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Tim Cook has the decidedly nontrivial distinction of being the first CEO of Apple since the very first to come to power without blood on his hands. For most of its history, Apple has had a succession problem: it had no internal mechanism for transferring power from one CEO to the next without descending into civil war in between. “Each time,” Cook says, “the way that the CEO was named was when somebody got fired and a new one came in.”

This clearly bothered Steve Jobs, because he spoke to Cook about it shortly before he died. “Steve wanted the CEO transition to be professional,” Cook says. “That was his top thing when he decided to become...