Supernatural events are one of the elements of tragedy.
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There are two in the very beginning. The first is the Feast of Lupercal, which is believed to encourage women to conceive, should they be touched during a ritual running through the streets. Both Antony and Ceaser seem to believe this will work. However, the more obvious example is Calpurnia's dream, which prefigures Caesar's death. Whereas she is a terribly frightened woman (probably partly due to having so domineering a husband) who puts great stock in the omen, he blows it off, showing his extreme arrogance and belief that he can control his own destiny.