350 11. POST-EUCLIDEAN GEOMETRY
that is, the intersections of the sphere with planes passing through its center. Let
one "line" (great circle) be the equator of the sphere. Describe the equidistant
curve generated by the endpoint of a "line segment" (arc of a great circle) of fixed
length and perpendicular to the equator when the other endpoint moves along the
equator. Why is this curve not a "line"?
11.7. Al-Haytham's attempted proof of the parallel postulate is fallacious because
in non-Euclidean geometry two straight lines cannot be equidistant at all points.
Thus in a non-Euclidean space the two rails of a railroad cannot both be straight
lines. Assuming Newton's laws of motion (an object that does not move in a
straight line must be subject to some force), show that in a non-Euclidean universe
one of the wheels in a pair of opposite wheels on a train must be subject to some
unbalanced force at all times. [Note: The spherical earth that we live on happens
to be non-Euclidean. Therefore the pairs of opposite wheels on a train cannot both
be moving in a great circle on the earth's surface.]
11.8. Prove that in any geometry, if a line passes through the midpoint of side
AB of triangle ABC and is perpendicular to the perpendicular bisector of the
side BC, then it also passes through the midpoint of AC. (This is easier than it
looks: Consider the line that does pass through both midpoints, and show that
it is perpendicular to the perpendicular bisector of BC; then argue that there is
only one line passing through the midpoint of BC that is perpendicular to the
perpendicular bisector of BC.)
11.9. Use the previous result to prove, independently of the parallel postulate, that
the line joining the midpoints of the lateral sides of a Thabit (Saccheri) quadrilateral
bisects the diagonals.
