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I've been recently thinking about what I've learned in mathematics, and I realised that in contrast to physics (or the other sciences), I tend to take the concepts and definitions for granted in that I assume they are "obvious" and have always been known. Of course this cannot be the case, and just as in physics, behind each concept or definition is probably a long history of concepts and definitions abandoned because they were not useful or simply incorrect. However, this history is rarely mentioned in classes, which motivates my question:

Are there examples of concepts, definitions, or entire (sub)fields of mathematics that were once influential and actively studied, but which were later abandoned because they were not useful, or because they were replaced by something more useful?

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There's always the classic example of the fluxion/infinitesimal being abandoned for the formal limit in the development of Calculus... – Isaac Solomon Nov 23 '12 at 6:30
Elaborate riffs on classical theorems of geometry. Substantial $19$th century work. Now mainly a staple of mathematical Olympiads. – André Nicolas Nov 23 '12 at 6:56
I might be mistaken, but I've heard it said that (classical) projective geometry is dead, and that it lives on in some form in more modern fields like algebraic geometry. – Isaac Solomon Nov 23 '12 at 7:03

The best known "abandoned" math that I know of are quaternions. They used to be everywhere, and Maxwell's equations were actually first stated using them. Quaternions eventually died off almost completely due to the development of vector analysis, coming back only in the very modern age for use in describing 3D rotations in computer graphics;

$$(\mathbb{H}-\{0\})/(\mathbb{R}-\{0\}) \cong SO_3 $$

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