The Lessons Non-Euclidean Geometry Can Teach

So YOU want to teach mathematics! Why you selected college geometry is more than I know because college geometry is about as useless as high school geometry. Seriously now, did the fact that an exterior angle of a triangle equals the sum of the two non-adjacent interior angles ever make any real difference in your behavior? And were you thrilled to find out that a line is tangent to a circle if it is perpendicular to the radius drawn to the point of contact? Let me warn you in advance t hat when you find out a line is perpendicular to a plane if it is perpendicular to all lines in the plane passing through its foot, you will be apathetic. Furthermore, when you prove that the volume of a rectangular solid is really the product of the length, width, and height you will probably be disgusted, for you knew t hat ever since you were in the seventh grade. The fact that you proved it is really silly, isn't it? You have gotten along pretty well so far just taking somebody's, a teacher's or author's, word for it. And after all, what difference does it make whether you assume this is so in the first place or whether you assume something else, and from this prove the theorem for finding this volume. None of you would try to make me believe that you can calculate the area of the floor of this room better than Fred Fable whose schedule conflicts prevented him from being one of your classmates in geometry in high school.