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Volume 5 - May 1962

Contents

Page 4 of 4

Other

  • Note on the Structure of Graphs

  • G.A. Dirac
  • Published online by Cambridge University Press:
    01 April 2019, pp. 221-228
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Book Reviews: Comptes Rendus Critiques

  • A Modern View of Geometry, by L. M. Blumenthal. Freeman, San Francisco, 1961. xii + 191 pages. $2.25.

  • H. S. M. Coxeter
  • Published online by Cambridge University Press:
    01 April 2019, pp. 203-204
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  • Problems for Solution

  • Published online by Cambridge University Press:
    01 April 2019, p. 70
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Book Reviews: Comptes Rendus Critiques

Other

  • Summer Session in Analysis and Geometry University of British Columbia

  • Published online by Cambridge University Press:
    02 July 2019, p. 103
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Notes and Problems

  • On the Fundamental Theorem of Affine Geometry

  • P. Scherk
  • Published online by Cambridge University Press:
    20 November 2018, pp. 67-69
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  • The fundamental theorem of affine geometry is an easy corollary of the corresponding projective theorem 2.26 in Artin's Geometric Algebra. However, a simple direct proof based on Lipman's paper [this Bulletin, 4, 265−278] and his axioms 1 and 2 may be of some interest.

    Lipman's [desarguian] affine geometry G determined a left linear vector space L={a, b,…} over a skew field F. We wish to construct 1−1 transformations γ of G onto itself such that γ and γ-1 map straight lines onto straight lines preserving parallelism. Designate any point 0 as the origin of G. Multiplying γ with a suitable translation, we may assume γ0=0. Thus γ will then be equivalent to a 1−1 transformation Γ of L onto itself which preserves linear dependence. Since Γ-1 will have the same properties, Γ must also preserve linear independence.

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  • Openings in Mathematics

  • Published online by Cambridge University Press:
    02 July 2019, p. 110
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Page 4 of 4