c. 1300, "onward movement, motion forward, a running in a prescribed direction or over a prescribed distance; path or distance prescribed for a race, a race-course" from Old French cors "course; run, running; flow of a river" (12c.), from Latin cursus "a running; a journey; direction, track navigated by a ship; flow of a stream;" from curs- past participle stem of currere "to run" (from PIE root *kers- "to run").
Also from c. 1300 as "order, sequence;" meanings "habitual or ordinary procedure" (as in course of nature) and "way of life, personal behavior or conduct" are from early 14c.
Most of the extended senses developed 14c. from notion of "line in which something moves" (as in hold one's course) or "stage through which something must pass in its progress." Thus, via the meaning "series or succession in a specified or systematized order" (mid-14c.) comes the senses of "succession of prescribed acts intended to bring about a particular result" (c. 1600, as in course of treatment) and the academic meaning "planned series of study" (c. 1600; in French from 14c.), also "that part of a meal which is served at once and separately" (late 14c.).
Meaning "the flow of a stream of water" is from mid-14c.; that of "channel in which water flows" is from 1660s. Courses was used for the flow of bodily fluids and 'humors' from late 14c.; specifically of menstrual flux from 1560s.
Adverbial phrase of course "by consequence, in regular or natural order" is attested from 1540s, literally "of the ordinary course;" earlier in the same sense was bi cours (c. 1300). Matter of course "something to be expected" is by 1739.