Holy Writ (by which, of course, I always mean the OED) traces "science" to the Latin word scire, "to know." Looking up "shit" in the OED gives a vaguer origin, along with a sense of intellectual shame and a fit of the giggles. The farthest they trace it is an Old Teutonic root, skit-, which they do not gloss but which was handed down unchanged into Swedish and still means "shit." The roots are very similar on the surface, but seem completely separate in terms of meaning. Are they really related? Oddly enough, yes.
Germanic and Latin share a common heritage; both of them are descended from the Indo-European family of languages (which also includes such far-flung members as Sanskrit, Celtic, Hindi, Yiddish, and the Romance and Slavic languages.) The American Heritage Dictionary's appendix of Indo-European Roots (available here) lists both "shit" and "science" as derivatives of skei-, meaning "to cut or split." It is related to the root sek-, which also means "to cut"; the Latin analogue secare, which gave us words like "transect," "secant," and "dissect," came from this second root.
So, conceptually, how do you get from "to cut" to "science" and "shit"? Scire, which furnished "science," began life meaning "to discern," "to tell one thing from another," logically linked to the division of knowledge into categories. (This same link led to the word "nice," another scire descendant-- it once meant "precise" or "exacting.") "Shit" is, once again, vaguer. The American Heritage Dictionary traces it, along with "blatherskite" (a Middle English term for a contemptible, long-winded person derived from skite, "diarrhea") and "shyster," as being "all from Germanic *sk

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