Abstract
In this paper we give a complete overview of test–problems by Viète from 1600, Harriot from 1631 and Oughtred from 1647. The original material is not easily accessible due to archaic language and lack of conciseness. Viéte’s method was gradually elucidated by the subsequent writers Harriot and Oughtred using symbols and being more concise. However, the method is presented in tables and from the layout of the tables it is difficult to find the general principle. Many authors have therefore described Viète’s process inaccurately and in this paper we give a precise description of the divisor used in the process which has been verified on all the test–problems. The process of Viète is an iterative method computing one digit of the root in each iteration and has a linear rate of convergence and we argue that the digit–by–digit process lost its attractiveness with the publications in 1685 and 1690 of the Newton-Raphson method which doubles the number of digits for each iteration.
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Notes
- 1.
The integer \(2\cdot 10^{28}\) in Viète is replaced by \(2\cdot 10^{34}\).
- 2.
M. Cantor, Vorlesungen über Geschichte der Mathematik, II, 1900, p. 640–641.
- 3.
Augustus De Morgan, Involution and Evolution, in The Penny Cyclopaedia of the Society for the Diffusion of Useful Knowledge, London 1846, Volume 2 p. 103.
- 4.
Translated by T. Richard Witmer [34].
- 5.
References
Florian Cajori. William Oughtred, a great seventeenth-century teacher of mathematics. Chicago Open Court Pub. Co, 1916.
Florian Cajori. A history of mathematical notations, Volume 1. Notations in elementary mathematics. The Open court publishing company, Chicago, 1928.
Jean-Luc Chabert (Ed.). A History of Algorithms. From the Pebble to the Microchip. Springer-Verlag, Berlin, 1999.
Herman Heine Goldstine. A History of Numerical Analysis from the 16th through the 19th Century. Springer-Verlag, New York, 1977.
Hermann Hankel. Geschichte der Mathematik in Alterthum und Mittelalter. Leipzig, 1874.
Thomas Harriot. Artis Analyticae Praxis. London, 1631.
Theophilus Holdred. A New Method of Solving Equations with Ease and Expedition; by which the True Value of the Unknown Quantity is Found without Previous Reduction. With a Supplement, Containing Two Other Methods of Solving Equations, Derived from the Same Principle. London, 1820.
William George Horner. A new method of solving numerical equations of all orders, by continuous approximation. Philosophical Transactions, 109:308–335, 1819.
Charles Hutton. A mathematical and philosophical dictionary. Volume 1. London, 1795.
Charles Hutton. Tracts on mathematical and philosophical subjects. Volume 2. London, 1812.
Samuel Jeake. A Compleat Body of Arithmetic. London, 1701.
Samuel Jeake, Senior. \(\varLambda \)O\(\varGamma \)I\(\varSigma \)TIKH\(\varLambda \)O\(\varGamma \)I’A, or Arithmetick surveighed and reviewed: in four books, etc. [Edited by Samuel Jeake, the Younger.]
John Kersey. The Elements of That Mathematical Art Commonly Called Algebra. London, 1673.
Jean Étienne Montucla. Histoire des mathématiques, Volume I. Paris, 1758.
Augustus De Morgan. Notices of the progress of the problem of evolution. The companion to the almanac, pages 34–52, 1839.
Isaac Newton. Arithmetica universalis; Sive de compositione et resolutione arithmetica liber. London, 1707. Edited by William Whiston.
Martin Andrew Nordgaard. A historical survey of algebraic methods of approximating the roots of numerical higher equations up to the year 1819. PhD thesis, Columbia University, New York City, 1922.
William Oughtred. Clavis mathematicae. London, 1631.
William Oughtred. The key of the mathematics new forged and filed. London, 1647.
Joseph Raphson. Analysis Æquationum Universalis, seu ad Æquationes Algebraicas Resolvendas Methodus Generalis, et Expedita, Ex nova Infinitarum serierum Doctrina, Deducta ac Demonstrata. London, 1690.
Roshdi Rashed. Résolution des équations numérique et algèbre: Šaraf–al–Din al-Tūsī, Viéte. Archive for History of Exact Sciences, 12(3):244–290, 1974.
Stephen Jordan Rigaud. Correspondence of Scientific Men of the Seventeenth Century, Vol.1. Oxford, 1841.
Muriel Seltman and Robert Goulding. Thomas Harriot’s Artis Analyticae Praxis. An English Translation with Commentary. Springer, 2007.
Jacqueline Anne Stedall. The Greate Invention of Algebra: Thomas Harriot’s Treatise on equations. Oxford University Press, Oxford, 2003.
Jacqueline Anne Stedall. Tracing mathematical networks in seventeenth-century England. In Eleanor Robson and Jacqueline Stedall, editors, The Oxford handbook of the history of mathematics, pages 133–151. Oxford University Press, 2009.
Jacqueline Anne Stedall. Reconstructing Thomas Harriot’s treatise on equations. In Robert Fox, editor, Thomas Harriot and His World: Mathematics, Exploration, and Natural Philosophy in Early Modern England, pages 53–63. Ashgate Publishing (Routledge), 2012.
Trond Steihaug. Computational science in the eighteenth century. Test cases for the methods of Newton, Raphson, and Halley: 1685 to 1745. Numerical Algorithms, 83:1259–1275, 2020.
Trond Steihaug and D. G. Rogers. Approximating cube roots of integers, after Heron’s Metrica III.20. Normat, 61(2):87–110, 2013.
Philip D. Straffin. Liu Hui and the first golden age of Chinese mathematics. Mathematics Magazine, 71(3):163–181, 1998.
David J. Thomas and Judith M. Smith. Joseph Raphson, F.R.S. Notes and Records of the Royal Society of London, 44(2):151–167, 1990.
Bartel Leendert van der Waerden. A History of Algebra. From al-Khwarizmi to Emmy Noether. Springer, Berlin, 1985.
François Viète. De numerosa potestatum ad exegesim resolutione. Paris, 1600.
François Viète. Opera mathematica, edited by Frans van Schooten. Leiden, 1646.
François Viète. The analytic art. Translated by T.Richard Witmer. The Kent State University Press, 1983.
John Wallis. A treatise of algebra, both historical and practical. London, 1685.
John Wallis. A discourse concerning the methods of approximation in the extraction of surd roots. Philosophical Transactions (1683-1775), 19:2–11, 1695.
John Ward. A Compendium of Algebra. London, 1695.
John Ward. The Young Mathematician’s Guide. Being a Plain and Easie Introduction to the Mathematicks. London, 1707. Reprinted 1709.
Derek Thomas Whiteside. The Mathematical Papers of Isaac Newton 1664-1666, Volume I. Cambridge University Press, Cambridge, 1967.
Tjalling J. Ypma. Historical development of the Newton-Raphson method. SIAM Review, 37(4):531–551, 1995.
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Steihaug, T. (2021). Computational Science in the 17th Century. Numerical Solution of Algebraic Equations: Digit–by–Digit Computation. In: Al-Baali, M., Purnama, A., Grandinetti, L. (eds) Numerical Analysis and Optimization. NAO 2020. Springer Proceedings in Mathematics & Statistics, vol 354. Springer, Cham. https://doi.org/10.1007/978-3-030-72040-7_12
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