A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space
The Russian edition of this book appeared in 1976 on the hundred-and-fiftieth anniversary of the historic day of February 23, 1826, when LobaeevskiI delivered his famous lecture on his discovery of non-Euclidean geometry. The importance of the discovery of non-Euclidean geometry goes far beyond the limits of geometry itself. It is safe to say that it was a turning point in the history of all mathematics. The scientific revolution of the seventeenth century marked the transition from "mathematics of constant magnitudes" to "mathematics of variable magnitudes. " During the seventies of the last century there occurred another scientific revolution. By that time mathematicians had become familiar with the ideas of non-Euclidean geometry and the algebraic ideas of group and field (all of which appeared at about the same time), and the (later) ideas of set theory. This gave rise to many geometries in addition to the Euclidean geometry previously regarded as the only conceivable possibility, to the arithmetics and algebras of many groups and fields in addition to the arith metic and algebra of real and complex numbers, and, finally, to new mathe matical systems, i. e. , sets furnished with various structures having no classical analogues. Thus in the 1870's there began a new mathematical era usually called, until the middle of the twentieth century, the era of modern mathe matics.
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A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric ...
Boris A. Rosenfeld
No preview available - 1988
absolute affine affine connection al-Tüsi analogous angle sum arbitrary axiom called century circle coincides complex numbers conic connection considered coordinates corresponding cross ratio curvature curve defined denoted differential dimensions distance elements elliptic equal equations equidistant Euclid Euclidean geometry Euclidean space Euler Figure finite formulas functions Galois Gauss geodesics given group of motions hyperbolic Ibn al-Haytham Ibn Qurra imaginary interpretation intersect isomorphic Jordan algebras Khayyám Leipzig Lie algebra line AC Lobačevskian geometry magnitudes manifold mapping mathematical mathematician matrices n-dimensional non-Euclidean geometry numbers obtained octaves orthogonal paper parallel axiom parallel lines parallel postulate perpendicular Poincaré points problem projective proof proposition prove the parallel pseudo-Euclidean space quadric quadrilateral quaternions radius rectangle representation respectively Riemann right angles rotation segment side simple Lie groups sine sphere spherical triangle spherical trigonometry straight line subgroup surface tangent term theorem theory of parallel topological treatise trigonometry vectors