Did Euler ever write $f(x)$, with parentheses?

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 $x$?

• Many papers of Euler can be found in the arxiv. – Alexandre Eremenko Jan 8 '18 at 14:21
• Already answered in the post: why-do-we-use-brackets-for-function-parameters – Mauro ALLEGRANZA Jan 9 '18 at 8:26
• @MauroALLEGRANZA it's not really answered there, even though that question is related. – Michael Bächtold Jan 9 '18 at 8:59
• See document E045, page 186. – Mauro ALLEGRANZA Jan 9 '18 at 9:37
• @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. – Michael Bächtold 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.

Edit:
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.

• 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". – Michael Bächtold Jan 17 '18 at 15:06
• @MichaelBächtold Yes, Cauchy for sure. – Francois Ziegler 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.

• Thanks. I don't understand what you mean by me excluding that notation for all his writings. – Michael Bächtold Jan 9 '18 at 10:06
• 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. – Otto Jan 9 '18 at 10:31
• 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. – Michael Bächtold Jan 9 '18 at 15:48