In the play "Julius Caesar" how do the conspirators convince others to help them overthrow Caesar? Are their reasons based on logic or emotions?
As is stated in the e-Notes character analysis of Cassius (see reference link below), he is the instigator of the plot against Caesar. No one else is shown trying to persuade anyone to help, although the conspiracy is partially underway at the time Cassius makes a brilliant and forceful attempt to involve Brutus. It would appear that the assassination of Julius Caesar was Cassius' idea and that it was he who persuaded others to join him. No doubt the other conspirators have had secret conversations among themselves, but these are not included in the play. Even Casca is not a member of the conspiracy at the time Cassius reveals his intentions to Brutus in Act 1, Scene 2.
Shakespeare must have realized that it would be too confusing and too time-consuming to show some conspirators talking to each other and approaching men who were not yet involved. This is a play that requires so many characters with speaking roles and so many others without speaking roles that it must have been hard to understand who were the conspirators, who were friends of Caesar, who were senators, and who were just neutral citizens, especially since they were all males, all dressed alike except for some soldiers in uniform, and all about the same age except for one or two elderly types. Shakespeare was forced to stage his play in this way because he was following history as recorded by Plutarch.
Caesar is surrounded with men when he is assassinated, and there are others present who are just observers or senators or petitioners. Then a whole mob of plebians gather to hear Brutus and Antony speak and to go on a rampage after Antony stirs them up. Then there must be soldiers present in some of the scenes, and especially at the Battle of Philippi. Shakespeare, or whoever directed this play, must have had a hard time just keeping order. No doubt some of the actors played several different parts. A mob member at Caesar's funeral might become a soldier at Philippi, and so on. But the orchestration of the assassination always revolves around Cassius.
In Act 1, Scene 2, we see from Cassius' dialogue that he will use any kind of argument, logical or emotional, to win people to his cause. He is a shrewd judge of people. He can tailor his pitch to suit the individual. His own motive is based on cunning, logic and prudence. He knows that Caesar wants to be king. He believes that Caesar will become a tyrant. He knows that Caesar dislikes him and will make life hard for him if and when he assumes absolute power. Cassius wouldn't care if Caesar actually did become king if he did not feel personally threatened. He is not patriotic and public-spirited but mean and selfish. He reveals himself in a soliloquy after he parts with Brutus.
Well, Brutus, thou art noble; yet I see
Thy honorable mettle may be wrought
From that it is disposed. Therefore it is meet
That noble minds keep ever with their likes;
For who so firm that cannot be seduced?
Caesar doth bear me hard, but he loves Brutus.
If I were Brutus now, and he were Cassius,
He should not humor me.
Brutus will help to murder Caesar for unselfish reasons, and since people always tend to judge others by themselves, he believes that Cassius is also acting for unselfish reasons. He will come to understand Cassius better later on when he has to ask him for money.