Skip to main content
Log in

Numerical solving of equations in the work of José Mariano Vallejo

  • Published:
Archive for History of Exact Sciences Aims and scope Submit manuscript

Abstract

The progress of Mathematics during the nineteenth century was characterised both by an enormous acquisition of new knowledge and by the attempts to introduce rigour in reasoning patterns and mathematical writing. Cauchy’s presentation of Mathematical Analysis was not immediately accepted, and many writers, though aware of that new style, did not use it in their own mathematical production. This paper is devoted to an episode of this sort that took place in Spain during the first half of the century: It deals with the presentation of a method for numerically solving algebraic equations by José Mariano Vallejo, a late Spanish follower of the Enlightenment ideas, politician, writer, and mathematician who published it in the fourth (1840) edition of his book Compendio de Matemáticas Puras y Mistas, claiming to have discovered it on his own. Vallejo’s main achievement was to write down the whole procedure in a very careful way taking into account the different types of roots, although he paid little attention to questions such as convergence checks and the fulfilment of the hypotheses of Rolle’s Theorem. For sure this lack of mathematical care prevented Vallejo to occupy a place among the forerunners of Computational Algebra.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atkinson K.E. (1989) An Introduction to Numerical Analysis. John Wiley & Sons, Singapore

    MATH  Google Scholar 

  2. Bails B. (1795) Instituciones de Geometría Práctica. Ibarra, Madrid

    Google Scholar 

  3. F. Cajori. Horner’s method of approximations anticipated by Ruffini. Bulletin of the American Mathematical Society, 17 (1911), pp. 409–414.

    Article  MathSciNet  Google Scholar 

  4. A.L. Cauchy. Cours D’Analyse de L’Ecole Royale Polytechnique. 1821. Edición facsimilar. Eds.: F.J. Pérez-Fernández & J.M. Díaz Moreno. S.A.E.M. THALES. Sevilla. 1998.

  5. A.L. Cauchy. Oeuvres complètes. Gauthiers-Villars. Paris. 1882–1974.

  6. E.B. Condillac, L’Abbé. La logique ou les premiers développements de l’art de penser. Paris. 1779. (Spanish translation in Editorial Losada, Buenos Aires, 1956)

  7. A.M. Cohen. Is the polynomial so perfidious? Numerische Mathematik 68 (1994). no. 2. pp. 225–238.

    Article  MATH  MathSciNet  Google Scholar 

  8. J.M. Díaz Moreno & F. Benítez Trujillo. Introducción a los métodos numéricos para la resolución de ecuaciones. Servicio de Publicaciones. Universidad de Cádiz. Cádiz. 1998.

    Google Scholar 

  9. S. Garma Pons. Las matemáticas en España a principios del siglo XIX, D. José Mariano Vallejo. Revista de Occidente 118 (1973). pp. 105–114.

    Google Scholar 

  10. J. M. Gentil Baldrich. Nuevos datos sobre la vida y la obra de José Mariano Vallejo y Ortega (1779-1846). LLULL 22 (1999). pp. 381–404.

    Google Scholar 

  11. M. Hormigón. Les mathématiciens dans la vie politique espagnole pendant la première moitié du XIXe siècle. Bolletino di Storia delle Science Matematiche 15 (1995). no. 1. pp. 27–47.

    MATH  Google Scholar 

  12. W.G. Horner. A new method of solving numerical equations of all orders by continuous approximation. Philosophical Transactions of the Royal Society of London 109 (1819). pp. 308–335.

    Article  Google Scholar 

  13. I. Newton. Analysis per quantitatum series, fluxiones, ac diferentias: cum Enumeratione linearum Tertii ordinis. Reprint of the 1711 original. Translated from the Latin by J. L. Arantegui Tamayo. Annotated by A. Durán Guardeño. Eds.: A. Durán Guardeño & F.J. Pérez-Fernández. S.A.E.M. THALES y R.S.M.E. Sevilla. 2003.

  14. P. Ruffini. Opere Matematiche. Ed. Cremonese della Casa Editrice Perella. Rome. 1953–1954. Ed. E. Bortolotti.

  15. P. Ruffini. Sopra la Determinazione delle Radici nelle Equazione numeriche di qualunque Grado. Modena, 1804. In [14], II, pp. 281–404.

  16. Smith D.E. (1959) A source book in Mathematics. Dover Publications, New York

    MATH  Google Scholar 

  17. C. Suárez Alemán. Aceptación en España de los criterios rigurosos del Análisis Matemático durante los siglos XIX y XX. Unpublished. Doctoral Dissertation. Universidad de Cádiz.

  18. J.M. Vallejo y Ortega. Aritmética para niños. Imp. Real. Madrid, 1804.

  19. J.M. Vallejo y Ortega. Adiciones a la Geometría de don Benito Bails. Imp. Real. Madrid, 1806.

  20. J.M. Vallejo y Ortega. Memoria sobre la curvatura de líneas . Imp. Real. Madrid, 1807.

  21. J.M. Vallejo y Ortega. Tratado Elemental de Matemáticas. Palma de Mallorca, 1812 (Vol.I y II) y Valencia, 1817 (Vol. III).

  22. J.M. Vallejo y Ortega. Compendio de Matemáticas Puras y Mistas. Valencia. 1819.

  23. J.M. Vallejo y Ortega. Ideas primarias que deben darse a los niños ...acerca de los números, Paris, 1826.

  24. J.M. Vallejo y Ortega. Tratado Elemental de Matemáticas. 2th Ed. Madrid. 1832–1832.

  25. J.M. Vallejo y Ortega. Compendio de Matemáticas Puras y Mistas. 4th Ed. Madrid. 1840.

  26. J.M. Vallejo y Ortega. Definiciones y extracto de las principales reglas y operaciones de la Aritmética, Madrid, 1840.

  27. J.M. Vallejo y Ortega. Explicación del sistema decimal, Madrid, 1840.

  28. J.M. Vallejo y Ortega. Tratado Elemental de Matemáticas. 3th Ed. Madrid. 1841–44.

  29. J.M. Vallejo y Ortega. Tratado completo de Matemáticas y Álgebra, Paris, 1856. (Posthumous work edited by his descendants).

  30. C.A. Valson. La vie et les travaux du Baron Cauchy. Paris. 1868. Réimpression augmentée d’une introduction per René Taton. Librairie Scientifique et Technique Albert Blanchard, París, 1970.

  31. F. Vea Muniesa. The influence of french mathematics textbooks on the establishment of liberal education system in Spain (1845–1868). Paradigms and Mathematics. Siglo XXI de España Editores, Madrid, 1995.

  32. M. A. Velamazán and E. Ausejo. De Lagrange a Cauchy: el cálculo diferencial en las academias militares de España en el siglo XIX. LLULL, 16 (1993). pp.327–370.

    Google Scholar 

  33. J. Wilkinson. The perfidious polynomial. Studies in numerical analysis. Ed. Gene H. Golub. MAA Stud. Math.. vol. 24. Math. Assoc. America. Washington D.C.. 1984. pp. 1–24.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to José-Miguel Pacheco Castelao.

Additional information

Communicated by U. Bottazzini.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pacheco Castelao, JM., Pérez-Fernández, F.J. & Suárez Alemán, CO. Numerical solving of equations in the work of José Mariano Vallejo. Arch. Hist. Exact Sci. 61, 537–552 (2007). https://doi.org/10.1007/s00407-007-0007-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00407-007-0007-5

Keywords

Navigation