The Pythagorean Theorem - Educational Outreach

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Lastly, throughout the book, the reader will notice a threesquaredtriangular figure at the bottom of the page. Onesuch figure signifies a section end; two, a chapter end; andthree, the book end.CreditsNo book such as this is an individual effort. Manypeople have inspired it: from concept to completion.Likewise, many people have made it so from drafting topublishing. I shall list just a few and their contributions.Elisha Loomis, I never knew you except throughyour words in The Pythagorean Proposition; but thank youfor propelling me to fashion an every-person’s updatesuitable for a new millennium. To those great Americans ofmy youth—President John F. Kennedy, John Glenn, NeilArmstrong, and the like—thank you all for inspiring anentire generation to think and dream of bigger things thanthemselves.To my two editors, Curtis and Stephanie Sparks,thank you for helping the raw material achieve fullpublication. This has truly been a family affair.To my wife Carolyn, the Heart of it All, what can Isay. You have been my constant and loving partner forsome 40 years now. You gave me the space to complete thisproject and rejoiced with me in its completion. As always,we are a proud team!John C. SparksOctober 2008Xenia, Ohio16
1) Consider the Squares“If it was good enough for old Pythagoras,It is good enough for me.” UnknownHow did the Pythagorean Theorem come to be atheorem? Having not been trained as mathematicalhistorian, I shall leave the answer to that question to thosewho have. What I do offer in Chapter 1 is a speculative,logical sequence of how the Pythagorean Theorem mighthave been originally discovered and then extended to itspresent form. Mind you, the following idealized accountdescribes a discovery process much too smooth to haveactually occurred through time. Human inventiveness inreality always has entailed plenty of dead ends and falsestarts. Nevertheless, in this chapter, I will play the role ofthe proverbial Monday-morning quarterback and execute aperfect play sequence as one modern-day teacher sees it.Figure 1.1: The Circle, Square, and Equilateral TriangleOf all regular, planar geometric figures, the squareranks in the top three for elegant simplicity, the other twobeing the circle and equilateral triangle, Figure 1.1. Allthree figures would be relatively easy to draw by our distantancestors: either freehand or, more precisely, with a stakeand fixed length of rope.17
Page 1: The PythagoreanTheoremCrown Jewel o
Page 4: The Pythagorean TheoremCrown Jewel
Page 7 and 8: Table of ContentsList of Tables and
Page 9 and 10: List of Tables and FiguresTablesNum
Page 11 and 12: Figures…continuedNumber and Title
Page 13 and 14: List of Proofs and DevelopmentsSect
Page 15 and 16: PrefaceThe Pythagorean Theorem has
Page 17: 1 : 2x2x2****22 13x 7 13x 7 0 : a
Page 21 and 22: s090045 0ss45450450090Figure 1.3: O
Page 23 and 24: Notice that the right-triangle prop
Page 25 and 26: The middle square (minus the donut
Page 27 and 28: Our proof in Chapter 1 has been by
Page 29 and 30: 2) Four Thousand Years of Discovery
Page 31 and 32: The proof Pythagoras is thought to
Page 33 and 34: cabcFigure 2.4: Annotated Square wi
Page 35 and 36: Figure 2.6 is the diagram for a sec
Page 37 and 38: 1x:b2:Area(CBA)3:Area( ABC)Area(ABC
Page 39 and 40: Euclid’s proof of the Pythagorean
Page 41 and 42: First we establish that the two tri
Page 43 and 44: 1 2y b b: y b c c2:A3:A4:Aunshaded
Page 45 and 46: XB'A090 CBFigure 2.12: Euclid’s C
Page 47 and 48: 2.3) Liu Hui Packs the SquaresLiu H
Page 49 and 50: One of my favorites was Boxel, a ga
Page 51 and 52: 2.4) Kurrah Transforms the Bride’
Page 53 and 54: Note: as is the occasional custom i
Page 55 and 56: 2.5) Bhaskara Unleashes the Power o
Page 57 and 58: 2.6) Leonardo da Vinci’s Magnific
Page 59 and 60: In Figure 2.24, we enlarge Step 5 a
Page 61 and 62: But, this is precisely the nature o
Page 63 and 64: 2.8) Henry Perigal’s TombstoneHen
Page 65 and 66: In the century following Perigal, b
Page 67 and 68: Returning to Henry Perigal, Figure
Page 69 and 70:
Figure 2.32 is President Garfield
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Even though the Cartesian coordinat
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B 2← γ →α y βx C-xC 2Figure
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Being polynomial in form, the funct
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Thus, the critical point ( C 2 ,0)i
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Sincexcpwas chosen on an arbitrary
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∂F/∂x = ∂F/∂y = 0x(C - x) 2
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Define a new function21 A2 A3 2xcp
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2.11) Shear, Shape, and AreaOur las
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Step 1: Cut the big square into two
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In Table 2.3, we briefly summarize
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We can formally state this similari
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What we will do in Section 3.2 is p
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There are three Pythagorean-triple
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Thus, if we multiply any given prim
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The last equality shows that the in
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A Pythagorean Quartet is a set of f
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3.5) Pythagoras and the Three Means
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yxy MHxab2ab2 y x FC y ab2ab 2ab y
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Hero was the founder of the Higher
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5 : Solve for area using the formul
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4 : Construct the parallelogram wit
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We already have introduced Thabit i
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Some historians believe Stewart’s
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In pre-computer days, both all four
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One example of a periodic process i
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The six trigonometric functions are
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The following four relationships ar
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As with any set of identities, the
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Figure 3.17 can be used as a jumpin
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The remaining two pillars are the L
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a2Law of Cosines c2 b2 2bccos()The
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An example of a Diophantine equatio
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However, Euler’s Conjecture did n
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4) Pearls of Fun and WonderPearls o
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We have achieved five-digit accurac
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140212213116)74370116)(74370(370)74
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X X X X O X X O X XO X X O O OO X O
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4.3) Earth, Moon, Sun, and StarsIn
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The four-step solution follows. All
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Figure 4.9 depicts Eratosthenes’
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---High above the earth, we see the
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The commonly accepted value is abou
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1BC : tan( BAC)BABC BAtan(BAC)0BC
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BD ACtan( 21 )Even for the closest
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3w 2h s w 5 s w 0.38196s&5 1s
Page 162 and 163:
Figure 4.16: Triangular Phi---The h
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What remains to be done is to devel
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3 : h32 4 h222 2 24C 2 h33162 2 2
Page 168 and 169:
Epilogue: The Crown and the Jewels
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Not only has it endured, but the Py
Page 172 and 173:
A] Greek AlphabetGREEK LETTERUpper
Page 174 and 175:
C] Geometric FoundationsThe Paralle
Page 176 and 177:
8. Congruent Triangles: Two triangl
Page 178 and 179:
3. Rectangle: A bh : P 2b 2h, b &
Page 180 and 181:
Recreational Mathematics11. Pickove
Page 182 and 183:
GGolden RatioDefined and algebraica
Page 184 and 185:
Puzzles (cont)Pythagorean Magic Squ
Page 186 and 187:
Pythagorean Theorem ProofsAlgebraic
Page 188:
Trigonometry (cont)Eratosthenes mea
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The Pythagorean Theorem - Educational Outreach

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