Benson Donald C. A Smoother Pebble: Mathematical Explorations

Oxford University Press, 2003. — 266 p. — ISBN10: 0195144368; ISBN13: 978-0195144369.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.Bridging the Gap
Ancient Fractions
The Egyptian Unit Fractions
Egyptian arithmetic
The greedy algorithm
The Babylonians and the Sexagesimal System
Sexagesimal fractions
Greek Gifts
The Heresy
Magnitudes, Ratio, and Proportion
Method 1—proportion according to Eudoxus
Method 2—Attributed to Theaetetus
The Music of the Ratios
Acoustics
The rotating circle
Waveforms and spectra
Psychoacoustics
Consonance versus dissonance
Critical bandwidth
Intervals, Scales, and Tuning
Pythagorean tuning
Approximating m octaves with n fifths
Equal-tempered tuning
The Shape of Things
Tubeland
Curvature of Smooth Curves
The inner world of the curve-bound inchworm
Curves embedded in two dimensions
Curves embedded in three dimensions
Curvature of Smooth Surfaces
Gaussian curvature — Extrinsic definition
Tubeland—A fantasy
Triangular excess
Euclidean Geometry
The parallels axiom
Non-Euclidean Geometry
Models of non-Euclidean geometries
The Calculating Eye
Graphs
The need for graphs
Materials for graphs
Clever people invented graphs
Coordinate Geometry
Synthetic versus analytic
Synthetic and analytic proofs
Straight lines 99
Conic sections 101
The Great Art
Algebra Rules
Algebra Anxiety
Arithmetic by Other Means
Symbolic algebra
Algebra and Geometry
Al-jabr
Square root algorithms
The Root of the Problem
Graphical Solutions
Quadratic Equations
Secrecy, Jealousy, Rivalry, Pugnacity, and Guile
Solving a cubic equation
Symmetry Without Fear
Symmetries of a Square
The Group Axioms
Isometrics of the Plane
Patterns for Plane Ornaments
Catalog of border and wallpaper patterns
Wallpaper watching
The Magic Mirror
Undecidability
The Magic Writing
IV A Smoother Pebble
On the Shoulders of Giants
Integration Before Newton and Leibniz
Archimedes' method for estimating pi
Circular reasoning
Completing the estimate of pi
Differentiation Before Newton and Leibniz
Descartes's discriminant method
Fermat's difference quotient method
Galileo's Lute
Falling bodies
The inclined plane
Six-Minute Calculus
Preliminaries
Functions
Limits
Continuity
The Damaged Dashboard
The broken speedometer
The derivative
The broken odometer
The definite integral
Roller Coasters
The length of a curve
Time of descent
Roller-Coaster Science
The Simplest Extremum Problems
The rectangle of maximum area with fixed perimeter
The lifeguard's calculation
A faster track
A road-building project for three towns
Inequalities
The inequality of the arithmetic and geometric means
Cauchy's inequality
The Brachistochrone
The geometry of the cycloid
A differential equation
The restricted brachistochrone
The unrestricted brachistochrone
Glossary
Notes
References
Index