Bibliography for Roots of Cubic Equations

unabridged

 

  1. On cubic CNS polynomials with three real roots.
    Brunotte, Horst
    Acta Sci. Math. (Szeged) 70 (2004), no. 3-4, 495--504, MathSciNet.  
  2. Geometrical solution of some cubic equations with non-real roots  
    Pathak, H. K.; Grewal, A. S.  
    Internat. J. Math. Ed. Sci. Tech. 33 (2002), no. 4, 575--596, MathSciNet.  
  3. Analysis of the roots of some Cardano cubics
    Leyendekkers, J. V.; Shannon, A. G.
    Notes Number Theory Discrete Math. 6 (2000), no. 4, 113--117, MathSciNet.  
  4. Cubic root of Klein-Gordon equation.
    Plyushchay, Mikhail S.; Rausch de Traubenberg, Michel
    Phys. Lett. B 477 (2000), no. 1-3, 276--284, MathSciNet.  
  5. On al-Tusi's discussion on positive roots of cubic algebraic equations. (Chinese)
    Bao, Fang Xun
    Qufu Shifan Daxue Xuebao Ziran Kexue Ban 25 (1999), no. 4, 52--55, MathSciNet.  
  6. The earliest correct algebraic solutions of cubic equations  
    Hughes, Barnabas
    Vita mathematica (Toronto, ON, 1992; Quebec City, PQ, 1992), 107--112, MAA Notes, 40, Math. Assoc. America, Washington, DC, 1996, MathSciNet.  
  7. A natural verification of the Sendov conjecture for the canonical cubic equation and other results for the location of its roots.
    Todorov, Pavel G.
    Acad. Roy. Belg. Bull. Cl. Sci. (6) 7 (1996), no. 7-12, 387--403 (1997), MathSciNet.  
  8. Cardano's 'Ars magna' and the solutions of cubic and quartic equations (Spanish)  
    C. Romo Santos  
    Rev. Acad. Canaria Cienc. 7 (1) (1995), 187-201.
  9. Khayyam, al-Biruni, Gauss, Archimedes, and quartic equations
    A. R. Amir-Moéz  
    Texas J. Sci. 46 (3) (1994), 241-257
  10. Approximate formulas for roots of cubic equations. The irreducible case. (Russian)
    Fedorov, F. I.
    Dokl. Akad. Nauk Belarusi 38 (1994), no. 6, 5--8, 121 (1995), MathSciNet.  
  11. A new approach to solving the cubic: Cardan's solution revealed
    R. W. D. Nickalls
    The Mathematical Gazette; vol 77, (1993), pages 354--359. 
  12. The roots of cubic equations over the 2 k-element field. (Chinese)
    Sun, Zong Ming
    Neimenggu Shida Xuebao Ziran Kexue Hanwen Ban 1993, no. 2, 21--26, MathSciNet.  
  13. The history of the formula for the roots of a cubic equation. (Chinese)
    Wang, Qing Jian
    Collected research papers on the history of mathematics, Vol. 3 (Chinese), 148--151, Inner Mongolia Univ. Press, Hohhot, 1992, MathSciNet.  
  14. An extension of Omar Khayyam's graphical solution of the cubic equation to the solution of the quartic
    Yardley, Peter D.
    Bull. Inst. Math. Appl. 27 (1991), no. 8-9, 173--174, MathSciNet.  
  15. On formulae for roots of cubic equation
    Lebedev, V.I.
    Soviet Journal of Numerical Analysis and Mathematical Modelling, v 6, n 4, 1991, p 315--324, Compendex.
  16. A unified approach for solving quadratic, cubic and quartic equations by radicals  
    Ungar, A. A.
    Comput. Math. Appl. 19 (1990), no. 12, 33--39, MathSciNet.  
  17. Graphical solution of the cubic equation developed from the work of Omar Khayyam  
    P. D. Yardley
    Bull. Inst. Math. Appl. 26 (5-6) (1990), 122-125.
  18. O formulakh dlya kornei kubicheskogo uravneniya. (Russian)
    [Formulas for the roots of a cubic equation]
    Lebedev, V. I.
    Akad. Nauk SSSR, Otdel Vychisl. Mat., Moscow, 1990. 14 pp., MathSciNet.  
  19. On the number of positive roots of cubic equations  
    Hogendijk, Jan P.
    Historia Math. 16 (1989), no. 1, 69--85, MathSciNet.  
  20. Geometric solution of the cubic equations in Raffaele Bombelli's 'Algebra' (Russian)  
    G. S. Smirnova  
    Istor. Metodol. Estestv. Nauk 36 (1989), 123-129.
  21. The exact probability that the roots of quadratic, cubic, and quartic equations are all real if the equation coefficients are random.
    Li, Hung C.
    Comm. Statist. Theory Methods 17 (1988), no. 2, 395--409, MathSciNet.  
  22. On the roots of the reduced cubic equation in the irreducible case. (Bulgarian)
    Deneva, Sonya; Diamandiev, Vasil
    Annuaire Univ. Sofia Fac. Math. Inform. 81 (1987), no. 2, 105--120 (1993), MathSciNet.  
  23. Estimates for the roots of the reduced cubic equation in the irreducible case, and application to the secular equation. (Bulgarian)
    Deneva, Sonya; Diamandiev, Vasil
    Annuaire Univ. Sofia Fac. Math. Inform. 81 (1987), no. 2, 223--235 (1993), MathSciNet.  
  24. Tartaglia, Archimedes and cubic equations  
    P. Schultz  
    Austral. Math. Soc. Gaz. 11 (4) (1984), 81-84.
  25. Simple transcendental expressions for the roots of cubic equations.
    McKelvey, J. P.
    Amer. J. Phys. 52 (1984), no. 3, 269--270, MathSciNet.  
  26. On root polynomials of cyclic cubic equations.
    Girstmair, Kurt
    Arch. Math. (Basel) 36 (1981), no. 4, 313--326, MathSciNet.  
  27. Luddhar's Method of Solving a Cubic Equation with a Rational Root  
    R. S. Luthar
    The Two-Year College Mathematics Journal, Vol. 11, No. 2 (Mar., 1980), pp. 107-110, Jstor.   
  28. Explicit roots of the cubic polynomial and applications.
    Miura, Robert M.
    Appl. Math. Notes 5 (1980), no. 1, 22--40, MathSciNet.  
  29. A method of solving the cubic equation. (Bulgarian)
    Lefterov, L. P.
    Godisnik Vis s. Ucebn. Zaved. Prilozna Mat. 11 (1975), no. 4, 35--38 (1977), MathSciNet.  
  30. Geometric algebra, and the solution of cubic equations in radicals. (Russian)
    Rozenfel\cprime d, B. A.; Cernova, M. L.
    Geometry collection, No. 12. Trudy Tomsk. Gos. Univ. 244 (1975), 96--101, MathSciNet.  
  31. An Interesting Relationship Among the Roots of a Cubic Equation  
    Mandelbaum, Joseph ; Schild, Albert
    Math Teacher 63, 5, 393-394, (May 1970), ERIC.
  32. Another Solution of the Cubic Equation  
    J. G. Campbell
    Mathematics Magazine, Vol. 35, No. 1 (Jan., 1962), pp. 43-44, Jstor.   
  33. A Graphical Determination of the Nature of the Roots of a Cubic  
    Morton J. Hellman
    Mathematics Magazine, Vol. 34, No. 4 (Mar., 1961), pp. 221-222, Jstor.   
  34. A Mechanical Device for Finding the Real Roots of the Cubic  
    Morton J. Hellman
    The American Mathematical Monthly, Vol. 68, No. 3 (Mar., 1961), pp. 278-279, Jstor.   
  35. The vanishing of the homogeneous product sum of the roots of a cubic.
    Ward, Morgan
    Duke Math. J 26 1959 553--562, MathSciNet.  
  36. A Method for Finding the Real Roots of Cubic Equations by Using the Slide Rule  
    Louis L. Pennisi
    Mathematics Magazine, Vol. 31, No. 4 (Mar., 1958), pp. 211-214, Jstor.   
  37. Omar Khayyam's Solution of Cubic Equations
    Eves, Howard
    Mathematics Teacher 51 (1958), 285--86.
  38. Omar Khayyam, Mathematician  
    Struik, D. J.
    Mathematics Teacher 51 (1958), 280--84.
  39. Contribution to the investigation of the roots of a cubic equation. (Russian)
    Nilov, G. N.
    Kabardin. Gos. Ped. Inst. Uc. Zap. 12 1957 28, MathSciNet.  
  40. The Cubic Equation and the Mannheim Slide Rule  
    H. A. Arnold
    The American Mathematical Monthly, Vol. 63, No. 9 (Nov., 1956), pp. 656-657, Jstor.   
  41. A Geometric Determination of the Nature of the Roots of the Cubic, Biquadratic, and Quintic Equations  
    M. S. Klamkin
    The American Mathematical Monthly, Vol. 61, No. 5 (May, 1954), pp. 340-342
  42. A variational problem for the roots of a cubic equation. (A contribution to the theory of servo-mechanisms.)
    Bückner, Hans
    Quart. Appl. Math. 8, (1950). 293--296, MathSciNet.  
  43. Diophantine Equations Connected with the Cubic Fermat Equation  
    J. D. Swift
    The American Mathematical Monthly, Vol. 56, No. 4 (Apr., 1949), pp. 254-256, Jstor.   
  44. Calculating machine solution of quadratic and cubic equations by the odd number method.
    Bleick, W. E.
    Math. Tables and Other Aids to Computation 2, (1947). 321--324, MathSciNet.  
  45. A Test for the Nature of the Roots of the Cubic Equation  
    E. E. Watson
    The American Mathematical Monthly, Vol. 48, No. 10 (Dec., 1941), p. 687, Jstor.   
  46. Hyperbolic Solution of the Cubic Equation   
    W. T. Short
    National Mathematics Magazine, Vol. 12, No. 3 (Dec., 1937), pp. 111-114, Jstor.   
  47. A Graphical Solution for the Complex Roots of a Cubic  
    George A. Yanosik
    National Mathematics Magazine, Vol. 10, No. 4 (Jan., 1936), pp. 139-140, Jstor.   
  48. The Graphical Interpretation of the Complex Roots of Cubic Equations   
    Garcia Henriquez
    The American Mathematical Monthly, Vol. 42, No. 6 (Jun., 1935), pp. 383-384, Jstor.   
  49. Using the Hessian to Solve a Cubic Equation
    Jas. A. Ward
    National Mathematics Magazine, Vol. 9, No. 8 (May, 1935), pp. 235-240, Jstor.   
  50. On the Vanishing of the Sum of the Nth Powers of the Roots of a Cubic Equation  
    Morgan Ward
    The American Mathematical Monthly, Vol. 41, No. 5 (May, 1934), pp. 313-316, Jstor.   
  51. A Note on the Roots of a Cubic  
    E. C. Kennedy
    The American Mathematical Monthly, Vol. 40, No. 7 (Aug., 1933), pp. 411-412, Jstor.    
  52. On the Roots of a Cubic and Those of Its Derivative  
    Raymond Garver
    Mathematics News Letter, Vol. 6, No. 7/8 (Apr., 1932), pp. 24-27, Jstor.    
  53. The History of the Solution of the Cubic Equation  
    Lucye Guilbeau
    Mathematics News Letter, Vol. 5, No. 4 (Dec., 1930), pp. 8-12, Jstor.   
  54. Discussions: A Criterion that a Cubic Equation has an Integral Root  
    H. S. Vandiver
    The American Mathematical Monthly, Vol. 33, No. 2 (Feb., 1926), pp. 94-95, Jstor.   
  55. Discussions: A Cubic Equation of Newton's  
    Norman Anning
    The American Mathematical Monthly, Vol. 33, No. 4 (Apr., 1926), pp. 211-212, Jstor.   
  56. An Algebraically Reducible Solution of the Cubic Equation  
    Glenn James
    The American Mathematical Monthly, Vol. 32, No. 4 (Apr., 1925), pp. 162-169, Jstor.   
  57. Addenda and Corrigenda: An Algebraically Reducible Solution of the Cubic Equation  
    Glenn James
    The American Mathematical Monthly, Vol. 32, No. 10 (Dec., 1925), p. 538, Jstor.   
  58. A Graphic Solution of the Cubic Equation  
    J. P. Ballantine
    The American Mathematical Monthly, Vol. 27, No. 5 (May, 1920), pp. 203-204, Jstor.   
  59. Discussions: Geometrical Construction of the Roots of a Cubic, and Inscription of a Regular Heptagon in a Circle  
    C. B. Haldeman
    The American Mathematical Monthly, Vol. 26, No. 9 (Nov., 1919), pp. 390-392, Jstor.    
  60. Discussions: Relating to the Graph of a Cubic Equation Having Complex Roots  
    Edwin S. Crawley
    The American Mathematical Monthly, Vol. 25, No. 6 (Jun., 1918), pp. 268-269, Jstor.   
  61. Discussions: Relating to Approximations to Nearly Equal Roots of a Cubic Equation  
    Paul Capron
    The American Mathematical Monthly, Vol. 25, No. 8 (Oct., 1918), pp. 342-347, Jstor.   
  62. On the Complete Logarithmic Solution of the Cubic Equation  
    R. E. Gleason
    The Annals of Mathematics, 2nd Ser., Vol. 13, No. 1/4 (1911), pp. 120-122, Jstor.   
  63. Cubic Congruences with Three Real Roots  
    Edward B. Escott
    The Annals of Mathematics, 2nd Ser., Vol. 11, No. 2 (Jan., 1910), pp. 86-92, Jstor.    
  64. A Simple Method for Graphically Obtaining the Complex Roots of a Cubic Equation  
    Rutherford E. Gleason
    The Annals of Mathematics, 2nd Ser., Vol. 11, No. 3 (Apr., 1910), pp. 95-96, Jstor.   
  65. A Generalized Trigonometric Solution of the Cubic Equation  
    W. D. Lambert
    The American Mathematical Monthly, Vol. 13, No. 4 (Apr., 1906), pp. 73-76, Jstor.   
  66. Approximation of the Greatest Root of a Cubic Equation with Three Real Roots  
    Charles Gilpin, Jr.
    The American Mathematical Monthly, Vol. 13, No. 6/7 (Jun., 1906), pp. 140-141, Jstor.   
  67. The Irreducible Case of the Cubic Equation  
    Alston Hamilton
    The Annals of Mathematics, 2nd Ser., Vol. 1, No. 1/4 (1899), pp. 41-45, Jstor.   
  68. A Graphical Method of Deducing the Criteria for the Nature of the Roots of Cubic and Quartic Equations  
    R. E. Gaines
    The Annals of Mathematics, 2nd Ser., Vol. 1, No. 1/4 (1899), pp. 111-112, Jstor.   
  69. A New Solution of the Cubic Equation  
    L. E. Dickson
    The American Mathematical Monthly, Vol. 5, No. 2 (Feb., 1898), pp. 38-39, Jstor.   
  70. On a Special Form of the General Equation of a Cubic Surface and on a Diagram Representing the Twenty-Seven Lines on the Surface
    H. M. Taylor
    Philosophical Transactions of the Royal Society of London. A, Vol. 185 (1894), pp. 37-69, Jstor.   
  71. A Conjecture concerning the Method by Which Cardan's Rules for Resolving the Cubic Equation  
    Scipio Ferreus; Francis Maseres
    Philosophical Transactions of the Royal Society of London, Vol. 70 (1780), pp. 221-238, Jstor.   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2006