Reincarnation

Plato believed in reincarnation. Of course, he was living in a small, stable, population, so he could speculate about it. I've always been skeptical, because the theory (doctrine? dogma?) of reincarnation can't handle population growth very well. If humans began with Eve about 200,000 years ago, then where did the "souls" that transmigrate originally come from? And where do they come from today as population continues to grow worldwide?
But what got me thinking about the whole thing again is the question of how the concept got started in the first place. At what point did one person say, "Our souls move from person to person. When you die, your soul goes into some newborn"?
Someone had to make that hypothesis. I wonder what caused her or him to do so. If we could see into that first instant when a person had the idea, we might get an idea of what it is that makes the concept of reincarnation so persistent, despite there being absolutely no evidence for it (Forget hypnosis and "channeling." They're bogus.)
Did that first theorist see his or her mother in the eyes of a newborn? Did a young man show the characteristics of an ancestor? Did a young woman act like her dead grandmother?
Reincarnation as a doctrine gives us lots of avenues for speculation and creativity (One of my favorite novels, Ursula LeGuin's The Tombs of Atuan, is based on reincarnation), but as far as it having any logical or scientific basis -- it ain't there.