c. 1300, "counting," especially "reckoning of money received and paid, detailed statement of funds owed or spent or property held," from Old French acont "(financial) account, reckoning, terminal payment," from a "to" (see ad-) + cont "counting, reckoning of money to be paid," from Late Latin computus "a calculation," from Latin computare "to count, sum up, reckon together," from com "with, together" (see com-) + putare "to reckon," originally "to prune," from PIE root *pau- (2) "to cut, strike, stamp."
From the first often in plural form; sometimes in late Middle English accompt (see account (v.)). Meaning "course of business dealings requiring records" is from 1640s; hence "arrangement to keep money in a business, bank, etc." (1833), also "customer or client having an account" (1937). Money of account (1690s), that used in reckoning but not circulating as coin or paper, preserves the "counting" sense of the word.
From the notion of "rendering an account" comes the sense "statement answering for conduct" (mid-14c.) and the general sense "narration, recital of facts," attested by 1610s. Phrase by all accounts is attested from 1798. From the notion of "statement of reasons" comes on no account "under no circumstances" (1704). Also from c. 1300 in reference to answering for one's conduct, especially at the Last Judgment. Meaning "estimation, consideration," especially in the eyes of others, is from late 14c.
On account in the financial sense "as an item to be accounted for at the final settlement" is from 1610s, hence on account of in the general sense "for the sake of, in regard to, in consideration of" (1640s, originally upon account of). Also on (my, your, etc.) account "on (one's) behalf." To give accounts "prepare or present a statement of funds and property" is from mid-15c; the older term was cast accounts (mid-14c.); to take account of originally was to make an inventory; take into account "take account of" is from 1680s.
The spellings accompt, accomptable, etc. are artificial forms used, not prevailingly, in the sixteenth and seventeenth centuries. They are now obsolete, or nearly so, though accompt and accomptant may still be used in the formal or legal style. The pronunciation has always conformed to the regular spelling, account, accountable, etc. [Century Dictionary]
Meaning "to reckon for money given or received, render a reckoning," is from late 14c. Sense of "to explain, justify" (c. 1300) is from notion of "present a detailed explanation of money, etc. held in trust." Transferred sense of "to value, to estimate" (to account as belonging to a certain class of quality) is from late 14c. Intransitive sense of "to render an account of particulars" is from late 14c.; hence transitive sense "give an explanation" (1670s, usually with to before a person and for before a thing).
In later Old French partly re-Latinized as acompter (Modern French accompter), hence late Middle English accompten. Related: Accounted; accounting.