The Elements of Non-Euclidean Geometry by Julian Lowell Coolidge Ph.D. - Harvard University Contents: CHAPTER I FOUNDATION FOR METRICAL GEOMETRY IN A LIMITED REGION Fundamental assumptions and definitions Sums and differences of distances Serial arrangement of points on a line Simple descriptive properties of plane and space CHAPTER II CONGRUENT TRANSFORMATIONS Axiom of continuity Division of distances Measure of distance Axiom of congruent transformations Definition of angles, their properties Comparison of triangles Side of a triangle not greater than sum of other two Comparison and measurement of angles Nature of the congruent group Definition of dihedral angles, their properties CHAPTER III THE THREE HYPOTHESES A variable angle is a continuous function of a variable distance Saccheri's theorem for isosceles birectangular quadrilaterals The existence of one rectangle implies the existence of an infinite number Three assumptions as to the sum of the angles of a right triangle Three assumptions as to the sum of the angles of any triangle, their categorical nature Definition of the euclidean, hyperbolic, and elliptic hypotheses Geometry in the infinitesimal domain obeys the euclidean hypothesis CHAPTER IV THE INTRODUCTION OF TRIGONOMETRIC FORMULAE Limit of ratio of opposite sides of diminishing isosceles quadrilateral Continuity of the resulting function Its functional equation and solution Functional equation for the cosine of an angle Non-euclidean form for the pythagorean theorem Trigonometric formulae for right and oblique triangles CHAPTER V ANALYTIC FORMULAE Directed distances Group of translations of a line Positive and negative directed distances Coordinates of a point on a line Coordinates of a point in a plane Finite and infinitesimal distance formulae, the non-euclidean plane as a surface of constant Gaussian curvature Equation connecting direction cosines of a line Coordinates of a point in space Congruent transformations and orthogonal substitutions Fundamental formulae for distance and angle CHAPTER VI CONSISTENCY AND SIGNIFICANCE OF THE AXIOMS Examples of geometries satisfying the assumptions made Relative independence of the axioms CHAPTER VII THE GEOMETRIC AND ANALYTIC EXTENSION OF SPACE Possibility of extending a segment by a definite amount in the euclidean and hyperbolic cases Euclidean and hyperbolic space Contradiction arising under the elliptic hypothesis New assumptions identical with the old for limited region, but permitting the extension of every segment by a definite amount Last axiom, free mobility of the whole system One to one correspondence of point and coordinate set in euclidean and hyperbolic cases Ambiguity in the elliptic case giving rise to elliptic and spherical geometry Ideal elements, extension of all spaces to be real continua Imaginary elements geometrically defined, extension of all spaces to be perfect continua in the complex domain Cayleyan Absolute, new form for the definition of distance Extension of the distance concept to the complex domain Case where a straight line gives a maximum distance CHAPTER VIII THE GROUPS OF CONGRUENT TRANSFORMATIONS Congruent transformations of the straight line,,, hyperbolic plane,,, elliptic plane,,, euclidean plane,,, hyperbolic space,,, elliptic and spherical space Clifford parallels, or paratactic lines CHAPTER IX POINT, LINE, AND PLANE TREATED ANALYTICALLY CHAPTER X THE HIGHER LINE GEOMETRY CHAPTER XI THE CIRCLE AND THE SPHERE CHAPTER XII CONIC SECTIONS CHAPTER XIII QUADRIC SURFACES CHAPTER XIV AREAS AND VOLUMES Volume of a cone of revolution, a sphere, the whole of elliptic or of spherical space CHAPTER XV INTRODUCTION TO DIFFERENTIAL GEOMETRY CHAPTER XVI DIFFERENTIAL LINE-GEOMETRY CHAPTER XVII MULTIPLY CONNECTED SPACES CHAPTER XVIII THE PROJECTIVE BASIS OF NON-EUCLIDEAN GEOMETRY CHAPTER XIX THE DIFFERENTIAL BASIS FOR EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY