When Cassius speaks to Brutus during the festival race, what does Cassius argue that Caesar is in Shakespeare's play Julius Caesar?
When Cassius speaks to Brutus in Act I, Scene 2, he argues that Caesar is a tyrant.
In this particular scene, often referred to as "the seduction scene," Cassius strives to persuade Brutus that Caesar is dangerous to the republic as the sole power. For, Caesar is the last of the triumvirate of which he was a part, having himself defeated the second triumvir Pompey, who was later murdered, while the other triumvir, Marcus Crassus, conqueror of Spartacus, has previously been killed by his enemies.
In Scene 2, after Caesar and all the others have left, Cassius approaches Brutus, telling him that he has noticed a change in Brutus, his friend. Brutus replies that his troubled look is entirely because of personal matters, and it has nothing to do with their relationship. Cassius then asks Brutus if he does not see himself objectively; Brutus responds that the eye only records a reflection. Cassius then uses this concept of reflection for his purpose:
And it is much lamented, Brutus,
That you have no such mirrors as will turn
Your hidden worthiness into your eye
That you might see your shadow. I have heard
Where many of the best respect in Rome,
Except immortal Caesar, speaking of Brutus
And groaning underneath this age’s yoke,
Have wished that noble Brutus had his eyes. (1.2.58-65)
In this passage Cassius tells Brutus that he needs to recognize his hidden noble qualities--"your shadow"-- because many of the noble Romans speak of him as they complain of the tyranny of Caesar's government--"this age's yoke"--and wish that Brutus could perceive better what is taking place in Rome. Cassius then flatters Brutus before he reaches his point that Caesar has "now become a god" (2.1.118), and he implies that he and Brutus and the others are merely like slaves under him:
...doth bestride the narrow world
Like a Colossus, and we petty men
Walk under his huge legs and peep about
To find ourselves dishonorable graves. (2.1.136-139)
Further, Cassius informs Brutus that before now, no one could contend that just one man was of importance in all of Rome. But, now it seems that there is only room for this tyrant, Caesar. He reminds Brutus that there was another Brutus once [his ancestor, Lucius Junius Brutus, who expelled the last king of Rome] who would never have allowed a tyrant to rule his Roman Republic:
There was a Brutus that would have brooked
Th' eternal devil to keep his state in Rome
As easily as a king. (2.1.186-188)
Brutus acknowledges that he understands what Cassius wants of him, and he asks Cassius to refrain from any more persuasive words. Later, after considering what Cassius has said, Brutus says he will find an opportune time to discuss the matter further with him.
During the celebration of the feast of Lupercalia, Cassius approaches Brutus and starts to try to undermine Caesar in Brutus's mind. Basically, Cassius tells Brutus that Caesar is being given too much power by the Romans. He also argues that Caesar is not fit to rule Rome.
Cassius appeals to Brutus to do something about Caesar. He says that Caesar is going to destroy the Roman Republic and that he (Cassius) would rather die than live under a king's rule.
This is not what causes Brutus to start to worry about Caesar's growing power -- he's already been worried about it. But it does increase his concern.