d, the two angles of the lune are congruent.The area of a sphere is closely related to finding the area of a circle in Euclidean geometry. To find the area of a circle, the formula, piR is used in Euclidean Geometry. In Spherical geometry, a similar formula is used, 4piR.Besides Euclidean Geometry and the Non-Euclidean Spherical Geometry, there is also the commonly known Non-Euclidean Hyperbolic Geometry. Hyperbolic Geometry is closely related to Einstein’s General theory of Relativity. Hyperbolic Geometry is a “curved” space, related to some Spherical Geometry. Einstein’s General Theory of Relativity can be understood by saying that matter and energy distort space, and the distortions of space affect the motions of matter and energy, being related to, “curved space.” Cosmologists today believe that “curved space” is the fourth dimension, although it hasn’t yet been proven by means of postulates, theorems, or proofs.It is very difficult to draw a mental picture of four-dimensional space. Although, it has been drawn in a way excepted by crazed mathematicians and cosmologists. Four-dimensional space is derived from flatland that contains sliding figures within. See Appendix 2-1. This flatland is then manipulated into a curved plane. See Appendix 2-2. Both the flatland and the manipulated curved flatland, is related to that of Mercury’s orbit, replacing Euclidean geometry with Hyperbolic.Hyperbolic geometry has an interesting “twisted” figure to it, the pseudosphere. The pseudosphere is the two-dimensional object, which is related to the normal sphere used in Spherical Geometry. However, the pseudosphere is smaller then the plane it lies on and tends to “bend back” on itself. Since the pseudosphere is bigger than the plane it lies on, it is hard to be drawn out and to be visualized. In Appendix 2-3, the fourth dimension is shown, for us to b...
