Preface, Contents, and Introduction

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The elements of geometry and the five groups of axioms

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Group I: Axioms of connection

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Group II: Axioms of Order

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Consequences of the axioms of connection and order

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Group III: Axioms of Parallels (Euclid's axiom)

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Group IV: Axioms of congruence

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Consequences of the axioms of congruence

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Group V: Axiom of Continuity (Archimedes's axiom)

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Compatibility of the axioms

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Independence of the axioms of parallels. Noneuclidean geometry

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Independence of the axioms of congruence

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Independence of the axiom of continuity. Nonarchimedean geometry

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Complex numbersystems

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Demonstrations of Pascal's theorem

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An algebra of segments, based upon Pascal's theorem

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Proportion and the theorems of similitude

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Equations of straight lines and of planes

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Equal area and equal content of polygons

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Parallelograms and triangles having equal bases and equal altitudes

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The measure of area of triangles and polygons

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Equality of content and the measure of area

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Desargues's theorem and its demonstration for plane geometry by aid of the axio…

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The impossibility of demonstrating Desargues's theorem for the plane with the h…

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Introduction to the algebra of segments based upon the Desargues's theorme

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The commutative and associative law of addition for our new algebra of segments

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The associative law of multiplication and the two distributive laws for the new…

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Equation of straight line, based upon the new algebra of segments

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The totality of segments, regarded as a complex number system

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Construction of a geometry of space by aid of a desarguesian number system

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Significance of Desargues's theorem

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Two theorems concerning the possibility of proving Pascal's theorem

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The commutative law of multiplication for an archimedean number system

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The commutative law of multiplication for a nonarchimedean number system

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Proof of the two propositions concerning Pascal's theorem. Nonpascalian geomet…

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The demonstation, by means of the theorems of Pascal and Desargues

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Analytic representation of the coordinates of points which can be so construct…

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Geometrical constructions by means of a straightedge and a transferer of segme…

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The representation of algebraic numbers and of integral rational functions as s…

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Criterion for the possibility of a geometrical construction by means of a strai…

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Conclusion

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Appendix

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