(Obscure) areas of mathematics that are largely inactive or forgotten today? [duplicate]

Question

I am looking examples of a mathematical theory (i.e. a body of knowledge, with its own definitions, results, principles etc., i.e., its own language) that is completely inactive or forgotten by today.

For example, it seems to me that sphere geometry is largely inactive today (although I am sure someone in the comments will correct, that, actually, somewhere there is a very active seminar on sphere geometry going on).

I would like such examples of theories to be fairly obscure, yet still readable, if I were to devote time to them. In this sense:

1. Bonus points of the books or papers where the theory is explained cannot be found on books.google.com.
2. Bonus points if this theory is not mentioned in the Princeton Companion to Mathematics.
3. Bonus points if the books are available only in languages that are not English.
4. Bonus points if there exist carefully written textbooks explaining the theory (no matter how old the books is).

It is OK if the theory is subsumed by a more general theory that is active today, as long as today's theory uses a different "language" (I will let you decide how exactly you define what "language" is). For example, if there was a lot of theory to explicitly solve certain types of equations in the 17th century, for various tricky cases, but that is all supplanted today by, say, a numeric approach, than that would be fine by me.

I have read this post, as well as these notes on lost mathematics, but they don't quite fit the bill.

Answer 1

The most difficult condition to satisfy is 1, since Google book "knows" most of books. All other conditions are easy to satisfy, and many examples can be given, but I will give only one.

Princeton Companion in fact does not mention MOST of mathematics, if you count by volume of output (books and papers). It only mentions what is "most important" on the view of the authors. And this view is inevitably subjective.

Here is an example of a theory which satisfies most of your criteria. It is called "Series of exponentials", and deals with representation of analytic functions in the form of $$\sum_{n=0}^\infty a_ne^{\lambda_nz},$$ where $a_n$ and $\lambda_n$ are arbitrary COMPLEX numbers. The theories with real or pure imaginary $\lambda_n$ are much better known.

The main developers of this theory were A. F. Leont'ev and his collaborators and students. Leont'ev published 5 books on the subject, all of them exist only in Russian (were never translated to any other languages). Google books only lists the titles. All his books are carefully written, and the smallest one:

MR0753827 Leontʹev, A. F. \cyr Tselye funktsii. Ryady èksponent. (Russian) [Entire functions. Series of exponentials] "Nauka'', Moscow, 1983. 176 pp.

is a good introduction to the subject. So this area satisfies all your criteria, except possibly 1.

Still I would not call it "obscure" or "inactive". There is a group of mathematicians, mostly in Russia which still pursues the subject, and some of their results are quite deep.

On the other hand, there are "obscure" and "inactive" areas which do not satisfy any of your conditions 1-4. An example is "arithmetic of probability distributions". The book which describes the state of the art is

MR0428382 Linnik, Ju. V.; Ostrovsʹkiĭ, Ĭ. V. Decomposition of random variables and vectors. Translated from the Russian. American Mathematical Society, Providence, R.I., 1977.

It contains many deep and beautiful results, but for some reasons this direction of research is inactive since the end of 1980s. And my experience shows that even knowledge of these results, even among specialists in statistics, gradually disappears.

Remark. Princeton Companion contains some interesting articles, but its claim that it gives a perspective of the WHOLE mathematics can be misleading and harmful for a young mathematician.

It does not even mention many large and vigorously developing modern areas of research. A survey of ALL mathematics of reasonable length is probably impossible at this time. The multi-volume German Encyclopedia of Mathematical Sciences published in the early 20 century comes closer to it, but this was in the early 20th century and very many volumes were required.