Book contents
- Frontmatter
- Contents
- Introduction
- Note to the Reader
- Acknowledgements
- Theory of Parallels — Lobachevski's Introduction
- Theory of Parallels — Preliminary Theorems (1–15)
- Theory of Parallels 16: The Definition of Parallelism
- Theory of Parallels 17: Parallelism is Well-Defined
- Theory of Parallels 18: Parallelism is Symmetric
- Theory of Parallels 19: The Saccheri-Legendre Theorem
- Theory of Parallels 20: The Three Musketeers Theorem
- Theory of Parallels 21: A Little Lemma
- Theory of Parallels 22: Common Perpendiculars
- Theory of Parallels 23: The π-function
- Theory of Parallels 24: Convergence of Parallels
- Theory of Parallels 25: Parallelism is Transitive
- Theory of Parallels 26: Spherical Triangles
- Theory of Parallels 27: Solid Angles
- Theory of Parallels 28: The Prism Theorem
- Theory of Parallels 29: Circumcircles or Lack Thereof (Part I)
- Theory of Parallels 30: Circumcircles or Lack Thereof (Part II)
- Theory of Parallels 31: The Horocycle Defined
- Theory of Parallels 32: The Horocycle as a Limit-Circle
- Theory of Parallels 33: Concentric Horocycles
- Theory of Parallels 34: The Horosphere
- Theory of Parallels 35: Spherical Trigonometry
- Theory of Parallels 36: The Fundamental Formula
- Theory of Parallels 37: Plane Trigonometry
- Bibliography
- Appendix: Nicolai Ivanovich Lobachevski's Theory of Parallels
- Index
- About the Author
Introduction
- Frontmatter
- Contents
- Introduction
- Note to the Reader
- Acknowledgements
- Theory of Parallels — Lobachevski's Introduction
- Theory of Parallels — Preliminary Theorems (1–15)
- Theory of Parallels 16: The Definition of Parallelism
- Theory of Parallels 17: Parallelism is Well-Defined
- Theory of Parallels 18: Parallelism is Symmetric
- Theory of Parallels 19: The Saccheri-Legendre Theorem
- Theory of Parallels 20: The Three Musketeers Theorem
- Theory of Parallels 21: A Little Lemma
- Theory of Parallels 22: Common Perpendiculars
- Theory of Parallels 23: The π-function
- Theory of Parallels 24: Convergence of Parallels
- Theory of Parallels 25: Parallelism is Transitive
- Theory of Parallels 26: Spherical Triangles
- Theory of Parallels 27: Solid Angles
- Theory of Parallels 28: The Prism Theorem
- Theory of Parallels 29: Circumcircles or Lack Thereof (Part I)
- Theory of Parallels 30: Circumcircles or Lack Thereof (Part II)
- Theory of Parallels 31: The Horocycle Defined
- Theory of Parallels 32: The Horocycle as a Limit-Circle
- Theory of Parallels 33: Concentric Horocycles
- Theory of Parallels 34: The Horosphere
- Theory of Parallels 35: Spherical Trigonometry
- Theory of Parallels 36: The Fundamental Formula
- Theory of Parallels 37: Plane Trigonometry
- Bibliography
- Appendix: Nicolai Ivanovich Lobachevski's Theory of Parallels
- Index
- About the Author
Summary
Through the ostensibly infallible process of logical deduction, Euclid of Alexandria (ca. 300 B.C.) derived a colossal body of geometric facts from a bare minimum of genetic material: five postulates—five simple geometric assumptions that he listed at the beginning of his masterpiece, the Elements. That Euclid could produce hundreds of unintuitive theorems from five patently obvious assumptions about space, and, still more impressively, that he could do so in a manner that precluded doubt, sufficed to establish the Elements as mankind's greatest monument to the power of rational organized thought. As a logically impeccable, tighty wrought description of space itself, the Elements offered humanity a unique anchor of definite knowledge, guaranteed to remain eternally secure amidst the perpetual flux of existence—a rock of certainty, whose truth,by its very nature, was unquestionable.
This universal, even transcendent, aspect of the Elements has profoundly impressed Euclid's readers for over two millennia. In contrast to all explicitly advertised sources of transcendent knowledge, Euclid never cites a single authority and he never asks his readers to trust his own ineffably mystical wisdom. Instead, we, his readers, need not accept anything on faith; we are free and even encouraged to remain skeptical throughout. Should one doubt the validity of the Pythagorean Theorem (Elements I.47), for example, one need not defer to the reputation of “the great Pythagoras. Instead, one may satisfy oneself in the manner of Thomas Hobbes, whose first experience with Euclid was described by John Aubrey, in his Brief Lives, in the following words.
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- Lobachevski Illuminated , pp. xi - xviPublisher: Mathematical Association of AmericaPrint publication year: 2011