The reasoning that Machiavelli uses in this passage is: Machiavelli uses deductive reasoning by first introducing the conclusion that new rulers must be cruel and then supporting it with evidence.
Deductive reasoning is a form of reasoning where broad generalizations are first made before specific observations are used to corroborate it.
In the passage, we see an example of deductive reasoning because a broad generalization about the cruelty of leaders was first made before examples were used to support it.
Learn more about deductive reasoning here:
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