how round is circle

8/7/2019 How round is circle

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How Round Is Your Circle?WHERE ENGINEERING AND MATHEMATICS MEET

John Bryant and Chris Sangwin

PRINCETON UNIVERSITY PRESSPRINCETON AND OXFORD


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Contents

Preface xiAcknowledgements xixCHAPTER1

Hard Lines1.1 C uttin g Lines1.2 The Pyth agore an Th eorem1.3 Broad Lines1.4 C uttin g Lines1.5 Trial by Tr ials

C H A P T E R 2How to Draw a Straight Line2.1 A pproxim ate-Straight-Line Linkages2.2 Exact-Straight-Line Link ages2.3 H art 's Exact-Straight-Line M echanism2.4 Guide Linkages2.5 O ther W ays to Draw a Straight Line

C H A P T E R 3Four-Bar Variations3.1 Making Linkages3.2 The Pan tograp h3.3 The C rossed Parallelogram3.4 Four-Bar Linkages3.5 The Triple G eneration Th eor em3.6 How to Draw a Big Circle3.7 Ch ebyshev 's Paradoxical M echanism

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IX


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CONTENTS

C H A P T E R 4Building th e W orld's First Ruler 654.1 Sta nd ards of Length 664.2 D M ding the Unit by Geom etry 694.3 Building the W orld's First Ruler 734.4 Ruler M arkings 754.5 Reading Scales Accurately 814.6 Similar Trian gles an d the Sector 84

C H A P T E R 5D M ding the Circle 895.1 Un its of A ngular M easurem ent 925.2 Con structin g Base Angles via Polygons 955.3 Co nstru cting a Regulr Pen tagon 985.4 Building th e W orld's First Pro trac tor 1005.5 Approximately Trisec ting an Angle 1025.6 Trise cting an Ang le by Other Means 1055.7 Tris ectio n of an A rbitrary Angle 1065.8 Origam i 110

C H A P T E R 6Falling A pa rt 1126.1 A dding Up Seque nces of Intege rs 1126.2 Duijvestijn's D issection 1146.3 Packing 1176.4 Plane D issec tions 1186.5 Ripp ing Pap er 1206.6 A Hom ely D issection 1236.7 Som ething More Solid 125

C H A P T E R 7Follow My Leader 127

C H A P T E R 8In Pu rsuit of Coat-Hangers 1388.1 W hat Is Area? 1418.2 Practical M easu rem ent of Areas 149


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C O N T E N T S XI

8.3 Areas Swept Out by a Line 1518.4 The Linear Planimeter 1538.5 The Polar Planimeter of Amsler 1588.6 The Hatchet Planimeter of Prytz 1618.7 The Return of the Bent Coat-Hanger 1658.8 Other Mathematical Integrators 170

C H A P T E R 9All Approximations Are Rational9.1 Laying Pipes u n de r a Tiled Floor9.2 Cogs and Millwrights9.3 Cutting a Metrie Screw9.4 The Binary C alendar9.5 The Harm onog raph9.6 A Little N onsense !

172173178180182184187

C H A P T E R 1 0How Roun d Is Your Circle?10.110.210.310.410.510.610.710.810.910.1010.1110.12

Families of Shapes of Constant WidthOther Shapes of Constant WidthThree-Dimensional S hapes of Constant WidthApplicationsMaking Shapes of Constant WidthRoundnessThe British Standard Summ it Tests of BS3730Three-Point TestsShapes via an Envelope of LinesRotors of Triangles with Rational AnglesExamples of Rotors of TrianglesModern and Accurate Roundness Methods

188191193196197202204206210213218220224

C H A P T E R 1 1Plenty of Slide Rule 22711.1 The Logarithmic Slide Rule 22911.2 The Invention of Slide Rules 23311.3 Other Calculations and Scales 23711.4 Circular and Cylindrical Slide Rules 240

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X U C O N T E N T S

11.5 Slide Rules for Special Pu rpo ses 24 111.6 The M agnam eta Oil Ton nage Calculator 24511.7 Non-Logarithmic Slide Rules 24 711.8 No mo grams 24911.9 Ou ghtred and Delam ain's Views on Education 251

C H A P T E R 1 2All a M atter of Balance 25512.1 Stacking Up 25512.2 The Divergence of the Harm on ie Series 25912.3 Building the Stack of Do mino s 26112.4 The Leaning Pencil an d Reaching the Stars 26512.5 Spiralling Ou t of Con trol 26712.6 Escaping from Dan ger 26912.7 Leaning Both Ways! 27012.8 Self-Righting Stacks 27 112.9 Two-Tip Poly hed ra 27312.10 Uni-Stable Po lyhed ra 274

C H A P T E R 1 3Finding Some Equilibrium 27713.1 Rolling Uph ill 27713.2 Pe rpend icu lar Rolling Discs 27913.3 Ellipses 28 713.4 Slotted Ellipses 29113.5 Th e Super-Egg 292

Epilogue 296References 297Index 303

how round is circle

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