A Smoother Pebble: Mathematical Explorations

Donald C. Benson

Oxford University Press, Oct 30, 2003 - Mathematics - 280 pages

This book takes a novel look at the topics of school mathematics--arithmetic, geometry, algebra, and calculus. In this stroll on the mathematical seashore we hope to find, quoting Newton, "...a smoother pebble or a prettier shell than ordinary..." This book assembles a collection of mathematical pebbles that are important as well as beautiful.

Contents

• Ancient Fractions: 5
• The Egyptian Unit Fractions: 6
• Egyptian arithmetic: 8
• The greedy algorithm: 13
• The Babylonians and the Sexagesimal System: 15
• Sexagesimal fractions: 16
• Greek Gifts: 18
• The Heresy: 20
• Magnitudes Ratio and Proportion: 22
• Method 1 proportion according to Eudoxus: 24
• Method 2 Attributed to Theaetetus: 26
• The Music of the Ratios: 33
• Acoustics: 35
• The rotating circle: 36
• Waveforms and spectra: 39
• Psychoacoustics: 44
• Consonance versus dissonance: 45
• Critical bandwidth: 46
• Intervals Scales and Tuning: 49
• Approximating m octaves with n fifths: 51
• Equaltempered tuning: 54
• Tubeland: 61
• Curvature of Smooth Curves: 62
• Curves embedded in two dimensions: 63
• Curves embedded in three dimensions: 64
• Curvature of Smooth Surfaces: 65
• Gaussian curvature Extrinsic definition: 66
• Tubeland A fantasy: 68
• Triangular excess: 71
• Euclidean Geometry: 73
• The parallels axiom: 75
• Models of nonEuclidean geometries: 77
• The Calculating Eye: 82
• Graphs: 84
• The need for graphs: 85
• Materials for graphs: 86
• Clever people invented graphs: 89
• Coordinate Geometry: 93
• Synthetic versus analytic: 94
• Synthetic and analytic proofs: 95
• Straight lines: 99
• Conic sections: 101
• Algebra Rules: 111
• Algebra Anxiety: 112
• Arithmetic by Other Means: 115
• Symbolic algebra: 116
• Algebra and Geometry: 120
• Aljabr: 121
• Square root algorithms: 122
• The Root of the Problem: 128
• Graphical Solutions: 129
• Quadratic Equations: 130
• Secrecy Jealousy Rivalry Pugnacity and Guile: 135
• Solving a cubic equation: 138
• Symmetry Without Fear: 142
• Symmetries of a Square: 145
• The Group Axioms: 148
• Isometrics of the Plane: 150
• Patterns for Plane Ornaments: 151
• Wallpaper watching: 158
• The Magic Mirror: 160
• The Magic Writing: 162
• On the Shoulders of Giants: 167
• Integration Before Newton and Leibniz: 168
• Circular reasoning: 170
• Completing the estimate of pi: 171
• Differentiation Before Newton and Leibniz: 172
• Descartess discriminant method: 173
• Fermats difference quotient method: 176
• Galileos Lute: 177
• The inclined plane: 179
• SixMinute Calculus: 184
• Preliminaries: 185
• Functions: 186
• Limits: 188
• Continuity: 189
• The Damaged Dashboard: 191
• The broken speedometer: 193
• The derivative: 194
• The broken odometer: 199
• The definite integral: 201
• Roller Coasters: 206
• Time of descent: 208
• RollerCoaster Science: 212
• The Simplest Extremum Problems: 213
• The lifeguards calculation: 215
• A faster track: 217
• A roadbuilding project for three towns: 219
• Inequalities: 220
• The inequality of the arithmetic and geometric means: 221
• Cauchys inequality: 223
• The geometry of the cycloid: 227
• A differential equation: 228
• The restricted brachistochrone: 230
• The unrestricted brachistochrone: 235
• Glossary: 243
• Notes: 249
• References: 257
• Index: 261

Popular passages

Page 1 - I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.‎

Bibliographic information

• Title: A Smoother Pebble: Mathematical Explorations
• Author: Donald C. Benson
• Publisher: Oxford University Press, 2003
• ISBN: 0198032978, 9780198032977
• Length: 280 pages
• Subjects: Mathematics / Applied; Mathematics / General; Mathematics / History & Philosophy