Who introduced the notation $y|_{x=a}$?

Question

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?

Answer 1

You can see :

• Giuseppe Peano , Lezioni di Analisi Infinitesimale, 2 vols., 1893, page 17 :

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

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

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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 $\text {Log} 1001$.

Answer 2

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.