# History question - why h in the definition of derivative?

Does anyone have a clue where the "h" came from?

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As you can see from the Cauchy clip at the MO question, "Why do we use $\epsilon$ and $\delta$?" (mathoverflow.net/questions/82302), the $h$ was not universal in 1850: Cauchy uses $i$ instead. – Joseph O'Rourke Jan 31 '12 at 19:02
And $h$ immediately precedes $i$ in the alphabet! – Mariano Suárez-Alvarez Jan 31 '12 at 20:45
A conjecture. The letter h (especially at the beginning of a word) has just the small effect of aspiration. In Latin, it is even mute, like e.g. in the English word honour, of Latin origin). So, h seems a natural choice for denoting a quantity that is small or tends to $0.$ – Pietro Majer Jan 31 '12 at 21:30
a, b, c are boring; d is for derivative, e is already taken, f and g are functions, so that leaves h as the first usable letter ;-) – Suvrit Jan 31 '12 at 21:33
@Pietro: H was not mute in Latin, it is only mute in modern Italian pronunciation of Latin. See e.g. en.wikipedia.org/wiki/Latin#Phonology and en.wikipedia.org/wiki/Latin_spelling_and_pronunciation . – Emil Jeřábek Feb 1 '12 at 10:26

I think that use of $h$ in the definition of derivative is linked to the relationship between Calculus of Finite Difference and Differential Calculus.

In the book Leçons sur le Calcul des Fonctions, Councier, 1806, Lagrange:

1. Assigns to Maclaurin and d'Alembert the origin of differentiation as the limit of finite differences, (pp. 1);
2. Writes "Considerons un fonction $fx$ d'une variable quelconque $x$. Si la place de $x$ on substitue $x + i$, $i$ étant une quantité quelconque indetermineé, elle divendra $f(x + i)$ ...", (pp. 8);
3. Develops $f(x + i)$ in series: $$f(x+ i) = fx + i f'x + \frac{i^2}{2} f''x + \frac{i^3}{2 . 3}f'''x + \frac{i^4}{2.3.4}f^{iv}+ \mbox{etc}$$ as we can see at (pp. 15), and
4. Writes "Nous appelerons la fonction $fx$ fonction primitive... Nous nommerous de plus la fontion dérivée $f'x$, primière fonction dérivée ou fonction derivée du primier ordre...", (pp. 15).

Notes B of Lacroix's book An Elementary Treatise on the Differential and Integral Calculus, Cambridge, 1816, pp. 599, , using $h$ instead $i$, is based on Lagrange's work.

In 1829, Dr. Martin Ohm, in Versuch eines vollkommen consequenten Systems der Mathematik, Vol. III, pp.53, Berlin, available here, uses $h$. He writes:

$$f(x+h)=f(x) + \partial f(x).h+\partial^2 f(x) .\frac{h^2}{2!} + \partial^3 f(x) .\frac{h^3}{3!} +.\ldots$$

Also, as we can see, Dr. Martin Ohm uses factorials!! (Martin and Georg Ohm were brothers. Georg discovered the Ohm's Law).

In G. Boole, Treatise on the Calculus of Finite Differences, MacMillan, London, 1880, pp.1, we can read: " The Calculus of Finite Differences may be strictly defined as the science which is occupied about the ratios of the simultaneous increments of quantities mutually dependent. The Differential Calculus is occupied about the limits to which such ratios approach as the increments are definitely diminished" Boole1.

At pages 2 and 3 , we can see the definition of derivative using $h$. All arguments are based on Finite Differences Boole2.

The differentiation was developed based on trigonometric assumptions (method of tangents). A very good history can be found in H.Sloman, The Claim of Leibnitz to the Invention of the Differential Calculus, MacMillan, 1860 Sloman1.

A possible explanation for the use of h in the definition of derivative (and the link between Differential Calculus and Calculus of Finite Differences) can be found in this book at page 127, second paragraph Sloman2.

PS: Although $h$ has been used in the books mentioned above (1816, 1829, 1860 and 1880), Milne-Thomson, in his recent book (1933), uses $\omega$ instead Milne-Thomson.

Milne-Thomson's book can be considered an example of Euler's notation use. In Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, Chapter 1, De differentiis finitis, pp.1, 1787, Euler writes "variabilis x capiat incrementum $\omega$" !

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It would seem reasonable to speculate that i stands for increment. It may have been changed to h later to avoid conflicts of notation. – Michael Renardy Apr 27 '12 at 13:59
@Michael I think you are correct. Please, see ref. of Lacroix's book, first paragraph, pp. 599: "... corresponding to the increment h ...". – Papiro Apr 27 '12 at 14:15
Laplace writes: "Nous désignerons ordinairements les variables des fonctions par les dernières letters de l'alphabet, x, y, etc., et las constantes par les premières a, b, c, etc..." – Papiro Apr 27 '12 at 14:23
Interesting, and thanks so much. One thing I would love to see is the original reference of Taylor which Sloman refers to... – Jeff McGowan Apr 27 '12 at 15:26
@Jeff: Taylor's work about increments is available at 17centurymaths.com/contents/taylorscontents.html. Also, there is a work from L. Feigenbaum, (tufts.edu/as/math/feigenbaum.html) published by Springer (springerlink.com/content/h720142152632171) but I have no access to it. – Papiro Apr 28 '12 at 11:42