My confusion came from an undergraduate lecture that I either remembered too dimly or didn't comprehend to begin with. You see, I had thought that those two Latin verbs were from the same Indo-European root because my instructor had made a play on words that centered on the phrase, "when the strike falls." But I've looked up the words in the Oxford English Dictionary and a few etymological dictionaries and can find no evidence to support my ill-founded conviction that "incident" and "incisive" are related.
The worst part of this discovery is that it is accompanied by the realization that I have done precisely what I rag on Martin Heidegger for doing: playing fast and loose with the history of words. I hope this isn't the start of something more serious. If I start looking at photos of Hannah Arendt with inexplicable longing -- already a possibility, given that she was definitely my type -- and am overcome with the desire to remove my shoes and dangle my feet in a Black Forest stream, I will know that I am infected. Maybe I should have a glance at Theodor Adorno's The Jargon of Authenticity in the hopes that it will serve as the zinc needed to forestall the worsening of this false etymology-inducing virus.