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A Motif of Mathematics

Scott B. Guthery
Docent Press
Publication Date: 
Number of Pages: 
[Reviewed by
Underwood Dudley
, on

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 F7 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"