How does Cassius convince Brutus to join the conspiracy?

In act one, scene two, Cassius tells Brutus to let him be his "glass," meaning his mirror to better see himself. He then proceeds to interpret, for Brutus, all of Brutus's reactions to what they can hear of Caesar's interaction with the citizens. When Brutus says that he fears that "the people / Choose Caesar for their king," Cassius replies that this must mean that Brutus recognizes that Caesar should not be "their king." Brutus, like Cassius, is concerned that Caesar is becoming too powerful.
Cassius then tells Brutus that they are as worthy as Caesar, that they "both have fed as well" and were "born free as Caesar." The implication that Cassius is trying to push here is that they are as worthy as Caesar and, therefore, Caesar should not be more powerful than they are. Cassius continues to mock Caesar's masculinity or supposed lack thereof. He relates a story in which Caesar was drowning and asked for Cassius's help and then says, incredulously, that this same Caesar "is now become a God." Again, Cassius is trying to persuade Brutus that Caesar has become too powerful and is underserving of this power. To emphasize the point, he calls Caesar a "coward" and compares him to "a sick girl."
As persuasive as Cassius's arguments are up to this point, he really hits a nerve, so to speak, when he questions what Caesar is doing to Rome. Brutus truly loves Rome and eventually agrees to join the conspiracy because he genuinely believes that Caesar must be removed for the benefit of his beloved Rome. Cassius points out that Rome has never been ruled by just one man before:

When could they say till now, that talk'd of Rome,That her wide walls encompass'd but one man?

Indeed, before Caesar, Rome was ruled by three men. Cassius suggests that now Rome is becoming something like a dictatorship. Finally, at the end of their conversation and just before Caesar re-enters, Cassius invokes the name of Brutus's relation, who, long ago, once fought to save Rome:

There was a Brutus once that would have brook'dThe eternal devil to keep his state in Rome

In summary, Cassius persuades Brutus to join the conspiracy against Caesar firstly by highlighting how much power Casear has accrued and then by questioning his right to this power given that he is no better than them. Cassius then says that Caesar is in fact less than them—he is cowardly and weak. He then suggests that Caesar is endangering Rome, which Brutus loves and is loyal to, and finally he implies that it is also, for Brutus, a question of family honor to protect Rome.