4
$\begingroup$

I have been reading about the history of Farey fractions and I am intrigued by the appearance of the mediant in the Farey tree and related objects.

After a bit of searching, Nicolas Chuquet's Triparty (drafted 1484, lost, rediscovered and published in 1880) is the earliest reference I can find featuring the mediant - what Chuquet calls the “règle des nombres moyens”.

Although such an "operation" appears natural from the point of view of modern algebraic notations, it is not clear to me if this operation would have received any attention from previous generations of mathematicians.

Q: When did the mediant first appear in mathematics?

$\endgroup$
5
  • $\begingroup$ The Oxford English Dictionary lists mediant (noun) only in the musical sense. $\endgroup$ Mar 5, 2018 at 21:30
  • $\begingroup$ @GeraldEdgar Yes, the term has a different meaning in music and no dictionary l've looked at gives its mathematical meaning - nor does Jeff Miller's site jeff560.tripod.com/mathword.html $\endgroup$
    – nwr
    Mar 5, 2018 at 21:38
  • 1
    $\begingroup$ Lots of music ideas (from the classical Greek times) became mathematics ideas. How can you be sure that "mediant" was not one of those? $\endgroup$ Mar 5, 2018 at 21:42
  • $\begingroup$ @GeraldEdgar Very good point! And being a musician myself - albeit a 10-thumbed, tone-deaf one - I'm surprised I hadn't considered a possible relationship. Musically, the mediant is the third degree of the (diatonic) scale, which itself has seven degrees. I'm struggling to see the relationship here. Maybe if I pluck my tuba it will come to me. $\endgroup$
    – nwr
    Mar 5, 2018 at 21:47
  • $\begingroup$ @GeraldEdgar A bit of snooping and we see that the mediant is half way between the tonic (1st degree) and dominant (5th degree), so one can indeed see how the word was chosen for its mathematical usage in terms of an "operation" on rational numbers. However, it is still not clear when it was first used in this sense, or indeed when this mediant first appeared in mathematics. $\endgroup$
    – nwr
    Mar 5, 2018 at 21:53

1 Answer 1

4
$\begingroup$

Precursors of continued fractions in a geometric guise go back to ancient Greece, geometric version of producing successive terms of a continued fraction expansion was known as anthyphairesis. Fowler is perhaps the most thorough scholar of ratios and anthyphairesis. Here is from his Approximation Technique, and its Use by Wallis and Taylor:

"This procedure can be used to approximate a wide class of numbers whose values are not known in advance, like roots and logarithms (as in Taylor's calculation, below), or it can be used to find more convenient approximations to a number for which some accurate but inconvenient approximation is given (as in Wallis' calculation).

This mediant has a long history going back to classical Greece: a special case of it is to be found at Plato's Parmenides 154b-d, and, while the inequality does not occur in Euclid's Elements, the case of equality was treated as V 12 and VII 12. It was then stated and proved in Pappus, Collection VIII 8. Chuquet rediscovered it and called it 'la rigle des nombres moyens" in his Triparty en la Science des Nombres (1484). It was, we shall see, adduced by both Wallis and, in a variant form, by Taylor. It was the first theorem enunciated in Cauchy's Cours d'Analyse (1821). It is also, inter alia, the basic generating property of Farey series. Thus we have an enduring motif of mathematics."

This second paragraph inspired Guthery to write a monograph A Motif of Mathematics dedicated specifically to the mediant, its significance and history, from ancient roots to Farey series and the Riemann hypothesis.

$\endgroup$
3
  • $\begingroup$ May I ask a technical question. I'm guessing that the conjectures of Franel and Landau (that the Farey Sequence is a good approximation of the "even distribution", thus giving an equivalence of RH) do not apply to the Stern-Brocot tree? Is it easy to state "why not". Also, the Guthery text is available on amazon at a very reasonable price, so I have ordered a "hard copy" - easier to digest. $\endgroup$
    – nwr
    Mar 6, 2018 at 17:47
  • $\begingroup$ I'm starting to think that my comment above will not be "easy" to answer. I'm guessing that it is related to the bound on Stern-Brocot denominators growing too fast when compared to the bound on denominators of the Farey sequence (namely "n"). $\endgroup$
    – nwr
    Mar 6, 2018 at 18:15
  • $\begingroup$ @NickR Sorry, Nick, I am afraid I am not well versed in these conjectures. $\endgroup$
    – Conifold
    Mar 6, 2018 at 18:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.