For example, mechanics and mathematical analysis were commonly combined into one subject during the 18th century, united by the differential equation concept; while algebra and geometry were considered largely distinct.
There is no major loss of content, with analytical mechanics in the old sense now expressed in terms of symplectic topology, based on the newer theory of manifolds.
The term theory is used informally within mathematics to mean a self-consistent body of definitions, axioms, theorems, examples, and so on.
A well-known example was the development of analytic geometry, which in the hands of mathematicians such as Descartes and Fermat showed that many theorems about curves and surfaces of special types could be stated in algebraic language (then new), each of which could then be proved using the same techniques.
The circular angle also has area interpretation when referred to a circle with radius equal to the square root of two.
[3] Early in the 20th century, many parts of mathematics began to be treated by delineating useful sets of axioms and then studying their consequences.
One started from the most general axioms, and then specialized, for example, by introducing modules over commutative rings, and limiting to vector spaces over the real numbers only when absolutely necessary.
In particular, this perspective placed little value on fields of mathematics (such as combinatorics) whose objects of study are very often special, or found in situations which can only superficially be related to more axiomatic branches of the subject.
A key theme from the "categorical" point of view is that mathematics requires not only certain kinds of objects (Lie groups, Banach spaces, etc.)
Saunders Mac Lane proposed that any concept with enough 'ubiquity' (occurring in various branches of mathematics) deserved isolating and studying in its own right.
Nevertheless, the methods of category theory have steadily advanced in acceptance, in numerous areas (from D-modules to categorical logic).
On a less grand scale, similarities between sets of results in two different branches of mathematics raise the question of whether a unifying framework exists that could explain the parallels.
A research area sometimes nicknamed Monstrous Moonshine developed connections between modular forms and the finite simple group known as the Monster, starting solely with the surprise observation that in each of them the rather unusual number 196884 would arise very naturally.
It had probably not seemed plausible that there would be 'enough' such rational factors, before the conjecture was enunciated; and in fact the numerical evidence was slight until around 1970, when tables began to confirm it.
Another case, which so far is less well-developed but covers a wide range of mathematics, is the conjectural basis of some parts of K-theory.