A Motif of MathematicsThe curious property that John Farey observed in one of Henry Goodwyn's tables has enduring pratical and theoretic interest. This book traces the curious property, the mediant, from its initial sighting by Nicolas Chuquet and Charles Haros to its connection to the Riemann hypothesis by Jerome Franel. |
Contents
| 1 | |
| 3 | |
| 5 | |
| 7 | |
15 Nicolas Chuquet and the Regle des Nombres Moyens | 14 |
16 Rational Approximation | 16 |
17 The Mediant and the Continued Fraction | 19 |
18 John Wallis Savilian Chair of Geometry | 22 |
225 Historys Grudge Against John Farey Sr | 112 |
The Table Makers | 115 |
31 Archibalds Mathematical Table Makers | 120 |
32 Lehmers Guide to the Tables in the Theory of Numbers | 122 |
33 Tables of Tables | 124 |
34 Nevilles Tables | 128 |
35 The Farey Series of Order 1025 | 129 |
36 Reviews of The Farey Series of Order 1025 | 133 |
19 Digit Generation | 23 |
110 The Möbius Transformation | 24 |
111 Mediant Convergents | 25 |
112 The Simpson Paradox | 32 |
113 A Motif of Mathematics | 35 |
History of the Farey Sequence | 39 |
22 Charles Haros Géometre | 44 |
23 Tables pour évaluer une fraction ordinaire | 45 |
24 Tables for evaluating a common fraction | 49 |
25 The Farey Sequence as the Argument of a Mathematical Table | 56 |
26 Instruction abrégée sur les nouvelles mesures | 58 |
27 Computing Logarithms | 61 |
28 General Purpose Root Finder | 64 |
29 Haros Publications | 65 |
210 The Bureau du Cadastre | 69 |
211 Grandes Tables du Cadastre | 76 |
212 Sources of Inspiration | 77 |
213 Bookends on the Era of Organized Scientific Computation | 79 |
214 Henry Goodwyn Brewer and Table Maker | 80 |
215 The Dispersal of Goodwyns Archive | 84 |
216 Goodwyns Publications | 88 |
218 Goodwyn and the Mediant Property | 90 |
219 Decimalization of the Pound Sterling | 95 |
220 John Farey Geologist and Musicologist | 98 |
222 Proof of a Curious Theorem Regarding Numbers | 102 |
223 Delambre and Tilloch Weigh In | 105 |
224 Fareys Publications | 108 |
37 Solving Diophantine Equations | 135 |
38 RectangularPolar Conversion Tables | 137 |
39 Reviews of RectangularPolar Conversion Tables | 141 |
310 Moritz Stern and Achille Brocot | 144 |
311 Gears and Rational Approximation | 146 |
Inventions and Applications | 151 |
41 Sampling Algorithm | 153 |
42 Dithering Algorithm | 155 |
43 DecimaltoFraction Conversion | 158 |
45 Slash Arithmetic and Mediant Rounding | 160 |
46 Patterns for Weaving | 165 |
47 Networks of Resistors | 169 |
The Mediant and the Riemann Hypothesis | 173 |
51 Jérôme Franel Chair for Mathematics in the French Language | 175 |
52 The Farey Series and the Prime Numbers Problem | 177 |
53 A Synopsis of Franels Proof | 182 |
54 Remarks Concerning the Earlier Paper by Mr Franel | 186 |
55 Nevilles Search for Structure | 191 |
56 Capturing Regularization | 195 |
Explorations and Peregrinations | 201 |
62 Mediant Factorization | 206 |
63 The MayerErdös Constant | 208 |
64 Ocagnes Recursion | 210 |
65 Primes and Twin Primes | 212 |
66 The Fractional Part Function | 214 |
67 Final Words | 220 |
Common terms and phrases
algorithm approximation arctan arithmetic Brocot Bureau des Longitudes Bureau du Cadastre calculated Cambridge Cauchy Charles Haros Chuquet Complete Decimal Quotients computation continued fractions curious property décimales Delambre dénominateurs denominators less digits division divisor equation Euler Iterations example Farey fractions Farey sequence Farey series Farey’s Figure Flitcon formulæ Fowler fraction ordinaire Fraction to Decimal Franel gear Géomètre Glaisher Henry Goodwyn Instruction abrégée integer part function irreducible fractions John Farey Journal Lehmer logarithms London Math Mathematical Tables mathematicians Mathématiques measures mediant property mediant rounding Nicolas Chuquet nombres moyens nouvelles mesures number theory Olinthus Gregory paper Paris patent application prime numbers Prony proof property of vulgar published rational numbers ratios resistors result Riemann hypothesis Royal Society says Science sequence of vulgar Series of Order Simpson's Paradox solution Stern-Brocot tree Tabular Series tangents theorem Theory of Numbers U.S. Patent valeur vulgar fractions Wallis whole number