The mediant is the wrong way to add two fractions but it has a surprising number of uses in mathematics other than trying to add fractions and it also encodes mathematics’ hardest outstanding problem, the Riemann Hypothesis. Guthery starts with the eponymous naming of the Farey sequence and traces the mediant back in time to its use in solving equations in the fifteenth century and forward in time to its appearance in twentieth century patent filings. “An engaging history of the mediant and its diverse range of applications in generating series from graph theory to the Riemann hypothesis, and a rehabilitation of Charles Haros and John Farey.” Historia Mathematica, November 2011
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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
| 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 |