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Viète, Descartes and the cubic equation

Published online by Cambridge University Press:  01 August 2016

R. W. D. Nickalls*
Affiliation:
Department of Anaesthesia, City Hospital, Nottingham NG5 1PB, UK, e-mail: dicknickalls@compuserve.com

Extract

An appreciation of the geometry underlying algebraic techniques invariably enhances understanding, and this is particularly true with regard to polynomials.

With visualisation as our theme, this article considers the cubic equation and describes how the French mathematicians François Viète (1540–1603) and René Descartes (1596–1650) related the ‘three-real-roots’ case (casus irreducibilis) to circle geometry. In particular, attention is focused on a previously undescribed aspect, namely, how the lengths of the chords constructed by Viète and Descartes in this setting relate geometrically to the curve of the cubic itself.

Type
Articles
Copyright
Copyright © The Mathematical Association 2006

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References

1. Cochrane, P. Virtual mathematics, Math. Gaz., 80 (July 1996) pp. 267278.Google Scholar
2. Nickalls, R. W. D. A new approach to solving the cubic: Cardan’s solution revealed, Math. Gaz., 77 (November 1993) pp. 354359.Google Scholar
3. Katz, V. J. A history of mathematics: an introduction (2nd edn.) Addison-Wesley (1998).Google Scholar
4. Busard, H. L. L. François Viète. In: Gillespie, C.C. (Ed.) Dictionary of Scientific Biography (1970–1990). Charles Scribner’s Sons, New York (1970).Google Scholar
5. Schmidt, R. On the Recognition of Equations, by Viète, François From the 1615 Anderson edition of De aequationum recognitione et emendatione. Including excerpts from On the Emendation of Equations, and a glossary of Viète’s technical terms. Translated by Schmidt, Robert In: The early theory of equations: on their nature and constitution. Translations of three treatises, by Viète, , Girard, , and De Beaune, . Golden Hind Press, Annapolis, Maryland, USA (1968).Google Scholar
6. Viète, F. Francisci Vietae Fontenaensis aequationum recognitione et emendatione tractatus duo (Two tracts by François Viète of Fontenay entitled ‘On the recognition of equations’ and ‘On the emendation of equations’). In: Anderson, (Ed.), Opera mathematica, volume 3; Paris, France (1615).Google Scholar
8. Viète, F. Ad Angularum Sectionem Analytica Theoremsta. In: Anderson, A. (Ed.), Opera mathematica, volume 4; 4to. Paris, France (1615).Google Scholar
[In this work Viète presents formulae for the chords of multiples of a given arc in terms of powers of the chord of the simple arc, equivalent to today’s usual formulae for sin and cos (see [9]).]Google Scholar
9. Hobson, E. W. Trigonometry. In: Encyclopaedia Britannica (11 th edn.) New York (1911); vol 27, pp. 272, 280.Google Scholar
10. Nickalls, R. W. D. Solving the cubic using tables. Theta; 10 (No. 2, Autumn, 1996) pp. 2124. (Pub: Mathematics Department, Manchester Metropolitan University, UK) ISSN 0953-0738.Google Scholar
11. Descartes, R. La Géométrie, Leiden (1637).Google Scholar
12. Smith, D. E. and Latham, M. L. The geometry of René Descartes, with a facsimile of the first edition. Dover, New York (1954).Google Scholar
[An English translation together with a complete facsimile of the 1637 French text including all Descartes’ original illustrations.]Google Scholar