"divine, spiritual, of or pertaining to a numen," 1640s, from Latin numen (genitive numinis) "divine will," properly "divine approval expressed by nodding the head," from nuere "to nod," from PIE *neue- "to nod" (source also of Greek neuein "to nod;" Old Irish asnoi "to promise," adnoi "to entrust") + -ous.
"divine spirit, presiding divinity," 1620s, from Latin numen "divine will, divinity," literally "a nod" (the notion is "divine approval expressed by nodding the head"), from nuere "to nod" (assent); see numinous.
"drooping or nodding, hanging with the apex downward," 1751, from Latin nutantem (nominative nutans), present-participle of nutare "to nod with the head," from PIE *neu- (2) "to nod" (see numinous).
1610s, "action of nodding," from Latin nutationem (nominative nutatio), noun of action from past participle stem of nutare "to nod," from PIE *neu- (2) "to nod" (see numinous). Astronomical use in reference to slight periodical oscillation of the earth's axis is from 1715. Related: Nutational.
Originally in English a legal phrase (1560s) from Medieval Latin, with the sense of "to wit," introducing an explanatory or parenthetical clause, it also introduced the derogatory meaning alleged in libel cases, which led to broader meaning. As a verb, from 1706.
The motto is often translated as "He (God) is favorable to our undertakings," but this is not the only possible translation. Thomson wrote: "The pyramid signifies Strength and Duration: The Eye over it & Motto allude to the many signal interpositions of providence in favour of the American cause." The original design (by William Barton) showed the pyramid and the motto Deo Favente Perennis "God favoring through the years."
The Latin elements are the perfective of annuere "indicate approval, agree to, grant," literally "nod to (as a sign)" (from assimilated form of ad "to;" see ad-, + nuere "to nod;" see numinous) + perfect passive of coeptus, past participle of coepere "to begin, commence."