Erland Samuel Bring's "Transformation of Algebraic Equations"
We translate Erland Samuel Bring's treatise Meletemata quaedam Mathematica circa Transformationem Aequationum Alebraicarum (Some selected mathematics on the Transformation of Algebraic Equations) written as his Promotionschrift at the University of Lund in 1786, from its Latin into English, with modern mathematical notation.
Bring (1736 - 98) made important contributions to algebraic equations and obtained the canonical form x^5+px+q = 0 for quintics before Jerrard, Ruffini and Abel. In due course, he realized the significance of the projective curve which now bears his name: the complete intersection of the homogeneous Fermat polynomials of degrees 1,2,3 in CP^4.
R. R. Harley, “A contribution to the history of the problem of the reduction of the general equation of the fifth degree to a trinomial form,” The Quarterly Journal of Pure and Applied Maths, vol. 6, p. 38, 1863–64.
F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. Leipzig: Teubner, 1st ed., 1884.
F. Klein and G. Morrice, Lectures on the Icosahedron and the solution of Equations of the Fifth Degree. Dover, NY: Dover Publications, 2nd ed., 1956.
M. Belardinelli, Mémorial des Sciences Mathématiques, vol. Fascicule 145, ch. Résolution Analytique des Equations Algebraiques Generales. Paris: Gauthier–Villars, 1960.
P. Doyle and C. McMullen, “Solving the quintic by iteration,” Acta Mathematica, vol. 163, pp. 151–180, 1989.
The reader is referred to “Erland Samuel Bring” at the MacTutor.
J. Sylvester and J. Hammond, “X. On Hamilton’s numbers,” Philosophical Transactions of the Royal Society of London, vol. 178, pp. 285–312, 1887.
J. Sylvester and J. Hammond, “IV. On Hamilton’s numbers.—Part II,” Philosophical Transactions of the Royal Society of London, vol. 179, pp. 65–71, 1888.
“The On-line Encyclopaedia of Integer Sequences: The Hamilton Numbers.” Available from http://oeis.org/A000905, 2021.
C. Hermite, “Sur la résolution de l’équation du cinquième degré,” Comptes rendus de l’Académie des Sciences, vol. 46, pp. 508–515, 1858.
O. Nash, “On Klein’s icosahedral solution of the quintic.” Available at arXiv:1308.0955, 2013.
H. W. Braden, “Bring's curve: its period matrix and the vector of Riemann constants,” Symmetry, Integrability and Geometry: Methods and Applications, vol. 8, 2012.
W. Edge, “Bring’s curve,” Journal of the London Mathematical Society, vol. 18, no. 3, pp. 539–545, 1978.
W. Edge, “Tritangent planes of Bring’s curve,” Journal of the London Mathematical Society, vol. 23, no. 2, pp. 215–222, 1981.
M. Weber, “Kepler’s small stellated dodecahedron as a Riemann surface,” Pacific Journal of Mathematics, vol. 220, no. 1, pp. 167–182, 2005.
D. Singerman and R. Syddall, “The Riemann surface of a uniform dessin,” Beiträge zur Algebra und Geometrie, vol. 44, no. 2, pp. 413–430, 2003.
Y.-H. He and J. McKay, “Sporadic and exceptional.” Available at arXiv:1505.06742, 2015.
S. Garibaldi, “e8, the most exceptional group,” Bulletin of the American Mathematical Society, vol. 53, no. 4, pp. 643–671, 2016.
L. Yang, “Modular curves, invariant theory and e8.” Available at arXiv:1704.01735, 2017.
How to Cite
Copyright (c) 2022 inSTEMM Journal
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.