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Why do we typically use $h$ for

$$\frac{\mathrm{d}f}{\mathrm{d}x}=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$

A student asked me this the other day. My guess was that it was originally height, because Newton was originally applying it to gravity and he was looking at change in height, and then it stuck.

Anyone have any insight into this?

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  • $\begingroup$ I do not see why is this question relevant for history of math and science. People use different letters. $\endgroup$
    – Alexandre Eremenko
    Commented Feb 7, 2015 at 21:01

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Here is an interesting list of first uses.

Small $h$ seems to have been used by Cauchy's Cours d'Analyse.

Both Cauchy's and Newton's papers are available online, so it should be possible to verify if it showed up there.

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  • $\begingroup$ It is no proof, but Cauchy's books seems to have been an influential one. Here is an earlier book by Lagrange: Théorie des fonctions analytiques, the link also from the above mentioned list. I could not spot an $h$ there so far. $\endgroup$
    – mvw
    Commented Feb 5, 2015 at 19:13
  • $\begingroup$ @charlotte That's not the right page. It does have an $h$, but not an $h \to 0$. This $h$ is constant! $\endgroup$
    – Robert Israel
    Commented Feb 5, 2015 at 19:15
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    $\begingroup$ This is a page which may be the origin of the symbol, though it's not being used in the same way; he's using $h$ where we would use $x$ and $n$ for $h$. visualiseur.bnf.fr/… $\endgroup$
    – charlotte
    Commented Feb 5, 2015 at 19:26
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Please, note that Newton did not used the "differential" notation, due to Leibniz, but the "fluxional" one : $\dot x$, etc.

See for example :

The moment of a fluent such as $x$ is the amount by which it changes in an indefinitely small length of time $o$, given by $\dot xo$ , and all products in an equation that involve orders of $o$ greater than one can be ignored.

We can see :

We can find the use of $h$ in :

My "humble opinion" is that there is no special reason for it ...

As long as the symbolism underwent the process of standardization, we can see that the (insufficient number of elements of the) alphabet partecipate to a process of "specialization" :

  • $x,y,z$ for the unknown; $a,b,c$ for the known terms, mainly in algebra; $i,j,k$ for numerical indices; $n,m$ for naturals;

  • in addition, $d$ was used for derivative, $f$ was chosen for the "typical" function; $e$ and $i$ acquired a status of proper nouns.

So, there were not many letters available ...

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  • $\begingroup$ Boole's text is a real treasure; thanks so much for sharing! The way he discusses the notation in that first chapter gives the impression that the notation was already fairly well established. I always assumed the $h$ was loosely related to $h$orizontal displacement, but really only because the same word is used in English, French & German and, of course, Latin... $\endgroup$
    – Rax Adaam
    Commented Nov 22, 2021 at 17:21
  • $\begingroup$ @RaxAdaam - you are welcome :-) $\endgroup$
    – Mauro ALLEGRANZA
    Commented Nov 23, 2021 at 7:22
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I have see many letters and symbols used, for example, it need not be $h$, but could also be \begin{align} f'\left(a\right)&=\lim_{\Delta x\rightarrow 0}\frac{f\left(a+\Delta x\right)-f\left(a\right)}{\Delta x},\tag{1}\\ f'_{+}\left(a\right)&=\lim_{\Delta x\rightarrow 0+}\frac{f\left(a + \Delta x\right)-f\left(a\right)}{\Delta x},\tag{2}\\ f'_{-}\left(a\right)&=\lim_{\Delta x\rightarrow 0-}\frac{f\left(a+\Delta x\right)-f\left(a\right)}{\Delta x},\tag{3} \end{align} as is shown in Widder's Advanced Calculus to denote the derivative of $f\left(x\right)$ at $x=a$, the derivative on the right of $f\left(x\right)$ at $x=a$, and the derivative on the left of $f\left(x\right)$ at $x=a$.

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  • $\begingroup$ Okay, and I notice Wikipedia uses $P$ and $\Delta P$. Obviously it doesn't have to be $h$, but it seems to be the most common. I don't recall ever reading a book which didn't use $h$. I'll check my old textbooks when I get home. $\endgroup$
    – charlotte
    Commented Feb 5, 2015 at 18:56
  • $\begingroup$ It doesn't matter that there are sources which don't use $h$.The question asks about why $h$ is used a lot, or rather why it has been used. This doesn't answer this question. $\endgroup$
    – Git Gud
    Commented Feb 5, 2015 at 18:57
  • $\begingroup$ @GitGud I'm merely pointing out that it doesn't have to be $h$, i.e. different sources use different symbols to mean the same thing or be more specific. $h$ is just the default. This is like asking why the default in many beginning Algebra courses is the variable $x$. It just depends on the author's or teacher's pick, and later when we throw $y$'s into the equation people have to get used to it, the same as I had to when reading Widder's book. $\endgroup$
    – bd1251252
    Commented Feb 5, 2015 at 19:04
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A quick look at "Analysis per quantitatum series, fluxiones, ac differentias ..." didn't find any $h$'s. In fact Newton didn't use the modern definition of derivative at all. The rigourous development of calculus in terms of limits didn't happen until the 19'th century.

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