East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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The reinterpretation and visualization for geometric methods of solving the cubic equation

삼차방정식의 기하적 해법에 대한 재조명과 시각화

  • Kim, Hyang Sook (Department of Computer Engineering & Institute of Natural Science Inje University) ;
  • Kim, Yang (Department of Computer-Aided Sciences Inje University) ;
  • Park, See Eun (Department of Computer-Aided Sciences Inje University)
  • Received : 2018.07.05
  • Accepted : 2018.08.28
  • Published : 2018.08.31
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Abstract

The purpose of this paper is to reinterpret and visualize the medieval Arab's studies on the geometric methods of solving the cubic equation by utilizing Apollonius' symptom of the parabola. In particular, we investigate the results of $Kam{\bar{a}}l$ $al-D{\bar{i}}n$ ibn $Y{\bar{u}}nus$, Alhazen, Umar al-$Khayy{\bar{a}}m$ and $Al-T{\bar{u}}s{\bar{i}}$ by 4 steps(analysis, construction, proof and examination) which are called the complete solution in the constructions. This paper is available in the current middle school curriculum through dynamic geometry program(Geogebra).

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References

  1. 교육부, 수학과 교육과정, 교육부 고시 제2015-74호[별책8].
  2. 김향숙.박진석.하형수, 원뿔곡선에 관한 Apollonius의 symptoms재조명과 시각화, 한국수학교육학회지 시리즈 A 제52권 제1호(2013), 83-95.
  3. 김향숙.박진석.이은경.이재돈.하형수, 중세 이슬람이 보인 입방배적문제 해결방법들의 재조명과 시각화, East Asian Mathematical J. 30(2014), 181-203.
  4. 박진석.김향숙, 수학사와 함께 떠나는 수학여행, 경문사, 2014(2015년 세종도서 우수학술도서).
  5. 박진석.김향숙, 해석기하학개론[제 2 판], 경문사, 2009.
  6. 반은섭, 삼차방정식의 기하학적 해결을 위한 수학적 지식의 연결 과정 분석, 한국교원대학교 박사학위논문, 2016.
  7. Berggren, J. L., Episodes in the Mathematics of Medieval Islam, Springer-Verlag, New York, 1986.
  8. Boyer, C. B. & Merzbach, U. C., A history of mathematics (2nd ed), McGraw-Hill, New York, 1991.
  9. Connor, M. B., A historical survery of methods of solving cubic equations, Unpublished master's dissertation, University of Richmond, Virginia, 1956.
  10. Coxford. A. F., The case for connections, In P. A. House & A. F. Coxford (Eds.), Connecting mathematics across the curriculum (pp.3 - 12), Reston, VA: National Council of Teachers of Mathematics, 1995.
  11. Eves, H., Omar Khayyam's Solution of Cubic Equation, Mathematics Teacher 51(1958), 285-286.
  12. Heath, T. L., Apollonius of Perga : Treatise on conic sections-The conics of Apollonius, Cambridge : at the university press, 1896.
  13. Hogendijk, J. P., On the trisection of an angle and the construction of a regular nonagon by means of conic sections in medieval Islamic geometry, the 2nd International Symposium of the History of Arabic Science, to be held in Aleppo, Syria, 1979.
  14. Hogendijk, J. P., Greek and Arabic Constructions of the Regular Heptagon, Archive for History of Exact Sciences 30(1984), 197-330. https://doi.org/10.1007/BF00328123
  15. Khayyam, O., An essay by the uniquely wise 'ABEL FATH BINAL-KHAYYAM' on algebra and equations, Translated by R. Khalil & Reviewed by W. Deeb. UK: RG1 4QS, 2008.
  16. Law, H. L., The comparison between the methods of solution for cubic equations in Shushu Jiuzhang and Risalah fil-barahin a'la masail ala-Jabr wa'l-Muqubalah, Mathematical Modley 30(2)(2003), 91-101.
  17. Looka, H. A., Polynomial equations from ancient to modem times: A review, University Bulletin 16(2)(2014), 89-110.
  18. Sinclair, Nathalie M., Mathematical applications of conic sections in problem solving in ancient Greece and medieval Islam, Simon Fraser Univ, 1995.
  19. Villanueva, j., The cubic and quartic equations: Intermediate algebra course, Electronic proceeding of ICTCM, Florida Memorial College, 2013. (Retrieved from http://archives.math.utk.edu/ICTCM/VOL16/C037/paper.pdf.)
  20. Wagner, R., A historically and philosophically informed approach to mathematical metaphors, International Studies in the Philosophy of Science 27(2)(2013), 109-135. https://doi.org/10.1080/02698595.2013.813257