Geometrical Researches on the Theory of Parallels

acute angle angle BAC angles equal angles opposite arcs assume axes AA axioms axis BB boundary line boundary-surface CD somewhere circle congruent consequently curvature cut the line dicular digon draw ef(b 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 CD line DC Lobatschewsky mid-point negative Non-Euclidean non-Euclidean geometry opposite angles opposite triangles parallax parallel to CD parallels AA pendicular perpen perpendicu perpendicular HK perpendicular sides pertain plane geometry point F Princeton College prolongation quadrilateral random rectilineal triangle right angles right-angled triangles sin II small soever soever the angle space sphere spherical triangle spherical trigonometry straight angle Theo Theorem 16 Theorem 23 THEORY OF PARALLELS three angles triangle ABC Fig triangle equal triangle the sum ular vertex whence follows