Euler is often credited with introducing the notation $f(x)$, and people cite the example $f(\frac{x}{a}+c)$, where he had to use parentheses around the function argument. On the other hand, when the argument was a single letter like $x$, I have mainly seen Johann Bernoulli and Euler just write $f\, x$ or $f\colon x$ (or $\phi\, x$), without the parentheses. If I recall correctly even Lagrange in his lectures introduced the function notation without parentheses.

Question: Did Euler (or Johann Bernoulli) ever write $f(x)$?

In case the answer is no, the follow up question is: when did it become standard to put parentheses around $x$?

  • $\begingroup$ Many papers of Euler can be found in the arxiv. $\endgroup$ Jan 8 '18 at 14:21
  • $\begingroup$ Already answered in the post: why-do-we-use-brackets-for-function-parameters $\endgroup$ Jan 9 '18 at 8:26
  • $\begingroup$ @MauroALLEGRANZA it's not really answered there, even though that question is related. $\endgroup$ Jan 9 '18 at 8:59
  • $\begingroup$ See document E045, page 186. $\endgroup$ Jan 9 '18 at 9:37
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    $\begingroup$ @MauroALLEGRANZA I see him write $f(\frac{x}{a}+c)$ there. Maybe I'm overlooking something. Certainly he cannot leave the parenthesis out in that case. $\endgroup$ Jan 9 '18 at 10:01

I’m guessing no. But how does one make sure? (Maybe with 85+ volumes of clean pdfs...)

Cajori, who started that $f(\frac xa+c)$ example, points out a $\varphi(z)$ in D’Alembert (1754, p. 50).

For “standard”, I would say Lacroix (1797, p. 87):

4. Pour représenter une fonction sans indiquer, en aucune manière comment elle peut être composée, je me servirai de la caractéristique $\mathrm f$; et il faudra entendre, par l'expression $\mathrm f(x)$, une fonction quelconque de $x$, en comprenant sous cette dénomination tout ce que comporte la définition du mot fonction (Intr. nº 1) : on doit donc bien se garder de prendre la lettre $\mathrm f$ pour un coefficient de $x$. J’indiquerai la substitution de $x+k$ aulieu de $x$ dans $\mathrm f(x)$, en écrivant $\mathrm f(x+k)$, et cela voudra dire que le résultat est composé en $x+k$, comme la fonction primitive l’est en $x$.

Side remark tying into your other question: This book of Lacroix writes “the function $f$” very often; e.g. pp. 93, 212, 258, 483–496, 502, mainly when describing results of Monge who also did this a lot (but avoided unnecessary parentheses). I think “$f$” all started with solutions of PDEs depending on “arbitrary functions” — though only Dedekind, I would say, made them “objects” in the sense you want at the other question.

In E213 “Remarques sur les mémoires précedens de M. Bernoulli” (1755), just quoted elsewhere, you can see Euler “forget” his evaluation colon and slip into writing $\Phi'(x)$ (p. 215) and eventually $\Phi(x)$ (p. 216). Same thing in E441 (1773, p. 429). So in the end, yes.

  • $\begingroup$ It's interesting that several pages before introducing the function notation $f(x)$, Lacroix uses extensively the notation $\mathrm{l}\, x$ to denote the logarithm of $x$, without parenthesis, and without worrying that one may confuse the letter l for a coefficient. It also raises the question if any of his contemporaries ever called l a "caractéristique d'une fonction". $\endgroup$ Jan 17 '18 at 15:06
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    $\begingroup$ @MichaelBächtold Yes, Cauchy for sure. $\endgroup$ Jan 18 '18 at 6:17

You can find all of Euler's original papers in the Euler archive. Glancing through his later papers does not yield a hit. But if you are interested enough you can probably exclude that notation for all his writings.

  • $\begingroup$ Thanks. I don't understand what you mean by me excluding that notation for all his writings. $\endgroup$ Jan 9 '18 at 10:06
  • $\begingroup$ I meant that you can look into all of his papers in the Euler archieve and probably you will find that Euler did never use f(x). But I am not sure and too lazy to do so myself. $\endgroup$
    – Otto
    Jan 9 '18 at 10:31
  • $\begingroup$ No problem. I was just hoping that maybe someone has already seen him write f(x). I'm also to lazy to search all his writings. I'll leave this open a few more days. $\endgroup$ Jan 9 '18 at 15:48

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