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A Short History


15th Century

ca. 2000 BC
  • Babylonians solve quadratics in radicals.

ca. 300 BC
  • Euclid demonstrates a geometrical construction for solving a quadratic.

ca. 1000
  • Arab mathematicians reduce:
           2p    p 
         ux  + vx = w
    to a quadratic.

  • Omar Khayyam (1050-1123) solves cubics geometrically by intersecting parabolas and circles.

ca. 1400
  • Al-Kashi solves special cubic equations by iteration.

  • Nicholas Chuqet (1445?-1500?) invents a method for solving polynomials iteratively.

16th Century

  • Scipione del Ferro (1465-1526) solves the cubic:
         x + mx = n
    but does not publish his solution.

  • Niccolo Fontana (Tartaglia) (1500?-1557) wins a mathematical contest by solving many different cubics, and gives his method to Cardan.

  • Girolamo Cardan (1501-1576) gives the complete solution of cubics in his book, The Great Art, or the Rules of Algebra. Complex numbers had been rejected for quadratics as absurd, but now they are needed in Cardan's formula to express real solutions. The Great Art also includes the solution of the quartic equation by Ludovico Ferrari (1522-1565), but it is played down because it was believed to be absurd to take a quantity to the fourth power, given that there are only three dimensions.

  • Michael Stifel (1487?-1567) condenses the previous eight formulas for the roots of a quadratic into one.

  • Francois Viete (1540-1603) solves the casus irreducibilis of the cubic using trigonometric functions.

  • Viete solves a particular 45th degree polynomial equation by decomposing it into cubics and a quintic. Later he gives a solution of the general cubic that needs the extraction of only a single cube root.

17th Century

  • Albert Girard (1595-1632) conjectures that the nth degree equation has n roots counting multiplicity.

  • Rene Descartes (1596-1650) gives his rule of signs to determine the number of positive roots of a given polynomial.

  • Isaac Newton (1642-1727) finds a recursive way of expressing the sum of the roots to a given power in terms of the coefficients.

  • Newton introduces his iterative method for the numerical approximation of roots.

  • Newton invents Newton's parallelogram to approximate all the possible values of y in terms of x, if:
                                    i j 
         Sigma(i, j = 0 -> n) [aij x y ] = 0

  • Ehrenfried Walther von Tschirnhaus (1646-1716) generalizes the linear substitution that eliminates the x^(n-1) term in the nth degree polynomial to eliminate the x^(n-2) and x^(n-3) terms as well. Gottfried Wilhelm Leibniz (1646-1716) had pointed out that trying to get rid of the x^(n-4) term usually leads to a harder equation than the original one.

  • Michael Rolle (1652-1719) proves that f'(x) has an odd number of roots in the interval between two successive roots of f(x).

  • Edmund Halley (1656-1742) discusses interative solutions of quartics with symbolic coefficients.

18th Century

  • Daniel Bernoulli (1700-1782) expresses the largest root of a polynomial as the limit of the ratio of the successive power sums of the roots.

  • Leonard Euler (1707-1783) tries to find solutions of polynomial equations of degree n as sums of nth roots, but fails.

  • Halley solves the quadratic in trigonometric functions.

  • Colin Maclaurin (1698-1746) generalizes Newton's relations for powers greater than the degree of the polynomial.


  • Etienne Bezout (1730-1783) tries to find solutions of polynomial equations of degree n as linear combinations of powers of an nth root of unity, but fails.
  • Euler tries to find solutions of polynomial equations of degree n as linear combinations of powers of an nth root, but fails.

  • Joseph Louis Lagrange (1736-1813) expresses the real roots of a polynomial equation in terms of a continued fraction.

  • Lagrange expands a function as a series in powers of another function and uses this to solve trinomial equations.

  • Lagrange shows that polynomials of degree five or more cannot be solved by the methods used for quadratics, cubics, and quartics. He introduces the Lagrange resolvent, an equation of degree n!.
  • Euler gives series solutions of:
          m+n    m    n
         x   + ax + bx = 0
  • John Rowning (1699-1771) develops the first mechanical device for solving polynomial equations. Although the machine works for any degree in theory, it was only practical for quadratics.

  • Alexandre Theophile Vandermonde (1735-1796) solves the irreducible cyclotomic equation:
         (z  - 1)    10   9   8   7   6   5   4   3   2 
         ------ = z  + z + z + z + z + z + z + z + z + z + 1 = 0 
         (z - 1)
    in radicals.
  • Gianfrancesco Malfatti (1731-1807), starting with a quintic, finds a sextic that factors if the quintic is solvable in radicals.

  • Lagrange finds a stationary solution of the three body problem that requires the solution of a quintic.

  • Erland Samuel Bring (1736-1798) proves that every quintic can be transformed to:
         z + az + b = 0

  • Jean Baptiste Joseph Fourier (1768-1830) determines the maximum number of roots in an interval.

  • Paolo Ruffini (1765-1822) publishes the book, General Theory of Equations, in which the Algebraic Solution of General Equations of a Degree Higher than the Fourth is Shown to Be Impossible.
  • Carl Friedrich Gauss (1777-1855) proves the fundamental theorem of algebra: Every nonconstant polynomial equation has at least one root.

19th Century

  • Gauss solves the cyclotomic equation:
         z  = 1
    in square roots.

  • William George Horner (1768-1847) presents his rule for the efficient numerical evaluation of a polynomial. Ruffini had proposed a similar idea.

  • Niels Henrik Abel (1802-1829) publishes Proof of the Impossibility of Generally Solving Algebraic Equations of a Degree Higher than the Fourth.

  • Carl Gustav Jacobi (1804-1851) studies modular equations for elliptic functions. The equation:
          6   6    2 2  2   2             4 4
         u + v + 5u v (u - v ) + 4uv(1 - u v ) = 0
    is fundamental for Hermite's 1858 solution of quintics.
  • Abel shows constructively that broad classes of higher-order equations can be solved in radicals.
  • Jacques Charles Francois Sturm (1803-1855) finds the number of real roots of a given polynomial in a given interval.

  • Augistin-Louis Cauchy (1789-1857) determines how many roots of a polynomial lie inside a given contour in the complex plane.

  • Evariste Galois (1811-1832) writes down the main ideas of his theory in a letter to Auguste Chevalier the day before he dies in a duel.
  • Friedrich Julius Richelot (1808-1875) solves the cycolotomic equation:
         z   = 1
    in square roots.

  • George Birch Jerrard (1804-1863) shows that every quintic can be transformed to:
         z + az + b = 0

  • Karl Heinrich Graeffe (1799-1873) invents a widely used method to determine numerical roots by hand. Similar ideas had already been suggested independently by Edward Waring (1734-1798), Germinal Pierre Dandelin (1794-1847), Moritz Abraham Stern (1807-1894), and Nickolai Lobachevski (1792-1856). Johann Franz Encke (1791-1865) later perfects the method.

  • Pafnuti Chebyshev (1821-1894) generalizes Newton's method to make the convergence arbitrarily fast and uses this to approximate the roots of polynomials.

  • L. Lalanne builds a practical machine to solve polynomials up to degree seven.

  • Gotthold Eisenstein (1823-1852) gives the first few terms of a series for one root of a canonical quintic.

  • Josef Ludwig Raabe (1801-1859) transforms the problem of finding roots to solving a partial differential equation, obtaining explicit roots for a quadratic.


1860, 1862
  • James Cockle (1819-1895) and Robert Harley (1828-1910) link a polynomial's roots to differential equations.

  • Carl Johan Hill (1793-1863) remarks that Jerrard's 1834 work is contained in Bring's 1786 work.

  • William Hamilton (1805-1865) closes some gaps in Abel's impossibility proof.

  • Johannes Karl Thomae (1840-1921) discovers a key ingredient for the representation of roots using Siegel functions.

  • Camille Jordan (1838-1922) shows that algebraic equations of any degree can be solved in terms of modular functions.

  • Ludwig Sylow (1832-1918) puts the finishing touches on Galois's proofs on solvability.

  • Hermann Amandus Schwarz (1843-1921) investigates the relationship between hypergeometric differential equations and the group structure of the Platonic solids, an important part of Klein's solution to the quintic.

  • Felix Klein (1849-1925) solves the icosahedral equation in terms of hypergeometric functions. This allows him to give a closed-form solution of a principal quintic.

  • Tables, nomograms, and various mechanical devices are constructed for solving trinomial equations.

1884, 1892
  • Ferdinand von Lindemann (1852-1939) expresses the roots of an arbitrary polynomial in terms of theta functions.

  • John Stuart Cadenhead Glashan (1844-1932), George Paxton Young (1819-1889), and Carl Runge (1856-1927), show that all irreducible solvable quintics with the quadratic, cubic, and quartic terms missing have the following form, with mu and v rational:
          5  5 mu^4 (4v + 3)   4 mu^5 (2v + 1) (4v + 3)
         x + --------------- + ------------------------ = 0
                 v^2 + 1               v^2 + 1

  • Carl Woldemar Heymann (1885-1910) solves trinomial equations using integrals.

1890, 1891
  • Vincenzo Mollame (1848-1912) and Ludwig Otto Hoelder (1859-1937) prove the impossibility of avoiding intermediate complex numbers in expressing the three roots of a cubic when they are all real.

  • Karl Weierstrass (1815-1897) presents an interation scheme that simultaneously determines all the roots of a polynomial.

  • David Hilbert (1862-1943) proves that for every n there exists an nth polynomial with rational coefficients whose Galois group is the symmetric group Sn. (The same is true for the alternating group An.)

  • Johann Gustav Hermes (1846-1912) completes his 12-year effort to calculate the 65537th root of unity using square roots.

  • Emory McClintock (1840-1916) gives series solutions for all the roots of a polynomial.
  • Leonardo Torres Quevedo (1852-1936) builds a machine for the mechanical calculation of the real and complex roots of an arbitrary trinomial equation.

  • Klein, Leonid Lachtin (1858-1927), Paul Gordan (1837-1912), Heinrich Maschke (1853-1908), Arthur Byron Coble (1878-1966), Frank Nelson Cole (1861-1926), and Anders Wiman (1865-1959) develop the fundamentals of how to solve a sextic via Klein's approach.

20th Century

  • Robert Hjalmal Mellin (1854-1933) solves an arbitrary polynomial equation with Mellin integrals.

  • R. Birkeland shows that the roots of an algebraic equation can be expressed using hypergeometric functions in several variables. Alfred Capelli (1855-1910), Guiseppe Belardinelli (1894-?), and Salvatore Pincherle (1853-1936) express related ideas.


  • Andre Bloch (1893-1948) and George Polya (1887-1985) investigate the zeros of polynomials of arbitrary degree with random coefficients.

  • Richard Brauer (1901-1977) analyzes Klein's solution of the quintic using the theory of fields.

  • Scientists at Bell Labs build the Isograph, a precision instrument that calculates roots of polynomials up to degree 15.

1938, 1942
  • Emil Artin (1898-1962) uses field theory to develop the modern theory of algebraic equations.

  • Vladimir Arnol'd, using results of Andrei Kolmogorov (1903-1987), shows that it is possible to express the roots of the reduced 7th degree polynomial in continuous functions of two variables, answering Hilbert's 13th problem in the negative.

  • Hiroshi Umemura expresses the roots of an arbitrary polynomial through elliptic Siegel functions.

  • Peter Doyle and Curt McMullen construct a generally convergent, purely iterative algorithm for the numerical solution of a reduced quintic, relying on the icosahedral equation.

1991, 1992
  • David Dummit and (independently) Sigeru Kobayashi and Hiroshi Nakagawa give methods for finding the roots of a general solvable quintic in radicals.