By Henry Parker Manning Ph.D.

A flexible creation to non-Euclidean geometry is suitable for either high-school and faculty periods. Its first two-thirds calls for only a familiarity with airplane and reliable geometry and trigonometry, and calculus is hired in basic terms within the ultimate half. It starts with the theorems universal to Euclidean and non-Euclidean geometry, after which it addresses the categorical adjustments that represent elliptic and hyperbolic geometry. significant themes contain hyperbolic geometry, unmarried elliptic geometry, and analytic non-Euclidean geometry. 1901 version.

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CHAPTER 5. ANALYTIC NON-EUCLIDEAN GEOMETRY 60 5. The angle between two lines: φ being the angle which a line makes with the radius vector at any point, we have cos φ = cos ip sin(θ − α), sin ip . sin φ = sin iρ For two lines intersecting at this point, sin φ1 sin φ2 = sin ip1 sin ip2 sin2 iρ = sin ip1 sin ip2 + sin ip1 sin ip2 . tan2 iρ Now, from the equation of the line sin ip1 = cos ip1 cos(θ − α1 ), tan iρ sin ip2 = cos ip2 cos(θ − α2 ); tan iρ so that sin φ1 sin φ2 = sin ip1 sin ip2 + cos ip1 cos ip2 cos(θ − α1 ) cos(θ − α2 ).

Theorem. In the elliptic geometry there are lines not in the same plane which have an infinite number of common perpendiculars and are everywhere equidistant. Given any two lines in the same plane and their common perpendicular. If we go out on these lines in either direction from the perpendicular, they approach each other. Now revolve one of them about this perpendicular so that they are no longer in the same plane. After a certain amount of rotation the lines will have an infinite number of common perpendiculars and be equidistant throughout their entire length.

5. Theorem. As the perpendicular distance varies, starting from zero and increasing indefinitely, the angle of parallelism decreases from a right angle to zero. Proof. In the first place the angle of parallelism, which is acute as long as the perpendicular distance is positive, will be made to differ from a right angle by less than any assigned value if we take a perpendicular distance sufficiently small. CHAPTER 3. THE HYPERBOLIC GEOMETRY 30 For, ADE being any given angle as near a right angle as we please, we can take a point, L, on DE and draw LR perpendicular to DA at R.