HG · "h2 2+ b2

, _ :+- sq . . __ . . 1 =:= a . • _

, . � . · Q � E • D,.

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! I . ,:. sqf upon<AB. = . -�q . uppn BH + sq . ... upon Af!.-'• '

. · · '

·:. h2 = a,2 + b�.. . · . . . . . . . . · ·s.- . :Creditec;i by ."Joh . Hoffma�, iil "Der .Pyth�- � .\· - _ ,

goraische· Lehrsatz , 11 182:1 , tb · Henry Boad oi' London; -see Jury Wipp�r, 1880, p; 19� : fig-. ��. ·

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GEOft{E�RIC ·, PROOFS 165

By dissection. Draw HL par . to AB, OF par . · to AH and XO par . to BH . Number parts as in figure_ . .: --

Whence : sq . AK = parts [ .(1 + 2 ) = (i + 2 ) in sq . HD ) ] · · · ·

+ pa1�ts · [ .(3 + 4 + 5 ) � · (3 + 4· + 5' in ·sq . H!J ) ] =:= sq; iiD + sq . HG •

. •. sq,. · upon ,AB = sq . upon 'HD . + sq . upon ·!:fA · �·. 112 = a2 + 'b2

•

Q .E .D . Fig. 209 a . Devised by the author

i ' -- - � to show a proof of ;Type-0 figure ,

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"'by dl8 !3ection,. Dec . 1933 . ,

D This type 'includes all proofs · derived from

., the f+gure in :vhich the squa�e con�tructed upon }.Pe shorter leg overlaps •. the given tr�angle and th� square. upon the hypotenuse . '• .

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I � ' In ffg . __ . 2±0, extend, ED G.� ., ' ,....

I , . to �, dr.aw Ht perp . to · OK an<;\ .

. ' "

· Sq ; · _ AK ;= i'ect . -B� + rect :' . · , �' H · draw HK. ,. · ·.\,1 • •. ·

·· ' , 1·. ;n .AL = (2' 'tri_: ·BHK = ���-! HD ) ·' • . , ' ..

-· '· \A·, ·' ., 1 y ·f 1 + (sq . HE by El.\cf.id 1 s pl'oof.1 •.

t v f \ · 1 • ,:. sq . :tnio�i' AB- -= .sq.. ·_upon ";I> • . ' ·• 2 , ·': 2 , ' 2 . • .. 'i ' _ I BH + s q •( upon AH •·· • • . h · = a + b • · : · · . .•' , � I · • · 1a . S�e ifury Wippe.r � 1880 ,

' � c·L � -- �1:.�·1<' p . 1_14, 4ig . 3 ; Versluys � p • . 12,

fi� . , . 1 given b� _Hoffmann . · · Fig . 210 I

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166

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THE PYTHAGOREAN PROPOSITION

Fig. ��1 ' •

In fig . 21�, extend ED -' ' · · to K, draw CL par . to AH, EM par .

to AB and draw FE . · sq . AK· = (quad. ACLN = quad. EFGM') . + ( tr:i . CKL = tr:t • �H = trap . BHEN + tri . EMA )

�· · i- (tri . KBD = ti�i-. FEH) "+ tri . BNJ? common to sq 1 s_ Af:. ai?-d HD = �q . HD + sq . AF.

· :. sq . upon AB = sq . upon BH. ' + ' "' AH h2 = a�+ b2 • sq. upon . • :.

a •. -See Edwards 1 Geom. , 1895_, p. 15.5, fig . . (2 ) •

, · 2n!-�int�t�-�l!�!n

. .In fig . 212 , ex-

i � I l Q j

.. t � . ,., . tend FB.'' and ' FG ;to t� and M . ., "' '-... makinS 'BL- = · AH �pd_ flM. ,_ ,

·- &·"' -; \ � =; BH, complete the rectan-� �� ', · " , -u . _>!J -g'le FO· and�-extend HA ·. t n - .N, -��· · �,. A ab.d ED· to K.

�

'', -�"'B 1Sq� AX '+ re'ct . -�MH ' , -� r 1 , � " " + . 4 tri . ABH = rect - FO -���,. ; : · . 'v � - • . \ = . sci:.

' liD + �q. HG '+ (rec_t . I' ' - 1 "'J), � J \� . NK = rect . MH ) + (rect • . ·MA ' , I U = 2 : tri . ··. ABH. ) + lrect : :PL . I '{',. I - / \

' 1 ' , 1 "' q = · 2 tri . ABH ) ; collecting ·.-C�;�.- -., -")( we have sq. AK = sq�. ' HD

...

. .. 11 - J. 1 HG·· .

\ '}? "' · • •. �q. :-"" . . ·· --------'1-------------::J . . �· : .. · sq. up.on AB = sq .

Fig,. 2� , , . ·up1bn B� + sq.� ,upbn AH. • • 0 • :h2 - a2 + b2 , . . . .. .._ · · r� - . .

. , a .: Cred:ited )to li�nri Boad· -·by Joh; · Hof�nn, �821 � see. Jury Wipper, 1880� p . 20, fig . 1� .�

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GEOMETRIC PROOFS 167

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Fig. 213

- -<rig. · 214

. In fig . 213 , extend ED to K, draw · HL par . t�AC , and draw OM.

Sq . AK � rect . BL + rect . AL = par�1 . HK + p�ra1 . HC = sq •

. JID· + sq . HG. ,

. :. sq. upon AB = sq. upon ' 2 2 2 BH + sq� upqn AH. • •• h = a. + ·b .

' . \ ( � '

a . Devised by the author, Aug . 1, 1900 .

..

2�!-��n����-l�lt!ttn . ·

! ' ,;j I� fig . 214, e.xtend ED to K and Q, draw dL perp . to EK, extend GA to M, tak� MN .= BH;

"-draw NO par . to AH, and draw . FE . · Sq. AX = -(tri . Clq, = tri •

. �·) + (tri . XBD = tri . EFQ ) _.

· + (trap . AMLP + �r� . AON = rect . GE ) ·.+ tr.i . BPD · common to sq ' s AX and -- BE + . . (trap . CMNO = trap . BHEP ) = sq . HD + sq ; HG. ·

. :. sq . upon AB = sq. upon . • . ' . 2 2 2 '

BH + . sq. upon· AH. · :. h I = a + b • · a . Original with the '

author; Aug . J 1, 1900 .· ' · . · : "

.

. ·1 . -An��tt�n�t!'5f_f2��!t:!!l "'- ', -. • - ' I

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·' # ,, •, • • . · E}nploy �ig • . 214, nilmoe.r�t:qe parts as there·· * numbered:; t�en, -·a� once : · sq._.,a = s'l:U� of 6 parts "

_·

[ (1 + 2 • sq. HD ) + (3 + 4 + 5. + 6 = sq. HG� = sq • . 1m + eq,. HG ].

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.. 168 THE PYTHAGOREAN PROPOSITION

. .h2 . . :. sq . upon AB = sq • . upon HB + sq . up�n JiA � 2 2 .

= a + b • Q . E . D •

'·

· a . Formulate.d by the· a4th�r , Dec . . 19, 1933 .•

. . . . . Ori� � u n d r e d F i f t e e n -------------------

In fig . 2�5 extend .HA , J' 'f to Q· making OA = HB� ED to K, and

, ' join OC , extend BD to P and ·join GJ'; ' , EP . Number parts 1 to ll _ as in

�, ·. �' H fig�e . Now: sq . AK = parts 1 \ · ' + 2 _-+ 3 + 4 + 5 ; · tl\9.pezoid EPCK

l ', L. -n .A. �o EK + P.C , " = X PD. ::: KD - .x PD = AH X :

J' �, ' 3 / 2 ' , . �

ll /; , , �� ; · I � <; N � 'I .-yT 1 1 AG = sq. H9" = parts � + � + 10

I · f / ' 1 + )1 + 1. . Sq.. HD = parts 3 + . 6; \.� / 1 ' , t . . .� .. sq. AK = 1 + 2. +_-�5 + 4

/ 'l�- - �·:- - .:. � k + 5 = 1'· + (2 = 6 + i + 8 ) . +" 3 + 4 / . iJ-· . . + 5 = l + -.·(6 + 3 ) + 7 + 8 +. 4 + '5

I Fig . 215 =· 1 + (6 + .3 ) + ·(7 + .8'''� 11 ) i- 4 I + 5 = 1 + ( 6 + 3 ) + 11 ·+ ·4 + 5

I , • • 1

I

= 1 . +. (6 + 3 )' + 11 + 4 + (5 = 2 - 4, since 5 + ·4 + 3 · -..

= . 2 + 3 ) = 1 +' ( 6 + 3 ) + 11 + 4 + 2 7 4 = 1 + ( 6 + � ) ' ' · +· 11, + 4 + (2 = 7 .+ 4 + +O ) � 4 = i· + (6 +. 3 ) + 11

1 + 4 + 7 + 10 = · (7 + 4· + lO +" 11 + 1 ) + (6 f 3 ) = · sq . 1 HG. + sq . � . · ·

-- "'

/. . :. sq . upon AB' = sq . upon HB +""sq . upon · HA . 2 '2 •2 � ; :. h = e. + b • . Q . E . D . . · . .

,o

_' + - - a·. 'This ffguZ.e and pr-oof .formulated by Joseph Zelso�, see proof- Sixt·y-Nine , !!, fig . 169 .. It. came

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to me ,on .-May 5,/ 1939 . . � . _

./ : • b . In th:fJs p:roof, � as· in all proofs recet·Jed :t onii tted the c'oftllJlil o:f "reasons l' .fo:r steps of the

· demonstration, artd reduce.d ·the .argumentati9n from many (in z·elso� ' s proof over , t.hirty) _s.teps , to. a cqm­pac't sequ�nbe of essentials/ thus. ·ieavi:ng', in all

. · ...... -ea ... s�s , . the reade� . rq :rae�ast t�e·. esse�t��1� , in fhe

. .forlll; ¥ given iil- · 9ur ac�epted moQ.ern ·texts . ·

· � B� so doing a savi�g · of as much as 60� of page s�ace results--also hours of time for thinker � � / . . and printer .

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C> GEOMET�IC PROOFS

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In �ig . 216, th�ough D ., .,-{ dr�� LN par . to AB, . extend ED t�

G , "' '' �� · and draw _Ht and Cll .

\ . '\ � j Sq . _ AH = (rect . AN . \ = 11aral . AD = sq . �H ) + (rect . MK

' ·= ·2 t1ri. . DCK = ' sq . GH ) . ' \ ' ) .

.A , . ., . :. sq. tipon AB· = sq. upon • · , "' 1 - \ ' " I B� + sq . upon AH .� :. h2 = a2 :t ·b� . ,..._ -'A''"!-��- � tJ · a . Contriv�d by--the L 1 · · / ;,o I . . author , August 1 , 1900 . . I I ;� \ · I ' ·

1 / " · \ t ' b . As ··in types ·A, -B and C.;.:' _ _ ;_ -· - � Ji C � ·many other �ro.oTs may be de­

ri v�d from the D type of figure .• , Fig . 216

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This typ.e· incllid.es .�;tl proofs ·del;'i'ved · f-rom �he figure in which the - squares const�ucted upon the-

\ 1, • -· - ·- - -- \ hypotenuse alid the long_er le_g' -.pverla:p .the gi�en tri-angle . . .

I J ' -

In..:.f�g-. . 2ti·, �.th�.o.��� H

1.,.;. · L . -·-u d.raw LM P,ar • . . to. KB, -and.,' d.raw GB , � .. - - ._ - ..., � u-v d HC ,., . I , , � an •. .

. : ' , . 1 ,' ;.,.� · Sq . A,K = rect • .IS + · r'ect . . : : · > ��-�!. _ - · ' ,. LA· = _ _ (2 . :t_r:t. . HBK .= sq . · HD ) + . (2

. 1 1 · ) ·tri � ·CAR = . 2 tri :· ·:BAG ::�:' sq • .AF ) ; -1

.-_ _:__L. -'-- .!___'l> . . :. �q � - �-qpon AB. � ' s-Q. • . upon ----·

A':. ;. '!-;:·' ' . • • - 2 . 2 • '-" � . -- .. BFI: + sn,. ' UP, on_ AH. • . : • . h I = a .If. )::>""' . '· ,� ' t '\. � �

. .

. . . . . ;·' "/.F _ · ,' . a . See, Jury: Wipper , 1880� ·

\ ,. · /. · p . : 1�� -VI'; Ed.W.�r4s '._Geom. ·, - �B95,

s? . p . ' 16�_ , ·fig . (38 ); Am. Math . Mo • . , v . ·"¢, 1898, p • . 74, proGf LXXV ; Fig : 217 iVersluys , p . 14, fig·. 9 ; one i of . .•

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170 THE PYTHAGOREAN PROPO�ITION

Hoffmann ' s collection, 1818; Fourrey, p . 71 , fig . g ; Math . Mo . , 1859, Vol . II , - No . 3. , Dem . i3 , �ig t 5 .

C ° K •.. - .. .. .. - �t

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' : , � " , (; ,.,"'

Fig. 218

In fig . 218, . extend DE to K and draw DL and OM par . re­s_pectivery· to AB .. and BH· •

, _ Sq; AK = (rect . �-= par�l . AD = sq .1 BE ) + (rect . LK - = paral . GD :::: trap . OMEK = trap . ·AGFB )· + . ( tri . KPN = tri : eLM) = :sci . :eE + · sq/ AF . • ' -

: .:._ �q . upon AB = ' sq . upon BH + sq . upon AH . :. l;l2 = �2 •

+ b2 . a ��see Am . Math . Mp . ,

V. V', 189B-;·p·: 74, LXXIX. -

) ..... .......... . '" ""Y " "'"''"''"''"'""'" I I� fig .- · 219� e�tend KB, \ - �o P� draw ON par . to HB, �ake Q 1 I - ·NM =· HB,, ahd dr�w ML par . -�o � ..

.'Sq • .AK· = (quad . - :tJOKO � · 1 � quad . GPBA ) + (trl • . · eLM·= tri . _ 1 : BPf ) +. · (� ra_p -. ANML, = trap . BDEO ) ' i _+ tri • . . ABH :common to sq ' s AK and 1 _AF + 'tri � BOH common to�_ sq ' s Alt \ --- \ and HD = sq . HD + sq. .AF . . - - I

--� - - - -- - �- · :.' sq. � upon AB .= sq. upon· I · BH + sq . upon AH . :�; .. -h2

= a2

Fig . 219 ..

- School:' V:i:si �pr, . ', 1 ( \ .

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+ b2 . I'

'- a. . Am.: Math • . Mo . , Vol . V, 1898, � . 1�i proof LXXVII ;' ! . \ \

Vol �. III , . p . ,?08, No . '410� ' · , -

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GEOMETRIC PROOFS 171

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G� ' - - · 1 Fig. 220

In fig . 220 , extend DE to K, GA to L,' ·draw GL par . to AH� and draw LD and HG.

Sq . AK = 2 [trap . ABNM = tri . · AOH common to sq ' s AK and AF + (tri . AHM : tri . AGO ) , + tri . HBN common to sq ' s AK

· ·' and H1) +. (tri .v BHO = tri . BDN ) ] ' \ = sq . HD + sq . AF . ·.

. \ :. sq. upon AB = sq . ·

upon BH + sq . UJ>On AH. . ,

a . See Am . � Math. Mo . , ··�v�o�l-�--v-,-'1898 , p . 74 , proof LXXVI .

E�'tend GF and Eb�""t� -0 , · anC.. comp1e��.-the rect . Mo; and extend DB to ·N .

Sq . .,. AK �= r�c t . MO (4 tri . ABH + rect . NO )

= { (rect . ' AL + rect . AO ) · (4 tri • .ARB + rect . NO )} = 2 (reb t • AO = rec t • AD _ _

+ rect � NO ) = (� rect . AD . , + 2 rect . NO - .rect . NO - - 4 tr.i ·: ABH J - (2 .J•ect . AD ·

/ c:� '

+ rect . NO � 4 tri . ABH )

(;:' = (2 rect . AB + -2 : rect . HD. ,.Fig . 221 " + rect . NF + rect . BO - 4 -

' · ·' ' tri . · ABH ) ··= [ rect . AB . . -.+ (rect . AB + rect . NF ) ' .+ rect . 'liD .+ (re�t . liD + rect .•

. ·B_Q )" - 4 ' tri . ABH ] ·= � tri � ABH + sq> HG . ·+ �q-� HD. . .

+ - 2 tri ABH - 4 tri • • ABH ) = sq.· HD + sq . HG . . :. · sq . upon AB = sq . upon BH + sq . upon AH •

. :. h2 = a2 + b2 .

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172 THE PYTHA(}OREAN P·ROPOSITION j -l -� � --

a . ·This fol'mula and �onvel'si.on is that of _the ::: __ - - - --�,......,..,Jll authol', Dec . 22, 1933 ��ut t�e' ffgul'e is as $iven lh

: �· Math . ·Mo..-, Vo1 2, 1898, p . , 7'4t. w�_r�_:,_��L _anoth�l' · somewhat -· diffel'ent-' pl'oof., No . LXXVIII . But �ame fig.-\ \ "' . .. . "'\ Ul'e fUI'ni shes : // ·. .

•.

. In · fig •.. -221, .extend: GF and ED to- 0 and com-

_ _pJ���->�-�4a,L.:ne,c;��7-MO-i·:..'. Exterid ·nB · to N. ·, .

\

. · · ;s� . AK = rect . NO. + 4 tl'i • . ABH = !'ect . MO = sq,. HD- + sq . AF + r.ect . BO +, [ I'ect . At. = . .(r�ct . HN = 2 tl'i . ABH )· �+ ('sq . HG "=:= 2 tl':L . ABH:· +-.I'ect . �F ) ) , . ' \ '

. ·which co11 ' d gives· flQ.. AXr·= s.q . HD + sq . HG� . :. s(i� · upon · AB = sq-. upon BH + ·sq . upon AH . '2 2 2 :. h =: a · + b • .,

_ . a . Ci'edited to Henry- Boa.d py Joh . Hoffms,nn, in "De:r Pythagor�isc�e·· Lehl'satz � " 1821 ; see Jul'y W:tp-.. pel', 1880� ·p � 21, fig �· +5 ._

' \ . . �n.!:.liY.!!.QtlQ.:..!�!!!!l:!l!.t.l' .

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. lin f:1-g . 222, �s.w CL· and ' 1' L. kL pal- .. re.spect1'\re1y� to AH land - · ,. . _,. I \ , . . "'"' "' ' JJ �II, . ·and -di-al( . trlito'Llglr'H;� • j - - -t - 1"

I " £ . _..

. Sq·. .AK = -hexagon .. :AHBXLC

. . : 1 �/.', · = pal' a� . -� + .para I . LA = �q . HD .,. I . - ,,��+ sq .• A'JJ . o · • •

• . I 1 . 4 .. . :._ !.'iq. upon .AB = s.q . · upon. / U ' . - · 2 2 2

A t · � .. BH + ·�q. upon All. :. h : = .a + b •

: )F _ _

·B(� Dey� sed by- �he a.utho:r, . ., .. .Mal'ch 12,. 1926 . ··

\ / Gv ., - Fig . 222

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. . ' . ·O.!!.!...:f!Y.!!.�t!Q_!lt!!!l:.f.21Lt

·-.

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Rect . LM = .[ sq . AK = (pa.l'�_s 2 common to sq. AK and sq . HD + 3 + 4 + 5 common to sq. �- an� sq. HG )

...._ • � ..- - • ._ .,..;:.,_,.•. r ·�

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GEOMETRIC PROOFS \ 173 -�--:r--�-_._..--...---·--:....,......:--..- -�� - .... "' _______ _ ', ' �

(} . 11 + parts 6 + {7 + 8 � sq • .HG ) c

� - ' • ·· - ,_- � --.Jr + 9 + 1 + 10 + 11 ::: {sq. AK � sq. '\ ' ' £ 16 HG + part s {(6 ::: 2 ). + 1 = sq .. , c \ 3 )�\"'1 ;· · liD} + :part s. (9 + 10 + 11 = 2 tri .. t lf ' "� -(. I \ 1 ABN + trr;_ ICP;E ] = [ (sq � .A:K = sq. t .$ _.

. 1 .1'-f, HD + aq . EG ) + (2 t:rai . ABH ( ..

,4 1 l + tri , Kl-'E:)l, or rect . . LM - 2 -. I\ .. �\ � I , tri , ABH +' t�·� . KPE ) _� ( sq .. AK- .'

t ' ' 7 /, ', t � sq . RD .+ · sq . R:A l· �- · • ,

M L �-� � _ . .r __ � N . . . • . s.q . AK � �q , .tiD + sq .

HA • :. sq. upon AB = sq.. upon Fig . 223 1ID + sq . upon HA: . :. h2 \ = a2 . + b2 • . Q.E .. ·D . · ' a . Orig-i-n.al with the·· author,- J'une 17· , 1939. .. b � See Am . Math ." Mo . · ., Vol. • . V, 1898, p . 74 , -- ·

proof -�!!� £or ano�her· p�oof, which is : (as per · e s sentials� .. : .. · · " · .. · .

. \ .

�- � . .., . In fig". .223 , extend OA:, HB , DE'· and OK to M,

0

N, K �nd L �� ape.ctiyely, and: draw MN; o LN· and co P.e -specti.v.�ly 'ptt·r .. .. t o AB, KB and HB .- ' _ ·

, ·

. 1 ·

. · � Sq. ·AX + 2 t�: L AGM + 3' tri . . GNF. + trap . AG� ·

= rect . ON = , s,q . :,an�,+ aq. HG + � tri . AGM +. 3 trl . GNF + tra� . ·COE�, whioh coll 1 d _g�ve s sq . AK = sq. HD :+ sq .· : JrG. ..

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• & · ·,, :. ,..

� . � :. sq . upon .AB = sq . · upon • ., � �.:. 2\..� • · BH + sq . upon AH . :. h��"· � a.•· + b � . . c r -:-· - :- ·0: -1 1< ·· ' . , · . .

. I . � .. · I t .��,e

I ; � . \ I , . ' '> , · f · . o } •

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- ' . ·. ' In fig : '224., e.xtend KB . . ' . ' il( . o-: . A . . ' �- I 0 I'\ • •

' I -\ - ·, 0 I . jf )F· and CA re spectiveli to o· and N, "I • {, • .., •

through H. ' draw .LM· .par'. -to ·KB; an<i I ' I _.( .

0 1 ' -� � 0

� (· · N: ..,� · �\, · Jo/ � � ·�.

clraw· GN a�d :MO l:'espectively par . .. . to Air a'tld 'BH. . . ,. - - ' . .

, o • · .sq . .AX = rect � LB' ·+ rec t .

LA = paral . ¥ffiMO + ·paral . HANM

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'174 THE _ PYT.HAGO� PROPOSITION-. . \ ' , ., :. s q . -upon AB ;=--- sg.. upon BH ·:f- sq . upon A..T{ .

2 - - 2 2 '----- -:. · h = a · + b • _ . -- -

.-, _

_ o a . Original with the a�thor, August 1, 1900 . b �,��any ·other ·p!'oofs ar� derivable from ·.this

type ·or · f·i'g1,1re •

· ·

. 'C . ·An. algebraic pr.oof is easi!-Y obta;l.ned fr_om :fig . .. 224 . .-

F

. ' • This type 1ncl�d& s all p�oofs de�ivea from .. �he fig_ur� in whiQh the .squares co'nstructed , �pon the ·h:r.Potenuae and the shorter leg ·.ove�lap tlie given tri- -angle .

<J

. __ . ; 1 . _ . �n� .. J!to.str.!��Il!�!l!t::�!��fl-( �· t;J\.. _ - _ _; In the fig . 225 , draw KM , . \ 6( . & . " I ' f .,-"' 1 par � t o AH.. , . ... � I �--/ 1 I . (

-. ' · 1 'I �" Scl,. 1\$. =- .tri � l3KM = tri , ' , t .\ · 1• . AC.G )- t (tri . !liM = tpi . BND )

�,' " - :·)- . . 'B- + quad. �c c<i,�on t'o·. sq ' s .AK . +

fi . _ ., . and · AK. + (tri .. A11.t1 = tl'i ., . CLF ) ,, - · '�, "' + t�ap·. NBHE' common to sq' s AX

and :EB :::; sq .• HD · + sq . HG. . _ : .. sq. upon A.B = sq. upon

BH I AH' • h2 2 + b2 , · + sq . upon • . . = a. · • ' . , a . The . Journal of Ed\t�.Ja.tl9P., _ V . · .XXVIII, i888,

p . _ �7t 24th p!!oof, oredlts thts prQ;Of' to J . M . Mc­Ch•eady� of 'Biack Hawlc, · Wis . ; see ·Edwa;tteis· • Geont4 , '

.- ' 1895, p . 89, . :srt .. 73 ; Heath'-'a IA..ath. Monogra.phs_, __ No .• , 2,

. ·1900, p . _ . ·32; .. proor XIX; Scie�tif�c :R�\ri�wi .Feb,. i6 , :· l_B8Si, p·. :31J .i�i:g�-:-:30J---R .. - A� :Bail, .Tuly 1 , �193'8-, one of }!:1,._�: 4Q . p:hoo:Ce . · · :, · o ·

• _ \ · . � · ·

.. - b . By nuxncering the

. 9-if:ssec'J;ed parts·, SI:l ob\1:1- . :, � --e-us--pztOo£.-.i.u seen .. . . .. ' '\. ---' ' ·'\.. --� --....:.. "

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GEOMETRIC PROOFS 175

In fig . 226 , . extend AH to N making HN = HE , through H draw LM par ._ to BK, ahd draw BN , HK and HC .

Sq . AK = rect . LB + rect . LA = (2� tri . HBK = 2 tri . HBN � sq. HD ) + (2 trf � CAH = 2 tri . AHC = sq . · HG ) = sq � HD. + sq . HG .

:. sq . upon AB = sq . upon · BH = sq . upon AH·. 1 :. h2 = a2 + b2 .

a . Original with the author, August 1 , 1900 . b . An algebraic proof may be res olved from

thi s figure . c . Other geometric proofs are easily derived

from thi s form of figure .

· Fig .

p . 14, fig . 1

In fig . 227 , _ draw LH perp . te AB and extend it to niee"f< ED· produced and draw MB, HK and. HC .

Sq . AK = rect . LB -t; rect . LA = (par�l . HMBK = 2 tri . MBH ' = sq . BE ) r+ (2 tri . CAR = 2 tri . ·AHC = sq . AF ) = sq .. BE + sq . AF .

-- :. sq . � upon AB � sq . - upon · · BH + sq . upon AH . , :. h 2 = a 2 + b2 .

227 . 1 a . .See Jury 1fipper , 1880, p j. 1 4 , i'ig . 7 ; Vert�luys ,

10 ; .. Fourrey, p . /71., fig . f . _ -

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\ GEOMETRIC PROOFS 177 '

V . V, 1898, p . 73, proof LXX; \A . R . Bell , Feb . 24 , 193� . . \\, b . In Sci . Am . Sup . , V . '70 , p . 359, Dec . 3 ,

19l0 , is a proof by A . R . Colburn, , by use of above � figure , but th� argument is not that given above �

In fig . 230 , e�tend FG to C and 'ED to K .

Sq . AK � - (tri . ACG = tri . ABH of sq . HG ) + (tri . CKL = trap . NBHE '+ tri . BMF' ) + ( tr·i . KBD __ = tri . BDN of sq . HD + trap . �D common to sq 1 s AK and HG-) + pentagon AGLDB conunon to sq 1 s AK and HG ) =:= sq . HD + sq . HG .

Fig,. 230 :. sq . upon AB = sq . upon BH AH h2 . 2 b2 + sq . upon . :. · = a + .,

a . See Edwards ' Geom. , 1895� p . 159 '- fig . (24 ) ; Sci·. Am . Sup . ; V . i 70 , p . 382 , Dec . 10 , . �910 , for· a proof by A . R . co!lourn on . same form of �-�gure .

'Fig . 231

The construction is obvious . Also that m + n = o + p ; al so that . tri . · ABH ·and tri . ACG are congruent . Then sq. AK = 4o + 4p + q = 2 (o + p ) + - 2 ( o + p ) + q =

-2 (m t n ) + 2 ( o + p )

+� q = 2 (m + o ) + (m +1 2n + o + 2p I I

+ q ) · = sq 1 HD + sq . HA . . I - T -j :. sq . up�n AB =-� sq . upon HD

+ sq . upon HA . : . . h2 .y,-a2 + b2 . Q . E . D . · . t

a . See Versluys , p . 48, fig . 49 , w�ere credited to R . Joan,

Nepomucen Reichenberger 1 Philosophia et Mathe si� Universa, Regensburg , 1734 . · ·

_

· b.. By u�ing congruent tri ' s and trap 1 s the -algebraic appearance will vanish .

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f I ! 178 . THE PYTHAGOREAN PROPOSITION

Fig . 232 1

b .\ By p , r and s\ in I re sult s .

.... : ,

Hav+ng the c onstruction, and the parts symbolizedj it i s ev:!-­dent that : sq . AK = 3o + . P + r + s = (3 o + p ) + ( o + p = s ) + r

- - 2 ( o + p ) + 2o + r = (m + o ) + (m + 2n + o + r ) = sq . HD + sq . HG .

:. sq . oupon AB = sq . upon HD + P HA . h2 2 + b2 : sq . u on • . . = a • .

a . See V�r.sluys , p . 48 , . fig . 50 ; Fourrey, p . 86 .

expre s sing the dimensions of m, n, o , terms of a, p , and h an algebraic proof

-.

- . ' I

Cohip'lete the. three sq ' s AK, HG and HD, dra� CG, KN, and vnL through G . Then - 1 ,

Sq.. AK = 2-[ �rap : AeLM = tri . GMA common to sq 1 s AF;. and--AF + (.tri . ACG = tri . AMH of sq . AF + - tri . HMB of .. _sq . liD ) + "(tri . C'LG = tri . BMD of' sq . HD ) ] = sq . HD + sq . H&. :. h2 ,; a2 + ·o2 .•

Fig . 233 :. sq . upon AB ·= sq . upon BH • - - I

+ sq . � upon AH . a . See Am . Math . Mo . , V . V, 1898, p . 73 ,

proof LXXII .

Draw CL and LK par . re spectively trr HB and HA, and -draw HL . · � / . I .

Sq . AK = hexagon ACLKBH - 2 � tri . ABH = 2 qu�d . ACLH - 2 tri . ABH = 2 tri .• ACG +· (2 tri . CLG · = sq . HD)

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GEOMETRIC PROOFS 179

+ (2 tri . AGH = sq . HG") -' 2 • �ri . ABH = sq . HD + sq . iiG -f: (2 tri . ACG = 2 tri . ABH - 2 tri . ABH = sq � HD - sq .

A 1 . HG : 4 ·_

� - K \ I ' i ' < :. sq . up9n AB = sq . upon iiD I \ ' /�""' ;' ) )... - - 2 2 2 \ � - r + sq . upon F..A . :. h = a + b . - , - ./.{ "' J , �.., Q .E . D . r � ' • I . a . _ Original by author Oc t .

·C I_,"' - ' ' t� 25 , :1:933 . " - .0:.. - - 7' 1\ '\ I / ' . • L "" \!.

Fig . 234

Fig . 235

c

In fig . 235 , extend FG t o C , ED to � and dr�w HL par . t o BK .

Sq . AK = Teet . . BL ' + rect . Jl..L . ­= (p�_r.al . �H . = sq . - HD ) + (paral . CMHA =· sq . HG ) �· sq . liD + sq . HG .

:. sq_�_JJpon AB = sq . upon BH + sq . upon A)H. :. h2 ::; a2 +- b2 • Q . E . D .

a � Journal of Education, V .

.•

XXVII, 1888, p . . 327 , fifteenth proof by M. Di-ckinsonu--W.inchester, N . H . ; .Edwards 1 Geom . , - 1895 : p . 158, fig .

· (22 ) ; ·Am . Math . Mo . , V . V , 1898, p . 73 , proof LXXI ; ----­Heath ' s Math . Monographs , No . 2 , p . 28, proof X:IV; Versluys , p . 13 , fig . 8--also p . 20 , fig . 17, �or same figure , but a s omewhat different proof, · a proof credited to Jacoo Gelder , 1810 ; - Math . Mo . , 1859, Vol . II, No . 2 , Dem . ll ; Fourrey, p . 70 , fig . d .

·.

b . An algebraic proof i s easily devised from this figure .

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180

I­I

THE PYTHAGOREAN PROPOSITION

- . ' .

Draw HL perp . to CK �nd ex-\ ' . ten� ED and FG t o K and C re sp ' ly .

. 1 I 'O c � � �� ���:.1<

Sq . AK = reqt . BL· + rect • . AL = (tri . MLK = quad . RDSP + �ri . PSB ) + [tri . BDK - (tri . SDM = tri . ONR ) = (tri . _ BHA - t;-±·. � )- = quad . . RBHE.] + [ (tri . C�==<i-t'r-i--.-�ABH) + (t:r;>i . CGA = tri . MFA ) + quad . GMfA ] = tri . . RBD + quad . RBHE + tri . APH + tri . MEH. + quad . GMPA = sq . HD + sq ._ HG.

:. sq· . upon ·AB = sq . upon BH - < .. G- . 2 2 + sq . upon All . · :. h = a + b . Q . E . D.

Fig . · 236 ' • • - ".!>

a . See Vers1uys , P� 46 , fig ' s 47 and 48 , as given by M. Rqgot,, and made knovp. by E . Fourrey in his "Curiosities of Geoinetr�, " on p . 90 . e

Fig . 237

In fig . 237 , extend. AG, ED , . . BD and FG to M·, K., L and C re spectiv_e-ly.

Sq . AK = . 4 tri • . -ALP + � quad . LCGP--+ sq . PQ + · tri . AOE - (tri . BNE = t�i . AOE ) = (2 tri . ALP + 3 quad . �CG� + �q . PQ + tri . AOE = sq . HG ) . -f; (2 trl . ALP + quad : LCGP - tri � AOE = sq . HD ) = _ sq . HD + sq � HG .

:. sq . upon AB = sq . upon BH?' T- sq . upon AH . :. h2 = a2 '+ b2 •

a . See Jury W;tpperi 1880 , p . 29, fig . 26 , a_s giv.e-n -by Refichenberge/r , in Philosoph- · ia et Mathe sis Universa , etc � , " Rati �bon?-e , 1774 ; ­Vers.iuys , p . 48, fig . 49 ; Fourrey, p . 86 .

b . Mr . Ri�hard A . Bell , of C leveland, 0 . , submitted, Feo � 28, 1938, 6 fig ' s and pro�fs of the type G, all found between Nov . 1920 and Fep . 28, 1938 . Some of his figures ar.e very simple .

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·' GE;OMETRIC PR.QGF.S------ --------- -------�---- �-------------- i81

..

\ ., · - 1 In tig: 238 , ex ten� E:r;>-· and . 1 H .FG. !tc K: and C respectively; dren.r HL

• 1 1\ pe�p . .. to OK, and draw H��. �pd !L.1( . _ _

I • . Sq . AK = re c t ; BL + rect . :AL · · · •\ = {.pa:rtal . MKBH = 2' tri . KBH - sq . HD )

i ' , A t'": , } �r .+ ( 1 cuu/1 2. t i CHA HG )' < , r para . t�lll.l1. -- . r . ::: sa •

I J. "}) ' "\ I = 1ID HG r

,.

, .. , �M I sq • + sq ; . :

I ' � � � 1 ' ·\ 1. . J :. sq . upon AB i = sq . .. upon BH .\ ·-�, L �� + s ' "l"\on AH - . h2 1 � + b2 -C . ::1. q . .....,.t' J

•

• • ;:: a.. •

< � "·..:. -- . · ,. a . See Jury Wi:pper , 1880 , l . p • 12 , fig . 4 •

. b-: Thi s proof is on1y a vari-ation of the one preceqin� . • 0 . ,

c . Fro� thi s figure an algebraic proof i s ob-tainable .

Q���tt�n�r��-fRt1Y�Qn�

· In fig . 139, extend FG to 0 , F� to L mdking FL = HB ,-·and 1tl:raaw KL and -KM respectively par •

. .A jto 1UI and. -BH . t ( .

'' \ � · I > F Sq. AK =. · [ tri . CKM ·1 ' , '� � � \ . = tri � ·BKL. ) ·- tri . . BNF = trap c - _ : \ · . 1!1� ·: � OBHE] + .\tri .;- KMN ; tX'i - BOD )

" I 1'1 ,� , = sq . HD � + [ t'ri . ACG = tri , ABH ) Cl. � � .. ... . . '_� � " + ( tr i . BOD· + hexagon AG�'BDO )

,. = ·sq . HG] :::· sq . JID· + ' sq: -HG.o Fig � 239 :. sq . upon .AB = . sq . upon

BH + sq . lipan AH . :. h2 = a2 + b2 • a . A s taken from "Philo'S'ophia e t, Mathesis

Universa, etc- . , " Rati sbonae , 1774 , by Reichenberger; see Jury Wipper , 1880 , p . 29, fig . 27 •

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182 THE PYTrt..i\.GOREAN PROPOSX':tliOr{ -------------��----·------------�--�--�---------

Fig . 24d

In f:!.g /· 240 , extend' HF and HA re spectively to N

\

and L, ana ·complete the r.q . HM, e.nd extend ED t o K and BG . to -C . � ' � . Sq . AK = s q .- HM . • 4 tri . ABH -:.:: (sq . l?K = sq . HD ) + sq . HG .±.--(.l!��t -. LG--= -2 tri . ABH ) + (.rect : OM = 2 tri . ABl:i 'i ,;,. ·· $q . HD + -sq . - HG + 1} , tr:L ABH - 4 ti�i . ABH = sq . 'liD ·+ sq . HG .

I, :. sq . upon AK = sq . upon BH + sq.. upon .AH . ' 2 2 ·· h --a2 -r' b . ' !" .1. .. a : Similar. -t"o Henry Boad·' s pl100 f , London,

1733 ; s.ee Jury ;,;:rpper , 1880 , p . 16, fig , - 9 ; Am. l<Iath . Mo . , V . V, 1898, p . 74 , proof LXXIV .

Fig . 241

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In fig . 241, exte�d FG- an� ED 'to C d K ti l dr . an , resp�c · ve y , aw FL _ par . t o AB , and draw RD and FK .

Sq . AK � (rect . AN = paral . MB ) + (rect . LK = 2 tri . CKF = 2 tri . OKO + 2 tri . F'OK = tri . F.MG _ + tri . ABH + 2 tri . DBH ) = sq . HD .. + sq_ . HG .

:. sq . '9PO!l AB = sq . upon Blf + sq . upon AJL :. h2 - = a,2 +. t2 • Q . E . D .

a . See Am . Ma{th . Mo . , Vol .. Y � 1898, p . - 74, pr·oo� LXXIII .

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0 G�OMETRIC PROOFS \ 1'83

· j " In fig . 242 , -P�.oduce FG t.o C3, through D �nd G draw LM and NO par . to AB, and drf!,w AI:( and ._ BG .

Sq . AK = rect . rnc + rect . AO = - (:rect . AM = 2· . �ri .- ADB = · 'sq . HD ) . + (2 _ �ri . GBA = �q . HG ) = sq . liD �

+ sq . HG. :. sqL .upon AB = sq . upon BH

+ sq . upon AH . :. h2 = a2 + b2 • : a . This · is __ No . · 15 of A . R .

Colburn '� 108 proofs ; see his proof in Sci . Am� Sup . , v. 70, p . 38?;

Dec-. 10 , 1910 . "' b . An· algebraic proof from t4i s figure is

easily obt-ained . -

2 q 2 tri . BAD = hx � a • - - - (1 ) 2 - tri . BAG = h (h - � ) = b2 • --- (2 ) ( · ( 2 2 2 I · ) 1 ) + (2 ) · .= 3 ) h - � .a .. -.+.::::h":':!"'--- -· � �·v:&-.-$.,.<L .• -

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In fig . 243 , p�_oduce HF and CK to L ;' ED to K, and AG to 0 , and draw KM and ON par ; t.C? AH .

Sq . AK = panal . AOLB . \ ( = [:tr·a.p·. - :A:<fFB + tri . OLM ·

=: tri·. ABIL1 = - -�q_. __ _H_G ] + { rect . GN = tri . CLF . - (t�i . CO� -= tri . KLM ) -· . (tr:l . O:J;.N = tri . CKP ) ] = sq . FK = sq . HD} = sq . HD t sq . HG .

:. sq . upon AB =· sq . upon BJI +. sq . upon AlL :. ·h2 = a2 _ +: b2 •

a • . Thi s proof i s due to ;I'rih . Geo . M . P}+il ... . . lips-, Ph .Ir. , of the West Chester State Normal School, p·a . , 1875.; see Heath ' s Math •. . Monographs , No . 21 p • . 36,

'I proof XXV . '

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{ ,.;;;.; 184 · TKr PYTHAGOREAN PROPOSITION . ,

·.

In· ffg . 244 , exte-nd CK -and HF to M, ED to K, : and AG to ·

· r 0 making GO = · HB , 1 draw ON p�:t> . H ··

to AH , and draw GN .· . Sq . AK � paral . ALMB A ] = paral . · GM + paral . . AN =., 'f (t'ri . I \ / \ • :t.·� HE' ) 1 , ' , "' l ,�o , NGO __ - tr . . NPO = trap . RB ·

1 ' � .,......, ' + (tri . KMN ·= tri . BRD ) ] = ·sq . liD ' "'V I -b \ G , ., � - - -.I->I'tl ... + sq . 'H . 1 _

1 " � t � \ · :. sq . upon AB ;:_ sg . upon C "' ' L ' , ., l:) -

- ,_ ' " ' \,i.J � BH + sq . upon AH • • ·. h2 -- a2 + b_2 • • � - - �.0:. 7 " -, -M -.. Fig . 244

" : -a . De�ised by the author , March 1·4 :. · .1926 .

I I I -�hrou$h D �raw DR par .

AB nieeting -HA a·t M., and/ through . G draw NO par . to AB me·'eting HB· . . I ' ., ' r at P , and draw � perp. . to AB .

_ _ -t , _\..,...::_ .�--" , Sq . AK � -(rec t . NK M ,Q_ ' -rJ., ' � , = .rec t . AR---=-pa-ra-1- . AMDB = sq . ' \ N .,_ - �- -cf, - --p; · - -HB ) + (rec.t . AO = paral . AGPB C � - - � - JJ( = sq . HG ) = s q . HD + sq . HG ..

:. s9- . · upc;m AB = sq . Fig . 245 upon HB + sq . upon HA . :. ·h2

= B2 + b2 • a . See Versluys , p . 28 , fig . 25 . By Werner .

. Produce· HA �hd HB tb . 0 a�ci N re·sp ' ly �aking AO = HB and BM = E:A;-and complete the _sq . HL .

Sq . AK = sq . HL - (4 tri . ABH = 2 rect . OG ) = _ [ ( sq . GL = sq-. ·HI) ) · + sq . HG + 2 rect . OG ] - 2 rect· . O.G = sq .· HD _ .+· s q . HG . :. �q . upon AB ::;:: sq . up'on BH + ' sq . upon AH . :. h2 = a2 + bf . . '

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GE0METRIC PROOFS 185

, .. , , F g., .J. \ 'l.. ' , " ' . I \ , , \ \ , " . ' I , ' I '· _ \ . � ,

, · -a .. :·see Ve.r sluys , p '. 52 , fig . 54 , as found ±n Hoff­mann 1 ·s list- . and in 11'9e s Pytha­gora.ische Lehrsatz , " 1821 .

I I I ' ' . i · Q.n�_!!.Y.!19.t��LE!H:.1:t:J!l!l! < ;. - . . . \ I - · A I . ,. L - : � I �· " t:7" , _J(' 'N \ . ,\� . ! . ,-:_ � - ' � I. ' �

c---r � ..J ... r - 1 I /"f¥: - .;... - \.. Produce � CK and HB to \.1. \ \. , ,_ " H \..,. "

Fig . 247 l·

L;·· AG .to M.,_ ED to :k, FG to C , and dra-W MN anq KO par . t o AH .

· . . - -Sq . r:AK = P?-ral . AMLB = quad . A:.GFB + rect .· GN. + {tri. MLN" = tr:f. . ABH ) = sq . GH- · ·· · . _· .

+ (reot . GN = sq . PO = sq . HD ) · = sq-. HG + sq ., HD . :. · sq . upon

. \. . AB = sq . ·upon HB � sq : upon HA . :. ·rr = a2 + l;>2 •

a .. . By ,n.r· •. _ _ Geo . M . ��il;J.ips,. of . . We st 1 Che ste�,. in 1875 ; Versluys , p . 58, 62 . I

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H -

Pa . , fig . '

This type include s -a. ... ll proofs devised from the figure. in which the square s con�tructed upon the hypotenuse and the two legs · overlap the given tri­angle .

. ' . Fig . 248

D:raw th.ro_ugh H,.. LN perp . to AB, and draw I;IK,· HC , NB �.nd NA . .

Sq. AK = rec t . � + rect .

· ..

LA = paral KN + paral . CN = 2 tri . KHB + 2 �ri . NHA = · sq . HD + sq . HG .

:. sq . upon AB = sq . upon HD + · sq . upon HA . :. l-12 = a2 + b2 • Q.E .D.

a . See Math . Mo . , 1859, Vol . II, No . 2 , Dem . 15 , fig . 7 .

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. � 1 86 THE PYTHAGOREAN PROPOSITION

__________ __ -�------!:to.Ltl..!tn_f!re d F i f !Y.::.Qn�--- - _ _ ___ _ _ __ _

Through H draw LM - perp . to -.AB . Extend FH to 0 making BO = HF, I dratv KO , CH, HN and BG.

Sq . AK = rect . LB + rect . LA = (2 tri . KHB = 2 tri . BRA = sq . HD ) + (2 tri . CAH = 2 tri . AGB = sq . AF ) = sq . HD + sq . AF .

:. sq . upon AB = sq . upon BH + sq . upon AH . :. h2 = a2 t ba .

a . Original with the author . Afterwards the fir_s t part of it was

Fig . 249 discovered to be the same as the solution in Am . Math . Mo . , V . V,

1898, p . 78 , proof LXXXI ; al so see Fourrey, p . 71 ; fig . h, in his "C-uri_o sitie s . " ,, b . This figure give s readily an algebraic proof .

Fig . 250

·-- ----��-� ------ --------- -- - -------------

Qo.�-tl�o.f!r�d F i f t�- TWQ

,.._�In";· fig . 250 , extend .:J�iD� to 0 , draw Ao;···OB , HK and HC , t.l1d draw, through H, LO perp . . to ··AB , and draw­CM perp . to AH .

Sq . AK = rect . LB + rec t . LA = (paral . ijOBK = 2 tri . OBH = sq . HD ) + (paral . CAOH =· 2 tri . OHA = ·sq . HG ) = sq . . HD + s q . HG • .

. :. sq . upon AB = sq . upon BH + . - AH . h2 2 + b2 sq . upon . . . = a . Q-:E<P . -j a . See Olney ' s Geom . ; 1872 , Part III , p . 251 , 6th method ; Jour-, . nal of Education, V . XXVI , 1887 ,

p . 21 , fig . . XIII ; ·Hopk:{.ns ' Ge om·. , �$96 , p . 91 , fig . VI ; Edw . Geom . , 1895, p . :}.60 , fig . (31 ) ; Am . Math . Mo .• , 1898, Vol . V , p . 74, proof LXXX ; Heath ' s Mah�h . Monographs , No . 1 , 1900 , p . ' 26 , proof XI . i

b . From thi s figure deduc e an algebraic proof .

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GEOMETRIC PROOFS i87

-- - --- -- - - �------- -- - -- -- - - --· - --- - - --- --- - ·Qn:�·.:!i!tnJ!'r:�9:..:-.Et:frtr::ftrr.-��

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Fig . 251

r 1 ' In fig . 251 , draw LM perp .

to AB through H, extend ED . to M , and draw BG, BM, HK and HC .

Sq . AK = rect . LB + rec t . LA = (paral. KHMB = 2 tri . MBP. = sq . HD ) + (2 tri . . . AHC = 2 tri . AGB = - sq .• HG ) = sq . HD. + s� . HG .

:. sq . upon AB = sq . ,, upon BH ·- -=f- sq . ·upon AIL :. h2 = a2 + b2 •

a . See Jury Wipp·er , 1880 , . p . 15 , fig . 8 ; Vcirsluys , p . 15� fig . 11 .

b . An algebraic proof follows the ''mean prop 1 1 " principle ;

_ __ ____ __ ___ -- -- _ -- --- · -- ---- -- - -- - - - - -- -----:: -- - - - - - --In --f'1:g-;·- 252·,---ext-errd--ED·-tt>--·Q;­CJ\- - -;71 J� BD to -R , . dra� HQ perp . to · AB , ON 1 ' � f perp . to AH , � perp . to ON and ex-

,: f He( H tend BH to L . · ·

, £ --' Sq . AK = t:r.i . ABH - c ommon to fi sq 1 s AK and HG + (tri ._ BKL = " trap .

HEDP of sq.. HD + tri . QPD of sq . HG ) + (tri . · KQM =. tri . BAR of sq . HG)

· + (t;r>i . CAlli = trap . QFBP of sq . HG + tri . P.BH of-- sq . ·HD) +· ( sq . MN = sq . RQ ) . = sq . ·HD + sq . HCi.

:. sq . upon AB = sq . 'upon .BH + s q . · upon- AH . :. h2 = a2 + b2

._

a . See Edwards ' Geom . , 1895 , p . 157 , fig . (13 ) ; Am . Math . Mo .·, V . V, · 1898, p . 74, proof LXXXII ..

In fig . 253 , extend ED to P , draw HP , draw OM perp . to AH , .and KL perp . to OM .

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188 THE PYTHAGOREAN PROPOSITION

_ Sq . AK = tri . A� common t o C J\- - - /.'�J< sq ' s AK and -NG

_+ trap . ENBH. c ommon

· \ /H.,/ lO to sq '- � AK and HD + (tri . BOH = tri . t ../ BND of sq. HD ) + (trap . KLMO = trap .

I . _ AGPN ) + (tri . KCL = tri . PHE of sq . fi )3 HG ) . + (tri . CAM = tri . l!PF of sq . HG )

\._ (, "' ) = sq . HD. + sq . HG. \ _ -.�], ,rp :. sq . upon AB .= sq . upon BH S�� + sq . upon AH . :. h2 = a2 + b2 • a . ·original with the author,

Fig . 253 .August 3 , l890 . b . Many other pr_o0fs may be

· ·.devis'ed f:r;>om this type of figure .

Sq . � = rect . MN - (rect . BN + . 3 tri . ABH· .

·

+ trap , AGFB ) � = (sq . HD = sq •. DH ) + �q . HG + rect . BN + [ rect . AL = (rect . HL = 2 tr i . ABH ) + (sq . AP " = tri . ABH +" trap . AGFB ) ] = �q . - liD + sq . HG + rect .,.J3N + 2 tri . ABH + · tri . - A.B!ff." trap . AGFB - rec t .

· ·BN -:-_;3 tri ·. ABH - ·trap_ • .AGFB = sq . HD + sq..- HG . - , :. s/q . upon AB =· sq . upon BH + ·sq . / upon AH .

Fig . 254

:. h2 = a2 + b2 • Q . E .D . . a . See Jury Wipper, 1880 , p . 22 , fig . 16,

credited by Joh . Hoffmann in "Der Pythagorai sche " Lehrsatz , " �821 , to Henry; Boac;l, of London, Englanc;l .

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GEOMETRIC PRQOFS 189 - - - -- -- _c.____ ---

'> fig . 255

40, like unto

In fig . 255 we have sq. AK = part s 1 + 2 + 3- + 4 _+ 5 + 6 ; sq . liD = parts 2 + 3 ' ; sq . HG = part s 1 + 4 1 + (7 = 5 ) + - (6 = 2 ) ; so sq . AK (l + 2 + . 3 + 4 + 5 + 6 ) = sq . HD[ 2 + (3 ' = 3) ] + sq . Hq [l + (4·• = 4 ) + (7 = �?+·�-- . . ....... ' "' + (:2 = 6 ) ] . --- .

:. sq . . J.IPOn AB ;.. sq. upon HD + sq . upon liN';-· :. h2 = a2 + b.2 • Q .E .D.

a . Richard A. �e ll , of Cl�ve ­lailq , 0 . , -devi sed above proof , Nov . 30, _ _ 1920 · �nd ga�e it to me Feb . 28, 1·938 . He has 2 other·s_, among hi s

it .

I ------------

Thi s type include s all proofs derived from a ' l figure in which there �as beeno a translation from it� normal position of one or more of t�e construc ted · '

squ�res . Symbolizing the hypotenuse- square by h, the

shorter�leg�square by a, and the longer-leg- square ' � ' v �

by b , we find, by inspection, t,hat there are s e v e-n di stinct __ case s pos sible in thi s: I-type figure, and th�t each of the firs� three nase s have fo u r pos sible arrangement s , each of the secopq three case s have t wo pos sible · arrangement s� and _ the seventh case has but . one arrangement , thus giving 19 sub-type s , as follows :

(1 ) Translation oT the h- square , wi�h The a- and b� square s constructed outwardly. The� a-sq . c onst 1 d out ' ly and the b-�sq . over­lapping .

(a ) (b ) (c )

,

(d )

The b.- sq . const ' d out ' ly and the a-sq . _ over­lapping . The a- and b-sq ' s const 1 d overlapping .

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190 THE PYTHAGO�EAN PROPOSITION

(2 ) Translation of the a-square , with (a ) The h- and b-sq 1 s cohst ' d out ' ly . (b ) The h- sq . c onst ' d out ' ly and the b-sq . over­

lapping .. (c ) l'he b-sq . c onst ' d out ' ly and the h: sq . -ove�---· lapping .

(d ) The h- �nd b-sq ' s c onst ' d overl�p�ing . .. - {3 )·�,ranslation of· the b-square , wi tl;l

(a: ) The h- · and a-sq ' s- c on.st 1 .d out ' ly . (b ) The. h-s.q . c onst ' d out ' ly and. tne a- sq . over-

la_pping . . -.

(c ) The a-sq . c·onst ' d out ' ly and the h- sq . over­lapping . (d ) The h- and a- sq ' s c onst ' d overlapping .

(4 ) TranS.J_ation of the h- and a-sq ' s , . w.tth (a ) The-b-sq . c onst ' d out 1 ly . (b )' The b-�q . overlapping .

(5 ) Translation of the h- and b-sq ' s with (a ) The a-sq . c onst ' d out ' ly . (o·)-Tfie a-sq . COnstTU-OV.EfrJ'ilpJf:fflg�· -·- --- --·-

(6 ) Translation of th� a- and b- sq ' s , with (a ) The h-sq . c onst ' d out ' ly .

· (b ) The h-sq . c onst ' d overlapping . (7 ) Tr�nslation of all three , h:.. , a- and b-square s ..

From the sources of proofs, qonsulted, I dis­c overed that only 8' out of the pos sible 19 . case's had

· received consideration . To complete the gap of .the 11 mis sing one s I have devi sed a proof fo� each mis s ­ing case, a 3 by the Law of Dis section (see fig . 111 , prbof �en ) ·a proof is readily produced t�r any posi­tion of the squares . Like Agas siz ' s student , after proper obs erva�+on he found the law, and then the ar� rangement ��paPts (scale s ) produced de sir�d result s .

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GEOMETRIC PROOFS - 0

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. 191

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.

Fig . 256 \

BH -t: sq . AlL

2!HLf!Y.��r.�g_Ei!i·r:::gighi Case (l ) , (a ) .

In {fg . 256 , the�q upon the \hypotenuse , hereaf\fer called the h- sq . has been trans-lated to the position HK . ·

,>

From P ' the middle pt . of AB draw PM making HM. =. AH ; draw

.LM, KM, and CM; draw KN = LM·, perp •. to LM produced, and CO = AB, p e t�p . to HM. I

Sq . . HK = (2 tri . HMC = HM � CO = sq . AH ) . + (2 tri . MLK = -�� x -�- = sq . ·BH ) = sq .

:. · 'Sq . ,upon AB = sq . upon BH + sq : upon :A.H-.-- -:. h2 = a2 + b2 . ,

a . Original with the author , August 4 , lz90�0�·------------------� �--�----------------�several other proofs from this figure i s possible .

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Case (1 ) , (b ) .

_ In fig.. 257, the

po sition of the · sq ' s are evident , as the b-sq . -overlaps and 'the h--sq. is trans-lated :to right·- bf _

normal po sition ; Draw PM ·'

Fig . 257

perp . · to AB ·through B , take KL = PB , draw LC , and BN and KO perp·. to LC , and FT perp . to BN.

Sq . BK = (trap . FCNT = trap . P.BDE ) + (tri . CKO = tri . ABH') + J:t:r:L .

� KLO = tri � BPH ) + (quad. BOLQ + tJi . BT:p----=. ·trap-. GFBA )

= sq . BH + sq . AH . : � . :. sq . upon AB = sq . upon BH + .sq . upon AH . 2 2 2 . :. h ·= a + b . ·

-· a1• One of my dis section device s . /

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192 THE PYTHP.GO.REAN PROPOSITION

.,_-�------------- ___ ___ _ ____ ___ ____ ___ _ _ _ ___ _ _ _ __ Q.n.� H u n Q.r.�fL2 i x !1. _ _ _ _ _

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"Fig . 258

Case (1 ) , (c ) .

In fig . 258 , draw' RA and produce it to Q, and draw CO ,. LM and KN each perp . - t o R� .

Sq . OK = (tri . GOA = tri . PDB ) + (tr>ap . CLMO + trap . PBHE') + (tri . NRK = tri . AQG ) + (quad . NKPA + tri . RML = t!ap . _ AHFQ ) = sq . HB + sq . CK .

:. s q . upon AH = sq- . upon BH + AH . h2 2 ' b2

_ sq . upon . . . = a -r • a . Devi sed , by autho� , -

to cover Case (1· ) , (c ) . o.

-------------------------------------,,----------P-rod·l:lc·e----H:A--,t-o·-P·-ma·k1:n-g--- ---� AP = Ha , draw P N par . to ' AB , and

/ \ through A draw . ON perp . to and �:_ 01 .- 'C- M1 = to AB , complete sq . OL, produce \ f I MO to ·a and draw HK perp . to AB .

\ j. i · I . Sq . OL = , (rect . ' AL :\• 1B· --=--pa;ra-l-:-P"D:B1\�-=- sq . HD ) + (rec t .

A · U �- I AM" = paral . ABCG :;= sq . HG:-1 = sq . llL_.iJL -��-1L. HB + sq,�_ HG .

:. sq . upon AB = sq . upon , HD + sq . upolf HA . :. ' h2 = a2 + b2 • Q . E . D .

Fig . 259 I I

� . See Ve� sluys , p . 27 , fig . 23 , as found in "Friend of Wi sdom, " 1887., as given by J . de Gelder, 1810 , in Geom . of Van Kunze , 1842 .

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_ GEOMETRIQ-f.cPROOFS 193 --------------·----------�---------------------

·" _ Case (1 ) , (d ) .

Draw HO perp . to AB and equal to HA , and KP 'par . to AB �nd equal to - HB ; draw ,CN par . to �AB , PL , EF, and extend ED to R and BD to Q . _

Sq . CK = (tri . LKP = trap . ESBH of sq . HD + tri . o ASE of· sq . .f HG ) :r (tri . HOB � . tri . SDB of si. HD

Fig . 26o + trap . AQ,DS of sq . HG ) +· (tri . CNH = tri . FHE of �q . �G ) + (tri . CLT

= ' tri . FER· of sq . ·_HG-) + sq . TO = sq . -DG of sq . HG = s q . HD· + sq . HG . 1

:. sq . upon AB = sq . upon BH + sq . upon AH . h2 2 2 n· :. = a + b . Q . E . · •

a . C onceived, by author , to cover case (1 )� (d ) .

. ·Fig . 261

Case (2 ) , (a ) .

In fig . 261 , with sq ' s placed as in the figure , draw HL perp .. t q CK, CO and· BN par . " to AH , making BN = BH, and draw KN�

. Sq . AK = · rect . BL + rect . AL = (paral . OKtH - s q . BD l + (paral . CORA = sq . AF ) = sq . BD + sq . H.G .

:. s q . upon .AB = sq . . 2 upon BH + s q . upon AH . :. h = a,2 + b2 .

a . Devised, �y autho� to cover 0�3e (2 ) , (a ) .

\-�--- -----�-- -------------- ---- -- --- - -- - - ---- -- --------------

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194. THE PYTHAGOREAN PROPOSITION

q��-tt[��t�[.[l�it�hQ�( ...__-�

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.,-·�- ---- -----

- ---

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f f . • � . r � r- ­l

�

Fig . 262

In fig . 26� , the sq . AK = / part s 1 + � + 3 +' 4 + 5 + 6 -+ 161• ,. Sq . HD = parts (12 � 5 ) + (13 = 4 ) of ·sq . AK. Sq . HG = parts (9. = 1) . + (10 = 2 ) + (11 = 6 ) + (�4 = 16) + (15 = 3 ) of -sq . AIL

:. sq . upon AB = sq . upon 2 2 b 2 HD -t: sq· . upon HA . :. h = a , + .

Q . E . D . a . Thi s di s section and

proof i s that of Richard A . Bell , devi sed by him July 13 , 1914, artd �iven �o m�· �eb � 28 . 1938 .

.

.. , __________ _ _ ____ �:Q_� s �--(.ZJ_, __ (b_)_.--=-.=.F.o.r_�whi.ch_a!!.e-------:_----------- ---------------.- - - ------�-�

more proofs extant than for any other of the se 19 cas� s �­Why? Because of the obv-ious di s section of the resulting

Fig . 263

+ trap . NKDF ) + AF . ·

figure s .

In fig . 263 , �xtend FG to C ., Sq . . AK = (pentagon AGMKB = quad . AGNB . conunon to sq ' s AK arid AF + tri . KNM common to sq ' s AK an� FK ) + (tri . ACG = tri . BNF

(tri . / CKM = tri . ABH ) = sq . FK + sq . ' . I

� / :. sq . upon �- = sq . upon BH + sq . upon AH .

: :. ·h 2 = -a 2 + b2 • . a . See- Hill !Is Geom . for Beg�nners ,. 1886 , p .

154 , pr9of I ; Be�an and Smitn ' s New Plane and Solid · · Geom. , 1899 , p .. J:04 , fig . 4 ; Versluys , p . 22 , fig . 20 , as given· �y SchlHmilc�, i849 ; al so F . C . Boon, proof ' 7 , p . 105 ; al so Dr . Leitzmann, p . 18, f!g . 20 ; a� so

•

I

\

i

--... .

GEOMETRIC PROOFS 195 ·.""" .

Joseph .Zelson, a 17 year-old boy in We �t Phila . , It, . ----- -�-�------- - -- -rrrgrl."�ooT�-T93_7_. -- --- - -----------�---------------

b . This figure is of spec ial intere st as ' the sq . · MD may occupy 15 other positions having c ommon ­vertex with sq . AK and its sides coincident with side or slde s pr'Oduc ed of sq . HG ; One such solution is that of fig . 256 .

.

In fig . '264 , extend FG to C . Sq . AK = quad . AGPB common t9 .sq 1 s AK and AF + (tri .. ACG = tri . ABH ) ,A . + (tri . CME = tri . BPF ) + (trap . \ I

• \ -p,�· EMKD common to sq 1 s AK and Ek ). . ··! \ Jl/ l f + (tri . KPD = tri . MLK ) = sq . DL I . �' I + sq ' .AF . . . ·E/ 9 �\ :. sq . upon AB = sq . upon BH C u � 2 2 - 1"\ + sq . upon AH . _ :. h = a + b •

,,

" '"'•

\�J,.""'.. a . .. See Edwards ' Geom . , 1895, ------ ------- -- ·-- ---- -- - - -· ----- �-�---�- �----p.---lel..,---f-1-g--. +35-)-;-Dr-T---be-1.-t.z-ma-nn-,---- ----------- --- ----_---------::��

• I

._.,,,

Fi;g . 264 p . 18, fig . 21 . 4th Edition .

In fig . 265 , extend FG t o C T and const . sq . HM =. sq . LD, the sq . -translated • .

� . Sq . AK = (tri . ACG = tri . I '\ " Ali.) ABH ) + (tri . coE · = tri . �PF ) + .(trap . I \. � / .... _�_1=" � EOKL comrilbn to both sq 1 s � ��d LD , 1 �� I or = trap . NQBH ) + (tri . KPL - tri . : £/ b- ' KOD -� tri . BQM ) + [ (tri . BQM + poly-

C �,- -�� gon AGPBMQ ) ·

= quad . AGPB common tp . "

\.. '� / sq ' s AK and _AFJ ·· = sq � LD -r sq . A� . \ :. sq . upon A:B = sq . upon/ BH

Fig,. 265 + sq . upon AH . :. h2 = a2 + ·b 2 • · a �See Sc i . Am . Sup . , V . 70 ,

P_!___359,· . pee . 3, 1910, ·by A . R . Colburn . b . � think it better to omit Colburn ' s sq . HM

(not nece s sary ) , and thus reduce it to proof above .

;..

(1

_._

- - - --- - - --------- -- ------ ------- --- -- - ----- ---�-- - - - -- - -- - ------------------�--

..

. "'

196 . THE PYTHAGOREAN PROPOSITION

Qo.�_!!Y.rr9.r.�Q._[i�!�:.�±9.h!. ________ __ _

- - -·-: ---- - ·---.

--�--'------------- -- --- l1' ----- - -------- ----- . ;_:� l .

I

-�----�0 � . I ' / I r-1 \ . 't I /;>( - I -y \

C 1..:; _ _ \L.� I< \ \/ " /1) �/ Fig . 266

In fig . 266 , extend ED_. to K anq draw KM par . to BH .

Sq . AK = quad . AGNB common to sq ' s AK and AF + (tri . ACG

·= tri . ABH ) + (tri . CKM = trap . CEDL + tri . BNF ) + (tri . KNM = tri . CLG ) = sq . GE + sq . AF .

:. sq . upon · AB = sq· . · upon BH -2 2 2 + sq . upon AH . :'· h = a + b _ .

a . See Edwards ' Geom . � 1895 , p . 156 , fig . . (8 ) .

" In · fig . 267 , e.xtend ED to C and draw KP par . to HB .

.,.....,. ' ' ' '

· - - .

,

·· '

A _ )3 Sq� _ AK � quad . _:.A·_-_G:;NB:=-·---=-C.::.o=m-- -------o-----:;�� ---------=-----· -----------·------ - ·- ---------- -- ,· --.. -·n1on�tcfsqts- AK and HG + tri . ACG f, / f.\ L ,v(� = tri . CAE · = t-rap . EDMA + tri .

,. ' '

!::-+-------- -- - - �- ---- - --- -- - -

' , 1' > . y / I BNF ) +� (tri . c'f{:p = tri . ABH ) . / \ \ (

. ) � .""' /� \ + tri . PKI'f = tri . LAM = s q . AD ' ·1) -' r ro/ Cl" I �4 _ _ �J( + sq . �F . C - · :. sq . upon AB = sq. upon

Fi 267 BH AH t..-a-- . :i a : - b\ 2 g . .+ s q . upon . : . .lJ. ' := at + •

- a ; See Am . Math . Mo . 1 V . VI , 1899� P . . 33 , proof LXXXVI �

/

qo_�-tl�o.Q.c��-��Y�rr!Y

In fig . 268, extend ED to C , DN to B , and draw . EO par . t-o AB , KL perp . to DB and HM perp . to EO .

Sq . AK = rect . AO + rect . Cu = para1 . AELB + para1 . ECKL :::: sq_. AD + sq . 'AF .

:. sq . upon AB = sq ; upon 't . . - • 2 2 ·- 2 BH = sq . upon AH . . . h = a + b .

I I �

-- - -- -- -- ------ ------ - ------- ------- ---------------- ---------------------H·· I <3 ·�

��O�TRIC PROOFS 197

a . See Am. Math . Mo . , Vol . VI , 1899, p . 33, LXXXVIII .

/ tvf/ Y !J' .t ' / , I ' I '\ /

In fig . 269 , extend HF to L and complete the sq . HE .

' , I. � I ��/ / \ I ll/ c-� - - - .,}''

Sq� AK = sq . HE - 4 tri . ABH � sq . CD + sq . HG + (2 rect . GL = 4 tri . ACG ) - 4 tri . ABH = sq . CD + sq . HC .

:. s q . upon AB = sq . '

, / � upon BH 1 + sq . upon AH . :. h2 = a2 + b2 . • �

_ a . _ Th+s is one of the conjectured proofs of Pytha­goras ; s ee Ball ' s Short Hist .

of Mat� . , 1888 , p . 24 ; Hopkins ' Plane Geom . , 1891 , p . 91 , fi-g . IV ; Edwards ' ·Geom . , 1895 , p . 162 , fig .

Fig . 2(?9

(39 ) ; Benian aPd Smith ' s New_ P lanb . Geoni ._ , 1899 , ·P · 103, fig . 2 ; Hea�h ' s MAth . Monographs ; No . 1 , 1900 , p . 18 , proof II .

Ffg . 270

CKM, HMF or BAL;

In fig . 270 , extend FG to C , draw HN perp . tb OK and KM pa� to _ _ !m_._---". _____ __ . _ _ _ -----: -------� ------ --- -·-· -

Sq . . AK = rect . BN + rect . AN· = paral . BHMK + paral . HACM = sq . AD + sq . AF .

:; s q . upon AB, = Bq . upon BH + sq .1 upon AH . :. h2 = a� + b2 • .

a . See Am . Math . Mo . , V . VI , 1899, p . 33, pro9f LXXXVII .

b . In this figure · the giv­en triangle may be either ACG,

taking either bf the se four triangle s

f

J I

-.

- -- - - - - - - - -- ----- -- -----� ---�-� '

' ,.

: \ .i I

---- -- � � - - -- �-- - - - - - - - -- - ---- � -- -

... '1, .. · ·._

. . .

198 THE PYTHAGOREAN PROPOSITION

several proofs �or each is possible . Again, by in­spection, we observe that the given triang�e may have any __ one of seven other positions within the square AGFH, right angles coinciding . Furthermore the square upon the hypotenuse may be constructed over­lapping; and for each different suppositiqn as t o the figure there will re sult several proofs unlike any, a s t o dissection, g_iven heretofore . - "

c . The simplic�ty and applicability of fig-ure-s under ·Case )2 ) , (b ) make s. it wo;rthy of n·o te .

) '�

In fig . 271 , sq . AK = sec ­tions [5 + (6 = 3 ) + (7 = 4 ) ] + [ (8 = 1 ) + (9 = � ) ] = sq . HG + sq . AE .

:. sq . upon AB = sq . upon BH + sg_ . upon HA . :. h2 = a2 + b2 • Q . E . D .

a . Devised by Rich�rd Bell, Cleveland, o : , 1 on Ju:J.y 4 , 1914 , _ one of his 40- proof,s-�

Case (2_) , (c ) . I . .

� - --- ·-·E- � - --\--- -- - -:--- ·--" - - -- - ----±-n---f-1-g-.:-2T-2·,---ED-be-ing-the-_�- - --�----Gf �L- _ \ sq . translated, the constr'\lction' · ' ' · c " \ /7J is evident . J{ " 1 '< 1 Sq . �- = quad . AHLC ' common

\ J "1> ' to sq_ 1 � AK and AF + (tri . ABC , = tri . ACG ) + (tri . BKD = trap . ' I LKEF + tri . CLF ) + tri . KLP common � to sq 1 s ' �K and ED = sq . ED + sq .

; Fig . 272 AF •

:. s q . upon AB = sq . upon BH + sq . upon AIL :. h2 .= a2 + b2 •

�

..

'I

· GEOMETRIC �ROOFS 199

l;l. . See Jury Wipper , 1880 , p . 22, fig . 17 ; _,as given by von Hauff, in "Lehrbegriff der reinen Ma:the­matik, " 1B03 ; Heath ' s Math . Monographs , 1900 , No . 2 , proof XX ; Versluys, p . 29, fig •. 27 ; Fourrey, p .- · 85- ­A . Marre , from· San'scr.it , "Yoncti Bacha" ; Dr . L·eitz­mann, p . 17 , fig . 19, 4th edition . .

Fig . 273

H�ving completed the three square s I Ak, HE and HG, draw, tlu'6Ugh H, LM perp . to ABC ·and join �C , AN and �E . .

Sq . AK = [ rect . LB = 2 (tri . �P- = tri � AEM ) = sq . HD} + [ rect . LA = 2 (tri . HCA = tri:. .

''ACH ) = sq . HG ] = sq . HD + - sq . , -HG.

:. sq . upon AB = sq . upon . HB HA. - 2 2 - 2 . + sq . upon • :. h = a _ + b •

a . See Math . Mo . (1859 ) , Vol . II , No . 2 , Dem. 14 , ' fig . 6 .

In fig . 274 , since part s 2 ·+ 3 = �q . on BH .= sq . DE� i t ;t s readily 'seen tha� the sq . upon AB = sq . upon BH + sq . upon AH . :. h2 ;, a2 + b2 .

a . ·· Devised by . Richard A .

J

Bell, _ Ju:I.:y __ :!:_'(,__1_2.18,_._'Qei11,g_ on� of __ ____ __ _________ _ ___ _ _ _ _ _ -,

- _ ___ __ _

-�his 4<?) proofs . .. He 'subni:t tted a

Fig . 274

. second dis section proof of same f�gure , also his .3 proofs of Dec . 1 and 2 , 1920 are similar to the above , as to ·figure ,.

I

0

I ,,

\

(f •

" -

2 00 THE PYTHAGOREAN PROPOSITION

C j.._ _ _ _ u

, -

\ . ' '

/ �"""

I '\ /

I My

Fig . 275

Case (2 ) , (d) .

In fig . 275, extend KB t o P , CA to R , BH to L , draw ·KM perp . to BL, tak� MN = HB , and draw NO par . to AH .

Sq . AK = tri . ABH c on:unon to sq-' · s AK and AF + .(tri . BON =·

.tri .. BPF ) + (trap . NOKM = trap . DRAE ) + ( tri . K;(k" ,; tri . ARQ ) + (quad . AHLC = quad:. AGPB ) = sq . AD + sq . AF .

:. sq . upon AB = s q . ".lpon BH + sq . upon AH . :. h2 = a2 + b2• . a . Se!3 Am . Math . Mo . , V .

VI, 1899, p. j4, proof XC .

In fig . 276 , upon CK const . tri . CKP ·= tri . ABH, draw CN pa� . to B�, KM par . to AH, draw ML and through H draw PO . .

S�. AK = rect . KO + r�ct . CO = (paral . PB = paral. CL � sq . AD ) + (paral . � PA = sq . AF ) = sq .

., t I An + sq . AF . , - - - - - -- - -- ·--�-�

A:-- - --�--- -- - - -------- ---- - -- ---:·�---s_cr

·.- --upoir·AB-�;- -sq-:� upon�-/ \ I ).. BH + sq . upon AlL : . . h ·= a + b •

£- . \ 1 / t: . a . Original wi\th the . \ >1L ., / author , Jul� ea·, 1900 . \.

l

\ / ' & ' ''Jl. ., _

b . An algebr'aic proof · · comes readily from th�s figur� .

Fig . 276

' 0

. GEOMETRIC PRQOFS . . , . . . ... . ' . - - ..... ""

Q.n.�_ttur:ulr.�!J. _§.�.y�n.!t.:..f!:l:.nt:- .. =·· · �; " '· '.�·;

-:' � . . ' .. -· .. , ...... ·�:- �- (::: . .

JVI/'f .... .I \ I " . ·· ; . � . / \

I' , - - - L)

, . \ -n. .... �, \ � J · \ / "\ \ I /0. \ I \ I / \ I \ rJ's -· '"\G_ .- � J (:.!:..:- _ ·-..:. _ _ Jk . F Fig . 277

= a� · + o2 •

· Case (3 )', (a') :

In j1� . 277 , · pr�duce DB to N, HB to' · T , KB to � , and draw PN, AO , KP an� RQ perp . to NB .

Sq . AI( = '(quad . CKPS -+ trf . BRQ =:: tr:a.p . BTFL ) + (tri . KBP ' = tri : TBG ) + (trap . OQRA·· = .trap . MBDE ) + (-tri . ASO _-:::: · tri . J3MH ) = sq . HD + sq.:· .PL !· .

_ �:. sq . upon AB = - �q . -upon BH -+ sq : upon AH . :. h2-

a . Devised for mi s sin� Case (3 ) , (a ) , 'March 17 , 1926 .

Q.nt���n�r.��-£Lg�!t.

� \ / / S� ..lL _ _ \D:_ _ Jtvl � I �!=' I ' I , ' I \ ,/ \

Cas·e (3 ) , . (b ) . . � l. "'

In ;fi&! 278, extend ED to K and through D draw . GM �ar . to AB·. . .

. s·q . AK := rect . AM + rect . . Q� = (para1 . GB = sq . HD ) + (paral . CD· = sq . . GF ) � sq. �HD + sq . GF .

:. sq . upon AB = sq . upoJ1

. ....

BH + sq . upon AH . . : .. h2 = a2 + b2• t-----------------· -�...:�----·F·igo-278----��--- - -- - - -a-;---Se e·-Am·.:·-Ma-�h-;--Mo·;-;---- --- ---:·-- - -� , . Vol . VI , 1899 , p . 33 , proof ' LXXXV .

�- - - �K

� ·

. b . This fig1,1r� furni shes an algebraic proof • . ; c . If any of the triangles congr�ent to tri .

ABH is ta'ken as the given triang+e.,� a figure eX!)ress­i'ng a diffe�ent relation t>f ,th� ·sq1,1are s is . obtained, hence covering some other case of· · the 19 pos sible case s .

.... ..

f 202 THE· PYTHAGOREAN PROPOSITION

' . Extend HA to G making

AG � HB� HB to M making. BM = .HA, c�mpiete the sq�re ' s

� Hp, EC , AK and HL . Number the

e.,../ I \ _.,{ / I \ dis sected parts , omitting the � tri ' s ' CLK and KMB . \ . 7 I If �'F" t I .\ ( \ I / , \ I )"'· Sq . AK = 1 + 4 + 5 \ 1 / 6 .. \ I ,..fj + 6 ) =:= · part s (1 c ommbrl t o sq ' s � - , - - ��K HD and AK ) + (4 connnon to sq 1 s

·

\ ,. EC and _ _ AK ) + (5 = 2 of sq. HD '

\lr-:' "' + 3 of _sq . EC ) + �6 = 7 of sq .

Fig . 279

··· Ec· ) = part_s .(1 + 2 ), + part s (3 + ·4 + 7 ) = sq . HD + sq . . EC •

:. sq . upon AB = sq . 2 2 . 2 . upon BH + sq . upon AH • . :. h- = a + b • . Q .E . D . · a . See "Geometric Exercise s in Paper Folding"

by T . Sundra Row, edited by Beman and Smith (1905 ) , ' p . ' 14 .

· In fig . 280, extend EF _ to K, _ and HL perp . to CK.

Sq . AK = �ect . BL + rect . AL = paral . BF + paral . AF = sq .

7 \ ,. 1 HD + sq . GF . � ... / I ./ AB

' l · .

y ' ,(. 1 :. sq. upon _ = sq . upon \ L -· ____ ":l>.).f _____ L_�:_!?_!! _�- �ct· ��E-��-�-�-----·:�--!12 = a 2 + _!)_:! ___ _ _ _ -�-�---�---�--- ·- �- --- --�-\;·- , - / I \ 1 . a . See Am . Math . Mo . , V .

.·

\ I ,. "' I l .\l i< VI , 1899, P . 33 , proof LXxxiV . c·\.4 - - .1.! - -·

Fig . 28o

In fig . 281 , ex.tend EF t o: K. Sq . AK = quad . ACFL common to sq ' � AK and GF

' • - - �

/ v , \ -:b )i= I I , ' I ' I / . t< � : - - -\�!K

Fig . 281

GEO�lliTRIC PROOFS 203

+ (tri . CKF = trap . LBHE + tri . ALE ) +

_ (tri . KBD = · tr:t . CAG ) ,

+ tri . BDL common to sq ' s � and HD = sq . HD + sq. AK .

:. sq . upon AB = sq . upon BH + AH. . h2 2 +" 'b2. sq . upon . . . = a •

a . See Olney ' s Geom � , _Part TII , 1872, p . 250, 2nd meth-od; Jury' Wipper, 1880, p . 23, fig. 18; proof by E . Forbe s , Winche s ­ter, N. H . , a s given in Jour . of

Ed ' n, V . XXVIII, 1888, p . 17, �5th proof ; Jour . of . Ed�nT V . XXV_, 18�7, p . 404 , fig . II ; Hopkins ' Plane

Geom . , 1891 , p . 91 , fig . III ;o Edwards 1 Geom . , ];895-, J> . 155 , 'fig . (5 ) ; Math . Mo . , V . VI:, 1899, p . 33 , proof LXXXIII ; Heath ' s Math . Monographs , No . 1 , 1900, p . 21 , proof V; Geometric Exercises in ·Paper Folding, by T . Sundra Row, fig . 13 , p . 14 of 2nd Edi��on of The Open Co4rt · -Pub . Co . , 19d5 . Every teaqher of

, .. geo�E?try s�ould use· · this paper 'folding prqo.f . ·· · " Als.o see Versluys , p . 29, fig . · 26 ,· 3rd_ para-

graph, .Clairaut, 1741 , an"d· found in "Yoncti Bacha" ; also Mat� . Mo . , 1858, Vol . I , p . 160 , Dem. 10 , and p . ' 46 , Vol • .:I., where . credited to Rev . A . ri . Wheeler .

b . By dis section ·an easy proof re sults . Also by algebra, as (in fig . 281 ) CKBHG = a� + b2 + ab ; whence readily h2 . = a2 + b2·. .

c . Fig . , ?80 is fig . 28+ �ith the extra line HL ; fig . 281 gives a proof 'by congruency, wh�le fig .

· 280 give s a . proof by equivalency, -and it.:-also · �ives a p�oof, by algebra, by the uae of the mean propor-

-- -----ti-ona-1-.�------��-- -- ------------�- -- ------ ------ ------ -- - --- -- --- -- ·--· d . Versluys ,

Macay ; Van Schoote1) , er ; and Clairaut .

} ' � � p . . . 20 , c onnect s this- - proof with !657 ; J. C . Sturm, 1689; Dobrin-

In fig . 2ff2, from the dissection it is obvi­ous kthat the sq . -upon AB = sq . upon BH + sq . u.t>on All.

. ,

. - .-

· ·'

\ 204

Fig . 282

THE PYTHAGOREAN PROPOSITION , _ _

:. AB2 = BH2 + HA2 , or h2 = a2- + b2 •

a . Devised by R . -A � Bell , Clevelanq.,: .0 .-· , on No.v . -30 , 1920, and given to the autnor Feb . 28, :J-938 .

Case (3 ) , ( c ) .

In fig . 283 , draw ·KL perp . to CG and extend BH to M .

Sq . AK = (tri . ABH = tri . / /'f, CKF ) + tri . BNH conunon to sq I s AK

4, / ; I ' U and IID + (quad . CGNK = sq •. IJI f\ - - - - � t l>J£ + trap .. MHNK + tri . KCL c ommon to I · '\ T t.'l,.. � J.'"·' sq 1 s AK and FG-) + tri \ CAG · = trap . \,.. � / /� ' BDEN + tri . KNE ) ·= sq . HD' + sq. ro . 1 1 L ,.. I Ii \ · 1 1 :. sq .. �pon AB = sq . upon

A ' · / "1> J3H + sq . upon AH . :. h2 = · a2 + b2 • Q . E .)) .

Fig . 283 a . See Sci . Am . Sup . ,. Vol . 70 , · � . 383 , Dec . 10 , 1910 , in .

wh+ch proof .A . R . Colburn makes 1T· the giv.en tri . , and then substitutes part .2 for part 1 , . part 3 for part s 4 and 5 , tnus show.ing sq . AK = ·sq . H1;> + sq . FG; also ' see ·Ver.sluys , p . 31, fig . 28, Geom . , of ·M . Sauvens , 1753 .t (1716 ) . /

Fig . 284

' In fig .. 284 , the construc -

ti9n is e:vident , FG · l?eing the translated o- square .

Sq . AK � quad . GLKC. common to sq ' s AK �nd CE + '<l'(tri . CAG .· = trap . BDEL + · tri . KLE ) + (tri . ABH = tri . CKF ) + tri .. BLH conunon to · sq ' s AK and HD = sq . HD + s·q . CE .

0

.. . '

GEOMETRIC PROOFS 205 .

· sq . upon AB = sq . upon BH + " sq . upon AH . � h2 = a2 + b2 . .

a . See Halsted ' s Element s of Geom . , 1895 , p . 78 , theorem XXXVII ; Edwards ' Geom. , l895 , p . 156 , fig . · (6 ) ; Heath-' s -Ma-th . Monographs , No . 1 , 1900 , p . 27 , proof XIII .

Fig .

· In fig . 22§ it i s ob­vious that the parts in the sq . HD and HF are the same · in number and �ongruent �? the parts in th� �quare AK-.

:. the sq . upon AB = sq . upon BH + sq . upon AH , or h2 a= a2 + b.2 .

tL One of R . A . Bell's proofs , of Dec . 3 , 1920 and rec�ived Feb . 28, 1938 .

Case (3 ) , (d ) .

In . fig . 286 , produce AH to 0 , draw CN par . to HB , and ex­tend CA . to G �

Sq . AK = trap . EMBH com­mon to sq ' s AK and HD + (tri . BOH = . tri. BMD ) + . (quad . NOKC. . = ,quad . FMAG) + (tri . CAN = tri . GAL )· + tri . AME common , to sq ' s ' AK and EG = sq . HD + sq . LF •

:. sq . upon· AB = sq . upon BH + sq . upon AH . :. h2 = a2 + b 2.

a . See Am . • Math . Mo . , · Vol . VI , 1899 ,. p . 34··, proof

b . As the relative position of · the given t�i­angle and the trans�ated square may be indefinitely

t i '

( ·�

•

..

·-' , � . ..

\ 1

' '

206 THE PYTHAGOREAN PROPOSITION

varied, s_o the number of proofs must be indefinitely gre�tJ of whic� the f�llowing two are examples .

Fig : 287

In fig . 287 , produce BH to Q, HA to L and ED to F , and draw KN perp . to QB- and connect A and G . �

Sq . AK = tri . AP� common · to sq 1 s AK and EG + trap . PBHE

common to sq ' s HD and � + (tri . ·BKN = tri . GAL ) + ( tri . .NKQ = tri . DBP ) + (quad . AHQC = quad . GFPA ) = sq . HD + sq . -HA .

:. sq . upon· AB = sq • . upon · : · 2 2 2 HD + sq . upon HA . :: . h = a. + b .

a � . Thi s fig . and proof . due to R . A . Bell of Cleveland, .o. He gave it to the author· Feb . 27, • 1938 . . .,

..... r,l In fig . 288, draw LM "' 1 ' through H . c., "" , ' r- - - '71 _ ..)1 1( Sq . AK -= rec t . KM + rect .

t 1 1 CM = paral . KH + pa�al. CH = sq .

: ��t:l 1 HD + (sq . _ on _ _ AI{ = sq . NF ) •

· n :. sq . - upon AB = sq . upon I oH + .

:AH.

h2 2 + b2. ¥ sq . upon · • :. = a )}, , fB . a . Original with the

�'�"' , . , � "' author , July 28, 1900 . � � "" · b . -An algebraic solution -, . , .., )F ' may be devi sed from thi s figure . \ / . /

"'-·"' / \

F�g . •288

/ ' j

I .

..... - .. •' \ f: . I

I ' I ' I '

J

GEOMETRIC PROOFS 207

,;> ��t-����(t�-�i�til:���

Fig . 289

c·ase (4 ) , (a ) .

In .fig . 289, extend • KH t,a T ma:ki:gg NT = AH , draw . TC , draw FR·, MN and PO perp . · to KH, and draw HS . par . to

AB . Sq . CK = (quad .. CMNH

+ tri � KPO = qua�. SUFq ) + tri . MKN = tri . 'HSA ) + (trap . FROP = trap . EDLB ) t (tri . FHR = tri . -ECB ) = sq . CD + sq. GH . _ - . _

:. sq . upon AB = sq . upon BH + s·q . upon AH. :. h2 = a2 + b2 • _

a . Devi sed by author for case (4 ) , (a ) March 18, 1926-.

Case (4 J, . (b ) .

In fig . 290 , draw GP par . to AB , take LS .= AH,

Sf\ • '>-F · draw KS , draw Lb , CN and � 1 ' , � perp . to KS , and . draw BR . �- ... "o� _ '�::_. _ _ _ J=t _

Sq . AK = (trL . CNK

- '-, - �t � :-�-��-:1:-�·;t �-:� �!i-�- :�-l · : -�!�; �

K:OM '�L ' � = trap . PGED ) + (tri . SOL I( 'V � =

_tri • GPR ) + . (quad . CNSA

= quad . AGRB ) = sq . GD + · sq . Fig . 290 AF .

. :. sq . · upon AB = sq . 2 • 2 2 upon BH + sq . upon AH . :._ , h = a + b • D

a . Devised by author f� Case (4 ) , (b ) .

' .

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I . .

I \ 208 THE PYTHAGOREAN' PROPOSITION

l I · t ' I 1 I 1 \ 1 ,-. GL ,.!N_ - .JF_,O

Fig . 291

Case (S ) , (a ) .

' In fig . 291 , CE and AF are the translated sq ' s ; pro­duce GF to 0 and complete the sq . Mb .; produce HE to S and complete the sq . ·US ; produce OB to Q, . draw MF, · draw WH, draw ST and UV perp . tp WH , and take TX = HB and draw XY perp . t o WH·. Since -sq � MO

. = sq . AF, and sq . US · = sq . CE , and since sq . RW ·� (quad . URHV + tri . WYX = trap . MFOB + (tri. HST = tri . BQH ) + (trap . - TSYX = trap . BDEQ ) + tri . uvw. = t:ri . MFN ) = sq.� . HD + (sq •

.. NB

= sq . AF ) . !. sq . RW = sq . upon AB = sq . upon BH + sq .

upon Jill . :. 112 = a.2 + b2 • a . De vi sed Mar0h 18, 1926 , for Case (5 ) , (a ) ,,

by: author .

· Extend HA to G mak­ing AG = HB ; extend HB t o D -mak:tn.g . . BD-= --HA .• - . . Complate . - ­

s q 1 s PD - a.nd PG-:- Dra.w· HQ. :pe1•p . to CK and -through P dl�aw LM and TU part . to AB �

· PR = QO = BW. "

The t:rB:_ns'lated sq 1 s are PD = BE t and PG =- RG 1 •.

. Sq . AK = parts (1 . + 2 + 3 + ; 4 + 5 + 6 + 7 + 8 ) = parts (3 + 4 + 5 + 6 -· sq . . PD ) + perta · (l + 2 + 7 + 8 ) = s.q . PG .

. •

0

I)

GEOMETRIC PROOFS-

:. sq . upon AB = sq . upon HB + -sq . upon HA . � h2 = a2 + b2 • Q . E . D •

a . See Versluis , p . 35 , fig . 34 . \ ' I I • I

· 1/\K Case ��5 ) , (b -) . : ,.If ', In fig . 293 , draw GL

- "- � , ,:..._�N� � \ thr_ough B , and draw PQ� CO and -�·'·J · • ' l - �- · � MN perp . to BL . , or- --- -· If � , . Sq . BK = (tri . CB0 = tri �

Q' I ,/ BGD ) + (quad . OC!a.J + tri . J3PO � �1f = trap . GFRB ) + (tri . MLN = tri .

\ J / f \ . BSD ) + (trap . PQNM = trap . SERB )

'v"" f \ = sq. HD + sq . DF . """ '\ I .1 '- :. sq . upon AB - = sq . upon .v L . ' · 2 2 2 \ f . /, F BH + sq . upon AH . :. h = a + b •

G�"'/� ... - --- --- - a . "Dev.i sed for Cas� (5 ) , (b ) , by the author , Ma�eh 28,

Fi�-. 293 1926 .

:-

Fig . 294

Case (6 ) , (a ) .

In fig . 294 , extend LE and FG to M thus completing the sq . HM, and draw DM :

Sq . AK + 4 tri . ABC = s,q . HM = sq . 'LD + sq . DF + (2 rect . HD = 4 tri . ABC ) , from which sq . AK = s q . LD + sq . DF . "

.�. sq . upon AB = sq . upon BH + sq . upon AH . :. h2 = cS. 2 + b2 .

a . This proof i s cred� ited to M . Mcintosh of Whitwater , Wi s . See Jour . of Ed ' n, ' 1888, v·ol . XXvii, p . 327 , seventeenth proof :

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{, ..

210 THE PYTHAGOREAN PROPOSITIO� �--�--------------------�- =- �-- --=--=·-=- =- ============-------

Fig . 295

Sq . AK = sq·. HM - (4. tri . ABH = 2 rect . HL = sq . EL + sq . �� + 2 r�ct . HL - 2 rect . HL = sq . EL + sq . LF .

:. s q . upon AB = sq . upon HB + sq . upon HA . :. h2' = a2 + b2 .

a . See J.ournal. of Edu­cation, 1887·, / Vol . XXVI , p . 21 , fig . XII ; Iowa Grand Lodge Bul­letin, F and A . M . , Vol . 30 , No . 2 , P � 44, fig . 2 , of Feb . - 1929 . Also pr . Leitzma��, p . 20 ," fig . 24 , 4th Ed ' n . ·

b .' An algebraic proof' i s h:2 = (a + b )2 - 2ab = a2 + b2 ·. ;

_ In fig . 296 , the .

translation is evident� ..... Take CM = KD . Draw AM; then draw GR , CN and BO

f_ .... _ .:\L / par . t o AH an� DU par . to -

. Qr \ , I :Y. £ BH . Take NP = BH a:nd I . /""Jt \QI T�\l draw PQ. par . to AH .

� / �� \ 1 · Sq . AK = (�ri . �... �� \,� ! t, � J CMN = tl1i-._ J)EU ) + (trap .. - - �- -- - �i. .... U.�--� CNPQ = trap . TKDU )

Fig . 296 + (quad . OMRB + tri . AQP ) = trap . FGRQ ) + tri . AOB

= tri . GCR ) = sq . EK + sq . FC . :. sq . upon AB = sq . upon HB + . sq . upon HA . -

2 2 v 2 .

:. h = a + b _. Q .� . D . a . DeyJs�d b_L.t_he autp.or , March 28, 1926 ..

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----- -·GE0METRIC-PR00Fs-----�

Fig . 297

Fig . 298

In fig . 297 , the trans ­lation· and construction is ev­ident .

Sq . AK = (tri . CRP = tri . BVE ) + (trap . ANST = trap . BMDV ) + (quad . NRKB + tri.. TSB = . trap . AFGC ) + tri. ACP c ommon to sq_. AK and AG = sq . ME + sq . � .

·

:. sq . - -�pon .AB = sq . upon_ BH · + sq . upon AH . :. h2 .;, a2 + b2 .

a . Devised �by author , �arch 26 , 1926 , 1 0 � . m . \ I

In fig . 298 , the sq . on AH i s translated t o po si­tion of GO , and the sq . on HB to position of . qD . _C omplete the figure and conceive the sum of the two sq ' s EL and GO as the two rect ' s EM + TC + sq . LN and the dis sec tion as numbered .

Sq . AK = (tri . ACP = tri . DTM ) + (tri . CKQ = tri . TDE ) + (.tri . KBR = tri . CTO ) + (tri . BAS

= tri . TCN f + (sq : SQ = sq . LN ).,' = sq . � EL + sq . GO .-­:. sq . upon AB = sq . upon BH + sq . up·on AH . .

:. h2 = a2 + b2 . : · a . Devi sed by authoi, ·�a�ch 22 , �926 . ,. .. · � b . As sq . EL, having a ve�tsx antl· a 1ide in

common with a vertex and .a side of sq . GO , either ex­ternally (as in fig . 298 ) , or internally, may have 12 different positions , and as sq . GO may have a vertex ·

: .

. . .

212 THE PYTHAGOREAN PROPOSITION

and a si�e in common with the fixed sq . AK, or in c onunon wi.th th!3 given triangle ABH, giving 15 differ­ent positions , there ls p·o ssible 1.80 - 3 = 177 dif­ferent figure s ! hehce 176 proofs �ther than the one given above , using the di s section as used here , and 178 more proofs by using the di ssection as given in proof Ten, fig .. lll .

c . Thi s proof is a varia�ion of. that given in proof Eleven� fig . 112 .

· In fig� 299 , the ' construction is e�ident , as FO is the translation of the sq . on AH , and KE is the· translation of the· •

·, \ W /. f . sq . on BH . '

f<<.. V Since re·c t . CN f · \ .> '· � = rect . QE , we hav-e sq . . I /1.l \.f u i AK = (tri . LKV ::.: tri . CPL) f- ..,...J;.t.�.:-:- _ .....;. �� - + (tri . KBW � . tri . ,LFC )

I -,/ t · I ', · 1 · +- (tri . BAT = tri . KQR )

. � / j ' \\ j + (tri . ALU' = tri . · RSK ) � - iJ'b �._L..:_·\-� + (sq . · TV = sq . MO ) . R -- - .&.e�- "'- � � = rect . KR + rect . FP

Fig . 299 + sq . MO = sq.' KE + sq . FO .

:. sq . upon AB =· sq . ' upon BH + sq . upon All. :. h2 = a 2 + b2 .

a . Devi sed by the author , March 27 , 1926 . . ,

In fig . 300 the translati�n and construction aT$ easily Seen . 1

Sq . AK = (tri . CKN = tri ;· LFG ) + (trap . OTUM = · tn,ap . RESA ) +; (tri . VOB ::; tri . RAD + (quad . ACNV

· · + tri . TKU = q��d . MKFL ) = sq . DS + sq . MF •

\ ·

. •'

.... ....

\

I t . � I'

. (

" GEOMETR�C PROOFS 213

•

. :. sq . upon AB = sq . upon HB + sq . upon H:A . :. h2

2 b2 = a , + .

Fig . 300

a . Devised by the author , Mar�h 27 , 1926 , 10 : 40 p . m . ·

· ·

AR = AH and AD = BH . C om�

f- F- plete sq ' s on AR and AD . Extend DE ---�-. to S and draw SA and TR. \ T ' i- / . s • f' /'T! ' · Sq . AK = (tri . Q,PB = · tri .

' )4 f VDR of sq . AF') + (trap . AI.PQ = t.rap . I '· · E ) ( . I � , ] · TAU of sq • . AE + tri . CMA = tri . · A· . SGA o._,'f sq . AF ) + (tri . CNM = tri . f \ l D...- 1 UAD of sq . AE ) + (trap . NKOL = trap . ' . l_v-�� k I . VRFS of sq . AF') +'Jtri . OKB = �ri . I 3 ')M , L.-:. DSA ' of sq . AF ) =. 'fpart s 2 + 4 = sq . ' . / \ . �-\ 4

· �.J / 4 '\. u' ' I< AE ) + (parts � + 3 + .5 + 6 = s q • .AF). O L -- --. ��- AB HB :. sq . upon = sq • . upon Fig . 301 + sq . upon HA . 1 :. h2 = a2 + b2 •

Q . E . D . a . B�vi sed, by author , Nov . 16 , 1933�

I·n fig . 302 ; c omplete the sq . on EH, draw BD P,ar . to

�- , / AH, and draw AL and KF perp .

�./ - . ', K 0 ' h t 0 DB . . . ' I >L , t ' , · Sq·. AK =

·

sq . HG - . (4 �vr/ '"" It (·� - - tri . ABH = 2 re.ct . HL ) = sq . � 'L ' EL + sq . D� + 2 rect . FM . - 2 r -- - -· -�

v ' / rect . ' HL = sq . EL +. sq . DK . ' / \ , t.'. sq . upon AB =t. sq . GV .upon HB + sq . upon HA . :. h2 . Fig . 302 2 2

"'' = a + .b • ,, .

. .

•

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" .

214

(19·) . a . • ' , . -

THE PYTHAGOREAN PROPOSITION

See Edwards 1 Geom. , 1895., p . 158, fig .

b . By changing position of sq . FG, many oth�r �

proofs might be obtained . c . This i,s a varia t_ion of proof� fig . -240 • .

I�2-����t!��El�! I

' -· Fig . 303·

In fig . 303, let W and X be sq 1 s wi.th sides equal re sp ' y to. _ _ AH and BH . Place them as in figure, A bei�g center of sq . W, �nd Q, middle of AB as cen­ter of FS . ST =--:E:H, . TF = AH . . Side s of s_q 1· s· FV and QS are perp . to side s- Aff , and BH .

It i s obvious that : Sq . _:AK = (parts '1 + 2 + 3 + 4 =:= sq . FY ) + sq .

QS = sq . X + sq . W . . --:. sq . upon AB = - sq . upon HB + sq. upon HA • .

·'· h2 = a2 + b2 . a . See Me s senger of Mat_h . , Vol . 2 , p . �03 ,

1873 , a'nd· there ·credited to Henry Perigal, F . R . S ;A . S .

Fig . 304

Case (6 )_, (b ) .

In ·f-ig . 304 , the construction is evident . Sq . AK = · (tri ; . ABH = trap . KEMN + tri . KOF) + (tri . BOH = tr� . KLN ). + quad . GO�C common . to ­sq 1 s AK and OF . + \t�i . CAG = tri . OKE )

= sq . MK + �sq . OF .

- . --------�-------i

. ' r

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• 'f!o

GEOMETRIC PROOFS I

215

:. sq . upon .AB = s_q . upon BH + sq . upon AH . = a2 + b2 • Q . E . D : \

a . See Hopkins ' Plane Geom. , . 1891 , p . 92 , fig fig . VIII . ' '

b . By drawing a line EH , ' a' proof through par- � allelog�am, may be obtained. Al so an algebraic proof .

c . Also any one of the other thrt?e· triangle s ; a s CAG may be �alled the given t�iangle , from which other proofs would foiloi� Furthermore since the trL ABH may have sev·en other positions leaving side of sq . AK. as hypotenuse ,. and the sq . � rna¥ have 12 po-.. sitions having a side and a vertex in common with sq. CF, we would have . ,8� proof s , some - -of -whfch have been

• <-

or will - be given; etc . , etc : , as to sq . CF, one of which 1 s the next .proof .

ln fig . 305 ,, through H .;o1\,!- draw LM,- and draw CN par . to BH / . ' Q / t \ · . and KO par_. to AIL �--r�_:,.�� Sq . ·AK- =: ' rect . KM + r7ct . I \ Ol./ f CM = paral. KH + paral . CH . = � t�" 'f x KO +· AH x CN = sq . on BH + s·q . : . • t on· AH = sq . MD + sq . MG . -eLL:lt : :. sq . upon AB = sq . upon

r •M :':1 £ BH + sq . upon AH . :. h� = ·a2 + b2 • 1t - ' t a . Original . with the ' I . t 6 1 _ t -1'1\ _ . author January 31 , 192 , 3 p . m . �s Q ...- --- _ .... .., .L.. - - �f ·

· Fig . 305

Case (7 ) , (a ) .

In fig . 3 6 , extend AB to X, draw WU and �S each = to AH and ar . t o AB, CV and HT p_erp . to AB, GR an� FP par . to AB, and LW and � p��p , to AB .

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216 THE PYTHAGOREAN PROPOSITlON

Fig . 306

Sq . WK = (tri . CKS = tri . FPL = trap . BYDX of sq . BD + tri . �ON ot sq . GF ) + (tri . TKH = tri . GRA = tri . BEX of sq . BTh + trap . WQRA of sq . • GF ) +' (tri . WUH = tri . LWG of . s q . GF ) + {tri . WCV = tri . WLN of sq:. GF) + (sq ;. VT = paral . RO of s q . GF ) = sq . BD + sq . GF .

:. sq . · upon AB = sq. upon HB + sq . upon HA .

� �2 = a2 + b2 • Q . E . D . � a . Original with the_ author , Aug . 8, 19QQ . _

b . As in fig . 305 many other arrangements are possible each of which will furnish .a proof or, proofs .

J . .._

( A ) --P roo fs �e t e rm t n e d b y a r,�m e n t s b a s e d up o n a s.qu a re .

Thi s. type include s all p�oofs derived from figure s in which ·one or more of the square s are not graphically repre sented . There are two leading c l�s s­e s or sub-ty:pes in this · ty:pe -:..f·J.r.st , t,he ·· class in which the determinati9n of the proof i's based upon a square ; �econ�; the class in which the determination of the proof i s based upon a triangle .

As in the I-type, so here , by inspection we find 6 sub-clas se s in our firs.t sub-type which may be symbplized 'thus·: 0

(1 ) The h-square omitted; �ith (a ) The a- and b- squares const ' d outwardly--

3 c·ase s . (b ) The a- sq . consi 1 d out ' ly and the b- sq .

� o�erlapping--3 cases . · (c ) The b-sq . const 1 d 'out 1 ly and the ·a- sq .

overlapping--3 cases . (d ) The a- and b-s�ua�es overlapping--3 cases .

. '

...

------��- ----�=====;;;;;;;;;!;;;;;;:;;;:::;:G::::E=O=ME=T=R=I=C==P=R=O=OF:....::.S::::::- ::...:-.=:.- -:..:;-:..:.:·---=- =:::.::.::.;;.;:::::;:.::=.:.::- ·2=.1:�1--- ----�-------1

; I I

- .

(2 ) The

(3 )

(a ) (b )

(c )

(d )

The .(a ) '(b ) ... ' "

(c ) I (4 ) I

: (4 ) THe I (a\) (b \) (c D (#) Th�

t� l . (c )

(6 ) ThJ (a ) (b ) (c ) j

a- sq . omitted, with The h- and b-sq ' s c on� ' d out ' ly- -3 case s . The h-sq . const 1 d out , ·ly and the b- sq . overlapping- -3 case s . The b- sq . const ' d out ' ly and the h-sq . overlapping� -3 case s . The · h- and b- sq ' s c onst ' d afid overlapping - -3 case s . b - sq . omi-tted, with The �- and a- sq ' s cpnst ' d oqt ' ly- -3 cases . The h-sq . const ' d out ' +Y anq the a-_sq . overlapping-�� case s .

I Tne · a- sq . corst ' d out ' ly and the h-sq . overlapping--3 case s .. �

The h- and a- sq ' s �onst ' d overlapping- -3 case s . h- and a-sq ' s omitted, with 0 The b - sq . const ' d out ' ly . The �- sq . cqnst ' d overlapping . The b- sq ; tran�lated--in all 3 case s . h- and b- sq ' d omitted, with The a- sq . const ' d out ' ly . _ · The .a-sq. const ' d overlapping •

The a-sq . translated--in all 3 case s . a- �nd�b-sq ' s omitted, with The h-sq . c onst ' d out ' ly . The h- sq . · const ' d overlapping . Th� h- sq . translated- -in all 3 case s .

I l Tl!J_e t0tal of the se enumerated c·ases is 45 . We

shall giv� but a f.ew of the se 45, leaving the re -. l mainder to the ingenuity of the inte�e sted student . I -� I

(7 ) ' Al� three square s omittE;d .

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Case (1 ) , . (a ) .

In fig . 307 , produce GF to N a pt .-, · on .the perp . to AB at B, and e�tend DE to L, draw HL ana. AM p�RP . to AIL •· ·

._..,. liP

The tri ' s AMG and ABH are equal .

Sq . HD +� sq . GH = (paral . HO ·= paral . LP )

�iig . 307 ,. + paral . MN = paral . MP = AM X AB =. AB X AB = (AB ) 2 •

! ' - 2 upon BH +· sq . upon AIL :. h :. sq . upon AB = sq .

= a2 + b2 . a . Devised by author

Maf'ch 20 , 1926 . for ca:se (1 ) , (a ) ,

b . See proof No . 88 , fig . 188. . By omitting lines CK and HN in said figure we have fig . 307 . There�ore proof No . 2Q9 i s only a variation of proof No . 88 , fig . 188 . 1 Analysi s of proofs given will show that

- supposedly new proofs are oniy modifications ot � '"inore. fundamental proof . t.. /

(Not a Pythagorean Proof . )

many some

While case (1 ) , (b ) may be proved in some other way, we have selected the fo1lowing as being

.. .-> � ..

quite· unique . It is due to the ingenuity of Mr . Arthur R . Colburn of Washington , D . C . , and is No . 97 of his · 108 proofs .

It re st s upon the following Theorem on ' Paral­lelogram, which i s : " " If from one end of the side of

·

. a par9-llelogram a straight line be drawn to an-y po·int in th� oppos�te- side , or the . opposite side extended, and a line from the other end of said first side be drawn perpendicular +.o the first line , or it s

·� \

. I � 1\

'

-----�-----------· _gE�T�IC . PROOPS 219 . . extension, the product of these two drRwn line s will measure the area df t;he parallelogr•am . 11 Mr• . Uolbu.r·n formulated this theorem and it f.s use is di scussed in

\ Vol . 4 , p . �5 , of the "Me.tnematics Teache1� ,. '' Dec . , 1911 . I have not seen hi s proof , but have .demonstrat -

I -�-:.-} I -- \ I I ._ -

- �

' Fig . 308

In the paral . ABCDJ from the end A of the side P� , draw AF to side D O pr•oduced, and from B , the other end of side AB , draw BG perp . to AF , Then AF x BG = . ar ea of paral .

, · .ABCD : Proof : From D lay off DE ··= CF, and draw AE

and BF fo�ming the paral � ABFE = paral . ABCD . ABF i s a triangl-e and i s one-half of ABFE : The area of tri . FAB ' = �FA x BG; therefore the area of paral .

· ABFE = 2 times the area of, the tri . FAB, or FA � BG . . � But the area of paral . ABFE = area of paral . - ABCD .

..

. Q . E . D . :. AF x · BG measures the ar�a of · paral . ABCD .

0 By means of thi s Parallelogram Theorem the

Pythagorean Theorem can be proved in many cases , of which here is one .

' ' '

Fig . 309

. I�Q_U�u4t��-�lt��n

Case (1 ) , (b ) .

In fig . , 309, extend GF and ED tC" L completing the paral . AL , draw FE and extend AB to M . Then by the paral . theorem :

( 1 ) EF X. AM = AE X A G . (2 ) EF X BM = FL X BF .

(l ) (2 ) = (3 ) E?{1rM - BM ) = AE X AG - FL X BF

..

...

' ·

220 THE PYTHAGOREAN PROPOSITION

(3 ) = (4 ) (EF = AB ) x AB = AGFH + BDEH, or sq . AB = sq . HG + sq . HD .

:. sq . upon AB = sq . upon BH + sq . upon AH . • , 2 _ a2 � b2 -

• • l! - • a . This is No . 97 of � . R . Colburn ' s 108

proofs . b . By inspecting this figure we di scover in

it the five dis sected parts as set forth ,by my Law of Dissection . �ee proof Ten, f�g . 111 .

Ca3e (2 ) , (b ) .

. Tri . HAC = �ri . ACH . .,__..._._.��J.. 'I'ri . HAC = -k sq . HG.

\ J· ;2, Tri . ACH = -k rect . AL . f I I /ft I - " 1/ : :. rect . AL = sq . HG. Similarly I V" l · j , G .. 1 : . . rect . BL = sq� on HB . But rect . AL

C r k + rect . BL = sq . AIL �<-t......, - --'•-J . :. s q . upon AK = sq . upon HB Fig . 310 + sq . upon HA . :. h2 = a2 + b2 •

Q . E . D . a . Sent t o me by J . Adams fro� �s-Hague ,

Holland . But_ the author no.t given . R§�.eived it March 2 , 1934 .

I

. '

Fig . .. 311 •,)

Case (2 ) , (c ) . · ··

In fig . 311 , pr.oduce GA to M making AM = HB , · draw BM, . . and d�w KL par . to AH and CO par . to BH .

Sq . AK = 4 tri . ABH + sq . .. AH X BH NH . = 4 . x 2 + (A!f - . BH ) � - ·

= 2AH X BH + AH2 2AH X BH .+ BH2 = BH2 + AH2 •

\

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J

. ..

..

...

GEOMETRIC PROOFS 221

:. sq . upon AB = sq . upon BH + sq •. upon AIL :. h 2 = a 2 + b 2 •

a . Original with author , March, 1926 . b . See ' Sci . Am. Sup . ; Vol . 7'0 , ' p . 383 , Dec .

10 , 191,0,. fig . 17, in which Mr . Colburn make s use of the tri . BAM .

c . Another proof, by author , i s obtained by comparison and' �ub stitut�.on of dis sected parts as numbered. ·

Case ( 4 ) , (b ) . -

In fig . 312 , prodpce FG. to P making GP = BH , draw AP and BP . - 2 Sq . _G]l = b = tri . BHA + quad . ABFG = tri . APG + quad . ABFG = tri . APB � + tri . PFB = !c2 + � (b + -a ) (b - a.) .

• b2 1 2 1b2 1 2 • 2 2 . . = 2c -+ 2 - 2a . . . c = a + b2 .

Fig . 312 :. s q . upon AB = s q . upon HB + sq . upon HA .

· a . Proof 4, on p . 104 , · rn "A Companion· of . Elementary School Mathematics , �' (lg24 ) by F . C . Boon, B .A . , Pub . by Longmans, Green and C,-.� •.

Fig . 313

In fig • ' 313 , and complete the sq . perp . to · AB , FM par .

. perp . to AB • ·

produce HB to F AF . Draw GL to AB and NH ...

Sq . AF = AH2 . = 4 AO x HO 2

+ [ L02 = (AO - HQ-)&f.·-= 2AO X HO + A02 2AO X HO + H02 = A0·2 + H02 = (Ab

= AH2 + AB )2 + (HO = AH X HB + AB )2 = AH4 + AB2 · + AJi2 X HB2 + AB2 = AH2

(AH2 .+ HB2 ) + AB� • . -.·. 1 = (AH2 + B1{2 ) + AB2 • :-":·:· ··AB� = BH2 + AH2 •

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1\ I I I

222 THE PYTHAGOREAN P�OPOSITION

:. sq . upon AB = sq . upon HB + sq . upon HA . :. h2 = a2 + b2 • • Q .E . D .

a . See Am . Math . Mo . , Vol . VI , 1899, p . 69, proof CIII ; D� . Leitzmann, p . 22 , fig . 26. ·

· b . The reader will ob serve that thi s proof prove s ' too much, as it first prove s that AH2 = A02 + H02 , which . i s the truth sought . Triangle s ABH and AOH are similar , .and what is· true , as to the relations of _the sides of tri . . AHO' must - be true , by the law of similarity, -�s to the relations of �he side s of the tri . ABH �

Fig . 315

' -

Case (6 )� · . (a ) . ·This· i s a popular figure with· author� .

· In fig . 314� dr.�w 0� and KD -par • . re spectively to AH and BH, draw AD and BD , and draw AF perp . to CD and BE perp . to KD extended .

· Sq ._ AK = 2 tri . · CDA + 2 tri . BDK = CD ' x AF + KD x EB = CD2 +' KD2 • <

:.o sq . upon AB = sq . upon BH 2 ' 2 2 -+ sq . upon AH . :. h = a + b •

· . �:. Original with the author , August · 4, _ 1900 .

In fig . 315, extend AH· and BH to E and F re spectively making HE = HB and HF = HA, and through, H draw·

. . LN per,p . to AB, draw CM and KM par . re spectively to AH and Blj, complete the rect . FE and draw LA, LB, Hq and HIC I I -

' Sq·. · 'AK = rect . BN +_ rect . AN = paral . BM + paral . AM = (2 tri . HMK = 2 tr·i . LHB = sq . BH ) + (2 tri . HAL · = 2·. tri . � = sq . AH ) .

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l ' GEOMETRIC -PROOFS 223

� sq . upon AB = sq . upon BH + sq . upon AH . . h2 = 2 b2 • • a + ...

a . Original with author March 26 , 1 926 , 9 p .m .

Fig . 316

In fig . 31?, coJnplete the' sq ' s HF and AK; in fig . 317 complete the sq ' s HF , AD and CG, and draw HC and DK. Sq. HF - 4 tri . ABH = sq . AK = h2 • Again sq ; HF - 4 -tri . ABH = a2 + b2 . :. h2 = a2 + b� . · . ..

:. s q . upon AB =.:= sq . upon BH + sq. upon AH . a . See Math . Mo . , .1858, Dern. 9, Vol . I , p .

159 , · and credited to Rev . A . · n . Wheeler of Brunswick, Me . , in work of Henry Boad, London, 1733 .

b . An algeb�aic proof : a2 � b2 + 2ab = h2 + . 2ab . :� h2 �= a2 + b2 .

sors .

sq . HM .•

\ - 2HB )( + AH2 .

c . Als� ,. two -equal squar.es of paper and scis-

In fig . �18 , .extend HB to N flP.d complete the .

HB X HA Sq . AK = sq . HM - 4 _2 = (LA + AH )2

HA = LA 2 + '2LA X AH + AH2 - 2HB X HA · = BH2 ' r

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··1\ ' I l 224 THE PYTHAGOREAN PROPOSITION

:. sq . upon AB = sq _. upon BH + sq . upon AH .

a . Credited to T . P . Stowell, . of Roche ster , · N . Y .

· ,see · The Math . Magazine , Vol. I , 1882 , p . 38 ; Olney ' s Geom . , Part III, 1872 , � . 251 , 7th method; Jour . , of Ed ' n, Vol . XXVI, 1877 , p . 21 , fig . IX ; also Vol . XXVII , 1888 , p . 327,, 18�h Rroof , by R . E . Binford, Independence , Texas ; The - · Fig . 318 School . Vi sitor , 'Vol . IX, 1888,

_ . p • 5 , proof II ; Edwards 1 Geom., 1895-: p . J,.59, _ fig . (27.) ; Am . Math . Mo . , Vol . VI , _ 1899 , p.. 70 , proo.f XCIV ; Heath 1 s Math . Monographs , No . 1 , 1900 , p . 23 , proof VIII ; 'Sci . Am. Sup . , Vol . 70 , p . _ 359� �ig . 4 , 1910 ; Henry Boad ' s work, London, 1733 .1 -

,b . For aigebraic solutions , se� � : 2 , in a pamphlet by Artemus Martin of �ashington, D . C . , Aug . 1912 , entitled " on Rational Right-Angled Triangles" ; an4 a solution by A . R-." C olbur.n, i:p. Sci . Am . Supple� ment , Vol . 10, p . 359,. Dec . 3 , 1910 .

c . By drawing the line _ AK, and considering

Fig . 319

the part of the figure to the right of said line AK, we have the . figu1•e from which the proof knbwn· as Garfield ' s So-r \ /lution ..follows-- see proof Two Hundred Thirty-One , fig . 330 .

In fig . 319, extend HA to L and complete the sq . LN .

� -

Sq ; AK = sq . LN

4 x HB x· HP::. = (HB + HA ) 2

2 - 2HB X HA = HB 2 .. + 2HB X �

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GEOMETRIC PROOFS

+ HA2 - 2HB x HA = sq . HB + sq . HA . �:._ :sq . upon AB = sq . upon BH + sq . upon AlL :. h2 = �2 + b2 •

a . See Jury Wipper , 1880, p . 35, fig . 32 , as . given in "Hubert 1 s Rudiment a Algebrae , '� Wurceb , 1762 ; Versluys , p . 70 , fig . 75 .

b . This fig, 319 is but a variation of fig . r 240 , -as als6 is " the pr6of .

Fig . 320

Case (6 ) , (b ) '.

In fig • . 320, comp�ete the sq . AK overlapping the tr.i . ABH, draw through H the line 'LM per.p .. to

· AB, ext�nd BH _ _ t() N making BN = AH, and draw KN perp . to BN, and CO perp . to AH . ' Then, by the paral­lelogram theorem, Case (1 h (b ) , ffg . 308, �q � AE = · paral . KM.

+ paral . CM = {BH + b2 .

x KN = a2 ) + (AH ' x CO = b2 ) = -a2 . - �

:. sq . upon AB = sq .· upon BH + sq . upon AH. a . See ·Math . Teacher, Vol . 4 ,. ' p . 45 ; 1911 ,

where the p:roof is credi,ted to· Arthur 1: .. Colburn . b . See fig . 324 ; . . whi_ch is more fundamental,

· · proof ;No . 221 or proof No . 225 ? c . ·see . fig ; 114 and fig . 328 .

Fig . 321

In fig . 321 , draw CL perp . xo AH, produce· BH to N mak�ng . BN - CL, and draw KN and CH . · · since CL '= AH and KN = BH, then -k sq . BC = tri . KBH + tri � AHC : t BH2 + -kAH� or -kh2 = la2 + �b2 • :. h.2 = a2 + b� •

. :. sq . upon AB = sq . upon HB + sq . upon HA . -

a . Proof 5 , on p . 104, in

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226 Tf!E P"TrHAGOREAN PRO.POSITIO�

"A Companion to Elementary Mathematic s " (1924 ) by F . C . Boon; A .B . , and credited to the late F·. C . Jack­son (" Slide Rule Jackson" ) .

In fig . 322 , draw 04 and KL . � par . to ' AH and BH re spectively, and 1\J,/ , \ through H draw- LM .

C � t � - Sq . AK = rect . KM + rect . OM r' """\ - -L, - lJ '\ " = paral . KH + paral . CH = BH x NL r ' 1 + AH x NH = BH2 + AH2 •

1 :. sq . upon AB = sq . upon BH + sq . upon AH . :. h2 = a2 + b2 • Q . E . D .

, a . Thi s is known a s Hayne s ' Fig . 322 Solution . See the Math . Magazine ,

Vol . I , p . 60 , 1882 ; also said to have b�en discovered in 1877 .by Geo . M . Phillips , Ph: Ph . D . , Prin . of the We st Che ster State · Normal School, Pa . ; see Heath ' s Math . Monographs , No . 2 , P r 3$, proof XXVI ; Fourrey, p . 76 .

b . An algebraic proof is easily obtained .

Fig . '323

In fig . 323 , construct sq� AK . Extend AH to G Thaking HG = HB ; on OK const . r't . tri . .., CKL (= ABH ) . and draw the perp . IJIM, and extend ·LK to G .

Now· LG = - HA , and it is obvi­ous that : sq . �K (= h2 -) = rect . MK + rect . MC = paral . 'HK + paral . HC = HB x HG + HA x CL = b2 + a2 , .or 2 2 2 . h = a + b . Q .E . D .

:. sq . upon AB -= sq . upon HB + sq . upon HA .

a . Thi s fig . (and proof ) was devised by Gus ­tav Cas s , a pupil in the Junior-Senior ' High School ,

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GEOMETRIC PROOFS 227

South Ben� , Ind . , and sent to the author , by his teacher , Wilson Thornton, May 16 , 1939 .

I I

Fig , ' 324a

dase (6 ) , (c ) . .

For convenience de signate the up�er part of fig . 324 , i . e . , the . . sq . AK, as fig . 324a , and the

, lower part as 324b . In fig . 324a, the con­

structio� - is evident , for 324b is made from the dis sected parts of 324a . · GH ' is a sq . each side of

� which = AH, LB ' is a sq . . each

J{ L.l �- --- 1 F side ·or which = BH . _ I -....-:�T �\ \ I . Sq . AK = 2 tri . ABH

1+ _2

f · ' f ' \ f tri . ABH + sq . MH = rec-t . B N m1 _ _Ai')J.o_ - �HI · + rect , OF' + sq . LM = sq . B ' L

Fig , 324b + sq .- A ' F .

· .•. sq . upon AB = sq . upon BH + sq . upon AH. :. h2 = a2 + b2•

a . - See Hopkins ' · Plane Geom . , 1891 , p . 91, fig . V; Am . Math . Mo . , Vol . VI , .1899, p , 69, XCI ; Beman an� Smith ' s New Plan� Geom . , 1899, p . 104 , fig� 3 ; Heath_' s Math . Monograpns , No . l , 1900 ,- p , 20 , _

proof IV . Also Mr . ;Bodo M: DeBeck, pf Cinc.innati , 0 . , abo'ut 1905 . without lmo�e;c:lge of· any previous so-

. WI'<· lution di scovered, above form · ·or figure and devi sed a proof from it . Also Versluys , p , 31, fig . 29 ; and "Curio sitie s of Geometrique s , Fo'urrey� p . 83 , fig . b, and p , 84 , fig . ft, by Sanvens , 1753 . . h. History ·relate-s that the Hindu Mathema­tician Bhaskara , born lil4 A . D . , discovered the above p"Poof and followed t,he figure with - the ·s :J,ngle word "Behold, " . not cpnde scending to give ci'ther _ than the figure and . thi s one word for proof . And hi story furthermore declare s that the Geometer s of Hindustan k,new the · · truth and proof of thi s theorem centur:J-es before the time of Pythagoras--may he not have learned about it while studying Indian lore at Babylon?

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THE PYTHAGOREAN PROPOSITION I •

Whether he gave fig . 324b as well as fig . 324a , as I am of th� opinion he did, many late authors think not ; -. with ;the - two figure s , 324a and 324b , side by side , the word "Behold ! " may be justified, e spe-

, c ially when we recall that _ the tendency of that age : was to . keep secret the di scovery of truth for certain 1 purposes and from certain c las ses ; but with the fig . i 324b omitted, the ac t is hardly defensible- -not any : more· so tha:n " See ? " would be after fig . 318 . 1 Ag�i�, authors who give 324a _and "Behold ! " : fail t-o tell their readers whether Bhaskara ' s proof 1 was geometric or algebraic . Why this silence on so e s sential a point ? For, if algebraic , the fig . 324a . . \ i s enougn as the next two proofs show . I now quote from Beman and Smith : ' "The inside square is evident­ly (Q - - a ) 2 , and each of the four triangle s is �ab ; • h2 4 1 b (b )' 2 h h2 2 ,L 2 II

• • - x 2a = - a , w_ ence = a + u •

It is conjectUred - that Pythagoras had di scov­erea it ·indepep.dently, as aiso did Walli s , -an English Mathematician, in the 17th century, and so rep@.�t�d;

,. • ) 'l. M.Y'.i� also Miss Coolidge , the blin� girl , a few years ago : see proof Thirty-Two , fig . 1�3 .

Fig . 325

Irr fig . 325, it is ob­vious that tri ' s 7 + 8 = rect . 'GL . Th.en it i-s easily seen,

'

from congruent parts , that : sq � upon AB = sq . upon BH + sq .

AH . h2 2 b2 upon . . . = a + . . a . Devi sed by R . A .

Be-1 1 , .Cleveland, 0 . , July· 4 , 1918 . He submitted three more of same typ� .

In fig . 326, FG 1 = FH ' = AB = h, DG 1 = EF /, \ = FN = OH 1 = BH = a, and DM ' - EH ' = ·G 1 N = FO = AH . = b .

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GEOMETRIC PROOF.S

. ,,

229

Then are tri ' s FG'D, G ' FN = FH ' E and H ' FO eacn �qual to FG ' D = tri . ABH .

Now 4 tri . FG ' D ' . + sq . G ' H ' = sq . EN . + sq . DO = a2 + b2 •

. .

�ut 4 tri . FG ' D + sq . G ' H ' = 4 tri . ABH + sq . '

' l I 2 2 GH = s q • AK = h • , :. h

- I

I I •'

'; = a2 + b2 • AB . :. sq . up on , = . AH .

sq . upon BH + sq . upon

a . Devi sed by author , Jan . 5, 1934 . b . See Versluys , p . 69, fig . 73 .

Draw AL · and BL par . . .. .

Fig . 328

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resp ' ly to BH and -AH, and com-�-P.lete the sq_. LN . ABKC = h2 = (b + a )2 � ·2ab ; but ABKC = (b - a ) 2 + 2ab:----

� 2h2 = 2a2 . + 2b2 , Or h2 = a2 + b2 . :. sq . upon . AB. = sq . upon BH + �q . upon AH .

a . See Ve)?.s.luys , p .� ·..,

72 , fig . 78, attributed to Saunderson (1682-1739 ) , and came probably from the Hindu Mathematician Bhaskara .

. . I��-�������-I�t�1r��i�t

_ In fi-g . 328·, draw CN par . to BH, KM par . to AH, and e�tend BH to L.

HB X HA Sq . A_K = 4 2 . + s� . MH = 2HB X HA + (AH - BH ) 2 = 2HB X HA + HA2 - 2HB X RA. + HB2 '= .HB2 + HA2 .

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230 THE PYTHAGOREAN PROP0:3ITION �------------------

:. sq . upon AB = sq . upon BH + sq . upon AH. a . See Olney ' s Geom . ., Part III , 18'72 , p . 250 ,

1st method; Jour . of Ed ' n, Vol . XXV , 1887 , p . 404, , fig ·. IV , and also fig .. VI ;_ Jour . of Ed ' n, Vol . ?CXVII , 1888, p . 327 , 20th proof, by R . E . Binford, of Inde� pendence , Texas ; Edwards ' Geom . � 189§ , p . 155 , fig . (3 ) ; Am . Math . Mo . , Vol . VI , 1899, p . 69 , proof XCII ; Sci . Am . Sup . , Vol . 70, p . 359 , Dec . 3 , �910 , fig . 1 ; Ver sluys , p . 68, fig . 7·2 ; D!1 , Leitzmann ' s work, 'U.930,· p . 22 , fig . 26 ; Fourr�y, p . 22 , f�g . a , as given by Bhaskara 12th century A .D . in Vi ja Ganit� . For ana algebraic proof see rig . 32 , pr?of No . 34 , under A1-gebraic Proofs .

b . A study of the many proofs by Arthur--R. Colbu�n, LL .M . , of Di st . of Columbia Bar , e stablishes the thesi s , so cften reiterated in this ·work, that figure s may take any form and po sition s9 long as they incluqe tri�ngle s whose side s ' bean- a rational algebra·ic relation to the side s of the gi ve.n triangle , or wno se dis sected areas are s o related, through �qui�alendy t�at h2 = a2 + b2 re sults . .

(B) �-P ro o fs b as e d upo � a t r t an � l e t h ro u � h t h e cal c u­l a t t on s and �omp a r t so n s o f e qu t u al en t areas .

Draw HC perp . t o AB . The three tri '·s ABH, BHC and HAC a:re sim-ilar .

We have _ three sim . tri 1 s erected upon the three sides o� tri •

• Fig . 329 ABH �hose hypot�nuse s are the three side s of tri . 'ABH . '

Now since .the · area of tri . CBH + area of tri . CHA = area of tri � - ABH, and since the areas of three sim . tri ' s are to each otner as the sq�ares of their corre sponding side s , (in thi s case the thl,ee hypote­nuse s ) , therefore the area of each tri . is to the sq . of it s hypotenuse as the areas of the other two t�i ' s are t o �he sq 1 s �f . their hypotenuse s . . -- : �

••

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. GEOMETRIC PROOFS 231 Now each sq . i s = to the tri . on whose hypote- :

nuse it i s erected taken a certain numbar o'f time s , this numb er· bethg the e·arne for all three . 'l'herefore Sil'l.Ce the hypo�eDUSeS On WhiCh the se Sq I S are eJ:•eCt •* ed are the sides of' the tri . ABH, and since the sum of tri ' s erected �n the iGgs i s = to the tri . erected on the hypotenuse . :. the sum of the sq 1 s erected on thE? legs =•the sq . erected on the hypotenuse , :

.

h2

= a 2 + b 2 • Q .• E • D •

. r . a : Original, ·oy Stanley .Jaohemski $ age 19, uf

Youngstown,- 0 . , Jtme 4- , · 1934, . a young man of ·s_uperior intellect .

. b • If m + n = p and m � then m + n : a� + b 2 = · n : b 2 =

n : p = a2 p : h2 .

m + n a2 + b 2 � - , or 1 =

a2 + b 2

•

h2

Fig . 330

p· h2

This algebraic proof h2

given b y E� . .

s . Loom-

In fig . 330 , extend HB t o :> zta.ktng BD = AH , through D draw DO par . to MI and equal to BH, and draw OB and CA .

'

Area of trap . CDHA = area ·of AL�B + 2 area of ABH .

:. � (AH + CD )HD = �AB 2 + 2 x �AH x HB or (AH + HB )2 = .AB 2 . + 2AH x HB , whence AB2 = BH2 + AH2•

:. sq. . upon AB = sq . upon BH + oq . upon AH . :. h2 = a2 + b2 •

a . This i s the "Garfield Demqns trat i or1, • � -.:hit upon by the Gene:Pal in a mathe­matical discussion. with other M . G . ' s about 1876 . See Jour . of Ed ' n, Vol . IIIJ) �.876 , p . 161 ; The Mat.h . lifag­azine , Vol . I , !882 , p . 7; The School Vi sitor � Vol . IX, J-888, p . 5 � proof III.; Hoyk1ns 1 P lane Geom.. , 1891, p . Ql , fig . VII ; Edrrards 1 Geom . , 1895·; p . 156, · fig .

. . - (11 ) ; -Heath 1 s M!lth . Monographs , No . 1 , 19CO, p . 251

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THE PYTHAGO�EAN -PROPQSITION ..

proof X ; Fourrey, .P . 95 ; School Visitor , Vol .- 20 , p . ;I.6T; Dr . Leitzmann, p . 23 , fig . 28a , and also fig . 28b for a variation .

b . For extension of any triangle , see V . J'el­inek, Casopis , 28 {1899 ) 79-- ; Fschr ·. Math . (1899 )

· 456 . � c . See No . 21� fig . 318 �

Fig . 331

�- - . :.• '-, ' .... . ,

By geometry, (see Wentworth ' s Revised · Ed-1 n; .1895 � p . 161, Prop 1 n XIX )-, we have AR.2 . + HB 2 = 2HM2 + -.2AM2.

2 .

But in a rt . tri . HM = AM . So b + a2 =

· 2AM2 + 2AM'� = 4AM2 = .4 (AB t · 2 .

= AB 2 = h 2 . :. h 2 = a 2 + b 2 . ., '

a . . See Versluy� , p : 89, r'1g . . 100� a s given by · Kruger·, 17 49 .

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I�2��[ndt��-Ihlr!Y�Iht��

0

· Given rt . trl . ABH . Extend · A1J\ BH to A 1 making HA'1 =· HA . Drop A 1 D

fl \.._ . perp . to AB in�ersecting AH a.t C . I I �' · Draw AA 1 and CB . .· i ¥ -· ,. \"· J./ . Since angle ACD = angle HCA 1 , , . r ·: .. · 3 th�n -angle CA �H :;: angie- BAH . There -

)r ) fore tr1 1 s CHA1 and BHA are equal . . Th�refore HC = HB .

Quad . ACBAf � (tri . CAA 1 Fig . 332 , , = tri . c�:) + tri . BHC + 'CHA I = !i '(:AD}

+ h (DB ) _ · h (AD .+ BD ) _ .!f _ a2 . b2 . 2 a2 +

2b2 • .

2 . . - 2 - 2 - ? + 2'' . :. h = . a . See Dr . W . Leitzm�n.Ii'1 s work, p . 23 , fig .

27., 193.�, 3rd edition, · credited to En� . , - whq discovered it 1 � 1909 .

b . _ See its algebraic proo� The above proof is truly algebraic areas . . The author .

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C • Hawkins , .o·f -� ' ;.

Fifty:, fig . 48 . throu,gh equal

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GEOM$TR�C PROOFS 233

Let C , D and E be the cen­ters of the sq ' s on AB , _ BH and HA . Then angle BliD = 45° , 'also angle EHA . · �·. line ED through H is a st .. line . Sifice _angle ARB = aogle BCA the quad . is inscriptible in a cir­cle whose center ls the middle pt .

·, of AB , · the angle CHB = angle BHD · · = 45° : :. CH is par .. to BD . · :. an"':' li'ig . 33.3 gle .cJID = angie IIDB = 90° . Dra.w

AG and BF perp . to ·Cir.� Sinc·e tri ' s AOC and CFB are congruent , CG = FB = DB and HG = AG = AE, then CH = EA + BD . .

..

Ndw area of ACBH. = HC (AG + FB ) = HC . x ED • ' 2 2

= area of ABDE . From each t�ke away tri . ABH, w.e get - 'tri . ACB = tri . BIID + tri:� HEA . 4 times this eq,'.J1

2 give s sq . upon AB ::£ sq . upon HB + sq • . _ �pon HA . :. h = a2 + "b� .

__ . a . See Fou�rey, p . 78, �%3 gi,ven by · M . Pi ton­

Bres sant j Versluis� . p . _ 90, fig . �03 , taken from Yan Piton-Bres-sant , per Fourrey, 1907 .

' ' I •

b .' See alg�br�ic proof No . 67, f:tg . 66 .

Fig : 333 a�d - 334 are same in· outline . Draw HF perp . to AB , and draw DC , DF ahd FC . A s in · proof� fig . 33� , HC is a st � line

• • <) par . to BD . Then ·tri . BDH = tri . · BJ?C . ·- - - {1 ) · ,J\s quad: · HFI:3ri· is in- ,,

� scriptible in a cfr'cle whose cent·er

Fig . 334 = tri . BCF .

,,

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i s the cehteD of HB , then angle .BFo · - · -

· . . 4 0 = angle DFH = _ 5 = angle FBC . :. FD . . ' �s par . to CB , ·whencE? tri . BCD · '

--- (2 ) .

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234 T� PYTHAGO�EAN PROPOSITION

:. tri . BCF = tri . BDH . In like manner tri . ACF = tri . AHE . :. tri . ACB = fri . BDH + tri . HEA • .; _ _ (3 )' . · 4 x (3 ) give s sq . upon AB = sq . upon HB + sq . upon HA .- . ·�· h 2 = a 2 + b 2 • '

. · a . See Fourrey, p . 79, as given by M . Piton-...

Bre s:sant of Vi tteneuve -Saint-Georges ; .· also Versluys , p . 91 , fig . 104 . .

. . ,

---�b . See algr braic proof No . 66, .. fig . ·67 .

-') l

In fig . 335, extenq BH to F making HF = AH, erect AG perp . to AB making AG = _ AB , draw GE par . to HBJO and GD par . - to AB .- Since tri ' s ABH and GDF are 'simi far , GD = li :(l - a/b ) , and FD = a (r - �a/b.J .

Fif? · 335

Area of fig . ABFG = area ABH + -area �G = area ABDG + area GDF . � � :. �ah· +_ tb[ b + : (b - a ) ] · = �h [h + � (1 :.. a/b) ] + �a (b - a ) (1 ·. - a/b ) ,

-� - - - (1.) . Whence h2 = . a2 + b2 • :: s.q . · upon · @ = sq . upon -BH + sq . upon AH .

� a . Thi s Proof i s due t o J . G . Thompson, of · Winchef,Jter , N .H . · see Jour . . of ,Ed·' n, . .. :ifol . XXVIII , - �888, .p . 17 , ·28� proof ; Heath ' s Math . Monographs , No . 2 , p .. 34 , pr of Xxiii ;"· Versluys , p . 78, fig . 87 , by Rupert_, 1900 ·

- · •

b . ' As . �re ar:e pos sible several figure s of above type, in each of which there will ·Te sult two

--; similar t�·ia gl,e s , there are pos sibl� many di'fferent .. I . \ ' proofs , differfng only ·· in s_hape of figure . The next proof 1 s one from the I!J.any. . ,

In fig . _ ·336 produce HB 'to F making- HF =- · HA , through A draw AC perp ._ to � making AC = AB, d:r;>aw -· , I , -CF, -AG par . - to HB, BE par . to All, and BD perp . to AB .

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GEOMETRIC PROOFS 235

Since tri ' s ABH and BDF are similar , . we ' find that DF = a (1 - a/b ) and HD

= h (1 - a/b ) . Area of trap . CFHA = 2 area

ABH + area trap . AGFB = area ABH + area trap . ACDB + area BDF .

Whence area ACG + area AGFB J= area . ACDB + area BDF or �ab + �b[b + (b - a ) ] = �h[ h + h (l - a/b )] + �a (b - a ) (1 - a/b ) .

· T-his e9.uation is_ ·equation (1 ) in the preced­ing solution, as it ought to be , since , if we draw BE .par . to AH a-nd:· con,sider only the figure below the

- line AB , calling the tri . ACG the given triangle , we . have idepti.cally fig . 335 , above .

·

:. sq .... upon AB = sq . upon BH + sq . upQn AH . :. h2"- = a2 + b2 •

·

a . ·original with t]:le author , August , 1900 . See· also Jour . , of Ed'' n� Vo;I. . �III , 1888 , p . 17', 28th proof , ·\

I "

Fig . . 3.3� ,

' In fig . 3�7 , extend HB to N �making .HN b AB·, draw . KN , KH ��d BG, ext�nd GA to M and drawi EL p�r . t� AH . ; Tri . KBA + trlL. ABH = quad . BHAK = (tr-i . � = tri· . GAB ) +c ._ ftri . tPGB = tri . HKB+ = q�ad : . {ABDG = tri . HBD + tri .

$ � ' , G� + t�i . ABH, whence tri . BAK = t :rh ; HBD + t r i . . GAH . - . - .. -, ! ' � k ' ' ,i :. sq . upon AB = sq . ' I 2 upon BI;I : + -sq . 1,1pon AH . :. h = a2 ·+ · b2 .

•• • ,1 a . See J.ur.y Wipper;. J.88o , p . 33 ,j fig . 30, as found 'ih the · wqrks of �oh: J . I . Hoffmann, Mayerice , 1821 ;- · Fourrey, p . 75 .

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236 � PYTHAGOREAN PROPOSITION

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In fig . 338, construct��e ·i three equilateral triangles upon�e· '" ' ,, , .... , H ""D th.!'ee sides of the-· given triangle I ' , ' - -1 . .

, . 1 . ABH, and draw ,EB and �1 , draw EG " I G 1 perp . t'o AH, and draw G� ,

, )l �-�.........;;;; .. �· Since EG and HB are paral.lei, \ 1

I tri . E.BH = tr i . BEG = -k �\ri:. "ABH .

I

\ 1 :. tri . GBH = tri . REG. ' 1 : (l ) Tri . HAF · = tri . EAB

\ ' 'J: ( , 1 ' '.� = 0tri . . EAK + tri . BGA = 2 tri . ABH )

Fig . 338 + ( tri . BKG = tri . EKH ) = tri . EAH + -k tri:-ABH. . . .

(2 ) In like manner , tri . BHF ;: tri . DHB + � tri . ABH . (1 ) + (2 ) = (3 ) ( ttai . HAF +\ tri . BHF = tri. BAF / + tri � ABH ) = trr. �AH + t�i . DHB + tri . ABH, whence tri . FBA = tri . EAH + tri . · DHB . •

• f I Bu� since areas of - similar surface s are t oe eaph other a� the �quares of their like dimensions ,

. w� �ave ·.,

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tri . FBA :

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tri . DHB :· tri . EAH = AB2 : BH2.

i AH2 , whence trf-�- -FBA · - ::-�tri ; oDHB + tri: . 'EAH = AB2 : BH2 + AH2 • But tri . FBA = tri •

DAH + tri . EAJr. :. AB2" = BH2 + AH2 .

:. sq . upon AB = � sq . upon HD + s.q . upon HA .

a : Devised 'by the l author Sept . 18, 1900 , for

similar regular po.lygone other than squares .

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·- - I�o.-t!u.o.g,r_�g__f.2t1i

�n fig . 339 ; from the ' \

middle point s of AB, BH �and HA draw the three perp ' s FE, GO and KD, making FE = .2AB ,

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GEOMETRIC PROOFS 237

GC · = ?BH. an� KD ·= '2HA, co_rhplete the thr�e i-soscele s tri ' s EBA, ORB �nd DAH, and draw EH, BK and DB .

·. · Sine� the se tr1. 1 s · a-re re spec.tively equal to the three sq ' s upon AB, BH and MA, it remains t o

·prove tri . EBA :::;; tri ; · CHB"- + tr·i . ·d )AH . The proof is . '

sam� · as that - in fig . 338, �e�ce proof for 339 is a variation of proqf f.or - 338 .

. a . . Devised by the autho� , because of the fig� , u:re , so as tb get · area at: tri . EBA = AB2 , etc . :. AB2 - ·BH2 + AH2 , -

r •

• ' • ,.. - \' �

. :. sq . upon AB = · sq . · upon BH + sq . 1upon AH . 2 2 2 .

:. h . = . a + b . . . 1 • •

b-. �his proof i s given by . Joh·. Hoffmann; see his. solution in Wippe.r '. s Py�hagora.i sche Lehrsatz , 1880 , pp . 45-48 .

· . SBe , al s9 , Beman a�d Smith ' s New Plane and Solid Geomet'ry, - 1899 , p . 105 , ex . - 207 ; Versluys , - . p . 59 � fig : : 63 . . � ' ' . .

I • • \.

·c . 'Since any polygon of three , four , five , or more side s , .regular ·or irregular , can be trans-· formed, (see - Beman and S��th, p . 109 ) , into an equiv­.alent triangle , and it into ari e·quivaient :i:s o sceles t'riangle who·se b.ase i s the assumed base of the poly­gon, then is · the sum of the are�s of two . such similar polygons , or s�micircle s ., etc . , con$tructed _upon the two legs of any rfght triangle equa_l to the- area of -!1 • • '

a similar ·polygon constructed upon the �ypotenuse of safd · right triangle , if the suni of the . two i.soscele s triapgle� s o . qonstructed, (be their �ltitudes what ' ' . \ ' '

they may ) , is eq,ml to the area of .the similar i sosce-les triangle constructed upon . the hypotenuse of the as sume-d triangle . AJ:so see Dr . Lei tzmann, (1930 ) , . ' , I p . 37, fig . 36 for semicirc�es __ . .

·

. d ._ See . proof J�w�o�H�un�d�r�e�d�F�o�r�t�y_-�O�ne _for the e stabiishment of above hypothe sis . . .

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I�2-'ti¥.tt[t�2.-Eo.r.i�::�Qo.� -'-,.., Let tri 1 s ·b:SA, DHB and EAH be similar isos-

celes tri ' s- upon the bases AB , BH and AH of the rt .

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238

Fig . 340

THE PYTHAGOREAN PROPOSITION

. _ ...

\ . tri . ABH, ami CF, DG and. . . EK their

altitudes fr0m thefr vertice s C , D and E , and L, M and N· the mid� die point s qf these alt;l.tud'es . •

Transform the tri 1 s DHB , EAH and CBA i"l'lto .. their re spective

· paral ' s BRTH, AHUW and OQBA . Prq1duce RT and WU to-· X,

and draw XHY . Thrpugh A and B - - · \

draw A ' AC ' and- ZBB ' par . to XY . " Thro�gh H dray Hb • par : to OQ and complete the paral . HF 1 • Draw ·� XD ' and E ' Z . Trf ' s E ' Yz and XHD ar� congruent , since � = HD ' and respective angle s are equal'. :. EY ( = � · · Dr�w E ' G ' par . to BQ, . and ·

paral . E ' G ' QB = paral . E 1 YZB· � paral . XHD ' F; also . � paral . HBRT = paral � HBB 1X . But paral . HBB 'X is S·ame as paral . XHBB ' which = paral . XHD ' F ' = paral . ' c E ' Y2B • ... .

:. J?a:r:al . E ' G ' � = tri.�. DJIB ,; · in lik� manner paral . AOG ' E ' = tri . ' EAH. As paral . AA ' Z� = paral . AO� ;;; tri . CBA, so tri , . . CBA =. tri . DHB + tr!t . EAR •. - - - (�� ' I

"' ' , ,.

' • *' . 2 2 Since tri . CBA : tri . DHB : tri . EAH = h : a b2 , . tri .

. CBA : .tri . DHB +' tri . EAR = h2 : a2 + b2 •

But tri . CAB = tri . DHB + tri . EAH . - - :.. (1) . . :. ·h2 = a2 + b2 .

. \ -

:. sq . upon AB = �q . up·on-·BH + sq . ' upon . AH . Q . E .D . .

a . Original- with author'. Fcirmulated Oqt .... 28; 1933 . The ,author has never see·n·� .nor read about , nor

· heard of , a proof fo� h2 = a2 + . b2 based on isosceie s triangle s h�ving any altitude o�whose equal side s ·

are unrel�ted �o a, b , and h . / . \ . · · . . · , . ' '· . • I I .. • , I , . , , · . I . .

�.- · . --- � Ib:2_f!U.o.d.r.�i-Eo.r.�t'�I'io. .

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Let X, Y a�d x' be three simi�ar pentagons on side s h, a and b . '.L'hen, if - X = ;Y +:. Z , :h2 � a2 + b2 ;

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GEOMETRIC PROOFS

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239

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Transform pentagon. X,. Y-and Z in-to equiv�lent tri ' s DQO ; RQT· and. �. Then, (by 4th proportional , E'ig . a ) ,. transform s_aid tri "'s _ into equivalent isos.ce­les tri '-s p r·�A, S ' HB and. V 'AH . . ·

· Ther1 proc�ed as in f:tg . ·34o • . :. h.2 . = a-2- + b2 • :. sq . upon·. AB = sq . 1lPOn .�H� + sq . upon. AH ... Q .E . D· .

-.or using .th� ·simil�r tri ' s XBH, Yiffi � a:q.d · ZAH, 'proving tri. XAB. = tri . YHB + tri . ZAH, whence 5. tri. XBA = 5 tri . YHB + 5 tr1 . ZAH; etc .

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240

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THE -PYTHAGOREAN PROPOSITION

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By argument e stablished under fig ' s 340 and 341 , if _regular polygons of any number of sides are const ' d - on the three side s of any rt . triangle , · the

' .

sum of the two les ser � -che greater , whence always h2 = a2 + b2 . -i

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a . Devi sed by the author , Oct . 29, 1933 . · b . In· fig . a , 1 -2 � HB ; 2-3 = TR ; 1-4 = GS ;

4-5 = SS ' ; 1-B = AH ; 6-7 ·= �; 1-8 = MV ; 8-9 = VV ' ; 1-10 = AB ; ll-11 = OQ ; 1 -12 = �D ; 12�13 = P ' P .

.

Fig . 342

·,

!�2-[�n�t��-EQtil�I�t��

. In fig . 342 , produce AH to E making HE = HB, prqduce BH to F making HF = HA, draw ···RB perp . to AB making BK = BA , KD;. par . to AH , and draw EB , KH , KA,, AD and AF • BD . = AB and KD = HB . -

\ Area" of tri . ABK = (�re& of tri . KHB = area of tri . EHB .) ·+ (ar�a of tri . AHK = area-of tri . AHD )

- + (area of ABH = area of ADF ) . · : . . area . of ABK =· area of t:ri . . EHB +, area of

tri . AHF . :: ,, sq . upon AB = sq . upon BH + -sq . upon AIL \:. h2 = a2 + b2 . .

a . See -Edwards ' Geom . , 1895 , p . 158, fig . (20) .

. . �-----,_._--: - -- • f

Fig· . • 343

+ -�ED X DB =

= 2AH X AB -

= BH2 . + AH2 •

. Irr fig . 343 , ta�e AD = AH, 0 ' dr�w ED perp . . to AB , and draw AE . Tri � ABH and BED are ' similar , whence

. ; DE :: AH x BD + HB . . ·But DB = AB . -. - AH . ' - '. . Area of t:r:J,.' . ABH· :;:= �AH x . 'BH' '

AD X ED I ' '> • •

= 2 --- + �ED x DB = AD x ED AH2 (AB - �) + 1. AH (NB � - AH )2

. BH 2 BH 2AR2 + AB2 + AH2 2AH X AB . :. AB2

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GEOMETRIC PROOFS

: .. sq . upon AB = sq . upon. BH + sq . upon AH . :. h2 = a 2 + b2 . ,

a . See Am . Math . Mo . , Vol . VI , 1899 , p . 70, : proof XCV ;

. .

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1 L See proof Five , fig . 5 ,· under I , Algebra;tc Proofs , for an �l�ebra�c proof .

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' . Fig . 344 ·

In .fig . 34.4 , prod�ce .BA to L making -AL = AH, at L draw EL perp . to AB, and pro­duce BH to E . The �r�t s ABH and EBL are similar .

1' ' , , Area of'. tri . ABH = 2AH . 1 -.."'7"'"';>:: X BH =· 2LE X LB - LE X LA _ 1.. AH (AH + AB )2 _· AH2 (AH + AB ), - 2 BH BH

whence AB2 = BH2 .+ AH2 • :. sq .\ upon AB = sq . .

·-·2' ' upon BH + sq . upon AH . :. h = a2· + b2 � .

a . ,se·� .Am . Math . Mo . ,· . Vol . Vt, ·�S99 , p . 70 , prbof XCVI .

b . This and the preced-ing_ proof :are th:e converse of e�ch oth.er . The two _. proofs teacp. tha� ±f two �riangle·s are similar and so related that .the -area of either �riangle may �e ex- ·

pre s sed principally -in terms of the sides of the . · other , then either triangle ·may be taken:·as ·the prin-... A 0 •

cipal triangle , giving, of . cour.se,, �s many solutions as it is possible to expres s the area of e�ther in terms ·of the side s of the other .

In fig . 345·, produce HA and HB and describe ·

the arc of a cirtcle tang . to HX , AB and HY . From 0 , .

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THE Pl'"�rHAGORE��� PROPOSITION

. Fig : 345

i t s center , draw to point s of t�ngency, OG, OE and OD , and draw OH .

Area . . oi' sq ., DG

= r2 = -kab +· (2 BE x r 2 ·

'AE x r) 1 .,.

+ 2 · = 2ab + hr 2 . � . •

But sinc e 2r = h + a + :. r = t (h -t a . + b ) . :. -k (h + a + ' b ) 2 = tab + h (h +. a + b )", whence

b , .

= a2 + b2 . :. sq ! upon AB == sq . upon BH + sq . upon AH .

:. h2 -= a2 + b2 -. a . This proof is original with Prof . � . F . ('I '� • �

Yanney� Woo ster University, 0 . See Am . Math . Mo . , Vol . VI , 1899 , p . 70 , XCVII·.

.-IlQ_[ln�t�d�E�ril:§�Y�n In fig . 346 , let AE

= BH . Since. the: area "of a circ le i s n:r2 , ii +t · can b.e proven ·that the circle who se radius i.s AB =· the-' -circ le whose radius AH + the circle ' . whose radius i s �, � the truth sought is e s tablished .

· · It i s evid�ntJ. if .. . the "� ------. - - - � - _., ... .... . ... t":" f::::J',.....::'., -4 -:__::_ �--�'\- · - ., . triangl-e 1\.BH revolve s· in the

Fig • .. 346

plane of' the paper about A as - · a center� that the area .of the circle generat�d b'y · AB will equal 'the area o� , the · -- . . • I

circ1e generated hy AH plus the area of' ·the · a.n.nulus - . generated by HF .

Hence it must be shown, a:rea of the a�ulus is equal . to [ . . cl� wqose r�dius i s AE .

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if possible , that the the area of" �he cir-

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GEOMETRIC PROOFS -------------------,,

Let AB = h = AF , AH = b , BH = a , AD = -kBH = r , HK -= KF , and AK = mr , whence

.. h -'·, h + b GH :::: h + b 1 AK = 2 = mr ,

b HF = h -· b , HK = .K:B, = 2 Now (GH = h + b ) : (BH = 2r ) = (BH = 2r )

:, (HF ·- h - b ) . - - - (1 ) � ·

and b == Vh2 - 4r2 • h . . . + b 2 w�ence h = vfo2 + 4r2

- Vb2 + 4r2_±.12 - whence b r (m 1 � ) , and h + b

·- 2 . - mr ,

- !1 + l:h2 - -·4;2 -- , 4fi 2 . - · mr ,

-- �

1-:hence

=

h = r (ni - 2

1 . a - b + m > . . . . 2

= r (m + . � ) - r (m - � )

2 r = -- = HK. m Now sinc e (P..D = r )

"I? •

(AK = mr ) = (HK = � ) : _(JI.D = r ) . - - - ( 2 ) ,

. . ••• AD : ... OJC = HF : .AE , or 2nAD ! 2ftAK = HF : AEt

2 2 1h + b ,H� :. 2:nAK x HF = n.AD x AE , or n \ 2 J _ � n.AE x AE .

But 'tn� .a:r•ea of the annulus equals � the sum of the circ umferenc e s where radii a.re h and b times the width of the · annulus or HF .

:. the area of the annulus HF = the · area of the circle where radius i s 'HB .

' :.· tho area of the circle with radius AB = the area of the circle with radius AH + area of the an-nul us .

• ,_, 2 .1:.. • • .LJ. -

11> :·. nh2 = n-a. 2 + ilb2· • • :. sq . upon AB -= sq •

a2 +, b2 • upon BH + �q . upon AH .

a . See Am·. Math . Mo . , ·Vol . I , 1894 ; p � 223 , the pro of by .Ar;drew Ingrahain 1 Pre sident · uf -the Swain Free School , New. Bedfqrd,:.._�a.s s .

b . This proof, like that of pr•oof Two Hundred , . / . . --- -Fifteen , fig . 313 prove s too much, since both equ�- -tioris (l ) and (2 ) imply ' the truth sought . The author , --

--�-· -..__-------·�-------..... \ Professor· Ingraham, d"Qe..s _ not , show his reader s how he r . � � .....

�

determined that HK = m 1 henc e . t�e implicati.on i s ' \ ·:;a 2 ' ( I 2 2 ) hidden; in (1 ; we have directly h · - b " = 4r = a •

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�44 THE PYTHAGOREAN PROPOSITION .. ..._ .,. _ , ,. - -· - - - · . "' - �

j, ' • ------

... � .

Having begged the que stion in both equations., (1 ) and (2 ) , Profes sor Ingrah_am has , no doubt , un­consciously, fallen under the ·formal fallacy of D e t t t i_o D r i n c t p i i_. 1 - �

1 ' c . From the :Pr·ece�ing qrray of proofs it is � evident that the algeb�aic a�d geometTic proofs of

· thi s mo st imp·ortant truth a:re as uniimi ted in m:unber . - .. as are the ingeniou� reso�rce s and ideas of the mathe� matical 1-nYe�tigator �

NO TRIGONOMETRIC PROOFS --r·r· --,

Facing forwa�d the thoughtful reader �ay rai se the. que stion : Are there any proofs based uporC the science of trigonometry or analy.tical geometr<y?

There are no trigonometric proofs , because : all the fundamental formulae at' trigonomet-ry- are them- ·, _ ' selves· ba�ed upon the truth of the P�hagu;ean _Theo-rem; b�cause of thi s theorem we say sin2A- + cos2.A = 1 , · etc . Triginomet'ry i s-, because the Pythagorean ' Theorem i' s . - . ,, 1 Therefore the so-s tyled Trigonometric Proof/' given by J . Versluys , in his. Book, Zes . en Negentig Bewi j zen, 1914 (� collecti·or{ ·of 96 proofs ), ·p:--·9·1r; ·

prq_of 95 , is · notf a p�oof since it ·employs the formula i 2 2A )1 s n A + co s = • . . \ :t

·� As De sc�rte� made t�e Pythagorean theorem the basis qf �is method of analytical geo�etry, no !nde ­pendent proof can here appear . Analytical �Geometry ·

is Euclidian Geometry treated algebraically and; hence involve s all principle s already establi shed .

Therefore in analytical geometry all rela­tions concerning - the sides of a right-angled triangle imply or re st directly upon the -Pythagorean· theorem as is shown in. the 'eq.uation, viz . , x2 + y2 = r2 : ·

And The Calculus being but an algebraic in­vestigation of geometric var�aples by the method of limit s it accept � th� truth of geo�etry . as e s tab­lished, and . theref.ore furn�shes .no new proof , other than nhat , if square s be constructed upon the three

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1596- 16�50

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GEOMETRIC PROOFS 245

side s of a vat•iable oblique triangle , as any angle of .... the three approachei a right angle the square on the . I side opposite<- approaches in area t:tle S tllll of the I I . ' I squares . up.on.- -the other two side s .

\ But not · so with quaternions , or vector p.naly­sis . It is a mathematical science which introduce s a new �concept !·not employed in any· �of the mathematical

I , . sciences ment�oned heretofo�e , --the cqncept ,of direc-tio:rL -

,. . And QY means of t�i s new concept the complex

.demonstrations of old truths are wonderfully simpli·­fied, · or new ways of .reaching the same truth are de­veloped .

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-·� , t - .

r .

' \ I I I I J I

III . QUATERNIONIC PROOFS

We here give four .quaternionic proofs of the I �ythagorean ?reposition . Other prpofs are possible .

. j ·

Fig . 347 ;

In fi,g . 347 de signate the . side � as to distance an4 direc tion by a., b and g (in place of the Greek al­pha a., oeta � and gamma y ) . Nmr; by the principle of directipn, a · = b + g; ·

also since the angle at H is a right angl� , 2sbg = 0 (s signif;Les Scalar .

'-See Hardy, 1�81 , p . 1 6 ) . . ( I ( 2 ( 2 . / 2 2 l ) ,1a + b = _ g 1 ) = 2 ). a . = ·/b + 2sbg +. g .

1 (2 ).· reduce¢ = (3 ) • :, a2 = b2 + g2 ' CQhSidered aS , j lengths . i'. sq . upon AB = sq . upon BH + sq•·: upqn AH . I :. h2 = a2 + b2 • Q . E . D . / .

a . 'See Hardy ' s Element � of Quaternion� , 1881 , a:rt . 84, ;I. ; al s_o_Jour . of Eduqation, Vol . · 1888, p . 327 , .Twenty-Sec ond Proof ; Versluys , 'fig . 108 .

p .� 82, XXVII . . ' p . 95 r

..

Fig . 348

In fig . 348, extend BH to C 1 ma:king HC � HB and draw AC � As vec­

tors AB = AH + BB , or A : B + G (1 ) . .. ) Al so AC = AH + HC , or A = B - G (2 ) .

( 1 1 Squaring (1 ) and 2 )- a�d J .a<;lding, . we have A 2 + A 2 = 2]3_�-�--:2G2 • Or as lengths , AB2 + AC2 = 2AH2 + 2AB2 • But AB = AC .

. :. AB2 ::; .AJ-i·2 + HB2 . 0 .

· · .:. sq . upon AB = sq . vpon AH � + sq . upon· HB . . h2 '• . = a2 + h2 .. . �

246

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,.

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. r" ''' ' I

� ��- -r f f

.. \

QUATERNIONIC PROOFS 247

a . This is Jame s A . Calde�head ' s s olution . See Am . Math . Mo . , Vol . VI , 1899, p : 71 , proof xcrx;

1 r Th r e e -.a � - - - c !' -· \ In fig . 3 49., complete 1 -che rect.

·Fig . 349

HC and draw . . HC-.- As vectors ,AB = AH + HB , or a = b + g (1 ). HG = HA + AC , , ;'or a = -b :r 'S ( 2 ) • ' . 0

. · Squaring (1 ) and (2 ) and add-ing, give s A2 + A 1 2 = 2B2 + 2G2 • Or consider�d as line s , AB2 + HC2 = aAH2

2 .

+ 2HB . But HC = AB •· · AB

. 2 - AH2 + HB2::: _ _ It . - • - - "" . .

2 . , 2 - .:. sq . upon AB = sq . upon AH +, sq . upon HB • -_.:=<-h2 = a 2 + _.b2 .

. �

� a . Another .of eJame s A . Calderhead ' s . solutions.

See Am . Math . Mo . , Vol . VI ,. 1$99, · p . 71 , _ proof C ; -�ersluys , p . 95 , fig . 108 .

. - �

I

. 1l . . �- - -- :-: .. .. .. �

� ,._.._; � ,, _.. .. ' ,._ /1'- 1 " ! • , ' c I � , -!, - � �- ' , \ ' . , ..

.,

I '\. -- . . ;--\ . . \ ' ,_ \

' ·fi. 1 ,. ,,, il'

_ r \ ,;, I 1 ' F ,., I . \ -.. . � I I ' . ,. I I ,....,., . I \ � 4 � , . ( ' I / \ , , £' ' �' I ,- ',J . .. , 1�. . .. . ' ...... � .,. - -- - --

Fig . 350

:. sq . - upon AB = · .-. · h2 = a2 + b2 .

.

E2Y.r. In fig . 350 , the con-

I I I

struction i s evident , as an-­�Ie · GAK _= -angle' BAK . The ra­dius being unity, LG and LB are sines of GAK and BAK .

As vectors , AB = AH + HB; or a = b + g ·(1 ) . Also · AG = AF �+ FG or a 1 = -b + g (2 ) . Squaring (1 ) and (2 ) ·

' 2 � , 2 ·and adding gives ;a + a = 2b2 + 2g2 . Or- � e·onsidering the vectors as distance s , AB2 + AG2 = 2AH2 + 2HB2 , or A8� '

2 . '2 = A'f:I . + HB . sq . �pon · AH + sq � upon BH .

a . Qr4g1nal with the author , August , 1900 . 0

b·. Other solutions. from· --the--trigonometric right line. function figure :(see Schuyler ' s T�ig_onome­try, 1873 , p . 78, art 85 ) are easily .devi sed through vector analysis .

. • I J ' J I

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I

t--�,..,._·� i> . -·

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_ ..

..

·�

........." ·-"� . '

\

.. ·-

IV . DYNAMIC PROOFS

. The Science of Dynamic s , since 1910, is a

c�a1mant for . a place as to a few proofs of the Pytha-... gorean Theorem .

A dynamic proof employing the principle of moment Qf a couple appears as proof 96 , on p . 95� in J . Versiuys ' (1914 ) c ollection of proofs .

•..

..

It is as follows : 1 � - '

0

In compliance with the theor,y of the· moment of couple, · .

- � - - - �-�-- -

. A , \ � . �/�' . in mechanisc { ( see "Mechanip.s

"-= .. ,. for- Beginne:r(3 , Part I , " 1891 , : ' , '�6.. �· � � ,-ft� � � by Rev . · J . B . Locke , p . 105 ) ,

..f,\ - -r the momen� of the sum �� two

rr.r � _ . conjoined coupl·e s in ·the same . '...._ ·: . ·� � Ci _,� . . flat plane i s the same as the � Q'\�( 1 1 \.f, . sum of th'e moment s of�-tllEf�tvo � �t-- � \ \ ' J ' .

. 1 , ·� � couple s , from which it follows •

. ' 'f > that h2 · = a 2 + b 2 •

C L _ _ .:.. _ _ ,.:1("' -;.> If F� and AG repre sent

Fig • . 351 two equal �ower � they form a

" couple wpereoT the moment _equal s FH 5< AH , or b2 •

If HE and DB repre sent two other equal . powers they form a couple whereof the moment equals DB X HB or a2 •

J To find the moment of the two c ouple s · j oin the t o powers AG and HE , also the tw� powe� s DB and FH . To join the power � AG and � ' tak� AM = HE . The diagonal AN of the parallelogram of the two powers AG and AM is equal to CA . To join the power s FH . and DB , ta-�e )3.0' = DB . The diagonaf. BK of the parallelogram of · the. two power·s (FH = �p ) 'and BO., i s · the second

' . '

0

24 8

· I . 0

. · . ' \ � ... - . . \ .

--..- -. ..,

• ,

• •

i r··

•:'

DYNAMIC PROOFS -

I I I

249 .

component Qf the re sulte.�1t couple who se moment i s CH x BK, or h? . Thus we have h2 = a2 + b2 • _

a . See J . Versluys , p . 95 , fig . 108 . He (Versluys ) ·says : I found th.e above proof in. 1877 , bY. considering the method of the , theo�y of the principle of mechanic s and to. the �resent (1914 ) I have never met· with a lik� -proof anpwhere . .

_ · In Seience , New �eries , Oct . 7 , 1910 , Vol . 3 2, pp . 863-4 , Profe ssor EdJin F . Northrup , Paimer Bhysi­cal Laboratory, ·Princeton, N . Jo . ,. through equilibrium of force.s , e stabli she s the formula h2 = a2 + b2 • ·rn. Yol . J3 , p . 457 , Mr·. Mayo D .: Hersey-? of

- the U . S . Bureau of Standards , Washington, D . c . ; says that , if we admit Pr-ofe s sor Northrup. ' s proof , then the same resul� may be e stablished by a much simpler I cour se of reasoning based on c ertain simple dynamic laws .

Then in Vol! 34 , pp . - 181�2 , Mr . Alexander Mac - · -Farlane , of Chatham, Ontario , Canada , come s to the

support of Professor Northrup , and then give s. two .·

very fine dyn�ic proofs througn the use of trigono­� -metr:i.c functions and quaternionic ( laws .

• Having obtained p·ermis sion from the editor of Science , Mr . J. McK . Cattell , on February ·18, 1926 , to make use of these c' proofs found in s_aid volumes ·32 , 3 3 ·and 34, of : · Science , they now fo],low . f':;::-r .

' .

In fig . 352 , o·-p is a -J;'Q·d, ·"'. \ . _.t without mas s which can be · revolv.eCi ·.

,I I ;

Fig-?52

in the plane of the paper about · O as a center . 1 -2 i s another such rod in the plane of the paper .of . which p is - it s ni:J_ddle point . Con­centrated at each end of the rod 1 -2 are equal mas se s m and m 1 each�di stant r from p .

Let R equal the 0 -p , X = 0-1 , y = 0-2 .

"

distance When the · / l

..

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_ __ c__ l

.·

J} . , : . > l

r '· . .

� �

\

--'

l l. · ---

� -'

..

.. . ?50 THE PYTHAGOREAN PROPOSITION . I

. i I •

system revolve s about 0 as ' a center , the point p will have a linear. ve.locity� r. = ds/dt = da/dt = RW, where ds is the element of the _arc described ·in time dt , da ia the differ.ential angle thr·ough which 0-tP turns , and W is �he angular velocity.

I .. � I · 1 . · As sume -the rod 1 -2--'free to turn ·on(, ...... a:·s a

center . . . -A<;;,ince m at 1 and m 1 · at 2 are equal and _equal-ly di stant from p , p i s the center of mas s . Under these conditions E 1 ·= � (2m )V2 = m:R2w2 • - - - · (1 )

2 . D o�ceive rod, 1-2 , to become rigorously . attached at p > · Then as 0 -p rev:olve s about 0 with an­gular velocity _W, 1-2 al so revblye s about p :with like angular .Yelocity. By .m.aking attacJ:unent a� p rigid

·

the s·ystem i s forced �o-·take _dn an additional kinetic energy, which can be �nly that , which is a re sult of the additional motion n�w pos se s sed _- �y in at 1 -and by m1 at 2, in virtu� of their rotation abo�t p as a center . Thi s added kinetic energy 1 s - E " = . � (2m )r2W2 = · mr2W2 • - - - (2 .) Hence total kinetic energy i s E-=-.. E·•- ·

+ E" = mW2 (R2 + r2 ) . -�. - (3 ) .

3 . ; Wi.th the attachment still - rigid at p , the kinetic energy of m at 1 i s , piainly, Eb d �mx2W2 • - - - '(4 ) Lik<?wi se E� = .!my2W2 • - - - (5 )

:. the total kinet'ic Etnengy must. pe E = Eb + · E.10r 1 2 ( 2 2 ) '(6 ) ,_ = 2mW X + y • - - - ,

:. (3 ) = (6') , or � (x2 + y2 ) = R2 + r2 • - - - (7 ) .. In (7 ) �e have a geometric re�ation of s ome ;I intere st , but in· a p�rticular . case when x = y, that

i s , when line 1-2 is perpendicular t-o line 0 -p , we have as a re sult x2 = R2 + �2 ; - - - (8 )

:. sq •. . upon hypotenuse = sum of square s upon the two legs of a . right tri�ngle .

Th�n in Vol . 33 , · p . 457, on March 24, 1911 , Mr . Mayo D ; Hersey says : · "while Mr . R . F . Deimal

· holds ···that · equation (7 ) above expre s se s a geometric _ fact--I am tempte4 to say 1 accident 1 -...:.w!l-ich tex.tbooks rai se to '· the dignity of a theorem . " He further says : "Why not let it be a simple one ? For instance , if ·the force � .whose rectangular compone�t s are x and y, . act s upon a parti�le of mas s m until that v2 'must be

' )

. · 1 , __ . ,., ' . . . ;r l

-I f

. I

(J

'

{ I DYNAMLC PROOFS

I '

I . I �5l

- posl ti ve ;, c onsequen,t).y, to hold· that tJ:le square of a simple vector i s negative i s to c ontradict th� e stab - _ li:shed� c'Onventfons of, ma:�hematicak analysis . . - - The qu�ternionist· t:r1.es.- td get out by saying · . 1 that after . all v i s fiot a velocity having direction, ·

but merely a speed . To this I r.epJ,.y that E = ·c o s Jmvdv = imv2 , and that "the se expresai0n� v and dv are both -vectors- having directions which are different ..

: · -Recently (in the Bulletin of the Quaterniom. . Association ) I have been considering what may he ' '

c&lled •the �eneral�zation of the Pytha� gorean Theorem . ·

� ;�.; �et A , t B�- C_, D ., etc . ; fig . 353-,

denote vectors having any direction in � it space ; aJld 1et- .�R de not� the v·ector from .

the origin of A to the terminal of th� last vector ; then the generalization of ­the· p·;T-� ± s · R2- =- A2 + B2 .JI + o:: + D_2 + .2· (c�� AB + cos AC; + co s· AD ) + 1 2 (co s BC

Fig . 353 + cos BD ) + 2 (co.s CD ) t e-tc . i where co s ·

AB deno tes the rectangle formed by A and; """-. " -the pro Jection of .E•_ parallel, :to (A . The theorem of- -·P ,·

i s linrl.ted to two ve-ctors A and B which are at right ' angle s to one : another , giving R2 = A2 + B2 • The ex-

. tension given i.n Euclid remove s the conditi�.ri of per­pendicularity, ·giving _ R2 = A2 + B2 + cos �-,=�

·

Space geometry give s _ R2 = A2 + B2 + t2 when A , B, 0 are othogon�f , �nd-�2 · = A2 + B2 + p2· + 2 co s

----�---�� 2 co s .AC + 2 cos BC when that condition- is re-moved . \

Further , space-algebra ·give s a complementary , v .,._ . -� - ....

theorem, nev�r dreamed of by either -Pythagoras or Euclid .

Let V denote in magnitude .. and direction the resultant of the �irected areas enc losed between the broken line s A + B + C + D and the resultant liQe R , and let sin AB denote in direction and magnitude the area enclosed between A and th� pro jection of B which i s perpendicular to A ; then the complementary theorem i s 4V = 2 ( sin AB + sin AC + sin AD + ) + 2 (sin BC + sin BD + ) + 2 (sin CD + ) + etc . . .

' '

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.. . � 252 _ . THE --F:ywHAGOREAli PROPO:JITIOU �----------�·--�---------�---�-----------��-·��·--·----

from ·a

· THg :PYTHAGOREAN 0-TJRIOSITY

The fcJ. lo�:ing · i�· 1·eport.ed to ha�,.re 'been taken

notebook of Ml· . Jo�l.'1 Watex•hou s e , en E:nginflG.r' of fLY��·'C1ty. I.t ' 1f-l app�ar-ed 1n print.{

I JJ in a-·1! . Y . · pa,pe:r· , . in July, 1899 . _ _ - --·

I Upon the side s of ; /_ I· ._:.

_ the !'ight tri5- ,

' angle , fig . 3 4 1 c onstruc t the square s AI , B!IT , �nd CE . C onne c t the pc;-1nts E and,.-· -: �H·· H, I er1d M 7 c<.ncl N and 11 . Upon therie line s - c on - ..

. :B.tpuct the squares EG, MJC and NP ,

and c onnect the · point s· P -an� F ;· -a---and ' K -� and L and o . The following

truths a:re demon­atre.bl.e .

• I ' . 1 . Square

BN = saua.re CE ]'iu , 354 • o + s quare AI . (Eu- ·

c lid ) 2 ,

2:. Triangle HAE = triangle IBM = triangle DCN = triangle CAB , sinc e HA = B I- and EA = MY, EA = DG and IfA = NZ , and HA : BA and EA = CA � ,

3 . Line s · HI and GK ·Bore paXlallel f. __ for , · since '

a.ngle GHI = angle _ IBM 1 · .

·: triapgle HGI = tri.angl9 BMI , , w·:q�nc e IG = IIvl = IK .__ Again ext end HJ; 't o H 1 making ·· IH 1 = IH , and dra"-�" H 1 K, '·rh ence triangle IHG � triangl e

IH ' I{, ea.Qh hav-ing two side s and the inc lu.ded angle ( re'spe c t;tvely equal . · .'.

,the di s t anc e s from G and K t o

'· the line HR 1 ar>e equal . - : . . the l.:tn� s- HI and GK ar e paral lel . In l ike manner it ma.y be shQW:fl that DE and

PF, al s o MN and LO , are :pe.l.,all e l . ..

_ I

' , ,

�--

II

• DYNAMIC PROOFS 253

4 . . GK = 4HI , for H-I-. = 'rU · = GT = .UV = VK (since VK is homologous to BI in the equal triangle s

VKI and BIM ) . · In +ike manner i t can "Q.e shown that 1 PF =' 4DE'. That LO = 4MN · i s proven as follows : tri­angle s LWM. and IVK are equal ; therefore the . homol.o­gous sides WM and VK are equal . Likewise OX and QD are equal each being equal to MN . Now in tri . WJX, MJ and XN � NJf therefore M- and N are· the middle

_ point s of WJ and XJ; theref.ore WX = /2MN; therefore LO = 4MN • . .

\ 5 . The thre� trap�zoids HIGK, DEPF and MNLO are each equal to 5 t�mes the t�iangle 'CAB . The 5 triangles compostng. the trapezoid HIGK &re each equal ----·

to the .triangle CAB , · each having the same bas� and · __ , . ____ . __ . . _ _ _ .?-_+t±�Y.Q..e as i;;r���le C.f\13 . J;n· ���7 ·.11l-�QPer . . ,� t may l?e

shown that tb.� ··:ttr@.pez.oi'd DEPF�--so �l so th�':"trapezold MNLO, equals 5 times .the . triangle CAB .

6 . The square MK + the _ square NP = 5 time s the square EG or BN . For the square on MI =. the

- - - --�- - --

·--· ----�--s-qu�e-on-MY-+-the--s-que;re-orrY-I-+--(2AB-).a_+_.A:e.a...·= 4XB2.----�-- · - �---�--�-�---'

�: . "

+ AC2 ; and the �quare on ND + the square on �Z + the s�uare ZD = AB2 + (2AC ) 2 = AB2 + 4AC2 • Th&refore the square MK + the square NP = 5AB2 + 5AC2 = · 5 (AB2 + AC2 ) = · 5BC2 = 5 times the square BN .

<? 7 . The bise�'tor· .of the angle ·A ' passes through the· ver�ex �; for .A.'S = NT . . But the bisector '" of th� angle B ' or C ' , does· not pass through the vertex B , or C-. Otherwise - .Bu· would equai Bti1 , whence NU" + U"M would equal NM :f. U"M ' ; that i s , th11 - sum of the t1{o

' le:gs C?f' � right triang:J..e wo�ld equal the hypotenuse + the perpendicular l!Pon . . tlie hypotenuse fr.om the right

. . a��e . But •. t�is :ts im�o¥sible ; Therefore the bise_c­tor of the angle B ' doey not pass through the vertex B • ! � . 8 .. The square on LO =,. the sum of the square� .on PF and GK; for LO ·: PF : GK = BC : CA : AB , ·

9 . Etc . , etc . • :. See Case'y ' s Sequel to Euclid, 190q , Part I ,

p . 16 .

I

;• . , l ' ·... • : �

i

' -

I . I ·i

I

. I

, .

• I

PYTHAGOREAN MAGIC SQUARES

The sum- of any row, column ·or diagonal of the square AK is 125 ; hence the o sum of all the number s ·in the square i s 625 . The sum -of any row, c olumh or diagonal of square GH is 46 , and of HD i s 1 4 T; hence ·the sum of all the numbers in the square GH is 184':, and in · the· square HD is 441 . The.refore the magic square AK (625 ) = the magic

:F'ig . 355 square HD (44'1 f + the magic --------� __ sq.uane....HG-.:{-1&4-)-. -------------------::11

Formulated by the author , July, 1900 .

I Fig . 356

!tt:Q -" The square AK is com­

posed of 3 magic square s , 52 , 152 an� 252 • The square HD is . a magic sqtia�e each number of which is a square· . The square - . HG is a magic sqrare formed from the first lp numbers . Fur-

• <

· thermore , observe that the sum of the nine squ�re numb�rs in the square HD equal s 482 or 2304 , R square number .

Formulated by the , author , July, 1900 .

254

, _

PYTHAGOREAN MA�IC SQUARES 255

The sum of all the num� bers (AK = 325 ) = the sum of

. all the numbers ln square (HD = 18g ) + the sum of all the numbers in square (HG + 136 ) .

Square AK is ·made up of 13 , 3 x (3 x 13 ) , and 5 x . (5 x 13 ) ; square HD is made up of 21 , 3 x (3 x 21 ) , and square HG is made up of 4 x 34 - each row, c olumn �nd dfag-

\ anal , and the sum of the four Fig . 357 inner numb�rs •.

Many o.ther magic squares of this type giving 325 , 189 and 136 for the sums of AK, HD and HG respectively may be formed .

· - ·

Thi s one was formed by Prof . Paul A . Towne ,

Fig . 358

type may be formed .­my own of thi s type .

E.�Y.r:. The sum of numbers in

sq . (AK = 625 ) = the sum of· numbers in sq . (HD = 441 ) + the sum·-·of number s in sq . - (HG = 184 ) .

Sq . AK give s 1 x (1 X 25 ) ; · 3 X (3 X 25 ) ; and 5 x (5 x 25 ) , a s element s ; sq . HD

- . giVeS 1 � (1 X 49 ) ·j 3 X (3 X 49) as element s ; and sq . HG gives 1 � 46 and 3 x 46 , as elem�nts .

. Thi s one also was . . form.e9J- . ' by Profe ssor Towne , of We st l Edme stqn, N . Y . . Mapy of this

See fig . 355 , aboie , for one of

. - !--·· --�-'-

<>

. ---- - . - - .. - -, j

. ! \ I I I

. ·

..

. .

..

..

I ...

256 THE PYTHAGOREAN PROPOSITION

Al so see Mathematical Es says and Recreations , ----- --by�He-rman-'-Schubert,--1:n-The--0p·err-cour·rPnbTisning--co . , -­

Chicago , 1898, _ p . 39, for ·'an extended theory of The Magic Square .

1 ElY�

Ob serve the following ser�es : The sum of the inner 4 n�bers is 12 x 202 ;

of· the 16-square, 2 2 x 202 ; of the· 36-square , 32 X · 202 ; of the 64- square , 42 x 202 ; and of xhe 100-square , 52 x �02 •

\

· Fig . 359

/ 0

1 1 0n the - hypotenuse · and legs of the right­angled triangle , · ESL, · a�e bon&tr�cted the . c encentric magic-fqua:t .. es of . lOO, 64, 36 an.d 16 . Tne sum or the two numbers at the extremities qf the diagonals , and

.,. .

- - z - --- - - • . ,

' • · . -� .. · .. · . . .:.. · " . ·,�

lj

I

I ' I . I I

PYTHAGOREAN MAGIC SQUARES · 257 :--- ------------------- --- ----� -

--�------�- ---�--------- ---- ------ --------·---·--+.11

\

1------ -

I , of all line s , horizontal and diago�al , and of th� two numbers equally dl"stant from the e4tremi tie s , i s 101 . . The _sum of the numbe�s in th�· diagon�ls and line� of each of the four concentric magtc square s _ i s 101 mul­·tiplied by half the number of cells in boundary line s; that i s , the summations a:re 101 x 2·; _ 101 x 3-; 101 x 4; 101 x 5 . The sum of the 4 central numbers is 101 x 2.

:. the sum of the numbers. ip the sqvare (SO

= 505 x 10 = 5050 ) � the slli� of the . numbers in the square. (EM = 303 x 6 = 1818 ) + the sum of the h�oers in .the square (EI' = ·464 x 8 = .3232 ) . 5052 . = 39)2 � + 4042 • .

. �-

Notice that in the above diagram the concen­tric magi_c squares on the legs i s identfcal with the central c9nc·entric magic square s on · the hypotenuse . " Profe s sor P�u� A . Towhe , �est Edmeston, N . Y .

II An -indefinite .number of magic squares of this type are readily formed .

. : I

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.,,

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--- - -----

. .

L-���---��-----· · ----�--- ·-- ':>---- ----· ----------��------- -------------�

r.

The follQwing proo..fs have c·ome to me sinye June 23 , 1939 , the day on which I

. � . \ finished page 257 of thi s 2nd edition.

' ' ' I�2-[[ng���-E�ril��l��i

" .

In fig . 360� extend HA to P �aking AP = HB , .and through P draw PQ par . to HB, making- CQ = ijB·; extend GA t o 0 , me:king AO �:: AG; draw FE., GE, GD ,. GB·, co·, ·QK, HC and BQ .

· Since , obvious , tri . tQQ = tri . ABC = tri . FEH, and since area of tri . BDG � �BD x FB� then area of quad . GBDE = BD x (FB :;:: HP ) = . area ·of paral . BHCQ . = sq . ·HE"--+ 2 tri . BHG,o then it follows<'­that : ·t·::::Y_,

Fig . 36o Sq . - AK = hexagon ACQKBH - 2 . tri . ABH = (tri . ACH = tr: L GAB ) + (paral ·. BHCQ

= sq ._ B� + 2 tri . BHG ) + (tri .. QKB = tri . GFE ) = hex-agon GABDEF -. 2 tri . ABH = sq . AF + /sq . aE . There� . - 2 fore sq . upon :AB = sq . upon HB + sq . upon HA . · :. h = a2 + b2· • Q . E . D . __

. -:..

· ·a . Deviaed, demonstrated with geometric , rea­sqn for each step , and submitted to me June, 29, 1939 . Approved and h�r_e_r_e_c_o.r..ded July 2 , 1939 , after ms . for - 2nd - edition was pompleted .

b . Its place , as to type and figure , is next . • I · �fter Proof Slxty-Nine , p . 14 J: , of thi s edition .

· 9 . Th�s proof is an Original , hi s No . V�I , by Joseph Zelson, of We st Phlla . '· High School , · Ph;tla . , Pa·.

• !

' ·- ..,--.,

258 .

. ·

I ..

I \

ADDENDA • 259

f--.--- ----- --- ----- ----------- -i-w'f\-H·••·n·d·re-d--F-o-rt'f.- H-i-n·e - - � �- � - - - - - - - - - - - - - - -� ---- - - --·---

- I

()• •

Fig . 361

It i s easily proven that : tri . GFY = tri . FHE = tr·i . ABH = tri . CMA = tri . AR9 = tri . CWK; also that tri . GAE -=-·t-ri-;-· CMH; tri . LGE: = tri . CXH; tri . _ FYL =· tri . EDN = tri . WKY; tri·.� ... GFY = tri . GFL + tri_ . NED ; that paral . BHWK = sq . HD . Then it follows that sq .. AK = pentagon. MCKBH - 29 tri . ABH = (tri . MCH = tri . · GAE ) + (tri . CXH = tri ! LGE ) -+ ( (q-uad . BHXK = pent • HBDNE ) = sq . BE + (tri . EDN = trt . WKX ) ] = hexagon GAHBDNL - 2 tri . ABH = sq . AF + sq . aE .

, :. sq .. ___ upon. AB. = sq . upon HB + sq . upon HA .

'• '

:. h2 a2 =t= b2 . . --- ---------�---------------�

a . Thi � .�r96f, . �ith figure , devised by Mas---- ' ' t ter Joseph Zels6n and submi·tted' Jilhe '29·, 1939 , and here ' recorded July 2 , l939 .

b . Its place· i s next ·after No . 247 , on p . 185 above .

Fig . 362

. . .

In· fig . 362 , draw GD . At A and B erect perp 1 s A9. and BK to. AB . · T:b..rough L and

·-0 draw FM, and EN = FM = AB . � Extend DE to K .

I t i s obvious that : , quaq . GMLC = quad . OBDE ; quad . OBDE + (tri . LMA = tri . OKE ) � tri . ABH; tri . BDK .

= tri . EDN = tr� . ABH = tri . EFH = tri . MFG � tri . CkG . �

,,

,; '

... ·•

\ 260 THE PY,THAGOREAN , PROPOSITIO�. ·

\ �------- - - --- -------

Then . . i.t��t-ol�ow�h�t : ----�

sq .-

_GH-+ sq . HD = he�-

agon GABDEF - 2 tri . ABH

= [ trap ,, FLOE = (FL + OE = FM =;/B ) x (FE = �)l [ ·= lu + BO = BA = · AB). X (AB ) ] · 2 + trap_.. -LABO 'e - - - = AB · 2 '

= sq. on Aa . :. sq . upon AB = sq . upon HB + sq . upon· HA .

:. h 2 = a 2 +. b? . a � Type J, -·cas-e (1 ) .1 (a.b�._.,...,So it s place is

next after proof Two Hundred '·tUne , p . 218 . · b ·. Proof and fig . devised .by Joseph Zelson .

Sent to me July 13, 1939 .

1<1 Construct tri . KGF •L--------��------------------�����,�------------------:�tp{.-A&H;-���-FE-to-b----------------•�

"Lr._ .... \ f: and· 0 , the poin:t at w�ich &

. ' I

. \

f)

;' ;;Itt, .... . - . perp ; from :0 intersects FE G lt'_ ' , ' '\ M and N where .perp 1 s from G

1 / .,; \ -'":'-�� exten�ed; al s_o extend AB to

(\'�---- ' t , . .- _ � - - ' and D �iil intersect AB ex-t \ -- - - ,, . . tended, draw GD . LM :a-:,� . :- By showing that : ·

tri . KLF -= tri . DOE = tri . Fig . 363 DNB; t:ri . ;FLG ::: tri . · AMG;

'\. tri . KGF = 'tri . EFH = tri . ; ABH ; then it follows that : sq . GH + sq . HD = hexagqn

LGMNDO - 4 tri . ABH

= [ tr.ip . LGDO = (LG + DO = KG = AB ) X :r = AB l +(2 X alt . FL�_ + [trap . GMNU,= (GM + ND = AB ) x (AB

,) +�2 x alt . {AM = FL} l]

. 2AB2 - 4 tri . ABH = - 4 tri . ABH � AB2 4 tri . ABH . 2 .

• h2 . . :. sq . upon AB = sq . upon iiB + sq . upon HA .

= a2 + b2 • Q .E .D •

- ,•

261 ADDEN.QA ----�- ---- ·-- ----- - -===:::--:=-===:::::-:-:��:;::. =· =-==::t--==-- =- ==-======::-:::::==:=::-::-::::--:::-::::. ___ ------- ----- - -- ----- --- -

� '

-. ' .. .

a • . Type J . Case (1 ) , . (� )- . So its place i s next after Pr�of Two Hundred .Fifty-One .

· ' b . This proof and -fig . also devi sed by Master Joseph- Zelson, a l�d with a superior intellect . Sent

:· -- . : t o me _.Jtily 13 , 1939 . . •

1 - - - -By dis section, as per

figure , and the numbering of corresponding_ parts· by same numera� , it follows, through

1 ' 7f�·.u f. superposition of co-ngruent ,., I , a 'l � ' • \� / ' ' part s (the most obvious proof) � : h t',·��• t'� •' ) that the sum of the fo'ur ,'\ I · .,." 1 ' , e .., � part s (2 tpi 1 s and 2 · quad ' ls ) � I " t iY

--.. \ _ , ,

. C{�- _ _ _ _ - �if\ in the �q . AK = the sum of .___________________ ·-----------------------�he-�hree-pa��--���i�and-------

Fig . 364 -_ 1 �uad � ) in the sq . PG +- the sum of the two part s (1 t-ri .

and 1 quad . ) in the sq i PD . That is the area of the· sum Cbf the p�rts 1 + 2

+ 3 + 4 in �q_ . · AK (on the hypotenuse AB f = the area of the sum of the part s , 1 1 + 2 ' + ·6 in the sq . PG'

. (on. the line : GF = line AH ) + the area of the sum of the part s 3 ' · + 4 ' �n the sq. PD (on the line PK = line HB ) , ob serving that par� 4 + (6' � 5 ) = part 4 ' .

a . Type I , Case (6 ) , (a ) . So it s place be- "' longs next ·after fig . 305, page 215 .

· b . Thi s figure and proof was devised by the author on March 9 , 1940 , 7 :30 p . m .

<) In "Mathematics for the Million, " (1937 )� by

Lancelot Hogben, F . R . S . , from p . 63 , was · taken the f >llowing photostat . The exhibi't i s . ·a proof which i·s credited. · to an early ,{before 500 B . C . ) Chinese mathe­matician . See also David Eugene Smith ' s Hi story of Mathematic s , Vol . I , p . 30 ,o I

v .

. ' -

,•

�·

262 THE PYTHAGOREAN. PROPOS�TION

·----------------- ··--·--- - ---- --�---- --

, '>

l •

·.

longs April

\

\ 1\ FIG. 1 9

The Boolc of Cho;p;;s,;:;;,--}(:;ng, probably written about.··A.D. .CO, is attributed by oral tradition to a .source befl•re the Greek geometer taught what we·call the Theorem of Pythagoras� i .e. that the !>quare on the longest s ide of a right-angled triangle is equi\·alerlt to the sum of the squares un·'the other two. This \'er>· early dcarnple of·block printing from an ancient edition of the Chou Pd, �s gi\•en' in Smith's Histciry of Matltemati.-s, demon�tratcs the truth of the theorem. By joining to any !right-angle" triangle like the blac}t figure . iBf. th_!ee o!h�r right-angled triangles just like it, a square can be .formed. :1\exl trace four oblongs (rectangleffTike eafB;-each of which is made up Q.f. two triangles like c/8. When you ljavc read Chapter 4 you will be able to put together the"'Chinese puzzle, which is much less puzzling than Euclid. These are the steps : I

Trian�le-e]B =· i reft�ngle eaf8 = i 8fc . eB-Square A8CD = Sq\lare t/::h + 4,'times triangle ef8

"· tf:. � :!Bf . e8 - -- - - ---------- Also Square A8CD ,.. Bf2 + e82 + 28f . t8 So ef2 + 2llf . eB . .- 8]2' ·! c8i + 28f . t8 \ Hence . tf2 = Bj2 + e8: / /

. _//I \ \

a . This believe-it-or-not after proof Ninety, p . 154,

" Chine se Pr�of" be­thi s book . (E . S . • L .,

9 ' 1940 ) .

\

.'

r -r \ I

�

. � ' ! ..

�

I ' d f i [· ' ' I -' �

-{'

!_ ' . -...

&

GEOMETRIC PROOFS 263 ·.""-.

!�2-tt[n�(��-Elfi��E�[(

· Fig . 365

-----------------------In the figure· ext·�·nd

GF. and ·DE to M, and AB and ED to L, and ·,number the part!? a-s app�ars in the quad . ALMG .

It is easily shown that : �ABH = 6ACG, "tBKN = �KBL and .�CNF = D.KOE ; whence OAK = (�ABH = �ACG -i'n sq , HG ) + quad . �C com. " to 0 1 s AK and HG + (�BKN -= ��L) = (�BLD + quad . BDEO

= .sq . HD ) +. (llOEK = �NFC ) = OHD + OHG . Q . E .D . · :. h2 = a2 + b2 . '

a • . This fig . and demonstration was for�ulated by Fred . W . Martin, � ·a pupil. in the Ce:ntral Junior­Senior High School at South Bend, Indiana , May 27 , · J.'940 . ---------�-----·

'

b . It should appear in this book at the end of the B-Type section, Proof Ninety-Two ..

' Fig . 366

1 - ·-

--- Draw CL perp . to AH , jein .CH and CE ; al so GB . · . C onstruct sq . HD ' = sq . HD . Then ob serve that �CAH = . �BAG, �CHE = �ABH; �CEK � �BFG and �MEK = �NE 'A .

. Then. it follows that sq . AK = (�AH9 = �AGB in sq . HG ) +. (6HEC = -�BHA in sq . AK ) + (tEKC = �BFG in sq . HG ) + (�BDK - 6MEK = quad . BDEM in sq . HD ) + �HBM . . com . to sq 1 s _ AK and HD = sq . HD + sq . HCL :. h2 = a2 + b2 .

·• · Thi s proof was ' dis­covered by Bob Chillag, a pupil

.�

•

------------- -�---·----------;

(

•

0

. ' ...___ ______ -------�--�- __________ -----�4.. -_---;::;--- � ____ _.THE . ..EYT!!AGOREAJLl�ROROSIT�ION--- --------· -- -�-- -- - - - - -- - - · ---- 1-·

I �------------------·--- - ---

/ ' ' '

in the ' Central Junior-Senior High School , of South Bend, Indiana, j_n his teacher 1 a (Wilson T4ornton 1 s ) Gec�try clas s , qeing t�e fourth proof I have re- . ·cei·•.red from pupils of that scho,Jl . I received this proof . on May 28, 1940 .

• b . These four p1'\oofs show high .tntell,ec·tual ability, and prove what boys and girls can do when

_ · pe:r•mi tted to think independently and logically . � E . S .•. Loomis .

c . This proof belongs in the book a t the end ' - - --or· the E-Type -section, One Hund-red ·Twen�y-Six •

..

- - ------ - - - -- - - - -------- - - - --- - - - - - - ---------- - - - --- - -

f '

I '-·

-.. ,' - -....:

I

- -- - r-1

'

,,

- --- ---- ---- --- - --1-; ____ 1

I I

GEOMETRIC PROOFS 265

J .,

Geometric .proofs are either Euclidian, as the preceding 255, or Non-Euclidian which are either Lobadhev skian (hyPothesi s , hyperbolic , and curvature , negative ) or Rieman�an (hypothesis. , Elliptic , and curvature , pos�tive ) .

The foliowing non-euclidian proof i s a liter-ai trans6ription of the one given in "The Elements · of Non-Euclidian G�ometry, " (1909 ) , . by Julian Lowell Cool�dge , Ph .D . , of Harvard University. It appears on pp . 55-57 of said 1-:::>rk . It pre sumes a surface of constant negative curvature , --a pseudo sphere , - -�ence L�...."bachevskian; and its· \.e stabli slunent at said pages was :p.ece s sary as a " sufficiept basis for trigonome­try, ir whose figures must 1:1.ppear on such a surface .

The complete exhibit in said work reads :

"Let us not fail to notice that si!fce 4:ABC i s a right angle we have ,' (Chap-: ·· III , frheorem/ 17 ) ,

lim . BC = co s (� - e ) = sin e . - - � (3: ) AC � , ·

·<>

" The extension of these functions · to angles ' ' . . n .

whose measures are great·er than 2 will afford no

difficulty, .for , on the one hand, the defining s.eries ' �

remains c onverge:p.t , and, on the other, the geometric extens1on° may be effected as in · t�e elementary pooks .

" Our next task is a m�st serious and funda-

. . .

mental one , t o 'find "the relations <which coni'l.ect the . .. . ' ------------ ----- --�-1-=--------W'eas.ul!es-And-s-!de-s--an�ng-J:e-s-of'-a.-ri-gtrt--t:rri-a;:rtgTe-;:--------Let thls �be the ..!lABC with 4-ABC · as ·the right ·angle}·· ·

·

Let . the measure of �BAC b� � while that of �BOA is e . - - - --- ·n-_We shall a ssume· that both � and e �re less than :2 , an obvious neces sity under th� euclidian or hyper�

, b�lic hypothe sis , while under the elliptic , such will s till obe the -case · if the side s of the triangle be not large_, and. the cas� where the inequalities do not. hold may be easily treated from the case where they do . Lfrt us also call a , b , c the measures ·of BC , CA, AB respect:tvely .

' " �)

-- \ -:

26o -- TliE PYTH1fGOREKN-F-ROPOS·ITION--

"We now ma�e rather an elaborate construc tion. -·

Take B1 in (AB ) as n!3a-r -to B as de sired, and A1 on the extension of (� ) beyond A , so that AA1 = BB 1 and cons true t �A1B 1 C 1 = ABC , C 1 lying ·

not far from C ; a con­struction which, by 1 (Chap . TV; Theorem 1 ) , i s easily pos sible if � be small enough . Let B1C1 meet (AC ) at 02 • A,�._...j��----...... �, �] �Cl.C2C will - differ but

��--------------�� � little from �CA, and we At.

Fig . 2

:B� may draw�C3 �erpendic ­ular to cc2 , wtiere c3 i s a point of (CC2 ) . Let us next find A2 on the

extension of (AC ) beyond -A ·s o that A2A = C2P . and E2 .. �-� on the ext�hsion .of , (C J.BJ. ) beyond B1 so _that B1B2

= C1C2 , which is certainly pos sible as C.J.c2· i s very small . � Draw A2B2 • . we saw that �CJ. C2C will differ from �BCA i>.Y §.:Q. infi:qi tesimal (as B1B decrease s ) and

� �CC 1B1 wi�l approach a right angle as a limit . We thus ·get two appro�imate eXp�e s sions of sin 9 whose

i.ld C1C3 CC1 c o s a/k BB1 c omparison y e s . = -=- + e 1 = · + €2 ,

c1c2 cc2 cc2 for CC1 � cos a/� BB1 is infinit� simal in comparison t o BBi or CCi · · Again, we see that a lin¢ through the middle point of (AA� ) perpendicular to AA2 yill also -

----�-�'·. -- -- -------------be .per-pendicular t o A1Q1 , "and the distanc·e of the in.-. , ter sections �ili _ differ infinite simally from sin ��J.•

We see that C J.C2 . differ s· by a higher ipfinite simal

! , I " I

� ' I �"'7 J,.� .. ,· (!' _ _.,.. ......... .,. ... ..,

r '

· c. I - ·- . . b AAi fro:p1· s�n � c�s b k AA1 , so - that cos k sin � 0 0

+ € 3 c o s a/K BB1 1 2

= --==--- + � 2 · del.

- ·- b "Next we see that A:A-1 = BB1 , and hence c o s -1 CiC2 k

= sin � co s a/k ·

002 + e 4 • . Moreo�er , by

• . .

. ..

----- ·-------- - - � '·

. . . . . �

-· .. . •

. ..

� � ; ... , .. ,• . GEOMETRIC PROOFS ·_ · ·

- ' ' �

267 ,.,.. .... . . -- .:-:-;-;:--� \ ' -�-- -----� ---

. constructiop. CiC2· = B1B2, 002 -��; AA2'·;:···: A perpend:I.cu-rar. to 1�i from the mid�le po1nt of (AA2 ) will be perpendiclii.ar -,to A2B2 , and t·lie dista:n�e of the inter­sec tions- , will ':dlff.er infinites.imally' ' from each of

t • .. ' t. •J � '

t��s� exp��s sl�n�. !:·�·si� 'ljl�·2if,��:�{�:�.cjk-:�J.B2 ·� Hence

• � • ' .c. •• • •

\ I _"

.a .... � ... "- >

b · · a . · c .. --·· · . ,-b: .. . : - · a 1 c ( C 0 s - - C 0 s - C 0 s - . < E • C 0 s ...- · = C,0 S· �- Cos- -, . --- 4 ) k . - k . k . ' ! k : k k ' . \ ' . . ".To get the special -formula :(or the euclidian

- . �

case , ve should develop all cbsines in power series , multiply through by k2 , anc{ th�n ·put l/k2 = o , get­ting b2 = a2 + · c2 , - the . usual Pythagorean ' formula . "

-a : This transcription was taken April 12, 1940, �hy E . S' . Loomis .

-

b . This proof shou;I.d come after e ,. p.. 244 . )

- - - - -

This famous Theorem, in Mathematical Litera-ture ; ha�r been called : ..

1 . The - Carpenter ' � Theorem � . 'The · Hecatomb Propos! tio'n , 3 . The P:ons A sino:rum 4 . The Pythagorean Proposition 5 . �TM 47th Propo'sit1on

Only .four kinds of proofs are pos�ible : 1 . Algebraic 2 . · GeQmettic--Euclidian or . non�Euclidian

· 3 . Quaternionic , \ . 4 . Dynamic

·,

, ...

--- fn:-my-:t:nve-s-t-±ga-t-1-ons-:-I�-ound-the-fci"l-l:'ow±-ng�---- ----------"11 Collections of Proof�: ·

1 . The · �eric�n Mathematical Monthl7 2 . The Colburn C ollection 3-.---The Edwards C ollection 4 . The - Fourrey C.olle�tion 5 . The. Hea�� Monograph Collection 6 . The Hoffmann Collection ·

7 . The · Richardson Collection9 • '" 8 . Tne Versluys Collection 9 . The Wlpp�;r � Q.ollec tion

lb . The Cr��er Collection 11 . The .. Runkle Collection

,�o •.

IOO 108 40 38 26 3,2 40 96 46 93 28

.. Year 0

1894\J901 1910 1895 ' . 1778 1900 1 1821 1858 1914 ' 1880 1837 1858

-- -- -�----

�- - -- -- � - - - - - � ·

I \ : , ·- I I,

..

.,. • I

Of the 370� qemonstrat1ons , for :

. :proof '"' 1 . ·The shorte s t , see p . _ 24, Legendre 1 s . . . . . . . . . . One

2 . The longest , · see p . 81 , Davie s Legendre . . ' Ninety

5 . ' 6 .. 7 .

3 ; The most popular, ·p . 109, • . . • • . • . . • . . . ; . Sixteen

4 . 0Arab1c , see p . 121 ; under proof . . . . Thirty-Three

�haskar,a, the Hindu, p . · 50 , • . . . . . . • . . Thirty�Six

The biirid girl , Coo11dge ,, p . � 118, . . _. . Thirty�Two

The Chinese--before 509 B . C . , p . 261 , ., . . . . . • • . :: . . .

.;>

8 .

9 .

10 .

11 .

12 .

_ . . . . . • . . , . • • • . . . . . • . . . • • Two ·Hundred Ftfty-_�hree ' ' ' I; ' '

•

Ann Condit , at age 16 , P � 140 (Unique ) . . • . . • . . . • .

� . . • . • . ' . • . . . . - . . . • • • � • • · • . , ' . . . � • . ·

• . . • . Sixty-Eight

�uclj.d ' s , p . 119 , , . . • • . • . • : . • � . • • • . . Thirty-Three

Garfield ' s (Ex-Pre s . ) , p . 231, . . . . . . . . . � . . . . . . . .

• • • • • • e • • • • • • • • • • •. • • • • • • Tw:o. l,Iundrerl Thir_ty-One - -. HuygE?ns ' - (b . 1629 ) , p . 1 18-, • . · . . .

-

. • • '

• • Thirty-One

Jashemski ·' s (ag� 18 ) , · p_, 230, o, • • _. _ ._ • • • •

-

• • ': • • • • • • •

. • • . • . . � . . . • . . • . • • . • • . • . . • . . . 'l'wo Hundred · Thirty

13 . Law of Dissection , p . 105 , . . • • • . . • . • . • . . . • . • . - Ten

14 . Leibniz ' s (b . 1646') , p . 59 , . - . • . • . . Fifty-Three ' '

15 . Non-Euclidi·an, p . 265 , . . • .. . Two Hundr-ed Fifty-Six ------------ ---�.--------- I ---�� ------ --

16 . -Pentagon, pp . 92 and 23.8 , . . . • . • • • . . • • � . . . • � .- • . . •

One Hundred Seven and Two Hundred �orty-Two " ----

17 . Reductio ad Absurdum, pp . _ 41 · �nd 48 , . . . � . . • . • • ; .

· . . • . . � • , • • • . . • .• • . ·-� . • • . . • Sixteen �nd Thirty-Two ..... .

18 . Theory of Lim! t s , p . · 86 , . . . • • • ;�· . . . Nine.ty-Eight

268

-., .

, 0

' ) .

' -'

..... --- -

. . · '

. I .

ADDENDA

They came to me from everJWhere . t> _..,. - .,.... -� • .... �

269 #

1 . In 1927, at the date of the p�1nting of· the 1st edition, it shows--No . of Proofs :

Algebraic , 58 ; Geometr!c ,· 167 ; Quater­nionic , 4 ; Dypamic., 1 ; i:n al� 230 dif­ferent p·roqfs .

2 . On Novemb�r 16 , 1933 , my manuscript for a second edition gave :

.

Algebraic , 101 ; Geometric , 211 ; Quater­nionic , 4 ; ' Dynam:t.£, 2 ; in all 318 dif­ferent proofs .

3 . Qn, May 1 , 1940 at the revis ed completion of the m�nuscript for :my ·2nd edition of The Pythagorean . .

. •

Propositio�, it ·contains -�proofs :

0

Algebraic , 109; Geometric , 255 ; Q11ater­nio!lic , 4.; Dyna.nrl:c , 2 ; in _all 370 dif- '

ferent proofs , each prqof calling for its ·ow� specific figure . And the · end is not yet .

·'

('

.- . E . _S . ,, Loomis , Ph. D . at age nearly 88,

May 1 , 1940

t ! l t

- I l

; .

B I BL I O G R A P H Y ' �.

The source s ,_ frq_l!l �hlch hl�-�-C?ri�al �ac t s , sugge stions ; illustratiqhs , demonstrations and other data _ concerning the 'preparation of thi s book, or fon

' � .

further investigation by any ofre who may wi sh to com-pare and consider original source s , _ are compri sed ln

- the following tabulation . . ..,

I . TEXTS ON GEOMETRY'

1 . Beman and Smith ' s Plane and Solid Geometry, 1895 . 2 . Bezout ' s Elements of Geometry, 1768 . 3 . Camerer , J . G . e t C . F . Hanbes-Berel , 1824 . 4 . Chauvenet ' s Elements of Geometry, 1887 . 5 ., Cramer� . . (C. ) , 1837 , 93 proofs . •

6 . Davies ' Legendre , 1858 . 7·. ·Dobriner ' ·s · (Dr . · H . ) Geometry. 8 . Edwards ' Geometry, 1895 . · 9 : Halsted ' s Elementary .Geometry, 1895 .

10 . Hill ' s Geometry for ··Beginners , 1886 . 11 : �opkins 1 . Plane Geometry, 1891 . 12 • . Kunze ' s (v�n ) Geometry, 1842 . 13;. Mac�ay ' s (J . . :s ·. )· Plane Geometry . . 14 . Mahler ' s (G. ) "Ebene 'Ge6metr1-.e , " 1897 . 15 . Milne ' s Plane and. So�id 9€ometry, 1899 . 16 . Olney ' s Element s of Geometry� 1872 , Univ ' y Ed ' n . '17 ; Pisano (Leonardo ) J>ractica Geometrlae . 18 . Ray ' s Geometry �

l · · ''

.19 . Rlchards 1 (Claudius ) "Euclide s Elementorum J-_:_ __ __________ ...:.u· - e eme-t-:ra-.1:-c-er�-:I:645,.-:--An1;r';(erp-±ae·-. --, --- - -

20 . Sauvens·' .(M . ) Geomei;;ry, 1753 . 21 . Schuyler-' s (Dr . A . ) Elements df Geometry, 1876 . 22 . Slmpson 1 s (Thos·. ) Elemental'•y Geometry. 23. . Todhunter ' s Element s of Euclid, 1897 . 24 . Wells ' E s sentials of Geometry, 1899 . 25 . Wentworth and Smith i. s Plane Geometry, . Revi sed Edi- ·,i:;:.

tlon, ·1910 . · " .!

26 . "Wentworth 1 s New Plane and Solid Geometry, 1895 . , ..

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1 ..

2 . 3 . 4 . 5 . 6 . 7 . 8 .

9 ! 10 . 11 . 12 . 13 . 1'4 .

15 . 16 . 17 . 18 . 19 . 20 . 21 �

· THE PYTHAGOREAN PROPOSITION

II . OTHER TEXTS

BerE.stein, .. (F . . ) The Axiomat-ic Simplici�y_ of Proofs , 1909 . '

Blanchard 1 s (Orlando ) Arithmetic . � Brandes 1 Dis·sec tiori Proofs , 1908 . Casey 1 s Sequel to Euclid, Part I , 1900 . Cours de Mathematiq�e s , 1768 . Cullen 1 s (P . c . } Pamphlet . Delboe.uf 1 .s work . Fourrey 1 s (E . ) Curiosities G�ometrique s , 2nd Ed 1 n, 430 page s , 1778, Paris � Geometric Exercise s , Paper Folding, 1905 . Hal�ted 1 s Mensuration . Handbook .der Jtiathematik, . 1869 . Hardy 1 s Element s of Quat�rnions , 1881 . Hubert 1 s Rudimenta AlgebPae ; 1792 . Leitzmann 1 ·s (Dr . W . } 11De s Pythagori sche Leh'rsatz ,U 1936 . Littrow 1 s (E . von ) 11Popular Geometr.ie , 11 1839 . Loomis · (Dr . E . S . ) , ,The Pythagorean Proposition . Row 1 s· Geometr�c Exer9·ises in P'aper Folding, 1905 . Schubert 1 s Mathematical Es says , 1898 . Schuyler 1 s (Dr . A . ) , Trigonometry, 1873 . Simon (M . ) , 1906 . "� Vogt · (H . ) , The Geo�etry of Pythagoras , Vol . . 3 , 1909 .

22 . Wallis 1 Treatise of Algebra , -1685 .

• �---!11:.--. --.M,._.,O .... NOG=RA�P=H�S�-, . . ' .

1 . Heath 1 s Mathematiqal Monographs , . Nos . 1 �nd 2 , 1900 . ..

2 . Hoffma� 1 s (J . J . � . ) , The Pythagorean Tneorem, 1821--32 Proofs . ·

·

3 . Martin 1 s (Artemus ) , 11 0n Rational Right-Angled Triangles (Pamphlet ) , 19�2 .

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4,. Mathematical Monograph, No . 16 , 191�, Diaphontine Analysi s des Pythagorai·sche Lehrsatz , by Leitz­mann.

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BIBLIOGRAPHY 273

.. 5 . Naber0 (Dr . H . A . ) , Das Theorem des Pythag·oras , 1908 .

6 . Versluys ' (J . ) , Theorem van Pythagoras , Zes en Negentig Bewi jzen, 1914, 96 Proofs .

. 7 . _ "Vriend de Wiskunde , " 1898 by Vaes . (F : J . ) .

8 .. Wipper, (Jury ) , " 46 Beweise des Pythagoraischen Lehrsates , "'l 1880, 46 Proofs . � �

IV . JOURNALS AND MAGAZINES

1 . ��rican Mathematical Monthly, 1�9�-1901 , Vols . I to· VIII inclusive , 100 proofs . T�e Editor, Dr . B . F � Yanney said : '-'-'!!here is no limit to the

, number of proofs·;.:-we just had to quit . " 2 .• Archiv der Mathematik and �hys . , Vol s . II , XVII ,

XX, XXIV, 1855 , _by Ph . Grunert , 1857 . 3 . Aumerkunge11 ube!' Hrn . 0geh . R . Woif 1 s Ausgug aus :

der Geometrie , " 1746 , by Kruger . , 4 . Bernstein (F . ) , The Pythagorean Theore�, 1924 . ·

5 . Companion to Elementary School Mathe�ati�s , 1924 . 6 . Heut ton' s (Dr_. ) , Tract s --H+story o

.f Algebra, 3

Vols·. , 1912 . 7 . Jo�rnal of Educat'ion, 1885-8, Vols . XXV , XXVI,

�II , XXVIII . , 8 . . Jour.nal de Mathein, 1888, by F . Fabre . 9 . Journal of .the Royal Society of Canada, 1904 ,

Vol . II , � p . 239 . 10 . Lehrbegriff ' s Science of Mathematik . 11 . Masonic Bulletin, Grand Lodg� of Iowa , No . 2 , o - :

February 1929 . .

�--��--------cl-2-.-Ma-t-hema-M:e-a-3.-E-s-s-a-ys�nd--·Reo-.c,...rn.•e=na�t�-"1h. o...,.n"'""s".---�-------------'111

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13 . Mathematical Magazine , The , Vols . I and II , by Martin; 1��91 .

1�. Mathematfcal ' Monthly, Vols . I and II , 1858, 28 Proofs , by J . • D . Runkle . ,

15 . Mathematic s Teacher , Vol . XVI., .-�9�5., . . Vol . XVIII, - --· "1."925 . .. - ' - · - - - - · .

16 . Me s senger of Mathematics , Vol . 2 , 1873 . 17 . Nengebrauer ' s (0 . ) , H�story of the Pythagorean

The.orem, 1928 .

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' ' -THE PYTHAGOREAN PROPOSITION

iS . _ _ .Notes and Queries , by W . D . Henkle·, l.BJ5 .,. 81 . 19 . Nouvelle s Annales , Vol . LXII . ·

. �. \

20 . Open C ourt Pub . C o . , 1898, p . 39- -Magic Squares . 21 . Periodico di Mathematicke (Italian ) , � Hisiory of

'· che Pythagorean T�eorem . 22· . Philo s ophia et Matlie-sis 1frii ve-rsa . - -23 . �hilosophicai Tr�nsactions , 1683 . 24 .· Pythagorean Theo�em, The , by Cullen . 25 . Quarterly Journal ·of Mathematics , Vol . I . 26 . Recreations in Mathematics and Physic s , 1778 . 27 . R�nkle ' s Mathematical Monthly, No . 11, 1859 . 28'. School Visitor, 'The , Vol . - .-III ,- 1882 , · Vol . IX,,

1888 . . 2 9 . Scharer ' s. The Theorem of Pythagoras , � 1929 ..

. 30 . Scienqe, New Serie s , Vol . 32 , 1910 ; Vol . 33 , 1911 ; ��01 .1- 34 , 1912 . ' ' i. ' r-- � I 31 .. . cientia Baccalurem . 1 .

. 32 . he Sc�entif�c American ·supplement , Vol . 70 , 1910., ,.,pp . 359 , 382�3 . --

3� . Scientifique Revue , 1859� o

V . GENERAL REFERENCE WORKS

1 . �all ' s Short History of .Ma�hematic s , 1888 .

-.

2 . The E11cyclopedia Britannica--articles on Geometry, Magic .Square s , Dynamics , Quaternions or V�ctor An�lys:ts , Euclid, Pythagoras , e·tc .

3 . Encyclopadie, de s Elementar Mathematick, 1923 , by H . Weber and -J . Wellstein .

. 4 . Scartscrit Mathematic s . (1 ) "Viga Gani ta � 11

(2 ) "Yonc ti Bacha . �· . 5 . Wolf ' s (Dr . Rudolph ) , Handbook of Mathematic·s , 1869 6 . MU�ler (J . W . ) , N_Urnberg, 1819 . o

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VI .

The Royal Society of London- -Index 1800�1900, Vol . · I , Pure Mathematic s , 1908, lists the · following authors and article s : . 1 . Hu l l e r , H. K� F . vo n : Oken ! �i s (1826 ) , 763ff .

·2 . Giin d e r1i7:mn , C . : Grelle , -J • · 42 :(J:852 ) , - - 280-f-f .- -

3 . v t n·c e n·t , · A . J . N. : . N . A . .Mth . 11 (1852 ) , 5ff . 4 . DeKo r� an , A . : ·Q . J . Mth . 1 (1857 ) , 236ff . 5 . Sal w e n ,· G. : Q . J . Mth . 1 (i857 ) , 237ff . 6 . R.t c h ard�on '- · J . K .• : Camb . (M . ) Math . � - 2 (1860 ) ,

45ff . 7 . Azzare l l t , K . 1873

66ff . Rm . N . Line . · At . 2:-t (1874 ), .

8 . Harv e y , 11. : Edinb . Math . s : P . 4 (1886 ) , 17ff . · 9 . Zah radn t k , K . : Casopis 25 (1896 ) ,. 26lff . Sc.hr .

10 .

11 .

12 .

13 .

·14 .

15 .

Mth . (1896 ), 364ff . . C onfirmation . lit t t s t e t n , T . : Grunert Arch . 11 (1848 ) , i5.2 . . . -�.:·

.

Pythag . triangle . G ru n e r.t , J . ' A . : Arch . der . Mth . and Physik 31 (1858 ) , 472ff . Demonstration . Bo qu o y , G . vo n: Oken Isis 21 (1828 ) , lff . \

- - - - � - - - - - by di s section . P e r t d a l , H . : Mes s . Mth . 2 (1873 ) , 103ff . Extensipn to any triangle . J e l .t. n � k , V . : Casopis 28 · (1899-) , 7-9£f . , F . Schr .. Mth . '(:[899 ) , 456 . Extensions . A f! �-l t n , ·A . H . : Edinp . R. S . P . l,f (18�4 ) 703 . .

16 � (J Figtlre-s·' for proving . A nd r e , H . d ' : N . A . Mth . §-��46+�2��------------------------------�--------�

17 . Generalization . Ump fen b ac h ( Dr . ) : Grelle J . 26 (1843 ) , 92 .

18 . Puzzle s · connected with theorem . B r an d, .E • . : �v . :- - - -Sc . 2 (1894 ) , 274ff . ,

.. 809ff

·.

19 . Pythagorean ��iangles . Can t o r , K . : Schlonsilch, z . 4 (1859 ) , �06ff . . .

20 . Pythagorean. triangle s . E e w t s , J . : Am . Ph . S . . P . ·

9 (1865 ) , 4l5ff . 21 . Pythagorean · triangle s . llh t t wo r t h , 11 .• A . : Lpool ,

Ph . S . P . 29 (187.5 ) , 237ff .

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276 THE PYTHAGOREAN PROPOSITIOl'ii

22 . Pythagorean triang�es and applications to the · d:Lvision of the circumferen�,e . G ru e-b e r: Arch . Mth . und Physik ·l5 (1897 ) , 337ff �

23 . Pythagorean triangle , nearly i soscele s . Ha r t t n , A r t : D'e s _ Moi-nes Anal 3 . (1876 )., 47ff .. o

24 . Pythagorean tr.iangle , properties . B r e t o n: Ph . Le s Mondes 6 (1864 ) , 40lff . . · --/ 25·. :Right -S.ngl:ed triangle s . B an g m a , V . S. :. .Amst .

. - ·'Vh . 4 > (1818 ) , ' 28ff . �6 . Right-angled triangle s , squares on · side s (�mett ).

27 .

28 .

G e rg o n n e , J . D . : Gergonne. A . Mth . 14 (1823 ) , 334ff . Quadrangle s , complete . N e wb e rg , J . : Mathse s 11 .. (1891 ) , 33ff, 67ff, 8lff , · 189f'f . Quadrangles , . transf0rmations . Go b , A . : · 'As . Fr . C . R . (1884 ) , (Pt . 2 ) , 294ff .

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T E'S T I M O H I A L S

From letters of appreciation and printed Re­views the following four t� stify as to 'its wortl�.

has : 1927,

New Books . The Mathematic s Teacher 1928, The Pythagorean Theor�m, EliSha S . Lc ,mi s , Clev�Japd, Ohio , 214 pp . Price $2 . 0< .

. " One hundr.ed sixty-seven geometric pr_gpfs and �ifty-eight algebraic proo�s be sides several ot�er kinds of proofs- for the Pythagorean Theorem complled in detailed; authoritative , well�organized form will be- a rare ' find ' for Geometry teachers who are alive to the possibilities of their sub ject and for mathematic s clubs that are looking for int�rest­ing materia� .. Dr . Loomis has done ' a · scl!oJ�rly piece of work in collecting and arranging in such conven-• ient · .form this great number qf proof's of our his�toric theorem.

" The book however is more than a · ·me-re · cata;;. · loguing of proofs , valuable as that< may be , but pre­sent � an organized sugge stion for · many_ more· original proofs . The object pf the treati se is two.f?ld, ' to pre sent -to t�e f�ture i��e stigator , under . one co!er ,

, simply and concisely, what i s known relative t o the 0 . Pythagorean proposition, and to set forth certain e s -. - ) . '

tablished fact s concerning the proofs and geometric figure s pertaining thereto ."'.v'

"Ther e are fo�;r kip.ds of proofs , (l ) ' those baseq upon - linear relations- -the algebraic· proof , (2 ) tho�e based upon comparison of areas--the geo- ·

metric proofs , (3 ) those b�sed.upon vector operations

--the .quaternionic proofs , · (4. ) those based upon mass and velopity--the dynamic proofs . Dr . Loomis con-

,� -tends tTt the numbe� of. algebraic and geometric proof� are .each limitles s ; but that ' n��proof b� trig­onometry, analytic s o'r · calculus :l's pos sible due to

277 , ·>;>--'

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278 THE PYTHAGOREAN PROPOSITION -·

the fact tha� these sub ject s are based upon the right-triangle proposition .

"Thi s book i s a treasure che st for any mathe­matics teacher . 0 The twenty- seven yeaps which Dr . 41 ·

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Loomis has �played with this theorem i s one of hi s hobbie s , while he was Head of the Mathematics Depart-' ment of West High s·chdol , Cleveland, ·· Ohio, have been well ·spent- since he has gleaned such trea.sure s _ fr9m

,, the· archives . It is imp<;)s sible in a short r�view to do justice to - this«' splendid bit of researc-h wBrk so unselfishly _ done for the love of mathema�ics . Thi s

. boqk s�o�ld be highly prized by �verY .�&thematic s teacher and should find . a prominent place in -every school . and public library . "

· H . C . Christoffenson-'.

Teachers College Columbia U�iversit�, N . �. City

· From another review this appears :

" It (this work ) present s· all that the ·litera­ture of-' · 2400 years gives relative to the historically re�owned and mathematically fundamental ·Pythagor.ean ·proposition- -the proposition on which rest s the sci­ence s of c ivil englne�r+ng, navigation and astronomy, and to which Dr . Einstein conformed in formulating and posi-t-ing hi s. ge�eral ·theory of relativity in 1'915 .

"-It e stablishes that but four kinds of proofs are· pos siole�-the .Algebraic , the Geometric the Quater� nionic and · the Dynamic .

. · " It sh<;>ws that the number of Algebraic p�oofs is limitle s s �

" It depicts 58 algebraic and 167 geometric proofs .

'rit· declares that no trigonometric , -�nalytic g�ometr�, or calculus proof is possible .

" It c ontains 250 geometric figures for each .of which a demonstration i s . �iven .

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279 1 "rt contains a-c'omple-te bibliography of all

references to this celebrated theorem. "And lasj;ly tnis work·· of Dr'. Loomis is so

complete in its mathema-tical survey and analysis that it is destined to become the reference book of alr future investigator� , a�d to this end its sponsors are sending a complimentary copy . to each .of the great . mathematical librari�s of the United States and Europe . 11

Masters and Wardens As sociation of the

22nd Masonic District of Ohio

Dr . Oscar Lee Dustheimer , Prof . of �athe­matics and Astronomy in Ba�dwin-Wallace College , Berea, Ohio , under dat� of December 17, 1927 , wi·ote : "Dn . Loomis , I c onsider this" book a rea-l co11tribution to Mathematical Literature and one · that you can be Justly· proud of . . . • I am more than pleased wit.P. the book . "

O scar L . Dustheimer

Dr . H • . A . Naber , of Baarn, Holiand, in a weekiy paper for secondary instructors , printed, _ 1934, l n Holl.arid Dutch, has (as traD:slated ) : "The Pytha­gorean Proposition, , by Elisha S . Loomis , Profe s s or Emeritus 91.' {'iathematic s , Baldwin-Wallace College , " (Bera, 0 . ) • • • • •

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...-.:--Dr . Naber s tates • • • • "The author has classi­fied hi s (237 ) proofs in gro�p s : algebraic , geometri� quaternionic and dynamic proofs ; and these groups are further subdivided�"' 11 • • • • Prof . Loomis himself has wrought , · in his book, · a wo�k that . i s_ more durable than bronze and that tower higher even than the pyra­D;lids . '� " • • � . Let us . hope--until ·;re know · more com� plE(tely--th&t by thi-s� i>roce.dure , as o�r mentali·ty · grows -deeper , 1� will qecome as in pf.m: The Philo-sophic Insight . " ·• �);''f._i,

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I N D E X O F H AM E S

1 . Names of all ·authors of works referred to or consulted in the preparation .2!_ this book :may be fo�d in tlle bibliogra­phy on -pp . 271-76 . - '

2 . Names of Texts , Journals, Magazines and other publications consulte� or referred to also -appear in said bibliography.

3 . Ifames of persons for . whom. a proof bas been :riamed, or. to whom a proof has been credited, or from or through whom a proof has come, as well as authors of workS consulted, �e arranged alphabetically in thia Index of personal names .

4 .- .some names occur two or more t1mes , but-the e�liest occur­rence ia the only one _ paged.

Ad8lllB, j • , 6o Agassiz, 190 Amasis, 8

. Anaksimander, 8 .. Andre, H • d' , 98 �lin, A . H . , 275 .fwdronicus, 15 Annairiio of Arab ia,

900 N . C . , 51 Arabic proof, 121 As:Q., M • . v . (1683) , 1.50 Azzarelli, M. , 275

Ball, w. W. Rouse, 154 · Ba.ngma, V � S . , 276 Bauer, L . A . , 79 Bell, E . T ., (Dr. ) , vi Bell, RichaJ:d A . , 49 Bernstei�, F . , �7-2 Bhaskara, the Hindu, 50

. . Binford, R . E . , 230 . •

Blanchard, Orlando, 155 Boad, Henry, 1733, 137

Boon, F . c . , 6? Boquoy, G. von; 27-5 :Bottcher; J � E \ 112 , {/ Brand, E . , ·65 ·

., Brandes, 272 Bressert, M. Piton, 67

- Breton, 276 Bretschenachneider, 155� Brown, Lucius; 79 Buck, Jenny de , 149' · Bullard, L . J . , 83 Calderhead, Prof. J�s- ·A. ,

24.7. Camerer,-� 13 C�rs, Jules , 52 Cantor, M. , 5 Carmichael, Robert D . , 24

- Casey, 253 . ce.ss, Gustav, 226

, · C.attell, ·J . McK. , 249 . ,phase, Pliny Earle, 109

Cbauvene:t, .5.� ;I • . ' . . ...

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282 THE PYTHAGOREAN PROPOSITION

Chillag, Bob, 263 Chinese proof, 262 Chrietoffere6n1 H. C . , . 278 Clairaut, ·lJ41, 203 Colburn, Arthur E . , 53 Colline, Matthew, 18 Condit,"-� (f18e 16.) , 140 C9olidge, Mlee E . A.

(blind girl) , 119 Coolidge, Julian Lowell, 265 Copernicus, 12 Q

Cramer; c . , 271 Crelle, J . , 275

1 Cullen, R . C . , 155

Darwin,·· 12la. Davies (Lege���)_, __

-�4

DeBeck, Bodo M . l 227 Deimal, R . F . , 250 Delboeuf, 134 DeMorgah, A . , 105

. Descartes', 244 Dickinson, M. , 179 Dickson, Leonard E . , 17 Diogenee., 6 Dieeectio�1 . Law of--Lo0m1e , 106 Dobrinezo_, Dr . H . , 111 Dulfer, A. E � B . , �7 Duetheimer, Dr . Oscar L . , 279

:1. \� ,Edison, r2 Edwards, '25 Englapd, W. , 120 .Epstein, Paul; , ll5 Erat9klee, 9 Euclid, . 13 Evan.S, Geo . w. , 57 ·

:Exeeit, Rev . J . G. , 49 ' ' \.

· Fabre, � . , 57

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Ferekid, 9 _ Forbes, E . , 203 Fourr.ey, ··E • , 49 French, C . , 1:33

Ga�il�o, 12 Garfield, Pree . James A � , 224 Gause , 16 Gelder, Jacob de, 121 ·

�rgorme, J . D �-' 276 .

Ginsburg, Dr . J�huthiel, :x:iv Glaehi.er, J . W(. L . , 17· . Gob, A . , 276 . I . Graap, F . , 11 J Grueber, 276 Grunderniann�, c � , 275 Grunert, J . A . , 53 I ' Gutheil, B . von--World War

Proof, 117

Halsted, Geo . B • . , 21 Hanil.et, 120 , Hardy, 272 Harvey, W. ,: , 2-75-Hauff, "Vori, 123 Hawkins, Cecil, 57 Haynes , . 226 �eath, 24 Henkle, . W. D . , 274 Hersey, 'Mayo D . , 249 Hill, .271 Hippiae; 9 Hoffma.rm., Joh . J . I . , 18181

29 Hogben, Lancelot, .261 Hoover, w.m. , 155 ' Hopkins, G. I . , 68 Horace, 285 Houston, Joseph, 145 Hoyt, David W. , 109

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INDEx .OF NA1'1ES 283

Huberti, 1762, 225 , . 48 Hutton, Dr . , ·

Huygens, - 1657, 118 Hypasos, 11 · Hyps:tplem,· 157

Ingraham, Andrew, 243 Isidorum., 156

Jackson, C . s . , 226 Jashemeki, Stanley, 84 J�linek, v. 1 232 " Jeng:1:s, Khan, 155

. -Joan, R . ·, 177 Jdhnstbn1 Theodore H . , xiii Jowett, 86

Kambis , 9 Kemper,. p . J . , 70 Kha.yy!un,. omr, 120a Klagel, 151 Kneer, Alvin, 4o · Kr�ger, -M. , llO K"r\leger, .1746, 1 Kruitbosch, ·n . J . , 139 Kunze, von, 192

Laertius, 6 ·Laisnez, Maurice , 50 Lamy 1 R •. P._, 16851 85 Lecchi"o, 125 Legendre, Andren · M. , 24 Lehman, Prof . D; A . , 4o · �ibniz , von, 59 Leonardo da Vinci, 129

· Lew;ts, J . , 275. :Lei tzm&m, Dr . Wm. , 49 Litt:;-ow, � . von, 113 ·Locke, . J . B . , 248 Ifoorirl.s, Elatus G. , xi v Loomis, Dr . E . S . , 1581

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Macay, 211 J1acFar lane, Alexander, Mah�er, von G. , 105 Marconi, 12 �re, A . ; 50 Martin, Ar:temas, 17 Me.z:tin, Fred w. , 263 Masares, 20 McCready, J • . M. , 174 Mcintosh, M. ,, 209 Meyer, P . Armand, 44 Milne, .271 Mnessa:rch, 7 Molllnan, E •. , 79 Muller, J . w. , ' 274

Naber., nr·. H. A. , 273 Nasir-Ed-Din, ·Persian Ast�on-

omer L 128 .. ,.

Nengebrauer, 273 1 Newb�rg, J . , '276 Newton, B� Isaac, 12 Nielson, �· ' 114 � ortbrup, Edwin . F . 1 249

Oliver, 83 Olney, 73 Oz�, M.

llO --de C . G. , 1778,

Pappus, a . 375 A .D . , 126 Perigal, Henry; 102 Philippi, III of Spain, 156 Phill:.ips:, Dr . Geo . M. , 183 Pisano, Leonardo, 52 P�they, 7 Piton-Bres�ant, 67 Plato, i7 Plutarch,·, 6 Polycr�t�s , 7

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284 THE PYTHAGOREAN PROPOSITION

Posthumu_g_,�cL!-,_9.2 _ _____ ..... T.._...a..._rquin,-9--Proc1os, 4 Te�lehoff, von, 176�, 129 Psammenit, 9 Thales, 8 Ftolemus (Ptolemy) , 66 Theana, 10 Pythagoras, 6 Thgmpson, J . G . , 234

Raub, M. , 66 RB.I, Prof. Sarad.arujan (In­

dia) , 155 .Reichenberger, Nep�mucen;�17 Renan, H . , 46

· R�chards , Claudius, 156 Richardson, Prof . John M. , 26 Rogot, M. , 18o . Row, T . Sundra, 100 Runkle, J . D . , 66 Rupert, w.m. · w. , 23

SaJ.wen, G . ; 275 Sarvonaro la, 12 Saunderson, 229 Sauveur, 227 Schau-Gao, 5 Schlomilch, 194 Schooten, van, 203 Scherer, 274 Schubert, :Herman, 256 Schuyler, Dr . "Aaron, 64

·-simon, 272 Simpson,. David P . , xiii Simp�on, Tho�s , �55 Ska ts chk:ow, 5 Smedley, Fred S . , 90 ' '

· Sm1 th, David E . , 271 � th, Dr . l:fm. B �·, 155 Socrates , 86 Sonchis , 9 Sterkenberg,. C . G. , 40 Stowell, T . P . , 224 stur.m, ·J. c � , 203 �underl,Bn:d, · 158 -

T�ornt�n, Wilson, 50 Todhunter, Dr . , 271 Towne , P�ul w. , 255 Trowbrid8e, David, 152 Tschoil-Guri, 5 Tyro�, C . w. , 110

Umpfenbach, Dr . , 275 Uwan, 5

Vaes, F • J . , 46 . Vers�uys, J . , 13 Vieth, van, 132 Vincent, A . J . N . , 275 Vinci, Leonardo da, 129

• Vogt, H . , 272 Vuibert, Jal de, 47

WallisJ John, 52 Warius, Peter, 148 , Waterhouse, John, 252 Weber, H . , 4o Wells , 271

_ 'W!3J.lstein, T . , 40 �emtworth, 5·2 Werner, Oscar, 53

·Wheeler, Rev • . D. A . , 49 . Whitworth·, 275 Wipper, Jury, 3 Wolf, Rudolph, 274

Yanney, Prof. B . F . , 242 Young, Dr . Charles-A-. , 70

" Zahradnik, 275

• Zelson, Joseph, 122 . Zelson, Prof. Louis G � , 141.

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� "Ex e ' t � o n u m e n t u m , a e r e p e r e nn t u s Re, a l t qu e s i t u p y r am t du m . a l t t u s , Quod n o n t m b e r edax , n o n a qu i l a tmpo t e h � Po s s t t d t ru e re au t t n n um e ra b t l fs A nn o r u·m s er t e·s· e t fu ' a t·emp o rulft .

� Non om n t s mo r t a r . "

--Ho rac e 30 o de i n

Book III

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