Table of Contents:
“…The attempts to prove Euclid’s parallel postulate : The Greek geometers and the parallel postulate ; The Arabs and the parallel postulate ; The parallel postulate during the renaissance and the 17th century -- The forerunners of non-Euclidean geometry : Gerolamo Saccheri (1667-1733) ; Johann Heinrich Lambert (1728-1777) ; The French geometers towards the end of the 18th century ; Adrien Marie Legendre (1752-1833) ; Wolfgang Bolyai (1775-1856) ; Friedrich Ludwig Wachter (1792-1817) ; Bernhard Friedrich Thibaut (1776-1832) -- The founders of non-Euclidean geometry : Karl Friedrich Gauss (1777-1855) ; Ferdinand Karl Schweikart (1780-1832) ; Franz Adolf Taurinus (1794-1874) -- The founders of non-Euclidean geometry (cont.) : Nicolai Ivanovitsch Lobatschewsky (1793-1856) ; Johann Bolyai (1802-1860) ; The absolute trigonometry ; Hypotheses equivalent to Euclid’s postulate ; The spread of non-Euclidean geometry -- The later development of non-Euclidean geometry : Differential geometry and non-Euclidean geometry : Geometry upon a surface ; Principles of plane geometry on the ideas of Riemann ; Principles of Riemann’s solid geometry ; The work of Helmholtz and the investigations of lie.Projective geometry and non-Euclidean geometry : Subordination of metrical geometry to projective geometry ; Representation of the geometry of Lobatschewsky-Bolyai on the Euclidean plane ; Representation of Riemann’s elliptic geometry in Euclidean space ; Foundation of geometry upon descriptive properties ; The impossibility of proving Euclid’s postulate -- The fundamental principles of statics and Euclid’s postulate : On the principle of the lever ; On the composition of forces acting at a point ; Non-Euclidean statics ; Deduction of plane trigonometry from statics -- Clifford’s parallels and surface, sketch of Clifford-Klein’s problem : Clifford’s parallels ; Clifford’s surface ; Sketch of Clifford-Klein’s problem -- The non-Euclidean parallel construction and other allied constructions : The non-Euclidean parallel construction ; Construction of the common perpendicular to two non-intersecting straight lines ; Construction of the common parallel to the straight lines which bound an angle ; Construction of the straight line which is perpendicular to one of the lines bounding an acute angle and parallel to the other ; The absolute and the parallel construction -- The independence of projective geometry from Euclid’s postulate : Statement of the problem ; Improper points and the complete projective plane ; The complete projective plane ; Combination of elements ; Improper lines ; Complete projective space ; Indirect proof of the independence of projective geometry from the fifth postulate ; Beltrami’s direct proof of this independence ; Klein’s direct proof of this independence -- The impossibility of proving Euclid’s postulate, an elementary demonstration of this impossibility founded upon the properties of the system of circles orthogonal to a fixed circle : The system of circles passing through a fixed point ; The system of circles orthogonal to a fixed circle -- The science of absolute space -- The theory of parallels.…”