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Elementary Geometry, Then and Now

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The Geometric Vein

Abstract

What is elementary geometry, and when did it originate? The first of these questions—the content of elementary geometry—is not at all simple, and a clear-cut answer is not possible. The most natural answer for present purposes would be the following: “Elementary geometry is the collection of those geometric concepts and theorems taken up in secondary school, together with immediate consequences of these theorems.” However, in spite of the seeming simplicity of this answer, it raises at once a host of objections. The appeal to the word “geometric” in the definition is in itself hard to interpret, since the question “what is geometry?” also admits no clear-cut answer (on that, more below); but in any case, the rapid rate of change in school curricula in all countries of the world, currently seeming to reach its maximum, would oblige us if we adopted that definition to accept the existence of indefinitely many elementary geometries. The concept would have to change not merely from country to country, but for each given country also from year to year if not even from school to school. In addition, such a definition clearly refers only to the content of the school subject “elementary geometry,” while we are here asking about the content of the corresponding science—or, since the word “science” here may seem pompous, about the corresponding direction of scientific thought.

Translated by Chandler Davis.

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References

  1. Johnson, R. A., Advanced Euclidean Geometry. Dover, New York 1960.

    MATH  Google Scholar 

  2. Yaglom, I. M., Complex Numbers in Geometry. Academic Press, New York 1968.

    Google Scholar 

  3. Golovina, L. I. and Yaglom, I. M., Induction in Geometry, Heath, Boston 1963.

    Google Scholar 

  4. Coxeter, H. S. M. and Greitzer, S. L., Geometry Revisited, Random House, New York 1967.

    MATH  Google Scholar 

  5. Efremov, D., Novaya geometriya treugol’nika. Matezis, Odessa 1903.

    Google Scholar 

  6. Coolidge, J. L., A Treatise on the Circle and the Sphere. Clarendon Press, Oxford 1916.

    MATH  Google Scholar 

  7. Simon, M., Über Entwicklung der Elementargeometrie im XIX Jahrhundert. Berlin 1906.

    Google Scholar 

  8. Bieberbach, L., Theorie der geometrischer Konstruktionen. Birkhäuser Verlag 1952.

    Google Scholar 

  9. Yaglom, I. M., Geometric Transformations III. Random House, New York 1973.

    Book  MATH  Google Scholar 

  10. Klein, F., Vorlesungen über nicht-euklidische Geometrie. J. Springer, Berlin 1928.

    MATH  Google Scholar 

  11. Coxeter, H. S. M., Introduction to Geometry. Wiley, New York 1969.

    MATH  Google Scholar 

  12. Pedoe, D., A Course of Geometry for Colleges and Universities. University Press, Cambridge 1970.

    MATH  Google Scholar 

  13. Baer, R., Linear Algebra and Projective Geometry. Academic Press, New York 1952.

    MATH  Google Scholar 

  14. Bachmann, F., Aufbau der Geometrie aus dem Spiegelungsbegriff. J. Springer, Berlin 1973.

    MATH  Google Scholar 

  15. Prenowitz, W. and Jantosciak, J., Join Geometries. Springer, New York 1979.

    MATH  Google Scholar 

  16. Bachmann, F. and Schmidt, E., n-Ecke. Bibliographisches Institut, Mannheim 1970.

    MATH  Google Scholar 

  17. Rouché, E. and Comberousse, Ch., Traité de géométrie. Gauthier-Villars, Paris 1899.

    Google Scholar 

  18. Hadamard, J., Leçons de géométrie élémentaire, I, II. Gauthier-Villars, Paris 1937.

    Google Scholar 

  19. Dieudonne, J., Algèbre linéaire et géométrie élémentaire. Hermann, Paris 1968.

    Google Scholar 

  20. Birkhoff, G. D. and Beatley, R., Basic Geometry. Chelsea, New York 1959.

    Google Scholar 

  21. Koo, D., Elements of Optimization. Springer, New York 1977.

    MATH  Google Scholar 

  22. Lebesgue, H., La mesure des grandeurs. Université, Geneve 1956.

    Google Scholar 

  23. Mandelbrot, B., Fractals. Freeman, San Francisco 1977.

    MATH  Google Scholar 

  24. Levinson, N., Coding theory: a counterexemple to G. H. Hardy’s conception of applied mathematics. American Math. Monthly 77 (No. 3, 1970), 249–258.

    Article  MathSciNet  MATH  Google Scholar 

  25. Hardy, G. H., A Mathematician’s Apology. University Press, Cambridge 1941.

    MATH  Google Scholar 

  26. Kemeny, J. G., Snell, J. L., and Thompson, G. L., Introduction to Finite Mathematics. Prentice-Hall, Englewood Cliffs, N.J. 1957. Kemeny, J. G., Mirkil, H., Snell, J. L., and Thompson, G. L., Finite Mathematical Structures. Prentice-Hall 1959.

    MATH  Google Scholar 

  27. Beckenbach, E. E., (editor) Applied Combinatorial Mathematics. Wiley, New York 1964.

    MATH  Google Scholar 

  28. Dembowski, P., Finite Geometries. Springer, Berlin 1968.

    MATH  Google Scholar 

  29. Shannon, C., Communication in the presence of noise. Proc. IRE 37 (No. 1, 1949), 10–21.

    Article  MathSciNet  Google Scholar 

  30. Rogers, C. A., Packing and Covering. University Press, Cambridge 1964.

    MATH  Google Scholar 

  31. Fejes Tóth, L., Lagerungen in der Ebene, auf ker Kugel und im Raum. Springer, Berlin 1972.

    Google Scholar 

  32. Shklarsky, D. O., Chentzov, N. N., and Yaglom, I. M., The USSR Olympiad Problem Book. Freeman, San Francisco 1962. Shklyarsky, D., Chentsov, N., and Yaglom, I., Selected Problems and Theorems in Elementary Mathematics. Mir, Moscow 1979.

    MATH  Google Scholar 

  33. Yaglom, A. M. and Yaglom, I. M., Challenging Mathematical Problems with Elementary Solutions, I, II. Holden Day, San Francisco 1964, 1967.

    MATH  Google Scholar 

  34. \( \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} klyarski\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} \), D. O., Čencov, N. N., and Yaglom, I. M., Geometričeskie neravenstva i zadači na maksimum i minimum. Nauka, Moscow 1970.

    Google Scholar 

  35. \( \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} klyarski\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} \), D. O., Čencov, N. N., and Yaglom, I. M., Geometričeskie ocenki i zadači iz\( kombinatorno\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} \)geometrii. Nauka, Moscow 1974.

    Google Scholar 

  36. Hadwiger, H., Eulers Charakteristik und kombinatorische Geometrie. J. reine angew. Math. 194 (1955), 101–110.

    Article  MathSciNet  MATH  Google Scholar 

  37. \( Boltyanski\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} \), V. G. and Soltan, P. S.,. Kombinatornaya geometriya različnyh klassov vypuklyh množestv. Štinca, Kisenev 1978.

    Google Scholar 

  38. Danzer, L., Grünbaum, B., and Klee, V., Helly’s theorem and its relatives. In Convexity, Proceedings of Symposia in Pure Mathematics, Vol. VII, edited by V. Klee. American Math. Soc, Providence, R.I. 1963; pp. 101–180. Russian translation: Dancer, L., Gryunbaum, B., and Kli, V., Teorema Belli i ee primeneniya. Mir, Moscow 1968.

    Google Scholar 

  39. Coxeter, H. S. M., An upper bound for the number of equal nonoverlapping spheres that can touch another of the same size. In Convexity (see [37]). Also in Coxeter, H. S. M., Twelve Geometrical Essays. London 1968.

    Google Scholar 

  40. Yaglom, I. M., Problema trinadcati šarov. Višča Škola, Kiev 1975.

    Google Scholar 

  41. Fejes Tóth, L., On the number of equal discs that can touch another of the same kind. Studia Scient. Math. Hungar. 2 (1967), 363–367.

    MATH  Google Scholar 

  42. Grünbaum, B., On a conjecture of Hadwiger. Pacific J. Math. 11 (1961), 215–219.

    MathSciNet  MATH  Google Scholar 

  43. Baston, V. J. D., Some Properties of Polyhedra in Euclidean Space. Oxford 1965.

    Google Scholar 

  44. Croft, H., 9-point and 7-point configurations in 3-space. Proc. London Math. Soc. 12 (No. 3, 1962), 400–424; 13 (1963), 384.

    Article  MathSciNet  Google Scholar 

  45. Harazišvili, A. B., Izbrannye voprosy geometrii Yevklidovyh prostranstv. \( Tbilisski\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} \) Universitet, Tbilisi 1978.

    Google Scholar 

  46. Blumenthal, L., Theory and Applications of Distance Geometry. University Press, Oxford 1953.

    MATH  Google Scholar 

  47. Szekeres, G., On an extremal problem in the plane. Amer. J. Math. 63 (1941), 208–210.

    Article  MathSciNet  Google Scholar 

  48. Erdös, P. and Szekeres, G., On some extremum problems in elementary geometry. Annales Universitates Scientiarum Budapestinesis de Rolando Eötvös Nominantae 3-4 (1960/61), 53–62.

    Google Scholar 

  49. Yaglom, I. M., and Faǐnberg, E. I., Ocenki dlya \( veroyatnoste\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} \). In Trudy VI Vsesoyuznogo Soveščaniya po Teorii Veroyatnostei i Matematičeskoi Statistike. Vilnius 1962; pp. 297–303.

    Google Scholar 

  50. Pirogov, S. A., Veroyatnosti složnyh \( Veroyatnoste\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} \) programmirovanie. Teoriya \( Veroyatnoste\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} \) i ee Primeneniya 13 (No. 2, 1968), 344–348.

    Google Scholar 

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Author information

Authors and Affiliations

  1. 1-ǐ Goncharnyǐ, apt. 17, Moscow, 109172, USSR

    I. M. Yaglom

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  1. I. M. Yaglom
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Editors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, M5S 1A1, Canada

    Chandler Davis  & F. A. Sherk  & 

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Branko Grünbaum

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© 1981 Springer-Verlag New York Inc.

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Yaglom, I.M. (1981). Elementary Geometry, Then and Now. In: Davis, C., Grünbaum, B., Sherk, F.A. (eds) The Geometric Vein. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5648-9_17

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  • DOI: https://doi.org/10.1007/978-1-4612-5648-9_17

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-5650-2

  • Online ISBN: 978-1-4612-5648-9

  • eBook Packages: Springer Book Archive

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