A new geometric characterization is presented for a function convex of order n on an open interval, distinct from the whole of R. We shall prove that if /: (a, b)~^R with b < + oo, if a is an arbitrarily fixed number, a ^ 6, and if F(x) denotes the ordinate of the point of intersection in the x, y-plsme between the vertical line x = a and the osculating parabola of order n to the graph of / at the point (x, /(#)), then / is convex of order n on {a, b) iff F is increasing thereon.