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Euclid’s Algorithm

  • Chapter
A History of Algorithms

Abstract

We have already remarked in the introduction to this book, that Euclid’s algorithm often represents for the mathematician the prototype of algorithmic procedure, and that it has relevance right up to today. It can be of use, not only in the search for the greatest common divisor, as described by Euclid himself (Section 4.1), but also, by adapting the procedure, in the solution of indeterminate equations, leading to Bézout’s identity (Section 4.3). It allowed al-Khayyam to compare two ratios, or to show that they were equal (Section 4.2); this appears even more clearly in the writing of continued fractions which were systematically studied by Euler (Section 4.4). Finally, what may appear surprising, the algorithm can be used in Sturm’s method for determining the number of real roots of an algebraic equation (Section 4.5).

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Bibliography

  1. Archimedes, Measurement of a Circle, in The Works of Archimedes Edited in Modern Notation (1897) with supplement The Method of Archimedes (1912), tr. T. L. Heath, Cambridge: Cambridge University Press, 1912, repr., New York: Dover, n.d.

    Google Scholar 

  2. Aristarchus, Treatise on the sizes and distances of the sun and the moon, in Aristarchus of Santos, The Ancient Copernicus, ed. & tr. T. L Heath, Oxford: Clarendon Press 1913, repr. New York: Dover, 1981.

    Google Scholar 

  3. Aristotle, Topics, tr. W. A. Pickard-Cambridge, in vol. 1, The Works of Aristotle translated into English, Ross W. D. ed., 12 vols., Oxford: Clarendon Press, 1908-52.

    Google Scholar 

  4. Artin E. and Schreier, O., Algebraische Konstruktion reellen Körper, Abhandlungen aus dem Mathematischen Seminar der hamburgischen Universität, 1927.

    Google Scholar 

  5. Bachet, Cl.-G., Sieur de Meziriac, Problèmes plaisons et délectables qui se font par les nombres (1612, 1624), Lyon: Pierre Rigaud, simplified edn, Paris: Blanchard, 1959.

    Google Scholar 

  6. Bézout E., Cours de Mathématiques à l’usage des Gardes du Pavillon et de la Marine, vol. III Algèbre, Paris: Musier, 1766.

    Google Scholar 

  7. Bochnak, J., Coste, M. and Roy, M.-R, Géométrie algébrique réelle, New York: Springer, 1987.

    Google Scholar 

  8. Brezinski, Cl., History of Continued Fractions and Padé Approximants, New York: Springer, 1991.

    Book  MATH  Google Scholar 

  9. Budan, Mémoire contenant la démonstration de quelques Théorèmes nouveaux, relatifs aux successions de signes, considérées dans les termes des suites et dans les coefficients des équations, in Nouvelle méthode pour la résolution des équations numériques d’un degré quelconque, Pans: Dondey-Dupré, 1822.

    Google Scholar 

  10. Cauchy, A.-L., Mémoire sur la Détermination du nombre de Racines réelles dans les Equations algébriques, Journal de l’Ecole Polytechnique, vol. 10 (1815), 457-548, repr., in Oeuvres, 2nd series, Paris: Gauthier-Villars, vol. 1, 1905, pp. 170–257.

    Google Scholar 

  11. Caveing, M. C, La constitution du type mathématique de l’idéalité dans la pensée grecque, Lille: Atelier National de Reproduction des Thèses, 1982.

    Google Scholar 

  12. Chou S.-C, Mechanical Geometry Theorem Proving, Dordrecht: Reidel Publishing Company, 1988.

    MATH  Google Scholar 

  13. Descartes, R., La Géométrie, Leyde: Jan Maire, 1637, English tr. D. E. Smith & M. L. Latham, with a facsimile of the first edition, La Salle, Illinois: Open Court Publishing Company (1925), repr. New York: Dover, 1954.

    Google Scholar 

  14. Euclid: Heath, T. L. The Thirteen books of Euclid’s Elements, 2nd edn, 3 vols., Cambridge: Cambridge University Press 1926, repr. New York: Dover, 1956.

    MATH  Google Scholar 

  15. Euler, L., De Fractionibus continuis Dissertatio, Commentarii academiae scientiarum Petropolitanae, 9 (1737), 98–137, repr. in Opera Omnia, I xiv, pp. 187-215, English translation in Math. Systems Theory, 18 (1985), pp. 295-328.

    Google Scholar 

  16. Euler, L., Introductio in Analysin infinitorum, Lausanne, 1748, repr. in Opera Omnia, I viii & ix, English tr. J. D. Blanton, Introduction to Analysis of the Infinite, New York: Springer, 1988, 1990.

    Book  Google Scholar 

  17. Fourier, J., Sur l’Usage du Théorème de Descartes dans la recherche des limites des racines, Bulletin des Sciences par la Société philomatique de Paris, (Oct. 1820), 156-165 & (Dec. 1820), 181-187, repr. in Oeuvres, Paris: Gauthier-Villars, vol. II, 1890, pp. 289–309.

    Google Scholar 

  18. Fowler, D. H., The Mathematics of Plato’s Academy: A new Reconstruction, Oxford: Clarendon Press (1987), repr. with corrections, 1990.

    MATH  Google Scholar 

  19. Hermite, Ch., Sur la fonction exponentielle, Comptes Rendus de l’Académie des Sciences de Paris, 77 (1873), 18–24, 74-79, 226-233 and 285-293.

    MATH  Google Scholar 

  20. Huygens, C, De automato planetario, repr. in Oeuvres Complètes (1713), vol. xii, pp. 579–652.

    Google Scholar 

  21. Itard, J ., Les Livres Arithmétiques d’Euclide, Paris: Hermann, 1961.

    MATH  Google Scholar 

  22. Knuth, D.E., The Art of computer programming, 1st ed., Reading Massachusetts: Addison-Wesley, 1968.

    MATH  Google Scholar 

  23. Lagrange, J.-L., Sur la résolution des équations numériques, Mémoires de l’Académie de Berlin, 23 (1769) 311-352, repr. in Oeuvres, vol. ii, Paris: Gauthier-Villars, 1868, pp. 539–578.

    Google Scholar 

  24. Lagrange, J.-L., Addition au mémoire sur la résolution des équations numériques, Mémoires de l’Académie de Berlin, 24 (1770) 111-180, repr. in Oeuvres, vol. ii, Paris: Gauthier-Villars, 1868, pp. 581–652

    Google Scholar 

  25. Lagrange, J.-L., Additions à l’analyse indéterminée, in Euler, L., Elémens d’Algèbre, French tr. from the German (1774), vol. ii, 369-658, repr. in Oeuvres, vol. vii, Paris: Gauthier-Villars, 1877, pp. 5–180

    Google Scholar 

  26. Lagrange, J.-L., Sur l’usage des fractions continues dans le calcul intégral, Mémoires de l’Académie de Berlin, 23 (1776), repr. in Oeuvres, vol. iv, Paris: Gauthier-Villars, 1869, pp. 301–332.

    Google Scholar 

  27. Lambert, J.H., Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques, Mémoires de l’Académie de Berlin, 17 (1761), 265–322, repr. in Mathematische Werke, vol. ii, pp. 112-159.

    Google Scholar 

  28. Lazard, D., Le meilleur algorithme d’Euclide pour K[X] et Z , Comptes Rendus de l’Académie des Sciences de Paris, 284, Série A (1977), 1–4.

    MathSciNet  MATH  Google Scholar 

  29. Legendre, A.-M., Eléments de Géométrie, Paris: Didot, 1794.

    Google Scholar 

  30. Lindemann, C., Über die Zahl π, Mathematische Annalen, 20 (1882), 213–225.

    Article  MathSciNet  MATH  Google Scholar 

  31. Liouville, J., Sur des classes très étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des traditionnelles algébriques, Comptes Rendus de l’Académie des Sciences de Paris, 18 (1844), 883–885.

    Google Scholar 

  32. Martzloff, J.-C, Histoire des mathématiques chinoises, Paris: Masson (1987), English tr. S. S. Wilson, A History of Chinese Mathematics, New York: Springer, 1997.

    Google Scholar 

  33. Rolle, M., Traité d’Algèbre, Paris: Michallet, 1690.

    Google Scholar 

  34. Samuel, P., About Euclidean Rings journal of Algebra, 19 (1971), 282–301.

    MathSciNet  MATH  Google Scholar 

  35. Shallit, J., Origins of the Analysis of the Euclidean Algorithm, Historica Mathematica, 21 (1994), 401–419.

    Article  MathSciNet  MATH  Google Scholar 

  36. Sinaceur, H., Corps et modèles, Essai sur l’histoire de l’algèbre réelle, Paris: Vrin, 1991.

    Google Scholar 

  37. Sturm, Ch., Mémoire sur la résolution des équations numériques, Mémoires présentés à l’Académie Royale des Sciences, Sciences mathématiques et physiques, vi (1835), 271–318.

    Google Scholar 

  38. Sylvester, J., On a theory of syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s function, Philosophical Transactions of the Royal Society of London, 142 (1853), 407–548, repr. in Works, vol. 1, Cambridge, 1904, pp. 429-586.

    Article  Google Scholar 

  39. Wallis, J., Arithmetica infinitorum, Oxford, 1656, repr. in Opera Mathematica, vol. I Oxford 1695, pp. 355–478.

    Google Scholar 

  40. Wallis, J., Algebra, Oxford, 1685, repr. in Opera Mathematica, vol. II, 1693, Oxford, pp. 1–482.

    Google Scholar 

  41. Youschkevitch, A., Les mathématiques arabes (VIIIe-XVe siècles), Paris: Vrin, 1976.

    Google Scholar 

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  1. University of Picardy, France

    Jean-Luc Chabert

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    Jean-Luc Chabert

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© 1999 Springer-Verlag Berlin Heidelberg

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Chabert, JL. (1999). Euclid’s Algorithm. In: Chabert, JL. (eds) A History of Algorithms. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18192-4_5

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  • DOI: https://doi.org/10.1007/978-3-642-18192-4_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63369-3

  • Online ISBN: 978-3-642-18192-4

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