Computational science in the eighteenth century. Test cases for the methods of Newton, Raphson, and Halley: 1685 to 1745

  • Trond SteihaugEmail author
Original Paper


This is an overview of examples and problems posed in the late 1600s up to the mid 1700s for the purpose of testing or explaining the two different implementations of the Newton-Raphson method, Newton’s method as described by Wallis in 1685, Raphson’s method from 1690, and Halley’s method from 1694 for solving nonlinear equations. It is demonstrated that already in 1745, it was shown that the methods of Newton and Raphson were the same but implemented in different ways.


Newton-Raphson iteration Nonlinear equations 



Special thanks to Professor Andrew Wathen and the librarian at New College, Oxford University for giving access to perfect version of [18] and to Professor Patrick Farrell, Oxford University for the hospitality and Professor Benjamin Wardhaugh, Oxford University for discussion on [30].


  1. 1.
    Bailey, D.F.: A historical survey of solution by functional iteration. Math. Mag. 62(3), 155–166 (1989). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cajori, F.: Historical note on the Newton-Raphson method of approximation. Am. Math. Mon. 18(2), 29–32 (1911). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chambers, E.: Cyclopædia: or an universal dictionary of arts and sciences. Volume I, London (1728)Google Scholar
  4. 4.
    Diderot, D. (ed.): Encyclopédie, ou dictionnaire raisonné des sciences, des arts et des métiers. Paris (1751)Google Scholar
  5. 5.
    Fenn, J.: Second volume of instructions given in the drawing school established by the Dublin Society: history of mathematicks. Elements of numerical arithmetick. Elements of specious arithmetick. Dublin (1772)Google Scholar
  6. 6.
    Halley, E.: Methodus nova accurata et facilis inveniendi radices æquationum quarumcumque generaliter, sine prævia reductione. Philos. Trans. (1683–1775) 18, 136–148 (1694)CrossRefGoogle Scholar
  7. 7.
    Harris, J.: Lexicon technicum, vol. II, London (1710)Google Scholar
  8. 8.
    Holliday, F.: Syntagma mathesios: containing the resolution of equations, London (1745)Google Scholar
  9. 9.
    Hutton, C.: A mathematical and philosophical dictionary, London (1795)Google Scholar
  10. 10.
    Jones, W.: Synopsis palmariorum matheseos, or a new introduction to mathematics, London (1706)Google Scholar
  11. 11.
    Kollerstrom, N.: Thomas Simpson and ‘Newton’s method of approximation’: an enduring myth. Br. J. Hist. Sci. 25, 347–352 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lagrange, J.L.: Resolution des équations numériqué, Paris (1798)Google Scholar
  13. 13.
    Lord, N.: 95.44 Newton tackles an olympiad problem. Math. Gaz. 95(533), 334–341 (2011)CrossRefGoogle Scholar
  14. 14.
    Newton, I.: Arithmetica universalis; sive de compostione et resolutione arithmetica liber, London. Edited by William Whiston (1707)Google Scholar
  15. 15.
    Newton, I.: Analysis per quantitatum series, fluxiones, ac differentias: cum enumeratione linearum tertii ordinis, London. Edited by William Jones (1711)Google Scholar
  16. 16.
    Newton, I., Colson, J.: The method of fluxions and infinite series; with its application to the geometry of curve-lines. by the inventor sir Isaac Newton, Kt late president of the Royal Society. Translated from the author’s Latin original not yet made publick. To which is subjoin’d, a perpetual comment upon the whole work, consisting of annotations, illustrations, and supplements, in order to make this treatise a compleat institution for the use of learners, London. Translated and commented by J.Colson (1736)Google Scholar
  17. 17.
    Parsons, J., Wastell, T.: Clavis arithmeticæ: or, a key to arithmetick in numbers and species, London (1704)Google Scholar
  18. 18.
    Raphson, J.: Analysis æquationum Universalis, seu ad æquationes algebraicas resolvendas methodus generalis, et expedita, ex nova infinitarum serierum doctrina, deducta ac demonstrata, London (1690)Google Scholar
  19. 19.
    Raphson, J.: Analysis æquationum Universalis, seu ad æquationes algebraicas resolvendas methodus generalis, et expedita, ex nova infinitarum serierum doctrina, deducta ac demonstrata, editio secunda cum appendice. London (1697)Google Scholar
  20. 20.
    Reyneau, C.R.: Analyse démontrée, ou la méthode de résoudre les problèmes des mathématiques, Tome I, Paris (1708)Google Scholar
  21. 21.
    Ronayne, P.: A treatise of algebra in two books: the first treating of the arithmetical and the second of the geometrical part, London (1717)Google Scholar
  22. 22.
    Sault, R.: A new treatise of algebra. According to the late improvements. Apply’d to numerical questions and geometry. With a converging series for all manner of adfected equations. In: William Leybourn Pleasure with Profit, London. Reprinted in 1695 (1694)Google Scholar
  23. 23.
    Simpson, T.: Essays on several curious and useful subjects, in speculative and mix’d mathematicks, London (1740)Google Scholar
  24. 24.
    Simpson, T.: A treatise of algebra; wherein the fundamental principles are fully and clearly demonstrated, and applied to the solution of a great variety of problems. To which is added, the construction of a great number of geometrical problems, with the method of resolving the same numerically, London (1745)Google Scholar
  25. 25.
    Stewart, J.: Sir Isaac Newton two treatises of the quadrature of curves, and analysis by equations of an infinite number of terms, explained: containing the treatises themselves, translated into english, with a large commentary; in which the demonstrations are supplied where wanting, the doctrine illustrated, and the whole accommodated to the capacities of beginners, for whom it is chiefly designed, London (1745)Google Scholar
  26. 26.
    Vellnagel, C.F.: Gründliche und ausführliche erläuterungen so wohl über die gemeine algebra als differential- und integral–rechnung. Druckts u. verlegts Christian Franc. Buch Jena (1743)Google Scholar
  27. 27.
    Wallis, J.: A treatise of algebra, both historical and practical, London (1685)Google Scholar
  28. 28.
    Wallis, J.: De algebra tractatus, historicus [et] practicus, Oxford (1693)Google Scholar
  29. 29.
    Ward, J.: A compendium of algebra, London (1695)Google Scholar
  30. 30.
    Ward, J.: The young mathematician’s guide. Being a plain and easie introduction to the mathematicks, London. Reprinted 1709 (1707)Google Scholar
  31. 31.
    Wardhaugh, B.: Consuming mathematics: John Ward’s young mathematician’s guide (1707) and its owners. Journal for Eighteenth-Century Studies 38(1), 65–82 (2015). MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wells, E.: Elementa arithmeticæ numerosæ et speciosæ, Oxford (1698)Google Scholar
  33. 33.
    Whiteside, D.T.: The Mathematical Papers of Isaac Newton, vol II. Cambridge University Press, Cambridge (1667–1670)Google Scholar
  34. 34.
    Wolff, C.: Elementa matheseos universæ Qui commentationem de methodo mathematica, arithmeticam, geometriam, trigonometriam, analysin tam finitorum, quam infinitorum, staticam et mechanicam, hydrostaticam, aerometriam, hydraulicam complectitur. Renger, Halae Magdeburgicae (1713)Google Scholar
  35. 35.
    Wolff, C.: A treatise of algebra; with the application of it to a variety of problems in arithmetic, to geometry, trigonometry, and conic sections. With the several methods of solving and constructing equations of the higher kind. Translated J. Hanna, London (1739)Google Scholar
  36. 36.
    Ypma, T.J.: Historical development of the Newton-Raphson method. SIAM Review 37(4), 531–551 (1995). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of BergenBergenNorway

Personalised recommendations