Table of Contents: Elementary mathematics from an advanced standpoint

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Part 1. 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 transformation 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 Mobius’ 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 (curves, surfaces, point sets) — Difference between analytic and synthetic geometry — Projective geometry and the principle of duality — Plucker’s analytic method and the extension of the principle of duality (line coordinates) — Grassmann’s Ausdehnungslehre , n-dimensional geometry — Scalar and vector fields , rational vector analysis — Part 2. Geometric Transformations — Transformations and their analytic representation — I.. Affine 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 determinant) — Fundamental theorem of Pohlke — II.. Projective Transformations — Analytic definition , introduction of homogeneous coordinates — Geometric definition: Every collineation 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 conics — 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 Transformations — Genus and connectivity of surfaces — Euler’s theorem on polyhedra — IV.. Transformations with 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 circle-points and imaginary sphere-circle — Imaginary transformation — Von Staudt’s interpretation of self-conjugate imaginary manifolds by means of real polar systems — Von Staudt’s complete interpretation of single imaginary elements — Space relations of imaginary points and lines — Part 3. Systematic Discussion of Geometry and Its Foundations — I.. The Systematic Discussion — 1.. Survey of the Structure of Geometry — Theory of groups as a geometric principle of classification — Cayley’s fundamental principle: Projective geometry is all geometry — 2.. Digression on the Invariant Theory of Linear Substitutions — Systematic discussion of invariant theory — Simple examples — 3.. Application of Source.

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