## A Smoother Pebble: Mathematical ExplorationsThis 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. |

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### Contents

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 |

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 |

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 |

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 |

257 | |

261 | |

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### Common terms and phrases

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### 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.