The word, as a term of Mathematical English (which is of course quite a distinct dialect of English) is a calque of the Mathematical German word "Mannigfaltigkeit". Franklin Becher, in the first paragraph of the lead article in the October, 1896, issue of the American Mathematical Monthly, "MATHEMATICAL INFINITY AND THE DIFFERENTIAL", doesn't quite use the word yet, but makes its origin clear enough.
---begin--- Mathematics, as defined by the great mathematician, Benjamin Pierce, is the science which draws necessary conclusions. In its broadest sense, it deals with conceptions from which necessary conclusions are drawn. A mathematical conception is any conception which, by means of a finite number of specified elements, is precisely and completely defined and determined. To denote the dependence of a mathematical conception on its elements, the word "manifoldness," introduced by Riemann, has been recently adopted. --end-- In his article on the foundations of geometry, available at http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/Geom.html , Riemann distinguished two types of "Mannigfaltigkeit", the discrete and the continuous: ---begin--- cat Grössenbegriffe sind nur da möglich, wo sich ein allgemeiner Begriff vorfindet, der verschiedene Bestimmungsweisen zulässt. Je nachdem unter diesen Bestimmungsweisen von einer zu einer andern ein stetiger Uebergang stattfindet oder nicht, bilden sie eine stetige oder discrete Mannigfaltigkeit; | Google Translate > Size terms are only possible where there is a general concept, which allows different modes of determination. According as, according to these modes of determination from one to another, a continuous transition takes place or not, they form a continuous or discrete manifoldness; ---end--- In Riemann's (eventual) context, those sentences would be understood now (at least by topologists of my sort, which is to say, geometric topologists, cf. http://front.math.ucdavis.edu/math.GT) as sketching the modern concept of a (topological or differentiable) manifold as a "mathematical conception" that can "precisely and completely defined and determined" by a collection [called an "atlas"] of "modes of determination" [called "charts"] among (some pairs of) which there are also given "continuous" (i.e., topological) or perhaps *smooth* (i.e., differentiable) coordinate changes. I dispute, incidentally, the claim that 3-manifolds are too hard to understand; they're *just* at the edge of that, but not over it (whereas 4- and higher dimensional manifolds are DEFINITELY over that edge, in various well-defined mathematical ways; e.g., the problem of determining whether two explicitly-given n-manifolds, n greater than 3, has been known for a long time to be computationally intractable [you can embed the word problem for groups into the manifold classification problem for n greater than 3], and much more recently has been shown to be doable in dimension 3). The French word for (something a little more general than a) manifold is "varieté", by the way; same sort of reason, I assume. ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove
