Non-Euclidean geometry : a critical and historical study of its development
Roberto Bonola, H. S. Carslaw (Translator), Federigo Enriques (Writer of introduction), János Bolyai, N. I. Lobachevskiĭ
Authorized English translation with additional appendices by H.S. Carslaw. With an introduction by Federigo Enriques. With a suppl. containing the George Bruce Halsted translations of The science of absolute space, by John Bolyai [and] The theory of parallels, by Nicholas Lobachevski
xii, 268, xxx, 71, 50 pages illustrations, diagrams 21 cm
9780486600277, 0486600270
631464
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