The History of Polynomials
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:
ux2p + vxp = w
to a quadratic.
1079
Omar Khayyam (1050-1123) solves cubics geometrically by intersecting parabolas
and circles.
ca. 1400
Al-Kashi solves special cubic equations by iteration.
1484
Nicholas Chuqet (1445?-1500?) invents a method for solving polynomials iteratively
1515
Scipione del Ferro (1465-1526) solves the cubic:
x3 + mx = n
but does not publish his solution.
1535
Niccolo Fontana (Tartaglia) (1500?-1557) wins a mathematical contest by solving many different
cubics, and gives his method to Cardan.
1539
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 Artalso 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.
1544
Michael Stifel (1487?-1567) condenses the previous eight formulas for the roots of a quadratic
into one.
1593
Francois Viete (1540-1603) solves the casus irreducibilisof the cubic using trigonometric
functions.
1594
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.
1629
Albert Girard (1595-1632) conjectures that the nthdegree equation has n roots counting
multiplicity.
1637
Rene Descartes (1596-1650) gives his rule of signs to determine the number of positive roots of a
given polynomial
1666
Isaac Newton (1642-1727) finds a recursive way of expressing the sum of the roots to a given
power in terms of the coefficients.
1669
Newton introduces his iterative method for the numerical approximation of roots.
1676
Newton invents Newton's parallelogram to approximate all the possible values of y in terms of x,
if:
Sigma(i, j = 0 -> n) [aij xiyj] = 0
1683
Ehrenfried Walther von Tschirnhaus (1646-1716) generalizes the linear substitution that
eliminates the xn-1term in the nth degree polynomial to eliminate the xn-2and xn-3 terms as well.
Gottfried Wilhelm Leibniz (1646-1716) had pointed out that trying to get rid of the xn-4term usually
leads to a harder equation than the original one.
1691
Michael Rolle (1652-1719) proves that f'(x) has an odd number of roots in the interval between
two successive roots of f(x).
1694
Edmund Halley (1656-1742) discusses interative solutions of quartics with symbolic coefficients
1728
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.
1732
Leonard Euler (1707-1783) tries to find solutions of polynomial equations of degree n as sums
of nth roots, but fails.
1733
Halley solves the quadratic in trigonometric functions.
1748
Colin Maclaurin (1698-1746) generalizes Newton's relations for powers greater than the degree
of the polynomial.
1757
Johann Heinrich Lambert (1728-1777) gives series solutions of trinomial equations:
xp + x + r = 0
1762
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.
1762
Euler tries to find solutions of polynomial equations of degree n as linear combinations of powers
of an nth root, but fails.
1767
Joseph Louis Lagrange (1736-1813) expresses the real roots of a polynomial equation in terms of
a continued fraction.
1769
Lagrange expands a function as a series in powers of another function and uses this to solve
trinomial equations.
1770
Lagrange shows that polynomials of degree five or more cannot be solved by the methods used
for quadratics, cubics, and quartics. He introduces the Lagrangeresolvent, an equation of
degree n!.
1770
Euler gives series solutions of:
xm+n + axm + bxn = 0
1770
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.
1771
Gianfrancesco Malfatti (1731-1807), starting with a quintic, finds a sextic that factors if the quintic
is solvable in radicals.
1772
Lagrange finds a stationary solution of the three body problem that requires the solution of a
quintic.
1786
Erland Samuel Bring (1736-1798) proves that every quintic can be transformed to:
z5 + az + b = 0
1796
Jean Baptiste Joseph Fourier (1768-1830) determines the maximum number of roots in an
interval.
1799
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
1799
Carl Friedrich Gauss (1777-1855) proves the fundamental theorem of algebra: Every nonconstant
polynomial equation has at least one root.
1801
Gauss solves the cyclotomic equation:
z17 = 1
in square roots.
1819
William George Horner (1768-1847) presents his rule for the efficient numerical evaluation of a
polynomial. Ruffini had proposed a similar idea.
1826
WilNiels Henrik Abel (1802-1829) publishes Proof of the Impossibility of Generally Solving
Algebraic Equations of a Degree Higher than the Fourth.
1829
Jacques Charles Francois Sturm (1803-1855) finds the number of real roots of a given polynomial
in a given interval.
1829
Carl Gustav Jacobi (1804-1851) studies modular equations for elliptic functions which are
fundamental for Hermite's 1858 solution of quintics.
1831
Augistin-Louis Cauchy (1789-1857) determines how many roots of a polynomial lie inside a given
contour in the complex plane.
1832
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
1832
Friedrich Julius Richelot (1808-1875) solves the cycolotomic equation:
z257 = 1
in square roots.
1834
George Birch Jerrard (1804-1863) shows that every quintic can be transformed to:
z5 + az + b = 0
1837
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.
1838
Pafnuti Chebyshev (1821-1894) generalizes Newton's method to make the convergence
arbitrarily fast and uses this to approximate the roots of polynomials.
1840
L. Lalanne builds a practical machine to solve polynomials up to degree seven.
1844
Gotthold Eisenstein (1823-1852) gives the first few terms of a series for one root of a canonical
quintic.
1854
Josef Ludwig Raabe (1801-1859) transforms the problem of finding roots to solving a partial
differential equation, obtaining explicit roots for a quadratic.
1858
Charles Hermite (1822-1901), Leopold Kronecker (1823-1891), and Francesco Brioschi (18241897) independently solve a quintic in Bring-Jerrard form explicitly in terms of elliptic modular
functions.
1860, 1862
James Cockle (1819-1895) and Robert Harley (1828-1910) link a polynomial's roots to differential
equations.
1861
Carl Johan Hill (1793-1863) remarks that Jerrard's 1834 work is contained in Bring's 1786 work.
1862
William Hamilton (1805-1865) closes some gaps in Abel's impossibility proof.
1869
Johannes Karl Thomae (1840-1921) discovers a key ingredient for the representation of roots
using Siegel functions.
1870
Camille Jordan (1838-1922) shows that algebraic equations of any degree can be solved in terms
of modular functions.
1871
Ludwig Sylow (1832-1918) puts the finishing touches onGalois's proofs on solvability.
1873
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.
1877
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.
1884, 1892
Ferdinand von Lindemann (1852-1939) expresses the roots of an arbitrary polynomial in terms of
theta functions.
1885
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 a spezial form.
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.
1891
Karl Weierstrass (1815-1897) presents an interation scheme that simultaneously determines all
the roots of a polynomial.
1892
David Hilbert (1862-1943) proves that for every n there exists an nth polynomial with rational
coefficients whoseGalois group is the symmetric group Sn
1894
Johann Gustav Hermes (1846-1912) completes his 12-year effort to calculate the 65537th root of
unity using square roots.
1895
Emory McClintock (1840-1916) gives series solutions for all the roots of a polynomial.
1895-1910
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 (18651959) develop the fundamentals of how to solve a sextic via Klein's approach.
1915
Robert Hjalmal Mellin (1854-1933) solves an arbitrary polynomial equation with Mellin integrals.
1905-1925
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.
1926
Paul Emile Appell (1855-1930) and Joseph Marie Kampe de Feriet (1893-1982) recognize the
hypergeometric functions in the series solution of the quintic.
1932
Andre Bloch (1893-1948) and George Polya (1887-1985) investigate the zeros of polynomials of
arbitrary degree with random coefficients.
1934
Richard Brauer (1901-1977) analyzes Klein's solution of the quintic using the theory of fields.
1937
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.
1957
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.
1984
Hiroshi Umemura expresses the roots of an arbitrary polynomial through elliptic Siegel functions.
1989
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.
History of the Quartic
In 1540, Cardan was given the following problem:
Divide 10 into 3 parts: The parts are in continued proportion and the product of the first 2 is 6
This problem lead to a quartic which Cardan was not able to solve. He gave it
to Ferrari. Ferrari was the first to develop an algebraic technique for solving the general quartic.
He applied his technique (which was published by Cardano ) to the equation
x4 + 6x2 - 60x + 36 = 0