## How Round is Your Circle?: Where Engineering and Mathematics MeetHow do you draw a straight line? How do you determine if a circle is really round? These may sound like simple or even trivial mathematical problems, but to an engineer the answers can mean the difference between success and failure. John Bryant and Chris Sangwin illustrate how physical models are created from abstract mathematical ones. Using elementary geometry and trigonometry, they guide readers through paper-and-pencil reconstructions of mathematical problems and show them how to construct actual physical models themselves--directions included. It's an effective and entertaining way to explain how applied mathematics and engineering work together to solve problems, everything from keeping a piston aligned in its cylinder to ensuring that automotive driveshafts rotate smoothly. Intriguingly, checking the roundness of a manufactured object is trickier than one might think. When does the width of a saw blade affect an engineer's calculations--or, for that matter, the width of a physical line? When does a measurement need to be exact and when will an approximation suffice? Bryant and Sangwin tackle questions like these and enliven their discussions with many fascinating highlights from engineering history. Generously illustrated, |

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### Contents

Hard Lines | 1 |

11 Cutting Lines | 5 |

13 Broad Lines | 10 |

14 Cutting Lines | 12 |

15 Trial by Trials | 15 |

How to Draw a Straight Line | 17 |

21 ApproximateStraightLine Linkages | 22 |

22 ExactStraightLine Linkages | 33 |

87 The Return of the Bent CoatHanger | 165 |

88 Other Mathematical Integrators | 170 |

All Approximations Are Rational | 172 |

91 Laying Pipes under a Tiled Floor | 173 |

92 Cogs and Millwrights | 178 |

93 Cutting a Metric Screw | 180 |

94 The Binary Calendar | 182 |

95 The Harmonograph | 184 |

23 Harts ExactStraightLine Mechanism | 38 |

24 Guide Linkages | 39 |

25 Other Ways to Draw a Straight Line | 41 |

FourBar Variations | 46 |

31 Making Linkages | 49 |

32 The Pantograph | 51 |

33 The Crossed Parallelogram | 54 |

34 FourBar Linkages | 56 |

35 The Triple Generation Theorem | 59 |

36 How to Draw a Big Circle | 60 |

37 Chebyshevs Paradoxical Mechanism | 62 |

Building the Worlds First Ruler | 65 |

41 Standards of Length | 66 |

42 Dividing the Unit by Geometry | 69 |

43 Building the Worlds First Ruler | 73 |

44 Ruler Markings | 75 |

45 Reading Scales Accurately | 81 |

46 Similar Triangles and the Sector | 84 |

Dividing the Circle | 89 |

51 Units of Angular Measurement | 92 |

52 Constructing Base Angles via Polygons | 95 |

53 Constructing a Regular Pentagon | 98 |

54 Building the Worlds First Protractor | 100 |

55 Approximately Trisecting an Angle | 102 |

56 Trisecting an Angle by Other Means | 105 |

57 Trisection of an Arbitrary Angle | 106 |

58 Origami | 110 |

Falling Apart | 112 |

62 Duijvestijns Dissection | 114 |

63 Packing | 117 |

64 Plane Dissections | 118 |

65 Ripping Paper | 120 |

66 A Homely Dissection | 123 |

67 Something More Solid | 125 |

FOLLOW MY LEADER | 127 |

In Pursuit of CoatHangers | 138 |

81 What Is Area? | 141 |

82 Practical Measurement of Areas | 149 |

83 Areas Swept Out by a Line | 151 |

84 The Linear Planimeter | 153 |

85 The Polar Planimeter of Amsler | 158 |

86 The Hatchet Planimeter of Prytz | 161 |

96 A Little Nonsense | 187 |

How Round Is Your Circle? | 188 |

101 Families of Shapes of Constant Width | 191 |

102 Other Shapes of Constant Width | 193 |

103 ThreeDimensional Shapes of Constant Width | 196 |

104 Applications | 197 |

105 Making Shapes of Constant Width | 202 |

106 Roundness | 204 |

107 The British Standard Summit Tests of BS3730 | 206 |

108 ThreePoint Tests | 210 |

109 Shapes via an Envelope of Lines | 213 |

1010 Rotors of Triangles with Rational Angles | 218 |

101 1 Examples of Rotors of Triangles | 220 |

Plenty of Slide Rule | 227 |

111 The Logarithmic Slide Rule | 229 |

113 Other Calculations and Scales | 237 |

114 Circular and Cylindrical Slide Rules | 240 |

115 Slide Rules for Special Purposes | 241 |

116 The Magnameta Oil Tonnage Calculator | 245 |

117 NonLogarithmic Slide Rules | 247 |

118 Nomograms | 249 |

119 Oughtred and Delamains Views on Education | 251 |

All a Matter of Balance | 255 |

122 The Divergence of the Harmonic Series | 259 |

123 Building the Stack of Dominos | 261 |

124 The Leaning Pencil and Reaching the Stars | 265 |

125 Spiralling Out of Control | 267 |

126 Escaping from Danger | 269 |

127 Leaning Both Ways | 270 |

128 SelfRighting Stacks | 271 |

129 TwoTip Polyhedra | 273 |

1210 UniStable Polyhedra | 274 |

Finding Some Equilibrium | 277 |

132 Perpendicular Rolling Discs | 279 |

133 Ellipses | 287 |

134 Slotted Ellipses | 291 |

135 The SuperEgg | 292 |

Epilogue | 296 |

297 | |

303 | |

### Other editions - View all

How Round Is Your Circle?: Where Engineering and Mathematics Meet John Bryant,Chris Sangwin Limited preview - 2011 |

How Round Is Your Circle?: Where Engineering and Mathematics Meet John Bryant,Chris Sangwin Limited preview - 2011 |