In Act 1, Scene 2, Caesar reveals to the audience that he is deaf in his left ear. He tells Antony:
Come on my right hand, for this ear is deaf,
And tell me truly what thou think'st of him.
This may have been simply to shift the actors around for some purpose. For example, if Caesar and Antony were standing close together and sideways to the audience, Shakespeare may have wanted to feature Caesar first and then have Antony come around so that the audience could get a good look at him. This is a scene in which most of the important characters have to be introduced. Notice how they all keep calling each other by name. This is unnatural but necessary to inform the audience and avoid future confusion. We meet Brutus, Cassius, Casca, Caesar, Antony, and Calpurnia. The only important man Shakespeare could not introduce at this point is Octavius, who will not arrive in Rome until after Caesar's assassination. Brutus's wife Portia will also be introduced later.
In Act 1, Scene 2, we learn that Caesar has "the falling sickness," epilepsy.
But, soft, I pray you. What, did Caesar swound?
He fell down in the market-place, and foamed at the mouth.
'Tis very like: he hath the falling sickness.
In Act I, Scene 2, Julius Caesar's physical problems are revealed:
1. Caesar's virility - As part of the Lupercal festivities, there is a foot race, and the young Marc Antony prepares to participate. Known for his athleticism and licentious behavior, Marc Antony is called upon by Caesar.
Forget not in your speed, Antonius
To touch Calpurnia, for our elders say
The barren, touched in this holy chase
Shake off their sterile curse. (1.2.6-9)
Certainly, in the time of the Elizabethans, the lack of fertility was thought to be the fault of the woman; however, Caesar's asking the obviously virile Marc Antony to touch his wife clearly suggests that Caesar may be impotent.
2. Caesar's epilepsy - Later, as Cassius attempts to persuade Brutus to join him and the other conspirators, he argues that Caesar is an unfit leader because of his tyrannical tendencies and because of his physical weaknesses, such as the time he jumped into the river, but called to Cassius to help him. Also, Caesar suffers from "fevers" and "fits." Cassius tells Brutus that when Caesar was in Spain, he had a fever and was of a "feeble temper":
And when the fit was on him, I did mark
How he did shake....
His coward lips did from their color fly,
And that same eye whose bend doth awe the world
Did lose his luster: I did hear him groan;...(1.2.121-125)