M. : Science, 1966. 648 s. The book is intended for students of universities and pedagogical institutes, researchers, teachers and engineers interested in geometry . The book involves acquaintance with the courses of analytical geometry and higher algebra, as well as with the basic concepts of group theory (in the scope of the first chapters of A. g. Kurosha). The differential-geometric chapter of the book involves familiarity with the course of differential geometry. All the concepts associated with tensor analysis in the book are defined, but for better assimilation of the material of the relevant sections of the book, preliminary acquaintance with the corresponding chapters of “Rhymanic Geometry and Tensor Analysis” P. is useful. K. Rashevsky . Vectors and affine operations on them . Geometry and numbers. Euclidean space . Material numbers. Vectors. Generalization of the concept of space . Vectors. Linear space. Axioms of addition. Axioms of Multiplication by Number. Linear dependence and independence of vectors. Axiom of dimension. Models of linear space. Coordinates of vectors. Abbreviation for summation. Conversion of coordinates. Subspaces of Linear Space. Tensors. Covariant vectors. Tensors. Symmetric and Cossimmetric Tensors. Linear operators. Vector Linear Functions. Matrix. Linear operators. Addition of operators. Multiplication of operators. Action of operators on covariant vectors. Operator product of vectors. Own vectors. Rectangular operators. Affinity space. Affine space axioms. Transfers. Affinity Coordinates. Metric operations on vectors. Euclidean space. Euclidean space axioms. Models of Euclidean space. Distances between points. Inequality of the triangle. Angles between vectors. Scalar product in coordinates. Orthogonalization. Rectangular Coordinates. Converting Rectangular Coordinates. Euclidean tensors . Metric tensor . Raising and lowering indexes. Reciprocal basis. Calculation of vector coordinates. Covariant coordinates of vectors. Euclidean operators. Calculation of operator matrix elements. Operator product of vectors. Transposed operator. Symmetric operator. Cossimmetric operator. Orthogonal operator. Euclidean rectangular operators. Oriented Space. Orientation of space. Slanted work. Vector product. The oblique product in coordinates. Vector product in coordinates. The relationship of the oblique work with the scalar . The relationship of a vector product with a scalar. Straight and flat . Geometry of straight lines. Direct lines. Coordinate Equations of Direct. Equations of a straight line on two points . The condition of belonging of three points of one straight line. Angle between straight lines. Distance from point to line. Perpendicular lowered from point to line. Reflection from straight. Reciprocal arrangement of two straight lines. Shortest distance between two intersecting straight lines. General perpendicular of two straight lines. Geometry of planes. Planes. Coordinate plane equations. M-plane equations by m + 1 points. Case m n - 1. Vector plane equation. Coordinate plane equation. Equation of plane by point and normal vector. Basic Plane Theorem. Equation of plane by point and guide vectors. Equation of plane on n points. Condition of belonging n + 1 points of one plane. Angle between planes. Distance from point to plane. Reflection from the plane. Distance between parallel planes . Geometry of m-planes . Equations of m-plane. Operator equations of m-plane. Perpendicular dropped from point to m-plane. Distance from point to m-plane. Distance from point to m-plane, other form. Reflection from m-plane. Reflection from m-plane, other form. Reflection from point. Reciprocal arrangement of two non-overlapping planes. Reciprocal location of two intersecting planes. Calculation of the dimension of intersection or general direction of two m-planes. Total perpendicular of two intersecting m-planes. Shortest distance between two intersecting m-planes. Distance between parallel m-planes. Stationary angles between two m-planes. Isocline and fully perpendicular m-planes. Dimensional diversity of m-planes. Movements and Affinity Transformations. Affinity conversions. Geometric transformations. Affinity conversions. Affine transformations in coordinates. Centroaffin transformations and transfers. The Affinity Transformation Group. Affinity conversion task. Athens transformations of the first and second kind . Fixed points and invariant directions. Conversion of kinship. Gomothetia . Conversion of m-native. Movement. Movement and Congruence . Movements in coordinates. Rotations and Transfers. Movement Group. Movement Task. First and Second Kind Movement . Fixed points and invariant directions. Imaginary vectors. Isotropic vectors. The canonical form of the matrix of the orthogonal operator. Classification of rotations. Stationary cornering angles. Paratactic rotation . Classification of movements. Representation of motions as a product of reflections from planes. Similarity. Similar and similar figures. Similarities in coordinates. Similar to . Similarity Center. Homotetia and transfer group. Multifaceted . Straight-line segments. Rays and segments. Length of segment. Oriented segments. Ratio of segments. Division of the segment in this respect. The relationships of segments in affine transformations. Parallelepipeds. Semi-planes and parallelepipeds. Borders of parallelepiped. Volumes. Volume of rectangular parallelepiped. Volume of arbitrary parallelepiped. Oriented n-parallelepipeds. The affinity of n-parallelepipeds. Volumes of arbitrary cubable figures. Simplexes . Simplexes . Simplex Facets . Simplex volume. Oriented n-simplexes. The affinity of n-simplexes. Center of gravity of n-symplex. Orthocentric n-simplex. Polyhedrons of the zero genus . Multifaceted . Prisms and Pyramids . Convex polyhedrons. Polyhedrons of the zero genus . Euler's theorem . The right polyhedrons. Right polygons and 3-polygons. The right n-polygons. Center of correct polyhedron. Characteristic simplex of a regular polyhedron. Classification of correct n-polygons. The correct n-simplex . The volume of the correct n-symplex . Proper n-parallelepiped or n-cub. Mutually correct n-polygons. Multifaceted, reciprocal with n-cube. The correct 4-sided . Symmetries of regular polyhedra. Right (n - 1)-cells. Spheres. Geometry of spheres. Spheres. Equation of the sphere in coordinates. Spheres of Apollonius . Equation of the sphere by n + 1 points. Spheres described around polyhedra. The condition that n + 2 points lie on the same sphere. Degree of point relative to the sphere. Reciprocal Location of Sphere and Straight . Geometric sense of the degree of the point relative to the sphere. Reciprocal location of the sphere and m-plane. Touch plane to sphere. Spheres inscribed in the polyhedron. Reciprocal arrangement of two spheres. Sphere bundle. Angle between spheres. Geometry on the Sphere. Large and small circles and m-spheres. Spherical distances. Spherical Theorem of Cosines . Spherical Sine Theorem. Dual Cosine Theorem . Area of spherical triangle. Co-ordinates in the field. Spherical coordinates in space. Scope element. Volume element in spherical coordinates. Scope of scope. Ball volume. Spherical Simplexes . Spherical Simplexes . The Spherical Simplex . Multifaceted angles of spherical simplex. Alternated sum of simplex angles. The volume of a spherical simplex. Autopolar Simplex. Geometry of m-spheres. Equations of m-spheres. Reciprocal arrangement of two m-spheres . Quadrics. General theory of quadric. Quadric equations. Reciprocal quadricity and straight line. Asymptotic directions . Center of symmetry. Diameter planes and conjugate directions. Planes of symmetry and main directions. Relating to . Polar plane and pole . Reciprocal location of the quadric and m-plane. Classification of quadric. Own vectors of the symmetric operator. Bringing to center. Bringing to main directions. Adjusting to Reciprocal Directions. Classification of Central Quadrics. Cones. Ellipsoids and hyperboloids . Asymptotic cone . Flat forming hyperboloids. Flat forming maximum dimensions. Paraboloids. Flat forming hyperbolic paraboloids. Degenerate quadrics. Overview of quadric types. Affinity transformations and quadric motions. Affinity quadric transformations. Affinity transformation as a product of affinity transformation with symmetric operator and motion. Affinity transformations that translate the quadric into themselves. Elliptical turns. Hyperbolic turns. Parabolic turns. Quadric motions and metric invariants of quadric equations. Invariants at rotations. Invariants in transfers. Invariants in arbitrary movements. Investigation of quadric equations using metric invariants . Moving vectors. Moving vectors in space. Free and sliding vectors. Equivalent systems of sliding vectopoms. Moving vectors on the sphere. Theorems about moving vectors. Equivalence of moving vector systems. The main vector and the main point of the system. The main point of the system of moving vectors on the sphere. Main axis of the system. Geometric meaning of the operator of the main moment. Own vectors of the cossimmetric operator. Canonical system of vectors. Equivalence condition of sliding vector systems. Projective transformation. Projective space. Central Design. Projective n-space. Projective coordinates of points. Double ratio of four points. Planes. Equation of plane on n-points. The Principle of Duality . Double ratio of two points and two planes. Projective transformation. Collineation. Collineation Group. Collineation task. Projective transformations of direct. Collinear mapping of n-spaces. Stable collineation points. Correlations. Configuration Theorems. Configurations. Papp's theorem. Pappa's Dual Teorem . Desargues' theorem . Full Quadrilateral Theorem. The complete quadrilateral theorem. Homology. Affine homologs . Perspective mapping of planes. Non-degenerate m-homologies . Involution Collineations. Geometry of m-planes . Crossing and sum of m-plane and l-plane. Projective operator coordinates of m-plane. Projective transformations in projective operator coordinates. Dimensions of intersection of m-planes in projective operator coordinates. Designing on m-plane in the direction (n - m - 1)-plane. Reflection from m-pair. Double ratio of two m-pairs. Transversals of two m-pairs. Affine operator coordinates of m-planes. Projective transformations in affine operator coordinates. Dimensions of intersection of m-planes in affine operator coordinates. Double ratio of two m-pairs. Quadrics. Quadric equations. Reciprocal quadricity and straight line. Relating to . Polar plane and pole . Polar transformation. Flat Forming Quadric. Duality Quadric. Simplification of quadric equations. Projective properties of second-order lines and linear quadri. Pascal and Brianchon theorems . Projective quadric transformations. Involutional correlations. Zero planes of zero-system. Differentiation of vectors. Differentiation by scalar argument. Vector functions of scalar argument. Differentiation and integration of vector functions. Vector differential equations. Relating to the line. Contacting m-planes. Accompanying Base. Length of arc. First curve of the line. Fresne Formula(n - 1)-th curve of the line. Natural Line Equations. Operator functions of scalar argument. Infinitesimal movements. Operator's record of Frene formulas. Helical lines. Differentiation by vector argument. Scalar functions of vector argument. Vector functions of vector argument. Vector Equations of Surfaces . Touch m-plane to m-surface. Touch plane and normal to surface. First quadratic surface shape. Surface volume element. Crossing of the tangent plane with the surface . Line curvature on the surface. Linear surface operator. The Dupin Index . Main Curves. Plane Curves and Sphere-Orthogonal Surface Systems . Full surface curvature. Gauss and Weingarten Formulas . Absolute Differentiation . Vectors and tensors on the surface . Absolute Differentiation . Curve Tensor. Definition of the surface by its quadratic forms. Parallel Transfer. Geodetic curvature of the line. Geodetic lines. Surface curvature in 2-dimensional direction . GaussBonnet theorem . Riman spaces and affine connectivity spaces. Conformational transformations. Conform space and conformal transformations. Conformational transformations. Liouville Theorem . Inversion Regarding Sphere. Conformity space. Stereographic projection. Conformational transformations as works of inversion. Projective interpretation of conformal space . Angle between spheres. Conformity Transformation Group. Geometry of m-spheres. The operator equation of the m-sphere. Stationary angles between m-spheres. Application of complex numbers and quaternions. Algebras. Complex numbers and quaternions. Plane of complex variable and space of quaternions. Transferences and homothetes. Turns. Movements and similarities 2-plane. Movements and Similarities of 4-spaces. 3-space motion. Spinor representation of 3-space rotations. Spinor representation of 4-space rotations. Inversions with respect to circles and spheres. Circular transformations of 2-planes and conformal transformations of 4-space. Double ratio of four complex numbers or quaternions. Space and Time . Space - Time and Pseudo-Euclidean Spaces. Space - time of classical mechanics . Space - Time of Special Theory of Relativity . Axioms of pseudo-Euclidean spaces . The Law of Inertia . Models of pseudo-Euclidean spaces. Distances between points. Isotropic cone. Spheres. Angles between vectors. Cosines Theorem . Interpretation of the diversity of Euclidean spheres on the pseudo-Euclidean sphere. Interpretation of the diversity of Euclidean spheres in pseudo-Euclidean space. Rectangular Coordinates. Straight and flat . Pseudo-Euclidean Movements. Movement and Congruence . Rotations and Transfers. Movement Group. The canonical form of the matrix of the pseudo-orthogonal operator. Anti-movements. Similarity and antisimilarity. Conform transformations and pseudoconform space. Conformational transformations. Inversion Regarding Sphere. Pseudoconformity space. Stereographic projection. Interpretation of the diversity of pseudo-Euclidean spheres on the pseudo-Euclidean sphere. Geometry of m-spheres. Application of double numbers and antiquaternions. Double Numbers and Antiquaternions . Plane of the double variable and the space of anti-nwaternions. Turns and anti-turns. Movements, anti-movements, similarities and anti-similarities. Circular transformations of 2-planes and conformal transformations of 4-space. Double ratio of four double numbers or antiquaternions. Coupled pairs of 2-plane points. Conformational transformations of the pseudo-Euclidean 2-plane. Spinor representations of 3-space rotations. Spinor representations of 4-space rotations. Interpretation of the diversity of direct projective 3-space in pseudoconform 4-space. Physics and Geometry . Pseudo-riman spaces . Non-Euclidean space. Compounding of velocities in special relativity. Flat Electromagnetic Wave in Special Theory of Relativity . Spinor representation of the Lorentz group and electron spin. Space - Time of General Relativity . Quantum Physics and Geometry . Notes. Bibliography