etter visualize parts of the pseudosphere.In Euclidean Geometry, the theorem, “If two lines are parallel to a third line, then the two lines are parallel to each other,” is disproved in Hyperbolic Geometry. Appendix 2-4 shows how it is proved false in hyperbolic geometry by means of three lines being contained on the pseudosphere. Euclidean lines and hyperbolic lines are different in only one way. It is said that lines in Euclidean Geometry are straight and endless, although in hyperbolic geometry, lines are contained on the sphere and curve. Although the lines curve, they are still parallel to the sphere in hyperbolic geometry. Even though the lines do have some similarities, there are three theorems in Euclidean Geometry, which are false in Hyperbolic Geometry. These theorems are as follows:1) If two lines are parallel to a third line, then the two lines are parallel to each other.2) If two lines are parallel, then the two lines are equidistant.3) Lines that do not have an end (infinite), also do not have a boundary.One postulate used in Hyperbolic Geometry is the parallel postulate, which states, “Given a line and a point not on that line, there are at least two lines which contain that point, which are in the same plane as the line and are parallel to that line.”In Euclidean Geometry, the area of a triangle is calculated by multiplying the length of any side times the corresponding height, and dividing the product by two. However, in hyperbolic geometry, if a triangle is scribed onto a sphere, the area formula for Euclidean geometry will not work out properly.In Euclidean Geometry the area of a square is a side cubed. However, in Hyperbolic Geometry it is possible for a square on the sphere to contain three right angles and one acute angle. In this case the formula for a square in Euclidean Geometry will not pertain to that of Hyperbolic Geometry. In essence, area formulas for polygons in Euclidean ...
