한국수학사학회지 (Journal for History of Mathematics)

한국수학사학회 (The Korean Society for History of Mathematics)
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졸트 공리의 역사적 고찰

A historical study of de Zolt's axiom

  • Jo, Kyeonghee (Division of Liberal Arts and Sciences, Mokpo National Maritime Univ.)
  • 투고 : 2017.04.09
  • 심사 : 2017.09.30
  • 발행 : 2017.10.31
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초록

De Zolt's axiom which is a precise formulation of Euclid's Common Notion 5, "the whole is greater than the part", for the notion of 'content' holds in any Hilbert plane. In this article, we study the history of de Zolt's axiom which has its origin in Euclid's Common Notions, and introduce an example of a plane geometry in which de Zolt's axiom does not hold. We show that there is no area function in this geometry and every square is equidecomposable with a square which is properly contained in the first one. From this we also show that there are two equidecomposable rectangles which have the same base and do not have the same altitude, and there is a rectangle which is equicomplementable with an emptyset.

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참고문헌

  1. M. DEHN, M. PASCH, Vorlesungen uber neuere Geometrie, Verlag von Julius Springer, 1926.
  2. Duhamel, Des methodes dans les sciences de raisonnement, Premiere partie: Des methodes communes a toutes les sciences de raisonnement. Deuxieme partie: Applications des methodes generales a la science des nombres et a la science de l'etendue [les duex parties se trouvent dans un seul tome avec des paginations separees], Paris : Gauthier Villars, 1866.
  3. EUCLID, The thirteen books of Euclid's Elements, Translated with introduction and commentary by Sir Thomas L. Heath, Vols. 1,2,3, Dover Publications, Inc., New York, 1956.
  4. M. HALLETT, M. ULRICH, David Hilbert's Lectures on the Foundations of Geometry 1891-1902, ed. by Michael Hallett and Ulrich Majer, David Hilbert's Foundational Lectures, vol. 1, Berlin: Springer, 2004.
  5. R. HARTSHORNE, Geometry: Euclid and beyond, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2000.
  6. D. HILBERT, The Foundations of Geometry, 1st ed, Authorized translation by E. J. Townsend, The open court publishing company, 1902.
  7. D. HILBERT, The Foundations of Geometry, 2nd English Edition, Authorized translation by Leo Unger from the 10th German Edition, Revised and Enlarged by Dr. Paul Bernays, The open court publishing company, 1971.
  8. S.-D. YANG, K. Jo, On Hilbert's 'Grundlagen der Geometrie', The Korean Journal for History of Mathematics 24(4) (2011), 61-86.
  9. R. KAYA, Area formula for Taxicab triangle, ME Journal 4(12) (2006), 219-220.
  10. W. KILLING, Grundlagen der Geometrie, Vol. 2, Part 5, Section 5, 1898.
  11. I. KOCAYUSUFOGLU, E. OZDAMAR, Isometries of Taxicab geometry, Commun. Fac. Sci. Univ. Ank. Series A1 47 (1988), 73-83.
  12. W. PEJAS, Die Modelle des Hilbertschen Axiomensystems der absoluten Geometrie, Math. Annalen 143 (1961), 212-235. https://doi.org/10.1007/BF01342979
  13. F. SCHUR, Ueber den Flacheninhalt geradlinig begrenzter ebener Figuren, Sitzungsberichte der Dorpater Naturforschenden Gesellschaft, 1892.
  14. O. STOLZ, Die ebebenVielecke und die Winkel mit Einschluss der Beruhrung-Winkel als Systeme von absoluten Grossen, Monatschefte fur Mathematik und Physik, Jahrgang 5 (1984), 233-240.
  15. K. P. THOMPSON, The nature of length, area, and volume in Taxicab geometry, 2011.
  16. K. VOLKERT, Ist das Ganze groBer als sein Teil ? - Einigie Anmerkungen zur Geschichte eines scheinbar evidenten Prinzips, Histoires de geometries : textes du seminaire de l'annee 2003, Paris, Foundation Maison des Sciences de l'Homme, 2004.
  17. K. VOLKERT, Le tout est-il tourjours plus grand que la partie ?, Revue d'histoire des mathematiques 16 (2010), 287-306.
  18. de ZOLT, Principii della equaglianza di poligoni preceduti da alcuni cenni critici sulla teoria della equivalenza geometrica, Milano : Briola, 1881.
  19. de ZOLT, Principii della equaglianza di poliedri e di poligoni sferici, Milano, Briola, 1883.