从高观点看初等数学/Elementary mathematics from an advanced standpoint

Part One: The Simplest Geometric Manifolds
I. Line-Segment, Area, Volume, as Relative Magnitudes
Definition by means of determinants; interpretation of the sign
Simplest applications, especially the cross ratio
Area of rectilinear polygons
　　Curvilinear areas
　　Theory of Amsler's polar planimeter
　　Volume of polyhedrons, the law of edges
　　One-sided polyhedrons
　II. The Grassmann Determinant Principle for the Plane
Line-segment (vectors)
Application in statics of rigid systems
Classification of geometric magnitudes according to their behavior under trans formation of rectangular coordinates
Application of the principle of classification to elementary magnitudes
　III. The Grassmann Principle for Space
　　Line-segment and plane-segment
　　Application to statics of rigid bodies
　　Relation to MSbius' null-system
　　Geometric interpretation of the null-system.
　　Connection with the theory of screws
　IV. Classification of the Elementary Configurations of Space according to their
Behavior under Transformation of Rectangular Coordinates
Generalities concerning transformations of rectangular space coordinates
Transformation formulas for some elementary magnitudes
Couple and free plane magnitude as equivalent manifolds
Free line-segment and free plane magnitude ("polar" and "axial" vector)
　　Scalars of first and second kind
　　Outlines of a rational vector algebra
　　Lack of a uniform nomenclature in vector calculus
　V. Derivative Manifolds
　 Derivatives from points (carves, surfaces, point sets)
Difference between analytic and synthetic geometry
Projective geometry and the principle of duality
Pliicker's analytic method and the extension of the principle of duality (lin coordinates)
Grassmann's Ausdehnungsiehre;n-dimensional geometry
Scalar and vector fields; rational vector analysis
Part Two: Geometric Transformations
　Transformations and their analytic representation
　I. AtBne Transformations
　 Analytic definition and fundamental properties
Application to theory of ellipsoid
Parallel projection from one plane upon another
Axonometric mapping of space (affine transformation with vanishing deter- minant)
Fundamental theorem of Poblke
　II. Projective Transformations
Analytic definition; introduction of homogeneous coordinates
Geometric definition: Every coUineation is a projective transformation
Behavior of fundamental manifolds under projective transformation
Central projection of space upon a plane (projective transformation with　vanishing determinant)
Relief perspective
Application of projection in deriving properties of comcs
　III. Higher Point Transformations
　1. The Transformation by Reciprocal Radii
　 Peaucellier's method of drawing a line
　 Stereographic projection of the sphere
　　2. Some More General Map Projections.
　 Mercator's projection
　 Tissot theorems
　　3. The Most General Reversibly Unique Continuous Point Transformatins
　 Genus and connectivity of surfaces
　 Euler's theorem on polyhedra
　IV. Transformations wlth Change of Space Element
1. Dualistic Transformations
2. Contact Transformations
3. Some Examples
Forms of algebraic order and class curves
Application of contact transformations to theory of cog wheels
　V. Theory of the Imaginary
　 Imaginary cirde-points and imaginary sphere-circle
Imaginary transformation
Von Staudt s interpretation of self-conjugate imaginary manifolds by means oJ real polar systems
Von Staudt's complete interpretation of single imaginary elements
Space relations of imaginary points and lines
……
Part Three:Systematic Discussion of Geometry and Its Foundations
II Foundations of Geometry
Index of Names
Index of Contents