Etymology
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ad nauseam (adv.)
"to a sickening extent," Latin, literally "to sickness," from ad "to" (see ad-) + nauseam, accusative of nausea (see nausea). Especially of the disgust aroused by wearisome repetition.
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ad valorem 
type of customs duties based on the market value of goods at the original place of shipment, 1711, Modern Latin, "(in proportion) to the value," from ad "to" (see ad-) + Late Latin valorem, accusative of valor "value" (see value (n.)). Sometimes abbreviated ad val.
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ad hoc 
Latin phrase, "to this, with respect to this, for this (specific purpose)," from ad "to" (see ad-) + hoc, neuter accusative of hic "this." Hence, "appointed or enacted for some particular purpose" (1879).
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ad infinitum 
"endlessly," Latin, literally "to infinity" from ad "to, unto" (see ad-) + infinitum "infinity," neuter accusative of adjective infinitus "endless" (see infinite). English version to infinity is attested from 1630s.
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ad hominem 

c. 1600, Latin, literally "to a man," from ad "to" (see ad-) + hominem, accusative of homo "man" (see homunculus). Hence, "to the interests and passions of the person." Originally an argument or appeal to the known preferences or principles of the person addressed, rather than to abstract truth or logic.

Aristotle (Topics, viii 11) remarks that it is sometimes necessary to refute the disputant rather than his position, and some medieval logicians taught that refutation was of two kinds, solutio recta and solutio ad hominem, the latter being imperfect or fallacious refutation. [Century Dictionary]
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ad lib 

also ad lib., 1811 as a musical instruction, shortened from Latin ad libitum "to (one's) pleasure, as much as one likes" (c. 1600), from ad "to" (see ad-) + libitum "pleasure," accusative of libere "to please" (see libido). As a noun from 1825; as a verb by 1915.

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a deux 
French, à deux, literally "for two," from à, from Latin ad "to, toward; for" (see ad-) + deux (see deuce). By 1876 as a French term in English.
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al dente (adv.)
1935, Italian, literally "to the tooth," from Latin dentem (nominative dens) "tooth" (from PIE root *dent- "tooth"). Italian al represents a contraction of words from Latin ad "to" (see ad-) + ille "that" (see le).
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al fresco (adv.)
also alfresco, 1753, Italian, literally "in the fresh (air)." Italian al represents a contraction of words from Latin ad "to" (see ad-) + ille "that" (see le). Alfresco also meant "painted on plaster that was still fresh or moist" (1764; see fresco).
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a la 
from French à la, literally "to the," hence "in the manner of, according to," from à, from Latin ad "to" (see ad-) + la, fem. of definite article le "the," from Latin ille (fem. illa; see le). Attested in English in French terms from fashion or cookery since late 16c.; since c. 1800 used in native formations with English words or names.
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