A tangent is a mathematical term,
meaning a line or plane that touches a curved surface but doesn’t intersect it.
The non-mathematical meaning of tangent comes from this sense of barely
touching something: when a conversation (or day-dreaming) heads off on a
tangent, it’s hard to see how or why it came up.
Sometimes the most brilliant,
break-through ideas appear out of nowhere when people are trying to solve a
problem by combining unrelated concepts (eg. found haiku such as ‘disruptive
new stuff’ and ‘ecological footprint’) and somebody’s thoughts go off on a
tangent – this is the concept behind ‘brainstorming’ (and day-dreaming if there's only one person involved) and it
has resulted in a lot of spectacular-sounding new ideas, a tiny percentage of
which are actually useful.