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On square roots and their representations

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References

  • Brent, R. P. [1976], “Multiple-precision zero-finding methods and the complexity of elementary function evaluation,” Analytical computational complexity, ed. by J. F. Traub, New York.

  • Cantor, G. [1869], “Zwei Sätze über eine gewisse Zerlegung der Zahlen in unendliche Producte,” Zeitschrift für Mathematik und Physik, Bd. 14, 152–158.

    Google Scholar 

  • Chrystal, G. [1904], Algebra ..., 2 Vols., 6th ed., reprinted 1952 by Chelsea Publishing Co., New York.

  • Colebrooke, H. F. [1817], Algebra with arithmetic and mensuration from the Sanskrit of Brahmagupta and Bāhskara, London.

  • Datta, B. [1931], “Nārāyana's method for finding approximate value of a surd,” Bulletin of the Calcutta Mathematical Society, Vol. 23, 187–194.

    Google Scholar 

  • Dickson, L. E. [1919–1923], History of the theory of numbers, repr. 1971, Chelsea Publishing Co., New York.

    Google Scholar 

  • Dijksterhuis, E. J. [1957], Archimedes, New York.

  • Dutka, J. [1971], “The square root of 2 to 1,000,000 decimals,” Mathematics of Computation, Vol. 25, 927–930.

    Google Scholar 

  • Engel, F. [1913], “Entwicklung der Zahlen nach Stammbrüchen,” Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 190–191.

  • Escott, E. B. [1937], “Rapid method for extracting a square root,” American Mathematical Monthly, Vol. 44, 644–646.

    Google Scholar 

  • Euler, L. [1748], Introductio in analysin infinitorum, T.I., repr. in Opera Omnia, Ser. 1, Vol. VIII, Leipzig, 1922.

  • Euler, L. [1770a], “De inventione quotcumque mediarum proportionalium citra radicum extractionem,” repr. in Opera Omnia, Ser. 1, Vol. VI, Leipzig, 1921, 240–262.

    Google Scholar 

  • Euler, L. [1770b], Vollständige Anleitung zur Algebra, repr. in Opera Omnia, Ser. 1, Vol. I, Leipzig, 1911.

  • Euler, L. [1774], “Nova ratio quantitates irrationales proxime exprimendi,” repr. in Opera Omnia, Ser. 1, Vol. VI, Leipzig, 1921, 316–349.

    Google Scholar 

  • Fowler, D. H. [1979], “Ratio in early Greek mathematics,” Bulletin of American Mathematical Society, N.S., Vol. 1, 807–846.

    Google Scholar 

  • Günther, S. [1882], “Die quadratischen Irrationalitäten der Alten und deren Entwicklungsmethoden,” Abhandlungen zur Geschichte der Mathematik, Heft 4, 1–134.

  • Halley, E. [1694], “Methodus nova accurata & facilis inveniendi radices aequationium quarumcumque generaliter, sine praevia reductione,” Philosophical Transactions of the Royal Society, Vol. 18, 136–148.

    Google Scholar 

  • Heath, Th. L. [1898], The works of Archimedes ..., repr. 1953 with the Supplement of 1912, Dover Publications, New York, 1953.

    Google Scholar 

  • Heath, Th. L. [1910], Diophantus of Alexandria, 2nd ed., London; repr. 1964, New York.

  • Heath, Th. L. [1921], A history of Greek mathematics, Oxford, 2 Vols.

  • Hofmann, J. E. [1930], “Erklärungsversuche für Archimedes Berechnung von 38-1,” Archiv für Geschichte der Mathematik der Naturwissenschaft und der Technik, Bd. XII, 386–408.

    Google Scholar 

  • Hofmann, J. E. [1934], “Über die Annäherung von Quadratwurzeln bei Archimedes und Heron,” Jahresbericht der deutschen Mathematiker Vereinigung, Bd. 43, 187–210.

    Google Scholar 

  • Horner, J. [1860], “Approximation to the roots of algebraic equations in a series of Aliquot parts,” Quaterly Journal of Pure and Applied Mathematics, Vol. III, 251–262.

    Google Scholar 

  • Khovanski, A. N. [1963], The application of continued fractions, Groningen.

  • Knorr, W. R. [1975], The evolution of the Euclidean elements, Dordrecht.

  • Knorr, W. R. [1976], “Archimedes and the measurement of the circle: A new interpretation,” Archive for History of Exact Sciences, Vol. 15, 115–140.

    Google Scholar 

  • Lagny, Th. F. de [1723], “Méthode générale pour transformer les nombres irrationaux en séries de fractions rationelles ...,” Mémoires de mathématique et de physique tirés des registres de l'Academie Royale des Sciences, 55–69.

  • Lagrange, J. L. L. [1776], “Sur l'usage des fractions continues dans le calcul intégrale,” Oeuvres, Vol. IV, 301–332.

    Google Scholar 

  • Lambert, J. H. [1758], “Observationes variae in Mathesin puram,” repr. in Opera Mathematica, Zürich, Vol. I, 1946, 16–51.

    Google Scholar 

  • Lambert, J. H. [1770], “Verwandlung der Brüche,” repr. in Opera Mathematica, Zürich, Vol. I, 1946–1948, 133–188.

    Google Scholar 

  • Lambert, J. H. [1781–1785], Deutscher Gelehrter Briefwechsel, Bd. I–VI, Berlin.

  • Lorey, W. [1939], “Über ein Eulersches Verfahren zur Wurzelberechnung,” Monatshefte für Mathematik und Physik, Bd. 48 (1939), 190–197.

    Google Scholar 

  • Lucas, E. [1878], “Théorie des functions numériques simplement périodiques, I,” American Journal of Mathematics, Vol. I, 184–240.

    Google Scholar 

  • Mercator, N. [1668], Logarithmotechnia ..., London.

  • Neugebauer, O., & Sachs, A. [1945], Mathematical cuneiform texts, New Haven, 42–43.

  • Ostrowski, A. [1929], “Ueber einige Verallgemeinerungen des Eulerschen Produktes ...,” Verhandlungen der Naturforschenden Gesellschaft in Basel, Bd. XL, T.I., 153–214.

    Google Scholar 

  • Patz, W. [1955], Tafel der regelmassigen Kettenbrüchen ..., Berlin.

  • Perron, O. [1960], Irrationalzahlen, Vierte Aufl., Berlin.

  • Rubbert, F. K. [1948], “Zur radizierung mit der Rechenmaschine,” Zeitschrift für angewandte Mathematik und Mechanik, Bd. 28, 190–191.

    Google Scholar 

  • Schröder, E. [1870], “... Methoden zur Auflösung der Gleichungen,” Mathematische Annalen, Bd. 2, 317–365.

    Google Scholar 

  • Sierpinski, W. [1964], Elementary theory of numbers, Monografie Matematyczne, T. 42, Warsaw (translated).

  • Simson, R. [1753], “An explication of an obscure passage in Albert Girard's commentary upon Simon Stevin's works ...,” Philosophical Transactions of the Royal Society of London, Vol. 48, 368–377.

    Google Scholar 

  • Smith, D. E. [1929], A source book in mathematics, repr. in 2 vols., 1959, Dover Publications, Inc., New York.

    Google Scholar 

  • Sulamin, E. [1976], “Computation of ρ using arithmetic-geometric mean,” Mathematics of Computation, Vol. 30, 565–570.

    Google Scholar 

  • Thibaut, G. [1875], “On the Súlvasútras,” Journal of the Royal Asiatic Society of Bengal, Calcutta, Vol. 44, 227–275.

    Google Scholar 

  • Thiele, T. N. [1884], “Arkhimedes og \(\sqrt 3 \),” Tidsskrift for Mathematik, R. 5, A. 2, 151–153.

  • Traub, J. F. [1982], Iterative methods for the solution of equations, 2nd ed., New York.

  • Tropfke, J. [1980], Geschichte der Elementar-Mathematik, 4. Aufl., Bd. 1, Neubearbeitet von K. Vogel, K. Reich, & H. Gericke, Berlin.

  • van der Waerden, B. L. [1983], Geometry and algebra in ancient civilizations, Berlin.

  • Wall, H. S. [1948], “A modification of Newton's method,” American Mathematical Monthly, Vol. 55, 90–94.

    Google Scholar 

  • Wertheim, G. [1898], “Die Berechnung der irrationalen Quadratwurzeln und die Erfindung der Kettenbrüche,” Abhandlungen zur Geschichte der Mathematik, Achtes Heft, 147–160.

  • Woepcke, F. [1854], “Recherches sur l'histoire des sciences mathématiques ...,” Journal Asiatique ..., Ser. 5, T. 4.

  • Wynn, P. [1956], “On a cubically convergent process for determining the zeros of certain functions,” Mathematics of Computation, Vol. 10, 97–100.

    Google Scholar 

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Communicated by B. L. van der Waerden

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Dutka, J. On square roots and their representations. Arch. Hist. Exact Sci. 36, 21–39 (1986). https://doi.org/10.1007/BF00357439

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