A Motif of Mathematics


Scott B. Guthery


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Publication Date: Sep 11, 2010

ISBN/EAN13: 1453810579 / 9781453810576

Page Count: 264

Binding Type: US Trade Paper

Trim Size: 6" x 9"

Language: English

Color: Black & White

Table of Contents

Chapter 1 The Mediant
Non-Arithemetised Mathematics
Ratio, Proportion and Fraction
Definition of the Mediant
A Sequence of Vulgar Fractions
Nicolas Chuquet and the Règle des Nombres Moyens
Rational Approximation
The Mediant and the Continued Fraction
John Wallis, Savilian Chair of Geometry
Digit Generation
The Möbius Transformation
Mediant Convergents
The Simpson Paradox
A Motif of Mathematics
Chapter 2 History of the Farey Sequence
Mr. R. Flitcon and Question 281
Charles Haros, Géomètre
“Tables pour évaluer une fraction ordinaire . . .”
“Tables for evaluating a common fraction . . .”
The Farey Sequence as the Argument of a Mathematical Table
“Instruction abrégée sur les nouvelles mesures . . .”
Computing Logarithms
General Purpose Root Finder
Haros’ Publications
The Bureau du Cadastre
Grandes Tables du Cadastre
Sources of Inspiration
Bookends on the Era of Organized Scientific Computation
Henry Goodwyn, Brewer and Table Maker
The Dispersal of Goodwyn’s Archive
Goodwyn’s Publications
“On the Quotient arising from the Division of an Unit. . . ”
Goodwyn and the Mediant Property
Decimalization of the Pound Sterling
John Farey, Geologist and Musicologist
“On a Curious Property of Vulgar Fractions”
“Proof of a Curious Theorem Regarding Numbers”
Delambre and Tilloch Weigh In
Farey’s Publications
History’s Grudge Against John Farey, Sr.
Chapter 3 The Table Makers
Archibald’s Mathematical Table Makers
Lehmer’s Guide to the Tables in the Theory of Numbers
Tables of Tables
Neville’s Tables
The Farey Series of Order 1025
Reviews of The Farey Series of Order 1025
Solving Diophantine Equations
Rectangular-Polar Conversion Tables
Reviews of Rectangular-Polar Conversion Tables
Moritz Stern and Achille Brocot
Gears and Rational Approximation
Chapter 4 Inventions and Applications
Sampling Algorithm
Dithering Algorithm
Decimal-to-Fraction Conversion
Analog-to-Digital Conversion
Slash Arithmetic and Mediant Rounding
Patterns for Weaving
Networks of Resistors
Chapter 5 The Mediant and the Riemann Hypothesis
Jérôme Franel, Chair for Mathematics in the French Language
“The Farey Series and the Prime Numbers Problem”
A Synopsis of Franel’s Proof
“Remarks Concerning the Earlier Paper by Mr. Franel”
Neville’s Search for Structure
Capturing Regularization
Chapter 6 Explorations and Peregrinations
The Integer Part Function
Mediant Factorization
The Mayer-Erdos Constant
Ocagne’s Recursion
Primes and Twin Primes
The Fractional Part Function
Final Words
Appendix A Landau’s Proof of Franel’s Two-Dimensional Integral
Appendix B Some Consequences of the Riemann Hypothesis”
Biography
Index