c. 1600, "composition, act of composing, constructing, arrangement" (also, in early use, with many senses now given to compound (n.2)), from compose + -ure. Sense of "tranquility, calmness, composed state of mind" is first recorded 1660s, from composed "calm" (c. 1600). For sense, compare colloquial fall apart "lose one's composure."
he accepted their problems with composure and she with equanimity
composite
composition
compositional
compositor
compost
composure
compote
compound
comprehend
comprehendible
comprehensible