keh₂-d-eyo-
“to keep sharpening, inciting”cascade, case, cadence, chance
Iterative related to Latin cadere: cascade, case, cadence, chance, occasion.
Discussion
The PIE form *keh₂-d-eyo- (to keep sharpening, inciting) is an iterative derivative of *keh₂d- (to fall, to drop), with the suffix *-eyo- expressing habitual or sustained action. The root produced one of the most widely branching word families in the Latinate vocabulary, connecting falling with chance, occasion, rhythm, and legal proceedings.
Latin cadere (to fall) is the central reflex, and its derivatives in English are extraordinary in their range: case (cāsus, "a falling, an occurrence" — what has fallen out, what has happened), cascade (a falling of water), cadence (the fall of the voice, rhythmic fall in music or speech), accident (ad-cidere, "to fall upon" — what falls on you unexpectedly), occasion (ob-cadere, "to fall toward" — what falls in your path, an opportunity), coincidence (co-incidere, "to fall together" — events falling at the same point), incident (in-cidere, "to fall into" — what falls into your experience), and decay (dē-cadere, "to fall down/away").
The legal sense of case — a legal case, a court case — derives from cāsus in the sense of "what has occurred, the matter at hand." The grammatical sense of case (nominative case, accusative case) is a separate derivation from the same word: the Latin cāsus translated Greek ptôsis ("falling"), which the Stoic grammarians used for noun declensions — the different "fallings" of a noun from its base form.
The musical term cadenza (from Italian, ultimately from Latin cadentia, "a falling") denotes the passage where the soloist falls freely, improvising without the orchestra. The connection between falling and freedom — the cadenza as a moment of release from structure — is built into the etymology.
Chance itself enters through Old French cheance, from Vulgar Latin *cadentia (things falling, the fall of dice). To take a chance is, etymologically, to throw the dice and see how they fall. This connects to hazard (a different root) but the conceptual link is the same: fortune as the fall of objects.
The word cadaver (a body that has fallen — Latin cadāver from cadere) and the adjective deciduous (de-ciduous, "falling down" — leaves that fall) round out the most literal applications of the root.