Where It Actually Comes From:
Handshakes are a relic of an era when everyone was a paranoid wreck expecting to be murdered to death by anyone and everyone they saw ... maybe even more so than nowadays.
You see, in the distant past, extending an empty hand was more than a friendly gesture -- it was an indication that a person wasn't holding some sort of sharp rock/knife/wrist-mounted miniature catapult about to be used against you. As history progressed, the practice became more and more complicated to address increasing fears: When ancient Romans got together, for example, they latched onto each other's arms clear up to the elbow to feel for daggers hidden up sleeves.
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It's like Wolverine, but way, way weirder and more inconvenient.
And have you ever noticed that handshakes always involve the right hand? Obviously that's because it's the one most likely to be brandishing a surprise abdomen aerator. At least, that's why some cultures favored the right over the left -- others used the right hand because, in those pre-toilet paper days, your left hand was your ass-wipin' hand.