Preface xiii Acknowledgements xix

Chapter 1: Hard Lines 1

1.1 Cutting Lines 5

1.2 The Pythagorean Theorem 6

1.3 Broad Lines 10

1.4 Cutting Lines 12 1.5 Trial by Trials 15

Chapter 2: How to Draw a Straight Line 17

2.1 Approximate-Straight-Line Linkages 22

2.2 Exact-Straight-Line Linkages 33

2.3 Hart's Exact-Straight-Line Mechanism 38

2.4 Guide Linkages 39

2.5 Other Ways to Draw a Straight Line 41

Chapter 3: Four-Bar Variations 46

3.1 Making Linkages 49

3.2 The Pantograph 51

3.3 The Crossed Parallelogram 54

3.4 Four-Bar Linkages 56

3.5 The Triple Generation Theorem 59

3.6 How to Draw a Big Circle 60

3.7 Chebyshev's Paradoxical Mechanism 62

Chapter 4: Building the World's First Ruler 65

4.1 Standards of Length 66

4.2 Dividing the Unit by Geometry 69

4.3 Building the World's First Ruler 73

4.4 Ruler Markings 75

4.5 Reading Scales Accurately 81

4.6 Similar Triangles and the Sector 84

Chapter 5: Dividing the Circle 89

5.1 Units of Angular Measurement 92

5.2 Constructing Base Angles via Polygons 95

5.3 Constructing a Regular Pentagon 98

5.4 Building the World's First Protractor 100

5.5 Approximately Trisecting an Angle 102

5.6 Trisecting an Angle by Other Means 105

5.7 Trisection of an Arbitrary Angle 106

5.8 Origami 110

Chapter 6: Falling Apart 112

6.1 Adding Up Sequences of Integers 112

6.2 Duijvestijn's Dissection 114

6.3 Packing 117

6.4 Plane Dissections 118

6.5 Ripping Paper 120

6.6 A Homely Dissection 123

6.7 Something More Solid 125

Chapter 7: Follow My Leader 127

Chapter 8: In Pursuit of Coat-Hangers 138

8.1 What Is Area? 141

8.2 Practical Measurement of Areas 149

8.3 Areas Swept Out by a Line 151

8.4 The Linear Planimeter 153

8.5 The Polar Planimeter of Amsler 158

8.6 The Hatchet Planimeter of Prytz 161

8.7 The Return of the Bent Coat-Hanger 165

8.8 Other Mathematical Integrators 170

Chapter 9: All Approximations Are Rational 172

9.1 Laying Pipes under a Tiled Floor 173

9.2 Cogs and Millwrights 178

9.3 Cutting a Metric Screw 180

9.4 The Binary Calendar 182

9.5 The Harmonograph 184

9.6 A Little Nonsense! 187

Chapter 10: How Round Is Your Circle? 188

10.1 Families of Shapes of Constant Width 191

10.2 Other Shapes of Constant Width 193

10.3 Three-Dimensional Shapes of Constant Width 196

10.4 Applications 197

10.5 Making Shapes of Constant Width 202

10.6 Roundness 204

10.7 The British Standard Summit Tests of BS3730 206

10.8 Three-Point Tests 210

10.9 Shapes via an Envelope of Lines 213

10.10 Rotors of Triangles with Rational Angles 218

10.11 Examples of Rotors of Triangles 220

10.12 Modern and Accurate Roundness Methods 224

Chapter 11: Plenty of Slide Rule 227

11.1 The Logarithmic Slide Rule 229

11.2 The Invention of Slide Rules 233

11.3 Other Calculations and Scales 237

11.4 Circular and Cylindrical Slide Rules 240

11.5 Slide Rules for Special Purposes 241

11.6 The Magnameta Oil Tonnage Calculator 245

11.7 Non-Logarithmic Slide Rules 247

11.8 Nomograms 249

11.9 Oughtred and Delamain's Views on Education 251

Chapter 12: All a Matter of Balance 255

12.1 Stacking Up 255

12.2 The Divergence of the Harmonic Series 259

12.3 Building the Stack of Dominos 261

12.4 The Leaning Pencil and Reaching the Stars 265

12.5 Spiralling Out of Control 267

12.6 Escaping from Danger 269

12.7 Leaning Both Ways! 270

12.8 Self-Righting Stacks 271

12.9 Two-Tip Polyhedra 273

12.10 Uni-Stable Polyhedra 274

Chapter 13: Finding Some Equilibrium 277

13.1 Rolling Uphill 277

13.2 Perpendicular Rolling Discs 279

13.3 Ellipses 287

13.4 Slotted Ellipses 291

13.5 The Super-Egg 292

Epilogue 296

References 297

Index 303