What are some examples of failed mathematical concepts in history? By "failed", I mean not fruitful for whatever reason (and subsequently abandoned). I ask as a non-mathematician who arrived at the question from thinking about the philosophy of math.
 A: I wouldn't use the adjective "failed". There are domains of mathematics, often connected to mathematical education, that have been hypertrophied above necessity, and brought back later to a reasonable size, sometimes almost nothing.
For example, from the second half of the XIXth century to the first half of the XXth, an exaggerated importance was given to trigonometry, either plane of spherical, linked to an hypertrophy on the study of triangles' properties. Up to the point that nowadays, trigonometry isn't considered as a branch of mathematics. No more than a set of recipes.
It reminds me now of a principle that has been used by Poncelet, the creator of projective geometry, named the "principle of continuity" which has been discussed because it was rather fuzzy. Cauchy, while in the process of defining the modern $\epsilon-\delta$ definition of continuity, in a report on Poncelet's work in 1820, said that his principle of continuity was "capable of leading to manifest errors" (see for example https://shouyin.wordpress.com/2013/05/29/principle-of-continuity/).
