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I read that Descartes and some other mathematician figured out a 'double tangent' method (as I think it was called) for calculating a derivative of a conic or some curve without using the concepts of limits or infinitesimals. There was an article about this in the Mathematical Association of America. It was said this method could find derivatives on many types of curves and the methods of finding derivatives without limits could be applied to basic calculus, such as using the product rule. The problem is that calculus without limits or infinitesimals is not really mentioned in a regular curriculum. I've noticed other so-called "lost theorems" or rarely-mentioned but useful theorems, some of which have been mentioned in math journals. Are there a lot of rarely-mentioned but useful theorems in math and science?

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    $\begingroup$ Your question appears to be both strongly opinion based, and incredibly broad. Please edit it to make it a bit more appropriate for this site. $\endgroup$ – Danu Dec 27 '14 at 13:35
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    $\begingroup$ You might be asking about a "double root" method that can be found in old (1800s) textbooks, although I believe Fermat would be a more correct attribution for the method. I discussed the method in this 22 February 2009 sci.math post. $\endgroup$ – Dave L Renfro Dec 27 '14 at 15:59
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    $\begingroup$ By a lemma of Hadamard's, the derivative of polynomials $p(x)$ (and other nice functions) is given simply by expanding $\tfrac{1}{h}\left(p(x+h)-p(x)\right)$ and setting $h$ to zero. The difference kills constant terms and so there can't be a $h$ in the denominator. For example if $p(x):=2x^2−x+3$, the expression is $2(2x+h)−1$, and note that $p'(x)=4x-1$. This is purely algebraic (no $\mathbb R$ anywhere) and I've seen it being called "Fermat observation". Don't tell anybody. $\endgroup$ – Nikolaj-K Dec 27 '14 at 19:07
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What most mathematicians are doing at a given time is determined by current fashion to a very large extent. I do not know a (fashion-independent) criterion by which a theorem can be "useful", but theorems can be beautiful. Some deep and beautiful theories can be out of fashion for many years. Sometimes they came into fashion again.

Some examples. Holomorphic dynamics created by Fatou and Julia in 1906-1918. It was very well received at that time. Then it went out of fashion. For about 60 years. Then it suddenly came back to fashion approximately in 1982 and remains one of the hottest areas of mathematics to this day.

Another similar example is Schubert Calculus. Or knot theory. Or integrable systems. Or theory of invariants.

I only mentioned some theories which were out of fashion for long time and currently are intensively developing. There is no doubt that there are theories which are considered "finished" and forgotten, and are still waiting for their revival.

I was talking about research above. Teaching is a different matter. We teach Calculus, which is a universal method to solve many problems, and usually do not teach pre-calculus methods to solve the same problems. Simply because we are limited in time when we are teaching, so we teach the things which are most important (on our opinion). We do not teach the ingenious methods by which Archimedes and others before Leibniz found the areas. And many other similar things which can be done automatically with the powerful methods we have now.

EDIT. Unlike other sciences (except astronomy), mathematics is VERY old. Euclid is 23 centuries old now. By comparison, "physics of 300 years old" and chemistry of 200 years old is of interest only to historians. But mathematics of 2300 years old is of interest to mathematicians. It is the SAME mathematics that we are doing now. If you don't believe me, read Archimedes. Just try. So the amount of accumulated knowledge in mathematics is enormous. We cannot teach all these things in our regular courses. We teach only a few selected things which on our opinion are most important.

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  • $\begingroup$ Nice answer. I was afraid that the question might find some broad answers, but you cut this down pretty well. $\endgroup$ – HDE 226868 Dec 27 '14 at 21:08
  • $\begingroup$ +1 for nice examples of things that went out of fashion and then returned. $\endgroup$ – Dave L Renfro Dec 29 '14 at 14:35
  • $\begingroup$ Another example is modular forms. They were studied in the early part of the 20th century, than went out of fashion for a few decades, but from the 1970s onwards became central to research in number theory (proof of Fermat's Last Theorem, etc.). $\endgroup$ – KCd Dec 31 '14 at 8:59

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