  In this paper, we examine how well a rational point P on an algebraic variety
X can be approximated by other rational points. We conjecture that if P lies on
a rational curve, then the best approximations to P on X can be chosen to lie
along a rational curve. We prove this conjecture for a wide range of examples,
and for a great many more examples we deduce our conjecture from Vojta's Main
Conjecture.
