Origins
To divide is to force apart. The word comes from Latin dīvidere, likely from dis- ('apart') and a lost root related to separation. Roman surveyors used the word when splitting land into parcels, and this practical origin survives in mathematics: to divide a number is to split it into equal portions.
The word family reveals how fundamental the concept of division is to human thought. A dividend — in finance — is the sum to be divided among shareholders. It comes from Latin dīvidendum, 'the thing to be divided'. The term migrated from accounting to the stock market, but the meaning is unchanged.
Individual is the most philosophical descendant. Latin indīviduus meant 'indivisible' — an individual is the smallest unit that cannot be split further without ceasing to exist. The concept of the individual person, so central to Western thought, is built on a metaphor of division reaching its limit.
Proto-Indo-European Roots
Device and devise also descend from this root, through a Vulgar Latin shift from dīvidere to *dīvīsāre. A device was originally a plan or scheme — a way of dividing a problem into manageable parts.
The geographical sense — the Continental Divide, the Great Divide — treats natural ridgelines as places where water is forced apart, flowing to different oceans. The euphemism 'crossing the great divide' for death draws on the oldest sense: the final separation.