Does anyone have a clue where the "h" came from?
[Math] History question – why h in the definition of derivative
ho.history-overview
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Paolo Ruffini's work on the impossibility of solving the quintic by radicals did meet a strong passive resistance. Around 1800 he proved the theorem up to a minor gap, that himself or somebody else could have fixed soon, had his book met the attention that deserved. But times were not ready for a such a revolutionary idea as proving the impossibility; 20-30 years later this idea had slowly spread and become more natural, and Abel and Galois got more lucky (so to speak).
This is in my opinion a major example of a particular theorem that was met with resistance before being accepted, and in fact it also shows that resistance is not necessarily associated with controversials, but sometimes even with indifference (which may be even worse).
A short and well written account of the story is in J.J.O'Connor and E.F.Robertson's article for the History of Mathematics archive: http://www-history.mcs.st-and.ac.uk/Biographies/Ruffini.html
This is a good question, given the way calculus is currently taught, which for me says more about the sad state of math education, rather than the material itself. All calculus textbooks and teachers claim that they are trying to teach what calculus is and how to use it. However, in the end most exams test mostly for the students' ability to turn a word problem into a formula and find the symbolic derivative for that formula. So it is not surprising that virtually all students and not a few teachers believe that calculus means symbolic differentiation and integration.
My view is almost exactly the opposite. I would like to see symbolic manipulation banished from, say, the first semester of calculus. Instead, I would like to see the first semester focused purely on what the derivative and definite integral (not the indefinite integral) are and what they are useful for. If you're not sure how this is possible without all the rules of differentiation and antidifferentiation, I suggest you take a look at the infamous "Harvard Calculus" textbook by Hughes-Hallett et al. This for me and despite all the furor it created is by far the best modern calculus textbook out there, because it actually tries to teach students calculus as a useful tool rather than a set of mysterious rules that miraculously solve a canned set of problems.
I also dislike introducing the definition of a derivative using standard mathematical terminology such as "limit" and notation such as $h\rightarrow 0$. Another achievement of the Harvard Calculus book was to write a math textbook in plain English. Of course, this led to severe criticism that it was too "warm and fuzzy", but I totally disagree.
Perhaps the most important insight that the Harvard Calculus team had was that the key reason students don't understand calculus is because they don't really know what a function is. Most students believe a function is a formula and nothing more. I now tell my students to forget everything they were ever told about functions and tell them just to remember that a function is a box, where if you feed it an input (in calculus it will be a single number), it will spit out an output (in calculus it will be a single number).
Finally, (I could write on this topic for a long time. If for some reason you want to read me, just google my name with "calculus") I dislike the word "derivative", which provides no hint of what a derivative is. My suggested replacement name is "sensitivity". The derivative measures the sensitivity of a function. In particular, it measures how sensitive the output is to small changes in the input. It is given by the ratio, where the denominator is the change in the input and the numerator is the induced change in the output. With this definition, it is not hard to show students why knowing the derivative can be very useful in many different contexts.
Defining the definite integral is even easier. With these definitions, explaining what the Fundamental Theorem of Calculus is and why you need it is also easy.
Only after I have made sure that students really understand what functions, derivatives, and definite integrals are would I broach the subject of symbolic computation. What everybody should try to remember is that symbolic computation is only one and not necessarily the most important tool in the discipline of calculus, which itself is also merely a useful mathematical tool.
ADDED: What I think most mathematicians overlook is how large a conceptual leap it is to start studying functions (which is really a process) as mathematical objects, rather than just numbers. Until you give this its due respect and take the time to guide your students carefully through this conceptual leap, your students will never really appreciate how powerful calculus really is.
ADDED: I see that the function $\theta\mapsto \sin\theta$ is being mentioned. I would like to point out a simple question that very few calculus students and even teachers can answer correctly: Is the derivative of the sine function, where the angle is measured in degrees, the same as the derivative of the sine function, where the angle is measured in radians. In my department we audition all candidates for teaching calculus and often ask this question. So many people, including some with Ph.D.'s from good schools, couldn't answer this properly that I even tried it on a few really famous mathematicians. Again, the difficulty we all have with this question is for me a sign of how badly we ourselves learn calculus. Note, however, that if you use the definitions of function and derivative I give above, the answer is rather easy.
Best Answer
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:
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 $\omega$" !