pythagorean proposition
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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."
DD 0
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 • • • • • •
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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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xiv THE PYTHAGOREAN PROPOS£TIQN . ' .
• • ·::...- >J ... , .... � .... _. - �--;:_� ;. • :�·,4'; . To Dr . Jehutnfel Ginsburg ,
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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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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
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' \
, ,
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
$30, and a�y three in �� or � of the 33'0 se�·s.·
or
the
There-
_ ________ ,.tore any sub-set of two equatJgns will be found in 36, and any of thre�- equat�ons �n 8, of the 330 sets.
_But some orie or more. of the 8 may l;>e some\ one or more of the 36 sets; hence a sub-set ·of two and � sub-set of three will not nece�sarily cause a rejection of 36 + 8 = 44 of the 330 sets.
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The sub-sets �h±ch gave the 70 proofs are: --2, 6, f.or which 36 sets must be rejected; 7, 20, for which 35 sets must be rejected·, sinc·e 7, 20, is found in �ne of the .36 sets abovej 2, 3, 13, fo1� which 7 other sets must qe i>e jected,
since 2, 3, 13, is found· in one of the 36. sets above; 2, 4, 13, t:.or which 6 otl:le.:r sets must be rejected; 3., :4,- _ q,. for rhich 7 other sets must be rejec-ted; 4, _6, 13, for which 6 other sets must be .rejected; 3, 4, 13, for which 6 other sets m�st be rejected; 4, 7, 11, for which 7 other sets must be rejected;
�nd ' � . . { ' I
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ALGEBRAIC PROOFS 35
4, 11, 20, for which 7 other sets must be rejected; for ·all of which 117 se.ts must be reJected.
Similarly the dependent sets of three, which are 2, 3, 4; 3, 6, 13; 2, 7, 14; 6, 14, 20; 3, 11, l�; 6� 11, 24; and 13, 18, 24; cause a rejection of. 6 + 6 + 6 + 6 + 8 + 7 + 8, or 47 more sets.
,/ · Alse the �epen�ent sets of four, and not already reje·cted, which are, 2, 4, 11, 18; 3, 4, 7, i4; 3, 6, 18, 24; 3, 13, 14, 20; 3, 11; 13, 24; 6, 11, 13, �8; and 11, 14, 20, 24, cause a rejection of 7 more s�ts. The dependent sets of (ours 'are di�cove�ed
· as follows:· take any two dependent sets of threes having a common term as 2, 3, 4, and 3, 11, 18; drop the common term 3, �nd write the set 2, 4, 11, 1$; 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.
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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•
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_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 ) •
' \
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. . \' .
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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
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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 - ! -- - - .
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'\
'>¥ \ \. -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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.. 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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. . ....
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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 '
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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 \ 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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- 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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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.
l .
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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 •
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lig.--·�
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. • 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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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:
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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
-- - ________ __,
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-- --- ------ �--- --------�-- - - ---�--- --- ------ - --- - - -- ---- --- ---------- ----�--��--------- ----�------------------- -- -- - - -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 ..
.-.
- ' .
- --- "f
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., I
. .,
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-- - - \
•
' • -----
l.
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'• ' �7·� .. '
' ' y ' . �;; � .
,,;,; ,. �-- � .�:� ..... � ' ,..,;)-�·'
fi • I I
' . '
� .. -
"-
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' i -----. -------- --.ALOiifi\A:icpRQOps ________
_
_____ 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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--AfjGEBRA-ro--PROOPs--- ------- -- ---- -.-. -59 ___ _ _ __ _
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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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----�....,...--------�----- ·--- ----- --- ---··--:-·-- --- - --------···· -- -- -.--- - - --- -- - -- � - ---- - - . - --- � -- � · - : · ·· - - - - -- - - - -4 61
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-
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·-
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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l
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- - - - - --· -- - - -------------------·-----------------;-"��
J
62 THE PYTHAGOREAN PROPOSITION
_ _ _ _ .,... ____ _
-------
------ ------ -- -- - - . ---·-
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. . ' tJ r � -.,- -
- --��---\o.-- - -- - - -
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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 •
\
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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
---- ----------'-'---- - -� - - - - - -
<·
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- 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 :.
--'
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' . . .
' ,,
' '
:!>
' . . ' .
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
()
'
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. 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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--·
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-
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;,... . -· ··-· •' " - --·
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r
•
72
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-
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-
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--
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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---�-------- --------------
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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
. .
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---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
.,.
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.______,�-----�- - 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
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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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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 •'
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= 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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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-/
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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
..
,
i .
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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 ,
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.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 .
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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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Fig . 106
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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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· case is illu;strated· by ·fig.: : · 108, in whi�h BH = HA, show
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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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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 ) .
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.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 '
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"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
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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
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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
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· 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 .
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a. · - See · Math. Mo: , V. IV, - ).897, p . 169, pro or • r;.: ... -r-.. .... -..,._,....
XXXVIII. I . . .
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� -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 -
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,.
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 {- -
' . •
\
., ·
�
<•
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\.
-.
: --- --..
'•
/ /
,_....__- ,
, ,
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 -------�--- �----�----------------�-�-- --�----��----�----�- -�- -======------- ----�- ------ -------�
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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 . · " , . -
_____ ..
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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 � .
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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
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•
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 , . !
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1·------------ -�- ---- --
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/
' 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) �
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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 .
..
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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�-.
' .
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I !
- - --- --1',., --�-
�---�-�--------�--- ----�---- - ----�--�-- - :,_ ____ . ----- ------ �----.
--- -�-- - - -
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/.
. .
1
-�-
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-. �
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• • ..
. . " ·
i
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- � --� . -� ·-- ---'---,{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.
( /
. . " ' .
\ '·
.= �
. . . , · .
. ,_ ..
.. ... ·
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. I
,.·
GEOMETRIC PROOFS l29
;t . / ' ..... 1
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�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 • ••
' I . !
....... �-�-
- ..
' .,
/
·.
· . �{. ·�..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
-- ----- ---------
' '
..
----· ---
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·
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. . 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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· I IJ I t - I :_ ___ )}; • .a' 'a • • I/(' 'c -I ' I
/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 '
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b
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' 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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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
- ' -',
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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.
. ' • I
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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 ..
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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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156
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
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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. ) ·
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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 .
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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 .
-��-- ,1 .
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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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.. 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,.
l I I l
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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 .
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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 .
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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 ._
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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"
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. 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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Rect . LM = .[ sq . AK = (pa.l'�_s 2 common to sq. AK and sq . HD + 3 + 4 + 5 common to sq. �- an� sq. HG )
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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,
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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< ·· ' . , · . .
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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 _.( .
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� (· · 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 • _ . -- -
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_ 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 . .-
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. ' • 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 .
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. __ . ; 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 ·
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.. - b . By nuxncering the
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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 .
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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
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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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\ ., · - 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
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, .. , �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
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< � "·..:. -- . · ,. 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
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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 . ,
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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
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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 .
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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 . ·
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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
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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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·" _ 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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f f . • � . r � r- l
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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 .
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.. , __________ _ _ ____ �: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
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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 ) .
' .
' I
i '
.;.
'• '
,·
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 :
I ! I
:
{, ..
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 ..
' I '
..
. '
L r •I
l , I
----- -·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 • ,, .
. .
•
I
' f f
" .
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
' '
' .;
• '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 .
I
I .'
(I_
)
' '
. .
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 .
I
1 / ,/ .
I I I I . I
I l 1
, . .
O•
. \ i
I '
. /1 '
I · t
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 •
\
I
• >
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 •
. ' .
I ; ' \
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 .
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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
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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
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'; = 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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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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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 .
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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 .
,,
I , \ :
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 .
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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 , ·\
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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
. ' -;. ,. � �
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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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. .
/
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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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
'\
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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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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� ��- -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
- ..--
I
t--�,..,._·� i> . -·
I '
_ ..
..
·�
........." ·-"� . '
\
.. ·-
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
..
/
_ __ 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 . . .
' '
\
I I , I !
I {, .
-r •.
I I I
. .
. /
.. . � 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.� ··:[email protected]'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 . , ..
271 .
�.·
' I
- -- ·- ---- - - -- --
i . J . I
. . '
. .. �-· --... -� .. .;. - ._
- -
·�
272
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 . ·
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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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iS . _ _ .Notes and Queries , by W . D . Henkle·, l.BJ5 .,. 81 . 19 . Nouvelle s Annales , Vol . LXII . ·
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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 .
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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 . . . -�.:·
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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
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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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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
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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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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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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���)_, __
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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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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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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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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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