Farey Series, A Story

Mathematical theorems often have descriptive names like Isoperimetric Inequality or the Law of Cosines. Others bear the name of their discoverer: Euler's Formula, Abelian group, or Bertrand's Paradox. However, it is a recorded part of the mathematical folklore that many theorems are misnamed [K. O. May]. This is known as the Law of Eponyms:

If Theorem X bears the name of Y, then it was probably first stated by and/or proved by Z.

For example, not all mathematicians are happy with the customary attribution of Venn Diagrams to John Venn. Wilson's theorem was not proven by Wilson (1741-1793) but by J.L.Lagrange in 1770 and Stirling's formula was discovered by Abraham de Moivre. So I was not at all surprised to read in Hardy's Apology (p 81-82) the following remark concerning J. Farey of the Farey Series fame:

... Farey is immortal because he failed to understand a theorem which Haros had proved perfectly fourteen years before ...

In Hardy and Wright (p 36) there appears another note

The history of 'Farey series' is very curious. Theorems 28 and 29 (properties of the Farey series, ab) seem to have been stated and proved first by Haros in 1802. Farey did not publish anything on the subject until 1816, when he stated Theorem 29 in a note to Philosophical Magazine. He gave no proof, and it is unlikely that he had one, since he seems to have been at the best an indifferent mathematician.

Cauchy, however, saw Farey's statement, and supplied the proof. Mathematician's generally have followed Cauchy's example in attributing the results to Farey, and the series will no doubt continue to bear his name.

Farey has a notice of twenty lines in the Dictionary of National Biography, where he is described as a geologist. As a geologist he is forgotten, and his biographer does not mention the one thing in his life which survives.

I wholeheartedly concur with contemporary attempts to present mathematics as an outcome of human activity. For those struggling with various aspects of math it must be consoling to learn that the creators of mathematics are as much human as the next man and no less error prone than other mortals. Mathematics is not being developed by stating axioms and deriving conclusions one after another. Being on lookout for historical confirmations, I could not miss an opportunity to point out that a non mathematician had distinguished himself by noticing a beautiful regularity in a certain sequence of numbers without being able to prove it. I introduced Farey as

... a British geologist who in 1816 published the statement to the effect that in the Farey series the middle of any three successive terms is the mediant of the other two. The proof has been eventually supplied by Cauchy. The series nonetheless bears the name of Farey.

The reason I tried to soften Hardy's sentiment was that, candidly, I just did not know enough about Farey himself or the history of the series that bears his name. We shall never know why Cauchy found it important to attach Farey's name to the theorem he proved. He might have wanted to show appreciation of the fact that all its simplicity notwithstanding the series was not discovered until the beginning of the 19th century.

Authors took a clue from Hardy. For example, in The Penguin Dictionary of Mathematics, the corresponding entry starts with

Farey sequence (of order n) (C.Haros, 1802; J.Farey, 1816) The finite ...

Conway and Guy have a footnote with the quote from Hardy and Wright. Beiler contributes more information and a kind opinion which is still influenced by Hardy's remark:

John Farey was a somewhat versatile man who lived in the Napoleonic era. He was a surveyor who collected rocks and minerals and found time to write articles in the Philosophical Magazine on such diverse subjects as geology, music, decimal coinage, carriage wheels, comets - and Farey series! He did not consider his discovery particularly important, little realizing that his name would go down in mathematical lore because he had found something which had entirely escaped such acute observers as Fermat and Euler. One must report with regret, however, that once again the man whose name was given to a mathematical relation was not the original discoverer so far as the records go. One C. Haros apparently anticipated Farey by fourteen years, but this fact was unknown to the mathematician Cauchy, who attributed the discovery to Farey, and others repeated Cauchy's statement. Anyway, "Farey series" sounds better than "Haros series." Who knows - perhaps some Arabian mathematician anticipated Haros by a thousand years.

(In Beiler, I found another mentioning of Arab mathematics. Referring to the identity in Fermat's Last Theorem he says (p 280): The Arabs knew that it was impossible 700 years before Fermat's day and even possessed a proof, albeit a faulty one. A humane approach to mathematics in all its glory!)

Maxim Bruckheimer and Abraham Arcavi published an article in The Mathematical Intelligencer that originated with remarks in a couple of other books that referred to the Farey series story second handedly. They discovered the original letter (it was not an article) Farey wrote to the Philosophical Magazine

On a curious Property of vulgar Fractions.
By Mr. J. Farey, Sen. To Mr. Tilloch

Sir. - On examining lately, some very curious and elaborate Tables of "Complete decimal Quotients," calculated by Henry Goodwyn, Esq. of Blackheath, of which he has printed a copious specimen, for private circulation among curious and practical calculators, preparatory to the printing of the whole of these useful Tables, if sufficient encouragement, either public or individual, should appear to warrant such a step: I was fortunate while so doing, to deduce from them the following general property; viz.

If all the possible vulgar fractions of different values, whose greatest denominator (when in their lowest terms) does not exceed any given number, be arranged in the order of their values, or quotients; then if both the numerator and the denominator of any fraction therein, be added to the numerator and the denominator, respectively, of the fraction next but one to it (on either side), the sums will give the fraction next to it; although, perhaps, not in its lowest terms.

For example, if 5 be the greatest denominator given; then are all the possible fractions, when arranged, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, and 4/5; taking 1/3, as the given fraction, we have (1+1)/(5+3) = 2/8 = 1/4 the next smaller fraction than 1/3; or (1+1)/(3+2) = 2/5, the next larger fraction to 1/3. Again, if 99 be the largest denominator, then, in a part of the arranged Table, we should have 15/52, 28/97, 13/45, 24/83, 11/38, &c.; and if the third of these fractions be given, we have (15+13)/(52+45) = 28/97 the second: or (13+11)/(45+38) = 24/83 the fourth of them: and so in all the other cases.

I am not acquainted, whether this curious property of vulgar fractions has been before pointed out?; or whether it may admit of any easy or general demonstration?; which are points on which I should be glad to learn the sentiments of some of your mathematical readers; and am

    Sir,
    Your obedient humble servant,
J. Farey.
Howland-street.

So Farey himself never claimed to have proven anything. He appears to have discovered a curious fact and demonstrates a laudable desire to learn more by sharing the discovery.

Bruckheimer and Arcavi have also found the original paper by C.Haros from 1802. Haros was looking into approximating decimal fractions with common fractions whose denominators do not exceed 99. It was convenient for him to arrange the fractions according to their magnitudes: 1/99, 1/98, ..., 1/4, 1/3, 1/2, 2/3, ..., 96/99, 97/99, 98/99. He noticed and verified that in this sequence each fraction differs from its neighbor by the reciprocal of the product of their denominators. He also showed that if two fractions a/b and c/d satisfy bc - ad = 1, and x/y is another fraction between a/b and c/d that satisfies two similar conditions with regard to both a/b and c/d, then x/y is the mediant fraction of a/b and c/d. Farey did not prove that in his series any fraction is the mediant of its neighbors. It does not appear that C.Haros even stated this property explicitly.

Cauchy published his proof in 1816, immediately after the appearance of a French translation of Farey's letter. It appears unlikely that he was aware of C.Haros' work. Even if he were, he would probably still credit J.Farey with the statement whose proof he supplied.

Historical and social backgrounds are becoming a part of mathematics education and for a good reason so. For a mathematician writing a text book it may be easier to collect and present all the necessary mathematical facts than to unearth all the pertinent historical information. However, an effort should be extended to have the latter as well verified as the mathematical body of the text.

References

  1. A. H. Beiler, Recreations in the Theory of Numbers, Dover, 1966
  2. M.Bruckheimer and A.Arcavi, Farey Series and Pick's Area Theorem, The Mathematical Intelligencer v 17 (1995), no 4, pp 64-67.
  3. J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag, NY, 1996.
  4. G. H. Hardy, A Mathematician's Apology, Cambridge University Press, 1994.
  5. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oxford Science Publications, 1996.
  6. K. O. May, Historiographic Vices I. Logical attribution, Historia Math. 2 (1975), 185-187.
  7. The Penguin Dictionary of Mathematics, J.Daintith and R.D.Nelson (eds), Penguin Books, 1989

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