The Pythagorean Theorem - Educational Outreach

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---High above the earth, we see the moon hanging inthe evening sky at an angle and inclination that variesaccording to latitude, date, and time. The ancient Greekswere the first to measure the distance to the moon and themoon’s diameter. They actually did this using severaldifferent ways. We will follow suit by employing the modernLaw of Sines, utilizing it one of the same geocentric setupsthat the Greeks used 2000 years ago in order to make thesetwo astronomical determinations. Today, our instrument ofchoice would be a theodolite in order to make the neededprecision angular measurements. The Greeks actually hadan ancient version of the same instrument, called anastrolabe, which allowed them to make the neededmeasurements in their day. Figure 4.11 shows our setupfor both measurements.Obelisk for measuring the Moon’sshadow in the absence of an astrolabeBF0.23620 014.47228 0 0.23620 0 Moon14.47228 0A 3959miles0.5D0Surface distanceEarthfrom B to D is1000milesCMoonFigure 4.11: Measuring the MoonIn order to measure the distance to the moon, firstpick two points B and D on the earth’s surface a knowndistance apart. Figure 4.11 suggests 1000 miles, roughlythe distance that the Greeks used. Both points need tohave either the same latitude or the same longitude. In theUnited States, two points of equal latitude (such as 40 0 N)are probably a tad easily to locate than two points of similarlongitude. Point D should correspond to a time of nightwhere the full moon is straight overhead or squarely in themiddle of the ecliptic as sighted by use of an astrolabe withthe sighting point squarely in the center of the moon’s disk.150
Simultaneously, an observer at point B would sight thecenter of the moon’s disk and ascertain the exterior angleFBCat shown. Modern email would make this an excitingexercise for amateur astronomers or high school studentssince these measurements can be communicatedinstantaneously. The ancient Greeks had no such luxury inthat they had to pre-agree as to a date and time that twoobservers would make the needed measurements. Severalweeks after that, information would be exchanged and thecalculations performed.By the Law of Sines, we can find the distance BC tothe moon per the following computational sequence.BC ABsin( BAD)sin( BCD)BCsin(14.47228 3959milesBC sin(0.23620BC 240,002.5miles3959miles0) sin(0.23620 )000sin(14.47228 ) )Notice that the ratio of the distance to the radius of theearth is given by the expressionBCAB 60.62,a traditional value first obtained by the Greeks after severalrefinements and iterations.Once we have the distance to the moon, we caneasily calculate the radius of the moon. Stand at point Dand measure the sweep angle between the moon’s two limbssighting through a diameter.151
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The PythagoreanTheoremCrown Jewel o
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The Pythagorean TheoremCrown Jewel
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Table of ContentsList of Tables and
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List of Tables and FiguresTablesNum
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Figures…continuedNumber and Title
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List of Proofs and DevelopmentsSect
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PrefaceThe Pythagorean Theorem has
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1 : 2x2x2****22 13x 7 13x 7 0 : a
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1) Consider the Squares“If it was
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s090045 0ss45450450090Figure 1.3: O
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Notice that the right-triangle prop
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The middle square (minus the donut
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Our proof in Chapter 1 has been by
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2) Four Thousand Years of Discovery
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The proof Pythagoras is thought to
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cabcFigure 2.4: Annotated Square wi
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Figure 2.6 is the diagram for a sec
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1x:b2:Area(CBA)3:Area( ABC)Area(ABC
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Euclid’s proof of the Pythagorean
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First we establish that the two tri
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1 2y b b: y b c c2:A3:A4:Aunshaded
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XB'A090 CBFigure 2.12: Euclid’s C
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2.3) Liu Hui Packs the SquaresLiu H
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One of my favorites was Boxel, a ga
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2.4) Kurrah Transforms the Bride’
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Note: as is the occasional custom i
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2.5) Bhaskara Unleashes the Power o
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2.6) Leonardo da Vinci’s Magnific
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In Figure 2.24, we enlarge Step 5 a
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But, this is precisely the nature o
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2.8) Henry Perigal’s TombstoneHen
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In the century following Perigal, b
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Returning to Henry Perigal, Figure
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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
Page 101 and 102: A Pythagorean Quartet is a set of f
Page 103 and 104: 3.5) Pythagoras and the Three Means
Page 105 and 106: yxy MHxab2ab2 y x FC y ab2ab 2ab y
Page 107 and 108: Hero was the founder of the Higher
Page 109 and 110: 5 : Solve for area using the formul
Page 111 and 112: 4 : Construct the parallelogram wit
Page 113 and 114: We already have introduced Thabit i
Page 115 and 116: Some historians believe Stewart’s
Page 117 and 118: In pre-computer days, both all four
Page 119 and 120: One example of a periodic process i
Page 121 and 122: The six trigonometric functions are
Page 123 and 124: The following four relationships ar
Page 125 and 126: As with any set of identities, the
Page 127 and 128: Figure 3.17 can be used as a jumpin
Page 129 and 130: The remaining two pillars are the L
Page 131 and 132: a2Law of Cosines c2 b2 2bccos()The
Page 133 and 134: An example of a Diophantine equatio
Page 136 and 137: However, Euler’s Conjecture did n
Page 138 and 139: 4) Pearls of Fun and WonderPearls o
Page 140 and 141: We have achieved five-digit accurac
Page 142 and 143: 140212213116)74370116)(74370(370)74
Page 144 and 145: X X X X O X X O X XO X X O O OO X O
Page 146 and 147: 4.3) Earth, Moon, Sun, and StarsIn
Page 148 and 149: The four-step solution follows. All
Page 150 and 151: Figure 4.9 depicts Eratosthenes’
Page 154 and 155: The commonly accepted value is abou
Page 156 and 157: 1BC : tan( BAC)BABC BAtan(BAC)0BC
Page 158 and 159: BD ACtan( 21 )Even for the closest
Page 160 and 161: 3w 2h s w 5 s w 0.38196s&5 1s
Page 162 and 163: Figure 4.16: Triangular Phi---The h
Page 164 and 165: What remains to be done is to devel
Page 166 and 167: 3 : h32 4 h222 2 24C 2 h33162 2 2
Page 168 and 169: Epilogue: The Crown and the Jewels
Page 170 and 171: 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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