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The book’s subtitle is “History and Application of the Mediant and the Farey Sequence”, which describes its content. The mediant of two fractions a/b and c/d is (a+c)/(b+d). The Farey sequence of order n is a list of fractions in lowest terms from 0 to 1, whose denominators do not exceed n, in increasing order. The sequence F_{7} is
0/1, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1/1.
Both have interesting properties — for example, the 2-by-2 matrices formed from successive terms in a Farey sequence all have determinant –1 and each term in a sequence is the mediant of its neighbors.
The author is a mathematics Ph.D. from Michigan State University (1969) who has spent his career in industry, at Bell Labs, Mathematica, and Schlumberger, among other places. He is a co-author of The Smart Card Developer’s Kit (1998) and Developing MMS Applications (2003).
The book is a print-on-demand volume from the author’s own small publishing company, which is one reason for its welcome low price. However, we must take the bad with the good, in this case the bad being the lack of an editor. The book contains a large amount of material and has a list of references that contains two hundred and forty-seven items, but it struck me as being a grab bag, as if the author had used a search engine on “Farey sequence” and “mediant”, found things, and thrown them in. If I had been his editor, I would have advised against reprinting papers by Franel, Landau, Littlewood, and Haros, especially the last, which occurs both in its original French and in an English translation. I would have recommended omitting some illustrations, such as the illegible 1795 charter of the French bureau of longitudes (p. 70) that, though they are no doubt easily available on the internet, have little to do with the book’s subject. I would have replaced the thirteen instances of Mathematica code with a note to the effect that the author would happily furnish code if asked for it.
Of course, readers can always skip over things that they do not care about, and they should find many things of interest. There is an explanation of why the Riemann hypothesis is true if and only if Farey sequences are sufficiently regular, there is material on the once important art and science of table-making, we see how mediants can be used to calculate square roots: there is much that will be new to general mathematical readers and that is not hard to understand.
Editors can also fix things that need fixing, of which there are plenty in this book. I could see what the author was getting at in (p. 163) “while there a number of generally agreed upon metrics exist for error,” but my mind-reading skills were not up to (p. 125) “whose second edition was published at last light in 1962.” Pages 62 and 63 contain several new words: formulæto, formulæsuch, formulæwere, formulædo, and formulæthat. It is amusing to read of V. I. Arnold giving a lecture mentioning polynomial rings in Paris in 1797 (p. 6), but less so to see him referred to in the index as Arnol’l and elsewhere in the text (p. 33) with an extra apostrophe as “Arnol’l’ ”. “Fibbonacci” gets two mentions on p. 171 but he does not make it into the index, which is poor. I could go on at much greater length — the number of errors may be near two hundred — but you should get the idea. If you are a person who is annoyed by such things you will be very annoyed, but if such trifles do not bother you, you will be able to read the book with equanimity.
In any event, the book contains good stuff, and how far wrong can you go for $17.99?
Woody Dudley has written some things that made it into print and he knows what happens: when he sees his marvelous prose in type, selective blindness strikes and his errors all become invisible.
Chapter 1 | The Mediant |
Non-Arithmetic Mathematics | |
Ratio, Proportion and Fraction | |
Definition of the Mediant | |
A Sequence of Vulgar Fractions | |
Nicholas 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 | |
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-Erdös 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" |