What is the difference between the Usuli principle "Necessities permit the forbidden" and Machiavelli's saying "The end justifies the means"? What is the definition of necessity? And what are the incidents that confirm that the Messenger of Allah, peace and blessings be upon him, used the former and not the latter?

The rule "the end justifies the means" is not absolutely correct. It is valid under two conditions: that the end is legitimate, and that the harm of the prohibited means is less than the benefit of the end. If these conditions are met, it becomes equivalent to the rule "when two harms conflict, the greater harm is averted by committing the lesser one," such as lying to reconcile people. However, if the harm of the end and the means are equal, then "averting harms takes precedence over achieving benefits," or "harm is not removed by an equal or greater harm." As for the rule "necessities permit prohibitions," it is a legitimate principle derived from the Almighty's saying: "But whoever is forced, neither desiring [it] nor transgressing, there is no sin upon him." The difference between the two is that the legitimate rule restricts the prohibited to necessity, while the saying "the end justifies the means" goes beyond that to justify a prohibited means if the end is praiseworthy, even if the harm of the means is equal to or greater than the benefit of the end.