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How Round Is Your Circle? Where Engineering and Mathematics Meet

John Bryant and Chris Sangwin
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

A few years ago I had an eye-opening experience when I was involved with design and manufacture of some novel automotive pushrods. These pushrods are cylindrical with a center hollow steel tube surrounded by aluminum reinforced with ceramic fibers. (As a side note, when I was in graduate school I could not have imagined myself becoming intimately familiar with details of mechanical engineering, machining and metallurgy that were vital in this enterprise.) Although I have worked in applied areas for most of my career, this exercise surprised me. Pushrods should have round cross-sections, the central steel tube should be round, and its central axis should be the central axis for the pushrod. Deviations in geometry can reduce engine performance. But roundness and concentricity are amazingly hard to measure in practice without destroying the pushrods!

How Round Is Your Circle?: Where Engineering and Mathematics Meet doesn’t address this problem directly, but it does discuss some related questions. I think that it might offer a few interesting surprises to mathematicians. How do you draw a straight line or verify that a circle is round? When does the width of saw blade cut matter? These apparently trivial problems can be central to the correct execution of an engineering design. According to the authors (a chemical engineer and a mathematician), “…we have tried to illustrate why mathematicians should take the practical problems of engineering seriously… We have also tried to illustrate what happens to familiar and apparently simple things when you dig a little deeper, or change something slightly.”

The authors are fascinated by analog scientific instruments (for example, planimeters, linkages, and slide rules), and the book contains a number of physical models as well as instructions for making and using them. (For example, there are instructions for building a hatchet planimeter using a coat hanger and washers.) I suspect that the opportunity to write about these instruments and other physical models was the main reason the authors wrote this book.

This is a collection of cool stuff, at least for those who share the authors’ fascination, but there isn’t really much of a coherent story here. The mathematics employed includes some geometry, trigonometry and elementary calculus, and there are at least a few good examples and applications that could be of value in the classroom. However, the book just doesn’t live up to its title — we see only a very little piece of the ground where engineering and mathematics meet.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

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