early 15c., prosecuten, "to follow up, pursue with a view to carry out or obtain" (some course or action), from Latin prosecutus, past participle of prosequi "follow after, accompany; chase, pursue; attack, assail, abuse," from pro- "forward" (see pro-) + sequi "follow" (from PIE root *sekw- (1) "to follow"). Meaning "bring to a court of law, seek to obtain by legal process" is recorded from 1570s. The Latin verb in Old French became prosequer, vulgarly porsuir, which passed to English as pursue.
"filled with desire (for something), wishing to obtain," c. 1300, from Anglo-French desirous, Old French desirros (11c., Modern French désireux), from Vulgar Latin *desiderosus, from stem of Latin desiderare (see desire (v.)).
c. 1300, purchasen, "acquire, obtain; get, receive; procure, provide," also "accomplish or bring about; instigate; cause, contrive, plot; recruit, hire," from Anglo-French purchaser "go after," Old French porchacier "search for, procure; purchase; aim at, strive for, pursue eagerly" (11c., Modern French pourchasser), from pur- "forth" (possibly used here as an intensive prefix; see pur-) + Old French chacier "to run after, hunt, chase" (see chase (v.)).
Originally to obtain or receive as due in any way, including through merit or suffering; the specific sense of "acquire or secure by expenditure of money or its equivalent, pay money for, buy" is attested from mid-14c., though the word continued to be used for "to get by conquest in war, obtain as booty" up to 17c. Related: Purchased; purchasing.