Hamilton provided the pistols and they had been tampered with to ensure quick-fire. Hamilton fired and missed (the idea that he fired in the air is wrong). Burr fired and hit. The pistols were the subject of a Smithsonian magazine article of some years ago, and they were x-rayed for that article. It is generally known that the duel was not just one involving political enemies; it is very rarely written that Hamilton had implied that Burr had enjoyed sexual relations with his highly educated daughter.
Bookshelf,
In his book (novel, really) “Burr,” Gore Vidal has Burr explaining that Hamilton didn’t miss on purpose: “His hand shook; mine never does.” It was funny in context.