Casca, talking with Cassius in Act I scene 2, gives a hostile account of how Antony has offered a crown to Caesar, three times, only to have it refused each time. Casca clearly thinks that these refusals were insincere:
I saw Mark Antony offer him a crown...he put it by once: but...to my thinking, he would fain have had it. Then he offered it to him again: then he put it by again: but... he was very loath to lay his fingers off it. And then he offered it the third time; he put it the third time by....
Caesar was so moved, according to Casca, that he "offered them his throat to cut" following the third refusal, triggering still louder cheers. The strain of the occasion, and the emotion of the crowd roaring its approval, finally triggered an attack of epilepsy in Caesar:
He fell down in the market-place, and foam'd at mouth, and was speechless.
'Tis very like: he hath the falling-sickness.
No, Caesar hath it not; but you, and I,
And honest Casca, we have the falling-sickness.
Thus Cassius turns Caesar's infirmity into another rhetorical attack -- Caesar's rise has given them a true "falling sickness."
The incident shows the intensity of popular feeling for Caesar, increased still further when he apologizes for his "infirmity" after regaining consciousness. Brutus fails to notice it, but he will soon find out that the Romans feel the same way about Caesar even after he falls to the knives of the plotters.