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It's my understanding that the convention of using letters from the end of the alphabet ($x$, $y$, $z$) to represent $variables$, and letters from the start of the alphabet ($a$, $b$, $c$) to represent $constants$ came to us from F. Vieta (who proposed vowels and consonants) by way of Descartes.

However, I was curious to know if there was ever any debate around subscript notation for the same distinction, i.e. $x$ vs. $x_o$, where the first is considered a variable and the second, an unknown constant.

I can imagine many reasons for $x_a$ being more popular than $a_x$; but, does anyone know where this notation got its start? &nd was there any contention?

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    $\begingroup$ Not from Torricelli but from Descartes' Géométrie (1637). $\endgroup$
    – Mauro ALLEGRANZA
    Mar 24, 2017 at 6:56
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    $\begingroup$ Cauchy in his Résumé (1823) used $x_0, y_0$ to denote a chosen value for the variable $x,y$. $\endgroup$
    – Mauro ALLEGRANZA
    Mar 24, 2017 at 7:05
  • $\begingroup$ I see $x,$ $y,$ $z$ used as subscripts all the time in physics and multivariable calculus. But maybe these examples are cheating. $\endgroup$
    – Dave L Renfro
    Mar 24, 2017 at 14:38
  • $\begingroup$ @MauroALLEGRANZA thanks for the flag on that fact - of course you're right, except that it was Vieta (Viète) first with vowels / consonants and then Descartes' contribution was start / end of alphabet. Edited to reflect this point. $\endgroup$
    – Rax Adaam
    Apr 4, 2017 at 17:05
  • $\begingroup$ I don't understand the question. What is $x_a$ a notation for? What is $a_x$ a notation for? What convention are you referring to? In what context is $x_a$ "more popular than" $a_x$? $\endgroup$
    – user466
    Apr 6, 2017 at 2:42

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Like Ben Crowell in the comments, I'm not sure I fully understand the question. But interpreting it broadly about the origin of the index notation, I quote Spalt who in Die Analysis im Wandel und im Widerstreit (2015), p. 106 while referring to a figure in Leibniz De quadratura arithmetica circuli ellipseos et hyperbolae (p. 528) claims (my translation):

Leibniz is the inventor of the index notation. Today we would write $D_1, D_2$ etc. for the indexed points $D$. Leibniz does this differently: he writes the numbers in front of the letter and he does not set them lower. (In the case of this figure he does not write the numbers smaller [so he wrote $1D, 2D$ etc.]. But he does in other cases. Sometimes he also sets the numbers lower, but he never writes the numbers smaller and sets them lower at the same time. ...)

I don't know when the passage to lower and smaller and after the letter happened and if one can find explicit reasons for this change.

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  • $\begingroup$ Never received a notification for this answer, so my apologies for a slow reply(!): regarding the confusion -- I was specifically curious about the use of indexed notation to indicate a constant, unknown value (i.e. $x_o$ or $x_a$ as opposed to the variable $x$) and how this notation came to be (i.e. were there other contenders such as $a_x$ to indicate a constant, unknown $x$-value). Your answer is very interesting; thank you for sharing. $\endgroup$
    – Rax Adaam
    Dec 20, 2022 at 23:27
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If we write $0_x$ then it looks like a special kind of 0, since the 0 is above the baseline for typography. So we would expect $0_x+1_x=1_x$ etc. With the $x_0$ notation it is clear that $x_0$ is a special kind of $x$, or at any rate the same kind of thing as $x$.

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  • $\begingroup$ Are you suggesting this is what people thought when they started writing indices like this? $\endgroup$
    – Michael Bächtold
    Nov 15, 2018 at 9:09

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