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Denis Roegel
To cite this version:
Denis Roegel. A reconstruction of Neville’s Farey series of order 1025 (1950). [Research Report] 2011.
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Neville’s Farey series of order 1025 (1950)
Denis Roegel
22 October 2011
This document is part of the LOCOMAT project:
http://locomat.loria.fr
Eric Harold Neville (1889–1961) was born in London and entered Trinity College in 1907.
There he became acquainted with Bertrand Russell and G. H. Hardy.
In 1914, visiting India, he was instrumental in persuading Srinivasa Ramanujan to go back with him to England.
Neville worked on a variety of topics, and in particular on Jacobian elliptic functions.
He chaired the Mathematical Tables Committee of the British Association for the Ad- vancement of Science (BAAS) from 1931 to 1947. He contributed two sets of tables, one on the Farey series of order 1025 (1950) [22] reconstructed here, and another on rectangular-polar conversion (1956) [23].
2 Farey series
2.1 Introduction
The Farey sequence F
nof order n is the increasing sequence of fractions
pqwith 0 ≤ p ≤ q ≤ n. If three consecutive terms
abi−1i−1
,
abii
,
abi+1i+1
, of the sequence are given, the middle one can be obtained by the mediant property:
a
i−1+ a
i+1b
i−1+ b
i+1= a
ib
iThis property was described by John Farey (1766–1826) in 1816 [9].
An algorithm for the construction of this sequence had already been given by Haros in 1802 [14], and the sequence is nowadays called the Haros-Farey sequence [8].
Farey sequences are related to the construction of the Stern-Brocot tree of the ratio- nals [6, 27]. A Farey sequence is sometimes called a “Brocot table” [20].
1Such a table is useful to find rational approximations of a number.
2.2 The Farey series of order 1025
Neville’s main table [22] gives the fractions 0 <
pq≤
11in increasing order and with q ≤ 1025. Each fraction is given as a triplet (p, q, q − p), which allows to stop the table at
12. The fractions following
12can be read backwards from the end of the table.
Each page contains 400 fractions. The last page contains 282 fractions. This table covers exactly 400 pages, and contains therefore 159882 fractions. The total number of fractions, when the two parts of the table are separated, is 159881 × 2 + 1 = 319765, since the last fraction counts only for one. Each page also gives the value at the end of the previous page and the value at the beginning of the following page.
1
For an introduction to the construction of the Stern-Brocot tree and Farey sequences, see [15, 16, 12, 13].
i
and the next one in the series
y. For instance, the solution of 657x − 134y = 1 is found in the second column of page 164, right after the entry
134657, and is x = 103, y = 505.
Using this table is however not that convenient, because it appears difficult to locate a given ratio in the main table. In his introduction, Neville provides an additional table (not reproduced here), giving the location of specific fractions, but one reviewer regretted that decimal equivalents were not given at least for the last value on each page.
2.3 The Farey series of order 50
The main table is followed by three smaller tables. The first one gives the Farey series of order 50 over two pages. For each fraction
pq, a triplet (p, q, q − p) is given, as well as five-digit decimal equivalents of
pqand
q−pq.
2.4 The Farey series of order 64
The second appendix gives the Farey series of order 64 over one page. For each fraction
p
q
, a triplet (p, q, q − p) is given.
2.5 The Farey integer-series of order 100
The third appendix gives the denominators of the fractions of the Farey series of order 100.
2.6 Neville’s table of factors
This table gives the complete factorization of all integers from 1 to 1049.
3 Reconstruction
For the reconstruction, we have used Pˇatraşcu and Pˇatraşcu’s algorithm. They gave a very simple way for the computation of the next term of a sequence [24, 26]. If
ab<
dc<
pqare three consecutive terms, we have
ab++qp=
dc(mediant property) and there is a k such that kc = a + p and kd = b + q. It is easy to show that k = j
n+dbk where n is the greatest possible denominator. Once k is computed, p and q follow. This algorithm is easily adapted for a decreasing sequence.
ii
It is our pleasure to acknowledge the help of Scott Guthery who first drew our attention to Neville’s table.
iii
The following list covers the most important references related to Neville’s table. Not all items of this list are mentioned in the text, and the sources which have not been seen are marked so. We have added notes about the contents of the articles in certain cases.
[1] Raymond Clare Archibald. Scarce mathematical tables. Mathematical Tables and other Aids to Computation, 1(2):66–67, 1943. [On Brocot’s original table.]
[2] Raymond Clare Archibald. Scarce mathematical tables. Mathematical Tables and other Aids to Computation, 1(4):132, 1943. [On Brocot’s original table.]
[3] Raymond Clare Archibald. E. H. Neville: The Farey Series of Order 1025 (review).
Mathematical Tables and other Aids to Computation, 5(35):135–139, July 1951.
[4] Paul T. Bateman. Review of E. H. Neville, The Farey series of order 1025, displaying solutions of the diophantine equation bx − ay = 1. Bulletin of the American Mathematical Society, 57(4):325–326, 1951.
[5] Thomas Arthur Alan Broadbent. Obituary: Eric Harold Neville. Journal of the London Mathematical Society, 37:479–482, 1962.
[6] Achille Brocot. Calcul des rouages par approximation, nouvelle méthode. Revue chronométrique, 3:186–194, 1860.
[7] Achille Brocot. Calcul des rouages par approximation, nouvelle méthode. Paris, 1862. [not seen]
[8] Cristian Cobeli and Alexandru Zaharescu. The Haros-Farey sequence at two hundred years: A survey. Acta Universitatis Apulensis, Mathematics-Informatics, 5:1–38, 2003.
[9] John Farey. On a curious property of vulgar fractions. Philosophical Magazine, 47:385–386, 1816.
[10] James Whitbread Lee Glaisher. Report of the committee on mathematical tables.
London: Taylor and Francis, 1873. [Also published as part of the “Report of the forty-third meeting of the British Association for the advancement of science,” London: John Murray, 1874.
2
Note on the titles of the works: Original titles come with many idiosyncrasies and features (line splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore seemed pointless to capitalize works according to conventions which not only have no relation with the original work, but also do not restore the title entirely. In the following list of references, most title words (except in German) will therefore be left uncapitalized. The names of the authors have also been homogenized and initials expanded, as much as possible.
The reader should keep in mind that this list is not meant as a facsimile of the original works. The original style information could no doubt have been added as a note, but we have not done it here.
iv
[11] James Whitbread Lee Glaisher. Table, mathematical. In Hugh Chisholm, editor, The Encyclopædia Britannica, 11th edition, volume 26, pages 325–336. Cambridge, England: at the University Press, 1911.
[12] Scott B. Guthery. A motif of mathematics: history and application of the mediant and the Farey sequence. Createspace, 2010.
[13] Godfrey Harold Hardy and Edward Maitland Wright. An introduction to the theory of numbers. Oxford: Clarendon Press, 1938.
[14] Charles Haros. Tables pour évaluer une fraction ordinaire avec autant de décimales qu’on voudra ; et pour trouver la fraction ordinaire la plus simple, et qui approche sensiblement d’une fraction décimale. Journal de l’École Polytechnique, 4:364–368, 1802.
[15] Brian Hayes. On the teeth of wheels. American Scientist, 88(4):296–300, July-August 2000. [reprinted in [16]]
[16] Brian Hayes. Group theory in the bedroom, and other mathematical diversions.
New York: Hill and Wang, 2008. [This collection of articles contains a reprint of [15]. [17] is a French translation of this book.]
[17] Brian Hayes. L’horloge de l’éternité. Paris: Vuibert, 2010. [Translation of [16].]
[18] Walter James Langford, Thomas Arthur Alan Broadbent, and Reuben Louis
Goodstein. Obituary: Professor Eric Harold Neville, M.A., B.Sc. The Mathematical Gazette, 48(364):131–145, May 1964.
[19] Roger Mansuy. Achille Brocot, mathématicien à ses heures, 2008. [available on http://www.math.ens.fr/culturemath/histoire%20des%20maths]
[20] Henry Edward Merritt. Gear trains, including a Brocot table of decimal equivalents and a table of factors of all useful numbers up to 200,000. London: Sir Isaac
Pitman & Sons, Ltd., 1947. [The table of useful numbers was reconstructed in [25].]
[21] Eric Harold Neville. The structure of Farey series. Proceedings of the London Mathematical Society, 51 (series 2):132–144, 1949.
[22] Eric Harold Neville. The Farey series of order 1025, displaying solutions of the Diophantine equation bx − ay = 1, volume 1 of Royal Society Mathematical Tables.
Cambridge: University Press, 1950.
[23] Eric Harold Neville. Rectangular-polar conversion tables, volume 2 of Royal Society Mathematical Tables. Cambridge: University Press, 1956. [not seen]
v
(ANTS 2004), volume 3076 of Lecture Notes in Computer Science, pages 358–366.
Springer, 2004.
[25] Denis Roegel. A reconstruction of Merritt’s table of “useful numbers” (1947).
Technical report, LORIA, 2011. [This is a reconstruction of the table in [20].]
[26] Norman Routledge. Computing Farey series. The Mathematical Gazette, 92(523):55–62, March 2008.
[27] Moritz Abraham Stern. Über eine zahlentheoretische Funktion. Journal für die reine und angewandte Mathematik, 55(3):193–220, 1858.
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