Origins
Three very different meanings of decline β grammatical, social, and personal β all come from the same image: bending away.βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ Latin dΔclΔ«nΔre meant 'to slope downward' or 'to turn aside', from dΔ- ('away') and clΔ«nΔre ('to lean').
The grammatical sense arrived first in English. In Latin education, to 'decline' a noun was to run through its case endings β nominative, genitive, dative, accusative. The metaphor was of bending or inflecting the base form. This remains standard terminology in linguistics.
Latin Roots
The refusal sense followed: to decline an invitation is to turn aside from it, to bend away from acceptance. The deterioration sense ('the decline of the Roman Empire') draws on the physical image of a downward slope.
The root *αΈ±ley- ('to lean') produced an unexpectedly diverse family. To incline is to lean toward. To recline is to lean back. A clinic comes from Greek klinΔ, a bed β something to lean on. Most remarkably, climate derives from Greek klΓma, 'a slope or zone of the earth', referring to how different latitudes lean toward the sun. The weather itself is a matter of leaning.