Etymology
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Words related to adagio

ad- 

word-forming element expressing direction toward or in addition to, from Latin ad "to, toward" in space or time; "with regard to, in relation to," as a prefix, sometimes merely emphatic, from PIE root *ad- "to, near, at."

Simplified to a- before sc-, sp- and st-; modified to ac- before many consonants and then re-spelled af-, ag-, al-, etc., in conformity with the following consonant (as in affection, aggression). Also compare ap- (1).

In Old French, reduced to a- in all cases (an evolution already underway in Merovingian Latin), but written forms in French were refashioned after Latin in 14c. and English did likewise 15c. in words it had picked up from Old French. In many cases pronunciation followed the shift. Over-correction at the end of the Middle Ages in French and then English "restored" the -d- or a doubled consonant to some words that never had it (accursed, afford). The process went further in England than in France, where the vernacular sometimes resisted the pedantic, resulting in English adjourn, advance, address, advertisement (Modern French ajourner, avancer, adresser, avertissement). In modern word-formation sometimes ad- and ab- are regarded as opposites, but this was not in classical Latin.

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adjacent (adj.)
early 15c., "contiguous, bordering; close, nearby," from Latin adiacentem (nominative adiacens) "lying at," present participle of adiacere "lie at, border upon, lie near," from ad "to" (see ad-) + iacēre "to lie, rest," related to iacere "to throw; lay ('cast (oneself) down')," from PIE root *ye- "to throw, impel." Only of things, never of persons or animals. Adjacent, properly, is near but not necessarily in contact; adjoining is so as to touch. Latin adiacentia meant "the neighborhood."
*ad- 
Proto-Indo-European root meaning "to, near, at."

It forms all or part of: abate; ado; ad-; ad hoc; ad lib; adage; adagio; add; adjective; adore; adorn; adult; adverb; advertise; agree; aid; alloy; ally; amontillado; amount; assure; at; atone; exaggerate; paramount; rapport; twit.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit adhi "near;" Latin ad "to, toward;" Old English æt.
ease (n.)

c. 1200, "physical comfort, undisturbed state of the body; tranquility, peace of mind," from Old French aise "comfort, pleasure, well-being; opportunity," which is of uncertain origin. According to Watkins is ultimately from Latin adiacens "lying at," present participle of adiacere "lie at, border upon, lie near," from ad "to" (see ad-) + iacere "to lie, rest," literally "to throw" (from PIE root *ye- "to throw, impel"). Recent dictionaries in English and French seem to support this derivation, though older sources found objections to it. Compare adagio.

At ease "at rest, at peace, in comfort" is from late 14c.; as a military order (1802) the word denotes "freedom from stiffness or formality."