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When a variable $y$ depends on other variables, say $y=c x^3$, one often writes $$y|_{x=2}$$ to say "$y$ when $x$ has value $2$". This might be more familiar in the context of derivatives where we find $\left. \frac{dy}{dx}\right|_{x=2}=12c$. It also appears in other areas of mathematics. For example the fiber of a bundle $E$ over some point $p$ is often denoted with $E|_p$. If we interpret a fiber bundle as a space depending on points of the base this seems to be analogous to the previous example.

Notice that in the example $y=cx^3$ it would be wrong to write $y(2)=8c$ since $y$ is not a map of type $\mathbb{R} \to \mathbb{R}$.

Question 1: Who introduced this bar notation?

Question 2: How did people in the early days express "$y$ under the condition that $x=2$"? I'm guessing they just used words, but was there an alternative notation? Or did they abuse the notation for function application, as in $y(2)$, once Euler/Bernoulli had introduced it?

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  • $\begingroup$ I've not found (yet) in Euler an example of e.g. $f(2)$ but the use of brackets for "grouping" is quite natural... He writes $\sqrt x$ but of course he needs $\sqrt {(x+2)}$. He use $\cos \Phi$ but I'm quite sure that he uses also $\cos (\Phi + \pi)$ ... The next step (Cuachy) is to use $f(x)$ and $f(x+i)$. $\endgroup$ – Mauro ALLEGRANZA Nov 3 '16 at 14:33
  • $\begingroup$ Thanks @MauroALLEGRANZA, I'm having difficulties placing your comments into the context of this question. For example $\cos$ and $l$ are functions, so it seems correct to write $l(e)$ and not $l|_{x=e}$. But maybe you wanted to convey another message. $\endgroup$ – Michael Bächtold Nov 3 '16 at 14:49
  • $\begingroup$ I appreciate your conjectures. Concerning the remark about being natural: if I write $y=f(x)=cx^3$ and $y=g(c)=cx^3$, hence $f(2)=8c$ and $g(2)=2x^3$, would you still find it natural to write $y(2)=8c$? I would argue that writing any of the above does still not make $y$ a function of type $\mathbb{R}\to\mathbb{R}$. $\endgroup$ – Michael Bächtold Nov 3 '16 at 15:34
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    $\begingroup$ Some comments were deleted from this discussion. For completeness: $l$ was an older notation for $\log$. And it was suggested that it might be natural to write $y(2)=8c$ if one had previously written $y=f(x)=cx^3$. $\endgroup$ – Michael Bächtold Nov 4 '16 at 7:22
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You can see :

$$[f(x)]_{x=a}=f(a).$$

Not sure it is the earliest... but Peano was a prolific "inventor of notations".


Regarding : how they express "$y$ under the condition that $x=2$"

see e.g. page 34 [shortened] :

let $y$ the natural logarithm of $x$ [...] and $f(x) = \text {Log} x$. Compute $\text {Log} 1001$.

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  • $\begingroup$ Nice. This actually reminds me that $y|_{x=a}$ is very similar to logicians substitution, which you already answered here mathoverflow.net/questions/243084/… $\endgroup$ – Michael Bächtold Nov 3 '16 at 15:40
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    $\begingroup$ @MichaelBächtold - Logicians "codified" substitution between 1930 and 1950 and Peano' book is 1893. It is worth noting that Peano (as well as Frege) is one of the main source for Russel & Whitehead's Principia Mathematica (1910-1913) : the first modern codification of math log. $\endgroup$ – Mauro ALLEGRANZA Nov 3 '16 at 15:50
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    $\begingroup$ Understood... It is also worth noting that Alonzo Church who provided the first precise definition of "formal substitution" in math log is the modern "inventor" of funcional application. $\endgroup$ – Mauro ALLEGRANZA Nov 3 '16 at 15:56
  • $\begingroup$ It's interesting that you say that functional application was "invented" with the Lambda calculus. I would have said that Eulers $f(x)$ is already functional application. But if we can't find $f(2)$ in Euler, than maybe he did indeed not think of it the way that it was later formalized. Also (going of in tangents), do you know when the alternative to the Lambda notation $x\mapsto x^2$ was first used? Or maybe I should ask this as a separate question... $\endgroup$ – Michael Bächtold Nov 3 '16 at 16:08
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Concerning alternative notations for $y|_{x=0}$: Lagrange in Théorie des fonctions analytiques, 1797, p.57 writes:

(...) et si on désigne par $(y), (y'), (y''),$ etc. les valeurs de $y,y',y'',$ etc. lorsque $x=0$, on a en général $$ y = (y) + x\,(y') + \frac{x^2}{2}(y'') +\ldots $$

This is developed further to allow other values than zero in 1825 in Martin Ohm's Die Lehre vom Grössten und Kleinsten, p.5:

If $V$ is a function of $x$, and $a$ a value of $x$, then we denote with $V_a$ or $(V)_a$ that which becomes of $V$ when we write $a$ everywhere instead of $x$.

(my translation)

This notation allows for ambiguities if there are other variables involved one might want so substitute for. (Moreover it is very close to the abuse of notation $V(a)$.)

The earliest use of the "modern" notation, but without the bar, I've so far found in Todhunter's A History of the Calculus of Variations 1861, p.23 ff.

(...) thus the first line in for example (2) will really when written at full become \begin{align} &\left\{ I\left( P' -\frac{dQ'}{dx} + \frac{d^2R'}{dx^2} - \ldots \right)\omega \right\}_{x=\beta} \\ -& \left\{ I\left( P' -\frac{dQ'}{dx} + \frac{d^2R'}{dx^2} - \ldots \right)\omega \right\}_{x=\alpha} \end{align} and as the value of $I$ when $x=\beta$ will not generally be the same as when $x=\alpha$, we cannot as Lacroix does put $A$ for $I$ in the terms include in (2).

Although the notation is quite self explanatory, Todhunter does not define it explicitly. So I suspect it was already being used at that time by someone else. But I haven't yet checked all the sources cited by Todhunter.

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