In order for the sun's rays to refract so we can see them, the sun has to be within 42 degrees of the horizon either rising or setting. That's pretty low, which means we can never see the full circle of a rainbow - we just see the top part of the arc.
As far back as the 17th century, the Irish understood this. And they were fond of saying that one was as likely to find a pot of gold as to find the end of a rainbow. The expression somehow changed and became a part of fairy mythology where Leprechauns bury pots of gold at the end..
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