Resolution’s earliest 14th century definition drew from its direct Latin source resolutionem (perhaps via the Old French resolution), which meant “a process of reducing things into simpler forms,” drawing from the notion of resolvere as a word for “loosen” or “untie.” Like the English word today, resolvere held a diverse array of additional meanings as well: “unyoke, undo; explain; relax; set free; make void, dispel.” Its root is the PIE leu- “to loosen, divide, cut apart.” “Solve” without the intensifying prefix has also carried similar meanings as it evolved.
The reductive/simplifying definition of “resolution” is still in use today, mostly via the notion of “solution” in scientific settings, though it is otherwise less common than other meanings, and most often in the form “resolve.” Notably, we see it in one of Hamlet’s existential soliloquies: O, that this too too solid [or sullied] flesh would melt, / Thaw, and resolve itself into a dew. (Here “resolve” is used to mean “dissolve, reduce to liquid.”)
In the 1540s, the word came to suggest the solving of a problem, first in terms of mathematical problems and later in terms of the resolution of a conflict (or, say, a political resolution or bill). The sense of “resolute” meaning firm or determined arose simultaneously, the notion being that one who is resolute has solved any dilemma and come to a firm decision. The optical sense of resolution (photo resolution, for example) arose in the 1860s, and also comes from this concept of fixedness and determination, suggesting a clear and unmuddled image, though this meaning is also thought to draw upon the concept of “resolving” as a word for separating something into components (as in dpi or pixels).
The word’s use in relation to New Year’s arose in the 1780s (or perhaps earlier) and suggested a firm plegde or decision to better oneself in the new year. As with many holiday traditions, New Year’s resolutions are rooted in religion. Babylonians were thought to be some of the first to make such resolutions, making promises to the gods to return borrowed objects and pay debts at the start of each year. Drawing from this earlier tradition, Romans also made promises to the two-faced god Janus, namesake of the month January. The Roman practice is the most the source of our contemporary practice of setting resolutions in the new year. The earliest New Year’s resolutions in English-speaking cultures tended to be rooted in piety and religious promises.
During the Medieval Era, there was yet another New Years-resolution tradition, known as the Vow of the Peacock, that has fallen out of practice in modern times. In Charles Dickens’ periodical All the Year Round, he wrote about the Vow of the Peacock, explaining that peacocks (and occasionally pheasants) represented “by the splendour and variety of their colours, the majesty of kings during the middle ages”—and were thought to be “the peculiar diet of valiant knights and heart-stricken lovers.” Therefore, in the new year, a great feast was held with a roasted peacock as its centerpiece. Each knight would make a vow of chivalry to the bird, after which it would be carved and divided among all those present.
[Portions of this originally appeared in my work on Writer’s Digest.]
Useless Etymology 