幾何

內(nèi)容概要

　　本書是一部本科生水平的幾何教程。通過《幾何》可以了解作者的思想以及作者在該領(lǐng)域做出的重大貢獻(xiàn)。書中首先講述歐幾里得基礎(chǔ)知識，然后進(jìn)一步引導(dǎo)讀者了解歐幾里得幾何的關(guān)鍵性內(nèi)容、近期發(fā)展和更多的最新結(jié)果，許多證明可以加深對內(nèi)容的理解。內(nèi)容有坐標(biāo)的引入、區(qū)域理論、幾何學(xué)結(jié)構(gòu)和有限場擴(kuò)展、平行公設(shè)歷史、多種非歐幾里得幾何和規(guī)則半規(guī)則多面體。《幾何英文(影印版)》是數(shù)學(xué)專業(yè)中等及以上水平讀者很難得的一本入門書籍。

書籍目錄

chapter 1. euclid's geometry
　1. a first look at euclid's elements
　2. ruler and compass constructions
　3. euclid's axiomatic method
　4. construction of the regular pentagon
　5. some newer results
chapter 2. hilbert's axioms
　6. axioms of incidence
　7. axioms of betweenness
　8. axioms of congruence for line segments
　9. axioms of congruence for angles
　10. hilbert planes
　11. intersection of lines and circles
　12. euclidean planes
chapter 3. geometry over fields
　13. the real cartesian plane
　14. abstract fields and incidence
　15. ordered fields and betweenness
　16. congruence of segments and angles
　17. rigid motions and sas
　18. non-archimedean geometry
chapter 4. segment arithmetic
　19. addition and multiplication of line segments
　20. similar triangles
　21. introduction of coordinates
chapter 5. area
　22. area in euclid's geometry
　23. measure of area functions
　24. dissection
　25. quadrature circuli
　26. euclid's theory of volume
　27. hilbert's third problem
chapter 6. construction problems and field extensions
　28. three famous problems
　29. the regular 17-sided polygon
　30. constructions with compass and marked ruler
　31. cubic and quartic equations
　32. appendix: finite field extensions
chapter 7. non-euclidean geometry
　33. history of the parallel postulate
　34. neutral geometry
　35. archimedean neutral geometry
　36. non-euclidean area
　37. circular inversion
　38. digression: circles determined by three conditions
　39. the poincare model
　40. hyperbolic geometry
　41. hilbert's arithmetic of ends
　42. hyperbolic trigonometry
　43. characterization of hilbert planes
chapter 8. polyhedra
　44. the five regular solids
　45. euler's and cauchy's theorems
　46. semiregular and face-regular polyhedra
　47. symmetry groups of polyhedra
appendix: brief euclid
notes
references
list of axioms
index of euclid's propositions
index

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