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I heard or have read the following nice explanation for the origin of the convention that one uses (almost) always $x,y,z$ for variables. (This question was motivated by question Origin of symbol *l* for a prime different from a fixed prime?)

It seems this custom is due to the typesetter of Descartes. Descartes used initially other letters (mainly $a,b,c$) but the typesetter had the same limited number of lead symbols for each of the 26 letters of the Roman alphabet. The frequent use of variables exhausted his stock and he asked thus Descartes if he could use the last three letters $x,y,z$ of the alphabet (which occur very rarely in French texts).

Does anyone know if this is only a (beautiful) legend or if it contains some truth? (I checked that Descartes uses indeed already $x,y,z$ generically for variables in his printed works.)

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Roland Bacher
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    Roland, this is definitely CW! – Wadim Zudilin Jul 02 '10 at 14:00
  • Wadim, what is CW? – Roland Bacher Jul 02 '10 at 14:05
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    As I learned your story from high school, I am pretty sure it's a legend. From wiki (standard convention): http://en.wikipedia.org/wiki/Variable_%28mathematics%29 – Wadim Zudilin Jul 02 '10 at 14:05
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    Wadim, thank you for the link which gives a more likely explanation. By the way, I find it amazing how useful notational conventions are. Illustration: A function $\epsilon$ is continuous at $f$ if for arbitrarily small positive $N$ there exists a positive $x$ such that $\vert\epsilon(f)-\epsilon(A)\vert<N$ if $\vert A-f\vert\leq x$. (Writing such an example makes me empathic to students.) – Roland Bacher Jul 02 '10 at 14:23
  • The Oxford English Dictionary credits Descartes with the introduction (in their entry for "X"), but makes no mention of the issue with the printers. The OED also says "no evidence in support of the hypothesis that x is derived ultimately from the mediæval transliteration" from Arabic. – PersonX Jul 02 '10 at 14:27
  • @Roland: I'll check your newer convention on my most experienced students. :-) I've tried to dig something about your legend but it looks coming from nowhere... I am definitely confused as I heard it in exactly the same from many years ago. – Wadim Zudilin Jul 02 '10 at 14:40
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    I'm afraid the concentration of non-mathematics questions is getting out of hand here, so I'm closing this one. Incidentally, the OED also mentions that Descartes started with $z$ and progressed backwards. – S. Carnahan Jul 02 '10 at 14:44
  • Scott, aren't there holidays? (from maths) $\ddot{\smile}$ – Wadim Zudilin Jul 02 '10 at 14:52
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    You'll find details on this point in Cajori's History of mathematical notations, ¶340. He credits Descartes in his La Géometrie for the introduction of $x$, $y$ and $z$ (and more generally, usefully and interestingly, for the use of the first letters of the alphabet for known quantities and the last letters for the unknown quantities) He notes that Descartes used the notation considerably earlier: the book was published in 1637, yet in 1629 he was already using $x$ as an unknown (although in the same place $y$ is a known quantity...); also, he used the notation in manuscripts dated – Mariano Suárez-Álvarez Jul 02 '10 at 16:44
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    (continued) earlier than the book by years.

    It is very, very interesting to read through the description Cajori makes of the many, many other alternatives to the notation of quantities, and as one proceeds along the almost 1000 pages of the two volume book, one can very much appreciate how precious are the notations we so much take for granted!

     – Mariano Suárez-Álvarez Jul 02 '10 at 16:47
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    Mariano, when the question is reopened, could you kindly post your comment as an answer? – Joel David Hamkins Jul 02 '10 at 17:49
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    It's a bit late to post this link, but I'll do so anyway. Meta thread: http://tea.mathoverflow.net/discussion/486/x-y-z – Anton Geraschenko Jul 02 '10 at 22:14
  • Similar question: http://mathoverflow.net/questions/104031/why-unknowns-are-usually-denoted-by-x-closed – Alexander Chervov Dec 07 '12 at 18:55

2 Answers2

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You'll find details on this point (and precise references) in Cajori's History of mathematical notations, ¶340. He credits Descartes in his La Géometrie for the introduction of $x$, $y$ and $z$ (and more generally, usefully and interestingly, for the use of the first letters of the alphabet for known quantities and the last letters for the unknown quantities) He notes that Descartes used the notation considerably earlier: the book was published in 1637, yet in 1629 he was already using $x$ as an unknown (although in the same place $y$ is a known quantity...); also, he used the notation in manuscripts dated earlier than the book by years.

It is very, very interesting to read through the description Cajori makes of the many, many other alternatives to the notation of quantities, and as one proceeds along the almost 1000 pages of the two volume book, one can very much appreciate how precious are the notations we so much take for granted!

Mariano Suárez-Álvarez
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    Mariano, I now understand why certain historical Qs mught be interesting. And this is definitely the case when both the question and the answer are good. – Wadim Zudilin Jul 03 '10 at 14:50
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    Cajori's book is a fantastic read, thank you for pointing it out! – Victor Protsak Jul 03 '10 at 23:53
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    Mariano - I quoted and linked to this answer in response to the same question over at Math.SE (http://math.stackexchange.com/questions/2936/why-are-x-and-y-such-common-variables-in-todays-equations-how-did-their-use-orig/2965#2965). Hope you're ok with that. – George Lowther Aug 21 '10 at 21:56
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I'm often amazed how legends propose completely preposterous "explanations". Why on earth would a typesetter have equal numbers of lead symbols for every letter, given that they have a very non-uniform distribution in ordinary text? And why would it take Descartes' math formulas to discover that the distribution is not balanced?

If on the other hand the typesetter had a distribution of lead characters that corresponds roughly to their use in non-math texts (which would seem to be a reasonable assumption), then surely it would disturb this balance much more profoundly to start systematically using the least-occurring letters in mathematical formulae than to stick to using those letters which are already in common use. I don't think such the theory put forward is even worth seriously investigating.

Marc van Leeuwen
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