During the feast of the Lupercal, Caesar orders Antony to spank Calphurnia (Caesar's "barren" wife) with his goatskin whip so she might become pregnant. Weird? Yes. But before your imagination runs too far, let us explain what's going on here.
Historically, during Lupercal festivities, it was traditional for young men to run naked through the streets, whipping everyone in sight. The idea was that touching women with the special whip would help them give birth to healthy babies. (We're not kidding. If you don't believe us, you can read Plutarch's biography of Julius Caesar, which describes in detail the kind of aforementioned whippings that went down at Lupercal festivals.)
So when Caesar tells Antony not to forget to "touch Calphurnia" when he's running through the streets, it's because the "elders say, / The barren, touched in the holy chase, shake off their sterile curse" (1.2.4). In other words, Caesar is hoping that Calphurnia will bear him children. This seems pretty random, don't you think? Why does Shakespeare go out of his way to include this bizarre moment in the play? Here are a few ideas:
1. Although Caesar blames Calphurnia's for being "barren," it's possible that Caesar could be the one who's impotent or sterile. (After all, it's not like they had fancy fertility doctors at the time.) We can't know for sure, but Shakespeare may be trying to plant the idea in the audience's mind that Caesar isn't as perfect as he thinks he is. There are lots of other references to Caesar's "shortcomings" in the play. In Act 1, Scene 2, Casca tells us how Caesar fainted when he was offered the crown (1.2.8), and Cassius happily reports that when Caesar was younger he became ill and acted like a "sick girl" (1.2.8).
2. When Caesar asks Antony to whip "barren" Calphurnia, we know that he's anxious about not having kids. For a guy who might become a king, it's important to have an heir to inherit the throne, right? Remember, Caesar pretends he doesn't want to be crowned king in Act 1, Scene 2, but he's lured to the Capitol in Act 2, scene 2 by Decius's promise that the Senate wants to crown him king. So the play raises the possibility that Caesar really does have dynastic ambitions. It's not only possible that Caesar wants to be a monarch; it also seems like he wants a little baby Caesar to inherit the throne.
3. Finally, this could be a not-so-subtle reference to the childless Queen Elizabeth I, who was way too old to have kids and hadn't yet named an heir to England's throne when Shakespeare wrote the play.