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.
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5This may be a better fit on [hsm.se]. – Sandejo Oct 22 '20 at 17:01
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1@Pumpal Hi Pumpal. Welcome to Mathematics Stack Exchange. As Sandejo mentions, this question will probably be better suited to the History of Science and Mathematics Stack Exchange. Note that opinion-based questions usually aren't well received here. Good luck! – Joe Oct 22 '20 at 17:03
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See Do mathematical objects disappear? (what follows is a repeat of my comment there) A good example is Eliakim Hastings Moore's program in "General Analysis" --- see Reinhard Siegmund-Schultze's 1998 historical survey paper Eliakim Hastings Moore's “General Analysis” and Moore's memoir Introduction to a Form of General Analysis in American Mathematical Society Colloquium Publications #2, Yale University Press, 1910. – Dave L. Renfro Oct 22 '20 at 19:59
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Very interesting. Thanks! – Oct 23 '20 at 21:30
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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/).
Jean Marie
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Why do you consider trigonometry's then perceived importance to have been exaggerated? – J.G. Oct 22 '20 at 18:43
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1@J.G. In secondary schools (I least in my country, France, but I think it was common in other countries), students ages 16-18 had a whole book entirely devoted to trigonometry with hundreds of exercises, some full of subtility for example the "Compléments de trigonométrie et méthodes pour la résolution des problèmes (1906)" by Frère Gabriel Marie. – Jean Marie Oct 22 '20 at 18:52
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FYI, an especially comprehensive trigonometry text in French is: Émile Gelin (1850-1921), Éléments de Trigonométrie Plane et Sphérique, Librairie Wesmael-Charlier (Namur, Belgium), 1888, ii + 252 pages. See pp. 58-62, for instance. – Dave L. Renfro Oct 22 '20 at 20:11
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@Dave L. Renfro Thank you very much, I didn't know it. I note that a common point with the other one is that it has been written by a religious person. I will try to find it somewhere in order to say the content of these specific pages. – Jean Marie Oct 22 '20 at 20:18
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1The google-books version is freely available where I'm at, but maybe not where you're at. Also, there's a 1906 2nd edition of this book, but I haven't been able to find a copy of it. While I'm at it, some advanced trig books in English are here, and others can be found in the comments here. – Dave L. Renfro Oct 22 '20 at 20:25
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@ Dave L. Renfro Thanks for all these indications. A non-scientific (though historical!) side remark about one of the authors of one of the documents you mention there: "Advanced Trigonometry" by Clement Vavasor Durell et al. The name "Vavasor" looks strange. In fact it is due to come from French: "Vavasseur" being a rather common name whose curious middle age origin meaning is "vassal of the vassal". – Jean Marie Oct 24 '20 at 05:19