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几何基础（英文版）

样章
作者：
David Hilbert, Authorized Translation by E. J. Townsend
定价：
69.00元
ISBN：
978-7-04-060492-4
版面字数：
150.000千字
开本：
特殊
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
出版时间：
2023-06-28
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
几何学
暂无
  • 前辅文
  • INTRODUCTION
  • CHAPTER I THE FIVE GROUPS OF AXIOMS
    • §1. The elements of geometry and the five groups of axioms
    • §2. Group I. Axioms of connection
    • §3. Group II. Axioms of order
    • §4. Consequences of the axioms of connection and order
    • §5. Group III. Axiom of parallels (Euclid’s axiom)
    • §6. Group IV. Axioms of congruence
    • §7. Consequences of the axioms of congruence
    • §8. Group V. Axiom of continuity (Archimedes’s axiom)
  • CHAPTER II COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS
    • §9. Compatibility of the axioms
    • §10. Independence of the axioms of parallels (Non-euclidean geometry)
    • §11. Independence of the axioms of congruence
    • §12. Independence of the axiom of continuity (Non-archimedean geometry)
  • CHAPTER III THE THEORY OF PROPORTION
    • §13. Complex number systems
    • §14. Demonstration of Pascal’s theorem
    • §15. An algebra of segments, based upon Pascal’s theorem
    • §16. Proportion and the theorems of similitude
    • §17. Equations of straight lines and of planes
  • CHAPTER IV THE THEORY OF PLANE AREAS
    • §18. Equal area and equal content of polygons
    • §19. Parallelograms and triangles having equal bases and equal altitudes
    • §20. The measure of area of triangles and polygons
    • §21. Equality of content and the measure of area
  • CHAPTER V DESARGUES’S THEOREM
    • §22. Desargues’s theorem and its demonstration for plane geometry by aid of the axioms of congruence
    • §23. The impossibility of demonstrating Desargues’s theorem for the plane without the help of the axioms of congruence
    • §24. Introduction of an algebra of segments based upon Desargues’s theorem and independent of the axioms of congruence
    • §25. The commutative and the associative law of addition for our new algebra of segments
    • §26. The associative law of multiplication and the two distributive laws for the new algebra of segments
    • §27. Equation of the straight line, based upon the new algebra of segments
    • §28. The totality of segments, regarded as a complex number system
    • §29. Construction of a geometry of space by aid of a desarguesian number system
    • §30. Significance of Desargues’s theorem
  • CHAPTER VI PASCAL’S THEOREM
    • §31. Two theorems concerning the possibility of proving Pascal’s theorem
    • §32. The commutative law of multiplication for an Archimedean number system
    • §33. The commutative law of multiplication for a non-archimedean number system
    • §34. Proof of the two propositions concerning Pascal’s theorem (Non-pascalian geometry)
    • §35. The demonstration, by means of the theorems of Pascal and Desargues, of any theorem relating to points of intersection
  • CHAPTER VII GEOMETRICAL CONSTRUCTIONS BASED UPON THE AXIOMS I–V
    • §36. Geometrical constructions by means of a straight-edge and a transferer of segments
    • §37. Analytical representation of the co-ordinates of points which can be so constructed
    • §38. The representation of algebraic numbers and of integral rational functions as sums of squares
    • §39. Criterion for the possibility of a geometrical construction by means of a straight-edge and a transferer of segments
  • CONCLUSION
  • APPENDIX

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