Geometrical Researches on the Theory of Parallels

a)+II acute angle angle BAC angles equal angles opposite assume axes AA axioms axis BF boundary line boundary-surface CD somewhere chord circle congruent consequently cos II curvature cut the line dicular draw end-points equal angles equations erected Euclid Gauss Greek Hence hypothenuse II(a II(b II(c imaginary geometry intersection point Kasan let fall line AA line AB line CD line DC Lobatschewsky mid-point negative non-Euclidean geometry opposite angles opposite triangles parallel to CD parallels AA pendicular perpen perpendicu perpendicular HK perpendicular sides pertain plane geometry point F Princeton College prolongation random rectilineal triangle right angles right-angled triangles sin C sin sin II small soever soever the angle space sphere spherical triangle spherical trigonometry Theo Theorem 16 Theorem 23 THEORY OF PARALLELS three angles triangle ABC Fig triangle equal triangle the sum ular vertex whence follows