If you take the news of your parakeet's death with equanimity, it means you take it calmly without breaking down. Equanimity refers to emotional calmness and balance in times of stress.
If equanimity reminds you of equal, that's because the words have a lot in common. The noun equanimity was borrowed from Latin aequanimitās, from aequanimus, "even-tempered, fair," formed from aequus, "even, level, equal," plus animus, "mind." The archaic phrase to bear with equal mind means "to bear with a calm mind," and is a translation from the Latin. The phrase a level mind also refers to calmness. A near synonym is composure.
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