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The quartic equation: invariants and Euler's solution revealed

  • R. W. D. Nickalls (a1)


The central role of the resolvent cubic in the solution of the quartic was first appreciated by Leonard Euler (1707-1783). Euler's quartic solution first appeared as a brief section (§ 5) in a paper on roots of equations [1, 2], and was later expanded into a chapter entitled ‘Of a new method of resolving equations of the fourth degree’ (§§ 773-783) in his Elements of algebra [3,4].



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1. Euler, L., De formis radicum aequationum cujusque ordinis conjectatio. Commentarii academiae scientiarum imperialis Petropolitianae 6 (1733), pp. 216231 = Opera Omnia, Series 1, 6 (Theory of equations) pp. 119. [Euler Archive, E30 (Latin):]
2. Bell, J., A conjecture on the forms of the roots of equations. arA7v:0806.1927vl [math.HO] (2008). [An English translation of Euler’s De formis radicum aequationum cujusque ordinis conjectatio (E30)].
3. Euler, L., Vollständige Anleitung zur Algebra (Elements of algebra), 2 vols, Royal Academy of Sciences, St. Petersburg (1770). [Euler Archive, E387 (English): docs/originals/E387e.PlS4.pdf]
4. Sangwin, C. R. (ed.). Euler’s Elements of algebra, Tarquin Publications, St Albans, UK (2006). [English translation of Euler 1770 (E387)]
5. Burnside, W. S. and Panton, A. W.. The theory of equations: with an introduction to the theory of binary algebraic forms (7th edn.), 2 vols; Longmans, Green and Co., London (1912).
6. Cremona, J. E.. Reduction of binary cubic and quartic forms, J. Comput. Math. 2 (1999) pp. 6292. lms98007/ [the seminvariants G and H2 – 16a2/ are denoted here by R and 3Q respectively]
7. Olver, P. J., Classical invariant theory, London Mathematical Society Student Texts No. 44, Cambridge University Press (1999).
8. Nickalls, R. W. D., A new approach to solving the cubic: Cardan’s solution revealed. Math. Gaz., 77 (November 1993) pp. 354359.
9. Ball, R. S., Note on the algebraical solution of biquadratic equations, Quarterly Journal of Pure and Applied Mathematics 7 (1866) pp. 69, 358369.
10. Dalton, J. P., On the graphical discrimination of the cubic and of the quartic, Math. Gaz. 17 (July 1933) pp. 189196.

The quartic equation: invariants and Euler's solution revealed

  • R. W. D. Nickalls (a1)


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