early 14c., "separate into parts or pieces," from Latin dividere "to force apart, cleave, distribute," from assimilated form of dis- "apart" (see dis-) + -videre "to separate," which, according to de Vaan, is from PIE *(d)uid- "to separate, distinguish" (source also of Sanskrit avidhat "allotted," Old Avestan vida- "to devote oneself to"). He writes: "The original PIE verb ... (which became thematic in Latin) meant 'to divide in two, separate'. It lost initial *d- through dissimilation in front of the next dental stop, and was reinforced by dis- in Latin ...." Also compare devise.
It is attested from late 14c. as "sever the union or connection with," also "disunite, cause to disagree in opinion." Intransitive sense of "become separated into parts" is from 1520s. Mathematical sense "perform the operation of division" is from early 15c. Divide and rule (c. 1600) translates Latin divide et impera, a maxim of Machiavelli. Related: Divided; dividing.
1640s, "act of dividing," from divide (v.). Meaning "watershed, separation between river valleys" is recorded by 1807, American English.